Supercharge, invasion and mudcake growth in downhole applications 9781119283409, 111928340X

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Supercharge, Invasion and Mudcake Growth in Downhole Applications

Scrivener Publishing 100 Cummings Center, Suite 541J Beverly, MA 01915-6106 Handbook of Petroleum Engineering Series Series Editor: Wilson C. Chin Scope: Covering every aspect of petroleum engineering, this new series sets the standard in best practices for the petroleum engineer. This is a must-have for any petroleum engineer in today's changing industry. About the Series Editor: Wilson Chin earned his PhD from M.I.T. and his M.Sc. from Caltech. He has authored over twenty books with Wiley-Scrivener and other major scientific publishers, has more than four dozen domestic and international patents to his credit, and has published over one hundred journal articles, in the areas of reservoir engineering, formation testing, well logging, measurement while drilling, and drilling and cementing rheology. Submission to the series: Phil Carmical, Publisher Scrivener Publishing (512)203-2236 [email protected]

Publishers at Scrivener Martin Scrivener ([email protected]) Phillip Carmical ([email protected])

Supercharge, Invasion and Mudcake Growth in Downhole Applications

by

Tao Lu, Xiaofei Qin, Yongren Feng, Yanmin Zhou and

Wilson Chin

This edition first published 2021 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA © 2021 Scrivener Publishing LLC For more information about Scrivener publications please visit www.scrivenerpublishing.com. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. Wiley Global Headquarters 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no rep­ resentations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchant-­ ability or fitness for a particular purpose. No warranty may be created or extended by sales representa­ tives, written sales materials, or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further informa­ tion does not mean that the publisher and authors endorse the information or services the organiza­ tion, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Library of Congress Cataloging-in-Publication Data ISBN 978-1-119-28332-4 Cover image: Downhole Logging, Aleksei Zakirov | Dreamstime.com Cover design by Kris Hackerott Set in size of 11pt and Minion Pro by Manila Typesetting Company, Makati, Philippines Printed in the USA 10 9 8 7 6 5 4 3 2 1

Contents Prefacexiii Acknowledgementsxvii 1 Pressure Transient Analysis and Sampling in Formation Testing1 Pressure transient analysis challenges 1 Background development 3 1.1 Conventional Formation Testing Concepts 5 1.2 Prototypes, Tools and Systems 6 1.2.1 Enhanced Formation Dynamic Tester (EFDT®) 9 1.2.2 Basic Reservoir Characteristic Tester (BASIC-RCT™) 13 1.2.3 Enhancing and enabling technologies 15 Stuck tool alleviation 16 Field facilities 17 1.3 Recent Formation Testing Developments 17 1.4 References 20 2. Spherical Source Models for Forward and Inverse Formulations21 2.1 Basic Approaches, Interpretation Issues and Modeling Hierarchies23 Early steady flow model 23 Simple drawdown-buildup models 23 Analytical drawdown-buildup solution 25 Phase delay analysis 26 Modeling hierarchies 28 2.2 Basic Single-Phase Flow Forward and Inverse Algorithms 36 2.2.1 Module FT-00 36 2.2.2 Module FT-01 37 2.2.3 Module FT-03 38 2.2.4 Forward model application, Module FT-00 39 v

vi  Contents 2.2.5 2.2.6 2.2.7 2.2.8 2.2.9

Inverse model application, Module FT-01 Effects of dip angle Inverse “pulse interaction” approach using FT-00 FT-03 model overcomes source-sink limitations Module FT-04, phase delay analysis, introductory for now 2.2.10 Drawdown-buildup, Module FT-PTA-DDBU 2.2.11 Real pumping, Module FT-06 2.3 Advanced Forward and Inverse Algorithms 2.3.1 Advanced drawdown and buildup methods Basic steady model Validating our method 2.3.2 Calibration results and transient pressure curves 2.3.3 Mobility and pore pressure using first drawdown data 2.3.3.1 Run No. 1. Flowline volume 200 cc 2.3.3.2 Run No. 2. Flowline volume 500 cc 2.3.3.3 Run No. 3. Flowline volume 1,000 cc 2.3.3.4 Run No. 4. Flowline volume 2,000 cc 2.3.4 Mobility and pore pressure from last buildup data 2.3.4.1 Run No. 5. Flowline volume 200 cc 2.3.4.2 Run No. 6. Flowline volume 500 cc 2.3.4.3 Run No. 7. Flowline volume 1,000 cc 2.3.4.4 Run No. 8. Flowline volume 2,000 cc 2.3.4.5 Run No. 9. Time-varying flowline volume inputs from FT-07 2.3.5 Phase delay and amplitude attenuation, anisotropic media with dip – detailed theory, model and numerical results 2.3.5.1 Basic mathematical results Isotropic model Anisotropic extensions Vertical well limit Horizontal well limit Formulas for vertical and horizontal wells Deviated well equations Deviated well interpretation for both kh and kv Two-observation-probe models 2.3.5.2 Numerical examples and typical results Example 1. Parameter estimates

41 43 46 49 52 55 59 61 61 63 65 67 68 69 71 73 74 74 76 77 78 79 81 82 82 82 83 83 83 84 85 86 88 89

Contents  vii Example 2. Surface plots 90 Example 3. Sinusoidal excitation 91 Example 4. Rectangular wave excitation 94 Example 5. Permeability prediction at general dip angles 96 Example 6. Solution for a random input 98 2.3.5.3 Layered model formulation 99 2.3.5.4 Phase delay software interface 100 2.3.5.5 Detailed phase delay results in layered anisotropic media 103 2.3.6 Supercharging and formation invasion introduction, with review of analytical forward and inverse models 110 2.3.6.1 Development perspectives 111 2.3.6.2 Review of forward and inverse models 113 FT-00 model 113 FT-01 model 117 FT-02 model 118 FT-06 and FT-07 models 119 FT–PTA–DDBU model 122 Classic inversion model 123 Supercharge forward and inverse models 123 Multiple drawdown and buildup inverse models129 Multiphase invasion, clean-up and contamination133 System integration and closing remarks 138 2.3.6.3 Supercharging summaries - advanced forward and inverse models explored 139 Supercharge math model development 139 Conventional zero supercharge model 141 Supercharge extension 142 2.3.6.4 Drawdown only applications 144 Example DD-1. High overbalance 144 Example DD-2. High overbalance 150 Example DD-3. High overbalance 154 Example DD-4. Qualitative pressure trends 158 Example DD-5. Qualitative pressure trends 161 Example DD-6. “Drawdown-only” data with multiple inverse scenarios for 1 md/cp application 163

viii  Contents Example DD-7. “Drawdown-only” data with multiple inverse scenarios for 0.1 md/cp application 168 2.3.6.5 Drawdown – buildup applications 173 Example DDBU-1. Drawdown-buildup, high overbalance 173 Example DDBU-2. Drawdown-buildup, high overbalance 177 Example DDBU-3. Drawdown-buildup, high overbalance 180 Example DDBU-4. Drawdown-buildup, 1 md/cp calculations 184 Example DDBU-5. Drawdown-buildup, 0.1md/cp calculations 188 2.3.7 Advanced multiple drawdown – buildup (or, “MDDBU”) forward and inverse models 193 2.3.7.1 Software description 193 2.3.7.2 Validation of PTA-App-11 inverse model 200 2.3.8 Multiphase flow with inertial effects – Applications to borehole invasion, supercharging, clean-up and contamination analysis 217 2.3.8.1 Mudcake dynamics 217 2.3.8.2 Multiphase modeling in boreholes 220 2.3.8.3 Pressure and concentration displays 222 Example 1. Single probe, infinite anisotropic media 223 Example 2. Single probe, three layer medium228 Example 3. Dual probe pumping, three layer medium230 Example 4. Straddle packer pumping 231 Example 5. Formation fluid viscosity imaging 233 Example 6. Contamination modeling 234 Example 7. Multi-rate pumping simulation 234 2.4 References 236 3 Practical Applications Examples 3.1 Non-constant Flow Rate Effects 3.1.1 Constant flow rate, idealized pumping, inverse method 3.1.2 Slow ramp up/down flow rate 3.1.3 Impulsive start/stop flow rate Closing remarks

237 238 239 245 250 255

Contents  ix 3.2 Supercharging – Effects of Nonuniform Initial Pressure Conventional zero supercharge model Supercharge “Fast Forward” solver 3.3 Dual Probe Anisotropy Inverse Analysis 3.4 Multiprobe “DOI,” Inverse and Barrier Analysis 3.5 Rapid Batch Analysis for History Matching 3.6 Supercharge, Contamination Depth and Mudcake Growth in “Large Boreholes” – Lineal Flow Mudcake growth and filtrate invasion Time-dependent pressure distributions 3.7 Supercharge, Contamination Depth and Mudcake Growth in Slimholes or “Clogged Wells” – Radial Flow 3.8 References

256 256 258 264 273 281 289 289 292 292 294

4 Supercharge, Pressure Change, Fluid Invasion and Mudcake Growth 295 Conventional zero supercharge model 295 Supercharge model 296 Relevance to formation tester job planning 298 Refined models for supercharge invasion 299 4.1 Governing equations and moving interface modeling 300 Single-phase flow pressure equations 300 Problem formulation 303 Eulerian versus Lagrangian description 303 Constant density versus compressible flow 304 Steady versus unsteady flow 305 Incorrect use of Darcy’s law 305 Moving fronts and interfaces 306 Use of effective properties 308 4.2 Static and dynamic filtration 310 4.2.1 Simple flows without mudcake 310 Homogeneous liquid in a uniform linear core 311 Homogeneous liquid in a uniform radial flow 313 Homogeneous liquid in uniform spherical domain 314 Gas flow in a uniform linear core 315 Flow from a plane fracture 317 4.2.2 Flows with moving boundaries 318 Lineal mudcake buildup on filter paper 318 Plug flow of two liquids in linear core without cake 321 4.3 Coupled Dynamical Problems: Mudcake and Formation Interaction323

x  Contents Simultaneous mudcake buildup and filtrate invasion in a linear core (liquid flows) 323 Simultaneous mudcake buildup and filtrate invasion in a radial geometry (liquid flows) 327 Hole plugging and stuck pipe 330 Fluid compressibility 331 Formation invasion at equilibrium mudcake thickness 335 4.4 Inverse Models in Time Lapse Logging 336 Experimental model validation 336 Static filtration test procedure 337 Dynamic filtration testing 337 Measurement of mudcake properties 338 Formation evaluation from invasion data 338 Field applications 339 Characterizing mudcake properties 340 Simple extrapolation of mudcake properties 341 Radial mudcake growth on cylindrical filter paper 342 4.5 Porosity, Permeability, Oil Viscosity and Pore Pressure Determination345 Simple porosity determination 345 Radial invasion without mudcake 346 Problem 1 348 Problem 2 350 Time lapse analysis using general muds 351 Problem 1 352 Problem 2 353 4.6 Examples of Time Lapse Analysis 354 Formation permeability and hydrocarbon viscosity 355 Pore pressure, rock permeability and fluid viscosity 357 4.7 References 360 5 Numerical Supercharge, Pressure, Displacement and Multiphase Flow Models 363 5.1 Finite Difference Solutions 364 Basic formulas 364 Model constant density flow analysis 366 Transient compressible flow modeling 369 Numerical stability 371 Convergence371 Multiple physical time and space scales 372 Example 5-1. Lineal liquid displacement without mudcake 373

Contents  xi Example 5-2. Cylindrical radial liquid displacement without cake 380 Example 5-3. Spherical radial liquid displacement without cake 383 Example 5-4. Lineal liquid displacement without mudcake, including compressible flow transients 385 Example 5-5. Von Neumann stability of implicit time schemes388 Example 5-6. Gas displacement by liquid in lineal core without mudcake, including compressible flow transients 390 Incompressible problem 391 Transient, compressible problem 392 Example 5-7. Simultaneous mudcake buildup and displacement front motion for incompressible liquid flows 396 Matching conditions at displacement front 399 Matching conditions at the cake-to-rock interface 399 Coding modifications 400 Modeling formation heterogeneities 403 Mudcake compaction and compressibility 404 Modeling borehole activity 405 5.2 Forward and Inverse Multiphase Flow Modeling 405 Problem hierarchies 406 5.2.1 Immiscible Buckley-Leverett lineal flows without capillary pressure 407 Example boundary value problems 409 General initial value problem 410 General boundary value problem for infinite core 411 Variable q(t) 411 Mudcake-dominated invasion 412 Shock velocity 412 Pressure solution 414 5.2.2 Molecular diffusion in fluid flows 415 Exact lineal flow solutions 416 Numerical analysis 417 Diffusion in cake-dominated flows 419 Resistivity migration 419 Lineal diffusion and “un-diffusion” examples 420 Radial diffusion and “un-diffusion” examples 423 5.2.3 Immiscible radial flows with capillary pressure and prescribed mudcake growth 425

xii  Contents Governing saturation equation Numerical analysis Fortran implementation Typical calculations Mudcake dominated flows “Un-shocking” a saturation discontinuity 5.2.4 Immiscible flows with capillary pressure and dynamically coupled mudcake growth Flows without mudcakes Modeling mudcake coupling Unchanging mudcake thickness Transient mudcake growth General immiscible flow model 5.3 Closing Remarks 5.4 References

Cumulative References

426 427 429 429 435 438 441 441 450 451 453 457 458 464

467

Index481 About the Authors

498

Preface Formation testing, unlike conventional logging methods focused on resistivity, acoustic, nuclear or magnetic resonance approaches, provides direct results as opposed to indirect inferred properties. In sampling, actual in-situ fluids are collected for surface evaluation. And in pressure transient analysis, properties that pertain to production economics like mobility, compressibility, anisotropy and pore pressure are obtained directly from the underlying Darcy flow equations. By and large, the conventional subject matter deals with single, dual and multiprobe tools where pad nozzles are displaced axially relative to each other and along the same azimuth. This being so, idealized spherical “source” or “sink” methods are used in formulating forward and inverse problems. Even so, few models have proven useful. An early steady model for spherical flow no longer applies to the lower mobility formations encountered in practice. Later transient models contain complicated Bessel functions and integrals whose effective use in the field is questionable. And then, a rapid, early-time prediction method for “effective permeability” and pore pressure, addressing the low mobility and “not so low” flowline volume limit – while significant in the 1990s and, in fact, invented by the last author, does not address all-important supercharging effects uncovered in recent field-based publications. Fortunately, progress in source methods has been made, but at such an unusual pace that any presentations at industry meetings would have been rapidly dated. In support of our work, John Wiley & Sons has published our research in three volumes during 2014 – 2019, introducing the latest ideas and techniques to the industry, complete with derivations, equations and software. The present work, our latest formation testing addition to Wiley-Scrivener’s Petroleum Engineering Handbook Series, serves several purposes. While “handbooks” normally refer to summaries of decades-old technologies, this edition is timely because numerous new advances have been made in related and interdependent areas. These include pressure transient analysis, forward and inverse modeling, supercharge, mudcake xiii

xiv  Preface growth and fluid invasion formulations, and contamination and cleaning multiphase methods – and all during the past two decades by the present authors. While China Oilfield Services Limited (COSL) does manufacture its own conventional single and dual probe tools, it is the availability of our complete suite of software models that allows its tools to be used in many more innovative ways. For example, methods are available to predict permeability and pore pressure rapidly from early time data in low mobility formations with strong flowline volume. But what if significant supercharging exists? Most inverse methods require constant flow rate drawdowns. What if this is not possible? And unacceptably, few authors have ever rigorously studied mudcake growth and fluid invasion, which produce the thick cakes responsible for stuck formation testers – the same phenomena associated with supercharge. Nor do they address the thin cakes that wreak havoc on nozzle pad sealing – leakages that would doom any formation testing job. Numerous related questions are treated in this comprehensive volume. And so this handbook, which addresses all of these problems from source model perspectives, provides unified discussions in forward and inverse formation testing analysis, supercharge in pressure evolution and permeability prediction, plus related topics in fluid invasion, mudcake growth and displacement front prediction. It is our hope that this work stimulates continuing research and enhances the innovative use of conventional tools in the field. During the past several years, other high risk research and development projects were undertaken at COSL. In the early 1990s, an innovative “multiprobe” formation tester was introduced by a major service company that has greatly benefited the industry. This tool, consisting of an active “sink probe” and a passive “horizontal” observation probe displaced at 180° azimuthally from the sink, would provide measurements for horizontal and vertical permeability. However, in low mobility applications, measured pressure drops at the latter probe were often orders-of-magnitude less than those obtained at the pumping probe. This limitation attracted the interests of COSL engineers, who raised several unusual design challenges. “What if three azimuthally displaced probes, each separated by 120° from the others, were used?” And further, “What if each probe in the triple multiprobe tool were capable of operating independently from the others?” What would be the logging advantages? What additional parameters of formation evaluation interest could we predict? Is it possible to detect heterogeneities? Dip angle? Can we pump at high rates without releasing dissolved gas? In order to design such a multiprobe tool, a fully three-dimensional transient model would be required to guide mechanical design

Preface  xv as well as to support interpretation procedures at the rigsite. Can a rapid, stable, accurate and easy-to-use computational method be devised? Is it possible to develop a robust procedure that supports field work in horizontal and vertical mobility definition? How would we apply “big data” statistical approaches using advanced algorithms? Can inverse procedures be solved accurately and rapidly at the rigsite and in field offices? These questions are addressed in a companion 2021 volume in John Wiley’s Advances in Petroleum Engineering series, entitled Multiprobe Pressure Analysis and Interpretation, by Tao Lu, Minggao Zhou, Yongren Feng, Yuqing Yang and Wilson Chin. This complementary volume contains math models entirely different from the present, but which are also applicable to conventional 180° dual probe tools. Both of our 2021 books, drawing on research and engineering developed over more than a decade, are essential to modern formation testing, and we hope that both will find permanent places on petroleum engineers’ bookshelves. In this time of great uncertainty, one truth prevails: now, more than ever, innovation is needed to explore and produce natural resources more efficiently. And innovation in engineering means nothing less than a thorough understanding of physics and mathematics and putting both to important practical use. The Authors, Beijing and Houston

Acknowledgements The authors wish to thank the management of China Oilfield Services Limited (COSL) for permission to publish this manuscript. Our research efforts hope to advance formation testing, algorithm design and well logging technology and bring greater efficiencies to exploration and production. We are also indebted to Xiaoying Zhuang for her interpretation and translation skills, and usual hard work and perseverance, which have been instrumental in communicating a wide range of engineering and technical ideas to English-speaking audiences over the past decade. And last but not least, we again thank Phil Carmical, Acquisitions Editor and Publisher, for his confidence and faith in our research activities. In times of economic uncertainty such as ours, it is imperative that “the show must go on” and oil and gas industry professionals continue to “push the envelope” despite the headwinds. This monograph describes our persistent and continuing efforts in this endeavor and we are pleased to present our ideas to our petroleum engineering colleagues.

xvii

1 Pressure Transient Analysis and Sampling in Formation Testing The formation tester is a well logging instrument with extendable pad nozzles which, when pressed against the borehole sandface, extracts in situ formation fluids for delivery to the surface for chemical examination. This process characterizes its fluid “sampling” function. By-products of this operation are pressure transient histories, which can be interrogated using Darcy math models for fluid and formation properties such as permeability, mobility, anisotropy, compressibility and pore pressure. This is referred to as “pressure transient analysis,” or simply, “PTA.” Both can be conducted as wireline or Measurement While Drilling, or “MWD,” applications, where these operations now represent invaluable elements of the standard well logging suite. Pressure transient analysis challenges. While collecting and transporting fluids is relatively straightforward, e.g., storing samples in secure vessels that maintain downhole conditions, the PTA process poses a greater design challenge. A well designed tool often begins with a good understanding of the environment, plus physics coupled with sound experience in mathematical modeling. Some ideas are obvious. For example, a single “source” or “sink” probe, serving both pumping and pressure observation functions, will at most provide the “spherical permeability” kh2/3kv/1/3, where kh and kv are horizontal and vertical permeabilities. Thus, “single probe” tools, while mechanically simple, will offer fewer logging advantages than “dual probe” or “multiprobe tools” which provide much greater formation evaluation information. 1

2 Supercharge, Invasion and Mudcake Growth

Figure 1.1. Drawdown-buildup pressure response with dynamic pumping action and flowline. But how are probe arrays configured and placed for optimal effect? Figures 1.1 and 1.2 illustrate the operation of a single probe tool that withdraws fluid and then stops, creating the expected “drawdown and buildup” shown. If a second probe is desired, should it be placed an axial distance apart but along the same azimuth? Or azimuthally apart, at 180o away along the borehole circumference? What about a “drawdown only” pumpout? Or perhaps, have the pump oscillate sinusoidally in place, thus mimicking the AC transmissions of an electromagnetic logging tool? How many probes are best? What are their flow areas? Do answers to these questions depend on fluid and formation properties?

Pressure Transient Analysis and Sampling 3

Figure 1.2. Downhole, surface and logging truck operations. Background development. The present book addresses these questions for “source” or “sink models” of the pumping nozzle, these terms referring to ideal representations of the flow where borehole and pad geometry are described using mathematically small closed surfaces. The recent books due to Chin et al. (2014) or Formation Testing: Pressure Transient and Contamination Analysis, Chin et al. (2015) or Formation Testing: Low Mobility Pressure Transient Analysis, and Chin (2019) or Formation Testing: Supercharge, Pressure Testing and Contamination Models, published by John Wiley & Sons, contain complete math derivations and detailed validations. However, the rapid pace of recent development suggests a separate volume in Wiley’s Handbook of Petroleum Engineering Series, focused on the main ideas

4 Supercharge, Invasion and Mudcake Growth

behind the recent works. These ideas are essential as they are also used in the design of newer COSL formation testing tools as well as in interpretation software now available to the petroleum industry. What engineers lack, at present, are job planning and PTA tools both useful at the rigsite and at engineers’ desktops. It is our purpose to support this pressing need.

Figure 1.3. Recent formation testing book publications.

Pressure Transient Analysis and Sampling 5

1.1 Conventional Formation Testing Concepts. Formation testing design concepts are rich and varied. A pumping probe, operating as a “sink” or (equivalently) a “source,” or both, also tracks pressure transient responses. Other pressure probes my reside along the tool body, displaced axially, azimuthally or both, which may actively pump or act as passive observers. While the primary formation tester function is fluid sampling, where in-situ reservoir fluids are collected and transported to the surface for analysis, pressure measurements represent critical by-products important to formation evaluation. Examples of testers offered by different manufacturers for wireline and MWD applications are given in Figures 1.4 – 1.7.

Figure 1.4. Conventional formation tester tool strings.

Figure 1.5. Formation testers, additional developments.

6 Supercharge, Invasion and Mudcake Growth

Figure 1.6. Conventional dual and triple probe testers.

Figure 1.7. Dual probe tester with dual packer. 1.2 Prototypes, Tools and Systems. In a “handbook” such as this, it is important to provide examples of prototypes, commercial tools and systems. The wide ranges in design parameters can be surprising to newcomers in formation testing. For example, the “vertical and sink probes” in Figure 1.6, which are displaced axially but lie along the same azimuth, can range from six or seven inches to as much as 2.3 ft (27.6 in) and 10.3 ft (123.6 in), where the latter two distances are obtained from the manufacturer’s figure in SPE Paper No. 36176. We might, for example, ask, “Just what does the distant observation probe “see” under different mobility backgrounds?” “Will the tool do the job for my formation?” This book attempts to answer the most obvious questions, but it also aims at providing the tools and software for readers to address those pressing questions that invariably arise in any new logging scenario. To provide a flavor of how hardware literature and specifications might appear, we have included discussion of COSL material related to its standard product lines. Note that COSL’s new “triple probe, 120o tool” (as opposed to a conventional 180o tool) is treated separately in our companion 2021 book.

Pressure Transient Analysis and Sampling 7

Close-ups of early single and dual probe prototype formation testers are shown in Figure 1.8. These photographs were obtained during field tests. The black pads shown perform an important sealing function, which prevents leakage of fluid through its contact surface with the sandface. However, they are not as “simple” as they appear. For instance, at any given pump rate, the pressure drop, which depends on nozzle diameter, may be excessive and allow the undesired release of dissolved gas – orifice sizes must be chosen judiciously, as suggested by the wide variety of choices shown in Figure 1.9. The shape of the hole or slot is also important; circular or oval shapes may be acceptable for consolidated matrix rock, but slotted models may be required for naturally fractured media or unconsolidated formations. Of course, in supporting PTA interpretation objectives, the size and shape of a formation tester’s pads must be incorporated into the host math model. More often than not, the model must be simple and mathematically tractable in order to obtain useful answers in a reasonable amount of time. This may require the use of idealized source or sink models, or numerical models with limited numbers of grids in the case of finite difference or finite modeling – consequently, questions related to calibration or geometric factors arise, along with test procedures, etc.

Figure 1.8. Early COSL single and dual probe prototype formation testers (details in 2014 and 2015 books).

8 Supercharge, Invasion and Mudcake Growth

Figure 1.9. COSL pad designs with varied sizes and shapes, for different applications, e.g., firm matrix rock, unconsolidated formations, fractured media, and so on.. Pressures obtained in PTA logging are used for multiple applications. For example, depending on the tool, permeability, anisotropy, compressibility and pore pressure are all possible (the term “mobility,” defined as the ratio of permeability to viscosity, is often interchangeably used, assuming that the viscosity is known). The pore pressure itself is used to identify fluids by their vertical hydrostatic gradients; this is possible because changes in pressure are affected by changes in fluid density. Sudden changes in pressure, for instance, may indicate the presence of barriers. However, the raw measured pressure, unless corrected for the “cushioning” effects associated with flowline volume, will not reflect pore pressures accurately. The correction depends, in turn, on the line volume as well as the compressibility and the mobility of the formation fluid. All said, the physics and math can be challenging, but solutions and analytical highlights are presented in the next chapter for a wide variety of tools and applications. Chapter 2 provides a broad state-of-the-art review for source and sink models.

Pressure Transient Analysis and Sampling 9

® ). The “Enhanced Formation Dynamic Tester” is an advanced wireline formation testing system that delivers: (1) Multiple, large-volume highpurity formation fluid samples with downhole fluid characterization, (2) Reliable formation pressure testing, and (3) Real-time downhole fluid analyze, and more. Typical tool string configurations and architectures are shown in Figures 1.10 and 1.11. For detailed specifications, the reader is referred to the latest updated manufacturer’s literature. 1.2.1 Enhanced Formation Dynamic Tester (EFDT

Figure 1.10. Tool string configurations. COSL’s EFDT is designed to obtain formation pressures and formation fluid samples at discrete depths within a reservoir. Analyzing pressure buildup profile and the properties of fluid samples helps provide a more complete description of reservoir fluids and behavior. The EFDT service provides key petrophysical information to determine the reservoir volume, producibility of a formation, type and composition of the movable fluids, and to predict reservoir behavior during production. THE EFDT is a modular formation testing system. It can be customized for the specialized applications. The modularity of EFDT ensures its ability to test and sample fluids in a wide range of geological environments and borehole conditions. For its basic configuration, the string includes a fully controllable Dual Probe Module for fluid intaking, a Flow Pump Module for variable-volume drawdown and pump out of contaminated fluids, a Fluid Sensor Module for dynamic properties of fluids, a PVT Carrier Module for monophase sampling, and a Large Sample Carrier Module for large-volume normal sampling. It can also be configured with a Straddle Packer Module, an Optical Analysis Module, a Focused Probe Module and a Multi-PVT Tank Module to meet the requirements of complex reservoir formation tests, such as low permeability rock or natural fractures.

10 Supercharge, Invasion and Mudcake Growth

The EFDT enables up to five properties of fluid and formation to be monitored during testing: fluid conductivity or capacitivity, fluid density, fluid dynamic pressure, fluid optical analysis and formation permeability and anisotropy. The EFDT provides up to four MonoPhase Sampling Tanks (MPST) for one run, which recovers high-quality pressurecompensated reservoir fluid samples during borehole formation testing operations. The new Multi-PVT Module can take up to 24-48 PVT samples in one run (6 X 350 ml per module). The EFDT uses standard EDIB telemetry protocol, is combinable with other EDIB logging tools, and requires the company’s ELIS surface acquisition system. Surface control interfaces and user output displays are given in Figures 1.12 and 1.13. Applications, benefits and features are summarized below. Applications Formation pressure measurements and fluid contact identification Repeatable formation fluid sampling Measurement of formation permeability and anisotropy Vertical interference testing ln-situ downhole fluid analysis Benefits Fast, high-accuracy pressure measurement using Quartz Pressure Gauges (QPG) with temperature compensation Conductivity/capacitivity, density, fluid dynamic pressure, NIR optical analysis and formation permeability anisotropy for real-time reservoir evaluation Savings of 50% sampling time using focus probe Multiple samples in one run, providing high quality PVT samples Features Modularity, offering expanded testing versatility Accurate pressure measurement using QPG Real time downhole fluid assessment PVT quality formation fluid samples

Pressure Transient Analysis and Sampling 11

Figure 1.11. Tool architectures.

12 Supercharge, Invasion and Mudcake Growth

Figure 1.12. Surface control interface.

Figure 1.13. Pressure measurement chart (left) and real-time fluid monitoring chart (right).

Pressure Transient Analysis and Sampling 13 TM

1.2.2 Basic Reservoir Characteristic Tester (BASIC-RCT ). COSL’s “Basic Reservoir Characteristic Tester” or “BASIC-RCT” is a third generation product of the formation tester family, characterized by its pump through function. BASIC RCT is a compact, convenient, safe and efficient tool. It can replace in part Drill Stem Testing (DST) operations in order to save rig time. BASIC RCT provides economical and reliable solutions to formation evaluation for oilfield exploration and engineering, representing a good means to reduce cost while solving difficult technical problems. BASIC RCT can be run on any service company logging unit, requiring only winch, cable head and depth measurement. All services, telemetry, gamma ray recording, test recording (digital, numerical listing, screen and printer graphics) are provided in real time. Tool configurations are shown in Figure 1.14. For latest specifications, the reader should refer to the manufacturer’s updates. Functions Measuring formation pressure accurately Taking multi-samples of formation fluids Taking large samples Pumping through contaminated formation fluids Monitoring formation fluid properties in real time. Flowing formation fluids at controlled rates Pumping through in reverse Making quick well site sampler transfer Providing real time and reliable data for analyzing permeability and formation damage Structure The BASIC RCT is a combination of surface system and downhole tools. The surface system includes the Acquisition and Data Process software, PC and DC control panel, and AC power supply. The downhole tools include the upper electronics section, mechanical/hydraulic section, sensor section, lower electronics section with a standard configuration, and also include the 2 520 cc large sampler with optional configuration (see Figures 1.15 and 1.16). www.cosl

14 Supercharge, Invasion and Mudcake Growth

Figure 1.14. Tool string configurations.

Figure 1.15. Tool architecture.

Pressure Transient Analysis and Sampling 15

Figure 1.16. Tool and surface system.

Figure 1.17. Pressure drawdown curve (left) and fluid contact curve (right). 1.2.3 Enhancing and enabling technologies. While we principally focus on pressure transient analysis in this volume, a number of enabling technologies contribute to the operational success of formation testers in general, and in particular the robustness of the tools mentioned in Sections 1.2.1 and 1.2.2. A critical problem is that associated with “stuck tools,” which results in expensive fishing jobs, lost tools and increased rig costs.

16 Supercharge, Invasion and Mudcake Growth

Stuck tool alleviation. Issues related to stuck pipe are as old as drilling itself. In “Development on Incongruous Pushing and Stuck Releasing Device of EFDT,” by Qin, X., Feng, Y., Song, W., Chu, X. and Wang, L. and appearing in Journal of China Offshore Oilfield Technology, Vol. 4, No. 1, April 2016, pp. 70-74, the authors analyze the causes of differential pressure sticking during openhole wireline logging. Their modular IPSRD releasing device, designed for EFDT formation tester applications, could be seamlessly assembled to the tool. “Stuck Release Arms” (SRA) are driven by hydraulic forces that free the dual probe tool from adhesive forces. In Chapters 4 and 5, we show how mudcake thicknesses can be accurately modeled and predicted – small values to reduce chances for tool loss are needed, while larger thicknesses are required to seal tester pads to the sandface – at the same time, providing excellent descriptions for supercharge pressure effects. The authors importantly point out that while measuring pressure and sampling, even at a single point in the well, duration times may last several hours or even tens of hours. In particular, for higher mud densities, the possibility of differential sticking – and the likelihood of expensive fishing jobs – is high. In extreme cases, loss of the tool downhole and well abandonment are possible. Figure 1.18 explains the conceptual ideas behind IPSRD. The left diagram illustrates the differential sticking process, with the following nomenclature: 1Wellbore fluid, 2-Backup, 3-EFDT, 4-Mudcake, 5-Probe, 6-Protector and 7-Formation. The right side outlines the tool architecture. Upper Stuck and Lower Stuck release modules USRM and LSRM are found at the top and bottom, with the Dual Probe Module (DPM) residing between the two. The “stuck release arms” (SRA) for each releasing module are designed in opposite directions for pushing separately. The paper describes several field applications and savings in logging costs.

Figure 1.18. IPSRD stuck tool release mechanism.

Pressure Transient Analysis and Sampling 17

Field facilities. Finally, we offer some snapshots of COSL logging trucks and rigsite facilities from which formation testing jobs are run. The photographs are self-explanatory.

Figure 1.19. Rigsite facilities. 1.3 Recent Formation Testing Developments. Conventional formation tester tools with single and dual probes are shown in Figures 1.8 and 1.9, noting that different testers may be outfitted with different pad designs depending on the application. For instance, small round nozzles may be used with firm matrix rock; in low permeability formations, larger nozzles may be preferable in order to prevent excessive pressure drawdowns that result in the undesired release of dissolved gas or increased mechanical demands. Larger slot nozzles are ideal when formations are lower in permeability or naturally fractured and higher pump rates are desired.

18 Supercharge, Invasion and Mudcake Growth

The right-side diagram in Figure 1.6 shows an active pumping “sink probe” mounted on the mandrel, with a passive “horizontal” observation probe located 180o circumferentially away around the borehole. A “vertical probe” is also shown displaced axially from the sink probe and lying along the same azimuth. This conventional 1990s designed “triple probe” tool has seen wide application since its introduction. However, in low mobility formations, questions related to weak pressure signal detection and large diffusion arise. These have motivated the design of a new and different type of “triple probe” tester, where three independently operated, closer probes are located about the borehole at 120o separations, all residing in the same axial plane and supporting pumping and pressure measurement. Axially displaced “vertical probes” also augment the new triple probe design. The new COSL tool offers advantages over conventional instruments and these are described in a companion 2021 book Formation Testing – Multiprobe Design and Pressure Analysis by Lu, Zhou, Feng, Yang and Chin (John Wiley & Sons). Because of the threedimensional nature of the physics, the complementary volume develops new analysis and interpretation methods that account for borehole size and shape, and without invoking symmetry assumptions, since the probes may differ during any logging run and pump with different flow rate schedules. Figures 1.20 – 1.22 show example graphics from the book.

Figure 1.20. New triple probe formation tester. Pads with “small round nozzle and slot probe” (top) and “all long slot nozzles” (bottom).

Pressure Transient Analysis and Sampling 19

Figure 1.21. New COSL triple probe tester, perspective view.

Figure 1.22. Simulator menu for Probes 3, 7 and 11 (top), sink Probe 7 pressure drop versus kh and kv at fixed rate (bottom).

20 Supercharge, Invasion and Mudcake Growth

1.4 References. Chin, W.C., Formation Testing: Supercharge, Pressure Testing and Contamination Models, John Wiley & Sons, Hoboken, New Jersey, 2019. Chin, W.C., Zhou, Y., Feng, Y. and Yu, Q., Formation Testing: Low Mobility Pressure Transient Analysis, John Wiley & Sons, Hoboken, New Jersey, 2015. Chin, W.C., Zhou, Y., Feng, Y., Yu, Q. and Zhao, L., Formation Testing: Pressure Transient and Contamination Analysis, John Wiley & Sons, Hoboken, New Jersey, 2014. Lu, T., Qin, X., Feng, Y., Zhou, Y. and Chin, W.C., Supercharge, Invasion and Mudcake Growth in Downhole Applications, John Wiley & Sons, Hoboken, New Jersey, 2021. Lu, T., Zhou, M., Feng, Y., Yang, Y. and Chin, W.C., Multiprobe Pressure Analysis and Interpretation, John Wiley & Sons, Hoboken, New Jersey, 2021. Qin, X., Feng, Y., Wu, L., Tan, Z., Zhou, Y. and Chin, W.C., “Permeability and Pore Pressure Prediction in Highly Supercharged FTWD Environments,” submitted for publication, 2020. Qin, X., Feng, Y., Song, W., Chu, X. and Wang, L., “Development on Incongruous Pushing and Stuck Releasing Device of EFDT,” Journal of China Offshore Oilfield Technology, Vol. 4, No. 1, April 2016, pp. 70-74. Zhou, M., Feng, Y., Xue, Y., Zhou, Y., Chen, Y. and Chin, W.C., “Multiprobe Formation Testing – New Triple Arm Logging Instrument,” submitted for publication, 2020.

2 Spherical Source Models for Forward and Inverse Formulations The 1990s sparked important innovations in formation tester design, e.g., single, dual and triple probe tools, straddle packer applications, optical fluid analysis, and so on. Very well received were the “early time, low mobility, non-negligible flowline volume” inverse methods used to predict mobility and pore pressure in formations where earlier steady state methods were no longer optimal – for example, Halliburton’s GeoTapTM method was successful in commercializing such models. One of its inventors, W.C. Chin, later went on to win two Small Business Innovation Research (SBIR) awards from the United States Department of Energy in 2004 to extend the early work and to embark on other promising avenues of pressure transient and sampling research. This work continued beyond the life of the DOE contracts, resulting in many new methods and algorithms that would see book publication in the 2000s. In particular, these were Chin et al. (2014) or Formation Testing: Pressure Transient and Contamination Analysis, Chin et al. (2015) or Formation Testing: Low Mobility Pressure Transient Analysis, and Chin (2019) or Formation Testing: Supercharge, Pressure Testing and Contamination, all with John Wiley & Sons. These research monographs introduced new methods and provided mathematical and algorithmic details, practical validations, approaches motivated by electromagnetic logging, and so on, which the present authors hope would stimulate further advances. Nonetheless, the very rapid pace with which the new models were introduced meant that the entire portfolio of ideal “source models” could not be understood in perspective, even by those actively engaged in research and engineering. A practical state-ofthe-art summary emphasizing key ideas, and less so the formal math, was long overdue and is presented in this “introductory” chapter. Our methods are applicable to all formation tester manufacturers’ tools. 21

22 Supercharge, Invasion and Mudcake Growth

However, this need was not driven by dissemination objectives alone. During the same time frame, China Oilfield Services Limited (COSL) would embark on several programs to develop leading edge formation testing tools, a technology dominated by leading oil service companies Schlumberger, Halliburton and BakerHughes. This effort was all-the-more ambitious because COSL, a newcomer to formation testing, would need to establish its competence in conventional tools before its long term objective could be achieved. What was this objective? The industry’s leading tester, at its most basic level, consisted of a source (or sink) probe nozzle which, when pressed against the sandface, would extract formation fluid samples for surface evaluation. By products of this extraction are pressure transients measured at the source nozzle – and also at a passive observation probe displaced 180o circumferentially about the borehole. But physical intuition and field observation would confirm extremely small pressures from the faraway probe, or weak signal to noise ratios, especially at low mobilities, that would lead to inaccurate predictions in demanding reservoir applications. COSL engineering staff asked, “What if 180o probe spacings were reduced?” What if three probes, each spaced 120o apart, were used? And what if each probe were capable of operating independently, playing active as well as passive roles, during the logging process? This clearly opens up new possibilities in formation tester interpretation. An accurate, robust and rapid full three-dimensional simulator accounting for borehole curvature and pad geometry was needed which would also support mechanical design and field operations. It would address practical questions. For instance, what pump rate and nozzle combinations would allow fluid withdrawal without releasing dissolved gas? How are pump characteristics specified? How can triple probe redundancies support determination of local heterogeneities? Dip angle? Before such a simulator could be developed, the limitations in existing state-of-art methods must be understood. Such a simulator has been developed and is reported in 2021 companion book. The present chapter summarizes our knowledge of existing models, in particular, the advanced spherical and ring “source models” derived in the three prior books, which will continue to be useful in ongoing developments related to the new triple probe formation tester. Our compilation of general algorithms, in a single volume, provides a comprehensive discussion of key formation testing interpretation methods applicable to all manufacturers’ tools.

Spherical Source Formulations 23

2.1 Basic Approaches, Interpretation Issues and Modeling Hierarchies. In this opening section, we review the main ideas and models developed in the books Formation Testing: Pressure Transient and Contamination Analysis, Formation Testing: Low Mobility Pressure Transient Analysis and Formation Testing: Supercharge, Pressure Testing and Contamination Models published by John Wiley & Sons in 2014, 2015 and 2019. Our discussions provide greater insight than existed at the time and our ideas are now presented from the perspective of developers who have designed a much broader three-dimensional model. This does not mean that the earlier works, based on idealized spherical and ring sources, are dated. In fact, the work is just as relevant to future testers, which will host circumferentially positioned sensors and also passive and active pressure displaced axially along the tool axis. Early steady flow model. What are formation testers? Simply said, they are borehole logging instruments with pad nozzles which, when pressed against the sandface, extract or “sample” formation fluids for detailed examination at the surface. By-products of the sampling process are flowline pressure transient histories (at one or more probes) associated with pumping actions, which can be interrogated for valuable information related to formation properties like mobility, permeability, anisotropy, compressibility and pore pressure. The earliest methods, now several decades old, are based on formulas like “ks = CQ /(2 rp P)” and gave only “spherical permeabilities” (subscripted by “s”). These approaches required long wait times for steady-state pressure drops P to develop. Later, more flexible approaches using Horner-type approaches were developed; while decreasing waiting times, they unfortunately required additional rock and fluid information, e.g., porosity and compressibility, introducing inconvenience and potential error. Simple drawdown-buildup models. The above formula, which again required steady conditions, was excellent for high mobility formations where pressure equilibrium could be achieved in minutes or seconds. However, it does not apply in the presence of larger flow line volumes when mobilities are low. Pressures normally indicative of the downhole reservoir environment are initially forced to compress or expand the fluid cushion residing in the line so that actual formation characteristics are obscured or hidden – an analogy can be made to gauging the power of a boxer’s punch with the boxer wearing heavily padded gloves. When flowline volume effects are large, bearing in mind

24 Supercharge, Invasion and Mudcake Growth

that “large” is relative and depends on unknown fluid compressibility and mobility, the measured pressures are distorted and cannot be used to calculate properties like mobility, permeability or viscosity. The Darcy component of pressure cannot be identified; this problem is akin to “wellbore storage” issues in well testing. In response to this, petroleum engineers simply waited for flowline effects to dissipate or subside, which in low mobility formations may require many hours. Not only did this increase logging time and expense, but the risk of stuck tools rose substantially. Flowline storage problems had been accepted as inevitable until a series of interesting breakthroughs achieved in the 1990s.

Figure 2.1. Early COSL single and dual probe formation testers (where “dual” refers to axially displaced probes). In petroleum engineering, students are taught that boundary value problems governing physical phenomena consist of partial differential equations constrained by boundary and initial conditions. Solve the relevant formulation and the problem is fully understood. But the real problem is practical: many important formulations cannot be solved in closed analytical form, so that any physical insights and convenient formulas that would have been useful remain hidden in numerical data. And alternative computational solutions are only partly reliable: “artificial viscosities” arising from truncation and round-off errors contribute to uncertainties in permeability. Mark Proett, working at Halliburton in the 1990s, developed an approximate “boundary condition only” analytical approach valid at early times when storage and flow

Spherical Source Formulations 25

effects were equally strong. Similar approaches developed for isotropic media are now available, e.g., one at BakerHughes evolved to become the company’s “formation rate analysis” or “FRA.” Proett’s simplified approach is discussed in United States Patent No. 5,602,334, “Wireline Formation Testing for Low Permeability Formations Utilizing Pressure Transients,” awarded to M.A. Proett and M.C. Waid in February 1997. From its Abstract, “An improved formation testing method for measuring initial sandface pressure and formation permeability in tight zone formations exhibiting formation permeabilities on the order of 1.0-0.001 millidarcies based on pressure transients which occur shortly after the tester enters its pressure buildup cycle and substantially before reaching final buildup pressure. The method makes an estimate of formation permeability based on fluid decompression transients which occur in the formation tester flowlines which occur shortly after the tester begins its buildup cycle. The method further estimates initial sandface pressure based on the change in pressure over time shortly after beginning the buildup phase. The method of the present invention thereby permits accurate estimates of formation permeability and initial sandface pressure to be made relatively early in the buildup cycle, thus substantially reducing the time required to make the pressure and permeability measurements.” Proett’s heuristic model, surprisingly, was very successful in predicting spherical mobility and pore pressure in low mobility environments from highly transient data. This assessment was also based on the availability of synthetic data obtained from detailed “forward” finite element calculations where timewise pressures (used to validate Proett’s scheme) were determined from given or known permeabilities. In retrospect, this is not altogether surprising. Many problems in mathematical physics can be studied, at least initially, without solving the complete formulation. As a case in point, consider classical mass-spring-damper systems: if a small mass is struck quickly, its initial motion is completely determined by auxiliary conditions, but only subsequently does the complete differential equation matter. Similarly, in formation testing, the differential equation would need to be solved if additional information is required. Analytical drawdown-buildup solution. Motivated by this need, Wilson Chin solved the complete transient anisotropic formulation in the late 1990s, with both flowline storage and skin effects in closed analytical form, and demonstrated how Proett’s (constant rate) solution

26 Supercharge, Invasion and Mudcake Growth

provided the leading term of an asymptotic, low mobility expansion whose application could be further extended. This “exact solution” forms the basis for Halliburton’s drawdown-buildup GeoTapTM model used in real-time mobility and pore pressure prediction in “formation testing while drilling” (FTWD) or Measurement While Drilling (MWD) tools. Further details are given in the prior cited books and in U.S. Patent 5,703,286, “Method of Formation Testing,” awarded to W. Chin, M. Proett and C. Chen, Dec. 30, 1997. Typical predictions in the field require less than one minute of tool test time, thus enabling higher density and more economical well logging. The term “exact,” at the time, referred to analytical descriptions obtained using spherical source models; however, we emphasize that the early solutions are approximate, with “exact” now reserved for methods accounting for borehole diameter and curvature, and the presence of circumferentially positioned probes. We emphasize that Chin’s method, or “exponential solution,” assumes a pumping nozzle modeled by ideal “sources” or “sinks,” and provides pore pressure and spherical (or ellipsoidal, in the case of transversely isotropic media) permeability predictions using early time data. It does not, however, give horizontal and vertical mobility or permeability individually, which can differ substantially in different directions, unless measurements from an additional probe are available. In the 1990s, this meant dual probes axially displaced along the tool axis, although a diametrically opposite “180o probe” was available for limited use at higher mobilities. The success of the new physics-based drawdown-buildup approach motivated other challenges. Are other host physical interactions possible? “Is it possible to use pressure diffusion in a completely different way that reduces test times significantly? It turns out that “pulse interactions” and “phase delay” approaches are viable. Phase delay analysis. The prior question seems counter-intuitive because high mobilities imply rapid pressure equilibrium – thus, low mobilities would suggest long test times. However, this conclusion is only the case if one restricts attention to conventional constant rate pressure drawdown processes (which are used to derive classically used formulas like “ks = CQ /(2 rp P)).” In fact, there are pumping actions for which the opposite is true – by focusing on mechanisms that depend strongly on diffusion, it is possible to develop fast algorithms for permeability and pore prediction. In this regard, Chin turned to possible analogies found in electromagnetic logging, that is, resistivity prediction in high conductivity diffusive formations. In electromagnetic well

Spherical Source Formulations 27

logging, a transmitter broadcasts constant frequency AC waves, whose amplitude decay and phase (that is, time) delay are recorded at neighboring coil receivers. These measurements are interpreted using Maxwell’s equations as the host mathematical model and anisotropic resistivities can be estimated – in fact, the greater the diffusion, the higher the signal-to-noise ratio and the better the predictions. Chin introduced his “phase delay” approach to formation tester mobility prediction by developing an analogy to electromagnetic logging as follows (e.g., refer to U.S. Patent 5,672,819, “Formation Evaluation Using Phase Shift Periodic Pressure Pulse Testing,” awarded to W.C. Chin and M.A. Proett in September 1997). The tester pump was taken as the “transmitter” while a second passive observation probe assumed the role of the “receiver.” When the pump piston oscillates sinusoidally, it creates an AC wave whose pressure amplitude and time delay can be measured at the observation probe. These measurements are interpreted using Darcy’s equations to give mobility estimates, thus completing the analogy to electromagnetic logging. Experiments performed at Halliburton were successful. Interestingly, time delays, in contrast to those observed in resistivity logging, are large and could be ascertained visually from strip charts, thus reducing demands on computational and electronics resources. And mechanical requirements were not demanding – pump frequencies on the order of 1 Hz were sufficient. But many questions remained unanswered at the time. Once a pressure signal leaves the pumping probe, its fate is completely determined by the formation – the “receiver,” so to say, “sees what it sees.” But what happens if what it sees is poor in quality? And what if the pump piston cannot execute pure sinusoidal waves as required by theory, but only limited numbers of wave cycles that are, say, rectangular in shape? It turns out, however, that the form of the created wave can be controlled by varying flowline volume in time, thus providing a means for customization, “active tuning” and quality control; these effects are considered in Chin et al. (2015). At the time the work was first performed, there was little incentive to commercialize the phase delay approach at Halliburton. The invention applied only to isotropic media – the required theoretical extensions to anisotropic formations, in which the effects of dip angle would figure prominently, were not available. To determine isotropic permeability, the single-probe early-time drawdown method was more cost-effective, simpler and additionally provided pore pressure. The phase delay

28 Supercharge, Invasion and Mudcake Growth

approach, while elegant and interesting, required dual probe tools and could not give pore pressure estimates. Now, some two decades later, the needed generalization to anisotropic media with dip has been completed, together with more powerful extensions to low-mobility, early-time, drawdown-buildup methods. The combination of the two, as we demonstrated in the 2015 book, allows both horizontal and vertical permeabilities – not just “spherical permeabilities” alone – to be predicted from early time data in very low mobility formations. We have summarized key interpretation approaches developed in the 1990s. They were useful in that detailed math models were solved analytically in closed form, also demonstrating which parameter groups were significant physically and how field procedures could be optimized. But proper understanding of these contributions requires us to understand the limitations of idealized mathematical modeling methods themselves, their physical implications, and of course, their consequences. Modeling hierarchies. Few innovations to pressure transient interpretation appeared until Halliburton’s sponsored research starting in the 1990s. These initial efforts, summarized in “Advanced Permeability and Anisotropy Measurements While Testing and Sampling in RealTime Using a Dual Probe Formation Tester,” SPE Paper 64650, presented at the Seventh International Oil & Gas Conference and Exhibition in Beijing by M. Proett, W.C. Chin and B. Mandal in November 2000, introduced several avenues of research which saw subsequent development. The first was the low-mobility, early-time drawdown-buildup method discussed earlier; the second, a completely analytical solution to the full boundary value problem developed by Chin; and the third, the phase delay method, also due to Chin, although restricted then to isotropic media. Difficulties with the analytical solution, which manifested themselves only years later, would motivate further work supported by the United States Department of Energy. In the two decades since the “exact solution” appeared, two dozen Halliburton papers continuing this line of development have appeared. And given the wide dissemination of these publications, appearing in journals and conferences associated with the Society of Petrophysicists and Well Log Analysts (SPWLA), the Society of Petroleum Engineers (SPE) and other organizations, it is important to additionally clarify what was meant by “exact” then and what it refers to in the present context. To understand this further, we need to understand the subtle differences between real-world tools and their mathematical idealizations.

Spherical Source Formulations 29

Figure 2.2. Single probe formation tester (courtesy, COSL).

Figure 2.3. Dual probe formation tester (courtesy, COSL). Figures 2.2 and 2.3 for single and dual probe testers provide exploded views showing what single and dual probe formation testers typically look like. When lowered into the borehole and pressed against the sandface, the Darcy flow schematics given in Figure 2.4 apply. In these diagrams, the areas to the right of the red dashed line are associated with the actual domains of flow. If these right-side pictures are copied and “flipped over” about the vertical line, so that they also appear on the left, we typically have a composite diagram with a “hole in the middle”

30 Supercharge, Invasion and Mudcake Growth

and no trace of the borehole. Only a centered “source” or “sink” remains, which is taken as spherical in isotropic flow and ellipsoidal in transversely isotropic applications. The effects of the hidden borehole and pad are largely ignored but retained only to the extent that a multiplicative correction factor, known as the empirical “geometric factor,” now modifies the actual nozzle orifice radius. This is the model used in practice to date. Although the model may appear excessively simplified, we emphasize that the math formulation still defied ready solution. The effects of formation invasion and supercharge, for instance, could not be introduced or solved until Chin’s book publication in 2019. For now, we continue our discussion of ideal sources and sinks. Ideal no-flow plane

Large borehole

Slim hole

Figure 2.4. Piston pad pressed against the sandface. Different source models exist which are not “created equal” by any means. Originally, decades ago, “point sources” (in three-dimensional space) were assumed at which fluid literally vanished into “r = 0,” where pressures became infinite; flowline storage and skin effects could not be modeled. The work of Proett, Chin and Mandal (2000) introduced spherical and ellipsoidal sources with nonzero radii as shown in Figure 2.5a (the descriptor “point” is no longer used). Although the hardware associated with a flowline does not appear in this figure, flowline volume “V” is accounted for by a term in the boundary condition formulation, that is, “VC P/ t,” where C and P represent fluid compressibility and pressure (this term vanishes at steady state). An “exact” transient closed form analytical solution for Darcy pressure, expressed in terms of complex complementary error functions, was given in Chin et al. (2014).

Spherical Source Formulations 31

Figure 2.5a. Idealized spherical flow for isotropic formations, ellipsoidal flow for transversely isotropic (anisotropic) media.

Figure 2.5b. Axisymmetric “ring” source.

32 Supercharge, Invasion and Mudcake Growth

Math models associated with point and nonzero radii spherical sources are posed in just two independent variables, namely, the spherical coordinate “r” and a time “t.” This renders the formulations tractable and amenable to analysis; however, the spherical nature of the flow does not allow the effects of cylindrical boreholes to be modeled, e.g., mud invasion and supercharging. When borehole effects must be considered, the cylindrical system shown in Figure 2.5b must be used. To retain the analytical simplicity offered by a two independent variable approach, models assuming again, only “r” and “t,” with “r” now denoting the cylindrical radii measured from the borehole axis, are used. Consequently, the source described in this approach is the “cylindrical ring source” shown, which at first does not appear to be related to point source approximations. However, it can be shown with some analysis that both probe representations share some attributes that make their timewise predictions meaningful. Cylindrical source models host the rapid miscible and immiscible multiphase contamination formulations used in the “finite difference” approaches in Chin and Proett (2005) now used at Halliburton for routine job planning operations. Ring source approaches are no longer used for the three-dimensional flows in our complementary 2021 book since detailed azimuthal modeling is possible. When a modeling situation demands complete adherence to the actual physical geometry, e.g., geometric pad and orifice details, borehole irregularities, inclined geological layers, and so on, “finite element” analyses must be used. Such formulations support mesh generation flexibility, and typical three-dimensional grid systems are shown in Figures 2.6 – 2.9 for different real-world scenarios.

Figure 2.6. Ellipsoidal anisotropic flow, skin layer, three-dimensional finite element, boundary conforming mesh. * *

From “Advanced Dual Probe Formation Tester with Transient, Harmonic, and Pulsed Time-Delay Testing Methods Determines Permeability, Skin, and Anisotropy,” by M.A. Proett, W.C. Chin and B. Mandal, SPE Paper 64650 presented at the SPE International Oil and Gas Conference and Exhibition in China held in Beijing, China, November 7-10, 2000.

Spherical Source Formulations 33

Figure 2.7. “Near-Wellbore, Finite-Element Simulator (NEWSTM)” from Halliburton Energy Services.**

Figure 2.8. Dual probe, pretest, simulation-pressure contours, 100 md isotropic formation (to the left are 1 psi color bands, and to the right are 10 psi color bands). * *

Figure 2.9. Pressure contours for the first drawdown with two probes and the second drawdown with one source probe, 100 md horizontal permeability, 0.1 kv/kh, and 10 psi color contour bands. ** **

From “New Wireline Formation Testing Tool with Advanced Sampling Technology,” by M.A. Proett, G.N. Gilbert, W.C. Chin and M.L. Monroe, SPE Paper 56711 presented at the 1999 SPE Annual Technical Conference and Exhibition held in Houston, Texas, October 3-6, 1999.

34 Supercharge, Invasion and Mudcake Growth

In concluding this section, we again raise the question, “How ‘exact’ is exact?” Although fluids no longer vanished at a point and pressures were no longer infinite, as in early “point source” models, the newer finite-radius models of the 1990s still implied significant physical limitations. For example, by “folding” the right side diagram about the vertical dashed line, in effect simplifying the mathematical problem, the resulting left-right symmetry about the vertical disallows flow perpendicular to the line. And, it is not possible to consider the effects of fluid invasion or dynamic mudcake growth unless coordinates like those in Figure 2.5b are permitted (later, we actually solve the invasion or supercharging problem in spherical coordinates by considering a related formulation with initial spatially varying pressure distributions). In addition, the effects of borehole size and formation tester pad geometries cannot be studied using simple models. Although far from ideal, it is clear that the practical need for usable math models drives our use of simpler coordinates such as that in Figure 2.5a. It is now also clear why steady formulas like “ks = CQ /(2 rp P)” contain “calibration constants” or “geometric factors” C that conveniently lump all non-ideal effects into one single coefficient. We emphasize that different size pads with different borehole radius combinations will have different constants. Furthermore, steady-state models may be calibrated differently from transient models, high-mobility calibrations will differ from those for low-mobility, and anisotropic problems require still more careful consideration. Hence, the development of robust interpretation models, in which permeabilities are predicted from pressure transient data, is far from trivial. Finally, we emphasize that the multiplicative nature of the correction C is completely arbitrary – there is nothing rigorous about this procedure. So to answer the earlier question, “exact” more appropriately means “approximate, but fortunately analytical.” Just how are calibration constants determined? Quite simply, one needs to have truly “exact solutions” in a physical sense. These can be obtained computationally using three-dimensional simulations or experimentally in test fixtures developed for formation testing applications. Examples of numerical solutions are shown in prior work in Figures 2.7 – 2.9, where realistic effects include mudcake modeling, cylindrical borehole radius effects, pad geometry influence, and so on. Despite the apparent geometric generality, such models are not exact in a true sense. Care must be taken to avoid gridding errors and related biases. All numerical models, whether they are finite difference or finite

Spherical Source Formulations 35

element in nature, are based on approximate derivatives using Taylor series and will neglect higher-order terms. This omission, together with computer round-off errors, results in “artificial viscosities” which effectively change the assumed input permeabilities or mobilities. In other words, forward simulation results will typically not correspond to the permeabilities entered into the input box; inverse permeability predictions, for this reason, may not be entirely correct if they are obtained by repeatedly running a forward simulator. Calibration can also be performed using the results of carefully monitored laboratory experiments in custom designed test fixtures; examples of COSL lab facilities appear in our complementary 2021 volume. Field results are also possibilities, but often, the conditions under which measurements are obtained are unknown or cannot be controlled precisely. The analytical models in the three cited books hold several advantages. First, even if they are approximate, their mathematical structures allow engineers to better understand the physics and compare related results. For example, since it is the mobility k/ that is dynamically important, where k is the permeability and is the viscosity, identical physical phenomena are produced whether we have the heavy oil situation “k = 1 md or a contrasting water flow with = 20 cp” as “k = 0.05 md and = 1 cp.” Second, analytical solutions are convenient to program and execute; they are operable on personal computers or smart phones. Three-dimensional simulators which host models like those in Figures 2.7 – 2.9, by contrast, are not ideal for other reasons. For one, they are difficult to set up; they require significant computation times, often hours. And finally, expensive hardware resources and licensed software are needed. As such, they are typically not used for inverse methods or engineering trend analysis. Thus, simpler source models such as those mentioned provide an effective compromise between viable solutions versus none at all. Such solutions, including those describing supercharging, are summarized in this chapter and book. We note that fully three-dimensional models are developed in our 2021 volume; that is, actual circular boreholes with prescribed radii with multiprobe nozzles mounted circumferentially, having any geometric shape and independently operating at any positive or negative rate. The complete gamut of source models is summarized in this chapter, and we emphasize that these tools remain an essential part of the petroleum engineer’s tool kit. In this addition to the our formation testing series, we will provide numerous, easily understood applied examples.

36 Supercharge, Invasion and Mudcake Growth

2.2 Basic Single-Phase Flow Forward and Inverse Algorithms. This chapter outlines and summarizes models developed in the previously cited 2014, 2015 and 2019 book publications. All are based on well-defined math formulations and are solved analytically in closed form whenever possible. Our discussions here are concise and describe software models that have been commercialized and optimized for speed and user-experience. All are available to the petroleum industry and many of the algorithms have been customized for oil service company applications. Readers interested in further applications should contact Stratamagnetic Software, LLC (see “About the Authors”). In the following exposition, we will simply list the software models by name, discuss the assumptions utilized and the results produced, display the screen user interface, and list example outputs. All of the models are developed and described in detail in the prior books, to include equations and boundary value problem formulations, solution procedures, analytical formulas, and so on. These references also provide detailed validation results over wide ranges of input parameters, e.g., small, medium and large flowline volumes, early, middle and late time pressure response, small versus large mobilities, and so on, and interested readers are encouraged to study that literature. Here, we will only reproduce limited numbers of calculated results for brevity. Our purpose is to summarize our collective capabilities in this chapter, although we will introduce the newer model in a limited number of multiprobe examples later. We emphasize that our models apply to tools from all oil service manufacturers, whether wireline or Measurement While Drilling, and are designed to work in Windows environments without the need of special hardware or add-on software – the models are self-contained and will produce line graphs and three-dimensional color graphics automatically as needed when results are available. 2.2.1 Module FT-00.

This addresses exact transient liquid response in homogeneous anisotropic media with flowline storage. It solves for the unsteady Darcy pressure field about an ellipsoidal source surface immersed in a transversely isotropic infinite homogeneous medium, allowing full skin effect and general flowline storage boundary conditions. The “nonzero radius source” model, much more powerful than limited point source approximations, handles nearfield boundary conditions without becoming singular (that is, “blowing up”) at the origin as do point-wise

Spherical Source Formulations 37

models. The exact analytical closed form solution is given in Chin et al. (2014), and different time limits will yield early straight line time pressure variations, later exponential behavior, solutions of the Horner type, as well as final steady-state results. Imposed volume flow rates need not be constant. Hand entry of piecewise constant rates is supported, allowing arbitrary sequences of production or injection flow. The solutions for this complicated scenario are again analytical and here obtained from the method of superposition. We emphasize that cylindrical borehole effects and drilling fluid invasion are not incorporated. And the model, because it uses linear superposition, does not apply to nonlinear gas flows. The results created by FT-00 are all-important because they are exact in the sense that they are mathematically precise so long as the spherical (or ellipsoidal) representation of formation tester nozzles is permitted. These solutions are superior to those obtained by finite element or finite difference analysis because they are not associated with truncation or round-off error. As such, they are used to validate our inverse procedures. FT-00 is a “forward solver” in the sense that pressures are calculated that correspond to given fluid, formation and tool parameter inputs, plus a prescribed flow rate schedule – this might be a simple constant rate that achieves steady-state conditions, a double-rate sequence to model drawdown-buildup behavior, or one more complicated. Inverse procedures solve for permeabilities, mobilities, compressibilities and farfield pore pressures when limited numbers of transient pressure points is available. This problem is far more challenging than forward procedures. “Synthetic (pressure) data” created by reliable forward algorithms such as FT-00 are used to validate inverse methods, which are almost always approximate in nature. As derived in Chin et al. (2014), for problems with and without skin effects, all solutions are expressed in terms of “complex complementary error functions,” which reduce to the expected algebraic and exponential solutions in certain respective limits. 2.2.2 Module FT-01.

FT-01, also assuming spherical sources as in FT-00, represents an exact steady-state inverse method. Although the pressure response in a liquid is linear and superposition applies, the asymptotic evaluation of that expression at large time yields nontrivial dependencies in permeability that lead to different (but mathematically consistent and correct) equations for kh, kv and the anisotropy kh/kv. In fact, Chin et al. (2014) shows that cubic equations like kh3 + ( ) kh + ( ) = 0, kv3/2 + ( ) kv

38 Supercharge, Invasion and Mudcake Growth

+ ( ) = 0 and (kh/kv) + ( ) (kh/kv)1/3 + ( ) = 0 are obtained where ( ) represents various lumped parameters depending on pressure drops, fluid and formation properties. Solutions for horizontal and vertical permeability, and also anisotropy, are therefore available once candidate measured time and pressure pairs are available. In actual applications, because kh generally exceeds kv, its use is more reliable with less noise contamination. As is known from algebra, the polynomial kh equation may have three real roots, or it may have one real and two complex conjugate roots. Only positive real solutions for kh are physically meaningful – if several real positive roots are found, other logging data will be needed for a unique determination. On the other hand, negative permeabilities are not meaningful. Small imaginary parts, however, do not rule out the usefulness of roots (with positive real parts) since these typically arise from the use of unequilibrated pressure data or data inconsistent with Darcy’s equations, e.g., pad slippage, inaccurate transducer calibration and thermal effects. In such cases, user judgements and additional logging data would prove useful. Exact cubic equation solvers are used in FT-01 and its inverse predictions are shown to be consistent with input results from FT-00. 2.2.3 Module FT-03.

FT-00 and FT-01 derive from the same high-level mathematical source formulation, and the forward and inverse solutions are exact in the sense that they follow from closed form analytical solutions. However, this does not mean that they are exact in a physical sense. Source solutions possess spherical symmetries (ellipsoidal in anisotropic media) that, while elegant theoretically, are not representative of real tools – that is, pad nozzles mounted on solid mandrels. Consider a real tester, say a single probe FTWD tool, logging a horizontal well in transversely isotropic media. A top-mounted pressure transducer will “see” essentially kv while one that is side-mounted will see basically kh (actually, complicated functions of the two apply at any specific azimuthal angle). It is clear that pressures obtained at any two angles (even at the same axial station) can be used to determine kh and kv if a three-dimensional model is available for history matching. This capability presently exists: as a drillstring torques and untorques, it winds and unwinds, taking it through a range of twist angles. We emphasize that, in contrast to conventional dual probe interpretation methods requiring axially displaced pressure measurements, it is possible

Spherical Source Formulations 39

to determine kh and kv with a single probe tool provided a threedimensional algorithm is used for azimuthal interpretation. Module FT03 provides this capability, allowing convenient representation for pad and packer sources, arbitrary azimuthal and axial placement for multiple probes and modeling of bedding plane effects. This module is based on finite difference solutions obtained on curvilinear meshes that host both bed boundaries and drill collar contours. We now temporarily interrupt our general introduction of capabilities to illustrate specific applications. 2.2.4 Forward model application, Module FT-00.

We summarize the required input parameters by reproducing the software screen shown in Figure 2.10a. Several blocks are apparent, namely, “Fluid and Formation Parameters,” “Tool Properties,” and “Pumping Schedule.” Figure 2.10a shows input parameters for a multirate pump schedule with mixed production and injection, allowing pumping times of any duration.

Figure 2.10a. Forward simulation assumptions.

40 Supercharge, Invasion and Mudcake Growth

This supports constant rate pumping, pulse interaction and phase delay modeling. Simulations are fast and require at most seconds. The pump schedule is plotted in Figure 2.10b. Source and observation probe transient pressure responses are given in Figures 2.10c,d. For single probe tools, the “Probe separation” input may be ignored. Observe the rapid equilibration in source pressure and close correlation between it and flow rate (compare Figures 2.10b and 2.10c). At the observation probe, as seen in Figure 2.10d, slower equilibration and smearing due to diffusion are found. Generally. the lower the permeability, the greater the diffusion, which is bad and good. It is “bad” when steady pressures are needed for input into steady inverse models for permeability prediction; however, it is “good” when specially designed transient interpretation approaches are available. For low permeability formations, dynamical interactions between short duration pressure pulses are strong and highly dependent on anisotropy. Also, phase delays are more apparent and useful in low mobility applications.

Figure 2.10b. Pumpout schedule, volume flow rate.

Figure 2.10c. Source probe pressure.

Spherical Source Formulations 41

Figure 2.10d. Observation probe pressure. 2.2.5 Inverse model application, Module FT-01.

In the following run, the fluid, formation and tool parameters of Figure 2.10a are retained, except that the dip angle is changed from 0 to 45 deg and a constant 10 cc/s pump rate is assumed for all time. Source (left) and observation probe (right) pressure transient responses appear in Figure 2.11a – again, observe how diffusion slows the equilibration to steady-state just 15 cm away from the source.

Figure 2.11a. Source (bottom) and observation probe (top) pressure responses.

42 Supercharge, Invasion and Mudcake Growth Time (s) 0.100E+02 0.200E+02 0.500E+02 0.100E+05

psource (psi) -0.24948E+04 -0.25001E+04 -0.25045E+04 -0.25116E+04

pobserv (psi) -0.85789E+02 -0.91263E+02 -0.96208E+02 -0.10421E+03

The computed p’s (probe minus a dynamically unimportant pore pressure) are shown above. We now consider the inverse problem and may think of our input pressure pair data (above table) as being obtained from a dual probe tool. The user screen in Figure 2.10a assumes that the skin coefficient is S = 0; thus, the assumptions in the software screen for FT-01 in Figure 2.11b consistently assume zero skin (note, our models also handle nonzero skin effects, as discussed in Chin et al. (2014)).

Figure 2.11b. Inverse steady-state solver. First consider our 10,000 sec (three hour) data. Exact calculation shows three possible solutions, namely, Tentative permeabilities (md) ... Complex KH root # 1: Complex KH root # 2: Complex KH root # 3:

-10.97 + 10.07 + 0.91 +

0.00 i, KV: 0.83 0.00 i, KV: 0.99 0.00 i, KV: 121.96

In this case, two of the roots are easily ruled out; the first kh is negative, while the third kh is substantially less than kv. The remaining kh and kv results, at 10.07 md and 0.99 md, are almost identical to the assumed 10 md and 1 md in the forward simulation creating the data. The method reproduces assumed permeability data exactly in this calculation with non-vanishing dip angle.

Spherical Source Formulations 43

This success is a nontrivial event. FT-00 solves a fully transient model (via a complex complementary error function formulation with flowline storage and dip angle) while FT-01 solves an analytically derived polynomial equation valid only at steady-state. Agreement and consistency between the two approaches ensures correct mathematics, physics and software logic. This large time validation case thus provides a demanding test of both models. In field applications, one might use unsteady data that is not consistent with the math model due to cost considerations and risks of tool sticking. Using ten-second pressure data in this case gives kh = 12.49 md and kv = 0.65 md. Twenty-second data yields 11.68 and 0.74, while fifty-second data leads to 11.02 and 0.83 – all acceptable, relative to the 10 md and 1 md assumed in FT-00. This accuracy is possible because the formation is relatively permeable. However, in low-mobility applications, steady conditions are almost never realized and methods like FT-01 may not be applicable. If in the screen of Figure 2.11b we had checked “nonzero skin,” a different less restrictive math model with increased degrees of freedom is used. In the present calculation, the algorithm would return a list of possible solutions, that is, (kh, kv, S) triplets, together with the corresponding spherical permeability ks listed at the far right of the table below (the zero skin solution obtained above is highlighted in red). kh(md) kv(md) S ks(md) 7.00 8.00 0.62 7.32 7.00 9.00 0.63 7.61 7.00 10.00 0.64 7.88 8.00 5.00 0.52 6.84 8.00 6.00 0.56 7.27 8.00 7.00 0.59 7.65 9.00 3.00 0.36 6.24 9.00 4.00 0.44 6.87 10.00 1.00 0.01 4.64 10.00 2.00 0.21 5.85

2.2.6 Effects of dip angle. The effects of dip angle are well known physically. For example, at zero dip, the tester “sees” kh from all directions, while at 90 deg, it “sees” kh from left and right, but kv from top and bottom (actually, a complicated function of both applies at each azimuthal angle). As an example, first consider the forward simulation in Figure 2.12a with zero dip angle and a pump rate fixed at 10 cc/s for all time. We vary dip from

44 Supercharge, Invasion and Mudcake Growth

0 to 90 deg with other parameters unchanged. Source pressure responses for all runs are identical since they depend on spherical permeability ks only, as shown in Figure 2.12b. But, as expected, transient observation probe responses (numbered by dip angle) vary in both magnitude and shape as seen from Figure 2.12c. For the kh = 10 md, kv = 1 md example here, pressure drops vary over a 200 psi range as dip angles increase. Again, these results are exact.

Figure 2.12a. Constant rate pumping example.

Spherical Source Formulations 45

Figure 2.12b. Source probe response (all runs).

Figure 2.12c. Observation probe response versus dip angle.

46 Supercharge, Invasion and Mudcake Growth

We examine the foregoing conclusions from the inverse perspective. Suppose a deviated well were drilled at 45 degrees dip. The 45 degree dip, zero skin forward simulation gives large time (100,000 sec) pressure drops of 2,512 psi at the source and 104.6 psi at the distant probe; the corresponding inverse calculation gives a consistently accurate kh = 10.02 md and kv = 1.00 md (which is in agreement with the permeabilities assumed in the forward FT-00 simulation). But what if, for this measured pressure pair, the exact FT-01 inverse solver were not available? If the conventional industry-standard formula (implicitly assuming zero dip) were used, one would instead calculate kh = 7.43 md and kv = 1.81 md with a kv/kh of 0.244 versus an exact value of 0.1. Such errors imply grave production and economic consequences. In the table below, inverse calculations for permeability using FT-01 are performed with the above pressures through a range of dips to show the significance of hole deviation. Dip kh 0 7.43 30 8.35 45 10.02 60 14.13 70 20.88 80 41.89 90 424.89

kv 1.81 1.44 1.00 0.50 0.23 0.06 0.00

kv/kh 0.244 0.172 0.100 0.035 0.011 0.001 0.000

2.2.7 Inverse “pulse interaction” approach using FT-00. The above inverse approach requires fully equilibrated steady pressure drop data at source and observation probes. In a high permeability environment, this is not severe; as seen earlier, 20 sec data may well suffice under certain conditions if exact data integrity is not an issue. In “tight” formations, however, steady observation probe responses may not be possible for hours or days. Even if rig costs were not a concern, the risks of tool sticking are – thus, one seeks permeability prediction methods that respond to earlier time dynamic data. Now, the use of steady formulas for permeability interpretation is an artificial limitation used only to render the mathematics tractable. As noted, steady conditions are usually achievable in higher permeability formations so that such models are sometimes useful. But in low permeability zones, field experience and exact calculations (using FT-00) fortunately show that source probe responses equilibrate very rapidly. Since they depend only on the spherical

Spherical Source Formulations 47

permeability ks (and not kh or kv individually), the value of ks inferred from the source probe pressure drop is an accurate one for most interpretation purposes. The steady flow permeability formula ks = Q /{4 Rw(P0 – Ps)} can be used which, again, only constrains the relationship between kh and kv (here, Q is volume flow rate, is viscosity, Rw is effective probe radius, and “P0 – Ps” is the source probe pressure drop). In general, to individually quantify kh and kv, additional information is required. Unlike the method underlying FT-01, we will not draw upon steady pressure data obtained at the observation probe. In the illustrative example below, a ks of 4.642 md (corresponding to our earlier kh = 10 md, kv = 1 md and S = 0 case) is fixed throughout and three simulations are performed with different combinations of kh and kv. Again, source probe results for the three runs are identical, but observation probe pressure transients are discernible from each other. Figures 2.13a and 2.13b are clearly different – the former is highly smeared while the pulses in the latter remain distinct; peak pressure drops (from printed FT-00 output not shown) are 19 psi for Figure 2.13a and 159 psi for Figure 2.13b. Again, these differences are seen from early time transient behavior. Observation probe pressure transient waveform shapes in Figures 2.13b and 2.13c are similar, at least on a normalized basis. However, they are very different in magnitude. From printed FT-00 output, peak pressure drops are 159 psi and 787 psi, respectively. Clear differences in observation probe characteristics suggest that permeability contrasts can be effectively examined using short duration pulse interference rather than long time steady-state drawdown. The dynamical interactions are strongly dependent on anisotropy. Under what circumstances is our “pulse interaction method” expected to perform well? Interestingly, the lower the permeability, the better the accuracy – a counter-intuitive situation at odds with our experience with steady-state methods. The explanation is simple: at low mobilities, diffusion predominates, so that dynamical interactions between short pulses with high frequency content are strongest; thus, high signal-to-noise ratios are achieved for history matching. Interference effects are most pronounced at low perms when diffusion is dominant, precisely the field condition associated with long wait times, high rig costs and increased risk of tool sticking. We emphasize that, at earlier times, the effects of porosity, compressibility and flowline volume do appear, so that calculations using different pulse types with varied durations, amplitudes and time separations is advisable.

48 Supercharge, Invasion and Mudcake Growth

Figure 2.13a. kh = 10 md, kv = 1 md (that is, kh > kv).

Figure 2.13b. kh = kv = 4.642 md (that is, isotropic).

Figure 2.13c. kh = 1 md, kv = 100.0 md (that is, kh < kv).

Spherical Source Formulations 49

Additional research is presently directed at optimizing the pulse sequence used, e.g., evaluating different combinations of pulse amplitude, width, separation and number. It is important to emphasize that no new hardware is required for pulse interaction analysis. Our FT00 may be used in infinite homogeneous media, but in applications where bedding plane effects are important, the FT-03 simulator below applies. Because we are evaluating flow differences associated with diffusion itself, it is important that the host math model does not introduce numerical diffusion effects related to truncation errors. These effects, referred to as “artificial viscosity,” are most prominent with finite difference and finite element simulators, even when second, up to fourth order, schemes are employed. For this reason, exact analytical models such as FT-00 should be used to interpret pulse interactions. In this discussion, we focused on a zero skin example; space limitations preclude similar discussions on nonzero skin results. However, the basic ideas and results for pulse interaction methods remain unchanged. 2.2.8 FT-03 model overcomes source-sink limitations.

The word “exact” often used in our earlier literature conveys a sense of confidence and accuracy that may not be deserved. As discussed previously, “exact” referred to closed form analytical solutions for forward and inverse models – a significant endeavor – but which applies only to approximate and idealized source formulations of the formation tester. While the source model used, which applies boundary conditions on an ellipsoidal surface (with nonzero minor and major radii, and which does not “blow up” as in point source models) is useful in this regard, the model nevertheless is approximate in a physical sense. In other words, the model “sees” a type of spherical symmetry when, in fact, real formation tester nozzles mounted on real mandrels “see” largely ahead and are shielded from behind by solid material. Thus, a real tool in an anisotropic formation will measure different pressure responses as the transducer is rotated azimuthally about the tool axis. A full three-dimensional model accounting for actual tool geometry plus bedding plane effects could be used as a history matching tool to determine kh and kv from pressure data collected at different angles even at the same axial location. Such data is possible in field practice. Since the drillstring torques and un-torques during operation, it winds and unwinds, offering the opportunity to record pressures at different azimuths (and axial locations if desired).

50 Supercharge, Invasion and Mudcake Growth

Whereas earlier examples emphasized the use of dual probe tools with axially displaced sensors, the above ideas apply to single-probe tools such as those in simple wireline and FTWD applications. The basic idea is simple: two pressure measurements are need to provide two permeabilities. These can be taken from axially or azimuthally displaced probes, or both, as in triple probe tools. This leads to the question, “How can we construct a simulator capable of addressing these inverse and job planning needs?” Module FT-03 solves the transient Darcy partial differential equation for transversely isotropic media on a boundaryconforming curvilinear mesh in the cross-plane intersecting the tool. The model is fully three-dimensional. High resolution is provided about the borehole using “glove tight, wrap-around” inner meshes while those in the farfield conform to bed boundaries. An algebraically expanding mesh with an origin centered at the pad nozzle or oval pad, or at the center of a packer, provides good resolution axially. The computational domain appears in Figure 2.14a. The transient equation is solved by finite difference, forward time marching using an Alternating-DirectionImplicit (ADI) scheme for fast speed and numerical stability.

Figure 2.14a. Three-dimensional computational mesh. We focus on key simulation results and consider the diagrams in Figure 2.14b. The left two show a horizontally oriented borehole in a homogeneous transversely isotropic formation, while the right one shows the same hole bounded by upper and lower planes (along which pressure

Spherical Source Formulations 51

or no-flow conditions may be prescribed). Again, A “sees” essentially kh, B sees primarily kv, while the general point C sees a complicated function of the two. If pressure measurements at any two of A, B or C are available, one can determine both permeabilities by history-matching using a three-dimensional simulator such as FT-03.

Figure 2.14b. Borehole orientation.

Figure 2.14c. Azimuthal pressure response in layered media.

52 Supercharge, Invasion and Mudcake Growth

For our purposes, we refer to Figure 2.14c for a low perm run with typical field parameters. The source probe p at A is 18 psi (because the kh it “sees” is high) while that at B is 54 psi (the kv it sees is low). This sizable difference is detected at the source probe and can be used to estimate anisotropy. We emphasize that a single-probe tool can be used to determine kh and kv provided azimuthal measurements are taken. Figure 2.14c shows observation probe results, assuming that source and observation probes are in line axially (this is not required in simulations). The effects of bedding plane proximity and centralization are subtle and general conclusions cannot be made. However, the simulator accounts for these as they affect azimuthal transducer placement. 2.2.9 Module FT-04, phase delay analysis (introductory, for now).

The pulse interaction method, most effective at low perms, is closely related to our dynamical model for “phase delay” analysis. In electromagnetic logging, wave amplitude differences and phase delays between transmitters and receivers are used to infer resistivity anisotropy using Maxwell’s equations. A simple formation testing analogy exists. If the pump piston is operated periodically and pressure amplitude differences and time delays are recorded between source and observation probes, then kh and kv can be obtained from history matching using a three-dimensional Darcy flow simulator. Permeability formulas as functions of amplitude and phase delay are given in Chin et al. (2014) for simple formations and Chin et al. (2015) for complicated applications. In general, the lower the permeability, the higher the diffusion and the lengthier the phase delay. Larger delays mean more accurate time measurement and hence better predictions. The model has since been extended significantly to layered anisotropic media for dipping tool applications, as shown in Figure 2.15a. This complication means that simple formulas are not available and the interpretation must be pursued numerically. Figure 2.15b shows typical transient pressure responses calculated at source (red) and observation (green) probes using the FT-04 simulator of Figure 2.15a. The large amplitude decreases are typical of low permeability rocks, but small pressure levels at the observation probe may be difficult to measure accurately. On the other hand, time delays are easier and more precise. Phase delays can be calculated using any number of models, e.g., our idealized source model FT-00, the “real tool” simulator FT-03, or the “source in layered media” model in FT-04. We will develop the subject of phase delays in greater detail shortly.

Spherical Source Formulations 53

Figure 2.15a. Layered anisotropic media with dipping tool.

Figure 2.15b. Pressure transient response, amplitude and phase contrasts clear (source probe, red; observation probe, green).

54 Supercharge, Invasion and Mudcake Growth

Figure 2.15c. Multiple receiver phase delay formation tester (see, Section 2.3.5 for math modeling details).

Figure 2.15d. Transmitter-receiver, receiver-receiver operational modes (see, Section 2.3.5 for math modeling details).

Spherical Source Formulations 55

2.2.10 Drawdown-buildup, Module FT-PTA-DDBU.

In deep offshore wells where narrow pressure windows can spell the difference between useful production and impending disaster, accurate real-time pore pressure and mobility prediction are essential to safety. Exact models like FT-00 provide forward analyses, calculating transient pressures when fluid, formation and tool properties are given. FTWD and “pressure-while-drilling” (PWD) require fast inverse methods that provide pore pressure for safety objectives and gradient analysis for fluid identification, plus mobility analysis for economic and production planning. These predictions are needed in real-time during drilling. Equations that support FTWD objectives can be derived from the exact, closed form, analytical solution underlying FT-00. Because “while drilling” data are typically early time, or are often obtained in low mobility applications, or both, derived formulas used must account for transient behavior and flowline storage effects, which can distort and mask mobility trends in drawdown and buildup data. In this section, we introduce early-time inverse methods, which have undergone continual improvement in our own development process. Later in this chapter, we summarize advanced models that also account for supercharge invasion. Existing methods use exponential, real or complex complementary error functions, complicated integrals, and so on, which are combined with regression methods for interpretation. For instance, “least squares” fits are often employed first; while reasonable, they introduce smoothing assumptions beyond those in Darcy’s diffusion laws. Additional approximations are not desired. Our rapid early time inverse solution is derived using “rational polynomials” that reduce the computational overhead associated with transcendental functions. This allows more data processing to be completed in a given time, thus freeing microprocessor resources for other important interpretation and control functions. Closed form analytical solutions are derived and used, and numerical regression and chi-square methods are never employed. Two key models developed for “drawdown only” and “drawdown-buildup” applications are reported in Chin et al. (2014). Both methods apply to transient pressure data distorted by flowline storage effects and are still used commercially. That prior work is briefly summarized below, noting that more general extensions are described later for problems with multiple drawdown-buildups, supercharge for overbalanced drilling, and “undercharge” for underbalanced drilling. Mathematical derivations are given in the cited prior books.

56 Supercharge, Invasion and Mudcake Growth

Figure 2.16a. Nomenclature for pressure transient analysis. Unlike conventional models, ours only requires three pressure-time data points along either drawdown or buildup curve, e.g., as shown in Figure 2.16a, plus auxiliary data related to test setup. For brevity only, we present examples using buildup data, noting that detailed write-ups and additional applications are available in the earlier books.

Figure 2.16b. Exact FT-00 forward simulation results from single pre-test (note large flowline volume assumed).

Spherical Source Formulations 57

In Figure 2.16b, we use our exact FT-00 forward solver to create the source probe pressure transient data shown. From the input screen, the pore pressure is 10,000 psi while the mobility is 0.1 md/cp.

Figure 2.16c. Predicted pore pressure and mobility (lower right).

Figure 2.16d. Exact FT-00 forward simulation pressures for two sequential pre-tests used for input to inverse model (next page).

58 Supercharge, Invasion and Mudcake Growth

Figure 2.16c displays inputs for FT-PTA-DDBU with pressures taken at approximately 10, 15 and 20 sec. Rapid calculation gives 9,951 psi and 0.11 md/cp, close to exact values in Figure 2.16b. Sensitivity analyses using other time data points yield only small variations. In Figure 2.16d, we create pressure data for two pre-tests having different flow rates.

Figure 2.16e. Predictions (first pre-test).

Figure 2.16f. Predictions (second pre-test). Figures 2.16e and 2.16f offer predictions of 9,988 psi and 0.11 md/cp, and 9,960 and 0.11 md/cp, using first and second pre-test data, values very close to assumed numbers on the prior page. Again, the solutions execute rapidly, are analytical and easily programmed, e.g., twenty lines of source code depending on the host compiler language. Also, because no iterations are involved, downhole computer microprocessor resources are utilized efficiently and available for other interpretation and control functions. Algorithms have been completed and validated for “drawdown only” as well as buildup test cycles – multiple drawdown and buildup combinations are summarized later.

Spherical Source Formulations 59

2.2.11 Real pumping, Module FT-06.

In real-world applications, flow-rate versus time functions are never constant or piece-wise constant as assumed in FT-00. Flow rates invariably ramp-up and down, with such effects being even more dramatic with highly compressible gas pumping. These features must be modeled numerically and analytical methods are not possible. Figure 2.17a shows the interface for a liquid-gas simulator which supports flowrate functions that may be, for instance, trapezoidal as in Figure 2.17b unlike the piecewise rectangular functions required of FT-00. Note that the interface for FT-06 is designed to closely mimic that of FT-00. Source and observation probe pressures are also given for the present application. Also, in lieu of kh and kv, one may specify ks and kv/kh.

Figure 2.17a. FT-06 liquid-gas simulator inputs.

60 Supercharge, Invasion and Mudcake Growth

Figure 2.17b. FT-06 pump rate and pressure solutions.

Spherical Source Formulations 61

2.3 Advanced Forward and Inverse Algorithms. With this section, we embark on models offering increased simulation capability, e.g., advanced drawdown-buildup models handling multiple constant flow rates, multiphase flow analysis and mud contamination algorithms, and rigorous methods that handle forward and inverse methods with significant borehole invasion and supercharge effects. Many of the models are based on spherical or ellipsoidal sources. Our multiphase invasion models are developed in cylindrical coordinates which naturally describe actual borehole contours. However, the cylindrical ring source discussed earlier is used to represent pumping actions. In contrast, the work in our companion 2021 book addresses actual boreholes with multiple discrete source or sink nozzles positioned at different circumferential positions. In this book, our fully threedimensional models will focus mostly on compressible liquids, with gas pumping being the emphasis of future development. 2.3.1 Advanced drawdown and buildup methods.

In conventional “repeat formation tester” (that is, Schlumberger RFTTM ) applications, permeabilities are predicted using steady-state spherical flow models for drawdown pressure. These assume higher mobilities which allow rapid equilibration. Independent analyses can then be performed because observed drawdowns performed sequentially do not interact with each other. When mobilities are small, however, the formulas are not valid and detailed history matching with transient Darcy flow simulators is required. In our approach, fast, rigorous methods are developed using rapidly changing early-time drawdown or buildup data which satisfactorily predict mobility, compressibility and pore pressure. Use of early data is crucial operationally because of reduced cost, decreased logging time and minimal risk of stuck tools. We emphasize that, unlike conventional inverse models that repeatedly run forward models in “brute force” manner using guessed values for permeability, ours provide direct solutions for both permeability and pore pressure – that is, rapid solutions are obtained in a single pass. Large flowline volumes are permitted as are unequally spaced pressure data points in time (only three pressure-time datasets are required). Moreover, because the algorithm is compact and fast, it is easily adapted for downhole microprocessor use to support real-time applications. The technique is illustrated with data from a well known industry example.

62 Supercharge, Invasion and Mudcake Growth

Basic steady model. The use of formation testers in permeability or mobility prediction is well established and numerous methods are available for different tool designs and operational procedures. The flow into the RFTTM, sketched in Figure 2.18, is typically assumed to be rapidly established and “substantially steady-state” (e.g., see Dussan, Auzerais and Kenyon (1994)), as indicated by the bottom arrow (ours) in the familiar diagram of Figure 2.19a. If P is the steady drawdown pressure, C is a flow shape factor, q is the volume flow rate, is the liquid viscosity, and rp is an “effective probe radius,” then the drawdown permeability can be calculated from the simple relation given by kd = Cq /(2 rp P). For convenience, the “C/(2 rp)” is usually represented as a single number, say kd = 5,660 q / P. Here kd, q, and P are expressed in md, cc/s, cp and psi. The constant 5,660 applies to the “standard” RFT probe. When the “large diameter” or the “fast-acting” probe is used, the constant should be 2,395; for the “large-area packer,” the constant becomes 1,107 (see Schlumberger’s Log Interpretation Principles/Applications (1989) for details – the notation used in this paragraph is retained for consistency with the existing literature and differs slightly with ours below). The previous values are calibration constants accounting for non-spherical effects like borehole wall curvature and pad geometry. These models have limitations, and to understand and overcome them, we have developed a general approach. Figure 2.19a illustrates a “double drawdown” – Bowles (2004), in the context of dual packers, notes that double-drawdown tests are effective in removing the need to perform repeated formation tests.

Figure 2.18. Repeat formation tester (RFTTM).

Spherical Source Formulations 63

Figure 2.19a. Test procedure from Schlumberger U.S. Patent 5,279,153 (flat pressures assume rapid equilibrium). As we show in Section 2.3.2 later, Figure 2.19a data are conventionally analyzed by twice application of the steady “kd = Cq /(2 rp P)” formula since the two (equilibrated) pressure traces act independently of each other. This redundancy is performed as a doublecheck on pretest results. For general low mobilities, however, this is not possible; we have developed multiple-drawdown methods for interacting drawdowns and will show how permeabilities are predictable from resulting buildups. Note that it is operationally preferable to work with closely spaced drawdowns, even if they interact strongly, in order to reduce expensive well logging times. The 1990s work of Chin and Proett at Halliburton provided direct calculations for isotropic mobility and pore pressure for problems with a single drawdown or a single drawdown followed by a single buildup. In this Section 2.3.1, which addresses one contribution of our 2015 Low Mobility book, the earlier Halliburton model is extended to double drawdown problems followed by a single buildup. The 2015 book provides detailed mathematical derivations used in the software model described below. These details, not duplicated here for brevity, motivate a more general method later. Validating our method. We now consider applications of our work. An important and obvious question that arises is, “How accurate are our claimed inverse benefits?” After all, laboratory results have yet to be performed – but fortunately, we can obtain physically useful pressure traces for inverse evaluation using results inferred from the published steady data in Figure 2.19a. In the 2015 book, three methods are described, namely, Method 1 – Drawdown alone test, Method 2 – Single drawdown and single buildup test, and Method 3 – Double

64 Supercharge, Invasion and Mudcake Growth

drawdown and single buildup test. We will soon show that useful interacting drawdowns can be developed from baseline Schlumberger results, and then how, using point-wise data taken from the unsteady curves, we can recover known input permeabilities. The new method applies to double-drawdown (left) and single-drawdown (right) of Figure 2.19b below, both in low mobility anisotropic applications. Additional use of “rational polynomial” (as opposed to exponential) solutions in our model, plus accurate “three data point” (without data smoothing) approaches permit fast calculation speeds useful in surface operations. This is particularly important in downhole microprocessor environments where computing resources like memory and speed are limited.

Figure 2.19b. New method for multiple drawdowns (refer to lower transient curves on both left and right sides).

Spherical Source Formulations 65

2.3.2 Calibration results and transient pressure curves.

Here we use Schlumberger’s steady published results to determine a geometric factor for use with our FT-00, which will be, in turn, used to produce transient pressure histories suitable for inverse analysis evaluation. We consider the well known example from Page 10-9 of the company’s Log Interpretation Principles/Applications. Data from drawdown analyses of two pretests (with 10 cc each) show that

and

P1 = 2,050 psi T1 = 15.4 sec q1 = 10/15.4 = 0.65 cc/sec P2 = 4,470 psi T2 = 6.1 sec q2 = 10/6.1 = 1.64 cc/sec

where the upper case T’s are defined in our Figure 2.19a (these parameters are used later in the FT-00 screen of Figure 2.20a). The well was drilled using mud with a viscosity of = 0.25 cp. Using the approximate spherical steady flow formula shows that and

kd1 = 5660

0.65

0.25/2050 = 0.45 md

kd2 = 5660

1.64

0.25/4470 = 0.52 md

In this example, the two values of permeability agree well. This consistency indicates that, for the tests conducted, steady-state pressures were found in both cases – reference to the log in Figure 10-19 of the original publication does, in fact, show a “flat part” or horizontal asymptote adjacent to the bottom arrow (ours) of our Figure 2.19a. We now proceed with an exercise showing how the above data and behavior are consistent with an exact FT-00 analysis for a suitably chosen geometric factor. We assume that the formation has a permeability of kd = 0.5 md as suggested by the average of 0.45 and 0.52 md above. As discussed in Chin et al. (2014), the exact solution to the full formulation has been incorporated in forward analysis algorithm in FT-00 which supports a general pump-out schedule using linear superposition methods. Shown in Figure 2.20a is the self-explanatory menu for the most general case, namely, transversely isotropic flow with flowline storage and skin – note that we have inputted the flow rates 0.65 and 1.64 cc/sec, the viscosity 0.25 cp, and two permeabilities of 0.5 md.

66 Supercharge, Invasion and Mudcake Growth

Figure 2.20a. Schlumberger mobility parameters assumed. Entering fluid, formation and tool parameters produces, within a fraction of a second on Intel i5 machines, complete solutions for source and, if applicable, multiple observation probes. Clicking on “Simulate” produces the “double bump” source probe drawdown response in Figure 2.20b for the pumping schedule shown (again, note how steady-state asymptotes appear for both tests). By trial and error, we have selected a geometric factor G of 0.38 which adjusts computed pressure drops seen from Figure 2.20b to the 2050 ( 20,000 – 18,000) and 4470 ( 20,000 – 15,100) psi values indicated in the Schlumberger data – only a single G value is required for this fit. Figure 2.20b does verify that the selected simulation parameters do in fact lead to rapidly equilibrating pressures. Note that exact agreement is not possible because of the approximate 0.5 md assumed; also, we indicate that the time 21.5 = 15.4 + 6.1 is used in the rate schedule. Our “0.38” geometric factor is analogous to the

Spherical Source Formulations 67

“5,660” used in the Schlumberger calculations. So far, we have shown that our transient FT-00 formulation and the steady spherical model are consistent at large times. We emphasize that our FT-00 model is exact and provides transient “synthetic data” valid for any point in time. This data is useful, as we will show, for evaluating our inverse methods. The question now is, “To what extent can input permeabilities be recovered from rapidly varying pressure transient points derived from low mobility extensions of Figure 2.20b?”

Figure 2.20b. Pressure response inferred from steady Schlumberger. 2.3.3 Mobility and pore pressure using first drawdown data.

Use of the above geometric factor ensures that pressure traces obtained using FT-00 are physically meaningful. Note that the source probe response in Figure 2.20b “levels off” horizontally twice, indicating steady behavior. We wish to extend the above baseline run transiently to those where the two drawdowns interact strongly. Families of extended pressure curves can be created by increasing decreasing mobility. This is easily done by increasing the viscosity from 0.25 cp to 1 cp. For the parameters indicated, this increase is enough to eliminate the appearance of any steady-state pressure response, so that all drawdown pressures are highly transient and overlap. This low-mobility effect, together with any nonzero flowline volume, can be detrimental to conventional steady-state formation permeability prediction. We will focus on the performance of the inverse method as flowline volume increases from “acceptable” to very large values for our 1 cp fluid.

68 Supercharge, Invasion and Mudcake Growth 2.3.3.1 Run No. 1. Flowline volume 200 cc.

(Software reference, pta-dd-3-run-with-rft-numbers.exe)

In this example, we consider a flowline volume of 200 cc. This and the calculated pressure response are shown in Figure 2.21a. It is clear that no steady state exists, so that “ks = CQ /(2 rp P)” is inapplicable.

Figure 2.21a. Flowline volume, 200 cc. Clicking on “Simulate” also produces the exact transient source probe pressure response below (again, at 15.4 sec, the initial drawdown ceases and is replaced by a stronger one). We will use pressure results from 3.6, 9.9 and 14.4 sec (highlighted in red) as inputs to our inverse model. This will attempt to recover the mobility of 0.5 md/cp and the pore pressure of 20,000 psi assumed in Figure 2.21a. In Runs No. 1-4, we will use drawdown data from the very left of our pressure curves. EXACT FT-00 Time (s) 0.000E+00 0.900E+00 0.180E+01 0.270E+01 0.360E+01 0.450E+01 0.540E+01 0.630E+01 0.720E+01 0.810E+01

FORWARD ANALYSIS RESULTS Rate (cc/s) Ps* (psi) 0.65000E+00 0.20000E+05 0.65000E+00 0.19084E+05 0.65000E+00 0.18275E+05 0.65000E+00 0.17559E+05 0.65000E+00 0.16926E+05 0.65000E+00 0.16366E+05 0.65000E+00 0.15871E+05 0.65000E+00 0.15433E+05 0.65000E+00 0.15045E+05 0.65000E+00 0.14702E+05

Spherical Source Formulations 69 0.900E+01 0.990E+01 0.108E+02 0.117E+02 0.126E+02 0.135E+02 0.144E+02 0.153E+02 0.162E+02 0.171E+02 0.180E+02 0.189E+02 0.198E+02

0.65000E+00 0.65000E+00 0.65000E+00 0.65000E+00 0.65000E+00 0.65000E+00 0.65000E+00 0.65000E+00 0.16400E+01 0.16400E+01 0.16400E+01 0.16400E+01 0.16400E+01

0.14398E+05 0.14129E+05 0.13891E+05 0.13680E+05 0.13493E+05 0.13328E+05 0.13181E+05 0.13052E+05 0.11688E+05 0.10337E+05 0.91422E+04 0.80852E+04 0.71500E+04

The inverse program was run with the red highlighted data. Screen outputs with iteration history are shown below. Convergence to 0.515 md/cp and 19,981 psi (shown in red) was achieved and agrees with input values of 0.5 and 20,000. It is worth noting that none of the input pressures used are even close to 20,000 psi. In general, pressure data near t = 0 should not be used. Computation time was approximately one second. We ask, “Can we duplicate this success with increased values of flowline storage, that is, with additional pressure distortion?” Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 1st Point Time T3 (sec): Pressure P3 (psi): Run 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. . . 130. 131. 132. 133. 134. 135.

0.650 0.500 0.380 3.600 16926.000 9.900 14129.000 14.400 13181.000

Error 52.3 % 52.0 % 51.8 % 51.5 % 51.2 % 51.0 % 50.7 % 50.5 % 50.2 % 49.9 %

P0(psi) 18532 18540 18548 18556 18564 18572 18580 18588 18596 18605

Md/Cp 0.009 0.018 0.027 0.036 0.044 0.053 0.061 0.070 0.078 0.086

2.9 2.4 1.9 1.3 0.7 0.2

19911 19925 19939 19953 19967 19981

0.511 0.512 0.513 0.513 0.514 0.515

% % % % % %

Stop - Program terminated.

2.3.3.2

Run No. 2. Flowline volume 500 cc.

Here we increase the flowline volume to 500 cc with all other parameters unchanged. The source probe pressure response in Figure 2.21b does not show any steady behavior at all.

70 Supercharge, Invasion and Mudcake Growth

Figure 2.21b. Flowline volume, 500 cc. EXACT FT-00 Time (s) 0.000E+00 0.900E+00 0.180E+01 0.270E+01 0.360E+01 0.450E+01 0.540E+01 0.630E+01 0.720E+01 0.810E+01 0.900E+01 0.990E+01 0.108E+02 0.117E+02 0.126E+02 0.135E+02 0.144E+02 0.153E+02 0.162E+02 0.171E+02 0.180E+02 0.189E+02 0.198E+02

FORWARD ANALYSIS RESULTS Rate (cc/s) Ps* (psi) 0.65000E+00 0.20000E+05 0.65000E+00 0.19620E+05 0.65000E+00 0.19258E+05 0.65000E+00 0.18913E+05 0.65000E+00 0.18585E+05 0.65000E+00 0.18273E+05 0.65000E+00 0.17976E+05 0.65000E+00 0.17693E+05 0.65000E+00 0.17423E+05 0.65000E+00 0.17167E+05 0.65000E+00 0.16922E+05 0.65000E+00 0.16690E+05 0.65000E+00 0.16468E+05 0.65000E+00 0.16257E+05 0.65000E+00 0.16056E+05 0.65000E+00 0.15865E+05 0.65000E+00 0.15683E+05 0.65000E+00 0.15509E+05 0.16400E+01 0.14828E+05 0.16400E+01 0.14116E+05 0.16400E+01 0.13439E+05 0.16400E+01 0.12794E+05 0.16400E+01 0.12180E+05

We again run our inverse program, as shown below. As expected, the pressure inputs taken at the same times as before are substantially changed. Here, we predict 0.512 md/cp and 19,995 psi – again close to the assumed values of 0.5 md/cp and 20,000 psi.

Spherical Source Formulations 71 Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 1st Point Time T3 (sec): Pressure P3 (psi):

0.650 0.500 0.380 3.600 18585.000 9.900 16690.000 14.400 15683.000

Run Error P0(psi) 1. 25.2 % 19673 2. 24.8 % 19678 3. 24.4 % 19684 4. 24.0 % 19689 . . 50. 2.3 % 19968 51. 1.8 % 19974 52. 1.3 % 19981 53. 0.7 % 19988 54. 0.2 % 19995 Stop - Program terminated.

Md/Cp 0.013 0.027 0.040 0.053 0.486 0.493 0.499 0.506 0.512

2.3.3.3 Run No. 3. Flowline volume 1,000 cc.

Here we repeat the above calculation, except that flowline volume in increased to 1,000 cc. Calculations again demonstrate accurate predictions using very early-time transient data. The mobility is 0.510 md/cp as opposed to 0.5 md/cp, and the pore pressure is almost identical to the 20,000 psi assumed in the FT-00 data generation.

Figure 2.21c. Flowline volume, 1,000 cc.

72 Supercharge, Invasion and Mudcake Growth EXACT FT-00 FORWARD ANALYSIS RESULTS Time (s) Rate (cc/s) Ps* (psi) 0.000E+00 0.65000E+00 0.20000E+05 0.900E+00 0.65000E+00 0.19807E+05 0.180E+01 0.65000E+00 0.19620E+05 0.270E+01 0.65000E+00 0.19436E+05 0.360E+01 0.65000E+00 0.19258E+05 0.450E+01 0.65000E+00 0.19083E+05 0.540E+01 0.65000E+00 0.18913E+05 0.630E+01 0.65000E+00 0.18747E+05 0.720E+01 0.65000E+00 0.18585E+05 0.810E+01 0.65000E+00 0.18426E+05 0.900E+01 0.65000E+00 0.18272E+05 0.990E+01 0.65000E+00 0.18122E+05 0.108E+02 0.65000E+00 0.17975E+05 0.117E+02 0.65000E+00 0.17831E+05 0.126E+02 0.65000E+00 0.17691E+05 0.135E+02 0.65000E+00 0.17555E+05 0.144E+02 0.65000E+00 0.17422E+05 0.153E+02 0.65000E+00 0.17292E+05 0.162E+02 0.16400E+01 0.16904E+05 0.171E+02 0.16400E+01 0.16493E+05 0.180E+02 0.16400E+01 0.16092E+05 0.189E+02 0.16400E+01 0.15701E+05 0.198E+02 0.16400E+01 0.15320E+05 Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 1st Point Time T3 (sec): Pressure P3 (psi): Run 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. . . 20. 21. 22. 23. 24. 25. 26. 27.

0.650 0.500 0.380 3.600 19258.000 9.900 18122.000 14.400 17422.000

Error 13.3 % 12.8 % 12.3 % 11.8 % 11.4 % 10.9 % 10.4 % 9.9 % 9.4 % 8.9 %

P0(psi) 19910 19913 19916 19920 19923 19926 19929 19933 19936 19940

Md/Cp 0.022 0.045 0.067 0.088 0.110 0.131 0.151 0.172 0.192 0.212

3.9 3.3 2.8 2.3 1.7 1.2 0.7 0.1

19974 19977 19981 19984 19988 19992 19995 19999

0.396 0.413 0.430 0.446 0.463 0.479 0.495 0.510

% % % % % % % %

Stop - Program terminated.

Spherical Source Formulations 73 2.3.3.4 Run No. 4. Flowline volume 2,000 cc.

Finally in Run No. 4, the flowline volume is increased to an extremely large value of 2,000 cc. Pore pressure prediction is again excellent at 19,999 psi, but mobility is still accurate and acceptable at 0.492 md/cp, as opposed to 0.5.

Figure 2.21d. Flowline volume, 2,000 cc. EXACT FT-00 Time (s) 0.000E+00 0.900E+00 0.180E+01 0.270E+01 0.360E+01 0.450E+01 0.540E+01 0.630E+01 0.720E+01 0.810E+01 0.900E+01 0.990E+01 0.108E+02 0.117E+02 0.126E+02 0.135E+02 0.144E+02 0.153E+02 0.162E+02 0.171E+02 0.180E+02 0.189E+02 0.198E+02

FORWARD ANALYSIS RESULTS Rate (cc/s) Ps* (psi) 0.65000E+00 0.20000E+05 0.65000E+00 0.19903E+05 0.65000E+00 0.19807E+05 0.65000E+00 0.19713E+05 0.65000E+00 0.19620E+05 0.65000E+00 0.19527E+05 0.65000E+00 0.19436E+05 0.65000E+00 0.19346E+05 0.65000E+00 0.19257E+05 0.65000E+00 0.19170E+05 0.65000E+00 0.19083E+05 0.65000E+00 0.18997E+05 0.65000E+00 0.18913E+05 0.65000E+00 0.18829E+05 0.65000E+00 0.18746E+05 0.65000E+00 0.18665E+05 0.65000E+00 0.18584E+05 0.65000E+00 0.18505E+05 0.16400E+01 0.18295E+05 0.16400E+01 0.18071E+05 0.16400E+01 0.17850E+05 0.16400E+01 0.17632E+05 0.16400E+01 0.17416E+05

74 Supercharge, Invasion and Mudcake Growth Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 1st Point Time T3 (sec): Pressure P3 (psi):

0.650 0.500 0.380 3.600 19620.000 9.900 18997.000 14.400 18584.000

Run Error P0(psi) 1. 6.7 % 19977 2. 6.2 % 19979 3. 5.7 % 19981 4. 5.2 % 19983 5. 4.6 % 19984 6. 4.1 % 19986 7. 3.6 % 19988 8. 3.1 % 19990 9. 2.6 % 19992 10. 2.0 % 19994 11. 1.5 % 19995 12. 1.0 % 19997 13. 0.4 % 19999 Stop - Program terminated.

Md/Cp 0.041 0.082 0.121 0.161 0.200 0.238 0.276 0.313 0.350 0.386 0.422 0.457 0.492

2.3.4 Mobility and pore pressure from last buildup data. In Runs 1-4, we used the far left of our computed pressure curves, that is, we effectively considered early time, single drawdown data. We showed that mobility and pore predictions were excellent despite large values of flowline volume. Now, in Runs No. 5-8, we will use the farright buildup portion of the pressure curve to predict mobility and pore pressure. This represents a far greater inverse challenge, since this buildup follows two interacting drawdown operations. We emphasize that our “late time” portion of the response is significantly earlier than typical times used in Horner plots. The model now includes four linear superpositions, but the iterative process is identical to that for early drawdowns and only slightly slower, typically one second per inversion on Intel i5 machines. In the examples below, we randomly select time points for inverse model input. 2.3.4.1 Run No. 5. Flowline volume 200 cc. (Software reference, pta-two-dd-2.exe)

We again assume a small 200 cc flowline volume. Exact pressure data computed from FT-00 are displayed corresponding to inputs in Figure 2.21a. Results in red are used in our inverse model – they are taken from the “zero rate” buildup part of the table. The inverse software now includes two added inputs relative to the prior drawdown model, namely, the drawdown end times for two pumping cycles. The predicted pore pressure and mobility, namely, 19,974 psi and 0.516 md/cp, are excellent when compared to the exact values 20,000 psi and 0.5 md/cp.

Spherical Source Formulations 75 Time (s) Rate (cc/s) Ps* (psi) 0.000E+00 0.900E+00 . .

0.65000E+00 0.65000E+00

0.20000E+05 0.19084E+05

[Buildup begins at 21.5 seconds.] . 0.189E+02 0.198E+02 0.207E+02 0.216E+02 0.225E+02 0.234E+02 0.243E+02 0.252E+02 0.261E+02 0.270E+02 0.279E+02 0.288E+02 0.297E+02 0.306E+02 0.315E+02 0.324E+02 0.333E+02 0.342E+02 0.351E+02 0.360E+02 0.369E+02 0.378E+02 0.387E+02

0.16400E+01 0.16400E+01 0.16400E+01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

0.80852E+04 0.71500E+04 0.63226E+04 0.58616E+04 0.74926E+04 0.89330E+04 0.10206E+05 0.11332E+05 0.12327E+05 0.13208E+05 0.13986E+05 0.14676E+05 0.15285E+05 0.15825E+05 0.16302E+05 0.16725E+05 0.17099E+05 0.17430E+05 0.17723E+05 0.17982E+05 0.18212E+05 0.18415E+05 0.18595E+05

Vol flow rate Q1 (cc/s): Vol flow rate Q2 (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st drawdown ends (sec): 2nd drawdown ends (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 1st Point Time T3 (sec): Pressure P3 (psi): Run 1. 2. 3. 4. . . 100. 110. 120. 130. 131. 132. 133. 134. 135. 136.

0.650 1.640 0.500 0.380 15.400 21.500 22.500 7492.600 27.900 13986.000 33.300 17099.000

Error P0(psi) 51.8 % 1213219 51.5 % 611986 51.3 % 411575 51.0 % 311371

Md/Cp 0.000 0.000 0.001 0.002

17.7 13.2 8.4 3.3 2.7 2.2 1.7 1.2 0.6 0.1

0.375 0.417 0.456 0.494 0.498 0.501 0.505 0.508 0.512 0.516

% % % % % % % % % %

23054 21990 21108 20365 20298 20231 20165 20101 20037 19974

Stop - Program terminated.

76 Supercharge, Invasion and Mudcake Growth 2.3.4.2 Run No. 6. Flowline volume 500 cc.

Here, the flowline volume is increased to 500 cc and Figure 2.21b applies. Exact FT-00 source probe pressures are given below and red values are taken as inputs to the inverse program. Calculated results give a pore pressure of 20,046 psi and a mobility of 0.509 md/cp, which again compare favorably with known values. Time (s) 0.000E+00 0.900E+00 0.180E+01 . .

Rate (cc/s) 0.65000E+00 0.65000E+00 0.65000E+00

Ps* (psi) 0.20000E+05 0.19620E+05 0.19258E+05

0.207E+02 0.216E+02 0.225E+02 0.234E+02 0.243E+02 0.252E+02 0.261E+02 0.270E+02 0.279E+02 0.288E+02 0.297E+02 0.306E+02 0.315E+02 0.324E+02 0.333E+02 0.342E+02 0.351E+02 0.360E+02 0.369E+02 0.378E+02 0.387E+02 0.396E+02 0.405E+02 0.414E+02 0.423E+02

0.16400E+01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

0.11595E+05 0.11147E+05 0.11572E+05 0.11975E+05 0.12359E+05 0.12724E+05 0.13072E+05 0.13403E+05 0.13718E+05 0.14018E+05 0.14303E+05 0.14575E+05 0.14834E+05 0.15081E+05 0.15315E+05 0.15539E+05 0.15751E+05 0.15954E+05 0.16147E+05 0.16330E+05 0.16505E+05 0.16671E+05 0.16830E+05 0.16981E+05 0.17125E+05

Vol flow rate Q1 (cc/s): Vol flow rate Q2 (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st drawdown ends (sec): 2nd drawdown ends (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 1st Point Time T3 (sec): Pressure P3 (psi): Run 1. 10. 20. 30. 40. 50. 51. 52. 53. 54.

Error 36.4 % 31.3 % 25.2 % 18.6 % 11.5 % 3.7 % 2.9 % 2.1 % 1.3 % 0.5 %

P0(psi) 374386 49401 31361 25358 22365 20575 20434 20300 20170 20046

0.650 1.640 0.500 0.380 15.400 21.500 23.400 11975.000 30.600 14575.000 40.500 16830.000 Md/Cp 0.000 0.030 0.106 0.210 0.330 0.457 0.470 0.483 0.496 0.509

Spherical Source Formulations 77 2.3.4.3 Run No. 7. Flowline volume 1,000 cc.

Here the flowline volume is increased to 1,000 cc and Figure 2.21c applies. The predicted values of 20,043 psi and 0.506 md/cp are again very accurate. Time (s) 0.000E+00 0.900E+00 0.180E+01 . . 0.189E+02 0.198E+02 0.207E+02 0.216E+02 0.225E+02 0.234E+02 0.243E+02 0.252E+02 0.261E+02 0.270E+02 0.279E+02 0.288E+02 0.297E+02 0.306E+02 0.315E+02 0.324E+02 0.333E+02 0.342E+02 0.351E+02 0.360E+02 0.369E+02 0.378E+02 0.387E+02 0.396E+02 0.405E+02 0.414E+02 0.423E+02

Rate (cc/s) 0.65000E+00 0.65000E+00 0.65000E+00

Ps* (psi) 0.20000E+05 0.19807E+05 0.19620E+05

0.16400E+01 0.16400E+01 0.16400E+01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

0.15701E+05 0.15320E+05 0.14948E+05 0.14639E+05 0.14769E+05 0.14896E+05 0.15020E+05 0.15140E+05 0.15258E+05 0.15373E+05 0.15485E+05 0.15594E+05 0.15701E+05 0.15805E+05 0.15906E+05 0.16005E+05 0.16102E+05 0.16196E+05 0.16288E+05 0.16378E+05 0.16465E+05 0.16550E+05 0.16634E+05 0.16715E+05 0.16795E+05 0.16872E+05 0.16948E+05

Vol flow rate Q1 (cc/s): Vol flow rate Q2 (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st drawdown ends (sec): 2nd drawdown ends (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 1st Point Time T3 (sec): Pressure P3 (psi): Run 1. 5. 10. 15. 20. 25. 26. 27.

Error 15.3 % 13.2 % 10.4 % 7.5 % 4.6 % 1.5 % 0.9 % 0.3 %

P0(psi) 146133 41374 28281 23918 21737 20430 20229 20043

Stop - Program terminated.

0.650 1.640 0.500 0.380 15.400 21.500 22.500 14769.000 28.800 15594.000 35.100 16288.000 Md/Cp 0.001 0.023 0.085 0.180 0.301 0.444 0.475 0.506

78 Supercharge, Invasion and Mudcake Growth 2.3.4.4 Run No. 8. Flowline volume 2,000 cc.

Here, the flowline volume is increased to a large 2,000 cc and Figure 2.21d applies. Nonetheless, the predicted 20,129 psi and 0.472 md/cp are in reasonable agreement with the known values of 20,000 psi and 0.5 md/cp. Time (s) 0.000E+00 0.900E+00 0.180E+01 . 0.198E+02 0.207E+02 0.216E+02 0.225E+02 0.234E+02 0.243E+02 0.252E+02 0.261E+02 0.270E+02 0.279E+02 0.288E+02 0.297E+02 0.306E+02 0.315E+02 0.324E+02 0.333E+02 0.342E+02 0.351E+02 0.360E+02 0.369E+02 0.378E+02 0.387E+02 0.396E+02 0.405E+02 0.414E+02 0.423E+02

Rate (cc/s) 0.65000E+00 0.65000E+00 0.65000E+00

Ps* (psi) 0.20000E+05 0.19903E+05 0.19807E+05

0.16400E+01 0.16400E+01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

0.17416E+05 0.17204E+05 0.17021E+05 0.17057E+05 0.17093E+05 0.17128E+05 0.17163E+05 0.17198E+05 0.17232E+05 0.17266E+05 0.17299E+05 0.17332E+05 0.17364E+05 0.17397E+05 0.17428E+05 0.17460E+05 0.17490E+05 0.17521E+05 0.17551E+05 0.17581E+05 0.17610E+05 0.17639E+05 0.17668E+05 0.17697E+05 0.17725E+05 0.17752E+05

Vol flow rate Q1 (cc/s): Vol flow rate Q2 (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st drawdown ends (sec): 2nd drawdown ends (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 1st Point Time T3 (sec): Pressure P3 (psi): Run 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Error 9.1 % 8.4 % 7.7 % 7.0 % 6.3 % 5.6 % 4.9 % 4.2 % 3.4 % 2.7 % 1.9 % 1.2 % 0.4 %

P0(psi) 54706 35976 29733 26611 24738 23490 22598 21929 21409 20993 20652 20369 20129

0.650 1.640 0.500 0.380 15.400 21.500 24.300 17128.000 30.600 17364.000 39.600 17668.000 Md/Cp 0.003 0.013 0.029 0.051 0.078 0.111 0.149 0.192 0.239 0.291 0.348 0.408 0.472

Spherical Source Formulations 79 2.3.4.5 Run No. 9. Time-varying flowline volume inputs from FT-07.

Here we introduce FT-07, a finite difference simulator for ellipsoidal sources in transversely isotropic media with time-varying flowline storage. Its interface is deliberately similar to FT-06 since both models solve similar problems. FT-07 differs in that volumes may depend on time, addressing inverse applications where the parameter is “tunable” in order to sharpen pressure responses. In this test of the inverse method, we create forward simulation pressure transient data using FT-07. The assumptions used appear in Figure 2.22a. The grid system was calibrated so that the transient solution for source probe pressure is nearly identical to that produced by exact simulator FT-00 when the flowline volume is 200 cc. As shown below, a time-dependent volume was assumed with 200 cc for the first two seconds, increasing to 5,000 cc by five seconds, and holding at 5,000 cc thereafter. The source solution and volume history are plotted in Figure 2.22b.

Figure 2.22a. Time-dependent flowline volume.

80 Supercharge, Invasion and Mudcake Growth

Figure 2.22b. Volume flow rate, flowline volume, source probe pressure plots. We ran the inverse model using the transient drawdown data at 1, 2 and 4 seconds shown below. Additional computing time relative to previous examples was required, that is, about 1 sec total time. C:\>pta-dd-3-run-with-rft-numbers-10000-iterations Use decimals after all integers! Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Run

Error

1500. 44.5 1600. 37.6 1700. 29.9 1800. 21.4 1900. 12.0 2000. 1.5 2004. 1.0 2005. 0.9 2006. 0.8 2007. 0.7 2008. 0.6 2009. 0.5 2010. 0.4 2011. 0.3 2012. 0.1 2013. 0.0 Stop - Program

1. 1. 1. 1. 24167. 2. 23949. 4. 23916.

P0(psi)

% 25144 % 25246 % 25360 % 25485 % 25624 % 25777 % 25784 % 25785 % 25787 % 25789 % 25790 % 25792 % 25794 % 25795 % 25797 % 25798 terminated.

Md/Cp 0.949 0.883 0.818 0.756 0.697 0.641 0.639 0.638 0.638 0.637 0.637 0.636 0.635 0.635 0.634 0.634

The predicted pore pressure of 25,798 psi (vs 25,000) and mobility of 0.634 md/cp (vs 1.0) are reasonable, considering the fact that our inverse drawdown model was never intended to work with large, rapidly increasing flowline volumes.

Spherical Source Formulations 81

2.3.5 Phase delay and amplitude attenuation, anisotropic media with dip – detailed theory, model and numerical results.

In electromagnetic logging, resistivity is determined by evaluating receiver amplitude attenuation or phase delay response of a sinusoidal transmitter signal using Maxwell’s equations. As simple as this idea sounds, its first MWD/LWD implementation only appeared in the 1980’s with NL Sperry Sun’s introduction of its Electromagnetic Wave ResistivityTM (EWR) tool, building on advances in wireline induction logging. The formation tester can be analogously used to evaluate permeability. A periodic pressure signal created by the pump piston is measured at a second probe and rock properties can be deduced from waveform changes using Darcy’s laws. And similar to resistivity methods employing multiple receiver coils, attenuations and delays between multiple observation probes can be used to determine permeability with analogous benefits. In this section, methods for isotropic uniform media are motivated and developed and then extended to anisotropic homogeneous media for tools oriented at arbitrary dip angles. We will demonstrate how both kh and kv can be obtained using combined drawdown-buildup and phase delay methods; and, in the case when beds are thick, how two phase delay measurements taken at different dip angles (corresponding to different bit locations while the drillstring changes direction) likewise yield both permeabilities. These methods assume “one transmitter, one receiver” tools. Additionally, we show how “multiple receiver” (that is, multiple observation probe formation testers) can produce more accurate results using relative receiver (observation probe) measurements, as in electromagnetic logging. These results are entirely exact and analytical. The math model and numerical approach, together with 3D extensions for layered media, are also presented. The phase delay approach first appeared in “Formation Evaluation Using Phase Shift Periodic Pressure Pulse Testing,” U.S. Patent No. 5,672,819, for W.C. Chin and M.A. Proett, issued Sept. 30, 1997. This focused on isotropic media and was never implemented because two probes, one source and the second observation, were required; for single spherical permeabilities characterizing such reservoirs, it was more convenient using single source probe drawdown-buildup methods. Here, we reconsider phase delay methods because, it turns out, it is possible to obtain both kh and kv in anisotropic applications. Moreover, this can be achieved in low mobility formations using very early time data.

82 Supercharge, Invasion and Mudcake Growth

2.3.5.1 Basic mathematical results.

For the remainder of Section 2.3.5, our discussions are short and analytical derivations are omitted for brevity. Complete details are offered in Chin et al. (2015). We provide results and interpretation approaches only, and focus on contrasting differences among the various phase delay methods developed previously. All models are listed with their well-defined assumptions. Analogiess between formation testing and electromagnetic logging are explained and exploited throughout. Isotropic model. For compressible liquids in isotropic homogeneous media, the model 2p/ r2 + 2/r p/ r = ( c/k) p/ t applies where , , c and k are porosity, viscosity, compressibility and isotropic permeability. We seek solutions in the form p(r,t) = P(r) exp(i t) where P(r) is a shape function and is frequency, in which solutions decay far from the source. If rw is the spherical source radius, it is possible to show that p(r,t) = A cos ( t – ) where the real quantities A and satisfy A = P0 (rw/r) exp[-(r-rw) { c /(2k)}] with the relation = (r-rw) { c /(2k)}. Here, A is the amplitude and is the phase, noting that the source will be ellipsoidal in transversely isotropic media. When either A or is known, the permeability k can be calculated assuming the remaining variables are available. As in resistivity or electromagnetic logging, larger frequencies give improved resolution with less depth of investigation; smaller frequencies provide less resolution but greater depth of investigation. Anisotropic extensions. For transversely isotropic media, the the effects of the dip angle can be significant. Here, the general Darcy equation for homogeneous media takes the form kh ( 2p/ x2 + 2p/ y2) + kv 2p/ z2 = c p/ t. In this model, kh is the horizontal permeability in the x-y plane and kv is the vertical permeability along the perpendicular z axis. This should be solved with boundary conditions on x2 + y2 + z2 = rw2 since this is a spherical source surface with clear physical meaning. However, the solution is possible only numerically. Since computational solutions are not convenient for interpretation, as tabulated values are cumbersome to use, an analytical approach is preferred assuming that disturbances originate at an ellipsoidal source. A detailed derivation is offered in Chin et al. (2015). There, the authors have shown that P(r*,t) = A cos ( t – ), A = P0 (rw*/r*) exp[- (r*- rw*) { c /(2)}] while the phase satisfies = (r*-rw*) { c /(2)}, noting that r*w2 = xw2/kh + yw2/kh + zw2/kv and r*2 = x2/kh + y2/kh + z2/kv. Note how kh and kv are coupled.

Spherical Source Formulations 83

Vertical well limit. For vertical wells, x = y = 0 so that the tool axis coincides with the z axis. This will lead to P(r*,t) = A cos ( t – ), A = P0 (zw/z) exp[-(z-zw) { c /(2kv)}] and = (z-zw) { c /(2kv)}. The subscript “w” denotes “well” or the surface of the spherical or ellipsoidal well that represents the source probe. It is also possible, although not necessary, to make the approximation z >> zw, in which case we obtain the simplified results A P0 (zw/z) exp[-z { c /(2kv)}] and z { c /(2kv)}. Thus, if Az is known at the observation probe position z, then amplitude can be used to derive the vertical permeability kv. Similarly, if z is known at z, then phase can be used to find kv. Horizontal well limit. Here assume that x = z = 0 so the tool axis and the y axis are coincident. Then, our results can be simplified and we have P(r*,t) = A cos ( t – ), A = P0 (yw/y) exp[-(y-yw) { c /(2kh)}] and = (y-yw) { c /(2kh)}, so that far away from the source, these become A P0 (yw/y) exp[-y { c /(2kh)}] and y { c /(2kh)}. Thus, if Ay is known at the observation probe position y, either amplitude equation can be used to derive the horizontal permeability kh. Similarly, if y is known at y, then either phase euqation can be used to find kh. Formulas for vertical and horizontal wells. For vertical and horizontal wells, simple formulas can be derived for permeability interpretation and microprocessor tool applications. Again, we use P = A cos ( t – ) = A cos [ (t – / )]. We can write P = A cos (t – t) where the time delay satisfies t = / . If we eliminate between this and L { c /(2k)}, where we have removed subscripts “h” and “v” for convenience, and also set = 2 f where f is the frequency in Hertz, we have the result k = L2 c /[4 f( t)2]. Note that the phase angle is large for small values of k; similarly, the time delay t is large for smaller k’s. This means that phase delay methods are useful for interrogating low mobility formations since they are easily and quickly measured. Later we offer numerical results which illustrate the robust nature of phase delay predictions. Amplitude methods are less accurate. Generally, two factors contribute to amplitude reduction with distance, an algebraic “geometric spreading” that is inversely proportion to distance, and an exponential effect related to formation effects. It is often difficult to separate the two effects from not-so-accurate field measurements. This problem is compounded by the fact that measurements will use only two-to-three pumping cycles, hardly enough to fulfill the periodicity assumptions implicit in P = A cos ( t – ).

84 Supercharge, Invasion and Mudcake Growth

Deviated well equations. The above results for vertical and horizontal wells can be extended to general deviated wells with dip angle . The dip angle is 0o for vertical wells and 90o for horizontal wells. Recall that our complete anisotropic results gave P(r*,t) = A cos ( t – ), A = P0 (rw*/r*) exp[- (r*- rw*) { c /(2)}] and = (r*-rw*) { c /(2)}. Without loss of generality, we assume that the tool is straight and located in the plane x = 0. If the observation probe is located a length L from the source probe, where the center of the ellipsoidal source is located at x = y = z = 0, we can write x = 0, y = L sin and z = L cos where L is a reference length parallel to the tool axis. We seek specific results for amplitude and phase. The amplitude is now governed by A = P0 (rw*/r*) exp[- (r*- rw*) { c /2}] with the coefficient rw*/r* derived from the square root of rw*2/r*2 = (kvxw2 + kvyw2 + khzw2)/(kvx2 + kvy2 + khz2). This becomes zw/z for vertical wells (when xw = yw = 0) and yw/y for horizontal wells (when xw = zw = 0), both independent of permeability. For deviated angles between 0o and 90o, however, both vertical and horizontal permeabilities remain and the coefficient rw*/r* remains complicated. Note that the approximation r* >> rw* will simplify the expression in the exponential, but rw*/r* itself remains problematic, at least for now. We will deal with this later. The phase function is easier to work with. We can use the approximation r* >> rw* to obtain usable results, that is, =(r*-rw*) { c /2} r* { c /2} (02/kh + L2 sin2 /kh + L2 cos2 /kv)1/2 { c /2}. This leads to the result that L { c /2}{sin2 /kh + cos2 /kv}] which correctly reduces to z { c /(2kv)} when = 0o (for vertical wells) and to y { c /(2kh)} when = 90o (for horizontal wells). Again, we note that the pressure response can be rewritten in the form P(r*,t) = A cos ( t – ) = A cos (t – t) where the measured time delay is t = / . Thus, the phase delay can be calculated from = t = 2 f t where f is the frequency in Hertz.

Spherical Source Formulations 85

Deviated well interpretation for both kh and kv. As noted, the expression for phase in deviated wells is considerably simpler than that for amplitude. In fact, if the phase L at the observation probe distance L from the center of the source probe is known, then L L { c /2} {sin2 /kh + cos2 /kv}]. This provides, for the dip angle assumed, one equation of two needed to find both kh and kv. There are two approaches that can be taken next, emphasizing that both apply to low mobility applications with data taken at early times. In the first, suppose we have a thick uniform layer where the formation tester resides in a drillstring that is changing direction. Then, the foregoing equation can be evaluated at two different dip angles a and b so that we have two separate linearly independent equations, that is, L { c /2} {sin2 a /kh + cos2 a /kv}] and, similarly, the equation L,a L { c /2} {sin2 b /kh + cos2 b /kv}]. These provide two coupled L,b equations in the two unknown permeabilities kh and kv which can be directly solved. The result, from simple algebra, is kh = (L2

c /2) (sin2

a

cos2

b

– sin2

b

kv = (L2

c /2) (sin2

a

cos2

b

– sin2

b

cos2 a) /( cos2 a) /(

2

cos2

2

sin2

a

b

b a

– –

b

2

2

a

cos2 a)

sin2 b)

We emphasize that these two equations are obtained at different dip angles and that the layer is assumed to be thick enough to allow measurements to be taken as the drillstring changes direction. As a check, setting a = 0o and b = 90o, and conversely, lead to the simpler formulas derived earlier. This method will not apply to thin layers as it is not possible to log at two different angles at the same point. An alternative approach is available, which also applies to thick layers. Recall that we need a second equation relating horizontal to vertical permeabilities in order to solve a “two equation, two unknown” system. Previously we developed a single-probe, drawdown-buildup method for low mobility applications which used early-time pressure transient data, which we refer to as our “DDBU” method. We will use this with the phase delay approach. In this second method, we use L L { c /2} {sin2 /kh + cos2 /kv}], which provides a relation connecting kh and kv for measured values of the phase L and dip angle . To determine kh and kv uniquely, we turn to our DDBU model where the spherical permeability ks is known from early time pressure transient data, that is, we know ks = kh2/3kv1/3. Combining this with the foregoing equation leads to relationships for the horizontal and vertical permeabilities kh and kv, namely,

86 Supercharge, Invasion and Mudcake Growth cos2 kh3 – {2 tan2 kv3/2 – {2 tan2

– {2

2 3 2 L ks /(L

c )} kh + sin2 ks3 = 0

2 3/2 2 L ks /(L

2 2 L ks/(L

c cos2 )} kv + ks3/2 = 0

c cos2 )}

2/3

+1=0

where = kv/kh represents the “anisotropy” or anisotropy ratio. The foregoing are cubic polynomial equations for kh , kv1/2 and 1/3 which can be solved exactly from standard algebraic formulas. Different equations apply in different situations. In some situations, it may be preferable to work first with kh if it is typically larger, as it would provide greater numerical accuracy. Once the kh equation is solved, the vertical permeability can be obtained from ks = kh2/3kv1/3 where the value of ks is known. In general, any of the cubic equations can be used, with ks = kh2/3kv1/3 applied in the final step to calculate the remaining permeability. We emphasize that in obtaining ks using our drawdown-buildup methods, we also predict the pore pressure from early time data. Thus, pore pressure and both permeabilities are available as a result. Other related methods are discuss in Chin et al. (2015) and interested readers are referred to that publication. Two-observation-probe models. In electromagnetic logging, two methods are available for resistivity prediction. In the first, amplitude attenuation and phase delay are measured at a single receiver displaced from a transmitter that broadcasts sinusoidal signals. The readings are interpreted using models developed from Maxwell’s equations. In the second approach, two receivers are introduced; amplitude ratios and phase delays relative to the two receivers only are used for interpretation. This provides several benefits. Amplitude attenuation due to conductivity or formation effects, which when measured relative to the transmitter, are typically “buried” in the rapidly decaying algebraic field associated with geometric spreading and is difficult to discern. This large source of error is eliminated by using two close receivers. Close receivers also provide less sensitivity to larger scale heterogeneities in the borehole, and allow increased accuracy. We expect similar advantages in analogous “two-receiver” or “two observation probe” approaches for formation testing. Such a tool will have one source (pumping) probe and (at least) two axially displaced passive observation probes. However, the latter do not require their own hydraulic pad thrusters, which would complicate mechanical design. The simple mechanical design in Figure 2.23 consists of an elongated pad driven by a single hydraulic thruster that presses the pad against the

Spherical Source Formulations 87

borehole wall. The pad contains two ports which independently measure pressure by means of transducers separated by inches. Finally, the use constant frequency test frequencies is assumed. A simple question arises – in measuring time delay, how does the receiver “know” when a signal begins and ends? This is easily solved by transmitting a constant frequency signal and having its initial amplitude suddenly change to a different level. This creates a “marker” signaling when the test signal departs the transmitter. When the observation probe “sees” this marker, as illustrated in Figure 2.24, the elapsed time recorded would be the required time delay (that is, the “clock starts running” when the transmitter sends its first signals). Formation tester tool body

Source (pumping) probe @ fixed frequency

Single hydraulic thrust unit, two pressure transducers

Two observation probes measure amplitude ratio and phase delay

Figure 2.23. Simple “two-receiver” observation probe. P

Single-frequency-signal, amplitude change identifies t = 0 for observation probe t

Figure 2.24. Transmitter “marker” defines instant of departure. To obtain usable results for our “two observation probe” approach, we approximate r* >> rw* to simplify previously derived mathematical expressions; that is, both observation probes are assumed to be far from the source pumping probe. In this case, A P0 (rw*/r*) exp[- r* { c /2}]. Now suppose that two measurements, A1 at r1* and A2 at r2* are available. Thus we have A1 P0 (rw*/r1*) exp[- r1* { c /2}] at “1” and similarly at “2,” the equation A2 P0 (rw*/r2*) exp[- r2* { c /2}]. Dividing one by the other, we obtain A1/A2 = r2*/r1* exp [(r2* - r1*) { c /2}] which is

88 Supercharge, Invasion and Mudcake Growth

independent of P0, and importantly, does not depend on the “ellipsoidal well radius” rw whose physical meaning is not clear. Next we recall that r2* = L2 (sin2 /kh + cos2 /kv) and r1* = L1 (sin2 /kh + cos2 /kv). It follows that A1/A2 = (L2/L1) exp [(L2 -L1) [{ c /2}{sin2 /kh+cos2 /kv}]], which provides one of the equations needed in our “two equation, two unknown” approach; the second is obtained from a DDBU test, where a spherical permeability ks is measured, thereby constraining ks = kh2/3kv1/2. To derive a complementary equation for phase, one might use = (r*-rw*) { c /(2)}, again assuming that r* >> rw*. This yields the equations 1 = r1* [{ } and 2 = r2* [{ } which, on division, leads to 1/ 2 = L1/L2 – this merely affirms that phase is linearly proportional to distance from the transmitter but otherwise no additional information is gained. The key lies in the calculation of phase differences between observation probes. This leads, together with the definition of dip angle, to a simple 2- 1 = (L2 - L1) [{ c /2}{sin2 /kh+cos2 /kv}]. This provides one of the equations needed in our “two equation, two unknown” approach; as before, the second is obtained from a DDBU test, in which the spherical permeability ks is measured, thereby constraining ks = kh2/3kv1/2. 2.3.5.2 Numerical examples and typical results.

So far, we have demonstrated how phase delay methods provide simple, fast and elegant alternatives to steady-state methods for permeability prediction in “low mobility” applications. However, we have not yet considered “how low is low” and also the pumping frequencies needed to excite the formation. Are these frequencies doable mechanically? Are they 10 Hz, 100 Hz . . . or perhaps 1,000 Hz? Also, the periodicity assumption behind “cos t” requires infinitely acting sinusoidal action – in practice, however, one can hope at best for two-tothree pump cycles which may not be perfect sinusoids. Numerical simulations have been defined to address these questions and representative results are described pointing to the practicality of the method. Both simple and more complicated examples are described that address these issues.

Spherical Source Formulations 89 Example 1. Parameter estimates.

The isotropic model given in the beginning of this section can be used to estimate the mechanical frequencies needed for low mobility applications. Again, because the governing equations are identical in form, this also applies to vertical and horizontal wells in transversely isotropic media. We have written two simple programs using our phase delay formula, namely, “phase-delay-estimates” and “phase-delaypermeability-calculation” to provide quick answers. These modules, based on a closed form solution, require no iteration, and results from typical simulations are shown in Figures 2.25a,b. C:\phase-delay-estimates Permeability ........... (md): Porosity .......... (decimal): Viscosity .............. (cp): Compressibility ..... (1/psi): Probe separation ....... (cm): Frequency .............. (Hz):

1. 0.2 1. 0.000003 15. 0.5

Phase delay ............(deg): Phase delay ........... (rad): Time delay ............ (sec):

0.1010E+03 0.1763E+01 0.5611E+00

Figure 2.25a. Estimating time delays for given parameters.

C:\phase-delay-permeability-calculation Probe separation ....... (cm): Porosity .......... (decimal): Viscosity .............. (cp): Compressibility ..... (1/psi): Frequency .............. (Hz): Time delay ............ (sec):

15. 0.2 1. 0.000003 0.5 0.5611

Permeability ........... (md): Phase delay .......... (rad): Phase delay ........... (deg):

0.1000E+01 0.1763E+01 0.1010E+03

Figure 2.25b. Predicting permeability from time delay.

The two programs yield consistent results. The numbers indicate a mobility of 1 md/cp is adequately resolved by a piston frequency of 0.5 Hz, which is doable in hardware. The not-too-small time delay of 0.5611 sec between dual probes with typical fifteen cm separation should be obtainable accurately. The method is practical from a sampling perspective. In present tools, pressures are measured every 0.25 sec, too slow to characterize transients in high mobility formations. However, this rate is useful for time delay measurements since piston pump cycles occur over durations of approximately one second. These numbers provide “ballpark estimates” that also apply to anisotropic situations.

90 Supercharge, Invasion and Mudcake Growth Example 2. Surface plots.

Again, consider an isotropic uniform medium with a permeability k and a porosity . The liquid viscosity and compressibility are and c. The formation tester piston movement is taken as a sinusoidally pulsating pressure source with strength Po, radius rw and frequency . Then, at any radial position r > rw, the transient pressure response satisfies the solution p(r,t) = A cos ( t – ) where A and are the “amplitude” and “phase delay” functions at r given by A = Po (rw/r) exp [– (r – rw) { c /(2k)}] and = (r – rw) { c /(2k)}. At the source position r = rw, we note that p(rw,t) = Po cos( t), the assumed form of the pressure excitation. But at r > rw, a phase delay is proportional to the product of probe separation and { c /(2k)}. For example, low permeabilities and high viscosities result in large phase shifts. As formation and fluid parameters are fixed, the excitation frequency can be used to control depth of investigation in a manner consistent with hardware sampling rates.

Figure 2.26. Amplitude (left) and phase delay (right) versus r and . In Figure 2.26, typical amplitude and phase delay plots versus r and are shown. The analytical phase result can be used to determine permeability when all other parameters are known – other formation properties are easily derived, e.g., c/k = (2/ ){ /(r-rw)}2. Note that k can also be determined from amplitude data – however, these may be more prone to error than time delays since the attenuation field due to a weak exponential decay is embedded in a rapidly falling “1/r” associated with geometric spreading. Amplitude measurements may also be undesirable because it is not possible to create a perfect sinusoid as only two-to-three wave cycles are possible. These amplitudes may vary significantly and lead to unacceptable error without further filtering.

Spherical Source Formulations 91 Example 3. Sinusoidal excitation.

We follow Figure 2.24 and define a constant frequency pump excitation with multiple amplitudes as in Figure 2.27a. Strictly speaking, this is not a periodic signal, but the intention is clear; the rapid amplitude change is introduced to provide a marker so that delays can be counted easily. The required pressure transient responses can be calculated from forward finite difference simulators FT-06 or FT-07. The assumed parameters for the simulation are shown more clearly in Figure 2.27b. Importantly, note that the mobility is 1 md/1 cp or 1 md/cp. To determine the time delay, we focus on the clearly discernible first group of small amplitude waves, and in particular, the center crest. From the circled values, we find (visually) that the time at the source probe is 3.8 sec while the time at the observation is 4.4 sec, yielding a time delay of 4.4 – 3.8 or 0.6 sec. Using one of the programs discussed above, we find C:\phase-delay-permeability-calculation Probe separation ....... (cm): Porosity .......... (decimal): Viscosity .............. (cp): Compressibility ..... (1/psi): Frequency .............. (Hz): Time delay ............ (sec): Permeability ........... (md): Phase delay .......... (rad): Phase delay ........... (deg):

15. 0.2 1. 0.000003 0.5 0.6 0.8745E+00 0.1885E+01 0.1080E+03

(very close to 1 md)

The calculated 0.8745 md is close to the known value of 1 md. Again, pressures were created from FT-06 or FT-07, both of which are approximate numerical simulators. From Figure 2.27c, it is clear amplitude methods are not ideal since these functions drift considerably. The flow rate in Figure 2.27a, defined by a single frequency but with amplitude changes taken at distinct intervals (in fact, every three cycles), is important for another reason. At low mobilities, phase delays with > 360o may be large; a single amplitude will not allow convenient phase delay measurement as it is difficult to distinguish one cycle from another. When amplitudes are “stepped” as shown, monitoring software used by the observation probe can count the number of 360o transitions that have transpired, so that delays associated with “phase wrapping” can be measured with minimal error. Of course, since is proportional to 1/2, phase wrapping may be avoided by decreasing frequency; this allows, at the same time, increased depth of penetration of the pressure signal so that a greater portion of the formation is sampled.

92 Supercharge, Invasion and Mudcake Growth

Figure 2.27a. Constant frequency pump excitation.

Spherical Source Formulations 93

Figure 2.27b. Input data and exploded view.

Figure 2.27c. Source and observation probe pressure.

94 Supercharge, Invasion and Mudcake Growth Example 4. Rectangular wave excitation.

In mechanical design, it may not be possible to pump exactly with sinusoidal precision. We consider an example showing phase delay results in one such instance. Recall that the underlying theory was derived assuming pressure disturbances proportional to exp (i t), that is, that they are perfect sinusoids. In practice, one can hope for two-to-three wave cycles at best that may not be perfectly sinusoidal. In the input screen for exact forward simulator FT-00 in Figure 2.28a, a pump schedule having three wave cycles in six seconds, or a frequency of 0.5 Hz, was formed from sequences of rectangular functions. Such functions are hardly representative of theoretical Fourier components and it is instructive to understand the consequences. To ensure numerical accuracy, we used FT-00 because it provides an exact solution that is not compromised by sharp transitions in the flow rate functions. Also, note that the mobility is a low 0.1 md/1 cp or 0.1 md/cp. Calculated results in Figure 2.28b show that both amplitude functions at source and observation probes drift considerably and are not useful for computation. As in the foregoing example, we therefore turn to time delay measurements. Interestingly, examine the two circled points located at the midpoints of the pumping cycles. The time delay, obtained visually, is clearly “4-3” or 1 sec. What does this imply? We use a software program discussed previously, that is, C:\phase-delay-permeability-calculation Probe separation ....... (cm): Porosity .......... (decimal): Viscosity .............. (cp): Compressibility ..... (1/psi): Frequency .............. (Hz): Time delay ............ (sec): Permeability ........... (md): Phase delay .......... (rad): Phase delay ........... (deg):

15. 0.2 1. 0.000003 0.5 1. 0.3148E+00 0.3142E+01 0.1800E+03

The calculated permeability of 0.3148 md, compared to the known input value of 0.1 md, is not bad, and acceptable when judged against present field logging standards. The discrepancy arises from two effects, namely, the length of the short duration test, and the use of rectangular as opposed to sinusoidal functions. These problems can be minimized by mechanical design.

Spherical Source Formulations 95

Figure 2.28a. Square wave assumptions and pressure responses.

96 Supercharge, Invasion and Mudcake Growth

Figure 2.28b. Pressure responses, exploded view. Example 5. Permeability prediction at general dip angles.

In the calculations below, we demonstrate the numerical consistency of three software programs highlighted in red. The first two are isotropic models developed previously. We choose “random non-zero” numbers and make sure the two are consistent in everything, whether or not needed now, since internal quantities may be needed later. Perfect consistency is seen for the two isotropic codes below. C:\phase-delay-estimates Permeability ........... (md): Porosity .......... (decimal): Viscosity .............. (cp): Compressibility ..... (1/psi): Probe separation ....... (cm): Frequency .............. (Hz): Phase delay ............(deg): Phase delay ........... (rad): Time delay ............ (sec):

1.234 0.2 1.1 0.0000034 15. 1.2 0.1573E+03 0.2745E+01 0.3640E+00

C:\phase-delay-permeability-calculation Probe separation ....... (cm): Porosity .......... (decimal): Viscosity .............. (cp): Compressibility ..... (1/psi): Frequency .............. (Hz): Time delay ............ (sec): Permeability ........... (md): Phase delay .......... (rad): Phase delay ........... (deg):

15. 0.2 1.1 0.0000034 1.2 0.3640 0.1234E+01 0.2744E+01 0.1572E+03

Spherical Source Formulations 97

Next, we evaluate the code “phase-delay-nonlinear.exe” for transversely isotropic media using any nonzero dip angle solving the prior cubic equations for kh, kv and . We enter the above parameters as inputs, and in the first case, take a zero (that is, 0.01 deg) dip angle. Comparing outputs from all three programs shows everything is consistent. In particular, the second “2” root (highlighted in red) gives the isotropic permeability assumed in the two simpler programs. C:\phase-delay-nonlinear (Return> Probe separation ......... (cm): Porosity ............ (decimal): Viscosity ................ (cp): Compressibility ....... (1/psi): Frequency ................ (Hz): Time delay .............. (sec): Spherical permeability ... (md): Dip angle ............... (deg):

15. 0.2 1.1 0.0000034 1.2 0.3640 1.234 0.01

Dip angle ............... (rad): Phase delay ............. (rad): Phase delay ............. (deg):

0.1745E-03 0.2744E+01 0.1572E+03

KH1 = -0.1234E+01 md, KV1 = 0.1234E+01 md Caution: KH permeability is negative KH2 =

0.1234E+01 md, KV2 =

0.1234E+01 md

KH3 = -0.2429E-05 md, KV3 = 0.3186E+12 md Caution: KH permeability is negative

Again, it is important to have recovered the isotropic permeability of 1.234 md. Additionally, there are two (unrealistic) negative kh roots cited in the warning. In the following, we consider the horizontal well limit (with 89.8 deg) and we similarly recover the isotropic permeability as required (root “3”). However, we also obtain an additional anisotropic solution with high horizontal and very low vertical permeability. C:\phase-delay-nonlinear (Return> Probe separation ......... (cm): Porosity ............ (decimal): Viscosity ................ (cp): Compressibility ....... (1/psi): Frequency ................ (Hz): Time delay .............. (sec): Spherical permeability ... (md): Dip angle ............... (deg): Dip angle ............... (rad): Phase delay ............. (rad): Phase delay ............. (deg): KH1 = -0.3541E+03 md, KV1 Caution: KH permeability KH2 = 0.3529E+03 md, KV2 KH3 = 0.1234E+01 md, KV3

= is = =

15. 0.2 1.1 0.0000034 1.2 0.3640 1.234 89.8 0.1567E+01 0.2744E+01 0.1572E+03

0.1499E-04 md negative 0.1509E-04 md 0.1235E+01 md

Additional solution.

98 Supercharge, Invasion and Mudcake Growth

Finally, we consider the same data but at 45 deg. We recover the isotropic solution in “root 2,” but as before, obtain an additional anisotropic permeability set in “root 3” that satisfies the input data. C:\phase-delay-nonlinear (Return> Numbers must have decimals ... Probe separation ......... (cm): Porosity ............ (decimal): Viscosity ................ (cp): Compressibility ....... (1/psi): Frequency ................ (Hz): Time delay .............. (sec): Spherical permeability ... (md): Dip angle ............... (deg): Dip angle ............... (rad): Phase delay ............. (rad): Phase delay ............. (deg):

15. 0.2 1.1 0.0000034 1.2 0.3640 1.234 45. 0.7854E+00 ok 0.2744E+01 0.1572E+03

KH1 = -0.1996E+01 md, KV1 = 0.4714E+00 md Caution: KH permeability is negative KH2 =

0.1234E+01 md, KV2 =

0.1235E+01 md

KH3 =

0.7627E+00 md, KV3 =

0.3230E+01 md

Additional solution.

Example 6. Solution for a random input.

In the three above examples using the “anisotropic, dip angle” algorithm, we recovered the isotropic permeability of “1.234 md” for dip angles 0o+, 90o- and 45o, and in addition, found other roots for permeability. Our recovering isotropic permeabilities was expected, of course, because we used data created from the isotropic model; thus, the foregoing cases served as validations for the more complicated anisotropic model. Now, let us repeat the above example, but change only the input for measured time delay from 0.3640 sec to 0.5 sec. In this sense, we have a “random” data set for which there is no reason to suspect isotropic properties. C:\phase-delay-nonlinear Probe separation ......... (cm): Porosity ............ (decimal): Viscosity ................ (cp): Compressibility ....... (1/psi): Frequency ................ (Hz): Time delay .............. (sec): Spherical permeability ... (md): Dip angle ............... (deg):

15. 0.2 1.1 0.0000034 1.2 0.5 1.234 45.

Dip angle ............... (rad): Phase delay ............. (rad): Phase delay ............. (deg):

0.7854E+00 0.3770E+01 0.2160E+03

Spherical Source Formulations 99 KH1 = -0.2546E+01 md, KV1 = 0.2898E+00 md Caution: KH permeability is negative KH2 =

0.2213E+01 md, KV2 =

0.3838E+00 md

KH3 =

0.3335E+00 md, KV3 =

0.1690E+02 md

Stop - Program terminated.

In fact, the above results give three real roots, one negative, one with kh > kv > 0 and the last with 0 < kh < kv. The latter two represent anisotropic formations and both are possible permeability solutions that are consistent with the phase delay inputs. The correct choice, of course, requires additional logging data or other qualitative judgement. 2.3.5.3 Layered model formulation.

The above capabilities have been extended to transversely isotropic media with any number of layers and to tools with arbitrary dip angle and source-to-observation probe separation. These extensions require a three-dimensional numerical modeling and interpretation approach. Detailed formulation, finite difference solution, inter-layer matching conditions and implementation notes are given in Chin et al. (2015) and will not be duplicated here. The work first focused on single layer modeling and then introduced the extensions needed to model multiple layers. The layered system represented by Figures 2.29a,b was addressed, where “s” and “o” denote source and observation probes. z (kv) Observer

(xo,yo,zo )

Source

(xs,ys,zs) y (kh)

x (kh)

Figure 2.29a. Layered anisotropic medium with dipping tool.

100 Supercharge, Invasion and Mudcake Growth zk

Observer

(io,jo,ko) Source

i=1

(is,js,ks) j=1

i = imax xi

k = kmax k = kb

k = ka

yj j = j max

k=1

Figure 2.29b. Discretized grid for finite difference solution. 2.3.5.4 Phase delay software interface.

Our phase delay capabilities are hosted by the input screen in Figure 2.30, where white boxes are reserved for data entry. For brevity, only the interface for homogeneous media is discussed and examples are limited to such applications. The diagram reminds users of the coordinate conventions assumed. At the left, gridblock sizes and indexes are selected, as are source and observation probe properties. As these change, automatic calculated results for probe separation and dip angle appear in the opaque boxes. Overall dimensions shown in bold font in the diagram also change automatically as gridblock sizes and indexes change. Formation properties include layer permeabilities and porosities, bed interface indexes, viscosity and compressibility; excitation characteristics include source probe peak-to-peak pressure and frequency. The source code implementing our pressure solution algorithm was written in Fortran. Its executable is called by a Windowsbased front-end developed using Visual Basic which conveniently processes input data. Output color contour and line plots are developed in C-code, which is executed by both Fortran and Visual Basic programs.

Spherical Source Formulations 101

Figure 2.30. Windows-based program interface. Once appropriate values are selected, the user – who does not require experience with programming or computational methods – clicks “Simulate” to automatically perform equation setup, matrix solution, all post-processing and output displays. For the parameters in Figure 2.30, the answer screens in Figures 2.31, 2.32 and 2.33 appear once a “Done simulating …” message box is acknowledged – typical simulations require just seconds on personal computers.

Figure 2.31. Rotatable plot of (P r 2 + P i 2) versus x and y for given layer (“r” and “i” superscripts denote “real” and “imaginary”).

102 Supercharge, Invasion and Mudcake Growth

Figure 2.32. Phase delay plot (-100 to +100 psi for source pressure).

Figure 2.33. Output text summaries. We refer interested readers to detailed notes related in output files and specialized user features in Chin et al. (2015). Illustrative computed results are offered next.

Spherical Source Formulations 103 2.3.5.5 Detailed phase delay results in layered anisotropic media.

We perform representative calculations of engineering interest and elaborate on computed results, demonstrating important features of the model. We turn to the input of Figure 2.30, but modified so that all six permeabilities are a large 1,000 md. In this range, phase delays should be small and this is confirmed in Figure 2.34a (both curves intersect the time axis at the same locations, so that the phase delay is minimal).

Figure 2.34a. Very high 1,000 md run. Next, we reduce all six isotropic permeabilities to 100 md, 10 md and 1 md, with results shown in Figures 2.34b, 2.34c and 2.34d, respectively.

Figure 2.34b. High 100 md run.

104 Supercharge, Invasion and Mudcake Growth

Figure 2.34c. Moderate 10 md run.

Figure 2.34d. Low 1 md run. For the isotropic runs considered above, only the lumped parameter c /k enters, so that it is not necessary to repeat calculations with other variables changed. The results for 1,000 md and 100 md are almost identical – phase delays are minimal because there is little diffusion and the amplitude change from source to observation probe is entirely due to geometric spreading. The results for 10 md and 1 md show significant differences. Here, time shifts are evident even visually. The additional amplitude decays arise from strong diffusion. If our source and observation probes are further displaced, distant amplitudes would be reduced and phase delays would increase. We next provide results for more complicated layering schemes. In Figures 2.35a to 2.35d, assumed horizontal versus vertical permeability distributions are displayed along with computed time delays. For brevity, the remaining parameters are not changed, but we note that only the lumped parameters c and “permeability/porosity” are important in the analysis.

Spherical Source Formulations 105

Figure 2.35a. Isotropic run, high permeability middle layer.

Figure 2.35b. Isotropic run, low permeability middle layer.

Figure 2.35c. Anisotropic run, high permeability middle layer.

106 Supercharge, Invasion and Mudcake Growth

Figure 2.35d. Anisotropic run, low permeability middle layer. In the next example, we fix all formation and fluid properties in the input screens of Figures 2.36a to 2.36c, noting that a layered anisotropic permeability distribution is assumed. As in the above examples, Figure 2.36a models a vertical tool with a source-to-observation probe separation of two inches. On the other hand, Figure 2.36b models the response of the same tool oriented horizontally – calculated amplitudes are larger although phase responses are similar (input dip angle is changed by editing the observation probe indexes). Lastly, we determined the response at 45o dip by changing probe indexes and modifying gridblock sizes to maintain the desired two-inch probe separation. The amplitude response, shown in Figure 2.36c, falls between those of Figures 2.36a and 2.36b, as expected. Computation times for all simulations reported were less than five seconds.

Figure 2.36a. Vertical tool (0o dip) in layered anisotropic medium.

Spherical Source Formulations 107

Figure 2.36b. Horizontal tool (90o dip) in layered anisotropic medium.

Figure 2.36c. Deviated tool (45o dip) in layered anisotropic medium. In the next example, we move with the tester as it progresses vertically down through a three-layer formation, taking a well logging perspective. We consider the isotropic layering in Figure 2.37a and a 1 Hz frequency. From Figure 2.37a, the upper interface index is kb = 16. From Figure 2.37b, the source probe is positioned at k = 17 while the observation probe is found at k = 19 – thus, the dual probe system lies entirely in the upper layer while the probe separation is 6 inches. The corresponding source and observation probe pressure traces are shown in Figure 2.37b. In Figure 2.37c, the source probe is found in the middle low permeability layer with k = 15 while the observation probe is positioned above the interface in the higher permeability layer at k = 17. The ten-fold reduction in permeability clearly decreases the amplitude response at the observation probe. In our Figure 2.37d, the formation tester is moved toward the center of the layer, so that the dual probe system is entirely within that layer. The larger time shift in our Figure 2.37d due to lower effective permeability relative to that in Figure 2.37b is clearly evident – pointing to the success of the method in resolving permeability contrasts between layers.

108 Supercharge, Invasion and Mudcake Growth

Figure 2.37a. Isotropic three-layer system.

Figure 2.37b. Dual probe system entirely in top layer.

Figure 2.37c. Source in middle layer, observation probe outside.

Spherical Source Formulations 109

Figure 2.37d. Dual probe system entirely in middle layer.

Figure 2.38a. Three layer example.

Figure 2.38b. Observation probe responses at 10 Hz (left) and 0.5 Hz (right).

110 Supercharge, Invasion and Mudcake Growth

In the above final example, we consider the three-layer formation described in Figure 2.38a. Both source (ks = 9) and observation (ko = 15) probes are located within the middle layer (6 < k < 16). The green observation probe response for the assumed probe frequency of 10 Hz is shown at the left of Figure 2.38b. The response at the right assumes the same parameters, but the source frequency is decreased to 0.5 Hz. These two examples show that different responses are obtained. In practice, as the formation tester traverses past different layers, multiple frequencies can be used to probe the background geology. Rapid history matching, possible because the computational model is extremely fast, allows similarly rapid identification of layer properties. This concludes our exposition on phase delay methods for formation testing. To our knowledge, the methods discussed are the only ones being pursued in oilfield research. To communicate the main ideas, we emphasized physical results and applications; the derivation of mathematical ideas and their numerical implementation are offered in Chin et al. (2015). 2.3.6 Supercharging and formation invasion introduction, with review of analytical forward and inverse models. The successful formation tester algorithms FT-00, FT-01, FT-03, FT-06, FT-PTA-DDBU and others, appearing in book publications Chin et al. (2014) and Chin et al. (2015) with John Wiley & Sons, motivated the solution of truly difficult problems that appeared in Chin (2019). While the first group remains state-of-the-art as of this writing, the newer 2019 models pushed the boundaries of ideal source and sink probe modeling even further. In Section 2.3.6, we will present forward and inverse models for drawdown and drawdown-buildup problems in low mobility and non-negligible flow line volume applications with general supercharging and invasion effects. The models are developed with exact, closed form, analytical solutions. Further, in Section 2.3.7, we will consider (without supercharging) the forward and inverse problems associated with multiple drawdowns and buildups, again providing analytical solutions. Before we focus on these newer models, we will review next the work of the earlier sections in this context. Note that the work in our companion 2021 book represents even more ambitious goals: a complete three-dimensional modeling of borehole diameter, flowline volume, formation fluid invasion or exit, and multiple pumping probes acting independently, although this is accomplished using a numerical (but extremely rapid and stable) algorithm.

Spherical Source Formulations 111

2.3.6.1 Development perspectives.

During the mid-1990s, Wilson Chin, working with his colleague Mark Proett at Halliburton Energy Services, in Houston, focused his efforts on rapid and efficient formation tester pressure transient interpretation methods. Since the 1950s, flow rate and pressure drop data had been routinely used during sampling operations to predict “effective” or “spherical permeability” (or, more precisely, mobility) – this single-probe measurement provided reservoir characterization information complementing the retrieval and analysis of actual fluid samples. However, the interpretation made use of a steady-state formula requiring complete pressure equilibrium – that is, steady flows that, in the environment of the 1990s and beyond, possibly required hours of expensive wait times at the rigsite and increased the risk of lost tools. We were tasked with the development of more rapid methods that would “roll out” with the introduction of a new formation tester. But disruptive technology is never easy. The obvious and economic use of early time data would be contaminated by pressure distortion effects associated with flowline storage volume, a problem compounded by tight zones, heavy oils, or both. An empirical method in use at the time seemed to work well; applications to synthetic and limited field data were successful, although why, unfortunately, was anyone’s guess. But rigorous mathematics would come to the rescue. The complete initialboundary value problem was formulated and laboriously solved exactly in its entirety. Closed form, analytical solutions for the “direct” or “forward problem,” in which transient pressure histories were sought given fluid, formation, tool and flow rate properties, were obtained in terms of complex complementary error functions. A special “exponential” limit of this solution was studied, which explained why our empirical method worked, and importantly, how it could be improved. This limit formed the basis for a new “inverse” model, in which permeability (or mobility), pore pressure and fluid compressibility could be predicted from a limited set of early pressure measurement data. The research resulted in a number of publications and contributions, some of which were later summarized in “Advanced Permeability and Anisotropy Measurements While Testing and Sampling in Real-Time Using a Dual Probe Formation Tester,” SPE Paper No. 64650, Seventh International Oil & Gas Conference and Exhibition, Beijing, China, November 2000 (for earlier work, refer to “Cumulative References”). In summary, the work led to three significant contributions –

112 Supercharge, Invasion and Mudcake Growth

A simpler “exponential” formula was developed which allowed rapid predictions of effective spherical permeability (or mobility) in tight zones, using early time data in the presence of strong flowline volume effects. Additional by-products of this approach included pore pressure and fluid compressibility. This method forms the basis of the company’s real-time GeoTapTM logging-while drilling service operable for single and also dual probe tools. A method to predict isotropic permeability (or mobility) using phase delay measurements was also developed. Basically, the travel time for sinusoidal waves created by an oscillating pump piston source and measured at a nearby observation probe would provide the desired predictions. However, while a patent award did result from this work, the method was not economically viable since two probes were required – unlike the drawdown-buildup approach above using the exponential formula and just a single source (or pumping) probe. For dual probe tools at zero dip angle (that is, operating in vertical wells), formulas were also given for kh and kv prediction using steady pressure drops obtained at source and observation probes – these measurements, of course, may require lengthy wait times. In 2004, the United States Department of Energy (DOE), through its Small Business Innovation Research (SBIR) program, awarded two hundred awards nationally in areas such as plasma physics, nuclear energy, refining, waste remediation, building and ventilation, and so on. Four grants were made for fossil fuel and well logging research – two of these awards, both won by Chin through his consulting firm Stratamagnetic Software, LLC, founded in 1999, related to formation tester interpretation and analysis. These grants, together with three additional DOE awards, carried stipends significant to any start-up organization and indirectly supported activities in Measurement-WhileDrilling, reservoir engineering, drilling and cementing rheology and electromagnetic logging. The freedom that the awards provided led to new methodologies which would dominate the Chin’s work for more than a decade. Many “loose ends” have since been resolved, and over the past several years, this work has been disseminated through John Wiley & Sons; in formation testing, in several volumes, the present one representing an additional contribution.

Spherical Source Formulations 113

Here we will recapitulate several key new industry capabilities applicable to all manufacturers’ tools, whether they address wireline or MWD applications. Then we address in this section “supercharge,” where high overbalance pressures distort formation tester measurements – a new interpretation model, suitable for desktop or downhole use, is developed for early time mobility, pore pressure and compressibility prediction in the presence of flowline storage. The following section develops new inverse methods for multiple drawdown and buildup applications for reservoir characterization, formation treatment and hydrate production. The final section in this chapter addresses (i) multiphase effects important in modeling borehole invasion and supercharging, (ii) clean-up and contamination analysis, (iii) cylindrical coordinate system applications, and (iv) math modeling issues. 2.3.6.2 Review of forward and inverse models. In this section, we discuss forward and inverse analysis methods that employ “simple” logging techniques such as steady-state drawdown, unsteady drawdown, and drawdown-buildup. The “forward” or “direct” problem solves for the pressure response when fluid, formation, tool and flow rate parameters are given; “inverse” or “indirect” formulations attempt to provide permeability (or, mobility), fluid compressibility and pore pressure when a limited number of time and pressure data points are given. With the exception of supercharge and multiple drawdown and buildup methods, the models here are developed in detail in Chin et al. (2014) and Chin et al. (2015). Section 3.3.6.2 introduces supercharge, developing detailed ideas in Section 3.3.6.3, and also multiple drawdown-buildup and multiphase models. The latter two are extensively treated later in Sections 3.3.7 and 3.3.8. FT-00 model. Our (initial) flagship forward simulator FT-00 is shown in Figures 2.39a,b,c. Its math model is the exact, analytical, closed form, analytical solution solving the complete initial-boundary value problem formulation for liquids in “Advanced Permeability and Anisotropy Measurements While Testing and Sampling in Real-Time Using a Dual Probe Formation Tester,” SPE Paper No. 64650, Seventh International Oil & Gas Conference and Exhibition, Beijing, China, November 2000. Although the solution is exact, the solution could not be used for real-time or even most desktop applications for two reasons. First, the “complex complementary error function” supplied in most scientific mathematical libraries was far too complicated for downhole use with microprocessors having limited capabilities.

114 Supercharge, Invasion and Mudcake Growth

Figure 2.39a. FT-00 (Main Interactive) exact forward liquid simulator. And second, transient pressure responses at observation probes could not be calculated for the entire range of logging applications because of very small and very large arguments. For these reasons, the “exponential model” was, and probably is currently used, although the authors at the time were satisfied that its scientific basis had been clearly established. In the early 2000s, however, Chin and other collaborators reworked the complex variables methods underlying the error function evaluation in order to render FT-00 fully functioning (details are offered in Chin et al. (2014)).

Spherical Source Formulations 115

Figure 2.39b. FT-00 (Batch Mode) exact forward liquid simulator. As a result, the Windows program will perform dozens or more simulations per minute (in batch mode) depending on the microprocessor used, and importantly, will provide transient pressure responses at both source probe and distant observation probes. Figure 2.39a displays all the required inputs for the “main, interactive” mode. Standard outputs include line graphs for assumed volume flow rate versus time, source and observation probe pressure responses versus time, and finally, normalized plots showing both pressure and flow rate responses. In addition, detailed tabulations are offered to support other user applications like report generation and spreadsheet plotting.

116 Supercharge, Invasion and Mudcake Growth

While the “main, interactive” mode is useful insofar as establishing one’s physical intuition for the flow variables at hand, it may be less convenient in history matching applications where, for example, numerous kh, kv, or other values need to be varied systematically to match calculated pressure responses to probe measurements. As shown in Figure 2.39b, our FT-00 software also supports an exact “batch mode” calculator. Here, at the bottom left, a convenient setup box can be “called” to define nested loop parameter ranges and increments for physical variables of interest. Line plots and tables can be displayed during batch calculations, or more conveniently, suppressed to the very end, at which time a single large tabulation is offered to the user.

Figure 2.39c. FT-00 (DOI) exact forward liquid simulator.

Spherical Source Formulations 117

In other applications, “depth of investigation” (DOI) is important in job planning and interpretation error assessment. Consider, for example, a low mobility situation – will the assumed pump rate, or the maximum mechanical rate the system is capable of, result in a measurable signal at the observation probe? Will pressure diffusion (smearing) be excessive? Rather than defining this quantity abstractly, as is commonplace in resistivity and electromagnetic logging, we use our ability to calculate probe responses at any distance from the source to advantage. Clicking the “DOI” button leads to the simplified menu in Figure 2.39c, which automatically and very rapidly supplies exact pressure results and plots at predetermined separation distances between zero and the “maximum probe separation” distance requested. FT-01 model. It is known that numerical methods, e.g., AnsysTM, ComsolTM, FluentTM and others, whether they are finite difference or finite element based, are influenced by truncation and round-off errors. In the historical context, these act as “artificial viscosities” in fluids problems. In formation testing applications hosted by Darcy’s equations, the calculated pressure response for a given inputted mobility may correspond to a different mobility whose value or even qualitative effect may be difficult to quantify. This is not acceptable for forward calculations. But the consequences are worse for the development in inverse methods because they cannot be properly validated. We noted that SPE Paper 64650 provided equations for kh and kv determination for dual probe tools, although using steady-state pressure drops in vertical wells. At the time, only AnsysTM synthetic data was available and applications were deferred. The book Chin et al. (2014) provides the exact, analytical, closed form solution for kh and kv determination assuming dual probe tools where steady-state, liquid assumptions are in place (transient flows are always supported). However, any dip angle is permitted. The screen for “FT-01” is shown in Figure 2.40. The method is validated by using synthetic pressure data generated by the fully transient FT-00 code (which does not suffer from truncation or round-off error), transferred to the first two boxes in Figure 2.40, and showing that predicted anisotropic permeabilities are consistent with those used in FT-00 to generate the pressure data.

118 Supercharge, Invasion and Mudcake Growth

Figure 2.40. FT-01, exact inverse liquid simulator. FT-02 model. In our description of FT-01, our exact inverse model for liquid flows using steady pressure data, we emphasized that it was validated by running forward liquid transient simulator FT-00 until steady-state conditions were achieved in order to obtain steady pressure inputs for inverse calculations. We have not yet introduced FT-02 in this book. FT-02 represents our exact inverse method for nonlinear gas flows based on exact, closed form, analytical solutions (details are offered in Chin et al. (2014)). Whereas FT-00 for liquids was constructed from simple exact solutions using linear superposition methods, an analogous forward simulator for nonlinear gas flows cannot be developed because superposition methods do not apply. Thus, a different validating forward simulator for gases was developed, in this case an exact one for steadystate nonlinear gas flows. This complementary pair of steady forward and inverse gas simulators is shown in Figure 2.41. The method allows simultaneous for kh and kv determination for dual probe tools using steady-state pressure drop data. It applies to all dip angles plus a range of thermodynamic effects, for instance, isothermal and adiabatic processes, and so on. We emphasize that inverse solutions need not be unique. In other words, more than a single horizontal and vertical permeability pair may be found for a given set of dual probe pressure drops. Additional logging information (outside the realm of formation tester analysis) is required to render the solution unique. Example calculations are offered in Chin et al. (2014).

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Figure 2.41. FT-02, exact, steady forward and inverse gas simulators. FT-06 and FT-07 models. Our exact FT-00 forward simulator for liquid motions is based on closed form, analytical solutions, and its versatile flow rate capabilities are founded on general linear superposition principles. For mathematical expediency, these required “piecewise constant” rate specifications, say “ +1 cc/s for two sec, + 5 cc/s for six sec, – 10 cc/s for three sec,” and so on. In many practical applications, pumps cannot achieve such constant rates because of excessive formation resistance or mechanical issues. In fact, timewise volume flow rate functions may take the form of triangles, trapezoids or non-ideal shapes. Thus, the need for a numerically based simulator capable of handling more general volume flow rate functions is apparent. A numerical option is also required for general transient nonlinear gas flows, for which closed form analytical solutions are not available, and for which, in any event, linear superposition methods are inapplicable. The FT-06 numerical finite difference simulator serves two combined functions. First, it solves liquid flow problems subject to arbitrarily defined flow rates, as is apparent from the flow rate schedule in Figure 2.42a. In fact, as shown, a numerical file read in by the user is also possible. Second, the computational engine is extended to nonlinear gas flows for a wide range of thermodynamic situations, e.g., isothermal, adiabatic or other processes of interest. Furthermore, anisotropy may be specified via “kh, kv” or “effective spherical permeability and kv/kh.” The same computational outputs as FT-00 are offered, that is, line plots for source and observation probe pressures, flow rate, and pressure-rate superposed plots versus time, plus detailed numerical tabulations. Example flow rate functions are displayed in Figure 2.42b.

120 Supercharge, Invasion and Mudcake Growth

Figure 2.42a. FT-06, numerical liquid and gas forward simulator. FT-06 assumes that flowline storage volume is constant for the duration of the simulation. In other applications, those focusing on hardware development efforts, the need for time-varying flowline volume simulation arises. It is known that when formations are low in mobility and flowline volumes are not small, pressure responses can be distorted or smeared. The need to dynamically “tune” flowline volume allows the field engineer to adjust the resolution in his pressure curve and permit more accurate interpretation using inverse prediction methods such as those offered in this book. The FT-07 numerical simulator provides a general means to define time-varying flowline volumes, as suggested in the bottom left menu shown in Figure 2.42c. Examples using FT-06 are offered in Chin et al. (2014), while applications using FT-07 are provided in Chin et al. (2015).

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Figure 2.42b. FT-06, general flow rate functions, forward simulator.

Figure 2.42c. FT-07, a FT-06 extension supporting general time-varying flowline volume.

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FT–PTA–DDBU model. Previously, we introduced two inverse models, namely FT-01 for liquids and FT-02 for gases, both requiring steady pressure drops from dual probe data. These models were based on exact, close form, analytical solutions of the respective steady Darcy formulations, and while impractical, do offer horizontal and vertical mobility predictions. In contrast, the FT-PTA-DDBU inverse model, for drawdown-buildup applications using buildup data, supports early time data usage for low mobility applications with non-negligible flowline storage effects. This model rapidly (within seconds) predicts the “effective” or “spherical mobility” kh2/3kv1/3/ where is the viscosity. This model was discussed in Chin et al. (2015) and its examples were duplicated earlier in this book. Its user interface has since been upgraded and the improved screen appears in Figure 2.43 below.

Figure 2.43. FT-PTA-DDBU, early time, low mobility, flowline volume non-negligible – buildup application shown). As indicated in Figure 2.43, only three time-pressure data points are required, together with the time “TDD1” at which drawdown ceases. Shown at the bottom right are pore pressure and mobility predictions. A “drawdown only model, using drawdown data” is also available. While both are still offered, they have been replaced by the more general inverse capabilities of the “multiple drawdown and buildup” system described later, which in addition to pore pressure and mobility, offers

Spherical Source Formulations 123

fluid compressibility. Note that the “multiple drawdown and buildup” options introduced later do not model supercharge due to overbalance effects, but a version of the simple drawdown-buildup code in Figure 2.43 with supercharge is available. Classic inversion model. Finally, we cite for historical purposes the original single-probe model offering spherical mobility when steady pressure drops are available assuming a continuous constant flow rate fluid withdrawal. The method is based on an exact analytical solution, but the main drawback with this approach is the possibility of long waits in low mobility environments, required so that steady conditions are achievable and flowline storage effects dissipate. The classic model has is incorporated in the software of Figure 2.44.

Figure 2.44. Classic inverse model. Supercharge forward and inverse models. In our prior discussion of inverse model FT-PTA-DDBU, we indicated that pore pressure, mobility and fluid compressibility were predicted from early time, single-probe, pressure transient data with non-negligible flowline storage effects. This zero-supercharge model, for drawdown-buildup applications utilizing buildup data, is again shown in the top of Figure 2.45. Mathematical details are offered in the formation testing book of Chin et al. (2014), explaining both exponential function as well as “rational polynomial” implementations (the latter, used in our work, is more robust, since exponentials are prone to compiler or microprocessor quality issues). This method is extended in Chin (2019) to include supercharge effects due to overbalance in the well. The screen at the bottom of Figure 2.45 contains one additional input box “Pover (psi)” for the over-pressure due to overbalance. Again, pore pressure, mobility and compressibility are predicted. Applications appear in Chin (2019).

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Figure 2.45. Both software modules apply to drawdown-buildup applications using buildup data. Pore pressure, mobility and compressibility predictions, zero supercharge (top), strong supercharge or overbalance pressure (bottom).

Spherical Source Formulations 125

In addition to the supercharge inverse model shown at the bottom of Figure 2.45, which applies to drawdown-buildup applications using buildup data (as shown in the yellow screen), a complementary supercharge inverse model for drawdown applications using drawdown data is also available and is shown in Figure 2.46 with essentially identical inputs as in Figure 2.45, except that TDD1 (for the time when drawdown stops) is not requested. Note that all the “black DOS screen” software items shown in the figures below represent completed and fully validated algorithms, except that, as of this writing, more attractive Windows user interfaces have not been written – all of the results generated use the “black screen” interfaces below as temporary “front ends.” In addition to Model SC-DD-INVERSE-2 for inverse calculations, a complementary forward solver, which calculates transient drawdown pressure responses at the source probe when fluid and formation properties, tool characteristics, volume flow rates, pore pressure and overbalance pressure are given, is available and shown in Figure 2.47. In fact, the forward or direct solver in Figure 2.47 was run to create synthetic transient (supercharged) pressure data, which was inputted into the inverse model Figure 2.46. Here, inverse calculations recovered the known mobility, pore pressure and compressibility.

Figure 2.46. Input screen for “Model SC-DD-INVERSE-2.”

Figure 2.47. Input screen for “Model SC-DD-FORWARD-3B.”

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For “drawdown only” applications, a special forward simulator was also written to calculate and plot a baseline “pore pressure and flow rate given” run assuming different values of overbalance pressure. This program is shown in Figure 2.48a and calculated pressure responses (with automated graphics displays) are given in Figure 2.48b.

Figure 2.48a. Input screen for “Model SC-DD-FORWARD-2-CREATETABLES-3B.”

Figure 2.48b. Pressure trends for selected overbalance pressures.

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Now, we return to supercharge models for drawdown-buildup applications using buildup data. The software shown in Figure 2.49a for the inverse solver is identical to that for the bottom screen in Figure 2.45 except for the user interface. The forward supercharge drawdownguildup solver is shown in Figure 2.49b, which solves the basic “pore pressure and flow rate given” model, plus a specified overbalance pressure. Pressure transients created by this forward solver are inputted into the inverse solver. Then validation is accomplished when mobility, pore pressure and compressibility are recovered.

Figure 2.49a. Input screen for “Model SC-DDBU-INVERSE-2.”

Figure 2.49b. Input screen for “Model SC-DDBU-FORWARD4NOPOR.”

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As of this writing, Windows user interface development is ongoing and subject to careful review given the rapid pace of development. We recognize that appearance, features and menu placement strongly affect productivity. For example, forward supercharge simulators for “drawdown only” and “drawdown-buildup” applications have been integrated and our “first pass” result is shown in Figure 2.50.

Figure 2.50. Input screen for integrated forward simulator for both “drawdown only” and “drawdown-buildup” applications.

Spherical Source Formulations 129 Multiple drawdown and buildup inverse models. We had explained how, using the exact forward solver FT-00 for liquids, we can calculate the complete pressure transient response corresponding to any multirate volume flow rate input provided “piecewise constant” rates were used (this allowed us to apply superposition methods to create exact solutions to the governing equations). Sample “curly, blue line” pressure traces are shown together with their corresponding “white box” flow rates in Figure 2.51. This menu also provides access to the two inverse routines in Figures 2.52 and 2.53 (bottom right black dots along curve represent required pressure-time data points).

Figure 2.51. Main interface, “multiple drawdown and buildup” inverse models (MDDBU) where three right-side dots indicate required pressure-time data points.

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Figure 2.52. Exact steady-state inverse solver (see “center button,” main menu).

Figure 2.53. Inverse method, Model 2 (same as FT-PTA-DDBU). There are different situations when multirate pumping can be used in practice, and an inverse calculation would be desirable using (three time-pressure point) data from the last flow rate cycle. Figure 2.54 shows the eleven pumping scenarios supported, noting that rates (in any order) can be positive, negative or zero. The “three black dots” in each diagram indicate where three time-pressure data points are to be selected for our exact inverse analysis.

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Figure 2.54. Eleven transient inverse situations supported.

132 Supercharge, Invasion and Mudcake Growth

Chin (2019) explains the exact analysis for calculating mobility, pore pressure and compressibility using three time-pressure data points from unsteady, possibly very rapid pressure responses. The models assume non-negligible flowline volume effects but do not include supercharge or overbalance pressures. Functionally, our multiple drawdown and buildup inverse model is equivalent to the simpler drawdown-buildup model FT-PTA-DDBU given previously. However, the algebra required for the underlying exact inverse solutions is extremely complicated. Detailed forward and inverse validations for all models are given in the 2019 book. Finally, we acknowledge what is perhaps the industry’s first use of “double drawdown” analysis (see Schlumberger Log Interpretation Principles/Applications, SMP-7017, Schlumberger Wireline & Testing, Sugarland, Texas, 1989). Figure 2.55 from that publication shows an important application of double drawdowns, but it is seen how both have achieved steady state and act independently. Our methodology, again, allows fully transient, interacting drawdowns and buildups, under low mobility conditions where flowline storage effects are non-negligible. This allows us to conduct multiple independent inverse tests, or even pretests, while reducing total test time significantly – other important petroleum engineering applications are described in Chin (2019).

Figure 2.55. Original Schlumberger double-drawdown application.

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Multiphase invasion, clean-up and contamination. Whereas the foregoing models deal with “clean” single phase liquid and gas fluids, flows in the reservoir are often multiphase in nature and require more complicated mathematical description. The early work of Chin and Proett (2005) describes a powerful transient, compressible liquid, miscible mixing model applicable to the formation testing process: invasion and mudcake buildup, overbalance and underbalance, onset of fluid pumping and fluid redistribution within the reservoir, and so on. This model was importantly extended to high inertia applications in Chin et al. (2015) and applications to detailed well interactions, supercharging and underbalanced effects are given elsewhere in this book. In our brief introduction here, we display the “main system menus” in Figure 2.56 while “run-time simulation menus” appropriate to a particular run appear in Figure 2.57 (a detailed presentation will follow in Section 3.3.8). Since any simulation requires dozens of inputs, the learning curve is steep – thus, an extensive library of prior runs (i.e., bottom of Figure 2.56) provides access to earlier setups that may be rerun at any time – and whose inputs may be simply modified by the user. These inputs include properties related to mud, mudcake, permeability, porosity, initial, pore and well pressure, and so on, as shown in separate categorized menus in Figure 2.57. As we will show later, as we have already done in Chin et al. (2015), many output options are available, from line plots, to static color plots, to dynamic color movies highlighting the mixing and pressure redistribution process. Although the math model and numerical solution are state-of-the-art, efficient coding and tightly integrated color graphics allow the software system to operate on modest Windows i5 machines rapidly and “right out of the box” – just as all of our other algorithms do. Also note that the initial run library at the bottom of Figure 2.56 consists of assorted runs and may be augmented by new runs saved by the user.

134 Supercharge, Invasion and Mudcake Growth

Figure 2.56. Main system level simulation menus and options.

Spherical Source Formulations 135

Figure 2.57. Run-time simulation menus for specific run.

136 Supercharge, Invasion and Mudcake Growth

The physical problem solved by our miscible flow simulator is readily explained. Figure 2.58 shows the initial invasion process (assuming overbalanced drilling) where mud penetrates the formation cylindrically. As the fluid invades and displaces or mixes with reservoir fluid, it leaves behind a growing mudcake. Figure 2.59 illustrates the onset of formation tester pumping. The red arrows show fluids entering the formation tester while blue denotes continuing invasion that is decreasing in rate. The withdrawn fluid, however, is not yet clean since it consists of mud and oil. If invasion eventually ceases, pumping at later times will lead to cleaner pumped fluids. To illustrate this process, screen shots typified by those in Figure 2.60 are offered.

Figure 2.58. Initial cylindrical invasion and mudcake buildup.

Figure 2.59. Pumpout (red) and simultaneous invasion (blue).

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Typical color plots for pressure and concentration, shown in Figure 2.60, appear periodically at different time intervals so that the physics in the reservoir can be monitored (for faster simulations, these screens can be turned off). Figure 2.60 contains two views, an early time screen shot at left and a later time view at right. Consider the left-most diagram in the left screen shot. This shows the pressure distribution in an angular section of the reservoir (red indicates high pressure in the well, while orange denotes lower pressure in the reservoir. The small “blue dot” represents low pressure at the probe where fluid is being withdrawn. The central diagram shows concentration, dirty “blue mud” at the left well interface invading clean red in situ fluid. The far right “striped” diagram indicates reservoir layer boundaries assumed for this simulation. The right side Windows screen displays analogous events at a much later time. Note how the “very colorful, dynamic” pressure plot still shows the high (red) pressure influence of the well. The nearfield is now blue (instead of orange) because pumping has continued for a significant time. The effect on concentration is more pronounced. Line plots showing viscosity and fluid contamination level (or concentration percentage) are offered as well and typical plots are shown in later calculations.

Figure 2.60. Early results (left) and later dynamics (right) times. During runs, different “snapshots” use different color scales; thus, “red” at one time denotes a different attribute value from “red” at a another time. After the simulations are completed, all color plots are collected, and color scales are normalized to the maxima and minima for the entire time collection of snapshots for “movie mode” playback. This provides a good understanding of the mixing process throughout all time, whereas the static plots provided during simulations lead to an understanding of properties as they distributed throughout in space.

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System integration and closing remarks. Finally, we indicate that effective productivity software provides more than individual models – an integrated platform hosting forward, inverse and multiphase software, together with manuals, papers, “best practice” tips, industry videos, marketing brochures, plus internal company and competitor information, and so on, is required. Figure 2.61 provides a glimpse of one such prototype that would support the needs of active users.

Figure 2.61. Integrated software platform, a beginning.

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2.3.6.3 Supercharging summaries - advanced forward and inverse models explored. The purposes behind formation testing pressure transient analysis are multifold, among them, pore pressure assessment for production planning and drilling safety assessment, gradient analysis for fluid identification, and identification of fluid contact interfaces and possible solid barriers. In addition, formation mobility is a key objective, with the focus being rock permeability, fluid viscosity and compressibility. Early on, we introduced new methods to extrapolate such information from early time pressure transient measurements, and demonstrated how we ensure our inverse methods are correct. We create “synthetic data” using analytical solutions in FT-00, analyze resulting pressure time histories using rapid inverse models, and predict fluid and formation quantities that agree with known FT-00 inputs. In this sense, our forward and inverse models are consistent. We will do the same with supercharging. Supercharge math model development. In both forward FT-00 and inverse FT-01 instances, spherical flow was assumed for isotropic media (while ellipsoidal models were used for transversely isotropic formations). This was done for mathematical expediency and the practical advantages offered by simple analytical models. There is no surprise here: in all areas of mathematical physics, geometric simplifications are sought to make algebraic manipulations manageable. However, sometimes these do not apply to the physical situation, and the wide variety of drilling scenarios contains examples where simple spherical or ellipsoidal models will not apply. One important class of applications deals with formation testing when wellbore fluids enter the reservoir, known as “overbalanced drilling,” and the second, when reservoir fluids enter the well, referred to as “underbalanced drilling.” The former builds mudcake seals to prevent fluid lost into the formation, while the second disallows cake formation, reducing skin damage and enabling production while drilling. We introduce three drilling situations in Figures 2.62 - 2.64. In Figure 2.62, the well is drilled underbalanced and reservoir fluid enters the well along the circular cylindrical surfaces of the wellbore. At the same time, the formation tester is injecting or withdrawing fluid, so that a spherical or ellipsoidal flow in superposed on the cylindrical flow. In Figure 2.63, with overbalanced drilling, the direction of the wellbore flow is reversed. In underbalanced drilling, the well pressure is less than the pore pressure, while in overbalanced drilling, the well pressure exceeds the pore pressure.

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Figure 2.62. Underbalanced drilling with reservoir outflow.

Figure 2.63. Overbalanced drilling with wellbore inflow. The geometric nature of the flow in Figures 2.62 and 2.63 precludes simple spherical flow modeling. A full cylindrical plus spherical or ellipsoidal flow overlap is required. This was the authors’ consensus up to the publication in Chin (2019), but since then, the work reported in our companion 2021 book, shows that the flows in Figures 2.62 and 2.63, and also that in Figure 2.64, can be accurately solved with a rapid and stable 3D numerical algorithm . . . and, in fact, with supercharge and multiple probes acting independently. Nonetheless, the work of Chin (2019), which provides exact, closed form solutions to a spherical (or ellipsoidal) source model of supercharge is significant. It applies, for instance, to dual probe tools where the pressure sensors are aligned axially – further, it extends prior forward and inverse models, and supports rapid, real-time interpretation in field applications.

Spherical Source Formulations 141

Figure 2.64. Overbalanced and underbalanced drilling applications with sealed borehole walls. In Figure 2.64, we consider a flow where mudcake has largely sealed the borehole walls against fluid influx or outflux. An overbalanced flow is assumed to have formed the mudcake, which is now thick or impermeable enough to allow at most local invasion speeds that are small compared to formation tester nozzle velocities. The figure would also apply to underbalanced drilling when fluid speeds leaving the formation are small compared to tester pumping speeds at the nozzle. For such cases, simpler spherical or ellipsoidal models apply. This is the model considered in this section. The spherical supercharging model in Chin (2019) is easily explained in terms of the conventional model underlying FT-00, FT-01, FT-PTA-DDBU and other related programs. Conventional zero supercharge model. The formation testing book by Chin et al. (2014) provides a number of zero-supercharge forward and inverse formulations in different physical limits together with exact solutions. One particular model, quite general, is given by Equations 5.1– 5.4 in that reference, here re-numbered as, 2P(r,t)/ r2 + 2/r P/ r = ( c/k) P/ t (A1) P(r,t = 0) = P0

(A2)

P(r = ,t) = P0

(A3)

(4 Rw2k/ ) P(Rw,t)/ r – VC P/ t = Q(t)

(A4)

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Equations A1 – A4 represent the complete isotropic, zero skin, spherical Darcy flow formulation for compressible liquids, which was solved exactly in terms of complex complementary error functions. This formulation was used to develop the forward simulation FT-00, as well as the inverse procedures FT-01 and FT-PTA-DDBU. Note that “spherical” implies mathematical idealization. The foregoing equations do not handle supercharge. The spherical source of radius Rw will not adequately describe borehole wall curvature or the effects of tester pads – to account for these non-idealizations, Rw often denotes the product of a corrective “geometric factor” (that is, “G,” determined empirically or through 3D finite element analysis) and the true nozzle radius. Supercharge extension. The effects of supercharge were highlighted not long ago by Halliburton and Chevron Thailand E&P, in a symposium article surveying several hundred wells. Rourke, Powell, Platt, Hall and Gardner, in “A New Hostile Environment Wireline Formation Testing Tool: A Case Study from the Gulf of Thailand,” SPWLA 47th Annual Logging Symposium, Veracruz, Mexico, June 4-7, 2006, summarize field experiences gained from over three hundred wells logged with a new formation tester. The key point from their paper is succinctly contained in one quote. In short, “While pressure testing in the infill development wells where depletion is often observed, it is not uncommon for the differential between hydrostatic and reservoir pressure to exceed 2000 psi.” This is also emphasized in a 2018 fact sheet, “Testing the Tight Gas Reservoir – Hostile Environment Wireline Formation Tester Reduces NPT in HPHT Boreholes,” available from Halliburton at www.halliburton.com. This importantly emphasized that, “Wells are drilled highly overbalanced because the differential between hydrostatic and reservoir pressure may exceed 2,000 psi.” Other studies also cite high overbalances, greatly exceeding the 200 psi found in earlier literature. Proett, Ma, Al-Musharfi and Berkane, in “Dynamic Data Analysis with New Automated Workflows for Enhanced Formation Evaluation,” SPE-187040-MS, Society of Petroleum Engineers Annual Technical Conference and Exhibition, San Antonio, Texas, October 9-11, 2017, consider overbalance pressures of 500-1,000 psi in several examples. Proett, Seifert, Chin, Lysen and Sands, in “Formation Testing in the Dynamic Drilling Environment,” SPWLA 45th Annual Logging Symposium, Noordwijk, The Netherlands, June 6-9, 2004, assume overbalance pressures of 1,000 psi in their examples.

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Now, we ask how “supercharging,” where the term will be used to represent both overbalanced and underbalanced effects, can be modeled by extending the framework of Equations A1 – A4. Our approach is straightforward. We solve these equations with a single important change, altering only the initial condition as shown in Equation B2, highlighted in red. In particular, we consider the model 2P(r,t)/ r2 + 2/r P/ r = ( c/k) P/ t (B1) P(r,t = 0) = P0 + Z/r, Z > 0, R > Rw

(B2)

P(r = ,t) = P0

(B3)

(4 Rw2k/ ) P(Rw,t)/ r – VC P/ t = Q0 (B4) Here Z = Pbh – P0 may be positive (overbalanced) or negative (underbalanced) with any magnitude, Pbh being the borehole pressure just outside the reservoir sandface, and P0 being the farfield reservoir pore pressure. In older references, “Pbh – P0” is often quoted in the 200-250 psi range. In fact, Halliburton and Chevron Thailand, as discussed previously, observe that overbalances exceeding 2,000 psi are not uncommon, particularly in infill drilling where the reservoir is depleting. For such problems, the use of inverse models based on Equations A1 – A4 will lead to incorrect pore pressures, mobilities and compressibilities. There are important physical differences from the math models used to construct our prior forward FT-00 and inverse FT-01 simulators. Both derive from Equations A1 – A4. These earlier methods assume Darcy’s partial differential equation, a general pumping boundary condition accounting for volume flow rate and flowline storage, a given farfield pore pressure P0 and a uniform initial condition in which the initial pressure is identical to P0. These conditions underlie the models in Chin et al. (2014, 2015) and earlier published papers by Chin and Proett. Our new model is described by Equations B1-B4. Whereas previously we assumed a uniform initial condition in which the initial pressure P0 is constant everywhere, now we allow the initial pressure to vary spatially with P(r, t = 0) = P0 + Z/r, where a constant Z > 0 for overbalanced applications, for R > Rnozzle , while for underbalanced drilling, we take Z < 0. The “1/r” decay seen in “Z/r” is an approximate description of monotonic pressure decrease away from the wellbore surface. However, it is interesting that “1/r” is also an exact solution of the steady spherical flow differential equation. We now provide illustrative applications.

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2.3.6.4 Drawdown only applications. Several Fortran computer programs have been written to implement our forward and inverse methods to “drawdown only” and “drawdown buildup” applications. A wide range of example calculations is offered in this section for drawdown-only problems. In each example, the forward model is used to create the pressure transient response seen by the formation tester probe in the presence of overbalanced pressure, with an assumed set of input parameters, e.g., permeability, viscosity, compressibility, pore pressure and so on. The forward pressure response results are given in sets of tabulated “time and pressure” numbers. Then, the inverse mode is evaluated, using three early-time data points arbitrarily selected from the above table and used to predict known pore pressure, mobility and compressibility. The inverse solver is operated in two modes, the correct supercharge mode requiring an overbalanced pressure input, and the incorrect mode – that is, the conventional inverse approach, which does not ask for overbalance inputs. The examples chosen show how supercharge can affect pore pressures (and hence, pressure gradients used for fluid, contact and barrier identification) and mobilities. Software references are given for readers wishing to duplicate or extend our published simulations. Note that authors’ comments are offered in the present Times Roman font, while computer screen results appear in Courier New font. Example DD-1. High overbalance.

We first create forward data, calculating the transient pressure history measured by the probe using the overbalanced data shown immediately below. Then, in two separate inverse examples, we select three (time, pressure) data points arbitrarily, and predict pore pressure, mobility and fluid compressibility. In the first instance, we assume that we know the overbalance pressure – the predictions are excellent. In the second, we assume that it is zero, that is, we will use the incorrect inverse method that does not account for supercharging – and we demonstrate that the predictions are (as expected) not very good. The latter mode, operating without knowledge of supercharge pressure, corresponds to present industry practice.

Spherical Source Formulations 145 C:\FT-PTA-SC>sc-dd-forward-3B

Software reference, sc-dd-forward-3B.for. Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc):

1 1 .00001 20000 2000 1 .5 1 1000

FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Forward pressure transient predictions for drawdown-only applications, for low mobility, isotropic, supercharged flows where flowline storage is not negligible. Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc):

0.1000E+01 0.1000E+01 0.1000E-04 0.2000E+05 0.2000E+04 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0

P(psi) 22000. 21818. 21644. 21477. 21317. 21164. 21017. 20877. 20742. 20613. 20489. 20371. 20257. 20149. 20045. 19945. 19849. 19758.

(selected for inverse input)

146 Supercharge, Invasion and Mudcake Growth 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 39.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0 500.0

19670. 19586. 19505. 19428. 19354. 19284. 19216. 19151. 19089. 19029. 18972. 18917. 18865. 18814. 18766. 18720. 18676. 18634. 18593. 18554. 18517. 18482. 18447. 18176. 17999. 17883. 17808. 17759. 17727. 17669. 17668. 17668. 17668.

(selected for inverse input)

(selected for inverse input)

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

2332.0621 2000.0000 4332.0621

Spherical Source Formulations 147

Figure 2.65a. Pressure transient response with overbalance. In the following we use our inverse solver, take pressure data from 10, 20 and 30 seconds, entering in the input screen an overbalance pressure of 2,000 psi known from forward simulator inputs. C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for.

Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1 .5 1 10 20489 20 19505 30 18865 2000

148 Supercharge, Invasion and Mudcake Growth OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 10.0000 20489.0000 20.0000 19505.0000 30.0000 18865.0000 2000.0000

The predicted results are excellent, namely, a pore pressure of 20,003 psi instead of 20,000 psi, a mobility of 1.0164 md/cp instead of 1 md/cp, and a compressibility of 0.0100 x (cc/FloLineVol) per psi or, since the flowline volume is 1,000 cc, 0.00001 1/psi, exactly as assumed. Note that, in our calculations, we used an oil compressibility that is ten-fold larger than that of water, which results to slower transient decays. The following output summarizes inverse results. Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20003.0000 1.0164 0.0644 0.0100 x (cc/FloLineVol)

We next consider a second inverse calculation, using an incorrect overbalance pressure input that does not correspond to the value used in the forward simulation creating the pressure transient data. C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for.

Here, we repeat the inverse calculation assuming an overbalance of 0 psi, that is, no overbalance – in other words, we are using the inverse method previously developed that does not account for supercharging. As expected, the results are not good. In particular, the output below shows a pore pressure of 22,003 psi instead of 20,000 psi, a mobility of 0.5463 md/cp versus an assumed value of 1 md/cp, and finally, a fluid compressibility of 0.0054 x (cc/FloLineVol)per psi instead of 0.0100 x (cc/FloLineVol)per psi. These unsatisfactory results are not surprising – they merely confirm our expected errors.

Spherical Source Formulations 149 Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1 .5 1 10 20489 20 19505 30 18865 0

OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 10.0000 20489.0000 20.0000 19505.0000 30.0000 18865.0000 0.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

22003.0000 0.5463 0.0346 0.0054 x (cc/FloLineVol)

150 Supercharge, Invasion and Mudcake Growth

Example DD-2. High overbalance.

The forward model, in this example, is similar to that used in Section 2.5.2 above except that the mobility is smaller by a factor of ten. C:\FT-PTA-SC>sc-dd-forward-3B

Software reference, sc-dd-forward-3B.for.

Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Plot every "NSEC" seconds:

0.1 1 .00001 20000 2000 1 .5 1 1000 50

Figure 2.65b. Pressure transient response with overbalance.

Spherical Source Formulations 151 FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Forward pressure transient predictions for drawdown-only applications, for low mobility, isotropic, supercharged flows where flowline storage is not negligible. Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc):

0.1000E+00 0.1000E+01 0.1000E-04 0.2000E+05 0.2000E+04 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0

P(psi) 22000. 21892. 21784. 21676. 21569. 21463. 21357. 21251. 21146. 21041. 20937. 20833. 20730. 20627. 20525. 20423. 20321. 20220. 20119. 20019. 19919. 19820. 19721. 19622. 19524. 19426. 19329. 19232. 19135. 19039. 18944. 18848. 18753. 18659.

(selected for inverse input)

(selected for inverse input)

(selected for inverse input)

152 Supercharge, Invasion and Mudcake Growth 34.0 35.0 36.0 37.0 38.0 39.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0

18565. 18471. 18378. 18285. 18193. 18101. 18009. 17114. 16256. 15434. 14647. 13893. 13170. 7420. 3674. 1235.

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

23320.6202 2000.0000 25320.6202

Following the illustrative Example DD-1 above, we perform a first inverse calculation using three (time, pressure) data points arbitrarily chosen, highlighted in red above, plus the known overbalance pressure of 2,000 psi. The predictions are excellent, in particular, a pore pressure of 20,000 psi in exact agreement with that assumed in the forward model, a mobility of 0.1040 md/cp versus an inputted 0.1 md/cp above, and a compressibility of 0.0100 x (cc/FloLineVol) per psi in agreement with the forward analysis. In the second inverse calculation, where we will assume a zero overbalance (that is, we use the older inverse model that does not account for supercharging), the results are questionable. The mobility is 0.0956 md/cp while the compressibility output is “0.0092” – these acceptable results compare with exact values of 0.1 md/cp and “0.01.” However, an incorrect pore pressure of 22,000 psi is obtained versus an assumed 20,000 psi. C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for.

Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible.

INPUTS, Inverse Model ...

Spherical Source Formulations 153 Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1 .5 1 10 20937 20 19919 30 18944 2000

OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 10.0000 20937.0000 20.0000 19919.0000 30.0000 18944.0000 2000.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20000.0000 0.1040 0.0646 0.0100 x (cc/FloLineVol)

C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for.

Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1 0.5 1 10 20937 20 19919 30 18944 0

154 Supercharge, Invasion and Mudcake Growth OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 10.0000 20937.0000 20.0000 19919.0000 30.0000 18944.0000 0.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

22000.0000 0.0956 0.0593 0.0092 x (cc/FloLineVol)

Example DD-3. High overbalance.

In the forward model below, we consider a higher mobility of 10 md/cp relative to the two prior examples. C:\FT-PTA-SC>SC-DD-FORWARD-3B

Software reference, sc-dd-forward-3B.for.

Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Plot every "NSEC" seconds:

10 1 .00001 20000 2000 1 .5 1 1000 1

Spherical Source Formulations 155

Figure 2.65c. Pressure transient response with overbalance. FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Forward pressure transient predictions for drawdown-only applications, for low mobility, isotropic, supercharged flows where flowline storage is not negligible. Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc):

0.1000E+02 0.1000E+01 0.1000E-04 0.2000E+05 0.2000E+04 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

P(psi) 22000. 21221. 20714. 20384. 20169. 20028. 19937. 19878.

(selected for inverse input)

156 Supercharge, Invasion and Mudcake Growth 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0 500.0

19839. 19814. 19797. 19787. 19780. 19775. 19772. 19770. 19769. 19768. 19768. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767.

(selected for inverse input)

(selected for inverse input)

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

233.2062 2000.0000 2233.2062

In our first inverse calculation, we assume we know the overbalance pressure of 2,000 psi, and we predict 20,014 psi, 9.5467 md/cp and “0.0094” for compressibility – these compare well with known values of 20,000 psi, 10 md/cp and “0.01,” providing excellent results. However, in the second inverse calculation assuming an overbalance of zero, that is, using an older inverse model that does not provide for supercharging, the results are poor. We have 22,014 psi as opposed to 20,000 psi for pore pressure. The mobility is 1.0512 md/cp versus a known value of 10 md/cp – a factor of ten discrepancy, although the compressibility is correct to the number of decimal places shown.

Spherical Source Formulations 157 C:\FT-PTA-SC>SC-DD-INVERSE-2

Software reference, sc-dd-inverse-2.for.

Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1 .5 1 5 20028 10 19797 20 19767 2000

OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 5.0000 20028.0000 10.0000 19797.0000 20.0000 19767.0000 2000.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20014.0000 9.5467 0.0606 0.0094 x (cc/FloLineVol)

C:\FT-PTA-SC>SC-DD-INVERSE-2

Software reference, sc-dd-inverse-2.for.

Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor:

1 .5 1

158 Supercharge, Invasion and Mudcake Growth 1st Point Time T1 Pressure P1 2nd Point Time T2 Pressure P2 3rd Point Time T3 Pressure P3 Overbalance pressure

(sec): (psi): (sec): (psi): (sec): (psi): (psi):

5 20028 10 19797 20 19767 0

Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 5.0000 20028.0000 10.0000 19797.0000 20.0000 19767.0000 0.0000

OUTPUT SUMMARY ...

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

22014.0000 1.0512 0.0067 0.0010 x (cc/FloLineVol)

Example DD-4. Qualitative pressure trends.

It is always instructive, before performing forward and inverse studies, to assess the qualitative effects of overbalance. The work reported here draws upon software reference sc-dd-forward-2-createtables-3B.for. This software automatically and conveniently produces transient pressure responses for different assumed overbalance pressures. For instance, in the following input listing, the “200 psi” is used to create pressure transient responses corresponding to overbalance pressures of 0, 200, 400, 600 and 800 psi. In this drawdown-only example, we observe that the initial pressure at t = 0 contains the effects of high overbalance, and that all pressure responses correctly tend to the same pressure at large times; in this particular example, that time is approximately 240 sec with pressures of 17,668 psi. From the graph in Figure 2.5d, it is clear that as overbalance effects disappear with time, all line graphs grow closer and closer. The rate of this convergence depends on tool, fluid and formation parameters. The effect of overbalance, as expected, is not a simple shift of the static “no overbalance” response.

Spherical Source Formulations 159 C:\FT-PTA-SC>sc-dd-forward-2-create-tables-3B

Software reference, sc-dd-forward-2-create-tables-3B.for. Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Plot every "NSEC" seconds:

1 1 .00001 20000 200 1 .5 1 1000 5

FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Forward pressure transient predictions for drawdown-only applications, for low mobility, isotropic, supercharged flows where flowline storage is not negligible. Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc):

0.1000E+01 0.1000E+01 0.1000E-04 0.2000E+05 0.2000E+03 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0

P0 (psi) 20000. 19902. 19808. 19718. 19632. 19550. 19471. 19395. 19323. 19253. 19187. 19123. 19062. 19003. 18947. 18894. 18842. 18793. 18746. 18700.

P1 (psi) 20200. 20094. 19992. 19894. 19801. 19711. 19626. 19543. 19465. 19389. 19317. 19248. 19182. 19118. 19057. 18999. 18943. 18889. 18838. 18789.

P2 (psi) 20400. 20285. 20175. 20070. 19969. 19873. 19780. 19692. 19607. 19525. 19447. 19373. 19301. 19233. 19167. 19104. 19044. 18986. 18931. 18878.

P3 (psi) 20600. 20477. 20359. 20246. 20138. 20034. 19935. 19840. 19749. 19661. 19578. 19497. 19421. 19347. 19277. 19209. 19144. 19082. 19023. 18966.

P4 (psi) 20800. 20669. 20543. 20422. 20306. 20196. 20089. 19988. 19890. 19797. 19708. 19622. 19540. 19462. 19386. 19314. 19245. 19179. 19115. 19055.

160 Supercharge, Invasion and Mudcake Growth 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 39.0 40.0 41.0 42.0 43.0 44.0 45.0 46.0 47.0 48.0 49.0 50.0 51.0 52.0 53.0 54.0 55.0 56.0 57.0 58.0 59.0 60.0 120.0 240.0 360.0 480.0 600.0 720.0 840.0 960.0

18657. 18616. 18576. 18538. 18501. 18466. 18433. 18401. 18370. 18340. 18312. 18285. 18259. 18234. 18211. 18188. 18166. 18145. 18125. 18106. 18088. 18070. 18053. 18037. 18021. 18007. 17992. 17979. 17966. 17953. 17941. 17930. 17919. 17908. 17898. 17888. 17879. 17870. 17862. 17854. 17846. 17682. 17668. 17668. 17668. 17668. 17668. 17668. 17668.

18742. 18697. 18654. 18612. 18573. 18535. 18498. 18463. 18430. 18398. 18367. 18338. 18310. 18283. 18257. 18232. 18209. 18186. 18164. 18143. 18124. 18104. 18086. 18069. 18052. 18036. 18020. 18005. 17991. 17978. 17965. 17952. 17940. 17929. 17918. 17907. 17897. 17888. 17878. 17870. 17861. 17683. 17668. 17668. 17668. 17668. 17668. 17668. 17668.

18827. 18778. 18732. 18687. 18644. 18603. 18564. 18526. 18490. 18456. 18423. 18391. 18361. 18332. 18304. 18277. 18251. 18227. 18204. 18181. 18160. 18139. 18119. 18100. 18082. 18065. 18048. 18032. 18017. 18002. 17988. 17975. 17962. 17949. 17938. 17926. 17915. 17905. 17895. 17886. 17876. 17684. 17668. 17668. 17668. 17668. 17668. 17668. 17668.

18912. 18859. 18809. 18762. 18716. 18672. 18630. 18589. 18550. 18513. 18478. 18444. 18411. 18380. 18350. 18322. 18294. 18268. 18243. 18219. 18195. 18173. 18152. 18132. 18112. 18094. 18076. 18059. 18042. 18027. 18012. 17997. 17983. 17970. 17957. 17945. 17934. 17922. 17912. 17902. 17892. 17685. 17668. 17668. 17668. 17668. 17668. 17668. 17668.

18996. 18941. 18887. 18836. 18787. 18740. 18695. 18652. 18611. 18571. 18533. 18497. 18462. 18429. 18397. 18366. 18337. 18309. 18282. 18256. 18231. 18208. 18185. 18163. 18143. 18123. 18104. 18085. 18068. 18051. 18035. 18020. 18005. 17991. 17977. 17964. 17952. 17940. 17928. 17917. 17907. 17686. 17668. 17668. 17668. 17668. 17668. 17668. 17668.

Note how, for the parameters chosen in this example, the effects of supercharge do not dissipate until about 15 minutes. Of course, the operational objective is not to wait for complete dissipation, but instead, to predict fluid and formation properties using early time pressure data that includes the distortive effects of supercharging.

Spherical Source Formulations 161

Figure 2.65d. Pressure trends for selected overbalance pressures. Example DD-5. Qualitative pressure trends. Software reference, sc-dd-forward-2-create-tables-3B.for.

Additional computations for a wider range of overbalance pressures are provided below without further comment. FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL

Forward pressure transient predictions for drawdown-only applications, for low mobility, isotropic, supercharged flows where flowline storage is not negligible.

Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc):

0.1000E+02 0.1000E+01 0.1000E-04 0.2000E+05 0.1000E+04 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04

162 Supercharge, Invasion and Mudcake Growth Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0

P0 (psi) 20000. 19919. 19866. 19831. 19809. 19794. 19785. 19778. 19774. 19772. 19770. 19769. 19768. 19768. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767.

P1 (psi) 21000. 20570. 20290. 20107. 19989. 19911. 19861. 19828. 19807. 19793. 19784. 19778. 19774. 19771. 19770. 19769. 19768. 19768. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767.

P2 (psi) 22000. 21221. 20714. 20384. 20169. 20028. 19937. 19878. 19839. 19814. 19797. 19787. 19780. 19775. 19772. 19770. 19769. 19768. 19768. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767.

P3 (psi) 23000. 21873. 21138. 20660. 20349. 20146. 20014. 19928. 19871. 19835. 19811. 19796. 19786. 19779. 19775. 19772. 19770. 19769. 19768. 19768. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767.

P4 (psi) 24000. 22524. 21562. 20936. 20528. 20263. 20090. 19977. 19904. 19856. 19825. 19805. 19791. 19783. 19777. 19774. 19771. 19770. 19769. 19768. 19768. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767. 19767.

Figure 2.65e. Pressure trends for selected overbalance pressures.

Spherical Source Formulations 163 Example DD-6. “Drawdown-only” data with multiple inverse scenarios for 1 md/cp application. C:\FT-PTA-SC>sc-dd-forward-3B

Software reference, sc-dd-forward-3B.for. As in prior examples, we first create the synthetic source probe pressure transient response, in this case using the following data . . . Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Plot every "NSEC" seconds:

1 1 .00001 20000 1000 1 .5 1 1000 50

FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Forward pressure transient predictions for drawdown-only applications, for low mobility, isotropic, supercharged flows where flowline storage is not negligible. Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc):

0.1000E+01 0.1000E+01 0.1000E-04 0.2000E+05 0.1000E+04 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0

P(psi) 21000. 20860. 20726. 20598. 20475. 20357. 20244. 20136. 20032. 19933. 19838. 19747.

(selected for inverse input)

164 Supercharge, Invasion and Mudcake Growth 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 39.0 40.0 41.0 42.0 43.0 44.0 45.0 46.0 47.0 48.0 49.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0 500.0

19660. 19576. 19496. 19419. 19346. 19275. 19208. 19143. 19081. 19022. 18965. 18911. 18859. 18809. 18761. 18715. 18671. 18629. 18588. 18550. 18513. 18477. 18443. 18411. 18380. 18350. 18321. 18294. 18267. 18242. 18218. 18195. 18173. 18152. 18131. 18112. 18093. 18076. 18058. 17922. 17834. 17776. 17738. 17714. 17669. 17668. 17668. 17668.

(selected for inverse input)

(selected for inverse input)

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

2332.0621 1000.0000 3332.0621

Spherical Source Formulations 165

Figure 2.65f. Pressure transient response with overbalance. In the first inverse application below, we select three pressure data points as shown, but assume that we know the overbalance pressure of 1,000 psi exactly. Inverse calculation #1 C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for.

Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1 .5 1 5 20357 15 19419 25 18809 1000

166 Supercharge, Invasion and Mudcake Growth OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 5.0000 20357.0000 15.0000 19419.0000 25.0000 18809.0000 1000.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20001.0000 1.0170 0.0644 0.0100 x (cc/FloLineVol)

Predicted results for pore pressure, mobility and compressibility are excellent, agreeing with the inputs assumed in the forward analysis that created the synthetic pressure data. But what if we did not use the inverse supercharge model, or equivalently, “What if conventional inverse methods with zero overbalance were applied?” To answer this question, we simply run our inverse supercharge model with a zero input overbalance pressure. The results are shown below. Inverse calculation #2 C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for.

Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1 .5 1 5 20357 15 19419 25 18809 0.

Spherical Source Formulations 167 OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 5.0000 20357.0000 15.0000 19419.0000 25.0000 18809.0000 0.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

21001.0000 0.7110 0.0450 0.0070 x (cc/FloLineVol)

The answer to the foregoing question is clear – pore pressures are not correct, while mobility and compressibility depart substantially from desired values (0.7110 vs 1.0 md/cp and “0.0070” vs “0.0030” 1/psi). We might also consider the follow-up question. “What if we used the inverse supercharge model, but with an incorrect overbalance guess?” This is considered in the calculation below. The assumed overbalance is closer to the correct value rather than being set to zero. Inverse calculation #3 C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for.

Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1 .5 1 5 20357 15 19419 25 18809 500

168 Supercharge, Invasion and Mudcake Growth OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

1.0000 0.5000 1.0000 0.5000 5.0000 20357.0000 15.0000 19419.0000 25.0000 18809.0000 500.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20501.0000 0.8369 0.0530 0.0082 x (cc/FloLineVol)

The predicted mobility and compressibility are slightly improved, but the pore pressure is still incorrect. It is, however, useful to note that predicted pressure gradient trends (from run to run) may be nonetheless useful in identifying fluid contacts. Example DD-7. “Drawdown-only” data with multiple inverse scenarios for 0.1 md/cp application. C:\FT-PTA-SC>sc-dd-forward-3B

Software reference, sc-dd-forward-3B.for. Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Plot every "NSEC" seconds: Stop - Program terminated.

0.1 1 .00001 20000 250 0.5 1 1 300 50

FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Forward pressure transient predictions for drawdown-only applications, for low mobility, isotropic, supercharged flows where flowline storage is not negligible.

Spherical Source Formulations 169 Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc):

0.1000E+00 0.1000E+01 0.1000E-04 0.2000E+05 0.2500E+03 0.5000E+00 0.1000E+01 0.1000E+01 0.1000E+01 0.3000E+03

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0

P(psi) 20250. 20079. 19912. 19750. 19593. 19440. 19292. 19147. 19007. 18871. 18738. 18609. 18484. 18363. 18245. 18130. 18018. 17910. 17804. 17702. 17602. 17506. 17412. 17320. 17231. 17145. 17061. 16980. 16901. 16824. 16749. 16676. 16606. 16537. 16470. 16405. 16342. 16281. 16222.

(selected for inverse input)

(selected for inverse input)

(selected for inverse input)

170 Supercharge, Invasion and Mudcake Growth 39.0 40.0 41.0 42.0 43.0 44.0 45.0 46.0 47.0 48.0 49.0 50.0 100.0 200.0 300.0 400.0 410.0 420.0 430.0 440.0 450.0 460.0 470.0 480.0 490.0 500.0

16164. 16108. 16053. 16000. 15948. 15898. 15850. 15802. 15756. 15711. 15668. 15626. 14519. 14190. 14171. 14170. 14170. 14170. 14170. 14170. 14170. 14170. 14170. 14170. 14170. 14170.

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

5830.1552 250.0000 6080.1552

Figure 2.65g. Pressure transient response with overbalance.

Spherical Source Formulations 171 Inverse calculation #1 C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for.

Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

0.5 1.0 1 5 19440 15 18130 25 17145 250

OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

0.5000 1.0000 1.0000 1.0000 5.0000 19440.0000 15.0000 18130.0000 25.0000 17145.0000 250.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

19999.0000 0.1014 0.0194 0.0030 x (cc/FloLineVol)

Predicted results are excellent for pore pressure, mobility and compressibility in this low mobility calculation. Next we examine the consequences of using an inverse model that does not account for supercharge, that is, one assuming that overbalance pressure is vanishing.

172 Supercharge, Invasion and Mudcake Growth Inverse calculation #2 C:\FT-PTA-SC>sc-dd-inverse-2

Software reference, sc-dd-inverse-2.for.

Inverse model for low mobility, isotropic, supercharged applications for "drawdown only" problems when flowline storage is not negligible. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

0.5 1.0 1 5 19440 15 18130 25 17145 0

OUTPUT SUMMARY ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance pressure (psi):

0.5000 1.0000 1.0000 1.0000 5.0000 19440.0000 15.0000 18130.0000 25.0000 17145.0000 0.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20249.0000 0.0972 0.0186 0.0029 x (cc/FloLineVol)

Predictions for mobility and compressibility are not too far from exact values, although the pore pressure is over-predicted.

Spherical Source Formulations 173 2.3.6.5 Drawdown – buildup applications. Example DDBU-1. Drawdown-buildup, high overbalance.

With a good degree of confidence from our “drawdown only” inverse supercharge model, we turn to drawdown-buildup applications. Here, our inverse model uses data from the buildup portion of the pressure transient curve. We have selected some interesting examples from numerous that we have tested. In the forward analysis below, with an overbalance of 2,000 psi, our assumed mobility is 10 md/cp and the time at which the piston stops withdrawing fluid is taken as 5 sec. In the pressure transient response, note how the expected buildup is not quite building up – in fact, the curve levels out and is almost flat. Now we turn to inverse calculations. The selected (time, pressure) data points at 5, 8 and 15 sec are 20,028 psi, 20,008 psi and 20,000 psi. In the first inverse calculation using our new supercharge model, we assumed an overbalance of 2,000 psi, and the predicted results 19,999 psi, 8.609 md/cp and “0.0090” for compressibility are excellent when compared with the known values of 20,000 psi, 10 md/cp and “0.01.” In the second inverse calculation, we assume that the overbalance is zero, using the older inverse model that does not account for supercharging. The results are not satisfactory. The pore pressure is correct at 19,999 psi versus a known value of 20,000 psi, however an unacceptable negative mobility of -71.849 md/cp and a negative compressibility are found. C:\FT-PTA-SC>SC-DDBU-FORWARD-4NOPOR

Software reference, SC-DDBU-FORWARD-4NOPOR.FOR. Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Time drawdown ends (sec):

10 1 .00001 20000 2000 1 .5 1 1000 5

174 Supercharge, Invasion and Mudcake Growth

Figure 2.66a. Pressure transient response with overbalance. FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc): Time drawdown ends ............. (sec):

0.1000E+02 0.1000E+01 0.1000E-04 0.2000E+05 0.2000E+04 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04 0.5000E+01

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

P(psi) 22000. 21221. 20714. 20384. 20169. 20028. 20019. 20012. 20008. 20005. 20003.

(selected for inverse input) (selected for inverse input)

Spherical Source Formulations 175 11.0 12.0 13.0 14.0 15.0 50.0 100.0 200.0 300.0 400.0

20002. 20001. 20001. 20001. 20000. 20000. 20000. 20000. 20000. 20000.

(selected for inverse input)

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

233.2062 2000.0000 2233.2062

C:\FT-PTA-SC>SC-DDBU-INVERSE-2

Software reference, SC-DDBU-INVERSE-2.FOR. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1 .5 1 5 5 20028 8 20008 15 20000 2000

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1.0000 0.5000 1.0000 0.5000 5.0000 5.0000 20028.0000 8.0000 20008.0000 15.0000 20000.0000 2000.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

Excellent results below!

19999.000 8.609 0.0583 0.0090 x (cc/FloLineVol)

176 Supercharge, Invasion and Mudcake Growth C:\FT-PTA-SC>SC-DDBU-INVERSE-2

Software reference, SC-DDBU-INVERSE-2.FOR. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1 .5 1 5 5 20028 8 20008 15 20000 0

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1.0000 0.5000 1.0000 0.5000 5.0000 5.0000 20028.0000 8.0000 20008.0000 15.0000 20000.0000 0.0000

Pore pressure and mobility predicted ... Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

Poor results below!

19999.000 -71.849 -0.4866 -0.0754 x (cc/FloLineVol)

Spherical Source Formulations 177 Example DDBU-2. Drawdown-buildup, high overbalance.

Here we alter the above forward analysis inputs, to emphasize the buildup part of the pressure transient response – the buildup curve here actually shows an increasing pressure with time. C:\FT-PTA-SC>SC-DDBU-FORWARD-4BNOPOR

Software reference, SC-DDBU-FORWARD-4BNOPOR.FOR (plots more densely, every second, as opposed to Version 4). Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Time drawdown ends (sec):

10 1 .00001 20000 2000 2 .5 1 1000 5

FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc): Time drawdown ends ............. (sec):

0.1000E+02 0.1000E+01 0.1000E-04 0.2000E+05 0.2000E+04 0.2000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04 0.5000E+01

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

P(psi) 22000. 21140. 20580. 20215. 19977. 19823. 19884. 19925. 19951. 19968.

(selected for inverse input)

178 Supercharge, Invasion and Mudcake Growth 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0

19979. 19986. 19991. 19994. 19996. 19998. 19998. 19999. 19999. 20000. 20000. 20000. 20000. 20000. 20000. 20000.

(selected for inverse input)

(selected for inverse input)

Figure 2.66b. Pressure transient response with overbalance. Inverse calculation #1 C:\FT-PTA-SC>SC-DDBU-INVERSE-2

Software reference, SC-DDBU-INVERSE-2.FOR INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec):

2 .5 1 5

Spherical Source Formulations 179 1st Point Time T1 Pressure P1 2nd Point Time T2 Pressure P2 3rd Point Time T3 Pressure P3 Overbalance

(sec): (psi): (sec): (psi): (sec): (psi): (psi):

5 19823 10 19979 19 20000 2000 . . .

Correct overbalance used

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

2.0000 0.5000 1.0000 0.5000 5.0000 5.0000 19823.0000 10.0000 19979.0000 19.0000 20000.0000 2000.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20000.000 9.901 0.0640 0.0099 x (cc/FloLineVol)

The foregoing predictsions are excellent, with a pore pressure of 20,000 psi exactly as assumed, plus a mobility of 9.901 md/cp versus an input value of 10 md/cp, while the compressibility is “0.0099” as opposed to “0.01.” Again, this first inverse calculation assumes an overbalance pressure of 2,000 psi. In the next calculation, we take a zero value, that is, use the older inverse model which does not account for supercharging. For this analysis, the pore pressure is accurate, however, the predicted mobility is 23.403 md/cp versus an assumed 10 md/cp, while the compressibility is “0.0235” as opposed to “0.01.” Inverse calculation #2 C:\FT-PTA-SC>SC-DDBU-INVERSE-2

Software reference, SC-DDBU-INVERSE-2.FOR

180 Supercharge, Invasion and Mudcake Growth INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

2 .5 1 5 5 19823 10 19979 19 20000 0

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

2.0000 0.5000 1.0000 0.5000 5.0000 5.0000 19823.0000 10.0000 19979.0000 19.0000 20000.0000 0.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20000.000 23.403 0.1513 0.0235 x (cc/FloLineVol)

Example DDBU-3. Drawdown-buildup, high overbalance.

In this example, the “buildup curve” actually decreases with time. This part of the complete pressure transient response, more precisely, represents the response when the pump piston has ceased withdrawing fluid from the formation. The decrease in time is a result of high overbalance pressures. C:\FT-PTA-SC>SC-DDBU-FORWARD-4BNOPOR

Software reference, SC-DDBU-FORWARD-4BNOPOR.FOR Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi):

1 1 .00001 20000 2000

Spherical Source Formulations 181 Volume flow rate (cc/s): 2 Pump probe, radius (cm): .5 Probe, geometric factor: 1 Flowline volume (cc): 1000 Time drawdown ends (sec): 5 FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc): Time drawdown ends ............. (sec):

0.1000E+01 0.1000E+01 0.1000E-04 0.2000E+05 0.2000E+04 0.2000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04 0.5000E+01

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0

P(psi) 22000. 21720. 21452. 21196. 20950. 20714. 20684. 20655. 20628. 20601. 20576. 20552. 20529. 20507. 20485. 20465. 20445. 20427. 20409. 20392. 20375. 20360. 20344. 20330. 20316. 20303. 20290. 20278. 20266. 20255. 20244.

(selected for inverse input)

(selected for inverse input)

(selected for inverse input)

182 Supercharge, Invasion and Mudcake Growth 40.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0

20159. 20104. 20068. 20044. 20029. 20019. 20012. 20000. 20000. 20000. 20000. 20000. 20000. 20000. 20000. 20000.

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

4664.1242 2000.0000 6664.1242

Figure 2.66c. Pressure transient response with overbalance. For our first inverse calculation, we assume that we know the overbalance pressure of 2,000 psi. The predictions are excellent, with the pore pressure being 20,002 psi as opposed to 20,000 psi, the mobility being 1.017 md/cp versus 1 md/cp, and the compressibility taking on the value of “0.01” exactly as inputted. In the second inverse calculation, the pore pressure is accurate, however, both the mobility and compressibility take on unacceptable negative values.

Spherical Source Formulations 183 Inverse calculation #1 C:\FT-PTA-SC>SC-DDBU-INVERSE-2

Software reference, SC-DDBU-INVERSE-2.FOR INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

2 .5 1 5 5 20714 10 20576 20 20375 2000

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

2.0000 0.5000 1.0000 0.5000 5.0000 5.0000 20714.0000 10.0000 20576.0000 20.0000 20375.0000 2000.0000

Pore pressure and mobility predicted ... Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

Excellent results!

20002.000 1.017 0.0645 0.0100 x (cc/FloLineVol)

Inverse calculation #2 C:\FT-PTA-SC>SC-DDBU-INVERSE-2

Software reference, SC-DDBU-INVERSE-2.FOR INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi):

2 .5 1 5 5 20714

184 Supercharge, Invasion and Mudcake Growth 2nd Point Time T2 Pressure P2 3rd Point Time T3 Pressure P3 Overbalance

(sec): (psi): (sec): (psi): (psi):

10 20576 20 20375 0

Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

2.0000 0.5000 1.0000 0.5000 5.0000 5.0000 20714.0000 10.0000 20576.0000 20.0000 20375.0000 0.0000

OUTPUT SUMMARY ...

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20002.000 ... Bad negative results! -1.287 -0.0815 -0.0126 x (cc/FloLineVol)

Example DDBU-4. Drawdown-buildup, 1 md/cp calculations.

In this example, we illustrate forward and inverse results. The models derived above are incorporated in software references sc-ddbuforward-4NOPOR.for for forward analysis and sc-ddbu-inverse-2.for for inverse analysis. C:\FT-PTA-SC>sc-ddbu-forward-4NOPOR

Software reference, sc-ddbu-forward-4NOPOR.for. Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Time drawdown ends (sec):

1 1 .00001 10000 250 1 .5 1 1000 20

Spherical Source Formulations 185 FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc): Time drawdown ends ............. (sec):

0.1000E+01 0.1000E+01 0.1000E-04 0.1000E+05 0.2500E+03 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04 0.2000E+02

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0

P(psi) 10250. 10142. 10038. 9938. 9843. 9752. 9664. 9580. 9500. 9423. 9350. 9279. 9211. 9147. 9085. 9025. 8968. 8914. 8861. 8811. 8763. 8815. 8865. 8912. 8958. 9002. 9044. 9084. 9122. 9159. 9194. 9228. 9261. 9292. 9321. 9350. 9377. 9403.

(selected for inverse input)

(selected for inverse input)

186 Supercharge, Invasion and Mudcake Growth 38.0 39.0 40.0 41.0 42.0 43.0 44.0 45.0 46.0 47.0 48.0 49.0 50.0 100.0 200.0 300.0 400.0 500.0 700.0 800.0 900.0 1000.0

9428. 9452. 9475. 9497. 9518. 9539. 9558. 9577. 9594. 9611. 9628. 9643. 9658. 9960. 9999. 10000. 10000. 10000. 10000. 10000. 10000. 10000.

(selected for inverse input)

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

2332.0621 250.0000 2582.0621

In the double-column listing above, we randomly select pressures at times t = 21, 30 and 40 sec for inverse analysis later (the “21” occurs immediately after tddend = 20, the time when piston withdrawal ends – our models, again, apply to the buildup cycle of the drawdown-buildup curve. The selected points are highlighted in red, while the dynamically significant portion of the entire pressure response is plotted below.

Figure 2.66d. Pressure transient response with overbalance.

Spherical Source Formulations 187

We now attempt to recover the inputs used in creating the above transient pressure response, however, using only three data values (t1, Pw, #1), (t2, Pw, #2) and (t3, Pw, #3), plus nozzle, pump and overbalance inputs. C:\FT-PTA-SC>sc-ddbu-inverse-2

Software reference, sc-ddbu-inverse-2.for. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1 .5 1 20 21 8815 30 9194 40 9475 250

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1.0000 0.5000 1.0000 0.5000 20.0000 21.0000 8815.0000 30.0000 9194.0000 40.0000 9475.0000 250.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

10000.000 1.011 0.0645 0.0100 x (cc/FloLineVol)

The pore pressure is correctly predicted as 10,000 psi, while the mobility is 1.011 md/cp, for a 1% error. Note that the inverse code does not request flowline volume as an input – it predicts the product VC. However, we had assumed (in the forward analysis) that it is 1,000 cc. The “cc/FloLineVol” term is therefore 1/1,000. If we multiply 0.0100 by 1/1,000 we obtain 0.00001/psi, the correct assumed compressibility. Our methodology applies to all manufacturers’ formation testers. If the flowline volume is available from the vendor, fluid compressibility is available from our model.

188 Supercharge, Invasion and Mudcake Growth Example DDBU-5. Drawdown-buildup, 0.1 md/cp calculations.

In the prior example, the forward analysis assumed a permeability of 1 md for a mobility of 1 md/cp and a commonly used overbalance pressure of 250 psi. In the illustration below, we consider a tighter formation with a mobility of 0.1 md/cp and a much higher (but less frequently encountered) overbalance of 1,000 psi. C:\FT-PTA-SC>sc-ddbu-forward-4NOPOR

Software reference, sc-ddbu-forward-4NOPOR.for. Fluid, formation, tool and pumping parameters ... Rock permeability (md): Liquid viscosity (cp): Compressibility (1/psi): Pore pressure (psi): Overbalance pressure (psi): Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Flowline volume (cc): Time drawdown ends (sec):

0.1 1 .000003 20000 1000 1 .5 1 1000 20

FORMATION TESTER, FORWARD PRESSURE TRANSIENT MODEL Fluid, formation, tool and pumping parameters ... Formation permeability .......... (md): Viscosity ....................... (cp): Liquid compressibility ....... (1/psi): Pore pressure .................. (psi): Overbalance pressure ........... (psi): Volume flow rate .............. (cc/s): Probe radius .................... (cm): Geometric factor ..... (dimensionless): Effective radius ................ (cm): Flowline volume ................. (cc): Time drawdown ends ............. (sec):

0.1000E+00 0.1000E+01 0.3000E-05 0.2000E+05 0.1000E+04 0.1000E+01 0.5000E+00 0.1000E+01 0.5000E+00 0.1000E+04 0.2000E+02

Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

P(psi) 21000. 20655. 20315. 19979. 19648. 19323. 19001. 18684.

Spherical Source Formulations 189 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 39.0 40.0 41.0 42.0 43.0 44.0 45.0 46.0 47.0 48.0 49.0 50.0 51.0 52.0 53.0 54.0 55.0 56.0 57.0 58.0 59.0 60.0 70.0 80.0

18372. 18064. 17761. 17462. 17167. 16876. 16589. 16307. 16028. 15754. 15483. 15216. 14953. 15025. 15095. 15165. 15233. 15301. 15368. 15433. 15498. 15562. 15625. 15687. 15748. 15809. 15868. 15927. 15985. 16042. 16098. 16153. 16208. 16262. 16315. 16367. 16419. 16469. 16519. 16569. 16618. 16666. 16713. 16760. 16806. 16851. 16896. 16940. 16983. 17026. 17068. 17110. 17151. 17530. 17859.

(selected for inverse input)

(selectef for inverse input)

(selected for inverse input)

(selected for inverse input)

190 Supercharge, Invasion and Mudcake Growth 90.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0

18144. 18391. 19615. 19908. 19978. 19995. 19999. 20000. 20000. 20000. 20000.

Q1*VISC/(4.*PI*RWELL*K), psi ........... Overbalance pressure, psi .............. Q1*VISC/(4.*PI*RWELL*K) + POVER, psi ...

23320.6202 1000.0000 24320.6202

The dynamically significant part of the transient pressure response is shown in Figure 2.6e.

Figure 2.66e. Pressure transient response with overbalance. Two inverse calculations are reported here. In the first illustration, we use pressures from times 21, 30 and 40 sec, noting that piston motions ceased at 20 sec. The predicted mobility and compressibility are excellent and agree with forward analysis inputs. However, the predicted pore pressure exceeds the input value by 26 psi.

Spherical Source Formulations 191

In the second attempt, we use pressure values from 21, 40 and 60 sec, covering a wider dynamic range for increased accuracy. Note that these values are obtained from the first 60 sec of tool operation, which is significant for the tight, overbalanced formation under consideration. Mobility and compressibility are still excellent – the pore pressure, now at 19,997 psi, is now extremely close to the input value of 20,000 psi. Inverse calculation #1 INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1 .5 1 20 21 15025 30 15625 40 16208 1000

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1.0000 0.5000 1.0000 0.5000 20.0000 21.0000 15025.0000 30.0000 15625.0000 40.0000 16208.0000 1000.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20026.000 0.100 0.0193 0.0030 x (cc/FloLineVol)

192 Supercharge, Invasion and Mudcake Growth Inverse calculation #2 C:\FT-PTA-SC>sc-ddbu-inverse-2

Software reference, sc-ddbu-inverse-2.for. INPUTS, Inverse Model ... Volume flow rate (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1 .5 1 20 21 15025 40 16208 60 17151 1000

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi): Overbalance (psi):

1.0000 0.5000 1.0000 0.5000 20.0000 21.0000 15025.0000 40.0000 16208.0000 60.0000 17151.0000 1000.0000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

19997.000 0.101 0.0194 0.0030 x (cc/FloLineVol)

Spherical Source Formulations 193

2.3.7 Advanced multiple drawdown – buildup (or, “MDDBU”) forward and inverse models. In an earlier section, we described an inverse method for early time, low mobility and non-negligible flowline volume applications. When three ‘pressure, time’ data points are selected on the buildup portion of the transient pressure curve, our results showed that predicted compressibility, mobility and pore pressure are correct and consistent with those in the exact forward model FT-00 used to generate the synthetic test data. The method, known as “FT-PTA-DDBU,” does not handle supercharge, although a prior supercharge discussion indicated that extensions to this model for buildup data as well as points on the drawdown curve were completed. In the present section, we describe significant extensions to the foregoing zero-supercharge algorithm, but do not restrict ourselves to a single drawdown-buildup cycle. 2.3.7.1 Software description.

Our validation methodology is straightforward. Using exact forward simulator FT-00, assumed with given fluid, formation and tool inputs, plus a range of complicated flow rate schedules as shown in Figures 2.67, 2.68a-2.68j and 2.69a, we construct transient pressure response curves. The schedules may consist of arbitrary piecewise constant rates, e.g., “withdraw, stop,” “withdraw, inject, stop, inject,” and so on. The rate schedule menus are illustrated with varying positive rates, however, any number of positive, negative or zero values are permissible in any order, and then, with arbitrarily chosen time durations (e.g., see the detailed validation for Figure 2.69a). Then, with any three ‘pressure, time’ data points selected from the final pressure segment of the curve, we show that inputted compressibility, mobility and pore pressure can be satisfactorily recovered in the inverse scheme. The use of the final pressure segment is customary in field logging practice. Mathematical details are offered in Chin (2019) and are not duplicated here, and neither will engineering applications be re-introduced here. In the interest of brevity, we will also not reproduce details of various forward-inverse validations. It suffices to replicate the software input screens for each of the supported flow rates, as in Figures 2.68a – 2.68j, as the corresponding capability will be obvious. However, to illustrate our approach, we will duplicate in complete detail all of the validations performed for the final, most complicated “PTA-App-11 Inverse Model” model.

194 Supercharge, Invasion and Mudcake Growth

Figure 2.67. Eleven general drawdown-buildup inverse models.

Spherical Source Formulations 195

Figure 2.68a. Model 1 rate function (black dots denote data points).

Figure 2.68b. Model 2 rate function (black dots denote data points).

196 Supercharge, Invasion and Mudcake Growth

Figure 2.68c. Model 3 rate function.

Figure 2.68d. Model 4 rate function.

Spherical Source Formulations 197

Figure 2.68e. Model 5 rate function.

Figure 2.68f. Model 6 rate function.

198 Supercharge, Invasion and Mudcake Growth

Figure 2.68g. Model 7 rate function.

Figure 2.68h. Model 8 rate function.

Spherical Source Formulations 199

Figure 2.68i. Model 9 rate function.

Figure 2.68j. Model 10 rate function.

200 Supercharge, Invasion and Mudcake Growth

2.3.7.2 Validation of PTA-App-11 inverse model.

Figure 2.69a. Model 11 rate function (three “black circles” show pressure data selected in final rate cycle) Our “inverse model” is an approximate one designed to predict pore pressure, spherical mobility and fluid compressibility in low mobility applications where flowline storage cannot be neglected. The method uses single-probe pressure transient data typically obtained during the first minute of logging – pressures which are highly unsteady and not steady-state. It is validated using pressure transient data created by our exact “forward model” FT-00 (immediately below) where formation and fluid properties, tool constants, reservoir conditions and pumping schedules are defined. Three time-pressure data points are arbitrarily selected (red from the output listing) and inputted into the approximate inverse model. Validations performed in the “black screen, DOS window” show that predicted pore pressure, spherical mobility and fluid compressibility are consistent with those assumed in FT-00. The reader is invited to use different time and pressure inputs to evaluate the method. Different data points will give slightly different predictions. Times should not be too close, but preferably, several seconds apart. The request in the “black screen,” that is, “Use decimals after integers” no longer applies to our latest software.

Spherical Source Formulations 201

Validation No. 1

Figure 2.69b. FT-00 exact inputs.

202 Supercharge, Invasion and Mudcake Growth

Figure 2.69c. Source probe pressure and pumpout schedule (all rates > 0). Flow rate diagram shows accelerating withdrawal of piston. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.140E+01

0.10000E+01

0.24622E+05

0.24970E+05 -0.37768E+03 -0.29935E+02

0.79261E-01

0.280E+01

0.10000E+01

0.24372E+05

0.24954E+05 -0.62804E+03 -0.46378E+02

0.73847E-01

0.420E+01

0.10000E+01

0.24204E+05

0.24946E+05 -0.79564E+03 -0.54407E+02

0.68382E-01

0.560E+01

0.10000E+01

0.24092E+05

0.24941E+05 -0.90845E+03 -0.58568E+02

0.64470E-01

0.700E+01

0.10000E+01

0.24015E+05

0.24939E+05 -0.98474E+03 -0.60719E+02

0.61660E-01

0.840E+01

0.10000E+01

0.23963E+05

0.24938E+05 -0.10366E+04 -0.61775E+02

0.59595E-01

0.980E+01

0.10000E+01

0.23928E+05

0.24938E+05 -0.10720E+04 -0.62225E+02

0.58048E-01

0.112E+02

0.20000E+01

0.23571E+05

0.24911E+05 -0.14291E+04 -0.88632E+02

0.62019E-01

0.126E+02

0.20000E+01

0.23289E+05

0.24893E+05 -0.17112E+04 -0.10700E+03

0.62530E-01

0.140E+02

0.20000E+01

0.23100E+05

0.24884E+05 -0.19003E+04 -0.11570E+03

0.60883E-01

0.154E+02

0.20000E+01

0.22972E+05

0.24880E+05 -0.20278E+04 -0.12007E+03

0.59213E-01

0.168E+02

0.20000E+01

0.22886E+05

0.24878E+05 -0.21143E+04 -0.12224E+03

0.57818E-01

Spherical Source Formulations 203 0.182E+02

0.20000E+01

0.22827E+05

0.24877E+05 -0.21732E+04 -0.12334E+03

0.56754E-01

0.196E+02

0.20000E+01

0.22786E+05

0.24876E+05 -0.22136E+04 -0.12437E+03

0.56185E-01

0.210E+02

0.30000E+01

0.22473E+05

0.24853E+05 -0.25268E+04 -0.14681E+03

0.58102E-01

0.224E+02

0.30000E+01

0.22173E+05

0.24833E+05 -0.28273E+04 -0.16747E+03

0.59233E-01

0.238E+02

0.30000E+01

0.21971E+05

0.24823E+05 -0.30288E+04 -0.17695E+03

0.58423E-01

0.252E+02

0.30000E+01

0.21835E+05

0.24818E+05 -0.31647E+04 -0.18159E+03

0.57381E-01

0.266E+02

0.30000E+01

0.21743E+05

0.24816E+05 -0.32568E+04 -0.18380E+03

0.56436E-01

0.280E+02

0.30000E+01

0.21680E+05

0.24815E+05 -0.33197E+04 -0.18472E+03

0.55642E-01

0.294E+02

0.30000E+01

0.21637E+05

0.24814E+05 -0.33629E+04 -0.18642E+03

0.55434E-01

0.308E+02

0.40000E+01

0.21372E+05

0.24796E+05 -0.36277E+04 -0.20438E+03

0.56340E-01

0.322E+02

0.40000E+01

0.21054E+05

0.24772E+05 -0.39465E+04 -0.22777E+03

0.57714E-01

0.336E+02

0.40000E+01

0.20840E+05

0.24762E+05 -0.41600E+04 -0.23816E+03

0.57249E-01

0.350E+02

0.40000E+01

0.20696E+05

0.24757E+05 -0.43040E+04 -0.24312E+03

0.56486E-01

0.364E+02

0.40000E+01

0.20598E+05

0.24755E+05 -0.44017E+04 -0.24539E+03

0.55749E-01

0.378E+02

0.40000E+01

0.20532E+05

0.24754E+05 -0.44684E+04 -0.24625E+03

0.55109E-01

0.392E+02

0.40000E+01

0.20486E+05

0.24752E+05 -0.45141E+04 -0.24834E+03

0.55014E-01

0.406E+02

0.50000E+01

0.20273E+05

0.24739E+05 -0.47273E+04 -0.26127E+03

0.55268E-01

0.420E+02

0.50000E+01

0.19935E+05

0.24712E+05 -0.50653E+04 -0.28785E+03

0.56827E-01

0.434E+02

0.50000E+01

0.19708E+05

0.24701E+05 -0.52915E+04 -0.29931E+03

0.56563E-01

0.448E+02

0.50000E+01

0.19556E+05

0.24695E+05 -0.54440E+04 -0.30465E+03

0.55960E-01

0.462E+02

0.50000E+01

0.19453E+05

0.24693E+05 -0.55474E+04 -0.30701E+03

0.55343E-01

0.476E+02

0.50000E+01

0.19382E+05

0.24692E+05 -0.56179E+04 -0.30782E+03

0.54793E-01

0.490E+02

0.50000E+01

0.19334E+05

0.24690E+05 -0.56663E+04 -0.31011E+03

0.54729E-01

0.504E+02

0.60000E+01

0.19176E+05

0.24682E+05 -0.58245E+04 -0.31761E+03

0.54530E-01

0.518E+02

0.60000E+01

0.18817E+05

0.24652E+05 -0.61830E+04 -0.34767E+03

0.56231E-01

0.532E+02

0.60000E+01

0.18577E+05

0.24640E+05 -0.64226E+04 -0.36038E+03

0.56112E-01

0.546E+02

0.60000E+01

0.18416E+05

0.24634E+05 -0.65840E+04 -0.36617E+03

0.55616E-01

0.560E+02

0.60000E+01

0.18307E+05

0.24631E+05 -0.66933E+04 -0.36865E+03

0.55078E-01

0.574E+02

0.60000E+01

0.18232E+05

0.24631E+05 -0.67679E+04 -0.36943E+03

0.54586E-01

0.588E+02

0.60000E+01

0.18181E+05

0.24628E+05 -0.68191E+04 -0.37170E+03

0.54509E-01

0.602E+02

0.00000E+00

0.18531E+05

0.24632E+05 -0.64686E+04 -0.36774E+03

0.56850E-01

0.616E+02

0.00000E+00

0.20641E+05

0.24825E+05 -0.43595E+04 -0.17512E+03

0.40170E-01

0.630E+02

0.00000E+00

0.22041E+05

0.24914E+05 -0.29590E+04 -0.86190E+02

0.29128E-01

0.644E+02

0.00000E+00

0.22979E+05

0.24959E+05 -0.20214E+04 -0.41430E+02

0.20496E-01

0.658E+02

0.00000E+00

0.23610E+05

0.24982E+05 -0.13905E+04 -0.17528E+02

0.12606E-01

0.672E+02

0.00000E+00

0.24036E+05

0.24995E+05 -0.96406E+03 -0.46272E+01

0.47998E-02

0.686E+02

0.00000E+00

0.24325E+05

0.25000E+05 -0.67466E+03

0.00000E+00

0.00000E+00

204 Supercharge, Invasion and Mudcake Growth

Figure 2.69d. Inverse model screen. Pore pressure here is predicted as 25,006 psi versus a known value of 25,000 psi. The mobility is 1.023 md/cp versus 1.000 md/cp and the fluid compressibility is obtained as 0.0000031/psi. Good results.

Spherical Source Formulations 205

Validation No. 2

Figure 2.69e. FT-00 exact inputs.

206 Supercharge, Invasion and Mudcake Growth

Figure 2.69f. Source probe pressure and pumpout schedule (mixed signs). DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.800E+00

0.10000E+01

0.24765E+05

0.24983E+05 -0.23494E+03 -0.17383E+02

0.73986E-01

0.160E+01

0.10000E+01

0.24580E+05

0.24967E+05 -0.41994E+03 -0.33153E+02

0.78947E-01

0.240E+01

0.10000E+01

0.24434E+05

0.24957E+05 -0.56650E+03 -0.42874E+02

0.75683E-01

0.320E+01

0.10000E+01

0.24317E+05

0.24951E+05 -0.68291E+03 -0.49246E+02

0.72112E-01

0.400E+01

0.10000E+01

0.24224E+05

0.24946E+05 -0.77557E+03 -0.53560E+02

0.69058E-01

0.480E+01

0.10000E+01

0.24151E+05

0.24943E+05 -0.84946E+03 -0.56524E+02

0.66541E-01

0.560E+01 -0.10000E+01

0.24454E+05

0.24965E+05 -0.54554E+03 -0.34611E+02

0.63443E-01

0.640E+01 -0.10000E+01

0.24800E+05

0.25000E+05 -0.20028E+03 -0.10084E+00

0.50352E-03

0.720E+01 -0.10000E+01

0.25073E+05

0.25021E+05

0.72520E+02

0.20736E+02

0.28594E+00

0.800E+01 -0.10000E+01

0.25289E+05

0.25034E+05

0.28878E+03

0.34214E+02

0.11848E+00

0.880E+01 -0.10000E+01

0.25461E+05

0.25043E+05

0.46060E+03

0.43308E+02

0.94026E-01

0.960E+01 -0.10000E+01

0.25597E+05

0.25050E+05

0.59735E+03

0.49579E+02

0.82998E-01

0.104E+02

0.10000E+01

0.25457E+05

0.25042E+05

0.45703E+03

0.41784E+02

0.91425E-01

0.112E+02

0.10000E+01

0.25128E+05

0.25004E+05

0.12762E+03

0.44179E+01

0.34617E-01

Spherical Source Formulations 207 0.120E+02

0.10000E+01

0.24868E+05

0.24982E+05 -0.13201E+03 -0.17991E+02

0.13629E+00

0.128E+02

0.10000E+01

0.24662E+05

0.24968E+05 -0.33750E+03 -0.32187E+02

0.95367E-01

0.136E+02

0.10000E+01

0.24499E+05

0.24958E+05 -0.50056E+03 -0.41661E+02

0.83229E-01

0.144E+02

0.10000E+01

0.24370E+05

0.24952E+05 -0.63019E+03 -0.48218E+02

0.76514E-01

0.152E+02 -0.10000E+01

0.24395E+05

0.24949E+05 -0.60481E+03 -0.51008E+02

0.84337E-01

0.160E+02 -0.10000E+01

0.24755E+05

0.24988E+05 -0.24503E+03 -0.12066E+02

0.49242E-01

0.168E+02 -0.10000E+01

0.25038E+05

0.25013E+05

0.38188E+02

0.13361E+02

0.34987E+00

0.176E+02 -0.10000E+01

0.25262E+05

0.25029E+05

0.26228E+03

0.29175E+02

0.11123E+00

0.184E+02 -0.10000E+01

0.25440E+05

0.25040E+05

0.44008E+03

0.39622E+02

0.90035E-01

0.192E+02 -0.10000E+01

0.25581E+05

0.25047E+05

0.58142E+03

0.46753E+02

0.80412E-01

0.200E+02

0.10000E+01

0.25694E+05

0.25052E+05

0.69398E+03

0.51958E+02

0.74870E-01

0.208E+02

0.10000E+01

0.25314E+05

0.25021E+05

0.31386E+03

0.20842E+02

0.66406E-01

0.216E+02

0.10000E+01

0.25016E+05

0.24992E+05

0.15550E+02 -0.81551E+01 -0.52444E+00

0.224E+02

0.10000E+01

0.24780E+05

0.24974E+05 -0.22022E+03 -0.25837E+02

0.11732E+00

0.232E+02

0.10000E+01

0.24593E+05

0.24963E+05 -0.40713E+03 -0.37380E+02

0.91815E-01

0.240E+02

0.10000E+01

0.24444E+05

0.24955E+05 -0.55562E+03 -0.45210E+02

0.81368E-01

0.248E+02

0.10000E+01

0.24326E+05

0.24949E+05 -0.67379E+03 -0.50822E+02

0.75427E-01 0.76177E-01

0.256E+02 -0.10000E+01

0.24595E+05

0.24969E+05 -0.40507E+03 -0.30857E+02

0.264E+02 -0.10000E+01

0.24912E+05

0.25002E+05 -0.87805E+02

0.22711E+01 -0.25865E-01

0.272E+02 -0.10000E+01

0.25163E+05

0.25022E+05

0.16270E+03

0.22127E+02

0.13600E+00

0.280E+02 -0.10000E+01

0.25361E+05

0.25035E+05

0.36118E+03

0.34911E+02

0.96657E-01

0.288E+02 -0.10000E+01

0.25519E+05

0.25044E+05

0.51881E+03

0.43518E+02

0.83879E-01

0.296E+02 -0.10000E+01

0.25644E+05

0.25050E+05

0.64422E+03

0.49579E+02

0.76959E-01

0.304E+02

0.00000E+00

0.25619E+05

0.25048E+05

0.61949E+03

0.47865E+02

0.77265E-01

0.312E+02

0.00000E+00

0.25491E+05

0.25031E+05

0.49104E+03

0.30706E+02

0.62534E-01

0.320E+02

0.00000E+00

0.25390E+05

0.25021E+05

0.39018E+03

0.20561E+02

0.52698E-01

0.328E+02

0.00000E+00

0.25311E+05

0.25014E+05

0.31059E+03

0.14192E+02

0.45693E-01

0.336E+02

0.00000E+00

0.25248E+05

0.25010E+05

0.24762E+03

0.99459E+01

0.40166E-01

0.344E+02

0.00000E+00

0.25198E+05

0.25007E+05

0.19769E+03

0.69607E+01

0.35209E-01

0.352E+02

0.00000E+00

0.25158E+05

0.25005E+05

0.15805E+03

0.46440E+01

0.29383E-01

0.360E+02

0.00000E+00

0.25127E+05

0.25003E+05

0.12652E+03

0.30039E+01

0.23742E-01

0.368E+02

0.00000E+00

0.25101E+05

0.25002E+05

0.10143E+03

0.18393E+01

0.18135E-01

0.376E+02

0.00000E+00

0.25081E+05

0.25001E+05

0.81423E+02

0.10135E+01

0.12447E-01

0.384E+02

0.00000E+00

0.25065E+05

0.25000E+05

0.65462E+02

0.43109E+00

0.65853E-02

0.392E+02

0.00000E+00

0.25053E+05

0.25000E+05

0.52713E+02

0.24554E-01

0.46582E-03

208 Supercharge, Invasion and Mudcake Growth

Figure 2.69g. Inverse model screen. The pore pressure is predicted exactly as 25,000 psi, and the mobility is 1.056 md/cp versus an input value of 1.000 md/cp. Very good. In the next validation, we will reduce the input mobility by a factor of 10.

Spherical Source Formulations 209

Validation No. 3

Figure 2.69h. FT-00 exact inputs.

210 Supercharge, Invasion and Mudcake Growth

Figure 2.69i. Source probe pressure and pumpout schedule (mixed signs). DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.800E+00

0.10000E+01

0.24738E+05

0.25000E+05 -0.26235E+03 -0.73748E-01

0.28110E-03

0.160E+01

0.10000E+01

0.24482E+05

0.24996E+05 -0.51775E+03 -0.42500E+01

0.82087E-02

0.240E+01

0.10000E+01

0.24233E+05

0.24982E+05 -0.76674E+03 -0.17797E+02

0.23211E-01

0.320E+01

0.10000E+01

0.23990E+05

0.24962E+05 -0.10096E+04 -0.37835E+02

0.37474E-01

0.400E+01

0.10000E+01

0.23753E+05

0.24939E+05 -0.12466E+04 -0.60813E+02

0.48782E-01

0.480E+01

0.10000E+01

0.23522E+05

0.24915E+05 -0.14779E+04 -0.84639E+02

0.57270E-01

0.560E+01 -0.10000E+01

0.23691E+05

0.24892E+05 -0.13088E+04 -0.10823E+03

0.82697E-01

0.640E+01 -0.10000E+01

0.23985E+05

0.24874E+05 -0.10151E+04 -0.12642E+03

0.12454E+00

0.720E+01 -0.10000E+01

0.24271E+05

0.24874E+05 -0.72914E+03 -0.12572E+03

0.17243E+00

0.800E+01 -0.10000E+01

0.24549E+05

0.24891E+05 -0.45051E+03 -0.10895E+03

0.24184E+00

0.880E+01 -0.10000E+01

0.24821E+05

0.24916E+05 -0.17883E+03 -0.83739E+02

0.46825E+00

0.960E+01 -0.10000E+01

0.25086E+05

0.24945E+05

0.86167E+02 -0.54988E+02 -0.63816E+00

0.104E+02

0.10000E+01

0.25080E+05

0.24975E+05

0.80491E+02 -0.25297E+02 -0.31429E+00

0.112E+02

0.10000E+01

0.24815E+05

0.25002E+05 -0.18472E+03

0.18834E+01 -0.10196E-01

0.120E+02

0.10000E+01

0.24557E+05

0.25012E+05 -0.44276E+03

0.12464E+02 -0.28150E-01

Spherical Source Formulations 211 0.128E+02

0.10000E+01

0.24306E+05

0.25005E+05 -0.69418E+03

0.136E+02

0.10000E+01

0.24061E+05

0.24987E+05 -0.93932E+03 -0.13155E+02

0.48804E+01 -0.70305E-02 0.14005E-01

0.144E+02

0.10000E+01

0.23822E+05

0.24964E+05 -0.11784E+04 -0.36026E+02

0.30571E-01

0.152E+02 -0.10000E+01

0.23721E+05

0.24939E+05 -0.12791E+04 -0.60666E+02

0.47429E-01

0.160E+02 -0.10000E+01

0.24014E+05

0.24915E+05 -0.98576E+03 -0.84778E+02

0.86003E-01

0.168E+02 -0.10000E+01

0.24300E+05

0.24904E+05 -0.70047E+03 -0.96174E+02

0.13730E+00

0.176E+02 -0.10000E+01

0.24577E+05

0.24912E+05 -0.42253E+03 -0.88359E+02

0.20912E+00

0.184E+02 -0.10000E+01

0.24848E+05

0.24932E+05 -0.15156E+03 -0.68480E+02

0.45183E+00

0.192E+02 -0.10000E+01

0.25113E+05

0.24957E+05

0.11273E+03 -0.42881E+02 -0.38039E+00

0.200E+02

0.10000E+01

0.25371E+05

0.24985E+05

0.37058E+03 -0.15145E+02 -0.40868E-01

0.208E+02

0.10000E+01

0.25098E+05

0.25013E+05

0.97500E+02

0.216E+02

0.10000E+01

0.24832E+05

0.25032E+05 -0.16770E+03

0.31797E+02 -0.18960E+00

0.224E+02

0.10000E+01

0.24574E+05

0.25031E+05 -0.42595E+03

0.31063E+02 -0.72926E-01

0.232E+02

0.10000E+01

0.24322E+05

0.25016E+05 -0.67766E+03

0.16114E+02 -0.23779E-01

0.240E+02

0.10000E+01

0.24077E+05

0.24994E+05 -0.92313E+03 -0.59986E+01

0.64981E-02

0.248E+02

0.10000E+01

0.23837E+05

0.24969E+05 -0.11626E+04 -0.31075E+02

0.26729E-01

0.12739E+02

0.13065E+00

0.256E+02 -0.10000E+01

0.23999E+05

0.24943E+05 -0.10013E+04 -0.56909E+02

0.56833E-01

0.264E+02 -0.10000E+01

0.24285E+05

0.24922E+05 -0.71526E+03 -0.77744E+02

0.10869E+00

0.272E+02 -0.10000E+01

0.24563E+05

0.24920E+05 -0.43673E+03 -0.79808E+02

0.18274E+00

0.280E+02 -0.10000E+01

0.24835E+05

0.24934E+05 -0.16527E+03 -0.65745E+02

0.39781E+00

0.288E+02 -0.10000E+01

0.25099E+05

0.24957E+05

0.99457E+02 -0.43118E+02 -0.43354E+00

0.296E+02 -0.10000E+01

0.25358E+05

0.24983E+05

0.35770E+03 -0.16798E+02 -0.46960E-01

0.304E+02

0.00000E+00

0.25478E+05

0.25011E+05

0.47758E+03

0.10628E+02

0.22254E-01

0.312E+02

0.00000E+00

0.25465E+05

0.25037E+05

0.46475E+03

0.36780E+02

0.79140E-01

0.320E+02

0.00000E+00

0.25453E+05

0.25054E+05

0.45265E+03

0.54290E+02

0.11994E+00

0.328E+02

0.00000E+00

0.25441E+05

0.25062E+05

0.44109E+03

0.62226E+02

0.14107E+00

0.336E+02

0.00000E+00

0.25430E+05

0.25064E+05

0.42998E+03

0.64385E+02

0.14974E+00

0.344E+02

0.00000E+00

0.25419E+05

0.25064E+05

0.41927E+03

0.63563E+02

0.15160E+00

0.352E+02

0.00000E+00

0.25409E+05

0.25061E+05

0.40890E+03

0.61304E+02

0.14992E+00

0.360E+02

0.00000E+00

0.25399E+05

0.25058E+05

0.39887E+03

0.58416E+02

0.14645E+00

0.368E+02

0.00000E+00

0.25389E+05

0.25055E+05

0.38914E+03

0.55319E+02

0.14216E+00

0.376E+02

0.00000E+00

0.25380E+05

0.25052E+05

0.37969E+03

0.52226E+02

0.13755E+00

0.384E+02

0.00000E+00

0.25371E+05

0.25049E+05

0.37052E+03

0.49244E+02

0.13291E+00

0.392E+02

0.00000E+00

0.25362E+05

0.25046E+05

0.36160E+03

0.46423E+02

0.12838E+00

212 Supercharge, Invasion and Mudcake Growth

Figure 2.69j. Inverse model screen. The pore pressure is 24,983 psi versus a value of 25,000 psi assumed. The mobility is 0.111 md/cp versus 0.100 md/cp inputted. Very good agreement is obtained. Next, we reduce inputted mobility 10 times.

Spherical Source Formulations 213

Validition No. 4

Figure 2.69k. FT-00 exact inputs.

214 Supercharge, Invasion and Mudcake Growth

Figure 2.69l. Source probe pressure and pumpout schedule (mixed signs).

DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s)

Rate (cc/s)

Ps* (psi)

Pr* (psi)

Ps**(psi)

Pr**(psi)

Pr**/Ps**

0.000E+00

0.10000E+01

0.25000E+05

0.25000E+05

0.00000E+00

0.00000E+00

-----------

0.800E+00

0.10000E+01

0.24734E+05

0.25000E+05 -0.26595E+03 -0.19886E-29

0.74773E-32

0.160E+01

0.10000E+01

0.24469E+05

0.25000E+05 -0.53094E+03 -0.39905E-13

0.75159E-16

0.240E+01

0.10000E+01

0.24205E+05

0.25000E+05 -0.79511E+03 -0.11820E-07

0.14866E-10

0.320E+01

0.10000E+01

0.23941E+05

0.25000E+05 -0.10585E+04 -0.67074E-05

0.63367E-08

0.400E+01

0.10000E+01

0.23679E+05

0.25000E+05 -0.13211E+04 -0.30878E-03

0.23372E-06

0.480E+01

0.10000E+01

0.23417E+05

0.25000E+05 -0.15830E+04 -0.40315E-02

0.25467E-05

0.560E+01 -0.10000E+01

0.23555E+05

0.25000E+05 -0.14451E+04 -0.25557E-01

0.17685E-04

0.640E+01 -0.10000E+01

0.23825E+05

0.25000E+05 -0.11751E+04 -0.10299E+00

0.87642E-04

0.720E+01 -0.10000E+01

0.24094E+05

0.25000E+05 -0.90616E+03 -0.30654E+00

0.33828E-03

0.800E+01 -0.10000E+01

0.24362E+05

0.24999E+05 -0.63807E+03 -0.73748E+00

0.11558E-02

0.880E+01 -0.10000E+01

0.24629E+05

0.24998E+05 -0.37082E+03 -0.15188E+01

0.40958E-02

0.960E+01 -0.10000E+01

0.24896E+05

0.24997E+05 -0.10437E+03 -0.27794E+01

0.26631E-01

Spherical Source Formulations 215 0.104E+02

0.10000E+01

0.24895E+05

0.24995E+05 -0.10491E+03 -0.46284E+01

0.44117E-01

0.112E+02

0.10000E+01

0.24629E+05

0.24993E+05 -0.37088E+03 -0.71220E+01

0.19203E-01

0.120E+02

0.10000E+01

0.24364E+05

0.24990E+05 -0.63586E+03 -0.10237E+02

0.16099E-01

0.128E+02

0.10000E+01

0.24100E+05

0.24986E+05 -0.89996E+03 -0.13865E+02

0.15406E-01

0.136E+02

0.10000E+01

0.23837E+05

0.24982E+05 -0.11633E+04 -0.17828E+02

0.15326E-01

0.144E+02

0.10000E+01

0.23574E+05

0.24978E+05 -0.14258E+04 -0.21903E+02

0.15362E-01

0.152E+02 -0.10000E+01

0.23446E+05

0.24974E+05 -0.15543E+04 -0.25862E+02

0.16639E-01

0.160E+02 -0.10000E+01

0.23716E+05

0.24970E+05 -0.12840E+04 -0.29517E+02

0.22988E-01

0.168E+02 -0.10000E+01

0.23985E+05

0.24967E+05 -0.10147E+04 -0.32756E+02

0.32280E-01

0.176E+02 -0.10000E+01

0.24254E+05

0.24964E+05 -0.74639E+03 -0.35562E+02

0.47645E-01

0.184E+02 -0.10000E+01

0.24521E+05

0.24962E+05 -0.47888E+03 -0.38007E+02

0.79366E-01

0.192E+02 -0.10000E+01

0.24788E+05

0.24960E+05 -0.21219E+03 -0.40228E+02

0.18959E+00

0.200E+02

0.10000E+01

0.25054E+05

0.24958E+05

0.208E+02

0.10000E+01

0.24787E+05

0.24955E+05 -0.21301E+03 -0.44651E+02

0.20962E+00

0.216E+02

0.10000E+01

0.24521E+05

0.24953E+05 -0.47859E+03 -0.47099E+02

0.98411E-01

0.224E+02

0.10000E+01

0.24257E+05

0.24950E+05 -0.74325E+03 -0.49748E+02

0.66933E-01

0.232E+02

0.10000E+01

0.23993E+05

0.24947E+05 -0.10071E+04 -0.52526E+02

0.52158E-01

0.240E+02

0.10000E+01

0.23730E+05

0.24945E+05 -0.12701E+04 -0.55298E+02

0.43539E-01

0.248E+02

0.10000E+01

0.23468E+05

0.24942E+05 -0.15323E+04 -0.57892E+02

0.37780E-01

0.53727E+02 -0.42392E+02 -0.78902E+00

0.256E+02 -0.10000E+01

0.23605E+05

0.24940E+05 -0.13947E+04 -0.60146E+02

0.43125E-01

0.264E+02 -0.10000E+01

0.23875E+05

0.24938E+05 -0.11250E+04 -0.61949E+02

0.55064E-01

0.272E+02 -0.10000E+01

0.24144E+05

0.24937E+05 -0.85636E+03 -0.63273E+02

0.73886E-01

0.280E+02 -0.10000E+01

0.24411E+05

0.24936E+05 -0.58854E+03 -0.64175E+02

0.10904E+00

0.288E+02 -0.10000E+01

0.24678E+05

0.24935E+05 -0.32154E+03 -0.64787E+02

0.20149E+00

0.296E+02 -0.10000E+01

0.24945E+05

0.24935E+05 -0.55337E+02 -0.65283E+02

0.11797E+01

0.304E+02

0.00000E+00

0.25077E+05

0.24934E+05

0.76995E+02 -0.65842E+02 -0.85515E+00

0.312E+02

0.00000E+00

0.25076E+05

0.24933E+05

0.76244E+02 -0.66604E+02 -0.87356E+00

0.320E+02

0.00000E+00

0.25076E+05

0.24932E+05

0.75624E+02 -0.67629E+02 -0.89428E+00

0.328E+02

0.00000E+00

0.25075E+05

0.24931E+05

0.75079E+02 -0.68891E+02 -0.91758E+00

0.336E+02

0.00000E+00

0.25075E+05

0.24930E+05

0.74595E+02 -0.70286E+02 -0.94224E+00

0.344E+02

0.00000E+00

0.25074E+05

0.24928E+05

0.74140E+02 -0.71658E+02 -0.96652E+00

0.352E+02

0.00000E+00

0.25074E+05

0.24927E+05

0.73725E+02 -0.72825E+02 -0.98780E+00

0.360E+02

0.00000E+00

0.25073E+05

0.24926E+05

0.73326E+02 -0.73623E+02 -0.10040E+01

0.368E+02

0.00000E+00

0.25073E+05

0.24926E+05

0.72960E+02 -0.73922E+02 -0.10132E+01

0.376E+02

0.00000E+00

0.25073E+05

0.24926E+05

0.72599E+02 -0.73652E+02 -0.10145E+01

0.384E+02

0.00000E+00

0.25072E+05

0.24927E+05

0.72261E+02 -0.72794E+02 -0.10074E+01

0.392E+02

0.00000E+00

0.25072E+05

0.24929E+05

0.71929E+02 -0.71375E+02 -0.99229E+00

216 Supercharge, Invasion and Mudcake Growth

Figure 2.69m. Inverse model screen. The predicted pore pressure is 24,968 psi versus 25,000 psi which is excellent. Here the mobility is 0.017 md/cp versus 0.010 md/cp, in an accepted accuracy range given this extremely low value. This example concludes our discussion of multiple drawdown-buildup inverse methods for early time, mobility, non-negligible flowline volume analysis.

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2.3.8 Multiphase flow with inertial effects – Applications to invasion, supercharging, clean-up and contamination analysis.

The study of mudcake formation, dynamics and growth within the borehole environment has long intrigued petroleum industry workers. Researchers early on focused on invasion consequences. For example, fluid influx was known to affect resistivity log interpretation; its effects were modeled using distinct cylindrical layered models as well as by resistivity distributions with continuous variations. Thick cakes, for instance, were a particular concern in drilling, since they were associated with stuck pipe and rigsite safety. Much of the literature, subject to speculation at first, was not reliable. In the 1980s, Chin and his NL Industries colleagues investigated the factors underlying dynamic mudcake growth using Catscan visualization in which cake formation and invasion were photographed in time and studied quantitatively. 2.3.8.1 Mudcake dynamics.

This work appeared in “Formation Evaluation Using Repeated MWD Logging Measurements,” by Chin, Suresh, Holbrook, Affleck and Robertson, presented at the SPWLA 27th Annual Logging Symposium, Houston, TX, June 9-13, 1986, and math models were developed years later in Formation Invasion, with Applications to Measurement-WhileDrilling, Time Lapse Analysis and Formation Damage (Chin, 1995). These works drew upon experimental results from linear and radial flow test vessels placed within Catscan imaging machines. Under constant pressure drops, invasion fronts and mudcake thickness were monitored in time, and analytical models were constructed using “moving boundary” methods from mathematical physics to describe dynamic boundary motion (laboratory test fixtures are shown in Figures 2.70a – 2.70f). Formation invasion and mudcake dynamics are also important in formation tester “sampling” and “pressure transient analysis (PTA).” A major tester objective is in-situ fluid collection – here, it is important to obtain clean reservoir fluids that are not contaminated by invaded drilling mud – and so, “job planning software” is required that estimates the pumping time required in order to extract an acceptably pure sample. In PTA, we have explained how the “mobility,” defined as the ratio of permeability to viscosity is actually predicted – an estimate value for effective multiphase viscosity due to multiphase mixing between reservoir and wellbore fluids is required to produce k. In what follows, we will describe our modeling efforts and display simulation results.

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Figure 2.70a. Catscan, linear test vessel with core sample (flow, top to bottom).

Figure 2.70b. Radial flow Catscan test vessel.

Figure 2.70c. Catscan, invasion in radial core sample (inner invaded white zone displacing outer dark fluid).

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Figure 2.70d. Linear flow Catscans, thin dark mudcake at center of core and invasion front at density contrast (flow, left to right).

Figure 2.70e. Linear flow Catscans, standard optical contrast.

220 Supercharge, Invasion and Mudcake Growth

Figure 2.70f. Linear flow Catscans, high contrast visualization. 2.3.8.2 Multiphase modeling in boreholes.

Rigorous math models describing dynamic multiphase fluid flows in downhole formation testing represent challenges that go beyond the usual tasks in conventional reservoir engineering. An original SPWLA work, sponsored by Halliburton, was developed in Chin and Proett (2005) – a decade later, major extensions of the model to include inertial effects via Forcheimer corrections would appear in Chin et al. (2015). Formulations, numerical analyses, solution methods and validations are described in the latter book and are not replicated here. It will suffice to cite the main hurdles in physical description, the computational method used, and key illustrations that highlight the main simulation features. In traditional reservoir engineering, the domain of flow is first partitioned into much smaller grid block or finite element arrays. Then the governing equations for Darcy flow, which may be single-phase, or miscible or immiscible multiphase, together with well production, farfield boundary and initial conditions, are discretized and written at each internal field node. The resulting algebraic system is typically massive and requires extensive resources in linear equation solution methods. Formation tester applications are challenging in the following respect. The same approach applies to the rock matrix adjacent to the

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sandface. However, this flow is dynamically coupled to the non-Darcy flow in the wellbore, whose radius must be specified so that well curvature effects are properly handled in the nearfield of the tester. This is further complicated by the need to describe a dynamically growing mudcake – at first, invasion is rapid, but as mudcake thickness increases, formation influx decreases (this model is described in terms of mudcake measurements obtained from filtration tests performed at the surface). In the final formulation, the wellbore fluid flow couples to that of the growing mudcake, which in turn, depends on the Darcy dynamics of the reservoir. Unlike vertical, deviated or horizontal wells, which are relatively simple to describe in reservoir modeling, the formation tester must be correctly modeled as a localized source or sink, and then, one introducing the distortive effects of flowline storage on pressure measurements. Additionally, layered media description is needed, since well logging runs will often traverse widely contrasting layers, barriers, fractures, and so on. The work of Chin and Proett (2005) addressed this problem, but the full transient, three-dimensional problem would not be simulated efficiently. Thus, a two-dimensional axisymmetric model was introduced, as shown in Figure 2.71. Axisymmetry meant that a “point source” would be modeled as a “ring source,” a requirement that was subsequently justified mathematically since the assumption, verified by detailed three-dimensional finite element analysis, did not significantly affect engineering accuracy. The calculated results shown later in this section pertain to the axisymmetric model of Chin et al. (2015), which extends the 2005 model to high velocity nozzle flows, such as those encountered in rapid pumping. The model of Chin and Proett (2015) addresses contamination clean-up applications and estimates the time needed to pump a clean sample. Its 2005 predecessor is presently used at Halliburton for job planning purposes while our PTA methods are still used in formation evaluation. However, we emphasize that these approaches are no longer necessary in single-phase flow modeling – in our companion 2021 book, we in fact model the wellbore in its entirety and are able to includes the effects of actual probes located at discrete positions along the circumference (the approach applies to holes of any radius). Moreover, the reservoir flow equations applicable to the rock matrix are now solved three-dimensionally in circular coordinates and need not be simplified using axisymmetric assumptions – supercharging effects are also successfully incorporated into the overall model, with computing speeds that are rapid and offering numerical stability.

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Figure 2.71. Single-probe supercharging and pumping model. 2.3.8.3 Pressure and concentration displays.

In this section, we focus on computed results and show that they are consistent with physical expectations. We consider a cross-sectional “slice” of the cylindrical domain in Figure 2.71, that is, a look at the r-z plane. Two coupled solutions are considered in Chin (2019), as shown in Figure 2.72 – one for pressure, the second for saturation or concentration.

Figure 2.72. Pressure and contamination profiles in r-z plane.

Spherical Source Formulations 223 Example 1. Single probe, infinite anisotropic media.

Here we describe graphical output results in detail and explain their uses and implications. Once all input quantities are saved (e.g., our menus appear in Figures 2.56 and 2.57) and “Simulate” in the user interface is clicked, displays of pressure and oil saturation (or formation fluid concentration) field are given periodically in time. Consider Figure 2.73a, which contains two plots, with pressure at the left and concentration at the right for a miscible run (for immiscible runs, concentration is replaced by oil saturation). For each of the diagrams in Figure 2.73a, the left vertical side corresponds to the sandface at the borehole wall – the right side corresponds to the radial farfield. The top and bottom horizontal lines coincide with the top and bottom of the reservoir. Therefore, these cross-sections display computed solutions in the r-z plane for the axisymmetric formulation. The left pressure plot is uniform vertically, with identical pressure profiles at all z stations. The red at the left represents high mud pressure, relative to the lower blue formation pressure at the right. The right concentration plot again indicates a purely radial flow without z variations. Invading blue mud is displacing red formation fluid. The multicolored zones between blue and red in either case represent events in the diffusive mixing zone. Cylindrical radial invasion occurs while drilling. Sometimes the invasion time is short – at other times, it can exceed a day. For long invasion times, it is not necessary to simulate extraordinarily long – we equivalently model the invasion associated with a higher permeability mud for a shorter time. Equivalence formulas are given in Chin (1995, 2002). The short times in Figures 2.73a and 2.73b mimic one day invasion. Note how mud pressure and invasion effects are deeper at “1 min” than at “0.33 sec,” as expected physically. At a time set by the user, the tester starts pumping – it can extract or inject fluid into the reservoir according to a multi-rate schedule. The left pressure plot in Figure 2.73c shows fluid withdrawal typical of sampling operations, in this case, by a single centered nozzle (single, dual and straddle packer probes are permitted which can be located arbitrarily along the sandface). The pressure plot shows a blue-green area associated with low nozzle pressure at. Above and below this zone are red colored pressures indicating higher pressures associated with supercharging – that is, as the nozzle withdraws fluid, high pressure mud invades the formation through the mudcake. Reverse flow streamlines are not displayed by the software.

224 Supercharge, Invasion and Mudcake Growth

Figure 2.73a. Pressure-concentration profiles, 0.33 sec.

Figure 2.73b. Pressure-concentration profiles, 1.00 min. The right plot in Figure 2.73c displays the concentration profile. The blue zone represents the mud filtrate that has penetrated the formation – it is now deeper than that in Figure 2.73b. Figures 2.73d and 2.73e illustrate similar phenomena at later times. Again, note the high supercharge pressures above and below the nozzle, indicating continuing filtrate invasion while the tester nozzle attempts to extract a clean sample. Whether or not this is possible for the input parameters assumed is one question the simulation addresses. Is a clean sample possible? If so, how long must pumping continue? If not, how might mud properties and weight to changed? The time scale for clean-up differs from that for pressure transient interpretation. How long must be tool stay in place to ensure good pressure data for permeability and anisotropy prediction – without risking a stuck tool? Good pressures for permeability prediction can be obtained even when filtrate has not been flushed. Thus, for tools that do not collect samples (e.g., formation-testing-while-drilling tools), the job planning simulator can be used to study pressure transients.

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Figure 2.73c. Pressure-concentration profiles, 3.33 min.

The left plot of Figure 2.73c indicates probe presence because its low (blue-green) pressures contrast with the high (red) ones due to supercharging. At the right, a small green zone associated with the probe is embedded in the blue filtrate – it is not red because the fluid is still contaminated. Variable meshes allow high resolution near the probe.

Figure 2.73d. Pressure-concentration profiles, 3.67 min.

Figure 2.73e. Pressure-concentration profiles, 5.67 min.

226 Supercharge, Invasion and Mudcake Growth

At the end of the simulation – a point in time set by the user, the pressure-concentration screens may be played back in “movie mode.” This feature enhances the field engineer’s physical intuition about the formation. In addition, three line graphs automatically appear that summarize computed results at source and observation probes. Figure 2.73f displays the concentration of formation fluid at the source probe. For the input parameters assumed in the present simulation, the results are not encouraging (we will show more optimistic results later). At first, the concentration is zero because the nozzle is not pumping. Once pumping commences, the formation fluid concentration increases to a maximum of 0.3 or 30% – not quite the 90-95% that is deemed adequate by petrophysicists. There may be several reasons for low dilute levels. Figures 2.73a–2.73e show high levels of supercharge pressure while the nozzle is pumping – our nozzle may, in fact, be pumping filtrate just entering the formation. But even if not, it is possible that rapid diffusion between filtrate and formation fluid – encouraged by the particular pore structure in the rock – mixes the two quickly in a detrimental manner. One purpose of the job planning simulator is to identify the reasons for sample contamination and to recommend fixes. Figure 2.73g displays the calculated pressure drawdown and buildup. Pressure transient predictions are always checked against exact analytical results to ensure that spatial and time grid parameters yield accurate results. In our case, the complex complementary error function solution of Proett, Chin and Mandal (2000) was used. This solution assumes a pure spherical (ellipsoidal) source model, so that fluid invasion at the borehole wall is not modeled. Assumed meshes were therefore calibrated against exact results for a perfectly sealed borehole.

Figure 2.73f. Formation fluid concentration at source probe.

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That our job planning simulator models fluid mixing and pressure transient analysis (due to fluid compressibilities) is desirable because it is convenient – but more important, both objectives are accomplished using the same software, thus reducing the uncertainties associated with using multiple simulators. In addition, because the same equations as those in reservoir engineering are used here, the same software can be used for reservoir engineering production predictions by sealing the borehole, expanding the flow domain to field scale, and lowering the borehole pressure to those characteristic of production scenarios.

Figure 2.73g. Source probe pressure transient history.

Finally, as shown in Figure 2.73h, the software also produces pressure transient outputs at any observation probe location defined by the user. We emphasize that it is not necessary to have an observation probe in the tool to use the software model – that is, the software can be used with single-nozzle tool FTWD applications to ascertain depth of investigation or penetration. Of course, in dual probe applications, the use of pressure drops available at source and observation probes enables prediction of horizontal and vertical permeability. For the present simulation, it is of interest to note the high level of diffusion in Figure 2.73h relative to that in Figure 2.73g. Such indicators are useful in ascertaining the probability of success that might be achieved in socalled “mini-DST” applications. The observation probe can be chosen at any distance from the source probe, e.g., seven inches or ten feet.

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Figure 2.73h. Observation probe pressure transient history. Example 2. Single probe, three layer medium.

In this simulation, we describe a situation without the upper-lower symmetries obtained earlier. Here the formation consists of three layers, with the lowest porosity layer at the top. Before pumping, there is strong cylindrical radial filtrate invasion into the formation, as is evident from the right-side plot of Figure 2.74a. The left-side pressure plot shows a small blue zone marking the lower pressures realized at the nozzle. That the entire plot is a single color indicates relatively little pressure variation otherwise, characteristic of the low permeabilities assumed.

Figure 2.74a. Initial pumping, highly invaded upper zone.

Figure 2.74b gives pressure-concentration plots later in time. The concentration plot shows continuing strong invasion in the low porosity layer. The pressure plot, with the high color contrast and the strong red zones above and below the probe along the sandface (left vertical boundary) indicates strong supercharging. The formation tester probe will measure high pressures, but the high values characterize more the

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high pressure in the mud than the pore pressure in the rock. The pressureconcentration behavior for Figure 2.74b continues with increased intensity in Figure 2.74c. In all the runs shown so far, note our use of variable spatial grids in the radial and vertical directions. Internal software logic also activates variable time gridding, enabling large time steps when flow gradients are small and smaller time steps when they are large. For example, smaller time steps are selected whenever a change in flow rate is imposed; higher grid densities are always used near nozzles.

Figure 2.74b. Supercharging seen in left pressure plot.

Figure 2.74c. Continued supercharging and invasion.

230 Supercharge, Invasion and Mudcake Growth Example 3. Dual probe pumping, three layer medium.

In this example, we consider a three layer medium again – the higher permeabilities here allow stronger pressure penetration as seen in the left pressure plot of Figure 2.75a.

Figure 2.75a. Initial cylindrical invasion before pumping.

Pumping has initiated in Figure 2.75b. The two small blue areas in the left pressure plot mark the low pressure zones associated with two pumping nozzles. The two small red areas in the right concentration plot mark the high formation fluid concentrations associated with continued pumping. From Figure 2.75c, at large times pumping has ceased and supercharging at the sandface is evident from the left red pressure zones.

Figure 2.75b. Dual probe pumping initiated.

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Figure 2.75c. Supercharging evident at large times. Example 4. Straddle packer pumping.

So far we have shown how pumpouts using single and dual probe nozzles can be realistically simulated. In many situations, particularly in unconsolidated sands and naturally fractured formations, pad nozzles may not be effective in reliably contacting producing zones. Then, straddle packer nozzles are employed. Whereas pad nozzles “see” a single point along the borehole wall, packer nozzles see axial extents that may be several feet in length and then pump from all azimuthal directions. Because they are associated with pump rates that may reach 1 gpm, they offer good depth of investigation and strong signal propagation – thus they are extremely useful in so-called “mini-DST” applications that seek to determine permeability over larger spatial scales than those normally possible with pad-type tools. For straddle packer applications, axisymmetric flow and ring sources are accurate models. Again, we consider a layered region that is initially invaded by mud filtrate. The left pressure plot in Figure 2.76a shows an elongated low pressure zone associated with the length of the straddle packer. The nonuniform vertical pressure variations indicate that the radial flux into the tool is not uniform – computational evidence that “uniform flux” pumping models are not correct even the packer resides entirely within a uniform layer. The formation fluid concentration plot in Figure 2.76b highlights the continuing invasion of filtrate into the near-sandface rock. The pressure plots in Figures 2.76c and 2.76d highlight the strong impact on local flow exerted by the straddle packer nozzle. Its long vertical extent allows it to withdraw large amounts of fluid into the tool. The low pressures at the bottom and bottom-right of the formation unfortunately encourage stronger invasion at the top, an effect clearly seen in the

232 Supercharge, Invasion and Mudcake Growth

concentration plot of Figure 2.76d. Not shown are pressure plots along various tool stations. As noted, while our algorithm allows nonuniform radial flux along the tool length, pressures along it do not vary although they do vary with time. Pressures fields away from the packer are deeper than for pad type nozzles because of higher pump rates.

Figure 2.76a. Initial pumping of cylindrical invaded region.

Figure 2.76b. Continued straddle packer pumping.

Figure 2.76c. Strong lateral pumping.

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Figure 2.76d. Lower formation strongly affected. Example 5. Formation fluid viscosity imaging.

Our examples apply to shallow wells drilled with oil base mud. Inputs are representative of typical downhole conditions, but due to space limitations, only key parameters and qualitative results are summarized. We asked if differences in formation fluid viscosity are detectable through pressure responses. Mud viscosity is fixed at 1 cp, while formation fluid viscosity is taken as 1, 3 and 5 cp for top, middle and bottom calculations in Figure 2.77. As expected, pressure drops increase going downward for source and distant probes.

Figure 2.77. Source and observation probe pressures.

234 Supercharge, Invasion and Mudcake Growth

The source probe response is interesting. At the top left, the minimum pressure bottoms out as a flat line – both viscosities are 1 cp. In the middle and bottom left figures, where formation fluid viscosities exceed mud viscosity, minimum pressures decrease with time because flow resistance at the probe increases with time (average viscosity increases due to miscible mixing). Example 6. Contamination modeling.

“Formation fluid concentration (or saturation) vs time” plots at the source probe indicate changing contamination levels. In Figure 2.78, plots starting with the upper left and proceeding counterclockwise show typical concentrations increasing as fluid diffusion decreases – diffusion strongly affects sample quality.

Figure 2.78. Source probe formation fluid concentration.

The upper left figure shows poor 30% sample quality obtained soon after pumping; when pumping stops, this decreases continuously as invasion continues. The upper right shows good 90% quality. Sample quality improves with decreasing mudcake permeability or well pressure. Example 7. Multi-rate pumping simulation.

Figure 2.79a shows three constant-rate pumping intervals separated by two quiescent periods over a thirty minute period. The red curve gives source probe pressure response and shows decreasing values with time (due to increasing viscosity at the probe as low mud and higher formation viscosity fluids mix).

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Figure 2.79a. Field log, multirate flow and pressure.

Figure 2.79b. Source and observation probe simulation.

The source probe response of Figure 2.79a is successfully simulated at the left of Figure 2.79b, requiring two minutes on typical personal computers. The observation probe response appears at the right. Both indicate that viscosity increases during transient mixing are detectable from time-varying pressure data. We have given an overview of validation and field examples illustrating our modeling approach – and shown that their predictions are consistent with the physics. We have developed a reservoir engineering approach to model formation testing in borehole environments with dynamic mudcake growth and supercharging. The simulators, useful in permeability prediction from pressure transient analysis and contamination studies, are invaluable in inverse applications for kh and kv permeabilities. They are also important in job planning. For example, if formation properties are roughly known, what flow rates and sequences are needed for detectable signals at the observation probe? What mudcake properties are required for effective sealing to overcome supercharging? What is the depth of investigation and the vertical resolution associated with a particular flow rate? Our work applies to vertical wells in layered media and is restricted to zero dip angle – it assumes that the permeability in a plane perpendicular to the well axis does not vary azimuthally. This does not mean that we cannot consider deviated and horizontal wells where both permeabilities change about the well. For example, one can study worst case events by considering an isotropic uniform medium controlled by the lower of kh and kv.

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2.4 References. Chin, W.C., Formation Testing: Supercharge, Pressure Testing and Contamination Models, John Wiley & Sons, Hoboken, New Jersey, 2019. Chin, W.C., Zhou, Y., Feng, Y. and Yu, Q., Formation Testing: Low Mobility Pressure Transient Analysis, John Wiley & Sons, Hoboken, New Jersey, 2015. Chin, W.C., Zhou, Y., Feng, Y., Yu, Q. and Zhao, L., Formation Testing: Pressure Transient and Contamination Analysis, John Wiley & Sons, Hoboken, New Jersey, 2014. Chin, W.C., Suresh, A., Holbrook, P., Affleck, L., and Robertson, H., “Formation Evaluation Using Repeated MWD Logging Measurements,” Paper No. U, SPWLA 27th Annual Logging Symposium, Houston, TX, June 9-13, 1986.

3 Practical Applications Examples Here we consider additional problems that arise in formation tester well logging. The topics listed below are not given in any particular order, but they address issues that represent non-ideal effects that are usually neglected because corrective models have not been available. These are varied. For instance, inverse solutions often assume constant drawdown flow rates – we’ve relaxed that assumption. Supercharging usually points to the need for corrections to predicted pore pressure due to overbalanced drilling. However, this excess pressure is related to mudcake formation and invasion, areas which are pertinent to stuck pipe issues and sealing effectiveness of formation tester pads. And, few practitioners realize that existing dual probe tools can be used to predict kh and kv at any dip angle. And so, we’ve addressed these “loose ends” and offer more comprehensive treatments in Chapters 4 and 5. In practical formation testing work, especially in pressure transient interpretation, repetitive operation of interactive simulators (e.g., FT-00, FT-06 and FT-07) is required for history matching. Labor demands are therefore significant. We will also introduce two rapid, “mass production” or “batch processes” which automate the algorithms behind our flagship simulator FT-00. The first, in “Rapid Batch Analysis . . ,” runs fully transient simulations (requiring all inputs) for any flow rate schedule automatically for different sets of nested do-loop parameters and provides a complete set of solutions. The second, in “Multiprobe DOI . . ,” focuses on responses at different observation probes. This feature assists designers in positioning pressure probes in array formation testing tools, and for such tools, allows users to compare measured to theoretical pressure transient profiles to assist in barrier detection. A simple “drawdown only” flow rate is programmed into the simulator. 237

238 Supercharge, Invasion and Mudcake Growth

3.1 Non-constant Flow Rate Effects. In this example, we study the effects of imperfect pumping on pressure transient development, maximum pressure drop, and inverse predictions for pore pressure, compressibility, and permeability or mobility. The constant flow rate in Figure 3.1 will define our “control problem,” serving as the basis for comparison. We can think of Case (a) as an idealization for the “slow ramp up/down” situation in Case (b), or the “impulsive start/stop” motion of Case (c).

Figure 3.1. Total pumpout of 5 cc, for all three piston scenarios. Fortunately, we can develop accurate solutions for each of the above problems, because the forward simulator FT-00 is based on an exact analytic solution; similarly, our inverse models (using our “multiple drawdown and buildup” algorithms) are analytically based. While the results are exact for the input fluid, formation and tool parameters assumed, they do not represent general fluid-dynamic results. However, the procedures we develop here are applicable to other problems and are useful in developing solutions for problems with complicated flow rates.

Practical Applications 239

3.1.1 Constant flow rate, idealized pumping, inverse method.

For this problem, we will create synthetic pressure data using exact forward simulator FT-00, as shown in Figure 3.2b, to evaluate our inverse method. Then, we will take three early pressure-time data points and, using Model 2 from our inverse “multiple drawdown-buildup” algorithms, predict mobility, compressibility and pore pressure.

Figure 3.2a. Constant rate pumping (idealization).

Figure 3.2b. FT-00 forward simulator input menu.

240 Supercharge, Invasion and Mudcake Growth

Figure 3.2c. Pumpout schedule.

Figure 3.2d. Source probe pressure.

Practical Applications 241

Figure 3.2e. Observation probe pressure. Pumpout schedule, source and observation probe pressures are given in Figures 3.2c,d,e. Notice that the source probe pressure has achieved steady-state equilibrium by 20 sec (see red font). We have arbitrarily selected pressure buildup data at 7, 10 and 15 sec to evaluate our inverse method. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flowrate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s) 0.000E+00 0.100E+01 0.200E+01 0.300E+01 0.400E+01 0.500E+01 0.600E+01 0.700E+01 0.800E+01 0.900E+01 0.100E+02 0.110E+02 0.120E+02 0.130E+02 0.140E+02 0.150E+02

Rate (cc/s) 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

Ps* (psi) 0.10000E+05 0.91426E+04 0.85071E+04 0.80335E+04 0.76797E+04 0.74148E+04 0.80734E+04 0.85597E+04 0.89209E+04 0.91900E+04 0.93908E+04 0.95410E+04 0.96534E+04 0.97377E+04 0.98010E+04 0.98486E+04

Pr* (psi) 0.10000E+05 0.99887E+04 0.99696E+04 0.99574E+04 0.99499E+04 0.99453E+04 0.99537E+04 0.99709E+04 0.99820E+04 0.99887E+04 0.99928E+04 0.99953E+04 0.99968E+04 0.99978E+04 0.99984E+04 0.99987E+04

Ps**(psi) 0.00000E+00 -0.85738E+03 -0.14929E+04 -0.19665E+04 -0.23203E+04 -0.25852E+04 -0.19266E+04 -0.14403E+04 -0.10791E+04 -0.81005E+03 -0.60920E+03 -0.45903E+03 -0.34659E+03 -0.26228E+03 -0.19898E+03 -0.15139E+03

Pr**(psi) 0.00000E+00 -0.11277E+02 -0.30351E+02 -0.42591E+02 -0.50077E+02 -0.54704E+02 -0.46328E+02 -0.29096E+02 -0.18044E+02 -0.11342E+02 -0.72458E+01 -0.47215E+01 -0.31609E+01 -0.21987E+01 -0.16104E+01 -0.12562E+01

Pr**/Ps** ----------0.13152E-01 0.20330E-01 0.21659E-01 0.21582E-01 0.21160E-01 0.24047E-01 0.20201E-01 0.16721E-01 0.14001E-01 0.11894E-01 0.10286E-01 0.91199E-02 0.83827E-02 0.80933E-02 0.82979E-02

242 Supercharge, Invasion and Mudcake Growth 0.160E+02 0.170E+02 0.180E+02 0.190E+02 0.200E+02 0.210E+02 0.220E+02 0.230E+02 0.240E+02 0.250E+02 0.260E+02 0.270E+02 0.280E+02 0.290E+02 0.300E+02

0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

0.98845E+04 0.99115E+04 0.99319E+04 0.99474E+04 0.99591E+04 0.99680E+04 0.99748E+04 0.99800E+04 0.99840E+04 0.99871E+04 0.99894E+04 0.99913E+04 0.99927E+04 0.99938E+04 0.99947E+04

0.99990E+04 0.99991E+04 0.99991E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99993E+04

-0.11555E+03 -0.88513E+02 -0.68090E+02 -0.52631E+02 -0.40905E+02 -0.31989E+02 -0.25194E+02 -0.19999E+02 -0.16015E+02 -0.12947E+02 -0.10576E+02 -0.87346E+01 -0.72968E+01 -0.61677E+01 -0.52752E+01

-0.10477E+01 -0.92883E+00 -0.86419E+00 -0.83127E+00 -0.81597E+00 -0.80958E+00 -0.80687E+00 -0.80479E+00 -0.80168E+00 -0.79676E+00 -0.78974E+00 -0.78067E+00 -0.76974E+00 -0.75722E+00 -0.74340E+00

0.90670E-02 0.10494E-01 0.12692E-01 0.15794E-01 0.19948E-01 0.25308E-01 0.32026E-01 0.40241E-01 0.50059E-01 0.61538E-01 0.74672E-01 0.89377E-01 0.10549E+00 0.12277E+00 0.14093E+00

Software reference: Multiple-DDBU-Solver.exe

On first running the above “exe” code, we obtain the screen in Figure 3.2f, which applies to drawdown pressures while pumping. We intended to analyze buildup data. Now, click “Agree,” so that the “Run” button becomes active, with captions turning from gray to black. Now select Model 2. The correct inverse in Figure 3.2g appears.

Figure 3.2f. Model 1, for drawdown “pressure-time” data.

Practical Applications 243

Figure 3.2g. Inverse pressure buildup problem (Model 2). Click “Run.” Then, the black MS-DOS worksheet appears, as shown in Figure 3.2h. We have replicated the remainder of the worksheet immediately below Figure 3.22. The copied material appears in Courier New font. Our predicted results are excellent (last group on the following page). From Figure 3.2b, our input mobility for forward simulator FT-00 used to create our synthetic data is (1 md)/(1 cp) or 1 md/cp, which compares well with the 1.042 value shown. The input pore pressure is 10,000 psi and our predicted value is 9,987 psi. The predicted value of compressibility is 0.0010 (cc/FloLineVol), and since the flowline volume is known as 100 cc, the compressibility is 0.00001/psi, exactly as given in the FT-00 menu. We emphasize that these predictions were obtained from early buildup data at 7, 10 and 15 sec. Note that the maximum pressure drop at the source is about 2,600 psi.

244 Supercharge, Invasion and Mudcake Growth

Figure 3.2h. Inverse worksheet. INPUTS, PTA-App-02 ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi):

1 1 0.34 5 7 8559 10 9390 15 9848

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): Stop time TDD1 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi):

1.000 1.000 0.340 0.340 5.000 7.000 8559.000 10.000 9390.000 15.000 9848.000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi): Type to close window:

9987.000 1.042 0.0067 0.0010 x (cc/FloLineVol)

Practical Applications 245

3.1.2 Slow ramp up/down flow rate.

In this example, we consider the a pump piston that is slow to ramp up and slow to ramp down. The flow rate function is far from the constant rate assumed in simple GeoTapTM models. The inputs to FT-00 are given in Figure 3.3b. The pressure transients created are inputted into the inverse code, and predicted formation parameters will be compared to known values.

Figure 3.3a. Slow ramp up/down rate pumping.

Figure 3.3b. FT-00 forward simulator input menu.

246 Supercharge, Invasion and Mudcake Growth

Figure 3.3c. Pumpout schedule.

Figure 3.3d. Source probe pressure.

Practical Applications 247

Figure 3.3e. Observation probe pressure. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s) 0.000E+00 0.336E+00 0.671E+00 0.101E+01 0.134E+01 0.168E+01 0.201E+01 0.235E+01 0.268E+01 0.302E+01 0.336E+01 0.369E+01 0.403E+01 0.436E+01 0.470E+01 0.503E+01 0.537E+01 0.570E+01 0.604E+01 0.638E+01 0.671E+01 0.705E+01 0.738E+01 0.772E+01 0.805E+01 0.839E+01

Rate (cc/s) 0.50000E+00 0.50000E+00 0.50000E+00 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.50000E+00 0.50000E+00 0.50000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

Ps* (psi) 0.10000E+05 0.98413E+04 0.96982E+04 0.95655E+04 0.92901E+04 0.90413E+04 0.88163E+04 0.86125E+04 0.84280E+04 0.82607E+04 0.81091E+04 0.79717E+04 0.78471E+04 0.77340E+04 0.76314E+04 0.75550E+04 0.76275E+04 0.76924E+04 0.77707E+04 0.79800E+04 0.81688E+04 0.83394E+04 0.84937E+04 0.86333E+04 0.87596E+04 0.88741E+04

Pr* (psi) 0.10000E+05 0.99998E+04 0.99977E+04 0.99943E+04 0.99906E+04 0.99852E+04 0.99789E+04 0.99730E+04 0.99678E+04 0.99633E+04 0.99595E+04 0.99562E+04 0.99535E+04 0.99511E+04 0.99491E+04 0.99475E+04 0.99463E+04 0.99473E+04 0.99498E+04 0.99527E+04 0.99575E+04 0.99631E+04 0.99684E+04 0.99730E+04 0.99770E+04 0.99804E+04

Ps**(psi) 0.00000E+00 -0.15870E+03 -0.30180E+03 -0.43450E+03 -0.70989E+03 -0.95866E+03 -0.11837E+04 -0.13875E+04 -0.15720E+04 -0.17393E+04 -0.18909E+04 -0.20283E+04 -0.21529E+04 -0.22660E+04 -0.23686E+04 -0.24450E+04 -0.23725E+04 -0.23076E+04 -0.22293E+04 -0.20200E+04 -0.18312E+04 -0.16606E+04 -0.15063E+04 -0.13667E+04 -0.12404E+04 -0.11259E+04

Pr**(psi) 0.00000E+00 -0.16166E+00 -0.22515E+01 -0.57109E+01 -0.94472E+01 -0.14801E+02 -0.21062E+02 -0.26997E+02 -0.32228E+02 -0.36721E+02 -0.40544E+02 -0.43787E+02 -0.46538E+02 -0.48873E+02 -0.50857E+02 -0.52547E+02 -0.53730E+02 -0.52654E+02 -0.50198E+02 -0.47284E+02 -0.42532E+02 -0.36931E+02 -0.31632E+02 -0.26970E+02 -0.22974E+02 -0.19582E+02

Pr**/Ps** ----------0.10187E-02 0.74602E-02 0.13144E-01 0.13308E-01 0.15440E-01 0.17793E-01 0.19457E-01 0.20501E-01 0.21113E-01 0.21442E-01 0.21588E-01 0.21616E-01 0.21568E-01 0.21472E-01 0.21491E-01 0.22647E-01 0.22817E-01 0.22518E-01 0.23408E-01 0.23226E-01 0.22240E-01 0.21000E-01 0.19733E-01 0.18522E-01 0.17392E-01

248 Supercharge, Invasion and Mudcake Growth 0.872E+01 0.906E+01 0.940E+01 0.973E+01 0.101E+02 0.104E+02 0.107E+02 0.111E+02 0.114E+02 0.117E+02 0.121E+02 0.124E+02 0.128E+02 0.131E+02 0.134E+02 0.138E+02 0.141E+02 0.144E+02 0.148E+02 0.151E+02 0.154E+02 0.158E+02 0.161E+02 0.164E+02 0.168E+02 0.171E+02 0.174E+02 0.178E+02 0.181E+02 0.185E+02 0.188E+02 0.191E+02 0.195E+02 0.198E+02 0.201E+02 0.205E+02

0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

0.89777E+04 0.90716E+04 0.91567E+04 0.92339E+04 0.93038E+04 0.93672E+04 0.94248E+04 0.94769E+04 0.95243E+04 0.95672E+04 0.96062E+04 0.96416E+04 0.96737E+04 0.97029E+04 0.97294E+04 0.97534E+04 0.97753E+04 0.97951E+04 0.98132E+04 0.98296E+04 0.98445E+04 0.98580E+04 0.98704E+04 0.98816E+04 0.98918E+04 0.99010E+04 0.99095E+04 0.99171E+04 0.99241E+04 0.99305E+04 0.99363E+04 0.99416E+04 0.99464E+04 0.99508E+04 0.99548E+04 0.99584E+04

0.99833E+04 0.99857E+04 0.99878E+04 0.99895E+04 0.99910E+04 0.99922E+04 0.99933E+04 0.99942E+04 0.99950E+04 0.99956E+04 0.99962E+04 0.99967E+04 0.99971E+04 0.99974E+04 0.99977E+04 0.99979E+04 0.99982E+04 0.99983E+04 0.99985E+04 0.99986E+04 0.99987E+04 0.99988E+04 0.99989E+04 0.99989E+04 0.99990E+04 0.99990E+04 0.99991E+04 0.99991E+04 0.99991E+04 0.99991E+04 0.99991E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04

-0.10223E+04 -0.92836E+03 -0.84326E+03 -0.76612E+03 -0.69618E+03 -0.63276E+03 -0.57524E+03 -0.52307E+03 -0.47573E+03 -0.43278E+03 -0.39380E+03 -0.35841E+03 -0.32629E+03 -0.29712E+03 -0.27064E+03 -0.24658E+03 -0.22472E+03 -0.20487E+03 -0.18683E+03 -0.17043E+03 -0.15552E+03 -0.14197E+03 -0.12965E+03 -0.11844E+03 -0.10825E+03 -0.98974E+02 -0.90534E+02 -0.82852E+02 -0.75859E+02 -0.69492E+02 -0.63693E+02 -0.58411E+02 -0.53598E+02 -0.49211E+02 -0.45212E+02 -0.41566E+02

-0.16712E+02 -0.14286E+02 -0.12232E+02 -0.10493E+02 -0.90179E+01 -0.77656E+01 -0.67017E+01 -0.57970E+01 -0.50275E+01 -0.43727E+01 -0.38156E+01 -0.33416E+01 -0.29385E+01 -0.25960E+01 -0.23052E+01 -0.20585E+01 -0.18495E+01 -0.16728E+01 -0.15236E+01 -0.13979E+01 -0.12922E+01 -0.12037E+01 -0.11297E+01 -0.10681E+01 -0.10170E+01 -0.97480E+00 -0.94013E+00 -0.91180E+00 -0.88878E+00 -0.87022E+00 -0.85536E+00 -0.84358E+00 -0.83431E+00 -0.82711E+00 -0.82158E+00 -0.81738E+00

0.16348E-01 0.15388E-01 0.14506E-01 0.13696E-01 0.12953E-01 0.12273E-01 0.11650E-01 0.11083E-01 0.10568E-01 0.10104E-01 0.96892E-02 0.93233E-02 0.90060E-02 0.87372E-02 0.85176E-02 0.83482E-02 0.82301E-02 0.81650E-02 0.81549E-02 0.82020E-02 0.83088E-02 0.84782E-02 0.87134E-02 0.90178E-02 0.93950E-02 0.98491E-02 0.10384E-01 0.11005E-01 0.11716E-01 0.12523E-01 0.13429E-01 0.14442E-01 0.15566E-01 0.16807E-01 0.18172E-01 0.19665E-01

We run the required inverse solver, Model 6. Results are given immediately beneath Figure 3.3f in Courier New font.

Figure 3.3f. Model 6 inverse problem.

Practical Applications 249 INPUTS, PTA-App-06 ... Volume flow rate Q1 (cc/s): Volume flow rate Q2 (cc/s): Volume flow rate Q3 (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st stop time TDD1 (sec): 2nd stop time TDD2 (sec): 3rd stop time TDD3 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi):

0.5 1 0.5 1 0.34 1 5 6 7.05 8339 10.1 9303 15.1 9829

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Volume flow rate Q2 (cc/s): Volume flow rate Q3 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st stop time TDD1 (sec): 2nd stop time TDD2 (sec): 3rd stop time TDD3 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi):

0.500 1.000 0.500 1.000 0.340 0.340 1.000 5.000 6.000 7.050 8339.000 10.100 9303.000 15.100 9829.000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

9993.000 1.036 0.0067 0.0010 x (cc/FloLineVol)

Type to close window:

Again, considering our use of early time buildup data at 7, 10 and 15 sec, our predicted pore pressure of 9,993 psi (versus a known 10,000 psi), mobility of 1.036 md/cp (versus a known 1 md/cp), and a compressibility of 0.00001/psi (exact) are excellent. The maximum source probe pressure drop is 2,500 psi, about 100 psi less than that in the prior problem.

250 Supercharge, Invasion and Mudcake Growth

3.1.3 Impulsive start/stop flow rate.

In this example, we consider a “stubborn” piston that may refuse to start, but then starts suddenly. The flow rate function in Figure 3.4a is definitely not “constant rate” as required in several industry inverse methods. We repeat the calculations in Sections 3.1.1 and 3.1.2.

Figure 3.4a. Impulsive start/stop rate pumping.

Figure 3.4b. FT-00 forward simulator assumptions.

Practical Applications 251

Figure 3.4c. Pumpout schedule.

Figure 3.4d. Source prove pressure.

252 Supercharge, Invasion and Mudcake Growth

Figure 3.4e. Observation probe pressure. Tabulated results are given below, and again, we use pressure buildup data from 7, 10 and 15 sec, the first point being just two seconds after the last drawdown terminate. The data used for inverse calculations is highlighted in red font. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s) 0.000E+00 0.333E+00 0.667E+00 0.100E+01 0.133E+01 0.167E+01 0.200E+01 0.233E+01 0.267E+01 0.300E+01 0.333E+01 0.367E+01 0.400E+01 0.433E+01 0.467E+01 0.500E+01 0.533E+01 0.567E+01

Rate (cc/s) 0.15000E+01 0.15000E+01 0.15000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.50000E+00 0.50000E+00 0.50000E+00 0.00000E+00 0.00000E+00 0.00000E+00

Ps* (psi) 0.10000E+05 0.95269E+04 0.91000E+04 0.87139E+04 0.85221E+04 0.83477E+04 0.81893E+04 0.80456E+04 0.79152E+04 0.77967E+04 0.76892E+04 0.75915E+04 0.75027E+04 0.75798E+04 0.76489E+04 0.77110E+04 0.79247E+04 0.81175E+04

Pr* (psi) 0.10000E+05 0.99995E+04 0.99934E+04 0.99831E+04 0.99726E+04 0.99650E+04 0.99601E+04 0.99565E+04 0.99536E+04 0.99513E+04 0.99493E+04 0.99476E+04 0.99462E+04 0.99451E+04 0.99461E+04 0.99486E+04 0.99515E+04 0.99561E+04

Ps**(psi) 0.00000E+00 -0.47308E+03 -0.89996E+03 -0.12861E+04 -0.14779E+04 -0.16523E+04 -0.18107E+04 -0.19544E+04 -0.20848E+04 -0.22033E+04 -0.23108E+04 -0.24085E+04 -0.24973E+04 -0.24202E+04 -0.23511E+04 -0.22890E+04 -0.20753E+04 -0.18825E+04

Pr**(psi) 0.00000E+00 -0.46858E+00 -0.66317E+01 -0.16915E+02 -0.27375E+02 -0.34952E+02 -0.39888E+02 -0.43495E+02 -0.46353E+02 -0.48712E+02 -0.50696E+02 -0.52381E+02 -0.53819E+02 -0.54895E+02 -0.53898E+02 -0.51379E+02 -0.48466E+02 -0.43875E+02

Pr**/Ps** ----------0.99048E-03 0.73689E-02 0.13152E-01 0.18522E-01 0.21153E-01 0.22029E-01 0.22255E-01 0.22233E-01 0.22109E-01 0.21938E-01 0.21748E-01 0.21551E-01 0.22682E-01 0.22924E-01 0.22446E-01 0.23353E-01 0.23307E-01

Practical Applications 253 0.600E+01 0.633E+01 0.667E+01 0.700E+01 0.733E+01 0.767E+01 0.800E+01 0.833E+01 0.867E+01 0.900E+01 0.933E+01 0.967E+01 0.100E+02 0.103E+02 0.107E+02 0.110E+02 0.113E+02 0.117E+02 0.120E+02 0.123E+02 0.127E+02 0.130E+02 0.133E+02 0.137E+02 0.140E+02 0.143E+02 0.147E+02 0.150E+02 0.153E+02 0.157E+02 0.160E+02 0.163E+02 0.167E+02 0.170E+02 0.173E+02 0.177E+02 0.180E+02 0.183E+02 0.187E+02 0.190E+02 0.193E+02 0.197E+02 0.200E+02 0.203E+02 0.207E+02 0.210E+02 0.213E+02 0.217E+02 0.220E+02 0.223E+02 0.227E+02 0.230E+02 0.233E+02 0.237E+02 0.240E+02 0.243E+02 0.247E+02 0.250E+02 0.253E+02 0.257E+02 0.260E+02 0.263E+02 0.267E+02 0.270E+02 0.273E+02 0.277E+02 0.280E+02 0.283E+02 0.287E+02 0.290E+02 0.293E+02 0.297E+02 0.300E+02 0.303E+02

0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

0.82918E+04 0.84496E+04 0.85924E+04 0.87218E+04 0.88390E+04 0.89453E+04 0.90416E+04 0.91289E+04 0.92082E+04 0.92800E+04 0.93452E+04 0.94044E+04 0.94581E+04 0.95069E+04 0.95512E+04 0.95914E+04 0.96279E+04 0.96610E+04 0.96912E+04 0.97186E+04 0.97434E+04 0.97661E+04 0.97866E+04 0.98053E+04 0.98223E+04 0.98378E+04 0.98518E+04 0.98646E+04 0.98763E+04 0.98869E+04 0.98965E+04 0.99053E+04 0.99133E+04 0.99206E+04 0.99273E+04 0.99333E+04 0.99388E+04 0.99438E+04 0.99484E+04 0.99526E+04 0.99564E+04 0.99599E+04 0.99631E+04 0.99660E+04 0.99686E+04 0.99710E+04 0.99732E+04 0.99753E+04 0.99771E+04 0.99788E+04 0.99804E+04 0.99818E+04 0.99831E+04 0.99843E+04 0.99853E+04 0.99863E+04 0.99873E+04 0.99881E+04 0.99889E+04 0.99896E+04 0.99902E+04 0.99908E+04 0.99914E+04 0.99919E+04 0.99924E+04 0.99928E+04 0.99932E+04 0.99936E+04 0.99939E+04 0.99942E+04 0.99945E+04 0.99948E+04 0.99950E+04 0.99953E+04

0.99618E+04 0.99672E+04 0.99720E+04 0.99761E+04 0.99796E+04 0.99826E+04 0.99851E+04 0.99872E+04 0.99890E+04 0.99906E+04 0.99919E+04 0.99930E+04 0.99939E+04 0.99947E+04 0.99954E+04 0.99960E+04 0.99965E+04 0.99969E+04 0.99973E+04 0.99976E+04 0.99979E+04 0.99981E+04 0.99983E+04 0.99984E+04 0.99986E+04 0.99987E+04 0.99988E+04 0.99988E+04 0.99989E+04 0.99990E+04 0.99990E+04 0.99990E+04 0.99991E+04 0.99991E+04 0.99991E+04 0.99991E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99992E+04 0.99993E+04 0.99993E+04 0.99993E+04 0.99993E+04

-0.17082E+04 -0.15504E+04 -0.14076E+04 -0.12782E+04 -0.11610E+04 -0.10547E+04 -0.95840E+03 -0.87106E+03 -0.79184E+03 -0.71997E+03 -0.65477E+03 -0.59559E+03 -0.54188E+03 -0.49312E+03 -0.44885E+03 -0.40864E+03 -0.37213E+03 -0.33896E+03 -0.30883E+03 -0.28145E+03 -0.25656E+03 -0.23394E+03 -0.21338E+03 -0.19468E+03 -0.17768E+03 -0.16222E+03 -0.14815E+03 -0.13535E+03 -0.12371E+03 -0.11311E+03 -0.10346E+03 -0.94675E+02 -0.86675E+02 -0.79388E+02 -0.72749E+02 -0.66700E+02 -0.61187E+02 -0.56161E+02 -0.51578E+02 -0.47398E+02 -0.43585E+02 -0.40106E+02 -0.36930E+02 -0.34030E+02 -0.31382E+02 -0.28963E+02 -0.26752E+02 -0.24731E+02 -0.22883E+02 -0.21192E+02 -0.19645E+02 -0.18228E+02 -0.16931E+02 -0.15743E+02 -0.14653E+02 -0.13654E+02 -0.12738E+02 -0.11897E+02 -0.11124E+02 -0.10414E+02 -0.97615E+01 -0.91612E+01 -0.86087E+01 -0.80998E+01 -0.76309E+01 -0.71984E+01 -0.67994E+01 -0.64309E+01 -0.60903E+01 -0.57754E+01 -0.54840E+01 -0.52140E+01 -0.49637E+01 -0.47316E+01

-0.38241E+02 -0.32823E+02 -0.28024E+02 -0.23897E+02 -0.20388E+02 -0.17414E+02 -0.14896E+02 -0.12763E+02 -0.10955E+02 -0.94197E+01 -0.81151E+01 -0.70055E+01 -0.60610E+01 -0.52568E+01 -0.45717E+01 -0.39881E+01 -0.34911E+01 -0.30678E+01 -0.27077E+01 -0.24014E+01 -0.21413E+01 -0.19206E+01 -0.17337E+01 -0.15756E+01 -0.14421E+01 -0.13298E+01 -0.12355E+01 -0.11565E+01 -0.10905E+01 -0.10357E+01 -0.99033E+00 -0.95294E+00 -0.92228E+00 -0.89731E+00 -0.87709E+00 -0.86086E+00 -0.84793E+00 -0.83773E+00 -0.82978E+00 -0.82364E+00 -0.81897E+00 -0.81546E+00 -0.81285E+00 -0.81094E+00 -0.80953E+00 -0.80848E+00 -0.80766E+00 -0.80697E+00 -0.80631E+00 -0.80562E+00 -0.80485E+00 -0.80395E+00 -0.80288E+00 -0.80162E+00 -0.80015E+00 -0.79846E+00 -0.79653E+00 -0.79437E+00 -0.79198E+00 -0.78935E+00 -0.78649E+00 -0.78341E+00 -0.78012E+00 -0.77662E+00 -0.77292E+00 -0.76904E+00 -0.76499E+00 -0.76077E+00 -0.75640E+00 -0.75189E+00 -0.74725E+00 -0.74249E+00 -0.73762E+00 -0.73265E+00

0.22387E-01 0.21170E-01 0.19909E-01 0.18696E-01 0.17561E-01 0.16511E-01 0.15543E-01 0.14653E-01 0.13835E-01 0.13083E-01 0.12394E-01 0.11762E-01 0.11185E-01 0.10660E-01 0.10185E-01 0.97594E-02 0.93813E-02 0.90506E-02 0.87675E-02 0.85324E-02 0.83461E-02 0.82097E-02 0.81247E-02 0.80929E-02 0.81164E-02 0.81976E-02 0.83391E-02 0.85439E-02 0.88153E-02 0.91568E-02 0.95721E-02 0.10065E-01 0.10641E-01 0.11303E-01 0.12056E-01 0.12906E-01 0.13858E-01 0.14917E-01 0.16088E-01 0.17377E-01 0.18790E-01 0.20333E-01 0.22011E-01 0.23830E-01 0.25796E-01 0.27915E-01 0.30191E-01 0.32630E-01 0.35237E-01 0.38016E-01 0.40971E-01 0.44104E-01 0.47420E-01 0.50920E-01 0.54605E-01 0.58476E-01 0.62533E-01 0.66773E-01 0.71195E-01 0.75796E-01 0.80570E-01 0.85514E-01 0.90620E-01 0.95881E-01 0.10129E+00 0.10684E+00 0.11251E+00 0.11830E+00 0.12420E+00 0.13019E+00 0.13626E+00 0.14240E+00 0.14860E+00 0.15484E+00

254 Supercharge, Invasion and Mudcake Growth

Figure 3.4f. Model 6, inverse solver. Again, we run Model 6, for our inverse modeling option. The relevant inputs and outputs are shown in Courier New font below. INPUTS, PTA-App-06 ... Volume flow rate Q1 (cc/s): Volume flow rate Q2 (cc/s): Volume flow rate Q3 (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st stop time TDD1 (sec): 2nd stop time TDD2 (sec): 3rd stop time TDD3 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi):

1.5 1 0.5 1 0.34 1 4 5 7 8721 10 9458 15 9864

Practical Applications 255 OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Volume flow rate Q2 (cc/s): Volume flow rate Q3 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st stop time TDD1 (sec): 2nd stop time TDD2 (sec): 3rd stop time TDD3 (sec): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi):

1.500 1.000 0.500 1.000 0.340 0.340 1.000 4.000 5.000 7.000 8721.000 10.000 9458.000 15.000 9864.000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

9987.000 1.042 0.0067 0.0010 x (cc/FloLineVol)

Type to close window:

Closing remarks. Excellent predictions using early time data are again obtained. We have 9,987 psi for pore pressure (versus a known 10,000 psi), a mobility of 1.042 md/cp (versus a known 1 md/cp, and a compressibility of 0.00001/psi (exact). Our overall conclusion for Section 3.1, using “multiple drawdownbuildup” inverse solver MDDBU, are clear: predictions are very good even for early time buildup data for rapidly changing flow rate data. Some manufacturers’ tools require constant speed pumping for proper permeability prediction, a luxury that may not be possible due to control issues, mechanical deficiencies or wear-and-tear. We have demonstrated that accurate assessments for pore pressure, mobility and compressibility are possible even for complicated flow rate functions, so long as rates are monitored and recorded. Once these are entered into the MDDBU input screen, predictions can be accurately given even using early time pressure buildup data. We emphasize that FT-PTA-DDBU, and its multirate extension MDDBU, both assume “zero superharge,” a condition that is not always realized in practice. Supercharge problems are considered next in Section 3.2.

256 Supercharge, Invasion and Mudcake Growth

3.2 Supercharging – Effects of Nonuniform Initial Pressure. When an inverse procedure such as FT-PTA-DD (for drawdown applications) or FT-PTA-DDBU (for buildup data) is used, information inputs such as probe radius, geometric factor, pump rate, ending drawdown time, plus discrete sets of pressure-time data, are entered. The pore pressure is not entered because the inverse algorithm does not require it: pore pressure, mobility and compressibility are products of the calculation. We often tend to think of these as accurate, which they are insofar as the mathematics is concerned. But more enters the application of inverse methods than is usually assumed. Very often, users are not aware that conventional methods offered by oil service companies assume zero supercharge. In fact, when drilling in depleted reservoirs, formation testers quite often function in a high supercharge environment with overbalance pressures as high as 2,000 psi. This was highlighted, as we noted in Chapter 2, in the Halliburton paper by Rourke, Powell, Platt, Hall and Gardner, in “A New Hostile Environment Wireline Formation Testing Tool: A Case Study from the Gulf of Thailand,” SPWLA 47th Annual Logging Symposium, Veracruz, Mexico, June 4-7, 2006, which summarized the results of a study covering several hundred wells. For completeness, we will explain in detail the differences between conventional “zero supercharge” inverse models and our more general “general supercharge” approach. Conventional zero supercharge model. The formation testing book by Chin et al. (2014) provides a number of zero-supercharge forward and inverse formulations in different physical limits together with exact solutions. Several from other service companies are also mentioned. One particular model, quite general, is given by Equations 5.1– 5.4 in that reference, here re-numbered as, 2P(r,t)/ r2 + 2/r P/ r = ( c/k) P/ t (A1) P(r,t = 0) = P0

(A2)

P(r = ,t) = P0

(A3)

(4 Rw2k/ ) P(Rw,t)/ r – VC P/ t = Q(t) (A4) Equations A1 – A4 represent the complete isotropic, zero skin, spherical Darcy flow formulation for compressible liquids, which was solved exactly in terms of complex complementary error functions. This formulation was used to develop the forward simulation FT-00, as well

Practical Applications 257

as the inverse procedures FT-01 (for steady dual probe applications) and FT-PTA-DDBU (for early time single probe problems). Note that “spherical” implies mathematical idealization. The foregoing equations do not handle supercharge since the (constant) farfield pore pressure is identical to the initial pressure – that is, the influence of high pressures near the sandface does not appear. The spherical source of radius Rw will not describe borehole wall curvature or the effects of tester pads – to account for these non-idealizations, Rw often denotes the product of a corrective “geometric factor” (that is, “G,” determined empirically or through 3D finite element analysis) and the true nozzle radius. The effects of fluid invasion and overbalanced pressures does not require the geometric presence of cylindrical well boundaries. An interesting mathematical device was used in Chin (2019) to mimic the effects of higher well pressures. Now, we ask how “supercharging,” where the term will be used to represent both overbalanced and underbalanced effects, can be modeled by extending the framework of Equations A1 – A4. Our approach is straightforward. We solve these equations with a single important change, altering only the initial condition as shown in Equation B2, highlighted in red. In particular, we consider the model 2P(r,t)/ r2 + 2/r P/ r = ( c/k) P/ t (B1) P(r,t = 0) = P0 + Z/r, Z > 0, R > Rw

(B2)

P(r = ,t) = P0

(B3)

(B4) (4 Rw2k/ ) P(Rw,t)/ r – VC P/ t = Q0 Here Z = Pbh – P0 may be positive (overbalanced) or negative (underbalanced) with any magnitude, Pbh being the borehole pressure just outside the reservoir sandface, and P0 being the farfield reservoir pore pressure. In older references, “Pbh – P0” is often quoted in the 200-250 psi range. In fact, Halliburton and Chevron Thailand, as discussed previously, overbalances exceeding 2,000 psi are not uncommon, particularly in infill drilling where the reservoir is depleting. In the above, the “Z/r” describes spatially varying pressures that decrease in magnitude away from the formation tester spherical source at early times. After large times, the effects of the initial condition term “Z/r” dissipate, as expected, with its influence restricted to the early time contact with the test probe.

258 Supercharge, Invasion and Mudcake Growth

Supercharge “Fast Forward” solver. In order to understand the effects of supercharge on Darcy flow development, it is important to have an analytically based forward simulator whose assumptions are consistent with those used in the development of the inverse simulator. The inverse method originally due to Proett and Chin considered time scales in which flowline storage effects were the same order of magnitude as those associated with low mobility. In this limit, the porosity does not appear. This simplification can be applied to a “fast FT-00 forward model” or simply “Fast Forward.” The resulting direct or forward model is hosted in the menu shown in Figure 3.5a.

Figure 3.5a. “Fast Forward” forward supercharge simulator.

Practical Applications 259

This simulator requires the usual “zero supercharge” inputs, but with one notable exception: the “overbalance pressure” is needed. The software supports “drawdown only” and “drawdown-buildup” applications. In the former, pressure-time data points are taken as fluid is withdrawn from the reservoir; in the former, data points are collected after pump withdrawal ends and pressures begin to build up. By clicking “DD only” or “DD-BU,” tabulated results and line graphs are generated immediately. What can we learn from calculated results? Let us first consider “DD-BU” for the parameters shown. We find the pressure response in Figure 3.5b. The initial pressure of 22,000 psi is the sum of the pore pressure of 20,000 psi and the assumed overbalanced pressure of 2,000 psi. After the drawdown ends, pressure buildup begins and the transient curve reaches a steady-state value of 20,000 psi, the given pore pressure. In other words, given enough time, the pore pressure is always achieved regardless of initial supercharge level (numerical values appear below).

Figure 3.5b. Drawdown-buildup with strong supercharge.

260 Supercharge, Invasion and Mudcake Growth Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0

P(psi) 22000. 20758. 19630. 18604. 17671. 16823. 17111. 17374. 17613. 17830. 18027. 19239. 19707. 19887. 19956. 19983. 19994. 19997. 19999. 20000.

Thus, if the only objective is to obtain pore pressure in the presence of supercharge, a pressure buildup test serves the purpose. The test, for example, can be shortened by performing a shorter drawdown. What can be said of drawdown tests? To answer this question, we assume identical inputs, but now press the “DD only” button. From Figure 3.5c, the pressure is seen tending to about 8,300 psi. Had we entered a flow rate of 0.5 cc/s (half as much), the asymptotic value would increase to 14,100 psi. Thus in a drawdown-only test, the large time pressure depends on the pump rate – but in the corresponding buildup test, the end pressure will be the pore pressure regardless of the volume flow rate. The next question that arises is, “Why perform an inverse calculation if everything can be accomplished using a buildup test?” The point is simple: formal inverse analyses are required if additional information related to mobility and compressibility are required. Our “drawdown only” inverse method with supercharge is shown in Figure 3.5d. When the correct overbalance pressure is entered as shown, our predictions appear in the lower right corner, that is, 19,999 psi (as opposed to an inputted 20,000 psi) for the pore pressure, 0.1013 md/cp as opposed to 0.1 md/cp for mobility, and a compressibility of 0.0009 x (cc/FlowlineVolume) = 0.0009 x (cc/300 cc) or 0.000003/psi – all three quantities extremely accurate!

Practical Applications 261

Figure 3.5c. Drawdown – only curve with supercharge. Transient time vs probe pressure response ... T(sec) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0

P(psi) 22000. 20758. 19630. 18604. 17671. 16823. 16052. 15351. 14713. 14134. 13607. 10371. 9123. 8642. 8456. 8385. 8357. 8346. 8342. 8341.

262 Supercharge, Invasion and Mudcake Growth

Figure 3.5d. Drawdown-only inverse supercharge model. The above accuracy was obtained only because we suspected that the formation tester fluid might have been withdrawn in the presence of supercharging. This suspicion might have arisen from knowledge that the reservoir in question was depleted. On the other hand, if we were completely naive, one might have used one of several “zero supercharge” inverse models offered by oil service companies. The predictions, in this case, would be similar to a run using our simulator in Figure 3.5d, but with the overbalance pressure “Pover” set to zero. The results would be grossly in error, that is, a pore pressure of 21,999 psi (versus a correct value of 20,000 psi), a mobility of 0.00865 md/cp versus a known value 0f 0.1 md/cp) and a compressibility of 0.0000027/ psi. Finally, we emphasize that while we have demonstrated our inverse capabilities with our forward and inverse drawdown-only algorithms, we could have obtained equally satisfying results for buildup data. Figure 3.5e shows the menu for our drawdown-buildup inverse algorithm. The predicted pore pressure is 20,000 psi (exact), while for mobility we have 0.1012 md/cp versus 0.1 md/cp, with compressibility being exact. We also emphasize that the example selected is very low in mobility.

Practical Applications 263

Figure 3.5e. Drawdown-buildup inverse supercharge model.

264 Supercharge, Invasion and Mudcake Growth

3.3 Dual Probe Anisotropy Inverse Analysis. In this section, we address important questions related to forward pressure calculations in the steady-state limit, and also, inverse predictions for kh and kv and “effective permeability” in transversely isotropic media. By “dual probe,” we mean axially displaced probes residing along the same azimuth. Although we have discussed combined “drawdown and phase delay” inverse methods in Chapter 2, we will not focus on such approaches here, as they are presently in research phase. Rather, we consider more mundane drawdown procedures that are at present simpler to perform in the field. The dual probe method introduced here is extended to “multiprobe observation” ports in Section 3.4, in the sense of “multiple receivers” in resistivity logging, that is, we will discuss pressure calculations for arrays of observation probes. The dual probe anisotropy inverse method considered here would apply to the source probe and any single probe in the observation probe array. Existing source model simulators. Before embarking on the above discussions, it is important to summarize our key forward and inverse simulation capabilities. Although these have been discussed separately in various sections of Chin et al. (2014), Chin et al. (2015) and Chin (2019), the combined “big picture” has not yet been published and we will do so for the first time here. We will highlight the role of FT-00, FT-01, FT-02, FT-PTA-DD, FT-PTA-DDBU and the supercharge extensions of the latter two algorithms, especially in applications where kh > kv and dip angles are large, noting that our discussions apply to all permeability and dip angle ranges and combinations. Our flagship forward simulator is FT-00, which is again based on an exact, closed form, analytical solution of the complete formulation with flowline volume effects. There are two limitations in the model. First, the pumpout schedule requires that each flow rate is “piecewise constant,” e.g., 1 cc/s for the first five seconds, -3 cc/s for the next two, 0 cc/s for the next four, and so on, with trapezoidal, triangular and similar rates disallowed. Second, the model does not allow supercharge effects. Let us first use FT-00 to generate nontrivial synthetic pressure data for input to our inverse algorithms. For this, we will use the parameters shown in Figure 3.6a. We emphasize that this run in not isotropic, in fact, we have kh = 10 md and kv = 1 md. We also note that we are considering 45 deg dip and not simpler vertical or horizontal wells. The tool configuration assumed corresponds to that typical of dual probe testers offered by several service company manufacturers.

Practical Applications 265

Figure 3.6a. Creating FT-00 pressure transient data for an anisotropic simulation at a high dip angle. Also note that the pumpout schedule is set up for a 1 cc/s constant fluid withdrawal rate for 1,200 sec. Multiple simulations can be performed for different run times by only changing the “1200” at the lower right. For example, by entering “20,” “100,” and “1000” in successive runs, the automatic line graph plotter will plot with different scales of time magnification. We now perform several such simulations.

266 Supercharge, Invasion and Mudcake Growth

Figure 3.6b. Source and observation probe pressures versus time at different magnifications.

Practical Applications 267

The results in Figure 3.6b are instructive because they show how rapid equilibration to steady-state is found at the source (or “sink”) probe, while the pressure at the observation probe, just 15 cm or 6 in away, requires a longer time to approach steady conditions. In fact, the tabulated output below shows that the end source and observation probe pressures are 19,749 and 19,990 psi, with the corresponding pressure drops from the hydrostatic pressure value of 20,000 psi being 251.03 and 10.254 psi. DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s) 0.000E+00 0.240E+02 0.480E+02 0.720E+02 0.960E+02 0.120E+03 0.144E+03 0.168E+03 0.192E+03 0.216E+03 0.312E+03 0.408E+03 0.504E+03 0.600E+03 0.720E+03 0.816E+03 0.912E+03 0.101E+04 0.110E+04 0.113E+04 0.115E+04 0.118E+04

Rate (cc/s) 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01

Ps* (psi) 0.20000E+05 0.19750E+05 0.19750E+05 0.19750E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05

Pr* (psi) 0.20000E+05 0.19991E+05 0.19991E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05 0.19990E+05

Ps**(psi) 0.00000E+00 -0.24994E+03 -0.25032E+03 -0.25048E+03 -0.25058E+03 -0.25065E+03 -0.25070E+03 -0.25074E+03 -0.25077E+03 -0.25079E+03 -0.25086E+03 -0.25091E+03 -0.25094E+03 -0.25096E+03 -0.25098E+03 -0.25100E+03 -0.25101E+03 -0.25102E+03 -0.25103E+03 -0.25103E+03 -0.25103E+03 -0.25103E+03

Pr**(psi) 0.00000E+00 -0.88945E+01 -0.93561E+01 -0.95618E+01 -0.96847E+01 -0.97686E+01 -0.98306E+01 -0.98788E+01 -0.99177E+01 -0.99499E+01 -0.10039E+02 -0.10095E+02 -0.10133E+02 -0.10163E+02 -0.10190E+02 -0.10208E+02 -0.10223E+02 -0.10235E+02 -0.10246E+02 -0.10249E+02 -0.10251E+02 -0.10254E+02

Pr**/Ps** ----------0.35586E-01 0.37377E-01 0.38173E-01 0.38649E-01 0.38973E-01 0.39213E-01 0.39399E-01 0.39549E-01 0.39674E-01 0.40018E-01 0.40233E-01 0.40382E-01 0.40495E-01 0.40602E-01 0.40670E-01 0.40727E-01 0.40776E-01 0.40818E-01 0.40828E-01 0.40837E-01 0.40846E-01

For example, we can assume that the foregoing numbers (also highlighted in red above) are available from dual probe measurements, say taken in the field. Can we obtain permeability from these numbers? One might argue that permeability can already be determined from the inverse “drawdown-only” method of Figure 3.5d, which requires only single probe data. This is not strictly correct, as the prior method will offer only the “spherical permeability” kh2/3kv1/3 when the formation is anisotropic. In order to determine both kh and kv, dual probe measurements are required – in the context of the present example, the dual probe measurements are obtained and needed at different axial flow stations.

268 Supercharge, Invasion and Mudcake Growth

To determine both horizontal and vertical permeabilities, we turn to exact inverse procedure FT-01, which applies to steady-state pressure drops. Again, our tabulated output shows that the end source and observation probe pressures are 19,749 and 19,990 psi, with the corresponding pressure drops from the hydrostatic value of 20,000 psi being 251.03 and 10.254 psi. We enter the pressure drop data

Figure 3.6c. FT-01 input screen. If we click “Solve,” the following output screen immediately appears. This is duplicated in Courier New font. INVERSE KH AND KV LIQUID SOLVER (ZERO SKIN) Copyright (2005), Wilson C. Chin, Ph.D., M.I.T. All rights reserved. Input parameters ... Dip angle ..................... (deg): Source probe delta-p .......... (psi): Observation probe delta-p ..... (psi): Effective source probe radius .. (cm): Probe separation ............... (cm): Volume flow rate ............. (cc/s): Liquid viscosity ............... (cp):

0.4500E+02 -0.2510E+03 -0.1025E+02 0.1000E+01 0.1500E+02 0.1000E+01 0.1000E+01

Practical Applications 269 Source probe delta-p is source probe pressure minus pore pressure .. negative for fluid withdrawal from formation (positive flow rate). It is positive for fluid injection into formation (negative flow rate). Similar definition for observation probe delta-p. Possible solutions ... Tentative permeabilities (md) ... Complex KH root # 1: -.111E+02 + 0.000E+00 i, KV: 0.809E+00 Complex KH root # 2: 0.103E+02 + 0.000E+00 i, KV: 0.953E+00 Complex KH root # 3: 0.878E+00 + 0.000E+00 i, KV: 0.130E+03 KH above strictly valid -- if real part is positive and imaginary part is zero ... sometimes imaginary part is allowed, if small compared to positive real part. KH with negative real part is never correct. CAUTION: KV above is computed using real part of KH even if KH has nonzero imaginary part .... Careful! Exact conclusions below ... Following based on strict adherence to requirements that real(KH)>0 and imag(KH)=0 ..... mathematically correct KH and KV pairs, if shown. Root No. 1: Kh = 0.103E+02 md, Kv = 0.953E+00 md Root No. 2: Kh = 0.878E+00 md, Kv = 0.130E+03 md Multiple permeabilities found, more log data needed.

As noted in Chin et al. (2014), the solution to the inverse problem generally possesses three solutions, since the governing equation (derived from the same parent model as FT-00) is a cubic polynomial. The software determines that two possible roots have been found, as noted immediately above, and we select the one with kh > kv, with “Root No. 1: Kh = 0.103E+02 md, Kv = 0.953E+00 md.” The 10.3 compares favorably with the 10 md assumed for horizontal permeability, while the 0.953 md compares well with the assumed 1 md for vertical permeability. Now, we used t = 1,180 sec late time data for our steady results, which could be an expensive proposition in logging. What if, on studying Figure 3.6b, we decided to use t = 120 sec data? In this case, our source and observation probe pressure drops are 250.65 and 9.7686 psi. In this case, we obtain Root No. 1: Root No. 2:

Kh = 0.108E+02 md, Kv = 0.858E+00 md Kh = 0.799E+00 md, Kv = 0.158E+03 md

The choice, Root No. 1 with kh > kv, gives kh = 10.8 md and kv = 0.858 md, which is satisfactory when compared to the known 10 md and 1 md.

270 Supercharge, Invasion and Mudcake Growth

One can now ask, is there still another way to validate the foregoing horizontal and vertical permeabilities? Or, what if our formation tester were a single probe tool? For these questions, we can still return to the prior drawdown inverse method given in Figure 3.6d, that is

Figure 3.6d. Drawdown inverse method. The data appearing immediately after Figure 3.6b, duplicated here for clarity, shows that DEFINITIONS Time ... Elapsed time (sec) Rate ... Drawdown flow rate (cc/s) Ps* .... Source pressure with hydrostatic (psi) Pr* .... Observation pressure with hydrostatic (psi) Ps** ... Source pressure, no hydrostatic (psi) Pr** ... Observation pressure, no hydrostatic (psi) NOTE: Ps* or Pr* < 0 means volume flow rate cannot be achieved in practice Time (s) 0.000E+00 0.240E+02 0.480E+02 0.720E+02 0.960E+02 0.120E+03

Rate (cc/s) 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01

Ps* (psi) 0.20000E+05 0.19750E+05 0.19750E+05 0.19750E+05 0.19749E+05 0.19749E+05

Pr* (psi) 0.20000E+05 0.19991E+05 0.19991E+05 0.19990E+05 0.19990E+05 0.19990E+05

Ps**(psi) 0.00000E+00 -0.24994E+03 -0.25032E+03 -0.25048E+03 -0.25058E+03 -0.25065E+03

Pr**(psi) 0.00000E+00 -0.88945E+01 -0.93561E+01 -0.95618E+01 -0.96847E+01 -0.97686E+01

Pr**/Ps** ----------0.35586E-01 0.37377E-01 0.38173E-01 0.38649E-01 0.38973E-01

If we click “Run” in Figure 3.6d, and enter the pressure data from t = 0, 24 and 120 sec above, the (black MS-DOS) Worksheet that appears, with the data filled in, is found to be –

Practical Applications 271 INPUTS, PTA-App-01 ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi):

1 1 1 0 20000 24 19750 120 19749

OUTPUT SUMMARY ... Volume flow rate Q1 (cc/s): Pump probe, radius (cm): Probe, geometric factor: Effective radius (cm): 1st Point Time T1 (sec): Pressure P1 (psi): 2nd Point Time T2 (sec): Pressure P2 (psi): 3rd Point Time T3 (sec): Pressure P3 (psi):

1.000 1.000 1.000 1.000 0.000 20000.000 24.000 19750.000 120.000 19749.000

Pore pressure and mobility predicted .. Pore pressure (psi): Spherical mobility (md/cp): FloLineVol*Comp (cm^5/lbf): Compressibility (1/psi):

20000.0000 4.7065 0.1116 0.0173 x (cc/FloLineVol)

Type to close window:

The predicted pore pressure is 20,000 psi and is correct. The “Spherical mobility” or “effective mobility” is predicted as 4.7065 md/cp, which is neither the horizontal mobility of 10 md/cp or the vertical value of 1 md/cp. The spherical mobility associated with “10” and “1” is simply 102/311/3 or 4.6416 md/cp, and the discrepancy in spherical mobility using early time data is only 1%. Of course, spherical mobilities say nothing about kh or kv, so such methods fall short of more powerful dual probe methods for anisotropy determination. Finally, for completeness, we ask if (for job planning purposes) there is a faster way to predict the steady-state pressure drops at source and observation probes needed to run inverse procedure FT-01 without running FT-00? The answer is, “Yes.” To do this, we turn to FT-02, which is an exact analytical model developed to study nonlinear gas flows with a thermodynamic exponent “m.” Clicking on the FT-02 system access button (not shown), we have two menus as shown in

272 Supercharge, Invasion and Mudcake Growth

Figure 3.6e below. One software application applies to inverse problems and is not used here (so it appears partially hidden). The second app is the “Exact Direct Gas Solver” shown in the forefront. Now, Chin et al. (2014) indicates that, if a value of m = 0 is used in this gas application, the system will run and provide exact liquid pressure solutions (fortuitously, the same equations appear and are solved). The inputs shown correspond to those for the present problem. If we click “Solve,” we obtain the tabulated output given immediately after Figure 3.6e.

Figure 3.6e. Exact direct gas solver for dual probe steady flows. DIRECT GAS SOLVER FOR PRESSURES (ZERO SKIN) Copyright (2005), Wilson C. Chin, Ph.D., M.I.T. All rights reserved. Input parameters ... Dip angle ..................... (deg): KV permeability ................ (md): KH permeability ................ (md): Pore pressure ................. (psi): Effective source probe radius .. (cm): Probe separation ............... (cm): Volume flow rate ............. (cc/s): Gas viscosity .................. (cp): Gas exponent m ...... (dimensionless): Newton-Raphson convergence history ...

0.45000E+02 0.10000E+01 0.10000E+02 0.20000E+05 0.10000E+01 0.15000E+02 0.10000E+01 0.10000E+01 0.00000E+00

Practical Applications 273 Iteration 1 2 3 4 5 6 7 8 9 10

Source Psi 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05 0.19749E+05

Pore Psi 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05 0.20000E+05

Observation probe (psi):

0.19990E+05

Note that the steady source probe pressure is calculated as 19,749 psi and the observation probe pressure is 19,990 psi, exactly as we had obtained earlier (see text immediately before Figure 3.6c) using FT-00. FT-00 solved a transient problem, and “marched” the solution to steadystate; although this required additional computing time, the total time was just seconds and not excessive at all. The advantage in using FT-00 is that it provides the time scale needed to achieve steady conditions (for example, see Figure 3.6b) – this time scale is important in job planning and informs drillers how long they may need to perform tests requiring steady state data. Long equilibrium times, of course, affect rig economics. For the present problem, the wait times are not long, although in low mobility applications, that may be hours or days. 3.4 Multiprobe “DOI,” Inverse and Barrier Analysis. There is no reason why “dual probes” should be restricted to only two axially displaced probes, residing along the same azimuth, as offered by several formation tester manufacturers, e.g., as shown at the left of Figure 3.7. In fact, as indicated at the right, observation probes opposite to the (pumping) sink probe can be displaced azimuthally; in addition, SPE Paper No. 36176 provides an example where two axially displaced vertical probes (residing along the same azimuth) are used, so that a total of four probes appear. We are interested in multiple probes as shown in Figure 3.8 for two important reasons. First, the steady anisotropic inverse method of Section 3.3 applies between the pumping probe and any single observation probe; thus, changes in anisotropy along the tool axis direction can be detected – not just vertically, but at any dip angle. Second, the existence of heterogeneities and barriers between receiver probes is easily inferred visually from pressure waveform analysis.

274 Supercharge, Invasion and Mudcake Growth

Figure 3.7. Conventional dual and triple probe testers.

Figure 3.8. Multiple “receiver” formation tester (having multiple spaced observation probes).

Figure 3.9. Transmitter-receiver, receiver-receiver operations modes (see Chapter 2 for phase delay interpretation details).

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To predict horizontal and vertical permeability (or mobility) successfully in the field using dual probe tools in concert with FT-01, it is important that both source and observation probes “see” steady-state conditions. There are two perspectives we address. For the field engineer engaged in job planning, he needs to know the probe spacing. Is it seven inches, two feet or ten feet? Then, some estimate for kh and kv is required, possibly obtained from delineation wells or inferred from other logging measurements. Whether or not steady flow is achievable is easily determined by examining a pressure transient curve.

Figure 3.10. Main FT-00 menu, see bottom right “Run” button.

276 Supercharge, Invasion and Mudcake Growth

Figure 3.11. Depth of investigation, DOI” analysis setup. The tool designer has a different perspective. For example, he must determine the probe spacings in Figure 3.8. Not only is he concerned with the possibility of steady states faraway, but also his ability to pump rapidly enough to obtain a detectable pressure signal. The FT-00 menu in Figure 3.10 provides a rapid means to produce pressure transient curves at pre-set 10 cm distance increments. Pressing the “Run” button at the bottom right of the screen activated a “Depth of Investigation” or “DOI” simulation mode hosted by the menu in Figure 3.11. This screen runs the same math engine but simplified for “drawdown only” inputs.

Practical Applications 277

Once the simulation parameters are entered and the run is initiated, screens automatically appear as shown in Figures 3.12a – 3.12g. For the drawdown-buildup sequence shown in Figure 3.12a, transient flow results up to the inputted 150 cm probe separation distance are given.

Figure 3.12a. Flow rate schedule.

Figure 3.12b. Source probe response.

278 Supercharge, Invasion and Mudcake Growth

Figure 3.12c. Pressure response at 10 cm (3.9 in).

Figure 3.12d. Pressure response at 20 cm (7.9 in).

Practical Applications 279

Figure 3.12e. Pressure response at 20 cm (7.9 in), continued.

Figure 3.12f. Pressure response at 50 cm.

280 Supercharge, Invasion and Mudcake Growth

Figure 3.12g. Pressure response at 90 cm (35 in). Although the required steady state (needed for horizontal and vertical permeability prediction) is achieved quickly at 10 cm, as shown in Figure 3.12c, Figure 3.12d indicates that for typical 7-8 inch dual probe separations, the flow is still transient at 30 sec. What happens at larger times? To answer this question, we re-run the DOI simulator with an extended simulation period, now set to 120 sec (or two minutes) at the bottom right of the DOI screen. Note that calculations are still extremely fast, requiring about 1-2 sec per display including screen plotting for Intel i5 machines. Now, from Figure 3.12e, which simply extends Figure 3.12d in time, we see that for 7-8 inch dual probe tools, a wait time of about 60 sec or one minute would suffice to create steady conditions. At 90 cm or 35 inches, the wait would be substantially increased. While we have addressed anisotropic permeability prediction in the context of FT-01, we stress that the pressure transient curves in Figures 3.12b – 3.12g are also useful in the detection of flow barriers normal to the tool axis. These diagrams show how the pressure curve should successively smooth as we proceed away from the source probe, a consequence of pressure diffusion. Any sudden deviation from the predicted pattern would clearly point to the existence of a heterogeneity that is not assumed in the homogeneous medium math model. For permeability and barrier detection, the observation probe “receiver arrays” in Figures 3.8 and 3.9 are extremely useful.

Practical Applications 281

3.5 Rapid Batch Analysis for History Matching. FT-00 was designed for rapid, accurate, easy-to-use and interactive forward simulations where the user can “randomly” change input variables to ascertain the effects of different variables like permeability, porosity, flowline volume, pumpout schedule, and so on. However, once the main parameters have been identified, a more automated simulation mode is convenient and preferable for history matching applications. One might, for instance, test the effects of different values of compressibility on different choices of permeability (see Figure 3.15 for the parameters that may be used to define “nested” do loops). Figure 3.13, as shown, applies to a single calculation only.

Figure 3.13. FT-00 host simulator.

282 Supercharge, Invasion and Mudcake Growth

Now, suppose we wish to explore the consequences of the drawdown-buildup pumping schedule in Figure 3.13 for a range of other parameters. Clicking the upper left “Batch Runs” button leads to the message in Figure 3.14, noting that the use of ? and ?? in the input fields of FT-00 triggers automated batch mode calculations. Eight parameters are available for single-loop or (at most) double-loop calculations, and the user is free to define minima and maxima as needed. As shown, for single-loops, ten calculations are shown, while for double-loops, a total of “five times five” is implied. These numbers may be changed as desired.

Figure 3.14. Batch mode information message.

Figure 3.15. Loop parameter setup.

Practical Applications 283

For illustrative purposes, let us consider calculations involving kh and kv only. For kh, we assume a range of 1 md to 500 md, while for each of the five selected values of horizontal permeability, we choose five values of kv ranging from 1 md to 100 md. The double-loop batch setup in this case is accomplished by replacing the upper left two permeabilities by ? and ?? as shown in Figure 3.16 below. Clicking “Simulate” leads to the query box in Figure 3.17, which asks if the user wishes to perform pressure transient plots at the end of each loop calculation (each requires 1-3 seconds, depending on the time duration assumed). For now, we select “Yes.” FT-00 will display only plots, with detailed tabulations stored in an output file. For brevity, we will give a few source and observation probe pressure pairs only. If the selected parameters are clearly not suitable, the batch run may be terminated by clicking “Exit Job.” When the batch calculations initiate, each single run is preceded by an input summary as shown in Figure 3.18a. In Figure 3.18a, note the title “Simulation No. 1” and the corresponding initial choice of the permeability pair kh = 1 md and kv = 1 md, as specified by Figure 3.15. Once all the plots associated with Simulation No. 1 are presented, the results for Simulation No. 1 appear. We then have Figures 3.19a,b,c. Finally, after Simulations No. 3-24, we arrive at Figures 3.20 a,b,c for Simulation No. 25, corresponding to the last values of kh = 500 md, kv = 100 md as anticipated from Figure 3.15. A series of runs like that just described starts with low mobility results such as those in Figure 3.18a.b.c and progresses to high mobility results like those in Figures 3.20a,b,c. The changes in the shapes of the pressure transient curves are obvious. These pictorials streamline the tedious work usually assorted with history matching. Again, we emphasize that detailed pressure tabulations are saved and can be viewed at the end of the simulations. Also, for single-loop calculations, where only a single ? would appear in Figure 3.16, the presentation just give is very much unchanged and should likewise aid the user in understand physical trends rapidly.

284 Supercharge, Invasion and Mudcake Growth

Figure 3.16. FT-00 running in automated batch mode (note, ? and ??).

Figure 3.17. Option to view pressure plots.

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Figure 3.18a. Simulation No. 1, input parameters.

286 Supercharge, Invasion and Mudcake Growth

Figure 3.18b. Simulation No. 1, Source probe response.

Figure 3.18c. Simulation No. 1, Observation probe response.

Practical Applications 287

Figure 3.19a. Simulation No. 2, with kh = 1 md again, kv increased.

Figure 3.19b. Simulation No. 2, Source probe response.

Figure 3.19c. Simulation No. 2, Observation probe response.

288 Supercharge, Invasion and Mudcake Growth

Figure 3.20a. Simulation No. 25, last kh = 500 md, kv = 100 md.

Figure 3.20b. Simulation No. 25, Source probe response.

Figure 3.20c. Simulation No. 25, Observation probe response.

Practical Applications 289

3.6 Supercharge, Contamination Depth and Mudcake Growth in “Large Boreholes” – Lineal Flow. The term “supercharge” is often discussed in the context of pressure error measurement in formation testing, that is, inaccuracies in pore pressure prediction as they are affected by overbalance in the nearby well. Although the older literature cites excess pressures in the 200 psi range, more modern field studies indicate that 2,000 psi levels are very likely, especially in the infill drilling of depleted reservoirs. An excellent reference on operational issues is provided in “A New Hostile Environment Wireline Formation Testing Tool: A Case Study from the Gulf of Thailand,” SPWLA 47th Annual Logging Symposium, Veracruz, Mexico, June 4-7, 2006, due to M. Rourke, B. Powell, C. Platt, K. Hall and A. Gardner. Additional information is found at Halliburton.com. Since this excess pressure is an important input to our inverse methods, it is necessary to understand the physics behind the phenomenon. Pressure differentials also play important roles in controlling mudcake thickness, which in turn affects issues in stuck pipe risks and nozzle pad sealing. The level of supercharge is correlated with other events in the well, for example, the depth of contamination due to mud filtrate invasion, the thickness of the mudcake, and relative permeability contrasts between cake and formation. Additionally, we have to account for formation porosity, pore pressure, plus mudcake properties like solid fraction and cake porosity. There are also transient effects involving “moving boundaries” or “fronts,” all of which derive from special solutions to Darcy’s equations. To formulate the reservoir problem properly, we need to distinguish between two geometric limits, as suggested in Figure 3.21. At the left is a flow where “Cake thickness/Hole radius 0” represents the positive cake thickness while “– xc” is the negative location of the caketo-mud interface to the left of the fixed origin x = 0. The term xf > 0 is the location of the moving interface separating the invaded from virgin rock. And lastly, x = L is the fixed location of the far reservoir boundary. Note that both xc(t) and xf(t) are functions of time. The formula for the latter is given Figure 3.22 and depends on the parameters shown. The corresponding cake thickness, following Equation 4-40, is given by the equation xc(t) = [ eff fs/{(1- c)(1-fs)}] (xf - xf,o). These formulas provide the desired results for filtrate (contamination) front position and mudcake thickness as functions of the overbalance (pm-pr) and time from initial invasion.

Figure 3.21. Mudcake thickness and hole radius considerations.

Practical Applications 291

DYNAMICALLY COUPLED LINEAL FLOW "1" p

where:

"3" p

mud x = -x

Exact:

"2"

c

x=0

x=x

f

res

x=L x

xf (t) = - H + {H2 + 2(Hxf,o + ½ xf,o2 + Gt)} G = - {k1(pm-pr)/( f eff )}/ { ok1/ f k3 - k1/k2 - eff fs /{(1- c)(1-fs)}} H = [xf,o eff fs /{(1- c)(1-fs)} - ok1L/ f k3] /{ ok1/ f k3 - k1/k2 - eff fs /{(1- c)(1-fs)}}

Nomenclature: xc ... xf ... xf,o ... L ... pm ... pr ... eff ...

c ... k1 ... k2 ... k3 ...

Transient cake thickness Transient invasion front Initial displacement (i.e., spurt) Lineal core length Constant mud pressure Pmud Constant reservoir pressure Pres Effective rock porosity Mudcake porosity Mudcake permeability to filtrate Rock permeability to filtrate Rock permeability to “oil”

f ... Mud filtrate viscosity o ... Viscosity of “oil” or formation fluid fs ... Mud solid fraction

Figure 3.22. Exact lineal invasion solution (Chin et al., 1986).

292 Supercharge, Invasion and Mudcake Growth

Time-dependent pressure distributions. The corresponding pressure distributions are easily evaluated. We will provide the procedure and leave details to the interested reader. Since there are three distinct regions of flow, and since the flows are assumed to be incompressible, each region is governed by a Darcy pressure equation of the form d2p(x)/dx2 = 0 with simple linear solutions in the form “ax + b.” In Figure 3.22, Region 1 (the mudcake) is defined by – xc(t) < x < 0, with Region 2 (the invaded zone) by 0 < x < xf(t) while Region 3 (the uninvaded virgin rock) satisfies L > x > xf (t). Thus, we can write the corresponding solutions in the form p1(x) = Ax + B, p2(x) = Cx + D and p3(x) = Ex + F. The six constants of integration A, B, C, D, E and F are determined by six conditions, namely, the end boundary pressures (i) p1(– xc(t)) = Pmud and (ii) p3(L) = Pres, the interfacial matching conditions for pressure, that is, (iii) p1(0) = p2(0) and (iv) p2(xf) = p3(xf ), plus interfacial matching conditions for velocity, that is, (v) k1/ 1 dp1(0)/dx = k2/ 2 dp2(0)/dx and (6) k2/ 2 dp2(xf )/dx = k3/ 3 dp3(xf )/dx. We remind the reader that this model assumes regions separated by distinct interfaces, which may be true initially, or whenever invasion speeds are rapid. In general, multiphase models are required, and then, the physics will depend on the miscible or immiscible nature of the flow. Chin and Proett (2005) provide detailed solutions and examples for such flows, and recent improvements to these are discussed in the prior chapter.

3.7 Supercharge, Contamination Depth and Mudcake Growth in Slimholes or “Clogged Wells” – Radial Flow. The identical problem describing supercharge, contamination depth, mudcake growth and transient pressure evolution, but for the slim holes shown in Figure 3.21, is similarly solved. The top diagram in Figure 3.22 applies, but with the cross-section now representing slices taken at a given angle down a cylindrical well. The incompressible cylindrical radial equation is d2p(r)/dr2 + (1/r) dp/dr = 0 and has the solution form “p(r) = a log r + b.” We may thus write p1(r) = A log r + B, p2(r) = C log r + D and p3(r) = E log r + F where these constants differ from those above. Analytical solutions for cake growth and filtrate front motion are given in a subsection of Section 4.3 entitled “Simultaneous mudcake buildup and filtrate invasion in a radial geometry (liquid flows).” The solution process for transient pressure and depth of contamination proceed in exactly the same manner as described in Section 3.6 above.

Practical Applications 293

Both of the lineal and radial solutions in Sections 3.6 and 3.7 were motivated by the Catscan experiments discussed originally in Chin et al. (1986). The work was motivated by the desire to understand the role of supercharging in affecting mudcake growth and the filtrate invasion that accompanies it. The radial flow test cell is shown in Figure 3.23a while a cross-section of the radial invasion profile appears in Figure 3.23b. The black outer ring represents the virgin rock, while the white ring shows the invaded zone saturated with mud filtrate. The thin mudcake, at the smallest radius next to the white hole, is barely perceptible at the scales shown in this photograph.

Figure 3.23a. Radial flow Catscan test vessel.

Figure 3.23b. Catscan invasion in radial core sample (inner invaded white zone displacing outer dark fluid shown).

294 Supercharge, Invasion and Mudcake Growth

3.8 References. Chin, W.C., Formation Testing: Supercharge, Pressure Testing and Contamination Models, John Wiley & Sons, Hoboken, New Jersey, 2019. Chin, W.C., Zhou, Y., Feng, Y. and Yu, Q., Formation Testing: Low Mobility Pressure Transient Analysis, John Wiley & Sons, Hoboken, New Jersey, 2015. Chin, W.C., Zhou, Y., Feng, Y., Yu, Q. and Zhao, L., Formation Testing: Pressure Transient and Contamination Analysis, John Wiley & Sons, Hoboken, New Jersey, 2014. Chin, W.C., Suresh, A., Holbrook, P., Affleck, L., and Robertson, H., “Formation Evaluation Using Repeated MWD Logging Measurements,” Paper No. U, SPWLA 27th Annual Logging Symposium, Houston, TX, June 9-13, 1986. Rourke, M., Powell, B., Platt, C., Hall, K. and Gardner, A., “A New Hostile Environment Wireline Formation Testing Tool: A Case Study from the Gulf of Thailand,” SPWLA 47th Annual Logging Symposium, Veracruz, Mexico, June 4-7, 2006.

4 Supercharge, Pressure Change, Fluid Invasion and Mudcake Growth In Section 2.3.6.3, “Supercharging summaries - advanced forward and inverse models explored,” we gave a mathematically elegant model for the “forward” or “direct” problem with supercharge effects considered, that is, the formulation where pressures are calculated when flow rate, tool, fluid and formation properties are given. We also solved the corresponding “inverse” or “indirect” problem, which predicts permeability (or, equivalently, mobility), compressibility and pore pressure, when any three “pressure, time” data points are given. Conventional zero supercharge model. The formation testing book by Chin et al. (2014) provides a number of zero supercharge forward and inverse formulations in different physical limits together with exact solutions. One particular model, quite general, is given by Equations 5.1– 5.4 in that reference, re-numbered in Section 2.3.6.3 as, 2P(r,t)/ r2 + 2/r P/ r = ( c/k) P/ t (A1) P(r,t = 0) = P0

(A2)

P(r = ,t) = P0

(A3)

(4 Rw2k/ ) P(Rw,t)/ r – VC P/ t = Q(t)

(A4)

295

296 Supercharge, Invasion and Mudcake Growth

Equations A1 – A4 represent the complete isotropic, zero skin, spherical Darcy flow formulation for compressible liquids, which was solved exactly in terms of complex complementary error functions. This formulation was used to develop the forward simulation FT-00, as well as the inverse procedures FT-01 and FT-PTA-DDBU. Note that “spherical” implies mathematical idealization. The foregoing equations do not handle supercharge. The spherical source of radius Rw will not adequately describe borehole wall curvature or the effects of tester pads – to account for these non-idealizations, Rw often denotes the product of a corrective “geometric factor” (that is, “G,” determined empirically or through 3D finite element analysis) and the true nozzle radius. Supercharge model. Now, we ask how “supercharging,” where the term will be used to represent both overbalanced and underbalanced effects, can be modeled by extending the framework underlying Equations A1 – A4 (in either case, invasion through the cylindrical borehole surface is assumed to have slowed significantly). Our approach is straightforward. We solve these equations with a single important change, altering only the initial pressure condition as shown in Equation B2, highlighted in red. In particular, we consider the model

2P(r,t)/ r2 + 2/r P/ r = ( c/k) P/ t P(r,t = 0) = P0 + Z/r, Z > 0, R > Rw

(B1) (B2)

P(r = ,t) = P0

(B3)

(4 Rw2k/ ) P(Rw,t)/ r – VC P/ t = Q0

(B4)

Here Z = Pbh – P0 may be positive (overbalanced) or negative (underbalanced) with any magnitude, Pbh being the borehole pressure just outside the reservoir sandface, and P0 being the farfield reservoir pore pressure. In older references, “Pbh – P0” is often quoted in the 200-250 psi range. In fact, Halliburton and Chevron Thailand, as discussed previously, observe that overbalances exceeding 2,000 psi are not uncommon, particularly in infill drilling where the reservoir is depleting. For such problems, the use of inverse models based on Equations A1 – A4 will lead to incorrect pore pressures, mobilities and compressibilities. Again, the foregoing formulations refer to the problem illustrated in Figure 4.1a. The solution to Equations B1 – B4 was given in an earlier chapter with examples and will not be repeated here.

Supercharge, Pressure, Invasion and Mudcake Growth 297

Figure 4.1a. Supercharge problem in formation testing.

Figure 4.1b. Linear flow Catscans, thin dark mudcake at center of core and invasion front at density contrast (flow, left to right).

298 Supercharge, Invasion and Mudcake Growth

Relevance to formation tester job planning. In this chapter, we introduced basic ideas central to formation testing, that is, supercharging, formation invasion, pressure diffusion and mudcake growth, all of which are related dynamically. Although the analytical solutions we give will shed considerable insight into which groups of parameters are important, closed form solutions are inherently limited in what they can offer to field practitioners. In formation testing, not only is supercharging relevant to the prediction of permeability (or, mobility) and pore pressure, but the pressure differential associated with overbalance is responsible for mudcake growth, ultimate slowdown and eventual blockage in the hole. Mudcake growth, illustrated in Figure 4.1b in Catscan experiments performed in Chin et al. (1986), is both bad and good. Thick cakes are undesirable because they are responsible for stuck pipe and, of course, stuck formation testers – the diagram in Figure 4.1c illustrates one mechanical concept that has been realized in hardware to reduce the possibility of tool loss. An excellent discussion on stuck tool remediation appears in “Development on Incongruous Pushing and Stuck Releasing Device of EFDT,” by X. Qin., Y. Feng, W. Song, X. Chu. and L. Wang., published in Journal of China Offshore Oilfield Technology, Vol. 4, No. 1, April 2016, pp. 70-74. The authors analyze the causes of differential pressure sticking during openhole wireline logging. Their modular IPSRD releasing device, designed for the EFDT formation tester applications, could be seamlessly assembled to the tool. “Stuck Release Arms” (SRA) are driven by hydraulic forces that free the dual probe tool from adhesive mud forces. The methods in the present chapter are useful in predicting mudcake growth in different borehole and overbalance environments.

Figure 4.1c. Stuck tool removal mechanism.

Supercharge, Pressure, Invasion and Mudcake Growth 299

While thinner mudcake are desirable insofar as tool loss considerations are concerned, thicker cakes are necessary for nozzle pads to seal effectively against the formation – this is shown in the same Figure 4.1c. For these reasons, we are emphasizing the importance of mudcake dynamics in this book and its central role in job planning. The present chapter deals with useful math solutions, in particular, those amenable to closed form analysis, while the models in Chapter 5 consider numerical methods. These are easily to operate at the rigsite and easily modified to incorporate real-world effects. For instance, flow barriers may be introduced with minor work, and the growth-limiting effects of dynamic erosion may be incorporated by a single Fortran “if, then” logical statement. Importantly, in this book, we explain how the dimensionless parameters obtained from simple filtration tests (on filter paper) can be used in sophisticated math models to predict mudcake behavior in boreholes with arbitrary radii, and also, predict corresponding pressure drops and invasion front displacements. Refined models for supercharge invasion. Despite the apparent generality shown in Figure 4.1a, the title of the present chapter, namely “Supercharge, Pressure Change, Fluid Invasion and Mudcake Growth,,” suggests that there is much more behind the notion of supercharge than the single word itself. In fact, this is exactly the case. In the present chapter and the next, we will explore refined descriptions and mathematical solutions for the supercharging process. Essentially, “supercharge” refers to the increase in pressure that results when fluid invades the reservoir. However, the processes are numerous and varied. For instance, “single phase fluid invasion” refers to displacements into the reservoir due to identical fluids, e.g., “fresh water into fresh water.” “Multiphase invasion” may be “miscible” or “immiscible.” Miscible may refer to “fresh water and salty water” or “oil mud into formation oil.” Immiscible displacements refer to processes in which oil and water do not mix, that is, those where surface tension and capillary effects are important. Then again, the borehole environment must be considered. Does mudcake exist at the sandface? If so, has it reached equilibrium thickness? Or is its growth ongoing, hence, with corresponding unsteady invasion front motions? As if these questions were not enough, we might ask if the situation under consideration can be modeled by linear flow or radial flow, compressible versus incompressible fluid effects, pad nozzle or straddle packer geometries, and so on. We will introduce this vast subject area in this book using a suite of idealized problems.

300 Supercharge, Invasion and Mudcake Growth

4.1 Governing equations and moving interface modeling. Petroleum engineers use partial differential equation models to simulate reservoir flows, to interpret well tests, to characterize formation heterogeneities, and to assist in infill drilling planning and secondary recovery. Many hierarchies for fluid flow modeling exist, ranging from simple single-phase oil alone or “gas only” flows to multiphase descriptions, encompassing both miscible and immiscible limits, to black oil and compositional models. In this book, we will address all but the latter two flow models. Since invasion modeling requires a very different perspective than that taken in elementary fluid mechanics, it is worthwhile to recapitulate the basic governing equations. Single-phase flow pressure equations. Fluid flows are governed by partial differential equations. For example, single-phase flows of constant density liquids in homogeneous, isotropic media satisfy Laplace’s equation for pressure, 2

p/ x2 +

2

p/ y2 +

2

p/ z2 = 0

(4-1)

an elliptic differential equation, while a slightly compressible liquid in the same medium satisfies 2

p/ x2 +

2

p/ y2 +

2

p/ z2 = (

c/k) p/ t

(4-2)

In Equations 4-1 and 4-2, the fluid pressure p(x,y,z,t) is the unknown dependent variable, and x, y, z, and t represent independent variables. The standard symbol denotes partial derivatives (subscripts will also be used). The quantities , , c, and k denote rock porosity, fluid viscosity, fluid-rock compressibility, and isotropic formation permeability, respectively. Equation 4-2 is the classical parabolic, diffusion or heat equation for pressure, so-called because it was first derived and solved in the context of heat transfer (Carslaw and Jaeger, 1946). On the other hand, for gas flows, under the same assumptions, 2pm+1/ x2 + 2pm+1/ y2 + 2pm+1/ z2 = = ( c*/k) pm+1(x,y,z,t)/ t

(4-3) where m is a nonzero exponent. In Equation 4-3, the pressuredependent, “compressibility-like” quantity c* = m/p(x,y,z,t) renders the boundary value problem nonlinear.

(4-4)

Supercharge, Pressure, Invasion and Mudcake Growth 301

DYNAMICALLY COUPLED LINEAL FLOW "1" p

where:

"3" p

mud x = -x

Exact:

"2"

c

x=0

x=x

f

res

x=L x

xf (t) = - H + {H2 + 2(Hxf,o + ½ xf,o2 + Gt)} G = - {k1(pm-pr)/( f eff )}/ { ok1/ f k3 - k1/k2 - eff fs /{(1- c)(1-fs)}} H = [xf,o eff fs /{(1- c)(1-fs)} - ok1L/ f k3] /{ ok1/ f k3 - k1/k2 - eff fs /{(1- c)(1-fs)}}

Nomenclature: xc ... xf ... xf,o ... L ... pm ... pr ... eff ...

c ... k1 ... k2 ... k3 ...

Transient cake thickness Transient invasion front Initial displacement (i.e., spurt) Lineal core length Constant mud pressure Pmud Constant reservoir pressure Pres Effective rock porosity Mudcake porosity Mudcake permeability to filtrate Rock permeability to filtrate Rock permeability to “oil”

f ... Mud filtrate viscosity o ... Viscosity of “oil” or formation fluid fs ... Mud solid fraction

Figure 4.2. Exact lineal invasion solution (Chin et al., 1986).

302 Supercharge, Invasion and Mudcake Growth

From a practical viewpoint, this means, say in well test interpretation, that superposition methods do not apply: the sum of individual solutions (for instance, that of a drawdown and a buildup) is itself not a solution. Analytically, nonlinearity implies that the possibilities for closed-form solutions are rare. But our solution methods for compressible flows, even for linear problems, will be numerical anyway. The nonlinear equation form as shown in Equation 4-3, first given by Chin (1993a,b), simply recasts Muskat’s exact equation in an analogous form preferable for numerical analysis (Muskat, 1937). The resulting equation is linear-like in appearance and thus allows us to readily adapt working linear numerical schemes to nonlinear problems. We observe, from classical heat transfer and fluid mechanics, that the constant exponent m describes the thermodynamics of the gas motion. In particular, it is known that m = 1, for isothermal expansion = Cv/Cp, for adiabatic expansion = 0, for constant volume processes = , for constant pressure processes

(4-5a,b,c,d)

where Cp is the specific heat at constant pressure, and Cv is the specific heat at constant volume. In inhomogeneous, anisotropic media, petroleum liquids are governed by (kx p/ x)/ x + (ky p/ y)/ y + (kz p/ z)/ z =

c p/ t

(4-6) where kx, ky, and kz are nonuniform permeabilities in the x, y, and z directions. Analogous equations for gases appear in Chin et al. (2017). There is much more to formation invasion modeling than Darcy’s law q = - (k/ ) p/ x. Equations 4-1 to 4-6 are derived using Darcy’s law, a low Reynolds number approximation to the Navier-Stokes momentum equations, in conjunction with a requirement for mass conservation. It is almost never correct to approach simulation by simply setting, say q = - (k/ ) p/ x = constant, to solve a problem, since this does not account for the underlying lineal, radial, or spherical geometry, for mass conservation, or for pressure boundary conditions. Yet, this is often done in the petroleum literature: Darcy fluid mechanics requires the solution of pressure boundary value problems, and with initial conditions in unsteady formulations.

Supercharge, Pressure, Invasion and Mudcake Growth 303

Problem formulation. Partial differential equations such as those in Equations 4-1 to 4-6 require auxiliary conditions that fix any and all degrees of freedom. Just as the ordinary differential equation

d2p(x)/dx2 = 0

(4-7a)

whose solution p(x) = Ax + B

(4-7b)

requires two boundary conditions to determine the constants A and B, boundary value problems require analogous boundary conditions, but specified along physical curves. In addition, for problems characterized by obvious time dependencies, initial conditions will also be needed. For the most part, the exposition in the remainder of this book, at least for analytical models, requires the reader only to appreciate the fact that the ordinary differential equation d2p(r)/dr2 + (1/r) dp/dr = 0

(4-8a)

for cylindrical radial flow has the solution p(r) = A log r + B

(4-8b)

(all logarithms in this book are natural logarithms), while the slightly altered “spherical flow” model d2p(r)/dr2 + (2/r) dp/dr = 0

(4-9a)

has the exact solution p(r) = A/r + B

(4-9b)

These solutions can be easily validated by back-substitution, but we emphasize that the arbitrary constants of integration A and B will vary from problem to problem. Equations 4-7 to 4-9 play important roles in constant density invasion problems. Again, Equations 4-1 to 4-6 model single-phase flows only, and as indicated, we will also address miscible flows where viscous diffusion is important and immiscible two-phase flows where capillary pressure and relative permeability cannot be ignored. In order to keep our early discussions elementary for now, we will defer the development of the latter models for now. Eulerian versus Lagrangian description. Equations such as those just given predict pressure as a function of x, y, z, and t. Once pressure solutions are available, the rectangular Darcy velocities are obtained as

304 Supercharge, Invasion and Mudcake Growth

u(x,y,z,t) = - (kx/ )

p/ x

(4-10)

v(x,y,z,t) = - (ky/ )

p/ y

(4-11)

w(x,y,z,t) = - (kz/ )

p/ z

(4-12)

The Eulerian velocities in Equations 4-10 to 4-12 represent speeds measured at a point in space (x,y,z). This description is useful to reservoir engineers because it provides flow rates at production wells, transient pressures at specific wells for history matching, and pump rates and pressures needed at injectors, among other quantities of interest. If these velocities do not vary with time, the flow is said to be steady. Otherwise, it is transient or unsteady. Thus, a constant velocity, single fluid flow through a linear core is steady, because it appears unchanged from one instant of time to the next, recognizing that numerous fluid elements are actually streaming through the pore spaces at any time. On the other hand, the Eulerian frame of reference is not ideal or convenient for every application. For example, radioactive and chemical tracers are often introduced into injection wells and monitored at production wells in order to study reservoir connectivity and sweep efficiency. This usage addresses the question of where a marked fluid element (or a tagged group of particles) is heading, an objective that requires us to follow the fluid. This is also important in environmental engineering, where the destinations of contaminants, as well as their origins and travel times, are of interest. For these purposes, a Lagrangian description is more suitable. Formation invasion, which deals with traveling fluid fronts, requires Lagrangian solutions to the equations of fluid motion which are not normally studied in college courses. Constant density versus compressible flow. A constant density, or incompressible fluid is a fluid consisting of elements that are not unlike infinitely rigid balls. Any disturbance to a single element is, therefore, instantaneously transmitted throughout the entire field of flow, so that the speed with which information propagates is infinite. Compressible fluids are characterized by elasticity. A fluid element that is disturbed will respond with minute volume changes and finite delay, before passing the disturbance to its neighbors. In borehole annular flows and drillpipe flows, sudden motions manifest themselves as sound waves governed by hyperbolic equations. In reservoirs supporting Darcy flow, compressibility allows pressure disturbances to slowly diffuse, similar to temperature diffusion in solids. Hence, petroleum engineers often model fluid flow and compressibility using heat equation models.

Supercharge, Pressure, Invasion and Mudcake Growth 305

Steady versus unsteady flow. In elementary reservoir flow analysis, simple single-phase flows are considered, that is, flows containing one and only one fluid species throughout the reservoir medium. For such constant density, incompressible flows, whenever applied pressures remain constant in time, the Eulerian pressure fields are steady and result in steady-state formulations. Only when the effects of compressibility are allowed in a single-phase flow can reservoir transients exist; thus, compressible flows can be both transient and steady. Other transients of importance are introduced in the drilling process. For example, changes in the mud used are often accompanied by changes in wellbore pressure and invading fluid viscosity, effects that can (and must) be modeled both analytically and numerically. Not all constant density flows are steady, of course. In two-phase flows of immiscible fluids, for example, in the Eulerian flow obtained by displacing oil with water, the relative saturations of each phase within any particular pore space will vary with time, and then, with position. This type of unsteadiness also exists when the two-phase flow is compressible. For rapid displacements, piston-like, plug, or slug-like flows will result, and the displacement process can be modeled by monitoring the progress of two different single-phase flows separated by a moving interface. (Later we will explain dimensionlessly what is meant by “rapid” or “slow.”) When fresh water displaces saline formation water of like viscosity in a core without mudcake under constant pressure, the flow is steady; but when water displaces oil, or conversely, the flow is unsteady because “total viscosity” changes with time. It is important to understand that different flow limits exist, which will be modeled by different boundary value problem formulations and fluid properties. In single-phase flows, it suffices to consider viscosity, but in two-phase flows, we require concepts like relative permeability and capillary pressure. The flow of a single fluid, which may be steady in an Eulerian description, is unsteady in a Lagrangian model because the individual fluid elements being monitored are always in motion. For this reason, the Eulerian relationships in Equations 4-1 to 4-12 provide only a partial solution to the Lagrangian problem. To complete the description, we turn to kinematic considerations that treat moving fronts and interfaces as distinct physical entities. Incorrect use of Darcy’s law. Darcy’s law states that the local velocity q in a direction s is given by the directional derivative q = - (k/ ) p/ s, where p is the transient or steady pressure, with k and

306 Supercharge, Invasion and Mudcake Growth

representing permeability and viscosity. Thus, in a lineal flow, we have q = - (k/ ) p/ x, whereas in a cylindrical or spherical radial flow, we have q = - (k/ ) p/ r, r being the radial variable. Equations 4-10 to 4-12 apply to three-dimensional flows in rectangular or Cartesian coordinates. Darcy’s law, a low Reynolds number approximation to the Navier-Stokes equations, does not embody the complete physical description of the invasion process. For example, it does not describe mass conservation. Only when the latter requirement is invoked, do we obtain partial differential equations for pressure such as Equations 4-1, 4-2, 4-3, or 4-6. These are solved with pressure (or Dirichlet) or flow rate (or Neumann) boundary conditions applied to inlet and outlet faces. Unfortunately, a number of published invasion models actually take (k/ ) p/ x = - q(t) as both starting and end points, where q(t) is specified, leading to an anticipated (but often incorrect) linear variation in pressure. Such approaches do not account for the cylindrical and spherical radial geometric spreading that Equations 4-8a and 4-9a automatically embody. Their results fail to satisfy these equations or their appropriate extensions, for example, it is clear from substitution that dp/dr = - q(t)/k does not satisfy d2p(r)/dr2 + (1/r) dp/dr = 0. Still other analyses invoke a “universal” t law at the outset, failing to appreciate that this limiting result, correctly derived decades ago by Outmans (1963), applies to lineal flows only – and then, only when cake compaction is insignificant, fluid compressibility is ignored, and formation permeability is high. Because all of this confusion proliferates throughout the invasion literature, this book will list all underlying assumptions used whenever new formulas are derived. We will state limitations and strengths clearly, and carefully document all the steps needed to arrive at solutions and numerical algorithms. Moving fronts and interfaces. The kinematics of moving fronts and interfaces has been studied in different physical contexts for over two hundred years. Most notable are the studies of free surfaces in ocean hydrodynamics and vortex sheets in free space (e.g., see Lamb, 1945), and more recently, flame propagation dynamics in combustion analyses. The following derivation, which applies to fluid fronts in porous media, is given in Chin (1993a). Let us consider a moving front or interface located anywhere within a three-dimensional Darcy flow (e.g., any surface marked by red dye), and let (x,y,z) denote the porosity. Furthermore, denote by u, v, and w the Eulerian speed components, and describe our interface by the surface locus of points –

Supercharge, Pressure, Invasion and Mudcake Growth 307

f(x,y,z,t) = 0

(4-13)

An interface, as in Figure 4.3, is defined by the kinematic property that fluid does not cross it. Hence, the velocity of the fluid normal to the interface must be equal to the velocity of the interface normal to itself. The velocity perpendicular to the surface, from vector algebra, is equal to -ft(x,y,z,t)/ (fx2 + fy2 +fz2), while the quotient (ufx + vfy + wfz)/ { (x,y,z) (fx2 + fy2 +fz2)} is the normal velocity of the fluid. The condition that these be equal is therefore given by the equality f(x,y,z,t)/ t + (u/ ) f/ x + (v/ ) f/ y + (w/ ) f/ z = 0 (4-14)

Figure 4.3. Any surface f(x,y,z,t) = 0 in a reservoir. Now, from calculus, the total differential df of any function f(x,y,z,t) is df = f/ t dt + f/ x dx + f/ y dy + f/ z dz

(4-15)

Division by dt yields df/dt = f/ t + dx/dt f/ x + dy/dt f/ y + dz/dt f/ z

(4-16)

Comparison with Equation 4-14 shows that we can set the so-called substantive, material, or convective derivative df/dt to zero, that is, df/dt = 0

(4-17)

provided we require that dx/dt = u(x,y,z,t)/ (x,y,z)

(4-18)

dy/dt = v(x,y,z,t)/ (x,y,z)

(4-19)

dz/dt = w(x,y,z,t)/ (x,y,z)

(4-20)

308 Supercharge, Invasion and Mudcake Growth

Thus, along the fluid particle trajectories defined by Equations 4-18 to 4-20, the function f(x,y,z,t) must be constant, since Equation 4-17 requires df/dt = 0. This proves that particles on a surface remain on it. Equations 4-18 to 4-20 define the fluid fronts and interfaces, but f(x,y,z,t) no longer plays an active role. Now, the Eulerian velocities u, v, and w in these trajectory equations are determined from a separate and 2 2 2 2 2 2 independent host formulation, for example, p/ x + p/ y + p/ x = 2 2 2 2 2 2 ( c/k) p/ t, or p/ x + p/ y + p/ x = 0, or still another flow model. While x, y, and z represent independent variables in the host Eulerian formulation, in the trajectory equations, the same x, y, and z become the dependent variables, with time now being independent. This role reversal, typically resulting in complicated mathematics, is standard in invasion modeling. The trajectory equations form coupled, nonlinear, ordinary (not partial) differential equations. When they are integrated in time, with all starting positions prescribed, the corresponding particles can be followed as they travel; final results include positions and travel times along the pathlines or streamlines. Also, when u(x,y,z,t), v(x,y,z,t), and w(x,y,z,t) are available, Equations 4-18 to 4-20 can be integrated backwards in time, under limited circumstances, to determine the origin of a particle or a group of particles. Observe that, while the solution of 2p/ x2 + 2p/ y2 + 2p/ x2 = 0, subject to pressure boundary conditions, does not involve the porosity (x,y,z), the Lagrangian description for the same flow does, as is clear from Equations 4-18 to 4-20. That this must be so is intuitively obvious. If we consider the steady-state flow through a kitchen sponge, it is clear that individual fluid particles must travel faster through smaller pore spaces in order to maintain the overall steady flow. Similarly, the traffic speed at the narrowed neck of a steady-state highway flow must exceed that found along multilane stretches. (If it does not, as is usually the case, it is because the flow is unsteady!) The changes in thinking here are critical to invasion modeling, but this philosophy aside, the algebraic manipulations required to produce sought solutions are relatively simple, although at times, quite tedious. For this reason, intermediate steps are retained. The primary products of this book are analytical results and algorithms, plus new philosophies and methodologies. Use of effective properties. While the fluid displacement process can be quite complex, for example, as in immiscible two-phaseflow mixing, very often, a piston-like, slug, or plug-like description suffices, at least in offering a qualitative but quantifiable model as the

Supercharge, Pressure, Invasion and Mudcake Growth 309

basis for preliminary discussion. For such flows, we can simplify the physical picture and formulate the problem as two idealized single-fluid regions separated by a distinct moving interface or displacement front such as that described above. This boundary is known as a mathematical discontinuity, across which certain physical quantities are conserved and other kinematic requirements enforced. Let us specialize our ideas to the displacement of hydrocarbons by water. The first region, ahead of the front, contains oil or gas together with immobile connate water. Behind the front is the second region, consisting of invading mud filtrate, and immobile residual oil (or gas) left behind the front by displaced hydrocarbons. The symbols denoting connate water and residual oil saturations are Sc and Sro (thus, the initial oil saturation is 1 - S c). If the geometric rock porosity is denoted by , then an effective porosity eff characterizing the invaded rock can be introduced with the definition eff = (1- Sro - Sc ). This definition can be used for the front trajectories defined by Equations 4-18 to 4-20. Again, this provides an approximate description for a rapid displacement process (later, immiscible two-phase flow theory will provide more precise simulations). Note that different definitions of porosity exist in well logging, depending on the type of instrument used for measurement. In this book, by porosity, we mean that associated with the connected pore spaces available for fluid transport, since these pore spaces are the ones implied by the equations governing fluid motion. While the rock under investigation may be uniform and homogeneous, characterized by a single permeability, we will at times derive our formulas allowing for two permeabilities, so that the results can be used on an ad hoc basis in modeling different permeabilities relative to different formation fluids. This usage is convenient in describing the differences in the flow of water relative to residual oil, versus the movement of oil relative to immobile connate water. This flexibility is consistent with our use of effective porosity and is again offered for convenience only. Finally, while we have emphasized the possibility of two coexisting formation fluids separated by dynamic interfaces, we stress that our results also apply to the case of a single fluid which we may envision as red water displacing blue water. This final example is useful in modeling the displacement of saline formation waters by invading mud filtrate, at least for short initial times, before ionic diffusion smears the separation boundary; it is important to resistivity interpretation applications.

310 Supercharge, Invasion and Mudcake Growth

4.2 Static and dynamic filtration. Here we introduce the ideas underlying quantitative formation invasion modeling, but restrict ourselves to isotropic Darcy flows dealing with piston-like, slug or plug-like displacements. Miscible and immiscible multiphase flows are considered once simpler techniques have been developed and their limitations are understood. We start with the simplest problems, tracking fluid fronts in cores without mudcakes, progress to mudcake-alone formulations, and finally to problems where the dynamics of the invasion front and the timewise growth of the mudcake are closely coupled. In this last class of problems, the enormous analytical complexities confronting mathematicians are aptly highlighted, complications that occur despite the simplicity of the fluid model used. Hence, we will be motivated to look for numerical methods that provide greater modeling flexibility, that is, have potential for greater expansion as we attempt to simulate invasion problems that more closely model reality. Computational finite difference methods are introduced in Chapter 5, where they are at first applied to the problems analytically addressed here. However, we extend these algorithms to classes of physical problems where the possibilities for closed-form solution are unlikely. We emphasize that the invasion solutions presented in this book also apply when the flow direction is reversed. For example, they are useful whether drilling overbalanced or underbalanced; they model influx into the wellbore from the formation once the obvious sign change in the pressure differential is made. Dynamic filtration in the borehole, when fluid flows parallel to the hole axis, is discussed, and the coupling of non-Newtonian annular to Newtonian reservoir flow is developed, in Chin (2017). 4.2.1 Simple flows without mudcake.

In this section, we study single-phase flow invasion into rocks, assuming that the influence of mudcake is negligible; the results model, for example, the use of brine or water as the drilling fluid without solids deposition. The formulations for such problems are simple; they highlight the basic differences between the reservoir flow problems covered in undergraduate curricula and the Lagrangian models needed to track moving fronts. Five problems appear in order of increasing complexity.

Supercharge, Pressure, Invasion and Mudcake Growth 311

Homogeneous liquid in a uniform linear core. The pressure partial differential equation governing transient, compressible, lineal, homogeneous, liquid flows having constant properties is 2p(x,t)/ x2 = ( c/k) p/ t. Here p is pressure, while x and t represent space and time; , k, , and c are rock porosity, rock permeability, fluid viscosity, and net fluid-rock compressibility, respectively. If we assume a constant density, incompressible flow, and ignore the compressibility of the fluid by setting c = 0, the right side of this equation identically vanishes. Then, the model reduces to the ordinary differential equation d2p(x;t)/dx2 = 0 where t is a parameter as opposed to a variable. P r

P l

Figure 4.4. Lineal flow. The solution to this equation is simply p(x;t) = Ax + B. In order to determine the integration constants A and B, boundary conditions for the pressure p(x;t) are required. Let us suppose that the left-side pressure at x = 0 is Pl, while the right-side pressure at x = L is Pr, as shown in Figure 4.4. That is, we take p(0;t) = Pl and p(L;t) = Pr, which completely determine A and B. If Pl and Pr are constants, then A and B are constants. However, if either or both are functions of time, as may be the case in drilling, then A and B may be functions of time. In this case, the pressure field p(x;t) = (Pr - Pl) x/L + Pl responds immediately to time changes in boundary pressure, since zero compressibility, implying absolute fluid rigidity, requires the instantaneous transmission of information. Henceforth we will omit t in the argument of pressure whenever we deal with constant density flows, understanding that time dependencies are parametrically allowed when warranted. The explicit use of t, as in p(x,t), will be reserved for transient compressible flows only. Now, the fluid velocity q is given by Darcy’s law q = - (k/ ) dp(x)/dx, which, in view of our solution, becomes q = - (k/ )(Pr - Pl)/L. This describes the fluid velocity at a fixed point in space. It is the velocity that an observer fixed to a particular pore space element measures (in our convention, we assume q > 0 if P l > Pr). For invasion

312 Supercharge, Invasion and Mudcake Growth

modeling, we are interested in the progress of an initial marked particle(s), and we prefer the alternative Lagrangian description. If we now let x denote the tag describing such marked particles, the particle velocity satisfies dx/dt = q/ where is the porosity, as required in Equation 4-18. This is correct physically, since smaller porosities create faster fronts for the same q, and vice versa. Using our expression for q, we have for the invasion front the ordinary differential equation dx/dt = {k/( )}(Pr - Pl)/L which, for constant porosity, integrates to x(t) = xo - {k/(

)}(Pr - Pl) t/L

(4-21)

where x0 is the initial marked position. For heterogeneous problems, = (x) and Pl = Pl(t), and the differential equation for x(t) can be integrated in the form (x) dx = - {(k/ )(Pr - Pl(t))/L} dt using table look-up techniques. Under the assumptions stated in the preceding paragraph, the displacement front in this single-fluid, cake-free, lineal liquid flow example varies, as expected, linearly with time. It is physically an uninteresting problem. However, the simplicity of the math allowed us to illustrate basic concepts. First, the porosity, which does not appear in Eulerian constant density flow problems, does appear in Lagrangian models. Second, the Lagrangian solution cannot be (easily) obtained without solving the Eulerian formulation first. Finally, in progressing from Eulerian to Lagrangian models, the independent variable x literally becomes the dependent variable for the front position. These observations also apply to multi-dimensional problems. We can use the solution x(t) = x0 - {k/( )}(Pr - Pl) t/L (assuming x0 = 0) to illustrate the basic ideas behind time lapse invasion analysis, that is, we will take as our host model the equation {k/( )}(Pr - Pl)/L = x(t)/t. Thus, if the position front x(t) can be monitored or measured as a function of the time t, say using resistivity, radioactive tracer, or Catscan methods, it follows that the quotient x(t)/t yields information about the quantity {k/( )}(Pr - Pl)/L. Of course, the greater the value of x(t) or t, the smaller the experimental error. This invasion front measurement will provide, at most, the value of the lumped quantity {k/( )}(Pr - Pl)/L. Thus, if any of its single constituent members k, Pr, Pl, or L are required, values for the remaining quantities must first be found separately using other means. For example, if the gradient (Pr - Pl)/L and the porosity are known, then the value of the mobility k/ is immediately available (but viscosity cannot be determined).

Supercharge, Pressure, Invasion and Mudcake Growth 313

Homogeneous liquid in a uniform radial flow. Now we repeat the same calculation for cylindrical radial flows. The pressure partial differential equation governing transient, compressible, radial, homogeneous, liquid flows having constant properties is 2p(r,t)/ r2 + (1/r) p/ r = ( c/k) p/ t. All of our quantities are defined as in the lineal flow, except that the radial coordinate r replaces the lineal coordinate x. If we assume an incompressible fluid with vanishing c, we obtain the differential equation d2p(r)/dr2 + (1/r) dp/dr = 0, whose solution is p(r) = A log r + B. For this radial flow, we impose pressure boundary conditions at the well and outer reservoir radii (that is, at rwell and rres) in Figure 4.5, in the form p(rwell) = Pwell and p(rres) = Pres. We emphasize that while mudcake effects are not yet included in this formulation, the example itself is not unimportant to formation invasion in real wells. In many shallow wells, muds only slightly thicker than water are used; at other times, wells may be drilled with watery brines that do not produce mudcakes.

Figure 4.5. Cylindrical radial flow. These conditions lead to Pwell = A log rwell + B and Pres = A log rres + B, and subtraction yields A = (Pwell - Pres)/(log rwell/rres). It is A, and not B, that is important when dealing with radial invasion. From p(r) = A log r + B, we find that the radial pressure gradient satisfies dp(r)/dr = A/r, so that the Eulerian velocity q satisfies q(r) = - (k/ ) dp(r)/dr = Ak/( r). The invasion front r(t) in the Lagrangian description, as in our first example, satisfies dr/dt = q/ , or r dr = - {Ak/( )}dt. Now consider an initially marked circular ring of tracer particles where r(t) = ro at t = 0. This initial condition leads to an integral in the form r2 = ro2 {2Ak/( )} t so that r(t) = [ro2 - {2Akt/(

)}]

(4-22)

314 Supercharge, Invasion and Mudcake Growth

If Pwell > Pres and rwell /rres < 1, the constant A is negative; for large times t, the above solution for r(t) can be approximated by r(t) {-2Akt/( )}. Thus, radial front positions will vary like t, even for a uniform liquid in a homogeneous rock without the presence of mudcake. To close this example, we address the meaning of large times. To do this, we rewrite the full solution in the form r(t) = [1 - ro2 2Akt ½

{-2Akt/( )}. Since (1 + ) 1 + ½ holds for | | 0. dxi/dt = - (kPref /( c2)) f( ) (xi- )/{(xi- )2 + yi2} d (4-27a) dyi/dt = - (kPref /( c2)) yi f( )/{(xi- )2 + yi2} d (4-27b) xf = xi + (dxi/dt) t (4-27c) yf = yi + (dyi/dt) t (4-27d) This applies recursively, starting with any initial value of (Xo/c,Yo/c), and may continue indefinitely; the finer the integration time step, the greater the physical resolution. More accurate integration schemes exist in the literature, which apply to all the fracture solutions in Chapter 2 of

318 Supercharge, Invasion and Mudcake Growth

Chin (2017). Unlike PDE-based finite difference schemes, where convergence and stability depend on the form of the truncation error, and notably the sign of t, Equations 4-27a,b,c,d can be integrated backwards in time taking t < 0. This provides the capability of tracing a particle’s origin in addition to its destination, an excellent resource in environmental applications where the source of contamination is desired. 4.2.2 Flows with moving boundaries.

Here we progress to flows with nontrivial external and internal moving boundaries. We first consider lineal cake buildup on filter paper, and then we examine the plug flow of two dissimilar liquids in a linear core without mudcake. These two examples set the stage for problems where mudcake growth, formation properties, and invasion front motion are all dynamically coupled, which will be treated rigorously in the following section. Lineal mudcake buildup on filter paper. In the previous section, we considered formation invasion without the retarding effects of mudcake. In order to understand the physics clearly, we now study the problem of isolated mudcake growth, as would be obtained in the laboratory lineal flow test setup in Figure 4.7. We consider a onedimensional experiment where mud, in essence a suspension of clay particles in water, is allowed to flow through filter paper. Initially, the flow rate is rapid. But as time progresses, solid particles (typically 6% to 40% by volume for light to heavy muds) such as barite are deposited onto the surface of the paper, forming a mudcake that, in turn, retards the passage of mud filtrate by virtue of the resistance to flow that the cake provides. Flow direction

Mud Mudcake Filter paper Filtrate

Figure 4.7. Simple laboratory mudcake buildup.

Supercharge, Pressure, Invasion and Mudcake Growth 319

We therefore expect filtrate volume flow rate and cake growth rate to decrease with time, while filtrate volume and cake thickness continue to increase, but ever more slowly. These qualitative ideas can be formulated precisely, because the problem is based on well-defined physical processes. For one, the composition of the homogeneous mud during this filtration does not change: its solid fraction is always constant. Second, the flow within the mudcake is a Darcy flow and is therefore governed by the equations used by reservoir engineers. The only problem, though, is the presence of a moving boundary, namely, the position interface separating the mudcake from the mud that ultimately passes through it and that continually adds to its thickness. The physical problem, therefore, is a transient process that requires somewhat different mathematics than that taught in partial differential equations courses. Here, the fluid itself is incompressible. Mudcakes in reality may be compressible, that is, their mechanical properties may vary with applied pressure differential, e.g., as in Figure 14-7. We will be able to draw upon reservoir engineering methods for subsidence and formation compaction later. For now, a simple constitutive model for incompressible mudcake buildup, that is, the filtration of a fluid suspension of solid particles by a porous but rigid mudcake, can be constructed from first principles. First, let xc(t) > 0 represent cake thickness as a function of the time, where xc = 0 indicates zero initial thickness. Also, let Vs and Vl denote the volumes of solids and liquids in the mud suspension, and let fs denote the solid fraction defined by fs = Vs/(Vs + Vl). Since this does not change throughout the filtration, its time derivative must vanish. If we set dfs/dt = (Vs + Vl)-1 dVs/dt - Vs (Vs + Vl)-2 (dVs/dt + dVl/dt) = 0, we can show that dVs = (Vs/Vl) dVl. But since, separately, Vs/Vl = fs/(1- fs), it follows that dVs = {fs/(1- fs)} dVl. This is a conservation of species law for the solid particles making up the mud suspension and does not as yet embody any assumptions related to mudcake buildup. Frequently, we might note, the drilling fluid is thickened or thinned in the process of making hole; if so, the equations derived here should be reworked with fs = fs(t) and its corresponding time-dependent pressure drop.

320 Supercharge, Invasion and Mudcake Growth

In order to introduce the mudcake dynamics, we observe that the total volume of solids dVs deposited on an elemental area dA of filter paper during an infinitesimal time dt is dVs = (1 - c) dA dxc where c is the mudcake porosity. During this time, the volume of the filtrate flowing through our filter paper is dVl = |vn| dA dt where |vn| is the Darcy velocity of the filtrate through the cake and past the paper. We now set our two expressions for dVs equal, in order to form {fs/(1- fs)} dVl = (1 - c) dA dxc, and replace dVl with |vn| dA dt, so that we obtain {fs/(1- fs)} |vn| dA dt = (1 - c) dA dxc. The dA’s cancel, and we are led to a generic equation governing mudcake growth. In particular, the cake thickness xc(t) satisfies the ordinary differential equation dxc(t)/dt = {fs/{(1- fs)(1 - c)}} |vn|

(4-28a)

Now, as in the first example of the previous section, we assume a onedimensional, constant density, single liquid flow. For such flows, the constant Darcy velocity is (k/ )( p/L), where p > 0 is the usual delta p or pressure drop through the core of length L. The corresponding velocity for the present problem is |vn| = (k/ )( p/xc) where k is the cake permeability, and is the filtrate viscosity. Substitution in Equation 4-28a leads to dxc(t)/dt = {kfs p/{ (1- fs)(1 - c)}}/xc

(4-28b)

If the mudcake thickness is infinitesimally thin at t = 0, with xc(0) = 0, Equation 4-28b can be integrated, with the result that xc(t) = [{2kfs p/{ (1- fs)(1 - c)}} t] > 0

(4-29)

This demonstrates that cake thickness in a lineal flow grows with time like t. However, it grows ever more slowly, because increasing thickness means increasing resistance to filtrate throughflow, the source of the solid particulates required for mudcake buildup. Consequently, filtrate buildup also slows. To obtain the filtrate production volume, we combine dVl = |vn| dA dt and |vn| = (k/ )( p/xc) to form dVl = (k pdA/ ) xc-1dt. Using Equation 4-29, dVl = (k pdA/ ) [{2kfs p/{ (1- fs)(1 - c)}}]-1/2( t)-1/2 dt. Direct integration, assuming zero filtrate initially, yields

Supercharge, Pressure, Invasion and Mudcake Growth 321

Vl(t) = 2(k pdA/ ) [{2kfs p/{ (1- fs)(1 - c)}}]-1/2( t)1/2 = {2k p(1- fs)(1 - c)/( fs)} t dA (4-30) Chin et al. (1986) and recent industry papers required detailed, tedious laboratory measurement of the cake parameters fs, c, and k. This could pose operational difficulties. It turns out that this procedure is unnecessary: their values can be inferred from the results of very simple field-implementable surface filtration experiments as discussed in Chapter 19 of Chin (2017). So far, we have encountered two types of t behavior, first for constant density, radial, single-liquid flows without mudcake, and then for lineal mudcake buildup and filtrate production without introducing any underlying rock, just the opposite problem. It turns out that there is still another type of t behavior, obtained by considering the constant density flows of two sequential fluids through a lineal core without mudcake (treated next). Thus, there are at least three types of t behavior each governed by different flow parameters or physical processes, and therefore, at least three different t time scales! Log interpretation, therefore, can be challenging, to say the least. Plug flow of two liquids in linear core without cake. We consider the Darcy flow through a single lineal core in which one liquid displaces a second in a piston-like, plug, or slug-like manner, as in Figure 4.8. We assume that the permeability to each fluid is the same, so that a single permeability k suffices. Pressures Pl and Pr are fixed at the left and right sides, with Pl > Pr , so the fluid system flows from left to right. No cake is present. For lineal liquid flows, 2p(x,t)/ x2 = ( c/k) p/ t describes transient, compressible liquids, with , , c, and k denoting rock porosity, fluid viscosity, fluid-rock compressibility, and permeability. We consider an invading liquid displacing a pre-existing formation liquid, the respective viscosities being 1 and 2. Region 1 P

Region 2 Pr

l

x axis x=0

x=x f

x=L

Figure 4.8. Simple linear flow of two dissimilar fluids.

322 Supercharge, Invasion and Mudcake Growth

A second objective of this exercise is the development of mathematical techniques that model internal moving interfaces, for example, the front x = xf (t) as indicated in Figure 4.8. For now, though, we may regard the pressure problem as a purely static one. For the incompressible fluids assumed here, the compressibility c vanishes, and the ordinary differential equations for pressure in Layers 1 and 2 become d2p1(x)/dx2 = 0 and d2p2(x)/dx2 = 0, which have the respective solutions p1(x) = Ax + B and p2(x) = Cx + D, where A, B, C, and D are integration constants completely determined by the end pressure boundary conditions p1(0) = Pl and p2(L) = Pr, and the interfacial matching conditions at x = xf, p1(xf ) = p2(xf ) and q1(xf ) = q2(xf ). The pressure continuity equation assumes that interfacial tension effects are negligibly small. Velocity matching, on the other hand, is a kinematic statement stating that the local velocity is single-valued, that is, it takes on one and only one value; the moving interface is convected with this velocity as demonstrated in Chapter 16 of Chin (2017). Now, since the k1 = k2 = k, the Darcy velocities satisfy q1 = (k1/ 1) dp1(x)/dx = - (k/ 1) A and q2 = - (k2/ 2) dp2(x)/dx = - (k/ 2) C, so that A/ 1 = C/ 2. This leads to the pressure solution for 0 < x < xf, p1(x) = ( 1/ 2)(Pr - Pl) x /{L + xf ( 1/ 2 -1)} + Pl

(4-31)

The pressure solution for xf < x < L is determined as p2(x) = (Pr - Pl)(x-L)/{L + xf ( 1/ 2 -1)} + Pr

(4-32)

The invasion front is determined, as in all our preceding examples, by setting dxf /dt = q1/ assuming that porosity is constant. We now use Equation 4-31 to obtain the speed relation dxf /dt = - (k/( 1)) ( 1/ 2)(Pr - Pl)/{L + xf ( 1/ 2 -1)}. If we follow the initial marked particle defined by xf (0) = xf,o, we obtain the exact integral ( 1/ 2 -1)xf + L = +{{( 1/ 2 -1)xf,o + L}2 + {2k (Pl - Pr)/( 2)}( 1/ 2 -1) t}1/2

(4-33)

Depending on the relative values of the 1 and 2, the displacement front may accelerate or decelerate (detailed calculations will be given in

Supercharge, Pressure, Invasion and Mudcake Growth 323

Chapter 5, under Example 5-1, “Lineal liquid displacement without mudcake,” where this problem is reformulated and solved using finite difference methods). The foregoing analysis is easily reworked to handle time-dependencies in the total differential pressure applied across the core. If (Pl - Pr) is a prescribed function of t, the differential equation should be integrated accordingly, for example, taking (Pr - Pl) dt = Pr t Pl (t)dt. Similar comments apply to situations where = (x). These changes lead to obvious analytical complications, which again motivate the need for numerical models. 4.3 Coupled Dynamical Problems: Mudcake and Formation Interaction. Here we derive exact, closed-form, analytical solutions for lineal and radial flows where the growth of the mudcake and the progress of the invasion front are strongly coupled. The first solution was given in Chin et al. (1986), but the radial solution available at the time did not model spurt, and also required numerical analysis. The full solutions are presented below. Simultaneous mudcake buildup and filtrate invasion in a linear core (liquid flows). We consider a realistic example where liquid mud filtrate displaces a preexisting formation liquid having a different viscosity. And while this process is ongoing, mudcake thickness is ever-increasing, so that filtrate influx rate is consequently decreasing. All the time, the filtrate-to-formation fluid displacement front moves to the right. In this problem, the dynamics of the mudcake growth are closely coupled to the invasion front motion. In our derivation, there is no assumption that the mudcake is significantly less permeable than the formation, an assumption usually taken to simplify the analysis. The work is exact in this regard, since the relative mobilities between cake, invaded zone, and virgin formation are left as completely free parameters for subsequent evaluation. This important formulation, its solution procedure, and the exact, closed-form, analytical solution for lineal liquid flow were presented in Chin et al. (1986). In the following, we will reconstruct the steps using the original authors’ published procedure and reproduce the earlier exact solution.

324 Supercharge, Invasion and Mudcake Growth

"1" p

"2"

"3" p

mud x = -x

c

x=0

x=x

f

res

x=L x

Figure 4.9. Three-layer lineal flow. In Figure 4.9, let Layer 1 denote the mudcake, and Layers 2 and 3, the filtrate-invaded and virgin oil-bearing formations, respectively. The origin x = 0 is the fixed cake-to-rock interface; also, x c > 0 represents the cake thickness, while xf > 0 is the displacement front separating invaded from un-invaded rock zones. The transient compressible flow equation assuming constant liquid and rock properties is the classic parabolic partial differential equation, for example, 2p1(x,t)/ x2 = ( 1 1c1/k1) p1/ t, for Layer 1. If we ignore all compressibilities, in effect considering incompressible liquids with c = 0, our layered equations reduce to the equations d2pi(x)/dx2 = 0, where i = 1, 2, 3. These are solved together with the pressure boundary conditions p1(-xc) = pm and p3(L) = pr, where pm and pr represent mud and reservoir pressures. We also invoke interfacial matching conditions for pressure, that is, p1(0) = p2(0) and p2(xf ) = p3(xf ), plus interfacial matching conditions for velocity, that is, k1/ 1 dp1(0)/dx = k2/ 2 dp2(0)/dx and k2/ 2 dp2(xf )/dx = k3/ 3 dp3(xf )/dx. Note that we have retained three separate permeabilities, k1, k2, and k3 in these equations, as explained in Chapter 16 of Chin (2017). The k1 represents, of course, the mudcake permeability. However, while we have but a single rock core, characterized by a single permeability, we will derive our results with two values k2 and k3. This flexibility allows us to set k2 = k3 = krock if desired, or allow them to differ, in order to represent separate permeabilities to filtrate (with residual oil) and oil (in the presence of immobile connate water). This ad hoc modeling permits us to mimic two-phase flow relative permeability effects within the framework of a simpler slug displacement approach. We also note that while three viscosities 1, 2, and 3 were explicitly shown for completeness, we in fact assume that 1 = 2 = f for the mud filtrate,

Supercharge, Pressure, Invasion and Mudcake Growth 325

since the liquid filtrates present in Layers 1 and 2 are identical. Also, we will later denote 3 = o to represent oil viscosity, that is, the viscosity of the displaced formation fluid. Now, the solutions to our ordinary differential equations for pressure are pi(x) = ix + i, i = 1, 2, 3. The constants can be determined as 1 = (pm-pr)/{( 3k1/ 2k3 - k1/k2)xf - 3k1L/ 2k3 -xc} 1 = pm + (pm-pr)xc/{( 3k1/ 2k3 -k1/k2)xf - 3k1L/ 2k3 -xc} 2 = (k1/k2)(pm-pr)/{( 3k1/ 2k3 -k1/k2)xf - 3k1L/ 2k3 -xc}

(4-34a)

2 = pm + (pm-pr)xc/{( 3k1/ 2k3 -k1/k2)xf - 3k1L/ 2k3 -xc} 3 = ( 3k1/ 2k3)(pm -pr)/

(4-34d)

{( 3k1/ 2k3 -k1/k2)xf - 3k1L/ 2k3 -xc} 3 = pm + (pm-pr)xc/{( 3k1/ 2k3 -k1/k2)xf - 3k1L/ 2k3 -xc} + xf { k1/k2 - 3k1/ 2k3 }(pm -pr)/ {( 3k1/ 2k3 -k1/k2)xf - 3k1L/ 2k3 -xc}

(4-34b) (4-34c)

(4-34e)

(4-34f)

Equations 4-34a to 4-34f completely define the spatial pressure distributions within Layers 1, 2, and 3. However, the solutions to the invasion problem are as yet incomplete because the positions xc and xf are unknown time-dependent functions that satisfy other constraints. Consider the mudcake first. Our previous differential equation for cake growth can be evaluated as dxc/dt = {fs/{(1- c)(1-fs)}} |vn| (4-35) = - [fs/{(1- c)(1-fs)}](k1/ 1)(pm-pr) /{( 3k1/ 2k3 - k1/k2)xf - 3k1L/ 2k3 -xc}

But this cannot be integrated since it depends on the front displacement xf (t), which satisfies its own dynamic equation. To obtain it, we evaluate the interfacial kinematic condition using the now known Darcy velocity as dxf /dt = - eff -1 (k2/ 2) dp2(x)/dx (4-36) = - (k1/ 2 eff) (pm-pr)/{( 3k1/ 2k3 -k1/k2)xf - 3k1L/ 2k3 -xc}

Here eff denotes the effective porosity that Layer 2 offers if immobile fluids are left behind once the filtrate front passes. This usage provides some degree of flexibility in modeling two-phase flow relative permeability effects within the framework of single-phase flow theory.

326 Supercharge, Invasion and Mudcake Growth

Still, Equations 4-35 and 4-36 are coupled; at first, recourse to numerical analysis appears necessary, but this is fortunately not the case. It turns out that exact analytical solutions can be obtained. If we assume the initial condition xf (t = 0) = xf,o > 0 for the mud spurt, and xc = 0, until xf = xf,o > 0 with xc(t) > 0, for xf > xf,o, we obtain the solution of Chin et al. (1986), xf (t) = - H + {H2 + 2(Hxf,o + ½ xf,o2 + Gt)} where G = - {k1(pm-pr)/( f eff )}/ { ok1/ f k3 - k1/k2 - eff fs/{(1- c)(1-fs)}} H =

[xf,o eff fs/{(1- c)(1-fs)} - ok1L/ f k3] /{ ok1/ f k3 - k1/k2 - eff fs/{(1- c)(1-fs)}}

(4-37) (4-38) (4-39)

Equations 4-37 to 4-39 completely describe the progress of the invasion front, as it is affected by filtrate and reservoir liquid viscosities, and mudcake properties and growth. The corresponding equation for mudcake growth is xc(t) = [ eff fs/{(1- c)(1-fs)}] (xf - xf,o)

(4-40)

for which dxc/dxf = eff fs/{(1- c)(1-fs)} > 0. This states that xf increases if xc increases; it is interesting that the proportionality factor depends on geometrical parameters only, and not on transport variables like viscosity and permeability. We emphasize that, in general, pure t behavior is not always obtained, although it does appear in the limit of very large t. The reader, following our earlier Taylor series exercise, should determine the exact dimensionless meaning of large time when pure t behavior is found. Finally, note that xf (t) - xf,o = eff-1 {2k1(1- c)(1-fs)(pm-pr)t/( f fs)}

(4-41)

is obtained in the limit when the mobility in the rock core greatly exceeds that of the mudcake. This is the restrictive limit typically considered in the literature; again, our solutions do not invoke any limiting assumptions about relative cake-to-formation mobilities. Finally, we emphasize that these results require us to characterize the mudcake by three independent parameters, namely, the solid fraction fs,

Supercharge, Pressure, Invasion and Mudcake Growth 327

the porosity c, and the cake permeability k. The work in Chin et al. (1986) and in recent industry studies requires such empirical inputs and elaborate laboratory. It turns out that all of this is unnecessary, and that a simple lumped parameter defined by convenient lineal filtrate tests on standard filter paper is all that is required. These ideas are pursued later. Simultaneous mudcake buildup and filtrate invasion in a radial geometry (liquid flows). Here, we will reconsider the simultaneous mudcake buildup and filtrate invasion problem just discussed, but we will use realistic cylindrical radial coordinates. Note that the exact linear flow solution in Chin et al. (1986) includes the allimportant effect of mud spurt. But while that paper alluded to progress towards a radial solution, the work at that time could not account for any spurt at all because of mathematical complexities and, furthermore, turned to numerical solution as a last resort. Thus, a useful solution was not available, and any applications to time lapse analysis would await further progress. Since then, the result of some significant efforts have led to a closed-form solution. The resulting solution and derivation are described in detail here. This availability, together with the simple recipe for mudcake properties alluded to, brings “time lapse” inverse analysis closer to reality. Cake Invaded zone "1"

Virgin rock r

"3"

"2"

Pmud

Pres

r = 0 R (t) 1

R 2

R (t) 3

R 4

Figure 4.10. Three-layer radial flow. We consider a realistic example where an incompressible liquid mud filtrate displaces a preexisting incompressible formation liquid having a different viscosity (gas displacement is discussed separately. Such fluids, flowing in homogeneous, isotropic media, satisfy Laplace’s equation for pressure. And while this process is ongoing, mud-cake thickness is ever-increasing, so that filtrate influx rate is consequently decreasing; all the time, the filtrate-to-formation fluid displacement front moves to the right. In this problem, as in our lineal one, the dynamics of the mudcake growth are closely coupled to the invasion front motion. In

328 Supercharge, Invasion and Mudcake Growth

our derivation, there is no assumption that the mudcake is significantly less permeable than the formation, an assumption usually taken to simplify the analysis. Also, t behavior is not presumed at the outset; doing so would be wrong. The work is exact in this regard, since the relative mobilities between cake, invaded zone, and virgin formation are left as completely free parameters for subsequent evaluation. In Figure 4.10, let Layer 1 denote the mudcake, and Layers 2 and 3, the filtrate-invaded and virgin oil-bearing formations, respectively. In this axisymmetric problem, the origin r = 0 is the borehole centerline. Here, r = R2 represents the fixed cake-to-rock interface; R2 is an absolute constant equal to the borehole radius. Note that r = R1(t) represents the time-varying radial position of the mud-to-mudcake interface, while R3(t) denotes the time-dependent invasion front position. Finally, r = R4 represents a fixed effective radius where the reservoir pore pressure Pr is specified. The driving pressure differential is (pm pr), where pm is the pressure in the borehole. The transient compressible flow equations for constant liquid and rock properties are of standard parabolic form, for example, 2p1(r,t)/ r2 + (1/r) p1/ r = ( 1 1c1/k1) p1/ t for Layer 1. But since we are ignoring all fluid compressibilities, in effect considering constant density liquids with c = 0, our equations reduce to the differential equations d2pi(r)/dr2 + (1/r) dpi/dr = 0, i = 1, 2, 3, which have the solutions pi(r) = i log r + i, i = 1, 2, 3. As in our earlier example, the integration constants can be determined from the end pressure boundary conditions p1(R1) = pm and p3(R4) = pr. Also, we will require the interfacial matching conditions p1(R2) = p2(R2) and p2(R3) = p3(R3) for pressure, and (k1/ 1) dp1(R2)/dr = (k2/ 2) dp2(R2)/dr and (k2/ 2) dp2(R3 )/dr = (k3/ 3) dp3(R3 )/dr for velocity. Note that we have retained three separate permeabilities, namely, k1, k2, and k3 in these equations. The k1 represents, of course, the mudcake permeability. However, while we have but a single radial rock core, characterized by a single permeability, we will derive our results with two values k2 and k3. This flexibility allows us to set k2 = k3 = krock if desired, or allow them to differ, in order to represent separate permeabilities to filtrate and oil. Note that we

Supercharge, Pressure, Invasion and Mudcake Growth 329

have also retained three viscosities 1, 2, and 3, even though the same liquid 1 = 2 flows through Layers 1 and 2 (in the previous example, we assumed that 1 = 2). This ad hoc modeling, consistent with our

introduction of eff earlier, permits us to mimic two-phase flow relative permeability effects within the framework of a simpler slug displacement approach. The six integration constants are easily found, using elementary algebra, as (4-42a) 1 = (k2/ 2)(pr-pm)/ k2/ 2 k1/ 1 k1k2 3/ 1 2k3 log[(R2/R1) (R3/R2) (R4/R3) ] (4-42b) 1 = pm - 1 log R1 2 = (k1 2/ 1k2) 1 2 = pm + 1 log (R2/R1) - 2 log R2 3 = ( 3k1/ 1k3) 1 3 = pm + 1log (R2/R1) + 2log (R3/R2) - 3 log R3

(4-42c) (4-42d) (4-42e) (4-42f)

where all logarithms are natural logarithms. It appears that we have defined the spatial pressure distributions within Layers 1, 2, and 3. However, the solutions to the invasion problem are incomplete because the position fronts R1(t) and R3(t) are unknown functions of t. As before, equations for cake growth and displacement front motion must be postulated. For mudcake growth, we have - dR1/dt = {fs/{(1= [fs/{(1-

c)(1-fs)}} |vn| c)(1-fs)}] (k1/ 1) dp1/dr = [fs/{(1- c)(1-fs)}] (k1/ 1) 1/r = [fs/{(1- c)(1-fs)}] (k1/ 1) 1(R1,R3)/R1

(4-43)

The analogous equation for displacement front motion is obtained from dR3/dt = - {k2/( 2 eff)} dp2/dr = - {k2/( 2 eff)} 2/r = - {k2/( 2 eff)} 2(R1,R3)/R3

(4-44)

These nonlinear ordinary differential equations, as in the lineal case, are coupled. But again, it is possible to integrate them in closed, analytical form for general initial conditions. If we assume that R3 = Rspurt R2,

330 Supercharge, Invasion and Mudcake Growth

when R1 = R2 (i.e., no cake) at t = 0, we find that the displacement front history R3(t) satisfies (k1R22/ 1)[ ½ (R3/R2)2 log (R3/R2) - ¼ (R3/R2)2 - ½ (Rspurt/R2)2 log (Rspurt/R2) + ¼ (Rspurt/R2)2 ]

+ (k1k2 3R42/ 1 2k3) [ ½ (Rspurt/R4)2 log (Rspurt/R4) - ¼ (Rspurt/R4)2 - ½ (R3/R4)2 log (R3/R4) + ¼ (R3/R4)2 ] + (k2R22/4 2 eff){(1- c)(1-fs)/fs}

[ log {1 + fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)}}

- fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)} + fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)} log {1 + fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)}}] = = {k1k2(pm - pr)/( 1 2 eff )} t (4-45)

which does not, we emphasize, in general follow t behavior (e.g., see Outmans, 1963). This exact formula is particularly useful in resistivity logging applications where the extent of formation invasion needs to be estimated prior to taking measurements. Equation 4-45 can be solved by assuming values for R3 and calculating the corresponding times. The associated cake radius function R1(t) is then obtained from R12 = R22 + (Rspurt2 - R32 )(fs eff )/{(1- c)(1-fs)}

(4-46)

It is also possible to show that dR12/dR32 = - [fs/{(1- c)(1-fs)}] eff < 0

(4-47)

This equation indicates that as our filtration front advances, with R32 increasing, the radius (squared) R12 decreases. This decrease, following the schematic shown in Figure 4.10, indicates that mudcake thickness is consistently growing. Equation 4-47 is a Lagrangian mass conservation law that is independent of transport parameters such as permeability and viscosity. Hole plugging and stuck pipe. Unlike the lineal cake problem studied earlier where, in principle, the mudcake can increase indefinitely in thickness over time, the maximum radial thickness that can be achieved in this radial example is defined by R1(tmax) = 0, and occurs at t = tmax. At this time, all fluid motions cease, at least within the

Supercharge, Pressure, Invasion and Mudcake Growth 331

framework of the piston-like displacements studied in this chapter, and molecular diffusion then becomes the dominant physical player. In order to determine the maximal radial displacement R3,max and its corresponding time scale tmax, we set R1(t) to zero in Equation 4-46, to obtain R3,max = [ Rspurt2 + {(1- c)(1-fs)/(fs eff )}R22 ]

(4-48)

Then tmax is obtained by substituting R3,max into Equation 4-45, that is, (k1R22/ 1)[ ½ (R3,max/R2)2 log (R3,max/R2) - ¼ (R3,max/R2)2 - ½ (Rspurt/R2)2 log (Rspurt/R2) + ¼ (Rspurt/R2)2 ]

+ (k1k2 3R42/ 1 2k3) [ ½ (Rspurt/R4)2 log (Rspurt/R4) - ¼ (Rspurt/R4)2 - ½ (R3,max/R4)2 log (R3,max/R4) + ¼ (R3,max/R4)2 ]

+ (k2R22/4 2 eff){(1- c)(1-fs)/fs} [ log {1 + fs eff {(Rspurt/R2)2 - (R3,max/R2)2}/{(1- c)(1-fs)}} - fs eff {(Rspurt/R2)2 - (R3,max/R2)2}/{(1- c)(1-fs)}

+ fs eff {(Rspurt/R2)2 - (R3,max/R2)2}/{(1- c)(1-fs)} log {1 + fs eff {(Rspurt/R2)2 - (R3,max/R2)2}/{(1- c)(1-fs)}] = = {k1k2(pm - pr)/( 1 2 eff )} tmax (4-49)

In reality, this “hole plugging,” important in stuck pipe considerations, is limited by borehole flow erosion, an essential element of the dynamic filtration process, a process that is discussed and modeled in detail in Chin (2017). Fluid compressibility. Here we examine the effects of fluid compressibility on invasion. This should not be confused with mudcake and rock compressibility, which represent different physical phenomena. We consider a simple lineal flow example in this introduction. We reconsider the liquid and gaseous lineal flows treated earlier, but this time, include the transient effects due to fluid compressibility in a homogeneous core without mudcake. The pressure P(x,t) now depends on both x and t. The relevant geometry is shown in Figure 4.11, where the left- and right-side pressure boundary conditions are P(0,t) = Pl and P(L,t) = Pr, and L is the core length. Let us assume that initially, P(x,0) = Po. We study liquids first, and then reformulate and solve the problem for gases.

332 Supercharge, Invasion and Mudcake Growth

P r

P l

Figure 4.11. Lineal flow. For compressible liquids, the partial differential equation governing pressure is 2P(x,t)/ x2 = ( c/k) P/ t where , , c, and k are porosity, viscosity, compressibility, and permeability. The auxiliary pressure conditions are P(0,t) = Pl, P(L,t) = Pr, and P(x,0) = Po. This formulation is identical to that of the classic initial and boundary value problem in heat transfer for a rod with prescribed end temperatures and arbitrary initial temperature (Carlsaw and Jaeger, 1946; Tychonov and Samarski, 1964). In reservoir applications, we typically have Po = Pr, but we will leave the formulation general, since the result may be useful in special experimental situations. This can be solved in closed form using separation of variables and Fourier series (Hildebrand, 1948), but we will not reproduce the standard derivation. The exact solution is P(x,t) = (Pr - Pl)x/L + Pl + (2/ )

(1/n)[Po - Pl + (Pr - Po )(-1)n] exp [- 2n2kt/(L2

(4-50) c)] sin n x/L

where a summation from n = 1 to is understood. The first line gives the steady-state response; the second is the transient compressible response. The largest transient contribution to Equation 4-50 arises from the n = 1 term, which has the amplitude decay factor exp [- 2kt/(L2 c)]. Only when 2kt/(L2 c) , that is, t >> L2 c/( 2k) does the effect of compressibility and initial conditions, through the amplitude factor [(Po Pl + (Pr - Po )(-1)n] with n = 1, vanish. If we consider the steady solution P(x,t) = (Pr - Pl)x/L + Pl only, the front satisfies dx/dt = - (k/ ) p(x,t)/ x = - (k/ )(Pr - Pl)/L. Its time scale is determined from the quotient L /(dx/dt), that is, L 2 /{k (Pl - Pr)}. Compressibility introduces a time scale proportional to L2 c/( 2k). The partial differential equation governing compressible, lineal, homogeneous, gaseous flows having constant properties is nonlinear, satisfying 2Pm+1(x,t)/ x2 = ( c*/k) Pm+1/ t where the terms have been defined earlier. While the initial and boundary value problem for the function Pm+1 in

Supercharge, Pressure, Invasion and Mudcake Growth 333 2Pm+1(x,t)/ x2 = { Pm+1(0,t) = Plm+1 Pm+1(L,t) = Prm+1 Pm+1(x,0) = Pom+1

m/(Pk)} Pm+1/ t

(4-51a) (4-51b) (4-51c)

(4-51d) superficially resembles the linear one for compressible liquids, with P replaced by Pm+1, the two formulations are different because the constant coefficient c/k in the liquid formulation is replaced by the function m/(Pk), which depends on the solution P(x,t). The liquid problem is linear, with the sum of individual solutions itself being a solution, rendering superposition using Fourier series possible (well test procedures similarly use superposition techniques). But the latter formulation, because of the pressure-dependence, is nonlinear, and closed-form solution is not possible except for the simplest problems. Nonetheless, we can develop some idea of the time scales that arise on account of compressibility if we approximate the nonlinear coefficient by the constant m/(Pavgk), where Pr < Pavg < Pl, and if we additionally assume that Po lies in the same range. Then, comparison of the two formulations leads us to infer a formal solution satisfying Pm+1(x,t)

(Prm+1 - Plm+1)x/L + Plm+1 (4-52) + (2/ ) (1/n)[(Pom+1 - Plm+1 + (Prm+1 - Pom+1 )(-1)n] exp [- 2n2ktPavg/L2 m] sin n x/L

which is a very crude approximation to the actual solution. But this formal procedure does provide some indication about the time scales governing transient decay. If we now raise each side of the above to the 1/(m+1) th power in order to solve for the transient pressure P(x,t), we would expect a term like exp [- 2ktPavg/{L2 m(m+1)}] to emerge from the algebra. The time scale suggested by this n = 1 term is quite different from that for linear liquids, and illustrates, through the constant m, the role of reservoir thermodynamics. Compressibility can be important for gas flows, but even for liquids, where our closed-form pressure solutions are relatively simple, the consequences related to front motion are difficult to determine. Consider, for example, liquid flows. Since lineal front trajectories satisfy dx/dt = - (k/ ) P/ x, we have, using Equation 4-50,

334 Supercharge, Invasion and Mudcake Growth dx/dt = - (k/ = - (k/ + (2/L)

(4-53)

) P(x,t)/ x ) {(Pr - Pl)/L

[Po - Pl + (Pr - Po )(-1)n] exp [- 2n2kt/(L2

c)] cos n x/L}

We examine a marked fluid element located at the inlet x = 0 initially. Then, the cos n x/L term in Equation 4-53 becomes unity, and if we retain only the leading n = 1 contribution, we have the approximation dx

for which x

- (k/

) {(Pr - Pl)/L

+ (2/L) [Po - Pl - (Pr - Po )]exp [- 2kt/(L2 - (k/

) {(Pr - Pl)t/L

- 2 (2Po - Pl - Pr)[L

(4-54)

c)]} dt

(4-55) c/( 2k)] (-1 + exp [- 2kt/(L2

c)] )}

This solution satisfies x(0) = 0. Therefore, for small times, the effect of compressibility, assuming Pr = Po, will be large or small accordingly as the product (Pl - Pr )[L c/( 2k)] is large or small. We emphasize that the exact, closed-form, analytical results obtained in this chapter reveal nontrivial dependences of our filtration front and cake growth formulas on numerous groups of parameters that may have been anticipated from dimensional analysis or dynamic similitude. In the mudcake radial flow analysis, our flagship problem, the derivation did not make any assumptions regarding the relative mobilities in the three different layered flows, and so, are completely general within the framework of their formulation. But the results are restricted to constant density liquid filtrates and liquid formation fluids, that is, not gases, and then, to incompressible mudcakes only. In general, we have found that t behavior is more the exception than the rule; example numerical calculations will be given later. Under these restrictions, once R1 and R3 are known as functions of time, we can evaluate the derived pressure formulas to provide complete spatial pressure distributions at any desired instant in time. In modern MWD and “time lapse” inverse analysis applications, pressure distributions are of lesser practical interest than the values of formation porosity, permeability, and mobility themselves. Of course, pressures and spatial pressure gradients are important in fluid production, that is, the reverse problem where the value of pm(t) necessary to produce at a prescribed flow rate is important.

Supercharge, Pressure, Invasion and Mudcake Growth 335 Formation invasion at equilibrium mudcake thickness. Earlier we considered the three-layer radial invasion problem consisting of mudcake, flushed zone, and uninvaded zone, and obtained a solution for coupled mudcake growth and displacement front motion. Here we revisit that problem, but now assume that the mudcake no longer grows in time because it has reached dynamic equilibrium. The problem nonetheless consists of three layers, these being, again, the mudcake, the flushed zone, and the uninvaded zone. A number of papers refer to a classic formula obtained by solving three coupled pressure equations, each taking the form d2p/dr2 + 1/r dp/dr = 0, as the invasion model (e.g., see Muskat, 1937) used as the basis for further development. However, that formula strictly applies to the concentric radial flow of a single fluid through three layers of nonmoving rock having different permeabilities. The formula does not apply when one of the internal boundaries is moving; for such problems, the pressure boundary value problem as cited is incomplete, as we have noted earlier in this chapter, since the interface equations at the moving boundary must be included in the formulation. Again, the earlier radial mudcake example deals with two moving boundaries, namely, the mud-to-mudcake interface, and the displacement front separating two possibly dissimilar fluids within the formation. The problem considered in this section is simpler, because the mudcake, having reached dynamic equilibrium, no longer grows. Its thickness, therefore, is to be regarded as statically fixed in time. The reader should return to our earlier derivation to review the basic assumptions and approach. There, the cake radius R1 in the coefficient function 2(R1,R3) was an unknown function of time that was to be determined as part of the solution. Here, we treat R1 as a constant that can be regarded as known, once the shear stress criterion discussed earlier is applied. Thus the integration of our radial displacement front equation proceeds more simply. After some algebra, we obtain the exact solution, ½ (k2/ 2) R32 log (R2/R1) (4-56) + (k1/ 1)R22 [½ (R3/R2)2 log (R3/R2) - ¼ (R3/R2)2] - (k1k2 3/ 1 2k3)R42 [½ (R3/R4)2 log (R3/R4) - ¼ (R3/R4)2] = = - k1k2 (pr - pm) t /( 1 2 eff ) + constant

The constant of integration in Equation 4-56 is determined from initial conditions as suggested by the equilibrium solution that would be

336 Supercharge, Invasion and Mudcake Growth

obtained from the static filtration solution as modified by erosive annular effects. In our discussion of the filtration process, we had taken the simple-minded view that static filtration will continue until the point at which mudcake thickness reaches the equilibrium thickness derived earlier. At that time, cake growth terminates, but front motions continue as determined by Equation 4-56. This view is approximate and was adopted for discussion only. In reality, the shear stress in the borehole continuously acts on the mudcake as it is formed, so that the interactions between mudcake growth, reservoir Darcy flow, and borehole annular flow can be complicated. We do not pretend to solve this more realistic problem, but we do believe that the principal elements of both static and dynamic filtration processes have been satisfactorily identified. 4.4 Inverse Models in Time Lapse Logging. Here we continue development of formation invasion models, and present experimental results as reported in Chin et al. (1986) to support our simulation efforts. Importantly, we introduce practical uses for the derived formulas describing coupled mudcake growth and displacement front motion, and in particular, we develop the physical principles and reservoir flow bases underlying time lapse analysis. This work is pursued within the framework of the plug-like displacement models treated so far. Extensions of these inverse prediction methods for miscible and immiscible multiphase flow effects will be considered in Chapter 5. There, powerful methods are developed first to “undiffuse” fronts that have diffusely smeared and distorted geometrically, in order to recover the original sharp step profiles so that the inverse plug flow models in this chapter apply. Also, methods to “unshock” saturation fronts developed from immiscible water-oil flows, in order to recover the original smooth flows, are developed so that they can be analyzed accurately for formation information. Experimental model validation. “Formation Evaluation Using Repeated MWD Logging Measurements” by Chin et al. (1986) summarized a multiyear effort aimed at assessing the viability of “time lapse” inverse analysis. The exact unsteady front equation given in that paper was simplified assuming high rock-to-cake permeability ratios for further evaluation, thus leaving rock porosity as the sole formation parameter. This being so, porosity could be solved for in terms of displacement front location, time, and mudcake properties. The objective of the work lay, in part, in determining the accuracy of the

Supercharge, Pressure, Invasion and Mudcake Growth 337

porosity thus obtained, in comparison to known core-measured values and values available from other types of porosity logs. Static filtration test procedure. In order to understand the static filtration process, Catscan measurements of flows in radial and lineal cores containing growing mudcakes were obtained over periods of hours, with pressure and mud weight systematically varied in sequential tests. Photographs of the plastic, translucent, radial, and lineal flow test fixtures appear in Chapter 14 of Chin (2017). Gel-like mud-to-cake boundaries, sometimes poorly defined due to weak density contrasts, a result of barite present in both mud and mudcake, were enhanced for visualization by adding lightening agents to the mud. The salt water in the Berea sandstone cores tested contrasted well with the fresh water filtrate used, so that special visualization methods were not necessary for the flow internal to the rock. A database of information for mudcake thickness, lineal and radial displacement front position versus time, for different mudcakes and differential pressures, was obtained over a period of months. Cake density and compaction were also measured as a function of distance from the rock and monitored in time (e.g., see Chapter 14 in Chin (2017)). The mudcake model used in the original work and here, described in Collins (1961), is a well-known buildup model for cement slurries. Cake thicknesses obtained by Catscan measurement agreed with predictions, so that the model is also potentially useful in differential sticking. Dynamic filtration testing. Dynamic filtration was studied using a closed-circuit recirculating flow loop. The test section consisted of a foot-long annular core of Berea sandstone, with a 2-in inner diameter, through which mud flowed. Under differential pressures ranging from 50 to 150 psi, a portion of this recirculating flow is lost as mud filtrate, passing radially into the core and into a collection tank open to a pressure-regulated chamber. The dimensions of the one-foot annular test section were selected to reduce end effects, thus allowing pure radial flow at the center of the core. The test fixture permitted independent control of the differential pressure across the rock and cake, the absolute pressure in the loop, and the mean flow speed. This speed was monitored ultrasonically, while constant system temperatures were controlled by heat exchangers. Pressures were regulated by an accumulator, and bubbles introduced by the replenishing mud supply that replaced lost filtrate were removed by a mechanical separator. The test apparatus was also size-constrained to allow convenient Catscan

338 Supercharge, Invasion and Mudcake Growth

recording while flowing. The preliminary reported results suggested that mudcake thickness ultimately remains constant with time under laminar flow conditions, showing that surface erosion due to shear stress does result in dynamic equilibrium. For turbulent flows, the cakes formed in the tests eroded, possibly because the low-pressure differentials used did not sufficiently compact the mudcake. Only limited data was obtained in these tests, and general conclusions were not drawn. Measurement of mudcake properties. The mudcake model used required independent lab measurements for permeability, porosity, and solid fraction. This implied the need for tedious, time-consuming tasks involving weighing, drying, sorting, and so on, procedures not unlike those reported later by other authors (e.g., Holditch and Dewan, 1991 and Dewan and Chenevert, 1993). The inaccuracies present in such tests pose hurdles to practical field implementation, since any formation predictions obtained would only be as accurate as the mudcake properties. Much of the early effort addressed sensitivities of predicted results as they depended on cake measurements. At the time, no solution to this problem was found, but it turns out that dynamically equivalent information can be obtained by a single measurement for filtrate volume and cake thickness at a single large value of time. This is explained later. Formation evaluation from invasion data. At the time of the experiments, a closed-form radial front solution as a function of mudcake properties, filtrate and formation fluid viscosity, mud and pore pressure, rock porosity and permeability, and spurt loss was not available. Thus, the precise conditions under which rock permeability, pore pressure, and oil viscosity can be predicted from front data could not be determined. Nor was the form of the ultimate methodology that would host such calculations known: the exact functional relationships were lacking. Thus, the original work focused on rock porosity only, since its role is obvious: when mudcake controls the net filtration rate, the invasion front depends only on porosity, a simple geometric volume variable. The early work, in this sense, evaluated more the uncertainties due to errors in mudcake characterization than it did the formation. To determine these sensitivities, the exact porosities of the Berea sandstone cores were independently measured by direct core analysis methods and were found to vary from 22% to 24%. Figure 4.12, reproduced from Chin et al. (1986), shows predicted porosities as a function of time after initial radial invasion. Errors at small times are due to two independent effects. The first is poor mudcake definition. The second arises from the neglect of

Supercharge, Pressure, Invasion and Mudcake Growth 339

spurt in the derived radial flow formulas, or more precisely, the incorrect assumption of zero spurt. The formulas used, in order to account for the large initial invasion due to spurt loss, responded by predicting porosities that are abnormally low, in the 10% range. This effect corrects for itself over time, since the limited volume of spurt becomes unimportant with time, as the radial front expands geometrically. At least in the runs reported, the time scale required for this correction is about one hour. If an exact radial flow solution had been available, and spurt loss could be estimated, the waiting time could have been reduced to ten minutes. Time (min)

Porosity (%)

Time (min)

Porosity (%)

1.2 3.9 9.0 16.1 25.6 36.1

10% 14% 17% 20% 21% 21%

49.1 64.1 81.0 100 121 144

22% 22% 23% 23% 23% 23%

Figure 4.12. Radial flow test, 15 ppg mud, p = 150 psi. As noted, unsteady cake growth and invasion fronts in time were monitored and captured in Catscan images. In Chapter 14 of Chin (2017), for example, a sequence of linear flow slides is displayed, obtained over several hours. A radial flow slide showing the central test cross-section containing a mudcake ring, together with a circular front moving into rock core, is given. Lineal and radial flow results and predictions, for our 9-15 ppg water-base muds, proved very repeatable. Field applications. Repeat resistivity data obtained from several MWD logging runs taken in the Woodbine Sand in Quitman, Texas, was analyzed using standard multilayer electromagnetic simulations. These calculations determined invasion front radii given resistivity time histories. Then, porosity could be determined using inverse fluid methods, with the procedure repeated at different depths. Typical permeabilities and porosities in the water sands were 200 md and 25%. Resistivity measurements were obtained at 30 minutes, 1 day, and 31 days. Samples of the 9 ppg water-base mud were retained for laboratory mudcake evaluation. The multilayer electromagnetic code was specialized to four layers, comprising the tool, the mud, and invaded and virgin rock, respectively. A radius Ri separated the latter two zones. This radius was obtained iteratively, with the correct value being the one that reproduces the known tool reading when the logging resistivities RT and Rxo are prescribed. The original paper gave logs calculated at several depth intervals, displayed side by side with the corresponding neutron

340 Supercharge, Invasion and Mudcake Growth

and density porosity logs (e.g., see Chapter 14 of Chin (2017)). The effective or invasion porosity logs were consistent with neutron porosity and density tool measurements, duplicating qualitative and quantitative features. Exact agreement was not expected, since the latter vertically averaged portions of their signals and were taken at rapid speeds; the high vertical resolution resistivity tool used took instantaneous formation snapshots and yielded accurate readings in this sense. Our porosity refers to connected pores that provide conduits for fluid flow. During the field tests, the classical t displacement law, strictly valid for lineal invasion, broke down after several days due to significant formation influx, making (simplified) radial flow modeling mandatory. Again, the early work did not address formation properties other than porosity, noting only that quantitative results may be possible for hydrocarbon viscosity, formation porosity, and permeability. The principal difficulties with these properties, it was realized, were subtle. In formations with permeabilities greater than a few millidarcies, mudcakes form rapidly and control the invasion process within minutes. As was the case with the work summarized above, the invasion front then depends largely on porosity. In order to determine hydrocarbon viscosity, rock permeability, and mobility ratio, it is clear that the mobility in the cake must be comparable to that of the formation, in order to create nontrivial dynamical coupling between the two flows, from which the information needed for use in inverse models can be derived or inferred. Loosely speaking, permeable formations should be probed using permeable mudcakes, while impermeable formations require comparably impermeable cakes. In this sense, permeability prediction stands the greatest chance for success when it is extremely low in value. Still, the matter concerning the time separation required between successive resistivity readings needs to be clarified. And finally, how the mudcake properties determined in lineal laboratory flow tests without underlying impermeable rock are to be used in radial time lapse analysis must be resolved. These questions are addressed next, where our exact radial mudcake invasion solution is taken as the basis for time lapse analysis. Characterizing mudcake properties. The invasion modeling results of Chapter 17 of Chin (2017) required us to characterize the mudcake by three independent parameters, namely, the solid fraction f s, the porosity c, and the cake permeability k. The theoretical work in Chin et al. (1986) and Collins (1961) required such empirical inputs, and

Supercharge, Pressure, Invasion and Mudcake Growth 341

elaborate laboratory procedures were developed to support the volume and Darcy flow resistance measurements needed. It turns out that all of this is unnecessary, if we apply the philosophy underlying time lapse analysis to mudcake properties prediction, using simple lineal filtrate tests performed at the surface, flowing through standard filter paper without underlying rock. The key idea lies in the fact that the foregoing parameters, for incompressible cakes anyway, only affect filtration by way of the two lumped parameters, namely /k and fs/{(1- fs)(1 - c)}, where is the filtrate viscosity. Simple extrapolation of mudcake properties. In our study of lineal cake buildup on filter paper, we found that the mudcake thickness can be written as xc(t) = {2kfs p/{ (1- fs)(1 - c)}} t. For simplicity, consider a collection vessel having the same area dA as the cross-section of the core sample. (For lineal flows, the complete area A can be substituted in place of dA.) Then, the filtrate height h(t) = Vl(t)/dA of the liquid column is simply h(t) = Vl(t)/dA = {2k p(1- fs)(1 c)/( fs)} t . Dividing the first equation by the second, fs/{(1- fs)(1 - c)} = xc(t)/h(t)

(4-57)

while the square of the equation for h(t) yields /k = 2 p{t/h2(t)}(1- fs)(1 - c)/fs

(4-58)

Thus, if x c(t) and h(t) are both known at some time t, the lumped quantities fs/{(1- fs)(1 - c)} and /k are completely determined. (Our definition of the filtrate height h(t) excludes mud spurt contributions.) We emphasize that k and fs/{(1- fs)(1 - c)} are material or constitutive constants intrinsic to the particular mudcake. The latter is a dimensionless number that depends only upon the packing arrangement of the solid particles making up the mudcake, which in turn depends on the instantaneous pressure gradient and the shearing effects of dynamic filtration, if present. These constants are not unlike others used in engineering analysis, for example, the viscosity of a lubricant or the yield stress of a steel test sample. This being so, their values can be obtained from the simple lineal buildup test just described, and are applied to more general cylindrical radial or spherical flow formulas derived for problems where mudcake and formation interaction are not weak. The question of mudcake permeability often arises in assessing formation damage, which, for example, manifests itself through reduced

342 Supercharge, Invasion and Mudcake Growth

production in reservoir engineering or by way of skin effects during transient well tests. Many researchers address this problem by forcing clean water through isolated mudcake (retrieved from filtration vessels) under pressure, thus ensuring a controlled test where the mudcake no longer grows; permeability is calculated by knowing the differential pressure, the cross-sectional area, the filtrate volume, and the water viscosity. This is the standard laboratory procedure used to determine rock core permeability, but its application to mudcake analysis is inconvenient, since it is laborious, and more often than not results in cake damage and tearing. This procedure can be circumvented if we observe that Equations 4-57 and 4-58 imply that the cake permeability k takes the value k=

h(t)xc(t)/(2t p)

(4-59)

which is completely determined using data from the foregoing test. Therefore, it is clear that separate flow tests for mudcake permeability prediction are unnecessary, since the test just described provides the needed information. In order to reduce the experimental error associated with mudcake characterization, the sample time t should be sufficiently large that errors due to initially nonuniform mudcake definition are minimized. This implies a wait of 30 to 60 minutes; in fact, a sequence of measurements corresponding to larger and larger wait times might be useful, to be terminated only when calculated results for mudcake properties converge to stable values. It is assumed, of course, that appropriate high temperature and pressure filtration vessels are used whenever necessary to model mudcake growth in deep holes. Experimentally, it has been observed that the mud-to-mudcake interface may be unclear and gel-like at times, thus introducing error into time lapse analysis. It may well be that special muds with easy-to-determine cake thicknesses will need to be formulated if inverse applications are to be successful. Radial mudcake growth on cylindrical filter paper. Many authors presume the universality of t mudcake-filtration behavior at large times; this may sometimes be valid in lineal flows. However, as we have seen from our general radial mudcake flow results, this assumption can be wrought with danger. The exact nature of mudcake growth is not only important to interpretation: cake thickness is a useful indicator for both formation damage and probability of differential sticking. While t behavior provides a “back of the envelope” guess, problems can arise when cake buildup is obviously radial, for example,

Supercharge, Pressure, Invasion and Mudcake Growth 343

when mudcake thickness is a substantial fraction of the hole radius, and in modern slimholes, where the buildup process may be uncertain. In this example, we will investigate the growth of mudcake in a radial flow vessel formed by thin cylindrically formed filter paper, as shown in Figure 4.13. Although it is possible to study this problem as a formal limit of our three-layer solution, it is instructive to reconsider its formulation from first principles. As shown earlier, the governing ordinary differential equation for an incompressible, isotropic, homogeneous, cylindrical radial Darcy flow is given by d2p(r)/dr2 + (1/r) dp/dr = 0. Then, the general solution to this equation takes the form p(r) = A log r + B.

Figure 4.13. Radial mudcake growth on filter paper. For this radial flow, we impose the mud pressure Pm at the edge of the growing cake interface and the external pressure Pext at the circularly wrapped filter paper. Thus, the differential pressure acting on the cake ring is Pm - Pext. Our boundary conditions are p(rc) = Pm and p(rext) = Pext. If we now substitute these into the general solution, we find that the integration constant A satisfies A = (Pm - Pext)/(log rc/rext). The differential equation for mudcake buildup takes the form drc(t)/dt = - {fs/{(1- fs)(1 - c)}} |vn| = + {fs/{(1- fs)(1 - c)}} (k/ ) dp(rc)/dr = + {fs/{(1- fs)(1 - c)}} (k/ ) A/rc

(4-60)

where A is again a function of log rc. This nonlinear ordinary differential equation can be integrated in exact closed form. To fix the constant of integration, we assume that no mudcake exists at t = 0; that is, the cake radius is the same as that shown in Figure 4.13, with rc(t = 0)

344 Supercharge, Invasion and Mudcake Growth

= rext. Then, we obtain the following exact implicit solution for radial cake growth as a function of time, ½ (rc/rext)2 log (rc/rext) - ¼ (rc/rext)2 + 1/4 = {kfs(Pm - Pext)/{ (1- fs)(1 - c)rext2}} t

(4-61)

In deriving Equation 4-61, we assumed Pm is constant. If it is instead a function of time, the integral Pm(t) dt will appear in place of Pmt. Now consider the conditions under which this general result reduces to the lineal t law. This is accomplished by introducing rc = rext - r, with r > 0; that is, rc/rext = 1 - r/rext = 1 - and > 0. Then, we expand the above left-hand side LHS in Taylor series for small , so that LHS = ½ (1- )2 log (1- ) - ¼ (1- )2 + ¼ ½ 2 = ½ ( r/rext)2. Substitution into Equation 4-61 and cancellation of common terms yield the cake thickness r

[{2kfs(Pm - Pext)/{ (1- fs)(1 - c)}} t] > 0

(4-62)

in agreement with lineal theory. Some indication of the extent to which ½ (1- )2 log (1- ) - ¼ (1- )2 + ¼ can be approximated by ½ 2 is found by tabulating these functions versus , noting that = r/rext. This is done in Figure 4.14. The results show that the t law is satisfactory for r/rext < 0.20. This applies to radial and lineal mudcake buildup on resistance-free filter paper only and does not apply to cake buildup on formations having comparable mobilities. r/rext LHS (exact) .0500 .0012 .1000 .0048 .1500 .0107 .2000 .0186 .2500 .0285 .3000 .0401 .3500 .0534 .4000 .0681 .4500 .0840 .5000 .1009 .5500 .1185 .6000 .1367 .6500 .1551 .7000 .1733 .7500 .1911 .8000 .2078 .8500 .2230 .9000 .2360 .9500 .2456

Lineal .0013 .0050 .0113 .0200 .0313 .0450 .0613 .0800 .1013 .1250 .1513 .1800 .2113 .2450 .2813 .3200 .3613 .4050 .4513

Figure 4.14. Radial versus lineal mudcake theory.

Supercharge, Pressure, Invasion and Mudcake Growth 345

Although thick mudcakes with large values of r/rext > 0.20 may be uncommon, at least in conventional drilling, with the lineal approximation found to be quite applicable, it may well be that thicker mudcakes are actually desired for accurate time lapse analysis applications. This is so because both the mudcake characterization tests discussed above, and the cruder, direct measurements for k, f s, and c alluded to earlier, ultimately require cake thickness measurements in some form or another. Having a thicker cake to measure, ideally formed from solids with good textural qualities that ensure discernible mud-tocake boundaries, barring the risks of stuck pipe, of course, reduces the level of experimental uncertainty. Finally, note that a time scale of interest in drilling is the time required for cake to completely plug the well, under the assumption of static filtration. The required formula is useful in evaluating experimental muds drilled in slimholes. When the hole is plugged, we obtain rc = 0. Then, substitution in Equation 4-61 yields the simple relationship 1/4 = {kfs(Pm - Pext)/{ (1- fs)(1 2 c)rext }} t. The “time to plug” the borehole is tplug = (1- fs)(1 - c)rext2/{4kfs(Pm - Pext)}

(4-63)

This provides an estimate of the time scale over which plugging may become important, and may be useful operationally in stuck pipe considerations. Again, Equations 4-61 and 4-63 appear as a result of exact radial flow theory. 4.5 Porosity, Permeability, Oil Viscosity and Pore Pressure Determination. Here we develop “time lapse” inverse analysis using three models in the order of increasing complexity. In the first, we address porosity prediction, when mudcake controls the overall invasion. In the second, we consider fluid invasion without the presence of mudcake, and we determine pore pressure, formation permeability, and hydrocarbon viscosity. In the third, the same formation properties are considered, except that the complicating effects of mudcake are not neglected. Numerical examples are given which illustrate the basic ideas. Simple porosity determination. In wells where mudcake controls the overall flow into the formation, and where r/rext < 0.20 is satisfied, a lineal mudcake model suffices. This being so, we unwrap the cake layer adhering to our wellbore and view the buildup process as a

346 Supercharge, Invasion and Mudcake Growth

lineal one satisfying the t law. But the invasion into the formation, of course, is highly radial: in this farfield, the effects of borehole geometry and streamline divergence must be considered in order to conserve mass. Now consider a well with a radius rwell and an axial borehole length L, for which the surface area dA of the unwrapped mudcake layer is 2 rwell L. Using previous results, the total filtrate volume passing through the mudcake at time t = t* is Vl(t*) = {2k p(1- fs)(1 - c)/( fs)} t* dA

(4-64)

= 2 rwell L [{2k p(1- fs)(1 - c)/( fs)}t*]

For incompressible flow, this equals the formation volume available for filtrate storage; that is, (rf 2 - rwell 2)L eff, where eff is the effective porosity. Thus, we can solve for the effective porosity as 2 -1 * *2 eff = 2rwell{rf (t ) - rwell } [{2k p(1- fs)(1 - c)/( fs)}t ] (4-65)

Ideally, the right side of Equation 4-65 will be independent of the time t*, but in reality, one anticipates larger measurement errors for small times because cake thicknesses are not yet well defined (e.g., the discussion for Figure 4.12). Radial invasion without mudcake. In the preceding example, we showed how formation porosity can be calculated from purely geometric considerations when mudcake controls the flow rate into the reservoir. Sometimes the opposite limit applies: in shallow holes and in special drilling applications, watery brines are sometimes used as the circulating fluid. We will develop the theory for such problems, and then demonstrate how formation properties can be straightforwardly predicted from time lapse analysis.

Figure 4.15. Radial invasion without mudcake.

Supercharge, Pressure, Invasion and Mudcake Growth 347

Let us now refer to Figure 4.15; here, R2 corresponds to the fixed borehole radius where a mud pressure pm acts, R4 is the fixed effective radius where the reservoir pore pressure pr acts, and R3 is the moving invasion front. The complete equation for compressible fluid flows in cylindrical radial coordinates takes the form 2p(r,t)/ r2 + (1/r) p/ r = ( c/k) p/ t. We will deal with constant density flows, so that d2pi(r)/dr2 + (1/r) dpi /dr = 0, i = 1, 2. Note that the subscript 1 refers to R2 < r < R3, while the 2 refers to R3 < r < R4. The solutions to our ordinary differential equations are p1(r) = A log r + B and p2(r) = C log r + D, where A, B, C, and D are determined by the boundary and matching conditions. These are p1(R2) = pm, p2(R4) = pr, p1(R3) = p2(R3), and (k1/ 1) dp1(R3)/dr = (k2/ 2) dp2(R3)/dr. In our derivation, we allow the possibility of unequal permeabilities, so that our results can mimic relative permeability effects in two-phase immiscible flow. One can show that A = (pr - pm)/log{(R3/R2)(R3/R4)k1 2/k2 1}, B = p - A log R , C = (k m 2 1 2/k2 1) A, and D = pr - C log R4. Then, the differential equation for radial front motion is as usual found from dR3(t)/dt = - {k1/( 1 )} dp1/dr = - {k1/( 1 )} A/R3 where A depends on R3. This nonlinear ordinary differential equation can be integrated exactly in closed analytical form. If we assume a pressure drop that is constant in time, together with the initial condition R3(0) = R2 (that is, we assume that the radial invasion front initially coincides with the wellbore radius), it follows that

[(pm - pr) t /( 1R22)] (k1/ ) + {½ (R3/R2)2 log (R3/R4) - ¼ (R3/R2)2 - ½ log (R2/R4) + ¼ }(k1 2/k2 1) = ½ (R3/R2)2 log (R3/R2) - ¼ (R3/R2)2 + ¼ (4-66)

for the radial front R3(t). This result could have been obtained as a limit of the three-layer radial solution, but its self-contained derivation from first principles is instructive and useful in its own right. We now develop the quantitative basis for time lapse analysis, within the framework of the plug-flow displacement model just discussed, for rock permeability, hydrocarbon viscosity, and pore pressure determination. In Problem 1, we consider simultaneous

348 Supercharge, Invasion and Mudcake Growth

reservoir permeability and hydrocarbon viscosity prediction, while in Problem 2, we will add to these unknowns the formation pore pressure. To fix ideas, we set k1= k2 = k in Equation 4-66, which is the situation of interest to reservoir engineers. If we multiply Equation 4-66 by the mud viscosity 1, we obtain the fundamental host equation [(pm - pr) t /R22] (k/ ) + {½ (R3/R2)2 log (R3/R4) - ¼ (R3/R2)2 - ½ log (R2/R4) + ¼ } 2 = 1 {½ (R3/R2)2 log (R3/R2) - ¼ (R3/R2)2 + ¼} (4-67)

Problem 1. Let us assume that the front position r = R3(t) is known at two instants in time, say r = R3* at t = t*, and r = R3** at t = t**, for example, as determined from multilayer electromagnetic analysis, as in Chin et al. (1986). Since the values of pm, pr, R2, R4, and 1 are known, we can evaluate the Equation 4-67 twice, using our data obtained at two points in time, to yield [(pm - pr) t* /R22] (k/ ) + {½ (R3*/R2)2 log (R3*/R4) - ¼ (R3*/R2)2 - ½ log (R2/R4) + ¼ } 2 = 1 {½ (R3*/R2)2 log (R3*/R2) - ¼ (R3*/R2)2 + ¼} (4-68a) and [(pm - pr) t** /R22] (k/ ) + {½ (R3**/R2)2 log (R3**/R4) - ¼ (R3**/R2)2 - ½ log (R2/R4) + ¼ } 2 = 1 {½ (R3**/R2)2 log (R3**/R2) - ¼ (R3**/R2)2 + ¼} (4-68b) In shorthand, letting RHS denote right-hand-side quantities, Equations 468a and 4-68b become [ ] * (k/ ) + { }* 2 = RHS* (4-69a) [ ] * * (k/ ) + { }** 2 = RHS**

(4-69b)

Thus, we have two linear equations in the unknowns k/ , a useful lithology indicator related to the well-known Leverett J-function, and the viscosity 2. This simple 2 2 system can be solved using elementary algebra. If the porosity of the formation were known from a separate logging measurement, or from late-time-invasion based porosity extrapolation, then these equations would yield solutions for formation permeability and hydrocarbon viscosity.

Supercharge, Pressure, Invasion and Mudcake Growth 349

In order to test our inverse time lapse ideas, let us first generate synthetic front displacement data versus time by assuming appropriate formation and fluid properties for our forward simulation. In Equation 467, the radius R3 is varied parametrically, and the corresponding invasion time t is computed. The results of a forward simulation are shown, where the parameters have been selected for illustrative purposes only (for brevity, only partial numerical results are given). INPUT PARAMETER SUMMARY: Rock core permeability (darcies): Rock core porosity (decimal nbr): Viscosity of invading fluid (cp): Viscosity, displaced fluid (cp): Pressure at well boundary (psi): Pressure, effective radius (psi): Radius of the well bore (feet): Reservoir, effective radius (ft): Maximum allowed number of hours: T T T T T T T T T T T T T T T T T T T

= = = = = = = = = = = = = = = = = = =

.0000E+00 .2978E+04 .6830E+04 .1148E+05 .1685E+05 .2292E+05 .2962E+05 .3693E+05 .4481E+05 .5323E+05 .6217E+05 .7161E+05 .8151E+05 .9187E+05 .1027E+06 .1139E+06 .1255E+06 .1375E+06 .1498E+06

sec, sec, sec, sec, sec, sec, sec, sec, sec, sec, sec, sec, sec, sec, sec, sec, sec, sec, sec,

Rf Rf Rf Rf Rf Rf Rf Rf Rf Rf Rf Rf Rf Rf Rf Rf Rf Rf Rf

= = = = = = = = = = = = = = = = = = =

.1000E-02 .2000E+00 .1000E+01 .2000E+01 .1000E+04 .9000E+03 .2000E+00 .2000E+01 .1000E+03

.2000E+00 .3000E+00 .4000E+00 .5000E+00 .6000E+00 .7000E+00 .8000E+00 .9000E+00 .1000E+01 .1100E+01 .1200E+01 .1300E+01 .1400E+01 .1500E+01 .1600E+01 .1700E+01 .1800E+01 .1900E+01 .2000E+01

ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft

We now apply the inverse method developed earlier, and in particular, assume the input parameters given below, INPUT PARAMETER SUMMARY: Cake-rock "delta pressure" (psi): Rock core porosity (decimal nbr): Viscosity of mud filtrate (cp): Radius of the well bore (feet): Reservoir, effective radius (ft):

.1000E+03 .2000E+00 .1000E+01 .2000E+00 .2000E+01

In a field situation, the foregoing (bold) inputs would represent best guesses. We next list the results of three separate calculations; additional best guesses for radial invasion front position versus time are shown in bold print, whereas predicted formation properties are shown in italicized type.

350 Supercharge, Invasion and Mudcake Growth TIME LAPSE ANALYSIS PREDICTIONS: Trial No. 1: Time of the 1st data point (sec): Radius of invasion front (feet): Time of the 2nd data point (sec): Radius of invasion front (feet): Formation permeability (darcies): Viscosity, formation fluid (cp): Trial No. 2: Time of the 1st data point (sec): Radius of invasion front (feet): Time of the 2nd data point (sec): Radius of invasion front (feet): Formation permeability (darcies): Viscosity, formation fluid (cp): Trial No. 3: Time of the 1st data point (sec): Radius of invasion front (feet): Time of the 2nd data point (sec): Radius of invasion front (feet): Formation permeability (darcies): Viscosity, formation fluid (cp):

.6830E+04 .4000E+00 .2962E+05 .8000E+00 .1000E-02 .2000E+01 .2962E+05 .8000E+00 .6217E+05 .1200E+01 .1000E-02 .2000E+01 .6830E+04 .4000E+00 .6217E+05 .1200E+01 .1000E-02 .2000E+01

Repeated runs for this example indicate that predictions for permeability and formation fluid viscosity are very stable with respect to errors in the input data. In fact, fluid viscosity remained stable to very large changes in assumed parameters, although we have not yet pinpointed the exact reasons for this fortunate circumstance. Problem 2. Next, suppose that the pore pressure p r was unknown and additionally desired. Here, we will rewrite the fundamental result of Equation 4-67 in the form (4-70) [pm t /R22] (k/ ) + (- t /R22 ) (pr k/ ) + {½ (R3/R2)2 log (R3/R4) - ¼ (R3/R2)2 - ½ log (R2/R4) + ¼ } 2 = 1 {½ (R3/R2)2 log (R3/R2) - ¼ (R3/R2)2 + ¼}

Evaluation of Equation 4-70 at three instances in time, say t*, t**, and t***, now yields a 3 3 system of algebraic equations, in particular, [ ]*

(k/ ) + ( ) *

+ { }*

*

2 = RHS [ ] ** (k/ ) + ( ) **(pr k/ ) + { }** 2 = RHS** *** [ ] *** (k/ ) + ( ) * * * (pr k/ ) + { }*** 2 = RHS (pr k/ )

(4-71a) (4-71b) (4-71c)

that is again easily solved using elementary algebra. These three linear equations completely determine the three unknowns k/ , pr k/ , and 2.

Once the values of k/ and pr k/ are known, the pore pressure pr can be

Supercharge, Pressure, Invasion and Mudcake Growth 351

obtained by simple elimination. Then, k/ , pr , and 2 are immediately available. We emphasize that the times t*, t**, and t*** and their corresponding radii R3(t*), R3(t**), and R3(t***) must be chosen so that the resulting simultaneous equations are not ill-conditioned, in the linear algebra sense. If any of the equations are too nearly identical, because the invasion data points are taken too closely in time, the determinant of the coefficient matrix will likely vanish and yield indeterminate or inaccurate solutions. For example, the solution to x + y = 4 and x + 1.01 y = 4, while mathematically unique, is unlikely to be physically useful because the result is unstable. One way to ensure correct conditioning is to suddenly change the mud pressure pm(t). But severe decreases or increases in pressure may lead to dangerous underbalanced drilling or undesired formation fracture, effects that outweigh the need for real-time formation information. Time lapse analysis using general muds. Now, we consider the complete radial flow invasion problem modeled earlier, where general mudcake and formation interaction is allowed. This model studied dynamically coupled mud filtrate invasion, simultaneous water or oil displacement, and time-dependent mudcake buildup. In that application, we had derived the closed-form solution in Equation 4-45 for the radial invasion front position as a function of differential pressure, mudcake, rock, displaced liquid properties, and time. We will use that solution as a host time lapse analysis model equation in a manner motivated by the foregoing inverse results. It will be convenient to first rewrite Equation 4-45 in the form

[{R22(1-

c)(1-fs)/4 2 eff fs} { log {1 + fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)}} - fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)} + fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)} log {1 + fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)}}} - k1(pm - pr) t /( 1 2 eff ) ] k2

+ {(k1k2R42/ 1 2k3) [ ½ (Rspurt/R4)2 log (Rspurt/R4) - ¼ (Rspurt/R4)2 = ( (-k1R2

- ½ (R3/R4)2 log (R3/R4) + ¼ (R3/R4)2 ] }

2/

)2

1) [ ½ (R3/R2 log (R3/R2) - ¼ (R3/R2

3

)2

- ½ (Rspurt/R2)2 log (Rspurt/R2) + ¼ (Rspurt/R2)2 ] )

(4-72)

352 Supercharge, Invasion and Mudcake Growth

In deriving Equation 4-45, we assumed k2 k3 and 1 2. This is not so in applications. Thus, we simplify and write Equation 4-72 in a more meaningful form, setting k2 = k3 = kr, k1 = kc, 1 = 2 = m, and 3 = o where kr is rock permeability, kc is cake permeability, m is

filtrate viscosity, and o is “oil” or displaced liquid viscosity. With this change, Equation 4-72 becomes

[{R22(1-

c)(1-fs)/4 m eff fs} { log {1 + fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)}} - fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)} + fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)} log {1 + fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)}}} - kc(pm - pr) t /( m2 eff ) ] kr

+ {(kcR42/ m2) [ ½ (Rspurt/R4)2 log (Rspurt/R4) - ¼ (Rspurt/R4)2

- ½ (R3/R4)2 log (R3/R4) + ¼ (R3/R4)2 ] }

= ( (-kcR22/ m) [ ½ (R3/R2)2 log (R3/R2) - ¼ (R3/R2)2

- ½ (Rspurt/R2)2 log (Rspurt/R2) + ¼ (Rspurt/R2)2 ] )

o (4-73)

As in the previous example, we will consider two specific time lapse analysis formulations. In the first, we assume that the applied differential pressure is known, and we seek formation permeability and hydrocarbon viscosity only. In the second, we attempt to determine reservoir pore pressure, formation permeability, and hydrocarbon viscosity simultaneously. Problem 1. Let us examine the physical quantities within the bold brackets [ ], { }, and ( ) of Equation 4-73. First, our (lumped) mudcake parameters will be regarded as known, since they can be obtained by the simple surface filtrate test defined earlier. That is, the parameter fs/{(1fs)(1 - c)} = xc(t*)/h(t*), the cake permeability kc = m h(t*)xc(t*)/(2 p t*), the p pressure differential used in the filtrate test, and the mud filtrate viscosity m are available from well site measurements. The borehole and effective radii R2 and R4 are also considered known, as is the initial spurt radius Rspurt. (This becomes less significant with time, as its effect on total invasion depth decreases, and need not accurately

Supercharge, Pressure, Invasion and Mudcake Growth 353

specified.) Finally, the effective porosity eff of the formation can be determined from the large time test in the first example or can be assumed as known from other log measurements, while the pressure differential (pm - pr) through the cake and formation is assumed as given. Thus, all of the quantities within our bold brackets are known parameters, with the exception of the time t and its invasion depth R3(t). As before we evaluate Equation 4-73 using invasion data from two instances in time, say the radius R3(t*) at time t*, and the radius R3(t**) at t**. The two numerical instances of Equation 4-73 are [ ] * kr + { }* o = ( ) * [ ] ** kr + { }** o = ( ) **

(4-74a) (4-74b)

which provide a 2 x 2 system of algebraic equations for the formation permeability kr and the hydrocarbon viscosity o. Again, our earlier comments on ill-conditioned equations apply; this means, in practice, that t* and t** cannot be too close together or too far apart. Problem 2. For this second problem, where the formation pore pressure pr is regarded as an additional unknown, we rewrite the host invasion equation in a form that separates out the effects of pore pressure, namely,

[{R22(1-

c)(1-fs)/4 m eff fs} { log {1 + fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)}} - fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)} + fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)} log {1 + fs eff {(Rspurt/R2)2 - (R3/R2)2}/{(1- c)(1-fs)}}} - kcpmt /( m2 eff ) ] kr

+ [[ kc t /( m2 eff ) ]] prkr

+ {(kcR42/ m2) [ ½ (Rspurt/R4)2 log (Rspurt/R4) - ¼ (Rspurt/R4)2 = ( (-kcR2

- ½ (R3/R4)2 log (R3/R4) + ¼ (R3/R4)2 ] }

2/

)2

m) [ ½ (R3/R2 log (R3/R2) - ¼ (R3/R2

o

)2

- ½ (Rspurt/R2)2 log (Rspurt/R2) + ¼ (Rspurt/R2)2 ] )

(4-75)

354 Supercharge, Invasion and Mudcake Growth

As in Problem 1, we evaluate Equation 4-75 using invasion radii data from three different instances in time, say R3(t*) at time t*, R3(t**) at t**, and the radius R3(t***) at t***. Therefore, the three instances of Equation 4-75 are [ ]* []

**

[]

***

kr + [[ ]] * kr + [[ ]]

**

kr + [[ ]]

***

prkr + { }* **

prkr + { }

***

prkr + { }

* o =( ) ** o =( )

o =( )

***

(4-76a) (4-76b) (4-76c)

which provide a 3 3 system of algebraic equations for the formation permeability kr, the product prkr, and the hydrocarbon viscosity . (Once kr and prkr are known, pr can be deduced.) As before, simple determinant inversion methods from elementary algebra can be used. And again, our earlier comments on ill-conditioned equations and on the stability of calculated formation parameters apply; this means, in practice, that the times t* , t**, and t*** cannot be taken too closely together. (More precisely, displacement fronts must not be spaced too closely together, in order that differences in dynamical effects clearly manifest themselves.) 4.6 Examples of Time Lapse Analysis. While we have demonstrated how quantities of interest, such as permeability, porosity, hydrocarbon viscosity, and pore pressure, can be uniquely obtained, at least from invasion depth data satisfying our equations for piston-like fluid displacement, the actual problem is far from solved even for the simple fluid dynamics model considered here. For one, the tacit assumption that invasion depths can be accurately inferred from resistivity readings is not entirely correct; invasion radii are presently extrapolated from resistivity charts that usually assume concentric layered resistivities, which are at best simplified approximations. And second, since tool response and data interpretation introduce additional uncertainties, not to mention unknown threedimensional geological effects in the wellbore, time lapse analysis is likely to remain an iterative, subjective, and qualitative process in the near future. With these disclaimers said and done, we now demonstrate via numerical examples how formation parameters might be determined from front radii in actual field runs.

Supercharge, Pressure, Invasion and Mudcake Growth 355

Formation permeability and hydrocarbon viscosity. In this and the following example, we will first use the exact forward invasion simulation model given by Equation 4-45 to compute dynamically coupled mudcake growth and radial displacement front motion, where the mud filtrate displaces a more viscous formation fluid. We will compute radial front position and mudcake boundary as a function of time, and subsequently, using this front information, we will attempt the backward inversion process where we extract formation permeability and hydrocarbon viscosity values. In other words, we will generate synthetic invasion front displacement data and invert the computed data in order to recover the original formation and fluid properties. This philosophical approach is well known in geophysics, where synthetic P-wave data for known geological structures is created by computer detonations and received surface signals are deconvolved in order to determine the prescribed geology. This validates both forward and backward simulations, demonstrating that the mathematics is at least correct and consistent. Clearly, this does not guarantee success in the field, since the predicted values should ideally be stable with respect to uncertainties in the input data. The input and output results of a typical radial flow forward simulation are displayed in Figure 4.16 (bold print denotes input quantities). We have assumed a 1 md, 20% porous rock, and a mudcake having 0.001 md permeability, 10% porosity, and 30% solid fraction. The mud filtrate is taken as water, with a viscosity of 1 cp, and the formation oil is assumed to be 2 cp viscous. Here, the well pressure is taken as 100 psi, and the formation pore pressure is assumed to be 0 psi, acting at wellbore and effective radii of 0.5 ft and 10 ft, respectively. (Only differences in pressure are important for this example.) The numerical calculations show that the borehole completely plugs with mudcake in 17,920,000 sec (that is, 4,978 hrs, or 207 days), at which point the invasion front radius terminates at 1.727 ft. INPUT PARAMETER SUMMARY: Rock core permeability (darcies): Rock core porosity (decimal nbr): Mud cake permeability (darcies): Mud cake porosity (decimal nbr): Mud solid fraction (decimal nbr): Viscosity of invading fluid (cp): Viscosity, displaced fluid (cp): Pressure at well boundary (psi): Pressure, effective radius (psi): Radius of the well bore (feet): Reservoir, effective radius (ft): Rspurt > Rwell radius @ t=0 (ft):

.1000E-02 .2000E+00 .1000E-05 .1000E+00 .3000E+00 .1000E+01 .2000E+01 .1000E+03 .0000E+00 .5000E+00 .1000E+02 .6000E+00

356 Supercharge, Invasion and Mudcake Growth Maximum allowed

number of hours: .1000E+07

T = .9521E+00 sec, T = .3242E+05 sec, T = .1270E+06 sec, T = .3132E+06 sec, T = .6284E+06 sec, T = .1121E+07 sec, T = .1856E+07 sec, T = .2921E+07 sec, T = .4445E+07 sec, T = .6630E+07 sec, T = .9862E+07 sec, T = .1525E+08 sec, T = .1565E+08 sec, T = .1607E+08 sec, T = .1653E+08 sec, T = .1704E+08 sec, T = .1763E+08 sec, T = .1776E+08 sec, T = .1792E+08 sec, Borehole plugged by

Rf = .6000E+00 Rf = .7000E+00 Rf = .8000E+00 Rf = .9000E+00 Rf = .1000E+01 Rf = .1100E+01 Rf = .1200E+01 Rf = .1300E+01 Rf = .1400E+01 Rf = .1500E+01 Rf = .1600E+01 Rf = .1700E+01 Rf = .1705E+01 Rf = .1710E+01 Rf = .1715E+01 Rf = .1720E+01 Rf = .1725E+01 Rf = .1726E+01 Rf = .1727E+01 mudcake ... run

ft, Rc = .5000E+00 ft, Rc = .4875E+00 ft, Rc = .4726E+00 ft, Rc = .4551E+00 ft, Rc = .4348E+00 ft, Rc = .4112E+00 ft, Rc = .3836E+00 ft, Rc = .3512E+00 ft, Rc = .3124E+00 ft, Rc = .2646E+00 ft, Rc = .2012E+00 ft, Rc = .9510E-01 ft, Rc = .8616E-01 ft, Rc = .7614E-01 ft, Rc = .6454E-01 ft, Rc = .5030E-01 ft, Rc = .2982E-01 ft, Rc = .2368E-01 ft, Rc = .1523E-01 terminated.

ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft

Figure 4.16. Numerical results, forward invasion simulation. Now let us apply the time lapse analysis methodology outlined in Equations 4-73, 4-74a, and 4-74b. We will assume the properties shown in bold print in the following calculations, and then determine formation permeability and hydrocarbon viscosity using invasion radii information taken at two different points in time, as computed in Figure 4.16. We will attempt this three separate times, in order to demonstrate the utility of the approach. INPUT PARAMETER SUMMARY: Cake-rock "delta pressure" (psi): Rock core porosity (decimal nbr): Mud cake permeability (darcies): Mud cake porosity (decimal nbr): Mud solid fraction (decimal nbr): Viscosity of mud filtrate (cp): Radius of the well bore (feet): Reservoir, effective radius (ft): Rspurt > Rwell radius @ t=0 (ft):

.1000E+03 .2000E+00 .1000E-05 .1000E+00 .3000E+00 .1000E+01 .5000E+00 .1000E+02 .6000E+00

TIME LAPSE ANALYSIS PREDICTIONS: Trial No. 1: Time of the 1st data point (sec): Radius of invasion front (feet): Time of the 2nd data point (sec): Radius of invasion front (feet): Formation permeability (darcies): Viscosity, formation fluid (cp):

.3242E+05 .7000E+00 .3132E+06 .9000E+00 .9573E-03 .1911E+01

Trial No. 2: Time of the 1st data point (sec): Radius of invasion front (feet): Time of the 2nd data point (sec): Radius of invasion front (feet): Formation permeability (darcies): Viscosity, formation fluid (cp):

.3132E+06 .9000E+00 .1856E+07 .1200E+01 .1059E-02 .2131E+01

Supercharge, Pressure, Invasion and Mudcake Growth 357 Trial No. 3: Time of the 1st data point (sec): Radius of invasion front (feet): Time of the 2nd data point (sec): Radius of invasion front (feet): Formation permeability (darcies): Viscosity, formation fluid (cp):

.1856E+07 .1200E+01 .3242E+05 .7000E+00 .1016E-02 .2033E+01

Figure 4.17. Numerical results, inverse invasion simulation. Figure 4.17 shows that, in the first attempt, we obtained 0.9573 md and 1.911 cp; in the second and third attempts, we have 1.059 md and 2.131 cp, and 1.016 md and 2.033 cp, respectively. These values compare favorably with the assumed 1 md and 2 cp shown in Figure 4.16. The disagreement arises because only four decimal places of information are used from Figure 4.16. Again, sensitivity studies must be performed to show that known values of formation properties remain stable to slight errors in input mudcake assumptions. When performing time lapse analysis in the presence of mudcake, significant differences between mudcake and formation mobility heighten this sensitivity. Only when the two are comparable, for example, as in the case where mudcake builds on likewise low permeability rock, can such predictions prove robust, repeatable, and accurate. Pore pressure, rock permeability and fluid viscosity. In this example, we will rerun the forward simulation exercise just performed, except that we will replace the pressure inputs by

Pressure at well boundary (psi): .1000E+03 Pressure, effective radius (psi): .0000E+00 Pressure at well boundary (psi): .5000E+03 Pressure, effective radius (psi): .4000E+03

Since the differential pressure (of 100 psi) in both cases remains the same, we would expect the same displacement front and cake buildup history. As Figure 4.18 shows, we do. INPUT PARAMETER SUMMARY: Rock core permeability (darcies): Rock core porosity (decimal nbr): Mud cake permeability (darcies): Mud cake porosity (decimal nbr): Mud solid fraction (decimal nbr): Viscosity of invading fluid (cp): Viscosity, displaced fluid (cp): Pressure at well boundary (psi): Pressure, effective radius (psi): Radius of the well bore (feet): Reservoir, effective radius (ft): Rspurt > Rwell radius @ t=0 (ft): Maximum allowed number of hours:

.1000E-02 .2000E+00 .1000E-05 .1000E+00 .3000E+00 .1000E+01 .2000E+01 .5000E+03 .4000E+03 .5000E+00 .1000E+02 .6000E+00 .1000E+06

358 Supercharge, Invasion and Mudcake Growth T = .9521E+00 sec, T = .3242E+05 sec, T = .1270E+06 sec, T = .3132E+06 sec, T = .6284E+06 sec, T = .1121E+07 sec, T = .1856E+07 sec, T = .2921E+07 sec, T = .4445E+07 sec, T = .6630E+07 sec, T = .9862E+07 sec, T = .1525E+08 sec, T = .1565E+08 sec, T = .1607E+08 sec, T = .1653E+08 sec, T = .1704E+08 sec, T = .1763E+08 sec, T = .1776E+08 sec, T = .1792E+08 sec, Borehole plugged by

Rf = .6000E+00 Rf = .7000E+00 Rf = .8000E+00 Rf = .9000E+00 Rf = .1000E+01 Rf = .1100E+01 Rf = .1200E+01 Rf = .1300E+01 Rf = .1400E+01 Rf = .1500E+01 Rf = .1600E+01 Rf = .1700E+01 Rf = .1705E+01 Rf = .1710E+01 Rf = .1715E+01 Rf = .1720E+01 Rf = .1725E+01 Rf = .1726E+01 Rf = .1727E+01 mudcake ... run

ft, Rc = .5000E+00 ft, Rc = .4875E+00 ft, Rc = .4726E+00 ft, Rc = .4551E+00 ft, Rc = .4348E+00 ft, Rc = .4112E+00 ft, Rc = .3836E+00 ft, Rc = .3512E+00 ft, Rc = .3124E+00 ft, Rc = .2646E+00 ft, Rc = .2012E+00 ft, Rc = .9510E-01 ft, Rc = .8616E-01 ft, Rc = .7614E-01 ft, Rc = .6454E-01 ft, Rc = .5030E-01 ft, Rc = .2982E-01 ft, Rc = .2368E-01 ft, Rc = .1523E-01 terminated.

ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft

Figure 4.18. Numerical results, forward invasion simulation. We wish to illustrate the use of the inverse time lapse analysis model inferred by Equations 4-75, 4-76a, 4-76b, and 4-76c. Once we input the known information found in the Input Parameter Summary printed in Figure 4.19, including the borehole pressure, we will attempt two predictions for simultaneous pore pressure, formation permeability, and hydrocarbon viscosity, using two different sets of time-dependent front displacement data. Unlike the inverse example in Figure 4.18, however, each set of data now consists of three readings, and not two, because of the additional unknown introduced. INPUT PARAMETER SUMMARY: Mud pressure in bore hole (psi): Rock core porosity (decimal nbr): Mud cake permeability (darcies): Mud cake porosity (decimal nbr): Mud solid fraction (decimal nbr): Viscosity of mud filtrate (cp): Radius of the well bore (feet): Reservoir, effective radius (ft): Rspurt > Rwell radius @ t=0 (ft):

.5000E+03 .2000E+00 .1000E-05 .1000E+00 .3000E+00 .1000E+01 .5000E+00 .1000E+02 .6000E+00

TIME LAPSE ANALYSIS PREDICTIONS: Trial No. 1: Time of the 1st data point (sec): Radius of invasion front (feet): Time of the 2nd data point (sec): Radius of invasion front (feet): Time of the 3rd data point (sec): Radius of invasion front (feet): Formation permeability (darcies): Viscosity, formation fluid (cp): Pore pressure in reservoir (psi):

.3242E+05 .7000E+00 .3132E+06 .9000E+00 .1856E+07 .1200E+01 .8404E-03 .1670E+01 .3999E+03

Supercharge, Pressure, Invasion and Mudcake Growth 359 Trial No. 2: Time of the 1st data point (sec): Radius of invasion front (feet): Time of the 2nd data point (sec): Radius of invasion front (feet): Time of the 3rd data point (sec): Radius of invasion front (feet): Formation permeability (darcies): Viscosity, formation fluid (cp): Pore pressure in reservoir (psi):

.3242E+05 .7000E+00 .1856E+07 .1200E+01 .1525E+08 .1700E+01 .1008E-02 .2017E+01 .4000E+03

Figure 4.19. Numerical results, inverse invasion simulation. Observe that the calculated values of permeability, 0.8404 md and 1.008 md, agree well with the assumed 1 md; the calculated hydrocarbon viscosities, 1.670 cp and 2.017 cp, agree with the assumed 2 cp; and finally, the calculated pore pressures, 399.9 psi and 400.0 psi, agree with the assumed 400 psi from Figure 4.17. Because the only error in the inversion process considered here is truncation error in our threedecimal-place accurate assumptions, the calculated results here and in the previous example provide some indication of computational sensitivities. The preceding results show that, at least in this limited study, pore pressure can be accurately obtained from time lapse analysis. Continuing research in sensitivity analysis will be required if time lapse analysis is to be successful. More than likely, those predicted parameters that prove to be unstable should be obtained by other logging means; such measurements can augment the capabilities developed here. We have developed time lapse analysis methods that are contingent upon the existence of sharp fronts and transitions. Log analysts who deal with resistivity interpretation and tornado charts, however, have rightly criticized those obvious deficiencies that arise in the modeling of resistivity variations using multilayer step and even straight-line ramped profiles. Radii used in the former are arbitrarily selected by eye, while the latter ramped profiles do not resemble real diffused ones with smoothed corners. Later, using ideas borrowed from seismic migration, where a parabolized wave equation is used to image underground formations, we address resistivity migration. An arbitrarily smeared, transient concentration profile is undiffused or migrated backward in time to produce the original step discontinuity. The distinct front radius obtained can be used with time lapse analysis formulas derived here for piston-like flows. (The method is tested using computer generated synthetic lineal and radial flow data.) Similarly, we can unshock saturation discontinuities in two-phase immiscible flow, to recover the original smooth flows for further study; this is demonstrated numerically.

360 Supercharge, Invasion and Mudcake Growth

4.7 References. Carnahan, B., Luther, H.A., and Wilkes, J.O., Applied Numerical Methods, John Wiley, New York, 1969. Carslaw, H.S., and Jaeger, J.C., Conduction of Heat in Solids, Oxford University Press, London, 1946, 1959. Chin, W.C., Borehole Flow Modeling in Horizontal, Deviated and Vertical Wells, Gulf Publishing, Houston, 1992. Chin, W.C., Petrocalc 14: Horizontal and Vertical Annular Flow Modeling, Petroleum Engineering Software for the IBM PC and Compatibles, Gulf Publishing, Houston, 1992. Chin, W.C., Modern Reservoir Flow and Well Transient Analysis, Gulf Publishing, Houston, 1993. Chin, W.C., 3D/SIM: 3D Petroleum Reservoir Simulation for Vertical, Horizontal, and Deviated Wells, Petroleum Engineering Software for the IBM PC and Compatibles, Gulf Publishing, Houston, 1993. Chin, W.C., Wave Propagation in Petroleum Engineering, with Applications to Drillstring Vibrations, Measurement-While-Drilling, Swab-Surge and Geophysics, Gulf Publishing, Houston, 1994. Chin, W.C., Formation Invasion, with Applications to MeasurementWhile-Drilling, Time Lapse Analysis and Formation Damage, Gulf Publishing, Houston, 1995. Chin, W.C., Computational Rheology for Pipeline and Annular Flow, Butterworth-Heinemann, Boston, MA, 2001. Chin, W.C., RheoSim 2.0: Advanced Rheological Flow Simulator, Butterworth-Heinemann, Boston, MA, 2001. Chin, W.C., Electromagnetic Well Logging: Models for MWD/LWD Interpretation and Tool Design, John Wiley & Sons, Hoboken, New Jersey, 2014. Chin, W.C., Wave Propagation in Drilling, Well Logging and Reservoir Applications, John Wiley & Sons, Hoboken, New Jersey, 2014.

Supercharge, Pressure, Invasion and Mudcake Growth 361

Chin, W.C., Quantitative Methods in Reservoir Engineering, Second Edition – with New Topics in Formation Testing and Multilateral Well Flow Analysis, Elsevier Science, Woburn, MA, 2017. Chin, W.C., Suresh, A., Holbrook, P., Affleck, L., and Robertson, H., “Formation Evaluation Using Repeated MWD Logging Measurements,” Paper No. U, SPWLA 27th Annual Logging Symposium, Houston, TX, June 9-13, 1986. Collins, R.E., Flow of Fluids Through Porous Materials, Reinhold Publishing, New York, 1961. Lamb, H., Hydrodynamics, Dover Press, New York, 1945. Lane, H.S., “Numerical Simulation of Mud Filtrate Invasion and Dissipation,” Paper D, SPWLA 34th Annual Logging Symposium, June 13-16, 1993. Lantz, R.B., “Quantitative Evaluation of Numerical Diffusion (Truncation Error),” Society of Petroleum Engineers Journal, Sept. 1971, pp. 315-320. Muskat, M., Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York, 1937. Muskat, M., Physical Principles of Oil Production, McGraw-Hill, New York, 1949. Proett, M.A., Chin, W.C., Manohar, M., Gilbert, G.N., and Monroe, M.L., “New Dual Probe Wireline Formation Testing and Sampling Tool Enables Real-Time Permeability and Anisotropy Measurements,” SPE Paper No. 59701, 2000 SPE Permian Basin Oil and Gas Recovery Conference, Midland, Texas, Mar. 21-23, 2000. Proett, M.A., Chin, W.C., Manohar, M., Sigal, R., and Wu, J., “Multiple Factors That Influence Wireline Formation Tester Pressure Measurements and Fluid Contact Estimates,” SPE Paper No. 71566, 2001 SPE Annual Technical Conference and Exhibition, New Orleans, LA, Sept. 30 – Oct. 3, 2001. Qin, X., Feng, Y., Song, W., Cbu, X. and Wang, L., “Development on Incongruous Pushing and Stuck Releasing Device of EFDT,” Journal of China Offshore Oilfield Technology, Vol. 4, No. 1, April 2016, pp. 70-74.

362 Supercharge, Invasion and Mudcake Growth

Rourke, M., Powell, B., Platt, C., Hall, K. and Gardner, A., “A New Hostile Environment Wireline Formation Testing Tool: A Case Study from the Gulf of Thailand,” SPWLA 47th Annual Logging Symposium, Veracruz, Mexico, June 4-7, 2006. Tychonov, A.N., and Samarski, A.A., Partial Differential Equations of Mathematical Physics, Vol. I, Holden-Day, San Francisco, 1964. Tychonov, A.N., and Samarski, A.A., Partial Differential Equations of Mathematical Physics, Vol. II, Holden-Day, San Francisco, 1967.

5 Numerical Supercharge, Pressure, Displacement and Multiphase Flow Models In this final chapter, we introduce numerical methods for the solution of complicated invasion problems, and in particular, we use modern finite difference equation modeling to study both single phase and multiphase formulations. We develop the basic ideas from first principles, initially for steady-state problems and then for problems with moving boundaries. Our discussions, mathematical, numerical, and physical, are self-contained and presented in an easy-to-read manner. Numerical analogies corresponding to the constant density flow analytical models given in Chapter 4 are derived first, coded in Fortran, explained, and executed. Then, computed results are given to illustrate the simulations and to demonstrate their physical correctness. These models include linear and radial incompressible flows, with and without mudcake. Once our basic approach to moving boundary value problems is understood, the numerical modeling is extended to include other realworld effects. These include transients that arise from fluid compressibility, gas displacement by liquids, and mudcake compressibility and compaction. We continue the discussion of piston, slug, or plug-like displacements initiated earlier for single-phase flows. The numerical concepts developed in the course of this modeling are generalized to miscible and immiscible flows later in this chapter. The work in this book on invasion dynamics and numerical simulation, fully self-contained, appears in few petroleum publications. While the modeling concepts used are powerful, the computer implementation is reasonably straightforward. Minor prerequisites include course work in elementary calculus and undergraduate petroleum reservoir flow analysis. 363

364 Supercharge, Invasion and Mudcake Growth

5.1 Finite Difference Solutions. Exact analytical solutions to practical engineering problems are rare, and recourse to numerical solutions is often necessary. Finite element, boundary integral (a.k.a., panel), and finite difference methods have been successfully used to solve complicated engineering problems. Recently, new finite difference technologies have been introduced to the petroleum industry. The work of Chin (1992a,b; 2001a,b) applies these methods to annular borehole flow and pipeline modeling, while the approaches of Chin (1993a,b) introduce rigorous modeling concepts to reservoir flow simulation. Chin (1994) applies finite difference methods to wave propagation problems such as drillstring vibrations, MWD telemetry, and swab-surge. In this chapter, we extend the finite difference techniques introduced in earlier publications to more difficult reservoir flow problems. These extremely powerful methods, which can be mastered with a minimum amount of higher math, in fact require no more than a background in simple calculus. Thus, we develop the fundamental ideas, and rapidly progress to state-of-the-art algorithms for steady and transient invasion problems. Basic formulas. Let us consider a differentiable function f(x) at three consecutive equidistant locations x i-1, x i , and x i+1, where i-1, i, and i+1 are indexing parameters. Here, we will assume that all grids are uniformly separated by the constant grid block distance x. Now, it is clear from Figure 5.1 that the first derivative at an intermediate point A between x i-1 and x i is df(xA)/dx = (xi - xi-1)/ x (5-1) while the first derivative at an intermediate point B between xi and xi+1is df(xB)/dx = (xi+1 - xi)/ x

(5-2)

Hence, the second derivative of f(x) at xi satisfies or

d2f(xi)/dx2 = {df(xB)/dx - df(xA)/dx}/ x

(5-3)

d2f(xi )/dx2 = {fi-1 - 2fi + fi+1}/( x)2 + O( x)2

(5-4)

Taylor series analysis shows that Equation 5-4 is second-order accurate in x. The O( x)2 notation describes the order of the truncation error. If x is small, then O( x)2 may be regarded as very small. Likewise, it is known that

Numerical Supercharge, Pressure and Multiphase Methods 365

df(xi )/dx = {fi+1 - fi-1}/(2 x) + O( x)2

(5-5)

is second-order accurate. Equations 5-4 and 5-5 are central difference representations for the respective quantities at xi because they involve left and right quantities at x i-1 and x i+1. Note that the backward difference formula d2f(xi )/dx2 = {fi - 2fi-1 + fi-2}/( x)2 + O( x) (5-6) for the second derivative is not incorrect. But it is not as accurate as the central difference formula, since it turns out to be first-order accurate, the error being only somewhat small. Similar comments apply to the forward differencing d2f(xi )/dx2 = {fi - 2fi+1 + fi+2}/( x)2 + O( x)

(5-7)

Alternative representations for the first derivative are the first-order accurate backward and forward difference formulas df(xi )/dx = {fi - fi-1}/ x + O( x)

(5-8)

df(xi )/dx = {fi+1 - fi}/ x + O( x)

(5-9)

f i +1 fi f i -1

x i -1

x i x i +1

Figure 5.1. Finite difference discretizations. Despite their lesser accuracy, backward and forward difference formulas are often used for practical reasons. For example, they are applied at the boundaries of computational domains. At such boundaries, central difference formulas (e.g., Equations 5-4 and 5-5) require values of i that are outside the domain, and hence, undefined. Although highorder accurate backward and forward difference formulas are available, their use often forces simple matrix structures into numerical forms that are not suitable for efficient inversion.

366 Supercharge, Invasion and Mudcake Growth

Model constant density flow analysis. The basic ideas behind the numerical solution of differential equations are reviewed using d2p(x)/dx 2 = 0 (5-10)

whose solution p(x) = Ax + B is determined by two side constraints. Suppose we supplement Equation 5-10 with the left and right boundary conditions p(0) = Pl

(5-11)

p(x = L) = Pr

(5-12)

The steady-state pressure solution, applicable to constant density, lineal, liquid flows in a homogeneous core, is p(x) = (Pr - Pl) x/L + Pl (5-13) Suppose that we wish to solve Equation 5-10 numerically. We introduce along the x-axis the indexes i = 1, 2, 3, ..., i max-1, i max, where i = 1 and i max correspond to the left- and right-side core ends x = 0 and x = L (e.g., see Figure 5.1). With this convention, the constant width grid block size x used takes the value x = L/(imax -1). Now, at any position xi (or simply i), the second derivative in Equation 5-10 can be approximated using Equation 5-4, that is, d2p(xi )/dx2 = {pi-1 - 2pi + pi+1}/( x)2 + O( x)2 = 0 (5-14) so that the finite difference model for our differential equation becomes pi-1 - 2pi + pi+1 = 0 (5-15) The pressures p1, p2, ..., and p imax at the nodes i = 1, 2, ..., and i max are determined by writing Equation 5-15 for each internal node i = 2, 3, ..., and imax-1. This yields imax -2 linear algebraic equations, two short of the number of unknowns imax. The two additional required equations are obtained from boundary conditions, in this case, Equations 5-11 and 5-12; in particular, we write p(0) = Pl and p(L) = Pr in the form p1 = Pl and pimax = Pr. To illustrate this, consider the simple case of five nodes (that is, four grid blocks), taking imax = 5 and with the grid size x = L/(imax -1) = L/4. We therefore have

Numerical Supercharge, Pressure and Multiphase Methods 367 p1 p1 -2p2 + p3 p2 - 2p3 + p4 p3 - 2p4 + p5 p5

i = 2: i = 3: i = 4:

= = = = =

(5-16a,b,c,d,e)

Pl 0 0 0 Pr

Equations 5-16a,b,c,d,e constitute five equations in five unknowns and easily yield to solution, using standard (but tedious) determinant or Gaussian elimination methods from elementary algebra. We could stop here, but we take the solution of Equation 5-16 one step further in order to develop efficient solution techniques. The simplicity seen here suggests that we can rewrite the system shown in Equations 5-16a,b,c,d,e in the matrix or linear algebra form | 1 | 1 | | |

0 -2 1

1 -2 1

1 -2 0

| | | 1 | 1 |

| | | | |

p1 p2 p3 p4 p5

| | | | |

= = = = =

| | | | |

Pl 0 0 0 Pr

| | | | |

(5-17)

The left-side coefficient matrix multiplying the unknown vector p is said to be banded because its elements fall within diagonal bands. The product shown equals the nonzero right side in Equation 5-17, which contains the delta-p pressure drop (Pl - Pr) that drives the Darcy flow. This delta-p, applied across the entire core, manifests itself by controlling the top and bottom rows of the governing tridiagonal matrix equation. It is also interesting to note that our use of central differences physically implies that the pressure at each and every point depends on its left and right neighbors, so that coupled equations necessarily appear. This is not true in certain supersonic flow problems in high-speed aerodynamics, governed by hyberbolic PDEs, where the time-like properties of some space variables may in fact require the use of backward differences! Also observe that the coefficient matrix in Equation 5-17 is sparse (or empty), with each equation containing at most three unknowns. If each equation had approached imax number of unknowns, the coefficient matrix would have been said to be full. Furthermore, note that our “banded matrix” possesses a simple “tridiagonal” (or three-diagonal) structure amenable to rapid solution. These are standard in linear algebra, and we simply note that our Equation 5-17 represents a special instance of

368 Supercharge, Invasion and Mudcake Growth | B1 C1 | A2 B2 C2 | A3 B3 C3 | ....... | | Aimax-1 Bimax-1 Cimax-1 | Aimax Bimax

| | | | | | |

| | | | | | |

V1 V2 V3 .. Vimax-1 Vimax

| |W1 | | |W2 | | |W3 | | = | ... | | | | | |Wimax-1 | | |Wimax |

(5-18)

for the unknown vector V which, when programmed in Fortran, is easily solved by a call to the subroutine TRIDI in Figure 5.2.

100 200

SUBROUTINE TRIDI(A,B,C,V,W,N) DIMENSION A(1000), B(1000), C(1000), V(1000), W(1000) A(N) = A(N)/B(N) W(N) = W(N)/B(N) DO 100 I = 2,N II = -I+N+2 BN = 1./(B(II-1)-A(II)*C(II-1)) A(II-1) = A(II-1)*BN W(II-1) = (W(II-1)-C(II-1)*W(II))*BN CONTINUE V(1) = W(1) DO 200 I = 2,N V(I) = W(I)-A(I)*V(I-1) CONTINUE RETURN END

Figure 5.2. Tridiagonal equation solver.

Thus, once the coefficient matrixes A, B, C, and W are defined in the main body of the computer program, with B1 = 1, C1 = 0, W1 = Pl; A2 = A3 = A4 = 1, B2 = B3 = B4 = -2, C2 = C3 = C4 = 1, W2 = W3 = W4 = 0; and, finally, A5 = 0, B5 = 1, W5 = Pr, the statement CALL TRIDI(A,B,C,P,W,5) will solve and store the pressure solution in the elements of P. For machine purposes, we will typically initialize memory using the dummies A(1) = 99 and C(IMAX) = 99, noting that these values do not affect the solution. In general, the internal coefficients are easily defined using the code fragment,

200

DO 200 I=2,IMAXM1 A(I) = 1. B(I) = -2. C(I) = 1. W(I) = 0. CONTINUE

which is followed by the subroutine call to the tridiagonal matrix solver (in our Fortran, IMAXM1 denotes IMAX-1). In this chapter, we will

Numerical Supercharge, Pressure and Multiphase Methods 369

study how the engine in the above Fortran will change from problem to problem. For d2p(x)/dx2 = 0, the exact linear pressure variation will always be obtained for any choice of grid number; unfortunately, this is not so with more complicated equations and formulations. The reader who is not familiar with Fortran should program, execute, and understand this simple example. The ensuing programs in this chapter and in Chin (2017) build upon this and are slightly more complicated. Transient compressible flow modeling. The governing equation for transient, compressible, single-phase, liquid flows through homogeneous cores is given by the classical heat equation 2p(x,t)/ x2 = ( c/k) p/ t (5-19) Equation 5-19 provides a useful vehicle for introducing basic ideas and for testing difference schemes for use in forward simulation or timemarching, that is, in modeling events as they evolve in time for given parameters and auxiliary conditions. As before, we will solve Equation 5-19 by approximating it with algebraic equations at the nodes formed by a net of coordinate lines, but now, the time coordinate must also be discretized at uniform time intervals. Hence, we deal with numerical solutions in the x-t plane. We replace our space-time continuum with independent variables formed by a discrete set of spatial points xi = i x, where i = 1, 2, 3, ... , imax, and a discrete set of time points tn = n t, where n = 1, 2, ... and so on. We will represent the function p(x,t) as Pi,n. We expect that, at any time tn, the function Pi,n at any point xi will be influenced by its left and right neighbors, so that the central difference formula pxx(xi,tn) = (Pi+1,n -2Pi,n + Pi-1,n)/( x)2 (5-20) holds. Central differences, however, cannot be used for time derivatives. Since causality requires that events must depend on past and not future history, backward differences apply. Thus, following Equation 5-8, we must write pt(xi,tn) = (Pi,n -Pi,n-1)/ t (5-21) Then, substitution of Equations 5-20 and 5-21 in Equation 5-19 shows that a difference approximation to the governing partial differential equation is (Pi+1,n - 2Pi,n + Pi-1,n)/( x)2 = ( c/k) (Pi,n - Pi,n-1)/ t (5-22)

370 Supercharge, Invasion and Mudcake Growth

which is O{( x)2} correct in space but only O( t) correct in time. Now, we can rewrite Equation 5-22 in the form Pi+1,n -2Pi,n + Pi-1,n = { c( x)2/(k t)}(Pi,n -Pi,n-1), so that Pi-1,n - [2 + {

c( x)2/(k t)}] Pi,n + Pi+1,n

(5-23)

c( x)2/(k t)}Pi,n-1 But Equation 5-23 for the tn solution is identical to Equation 5-15, that is, to pi-1 - 2pi + pi+1 = 0, except in two minor respects. The “2” in the simpler finite difference equation is replaced by “2 + { c( x)2/(k t)},” while the right-side “0” is replaced by the term -{ c( x)2/(k t)}Pi,n-1 = -{

assumed to be available from the computed solution in just one earlier time step. For n = 2, the Pi,2-1 or Pi,1 solution is simply the prescribed initial condition p(x,0). The tn level solution is obtained as in our foregoing example; that is, Equation 5-23 is written for each of the internal nodes i = 2, 3, ..., i max -1. Left- and right-side boundary conditions are introduced to supplement the resulting incomplete set of algebraic equations. The tridiagonal subroutine is used to solve for the tn level solution as a function of space. Once this solution is available, it is used to evaluate the right side of Equation 5-23, and the left side is solved once more in a recursive manner to produce pressure solutions at the subsequent time step. We emphasize that, in Equation 5-22 at time tn, both 2p/ x2 and p/ t are evaluated at the nth level. This leads to our use of matrix solvers, since all of the resulting nodal equations are algebraically coupled. Difference schemes requiring matrix inversion are known as implicit schemes. On the other hand, if we had approximated 2p/ x2 at the earlier (n-1) th time step, we would have obtained (Pi+1,n-1 -2Pi,n-1 + Pi-1,n-1)/( x)2 = ( c/k)(Pi,n -Pi,n-1)/ t (5-24)

It is clear from Equation 5-24 that Pi,n can be solved for explicitly by hand in terms of Pi-1,n-1, Pi,n-1, and Pi+1,n-1, thus making matrix inversion unnecessary. Then, Pi,n can be updated for every internal i index directly, using a simple calculator. Such explicit schemes, useful when computing machines were uncommon, are less stable, leading to “computer crashes,” than implicit ones, but exceptions can be found.

Numerical Supercharge, Pressure and Multiphase Methods 371

Numerical stability. To researchers and practitioners alike, nothing strikes greater fear about simulation than numerical instabilities. Computational instabilities manifest themselves through unrealistic oscillations in pressure buildup or drawdown curves, in unexpected wiggly spatial pressure distributions, computer screen freezing, and in O(1010 psi) overflow messages. How can instabilities be avoided in the development process? One useful tool is the von Neumann stability test. Numerical analysts employ stability tests to evaluate candidate algorithms before embarking on resource-consuming programming efforts. We will study stability in detail later, but for now, we consider the model heat equation u t = u xx for u = u(x,t). Let us presume that a discretized u can be approximated by v(x i,tn), where vi,n satisfies the explicit model (vi,n+1 - vi,n)/ t = (vi-1,n - 2 vi,n + vi+1,n)/( x)2, where

t and x are time and spatial increments. How robust is this obvious difference approximation? To obtain some mathematical insight, let us separate variables, and consider an elementary Fourier wave component vi,n = (t) ej x, where j = -1. Substitution then yields { (t + t) e j x - (t) e j x}/ t = (t)[e j x-

x -2e j x + e j x+ x ]/( x)2. Thus, (t + t) = (t) (1 - 4 sin2 x/2), where = t/( x)2. And since (0) = 1, we have the solution (t) = (1 - 4 sin2 x/2)t/ t. For stability, (t) must remain bounded (or finite) as t, and thus x, approaches zero. This requires that the absolute value |1 - 4 sin2 x/2| < 1, thereby establishing definite requirements connecting x and t. We need not have solved for (t), of course. For example, we could have defined an amplification factor = | (t+ t)/ (t)| from the original equation and determined that = |14 sin2 x/2| < 1, leading to the same requirement. Also observe that for large values of = t/( x)2, the time-marching scheme becomes unstable, that is, the explicit method is conditionally stable. Later in this chapter, an absolutely or unconditionally stable implicit scheme is devised for the heat equation for cylindrical and spherical radial coordinates, which reduces to Equation 5-23 in the lineal limit. We will prove its von Neumann stability at that time. Convergence. In our differencing of u(x,t), we denoted its numerical representation by vi,n. That u may not, in fact, tend to v, is suggested by this usage. In computational fluid dynamics, the exact

372 Supercharge, Invasion and Mudcake Growth

functional form of the formally small truncation error is all-important because it determines the type of passive higher-order derivative term that controls the structure of the solution. This determines how well computed solutions actually mirror those of the given partial differential equation. This point is well known in numerical analysis: without evaluating the kinds of derivatives characterizing the neglected terms, whose diffusive versus dispersive effects always remain with the computed solution, the extent to which an “obvious” difference scheme actually models a differential equation cannot be ascertained. It is also important that the tridiagonal structure in Equation 5-23 is diagonally dominant; that is, the absolute value of the middle diagonal coefficient, being 2 + { c( x)2/(k t)}) > 2 = 1 + 1, exceeds the sum of the (unity) coefficients of the side diagonals. This property lends itself to numerical stability, meaning that iterative solutions are not likely to “blow up” as a result of truncation and round-off errors. This does not guarantee that the computed solutions are correct, but it does buttress the accepted (but questionable) philosophy that any solution is better than no solution. As should be clear from Equation 5-23, only one additional time level of the solution needs to be stored at any given point, so that two levels of information are required in total. Thus, the Fortran associated with our scheme can be written using two-dimensioned scalar arrays PN(1000) and PNM1(1000) only, representing Pi,n and Pi,n-1, where the Fortran dimension of 1,000 might signify 1,000 closely spaced nodes. At the end of each time step, we copy PN into PNM1 and repeatedly apply the time-recursive procedure until termination. It is not necessary (or advisable) to have computer RAM memory allocated for a complete field P(1000,500), say, representing 1,000 nodes, and 500 time steps. Intermediate results, such as displacement front location, mudcake thickness, and pressure distributions, can be written to output files for subsequent post-processing and display. Also note that the coefficients A, B, and C need not be recomputed for subsequent time steps, since they are constants defined once and for all. The matrix solver TRIDI in Figure 5.2 will destroy A, B, C, and W at the end of each inversion, so that they require redefinition prior to each integration. (Other solvers are available which retain their input values at the expense of increased memory requirements.) Multiple physical time and space scales. Earlier we had considered transient front motions whose time scales depend on the relative viscosities of invading and displaced fluids. In addition to these

Numerical Supercharge, Pressure and Multiphase Methods 373

time scales, there now exist additional ones associated with the presence of multiple fluids having different compressibilities. In using computer programs such as those derived here and similar programs available in the industry, it is important to recognize that whether or not calculated solutions capture all the physics associated with these time scales will depend on the filtering effects of grids used, that is, on x, t, and their ratio. Unfortunately, there are no obvious answers, and it is the engineering evaluation of particular computed solutions as they relate to real-world problems that poses the greatest challenge. In this book, we only demonstrate how algorithms and programs are constructed. We will not delve into grid sensitivity studies and similar validation work, as our goals and objectives are strictly tutorial. Furthermore, our choices for parameters are motivated by simplicity and comparative purposes only, and are not intended to be representative of any particular oil reservoir. With these preliminary remarks completed, let us introduce the subject of numerical invasion simulation with a sequence of examples designed to cover a broad range of physical problems. Independent formulation parameters encompass (i) lineal, cylindrical, and spherical flow domains, (ii) constant density and compressible flow, (iii) possibly dissimilar fluids in formations, (iv) gas versus liquid problems, and finally, (v) the presence of mudcakes with or without compaction. Example 5-1. Lineal liquid displacement without mudcake. We have shown how d2p(x)/dx 2 = 0 is easily solved. We now return to an early example for the piston-like Darcy displacement of two constant density liquids with different viscosities in a homogeneous lineal core of given permeability k. The transient displacement depends on the relative proportions of fluid initially present and on which portions of the core (i.e., upstream or downstream) they occupy. Now d2p(x)/dx2 = 0 applies to constant density liquids, but parametric time dependence in the solution is permissible. In this problem, since two liquids are present, two such equations are needed, d2pi(x)/dx2 = 0, i = 1, 2 (5-25)

for the first (left) and second (right) sections. For numerical purposes, it is convenient to define an unknown, upper-case solution vector P(x) by P(x) = p1(x), 0 < x < xf = p2(x), xf < x < L

(5-26)

374 Supercharge, Invasion and Mudcake Growth

where x = x f (t) represents the position of the unsteady moving front. The boundary value problem for d2P(x)/dx2 = 0 satisfies the left- and right-side pressure boundary conditions p1(0) = Pl p2(L) = Pr

(5-27a) (5-27b)

which are easily programmed as demonstrated earlier. Now, the difference equation corresponding to d 2P(x)/dx 2 = 0 at x = xf does not apply, since the differential equation description of motion breaks down at the boundary separating two distinct fluids where pressure gradients need not be continuous. We therefore replace that equation with an alternative statement that encompasses the requirements posed by the interfacial matching conditions p1(xf ) = p2(xf ) (5-28a) q1(xf ) = q2(xf )

(5-28b)

This can be done in any number of ways, but the best choice is a technique that can be carried over to transient compressible flows without modification and that allows us to retain the diagonally dominant features of the time-marching scheme derived earlier. The final result is easily derived. First, Equation 5-28b requires that -(k1/ 1) dp1(xf )/dx = -(k2/ 2) dp2(xf )/dx as a consequence of Darcy’s law q = - (k/ ) dp(x)/dx. But since k1 = k2, this statement simplifies to (1/ 1) dp1(xf )/dx = (1/ 2) dp2(xf )/dx. Now, we will denote by if- and if+ the spatial locations infinitesimally close to the left and to the right of the front x = xf, which is itself indexed by i = if. (Note that this index satisfies if = xf / x + 1 in our nodal convention.) Then, in Section 1, we can approximate the pressure gradient dp1(xf )/dx using backward differences, while in Section 2, we can apply forward differences (again, differentiation through the interface itself is forbidden since the pressure gradient changes suddenly). This leads to (1/ 1) (pif- - pif-1)/ x = (1/ 2) (pif+1 - pif+)/ x, but since x cancels, (1/ 1) (pif- - pif-1) = (1/ 2) (pif+1 - pif+). Assuming that surface tension is unimportant, Equation 528a, which calls for pressure continuity, requires that pif- = pif+ or simply pif. Thus, at the interface,

Numerical Supercharge, Pressure and Multiphase Methods 375

(1/ 1) pif-1 - (1/ 1 + 1/ 2)pif + (1/ 2) pif+1 = 0 (5-29) applies. However, unlike the difference approximation to the differential equation, which is second-order accurate, our use of backward and forward differences in deriving Equation 5-29 renders it only O( x) accurate. In deriving Equation 5-29, we emphasize that we have used the same mesh size to the left and to the right of the front. This is physically permissible if the two viscosities are comparable, but incorrect if they are not; later, in modeling mudcake flows, we will find that significant mobility contrasts existing in the problem demand dual mesh systems. It is interesting, however, to observe that we can rewrite Equation 529 as pif-1 - (1 + 1/ 2)pif + ( 1/ 2) pif+1 = 0. In the single-fluid problem where 1= 2, this matching condition reduces to pif-1 - 2pif + pif+1 = 0, which is identical to Equation 5-15 for the exact differential equation. This fortuitous situation does not apply to compressible transient flows or radial flows. This completes our discussion for the solution of Equations 5-25 to 5-28 for the spatial pressure distribution, which assumes that the front location xf is prescribed. But the front does move with time, and our formulation needs to accommodate this fact. The physical problem is an initial value problem, a transient formulation in which an interface, initially located at x = x f,o moves with time – even though Equation 5-25 governing time-dependent pressure does not contain time derivatives! We can solve this unsteady problem by first producing the pressure distribution as just outlined, then updating the front location x = xf, and subsequently, repeating this process recursively, as required. The update formula is obtained from the kinematic requirement that dxf /dt= u/ = - (k/ 1 ) dp1/dx = - (k/ 1 )(pif - pif-1)/ x

(5-30)

in Section 1. This kinematic statement, formally derived earlier, was used extensively in the analytical invasion modeling pursued for lineal and radial flows. If we evaluate the right side of Equation 5-30 with the pressure solution just obtained, denoting existing solutions for p and xf as old, then the new xf is obtained by approximating Equation 5-30 as or

(xf,new -xf,old )/ t = - (k/ 1 )(pif - pif-1)old/ x

(5-31)

376 Supercharge, Invasion and Mudcake Growth

xf,new = xf,old - {k t/( 1 x)}(pif - pif-1)old (5-32) With this new front position available, we again solve for the pressure, followed by a front update, and so on. In Figure 5.3, the Fortran listing showing the structural components of the recursive algorithm is given. The front matching conditions and position updating logic are shown in bold print. Details related to dimension statements, interactive input queries, print statements, and so on, are omitted for brevity. Only those salient features that relate to the algorithm are replicated. Note that the Fortran statement IFRONT = XFRONT/DX +1, because IFRONT is a Fortran integer variable, will discard the fractional part of the right-side division. This means that the algorithm will not move IFRONT from one time step to the next unless it has advanced sufficiently. In this sense, the scheme is not truly boundary conforming; however, it is easily modified at the expense of programming complexity. Small meshes, in general, should be used in modeling invasion front motions. We will consider two computational limits that demonstrate the physics of piston-like fluid displacement, as well as the correctness of the program. For the first example, consider the simulation input and solution in Figure 5.3b. Note the high viscosity of the invading fluid relative to that of the displaced fluid. The plot and tabulated results correctly show that the front decelerates with time. This is so because fluid of increased viscosity displaces and replaces fluid having lower viscosity, with the relative proportion of the former increasing with time, as the low viscosity fluid is forced out the right side of the core.

Numerical Supercharge, Pressure and Multiphase Methods 377 C

C C

INITIAL SETUP IMAX = XCORE/DX +1 IMAXM1 = IMAX-1 IFRONT = XFRONT/DX +1 . N = 0 T = 0. NSTOP = 0 MINDEX = 1 TIME(1) = 0. XPLOT(1) = XFRONT

START TIME INTEGRATION DO 300 N=1,NMAX T = T+DT DO 200 I=2,IMAXM1 A(I) = 1. B(I) = -2. C(I) = 1. W(I) = 0. 200 CONTINUE A(1) = 99. B(1) = 1. C(1) = 0. W(1) = PLEFT A(IMAX) = 0. B(IMAX) = 1. C(IMAX) = 99. W(IMAX) = PRIGHT IF(VISCIN.EQ.VISCDP) GO TO 240 A(IFRONT) = 1./VISCL B(IFRONT) = -1./VISCL -1./VISCR C(IFRONT) = 1./VISCR W(IFRONT) = 0. 240 CALL TRIDI(A,B,C,VECTOR,W,IMAX) DO 250 I=1,IMAX P(I) = VECTOR(I) 250 CONTINUE PGRAD = (P(IFRONT)-P(IFRONT-1))/DX XFRONT = XFRONT - (K*DT/(PHI*VISCL))*PGRAD IFRONT = XFRONT/DX +1 IF(XFRONT.GE.XMAX.OR.XFRONT.LE.XMIN) NSTOP=1 . . WRITE(*,280) N,T,XFRONT,IFRONT 280 FORMAT(1X,'T(',I4,')= ',E8.3,' sec, Xf= ',E8.3,' ft, I= ',I3) MINDEX = MINDEX+1 TIME(MINDEX) = T XPLOT(MINDEX) = XFRONT 300 CONTINUE 400 WRITE(*,10) CALL GRFIX(XPLOT,TIME,MINDEX) STOP END

Figure 5.3a. Fortran source code (Example 5-1).

378 Supercharge, Invasion and Mudcake Growth INPUT PARAMETER SUMMARY: Rock core permeability (darcies): Rock core porosity (decimal nbr): Viscosity of invading fluid (cp): Viscosity, displaced fluid (cp): Pressure at left boundary (psi): Pressure at right boundary (psi): Length of rock core sample (ft): Initial "xfront" position (feet): Integration space step size (ft): Integration time step size (sec): Maximum allowed number of steps: Time (sec) .000E+00 .600E+02 .120E+03 .180E+03 .240E+03 .300E+03 .360E+03 .420E+03 .480E+03 .540E+03 .600E+03 .660E+03 .720E+03 .780E+03 .840E+03 .900E+03 .960E+03 .102E+04

Position (ft) .500E+00 .539E+00 .576E+00 .611E+00 .644E+00 .676E+00 .706E+00 .736E+00 .764E+00 .792E+00 .818E+00 .844E+00 .870E+00 .895E+00 .919E+00 .942E+00 .965E+00 .988E+00

| | | | | | | | | | | | | | | | | |

.100E+00 .200E+00 .100E+02 .100E+01 .100E+03 .000E+00 .100E+01 .500E+00 .200E-02 .100E+01 .200E+04 * * * * * * * * * * * * * * * * * *

Figure 5.3b. Numerical results (Example 5-1). INPUT PARAMETER SUMMARY: Rock core permeability (darcies): Rock core porosity (decimal nbr): Viscosity of invading fluid (cp): Viscosity, displaced fluid (cp): Pressure at left boundary (psi): Pressure at right boundary (psi): Length of rock core sample (ft): Initial "xfront" position (feet): Integration space step size (ft): Integration time step size (sec): Maximum allowed number of steps: Time (sec) .000E+00 .600E+02 .120E+03 .180E+03 .240E+03 .300E+03 .360E+03 .420E+03

Position (ft) .500E+00 .542E+00 .586E+00 .635E+00 .690E+00 .753E+00 .830E+00 .938E+00

.100E+00 .200E+00 .100E+01 .100E+02 .100E+03 .000E+00 .100E+01 .500E+00 .200E-02 .100E+01 .200E+04

______________________________ | * | * | * | * | * | * | * | *

Figure 5.3c. Numerical results (Example 5-1). Hence, continual slowdown is anticipated and is indeed obtained. In our second example, we reverse the role of the two fluids and allow a less viscous fluid to displace one having much higher viscosity. As the latter is forced through the core and emptied, fluid having lower viscosity replaces it, so that it naturally accelerates through the core (note red fonts above). Again, our computed results are physically correct; also note the differences in the time scales of the two problems.

Numerical Supercharge, Pressure and Multiphase Methods 379

It is clear that our calculations produce results that make physical sense. Of course, in the present problem where an analytical solution is available, there is no need to resort to numerical methods. But the solution is useful because it allows us to study the effects of grid selection, that is, the role of x and t in affecting computed solutions. We emphasize that the above calculations provide the time scales characteristic of the displacement flows. Both fronts start at the midpoint of the core, and both simulations terminate near the end of the core. Their total transit times are obviously different. These time scales, as our earlier closed-form solution ( 1/ 2 -1)xf + L = +{{( 1/ 2 -1)xf,o + L}2 + {2k (Pl - Pr)/( 2)}( 1/ 2 -1) t}1/2

(4-33)

shows, depend on numerous parameters, combined in well-defined groups. For example, both ( 1/ 2 -1) and 2k (Pl - Pr)t/( 2) are individually important. The power of well-formulated numerical models lies, of course, in their potential for simple extension. For example, if the left- and rightside boundary pressures PLEFT and PRIGHT are to be prescribed functions of time, these constants are easily replaced by Fortran function statements. Likewise, time dependences in the left side invading fluid viscosity VISCL are readily incorporated. These generalizations are not unusual to actual drilling situations. Changes in mud weight, which alter borehole pressure, are used for formation or well control; these changes are effected by varying both solids and viscosifier content. Finally, some notes on the computational efficiency of the scheme are in order. Using a Windows i5 machine, 1,000 time steps requires approximately one second or less for a 500 grid block problem, all the time printing intermediate solutions to the screen. (This is the slowest part of the process and can be omitted for increased speed). The compiled code, dimensioned for a maximum of 1,000 grid blocks, requires 40,000 bytes of RAM memory. By contrast, canned finite element simulators designed to solve general 3D problems, by contrast, can require orders-ofmagnitude more computing times for the same number of steps.

380 Supercharge, Invasion and Mudcake Growth

Example 5-2. Cylindrical radial liquid displacement without cake. We now rework the preceding problem and alter the formulation so that it handles cylindrical radial flows. Thus, we replace Equation 5-10 (that is, d2p(x)/dx 2 = 0) by Laplace's equation in cylindrical radial flows,

d2p(r)/dr2 + (1/r) dp(r)/dr = 0 (5-33) The required changes are minor. Using Equation 5-14, we find that a simple change of notation gives d2p(ri )/dr2 = {pi-1 - 2pi + pi+1}/( r)2 + O( r)2. Similarly, from Equation 5-5, dp(ri )/dr = {pi+1 - pi-1}/(2 r)

+ O( r)2. We will define the radial variable r by r = Rwell + (i-1) r so that i = 1 corresponds to the left boundary of the computational grid. Then, substitution in Equation 5-33 and minor rearrangement lead to [1 - ½ r/{Rwell + (i-1) r}] pi-1 - 2 pi + [1 + ½ r/{Rwell + (i-1) r}] pi+1 = 0

(5-34)

Recall that the matrix coefficients A, B, C, and W of the finite difference equation for the lineal flow model d2p(x)/dx2 = 0, extracted from the simple formula [1] pi-1 - 2 pi + [1] pi+1 = 0, were defined by the code fragment

200

DO 200 I=2,IMAXM1 A(I) = 1. B(I) = -2. C(I) = 1. W(I) = 0. CONTINUE

Comparison with Equation 5-34 shows that the only required change needed to model fully radial flow effects is a correction ± ½ r/{Rwell + (i-1) r} to the C and A matrix coefficients. That is, we replace the preceding code with

200

DO 200 I=2,IMAXM1 CORRECT = 0.5*DX/(WELRAD + (I-1)*DX) A(I) = 1. - CORRECT B(I) = -2. C(I) = 1. + CORRECT W(I) = 0. CONTINUE

Of course, there will be additional input and output nomenclature changes, calling for wellbore and farfield radii, starting front radii, and so

Numerical Supercharge, Pressure and Multiphase Methods 381

on. For readability, we have retained DX to represent the radial mesh length r, in order to limit the number of typographical changes; WELRAD represents the wellbore radius. The Fortran source code for this example, appearing in Figure 5.4a, uses the same front matching logic as does lineal flows. We will consider two computational limits that demonstrate the physics of radial displacement flows, as well as the correctness of the computer program. For the first example, we assume simulation input parameters that are identical to those of the first run in Example 5-1, plus wellbore and farfield radii of 100 ft and 101 ft, so that the net radial extent of 1 ft equals the core length of the previous example. This large radius allows the program to mimic purely lineal flows; we will compare the computed results with those obtained for exact lineal flow. For such large radii, the effect of the radial terms should be insignificant. If so, then the computed radial front positions should be identical to those in Figure 5.3c. The two-decimal-place bold numbers in Figure 5.4b, when compared to their three-decimal place counterparts in Figure 5.3c, demonstrate that exactly the same water-to-oil displacement results are obtained as we expected. This provides a useful computing and programming check. C

C C

200

INITIAL SETUP IMAX = (XCORE-WELRAD)/DX +1 IMAXM1 = IMAX-1 IFRONT = (XFRONT-WELRAD)/DX +1 . N = 0 T = 0. NSTOP = 0 MINDEX=1 TIME(1) = 0. XPLOT(1) = XFRONT START TIME INTEGRATION DO 300 N=1,NMAX T = T+DT DO 200 I=2,IMAXM1 CORRECT = 0.5*DX/(WELRAD + (I-1)*DX) A(I) = 1. - CORRECT B(I) = -2. C(I) = 1. + CORRECT W(I) = 0. CONTINUE A(1) = 99. B(1) = 1. C(1) = 0. W(1) = PLEFT A(IMAX) = 0. B(IMAX) = 1. C(IMAX) = 99.

382 Supercharge, Invasion and Mudcake Growth

240 250

280

300 400

W(IMAX) = PRIGHT IF(VISCIN.EQ.VISCDP) GO TO 240 A(IFRONT) = 1./VISCL B(IFRONT) = -1./VISCL -1./VISCR C(IFRONT) = 1./VISCR W(IFRONT) = 0. CALL TRIDI(A,B,C,VECTOR,W,IMAX) DO 250 I=1,IMAX P(I) = VECTOR(I) CONTINUE PGRAD = (P(IFRONT)-P(IFRONT-1))/DX XFRONT = XFRONT - (K*DT/(PHI*VISCL))*PGRAD IFRONT = (XFRONT-WELRAD)/DX +1 . WRITE(*,280) N,T,XFRONT,IFRONT FORMAT(1X,'T(',I4,')= ',E8.3,' sec, Rf= ',E10.5,' ft,I= ',I3) MINDEX = MINDEX+1 TIME(MINDEX) = T XPLOT(MINDEX) = XFRONT CONTINUE WRITE(*,10) CALL GRFIX(XPLOT,TIME,MINDEX) STOP END

Figure 5.4a. Fortran source code (Example 5-2). INPUT PARAMETER SUMMARY: Rock core permeability (darcies): Rock core porosity (decimal nbr): Viscosity of invading fluid (cp): Viscosity, displaced fluid (cp): Pressure at well boundary (psi): Pressure, effective radius (psi): Radius of the bore hole (ft): Reservoir effective radius (ft): Initial "Rfront" position (feet): Integration space step size (ft): Integration time step size (sec): Maximum allowed number of steps: Number spatial DR grids selected: COMPUTED T( 0)= T( 60)= T( 120)= T( 180)= T( 240)= T( 300)= T( 360)= T( 420)=

RESULTS: .000E+00 .600E+02 .120E+03 .180E+03 .240E+03 .300E+03 .360E+03 .420E+03

sec, sec, sec, sec, sec, sec, sec, sec,

Rf= Rf= Rf= Rf= Rf= Rf= Rf= Rf=

.100E+00 .200E+00 .100E+01 .100E+02 .100E+03 .000E+00 .100E+03 .101E+03 .101E+03 (i.e., 100.5) .200E-02 .100E+01 .200E+04 .500E+03

.10050E+03 .10054E+03 .10059E+03 .10064E+03 .10069E+03 .10075E+03 .10083E+03 .10094E+03

ft, ft, ft, ft, ft, ft, ft, ft,

I= I= I= I= I= I= I= I=

250 271 294 318 346 377 416 470

Figure 5.4b. Numerical results (Example 5-2).

Numerical Supercharge, Pressure and Multiphase Methods 383 INPUT PARAMETER SUMMARY: Rock core permeability (darcies): Rock core porosity (decimal nbr): Viscosity of invading fluid (cp): Viscosity, displaced fluid (cp): Pressure at well boundary (psi): Pressure, effective radius (psi): Radius of the bore hole (ft): Reservoir effective radius (ft): Initial "Rfront" position (feet): Integration space step size (ft): Integration time step size (sec): Maximum allowed number of steps: Number spatial DR grids selected: Time (sec) .000E+00 .600E+02 .120E+03 .180E+03 .240E+03 .300E+03 .360E+03 .420E+03 .480E+03

Position (ft) .600E+00 .647E+00 .695E+00 .745E+00 .796E+00 .849E+00 .906E+00 .967E+00 .104E+01

.100E+00 .200E+00 .100E+01 .100E+02 .100E+03 .000E+00 .100E+00 .110E+01 .600E+00 .200E-02 .100E+01 .200E+04 .500E+03

______________________________ | * | * | * | * | * | * | * | * | *

Figure 5.4c. Numerical results (Example 5-2). Next, we consider a physical situation where the geometric effects of radial spreading must be important, and accordingly we select a small slimhole radius of 0.1 ft and a farfield radius of 1.1 ft. These choices therefore fix the length of the core to one foot. Again, we initialize our front position to the center of the core sample. Computed results demonstrate important geometric effects. From t = 360 to 420 sec, the radial front has advanced from r = 0.906 ft to 0.967 ft, for a total extent of 0.061 ft. If we refer to Figure 5.3c for the lineal result, in the same time period, the front has advanced from x = 0.830 ft to 0.938 ft, for a total of 0.108 ft. The decrease in distance obtained in the radial case is clearly the result of geometric spreading, and the twofold change indicates that such effects can be significant for small-diameter holes. These changes are all-important to resistivity interpretation and modeling. Example 5-3. Spherical radial liquid displacement without cake.

Now let us rework the preceding cylindrical radial problem, and alter the analytical and numerical formulations so that they handle spherical radial flows. Such formulations model invasion at the drillbit and also point fluid influx into formation testers at small times. We will replace the governing equation for cylindrical radial flows, namely,

384 Supercharge, Invasion and Mudcake Growth

d2p(r)/dr2 + (1/r) dp(r)/dr = 0 in Equation 5-33, by the spherical flow equation d2p(r)/dr2 + (2/r) dp(r)/dr = 0

(5-35)

Again, we are restricted to constant density flows in homogeneous rocks. The required changes are minor, since we have merely substituted a “2/r” variable coefficient in favor of 1/r. Instead of Equation 5-34, we have [1 - r/{Rwell + (i-1) r}] pi-1 - 2 pi + [1 + r/{Rwell + (i-1) r}] pi+1= 0

(5-36)

The code fragment

200

DO 200 I=2,IMAXM1 CORRECT = 0.5*DX/(WELRAD + (I-1)*DX) A(I) = 1. - CORRECT B(I) = -2. C(I) = 1. + CORRECT W(I) = 0. CONTINUE

appearing in the cylindrical radial program requires only a one-line change in order to implement Equation 5-36, so that instead we have

200

DO 200 I=2,IMAXM1 CORRECT = DX/(WELRAD + (I-1)*DX) A(I) = 1. - CORRECT B(I) = -2. C(I) = 1. + CORRECT W(I) = 0. CONTINUE

As before, there are obvious input and output nomenclature changes, calling for bit and farfield radii, starting front radii and so on. (Again, for readability, we have retained DX to represent the radial mesh length r.) The source code is similar to that in Figure 5.4a, except for the single line change just described. In order to demonstrate the differences between cylindrical and spherical radial flows, we have assumed parameters identical to those in the second run of Example 5-2. At t = 480 sec, the cylindrical radial position is 1.04 ft, whereas at the same instant, the spherical radial position is 0.852 ft, which is significantly less. As the calculated results in Figures 5.5a and 5.5b show, the spherical front requires more time to reach the farfield boundary defined by the effective radius r = 1.1 ft. Its acceleration is less than that in the previous example as a result of increased geometric spreading.

Numerical Supercharge, Pressure and Multiphase Methods 385 INPUT PARAMETER SUMMARY: Rock core permeability (darcies): Rock core porosity (decimal nbr): Viscosity of invading fluid (cp): Viscosity, displaced fluid (cp): Pressure at "bit" boundary (psi): Pressure, effective radius (psi): Radius at the drill bit (ft): Reservoir effective radius (ft): Initial "Rfront" position (feet): Integration space step size (ft): Integration time step size (sec): Maximum allowed number of steps: Number spatial DR grids selected:

.100E+00 .200E+00 .100E+01 .100E+02 .100E+03 .000E+00 .100E+00 .110E+01 .600E+00 .200E-02 .100E+01 .200E+04 .500E+03

Figure 5.5a. Numerical results (Example 5-3). Time (sec) .000E+00 .600E+02 .120E+03 .180E+03 .240E+03 .300E+03 .360E+03 .420E+03 .480E+03 .540E+03 .600E+03 .660E+03 .720E+03 .780E+03 .840E+03 .900E+03 .960E+03 .102E+04 .108E+04

Position (ft) .600E+00 .637E+00 .673E+00 .706E+00 .738E+00 .768E+00 .797E+00 .825E+00 .852E+00 .878E+00 .904E+00 .928E+00 .952E+00 .975E+00 .998E+00 .102E+01 .104E+01 .106E+01 .108E+01

______________________________ | * | * | * | * | * | * | * | * | * | * | * | * | * | * | * | * | * | * | *

Figure 5.5b. Numerical results (Example 5-3). Example 5-4. Lineal liquid displacement without mudcake, including compressible flow transients.

In this example, we will revisit Example 5-1 but include the additional effect of nonvanishing fluid compressibility. This being the case, d2p/dx2 = 0 is no longer the governing equation. Instead, the governing partial differential equation is the heat equation 2p(x,t)/ x2 = (

c/k) p/ t

(5-19)

which requires initial conditions for spatial pressure distribution in addition to those for front position. Its finite difference approximation, as derived earlier, takes the form

386 Supercharge, Invasion and Mudcake Growth

Pi-1,n - [2 + {

c( x)2/(k t)}] Pi,n + Pi+1,n = -{

instead of the simpler equation (1) Pi-1 - 2 Pi + (1) Pi+1 = 0

(5-23) 2 c( x) /(k t)}Pi,n-1 (5-37)

derived for d2p/dx2 = 0. The finite difference program of Example 5-1 can be modified to handle transients due to fluid compressibility. First, the right-side of Equation 5-23 indicates that pressure information from one earlier time step is required before the tridiagonal equations can be solved. Thus, an initial condition is required, so that the program user must enter an initial pressure. When a new formation is penetrated, the initial pressure will always be equal to the reservoir pore pressure. However, in this book and in the code, we will leave this input completely general, if only for code flexibility and the possibility that the program will be used in special experimental situations. Once the pressure field in space is obtained for a particular time step, it must be copied into the pressure array for the earlier pressure before pressures can be recursively advanced and integrated in time. The bookkeeping of an earlier time pressure array means that an additional Fortran dimension statement, plus more allocated memory, will be required. Aside from new input statements required for fluid compressibilities, we will need to modify the matrix coefficients B and W as required by Equations 5-23 and 5-37. That is, the -2 of Equation 5-37 is replaced by the term - 2 - { c( x)2/(k t)} of Equation 5-23, while the 0 of Equation 5-37 is now to be replaced by the -{ c( x)2/(k t)}Pi,n-1 of Equation 5-23. (Note that the former change increases numerical stability by increasing diagonal dominance.) The interfacial velocity matching conditions derived in Example 5-1 do not change. But the meaning of the product c must be understood: it is different on either side of the front, which again moves from time step to time step. The required Fortran source code changes are shown in bold print in Figure 5.6a. The array Pi,n-1 is denoted by PNM1, and the initial pressure is PINIT. In order to determine the transient effects of fluid compressibility, we reconsider one of the data sets used in Example 5-1, where water displaces an oil with ten times the viscosity. The corresponding compressibilities are taken as 3 x 10-6 /psi and 50 x 10-6 /psi, while the initial pressure was assumed to be equal to the right-side reservoir pressure. If we compare computed results, which now include

Numerical Supercharge, Pressure and Multiphase Methods 387

time scales related to fluid compressibilities and moving fronts, with those in Example 5-1 (see Figure 5.3c), we find that in the present run, the effect of compressible flow transients on displacement front position with time is minimal. One Fortran subtlety deserves elaboration. The 200 do-loop defines two separate difference equations for the flows left and right of the front, but W(I) = -TERM*PNM1(I) refers to a single pressure. So long as the front does not move more than one mesh in a time step, errors due to copying water pressure as oil pressure, or conversely, do not exist (e.g., refer to the 260 loop); pressure continuity assures that both blocks contain equal pressures. C

100

C C

200

240 250

260

. .

INITIAL SETUP IMAX = XCORE/DX +1 IMAXM1 = IMAX-1 IFRONT = XFRONT/DX +1 N = 0 T = 0. DO 100 I=1,IMAX PNM1(I) = PINIT CONTINUE NSTOP = 0 MINDEX=1 TIME(1) = 0. XPLOT(1) = XFRONT START TIME INTEGRATION DO 300 N=1,NMAX T = T+DT DO 200 I=2,IMAXM1 IF(I.LT.IFRONT) COMP = COMPL IF(I.GE.IFRONT) COMP = COMPR IF(I.LT.IFRONT) VISC = VISCL IF(I.GE.IFRONT) VISC = VISCR TERM = PHI*VISC*COMP*DX*DX/(K*DT) A(I) = 1. B(I) = -2.-TERM C(I) = 1. W(I) = -TERM*PNM1(I) CONTINUE A(1) = 99. B(1) = 1. C(1) = 0. W(1) = PLEFT A(IMAX) = 0. B(IMAX) = 1. C(IMAX) = 99. W(IMAX) = PRIGHT IF(VISCIN.EQ.VISCDP) GO TO 240 A(IFRONT) = 1./VISCL B(IFRONT) = -1./VISCL -1./VISCR C(IFRONT) = 1./VISCR W(IFRONT) = 0. CALL TRIDI(A,B,C,VECTOR,W,IMAX) DO 250 I=1,IMAX P(I) = VECTOR(I) CONTINUE PGRAD = (P(IFRONT)-P(IFRONT-1))/DX XFRONT = XFRONT - (K*DT/(PHI*VISCL))*PGRAD IFRONT = XFRONT/DX +1 DO 260 I=1,IMAX PNM1(I) = P(I) CONTINUE

388 Supercharge, Invasion and Mudcake Growth 280 300 400

WRITE(*,280) N,T,XFRONT,IFRONT FORMAT(1X,'T(',I4,')= ',E8.3,' sec, Xf= ',E8.3,' ft, I= ',I3) MINDEX = MINDEX+1 TIME(MINDEX) = T XPLOT(MINDEX) = XFRONT CONTINUE WRITE(*,10) WRITE(4,10) CALL GRFIX(XPLOT,TIME,MINDEX) STOP END

Figure 5.6a. Fortran source code (Example 5-4). INPUT PARAMETER SUMMARY: Rock core permeability (darcies): Rock core porosity (decimal nbr): Viscosity of invading fluid (cp): Viscosity, displaced fluid (cp): Compr ... invading fluid (1/psi): Compr .. displaced fluid (1/psi): Pressure at left boundary (psi): Pressure at right boundary (psi): Pressure, initial time t=0 (psi): Length of rock core sample (ft): Initial "xfront" position (feet): Integration space step size (ft): Integration time step size (sec): Maximum allowed number of steps: Number spatial DX grids selected: Time (sec) .000E+00 .600E+02 .120E+03 .180E+03 .240E+03 .300E+03 .360E+03 .420E+03

Position (ft) .500E+00 .542E+00 .587E+00 .637E+00 .692E+00 .755E+00 .833E+00 .944E+00

.100E+00 .200E+00 .100E+01 .100E+02 .300E-05 .500E-04 .100E+03 .000E+00 .000E+00 .100E+01 .500E+00 .200E-02 .100E+01 .200E+04 .500E+03

______________________________ | * | * | * | * | * | * | * | *

Figure 5.6b. Numerical results (Example 5-4). Example 5-5. Von Neumann stability of implicit time schemes.

The implicit time scheme in Example 5-4 turns out to be stable numerically, and it is of interest to examine its von Neumann characteristics for a wider class of transient flow formulations. In particular, let us consider those encompassing lineal, cylindrical, and spherical radial limits, that is 2p(x,t)/ x2 = ( c/k) p/ t (5-38) 2p/ r2 + 1/r p/ r = ( 2p/ r2 + 2/r p/ r = ( and specifically examine

c/k) p/ t

(5-39)

c/k) p/ t

(5-40)

Numerical Supercharge, Pressure and Multiphase Methods 389

2p/ r2 + N/r p/ r = (

c/k) p/ t

(5-41)

where N = 0, 1, or 2 accordingly as the flow domain is lineal, cylindrical, or spherical. (Other values for N are related to nonconventional fractal descriptions that have been the subject of recent reservoir description studies.) We will now difference Equation 5-41 as suggested by Equation 5-22, and approximate p/ r by the central difference formula (Pi+1,n - Pi-1,n)/(2 r), while the reciprocal 1/r is evaluated at the center point i. This leads to

or

(Pi-1,n - 2 Pi,n + Pi+1,n)/( r)2 + (N/ri) (Pi+1,n - Pi-1,n)/(2 r) = (

(5-42) c/k) (Pi,n - Pi,n-1)/ t

{1 - N r/(2ri)} Pi-1,n - {2 + c( r)2/(k t)} Pi,n (5-43) + {1 + N r/(2ri)} Pi+1,n= - { c( r)2/(k t)} Pi,n-1

which immediately shows how the lineal flow algorithm given in the foregoing example can be modified to handle cylindrical radial and spherical flow effects. (That is, we now have the generalized matrix coefficients A = Ai = 1 - N r/(2ri) and C = Ci = 1 + N r/(2ri) instead of unit coefficients.) This represents the only required change. In order to determine its numerical stability, we will examine Fourier wave components having the form Pi,n = n e j (i r)

(5-44)

where j = -1, is a disturbance wavenumber (e.g., see Chin, 1994 for more detailed discussion) and represents the amplification factor introduced earlier. Substitution in Equation 5-43 gives = 1/[1 + {4k t/( c( r)2)} sin2 r/2 - j kN t/( cri r)] (5-45) For stability, we require that | | < 1. This is possible provided [1 + {4k t/(

c( r)2)} sin2

r/2]2 + [kN t/(

cri r)]2 > 1 (5-46)

Since sin2 r/2 > 0, the inequality always holds if t > 0, thus guaranteeing stability. (Of course, mesh sizes must be kept small in order to reduce truncation errors and to ensure convergence to solutions of the PDE.) Unlike the conditionally stable explicit scheme studied earlier, this implicit scheme, which requires only tridiagonal matrix inversion, is

390 Supercharge, Invasion and Mudcake Growth

unconditionally stable. We have tacitly assumed a positive time step t > 0 in arriving at this stability, which is the usual case. But in Chapter 21, we will introduce reverse time integration where we have t < 0. For such applications, the stability requirements are altered, and the nature of the numerical truncation errors changes. Example 5-6. Gas displacement by liquid in lineal core without mudcake, including compressible flow transients.

The piston-like displacement of formation gas by liquid filtrate, even without the complicating presence of mudcake, poses very difficult mathematical obstacles to solution. (More accurate two-phase immiscible flow modeling is pursued later in this chapter.) To the authors’ knowledge, the problem has not been correctly solved in the literature, despite its importance in studying flows in tight gas sands. Many investigators simply assume 2p(x,t)/ x2 = (

c/k) p/ t

(5-47)

which applies to liquids only, also applies to gases, with the appropriate value of c. (Note that cwater 0.000003 psi -1, whereas gas values, highly dependent on pressure, may be several hundred times this.) But in fact, as we have noted, the relevant equation for gases is 2pm+1(x,t)/ x2 = {

m/(kp)} pm+1/ t

(5-48)

with m being Muskat’s thermodynamic exponent. (Equations 5-47 and 548 both assume lineal, isotropic flow.) Note that m = 1 for isothermal problems, whereas for adiabatic flows, m = Cv/Cp 0.7 in the case of many gases. Let us demonstrate the nature of the complexities by examining the elementary case of incompressible gas displacement by incompressible liquids. Then, we will proceed directly to a formulation that models the general displacement of gas by liquid, with moving fronts and nonnegligible transient compressibility effects. This study will highlight the importance of numerical methods, and we will also, drawing on the von Neumann stability results of Example 5-5, demonstrate how seemingly unrelated pieces of information can provide insight into designing stable, robust, computational algorithms.

Numerical Supercharge, Pressure and Multiphase Methods 391

Liquid Left i= 1

Front

Gas Right

- + 2

if-1 if

if+1

imax-1 imax

Figure 5.7. Gas displacement by liquid. Incompressible problem. For reference purposes, we will consider the flow domain shown in Figure 5.7. In the incompressible limit, Equation 5-47 reduces to d2p1(x)/dx 2 = 0, and Equation 5-48 becomes d2p2m+1(x)/dx2 = 0. We have introduced the 1 and 2 subscripts to denote the left-side liquid and right-side gas flows, respectively; these subscripts also remind us that these flows satisfy very different ordinary differential equations. These second-order equations admit the solutions p1(x) = Ax + B and p2m+1(x) = Cx + D. For our gas flow, it is important to understand that it is not the pressure p2(x) that varies linearly, but the function p2m+1(x). Now, the solution to p1(x) satisfying the condition p1(0) = PL is p1(x) = Ax + P L, while the solution to p 2m+1(x) satisfying the right-side pressure boundary condition p2(L) = PR is p2m+1(x) = C(x-L) + PRm+1. (Again, L is the length of the core.) So far, A and C are unknown, but they are, in principle, fixed by invoking the continuity of pressure and velocity at x = x f. Since p2(x) = {C(x-L) + PRm+1}1/(m+1), the continuity of pressure requires that we have Ax f + PL = {C(xf -L) + PRm+1}1/(m+1). Next, we evaluate the derivative dp2(x)/dx = {C/(m+1)}{C(x-L) + PRm+1}m/(m+1). Thus, continuity of velocity requires (1/ ) dp (x )/dx = 1 1 f (1/ 2) dp2(xf)/dx, or A/ 1 = (1/ 2){C/(m+1)}{C(xf -L) + PRm+1}m/(m+1), since permeability is uniform throughout. In summary, we solve Axf + PL = {C(xf -L) + PRm+1}1/(m+1) (5-49a) A/ 1 = (1/ 2){C/(m+1)}{C(xf -L) + PRm+1}-m/(m+1) (5-49b)

392 Supercharge, Invasion and Mudcake Growth

analytically. It is clear that A can be eliminated between Equations 549a and 5-49b, but this leaves an intractable nonlinear equation for C. Even if explicit expressions for A and C are obtained, the integration of the displacement front equation dxf/dt = -k/( ) dp1(xf)/dx = -kA/( ) leads to complexities. These worsen when transient effects due to compressibility must be modeled. Thus, we are motivated to formulate the problem numerically, drawing on the success of Example 5-4 and the stability information obtained in Example 5-5. Transient, compressible problem. The finite differencing required to model Equation 5-47 has been discussed, and in Example 54, we in fact considered displacements by dissimilar liquids having different viscosities and compressibilities. Again, the transients that arise are of two types, namely, the usual compressible ones found in well testing and those due to front motions that depend on mobility contrasts. Equation 5-48, given its similarity to Equation 5-47, can be differenced in a like manner, provided we observe that the right-side coefficient m/(kp), or c*/k in our earlier notation, is not constant but dependent on p(x,t), which continuously evolves in time. (Numerically, this pressure can be evaluated at the previous time step, at any instant in the forward time integration.) Let us recall that Example 5-4 was successfully solved by approximating Equation 5-47 using Pi-1,n - [2 + {

c( x)2/(k t)}] Pi,n + Pi+1,n = -{

(5-23)

c( x)2/(k t)}Pi,n-1

This equation still applies to the left of the moving front in Figure 5.7, where the invading liquid resides. To the right of the front, however, Equation 5-48 applies. Since implicit finite difference equations of the form given in Equation 5-42 are unconditionally stable, we attempt to difference Equation 5-48 in such a way as to take advantage of that stability. To do this, we observe that our 2pm+1(x,t)/ x2 = { m/(kp)} pm+1/ t can be expanded as 2p(x,t)/ x2 + (m/p) ( p/ x)2 = {

m/(kp)} p/ t

(5-50)

If we difference all old terms exactly as before, and approximate our new ones with the O( x)2 accurate formulas

Numerical Supercharge, Pressure and Multiphase Methods 393

(m/p)( p/ x)2 = (m/Pi,n-1){(Pi+1,n-1 - Pi-1,n-1)/(2 x)} {(Pi+1,n - Pi-1,n)/(2 x)} m/(kp) =

m/(kPi,n-1)

(5-51) (5-52)

we obtain {1 - m( x)( p/ x)i,n-1/(2Pi,n-1)}Pi-1,n - [2 + {

(5-53)

m( x)2/(kPi,n-1 t)}] Pi,n + {1 + m( x)( p/ x)i,n-1/(2Pi,n-1)}Pi+1,n = - m( x)2/(k t)

Thus, the Fortran source code developed in Example 5-4 to model displacement by dissimilar liquids can be easily modified to handle gas displacement by liquids, if to the right of the front, we instead apply Equation 5-53. The front matching condition (1/ 1) pif-1 - (1/ 1 + 1/ 2)pif + (1/ 2) pif+1 = 0 (5-29) still applies at each time step; again, it embodies pressure and velocity continuity, and is not related to fluid compressibility. In the source code modifications shown in Figure 5.8a, the Muskat exponent m is denoted EM. The numerical results displayed in Figures 5.8b and 5.8c are obtained for two different values of porosity, with all other parameters otherwise held fixed. . . C

200

240

START TIME INTEGRATION DO 300 N=1,NMAX T = T+DT DO 200 I=2,IMAXM1 IF(I.LT.IFRONT) A(I) = 1. IF(I.LT.IFRONT) B(I) =-2.-PHI*VISCL*COMPL*DX*DX/(K*DT) IF(I.LT.IFRONT) C(I) = 1. IF(I.LT.IFRONT) W(I) =-(PHI*VISCL*COMPL*DX*DX/(K*DT))*PNM1(I) IF(I.GE.IFRONT) DPDX = (PNM1(I+1)-PNM1(I-1))/(2.*DX) IF(I.GE.IFRONT) A(I) = 1. -EM*DX*DPDX/(2.*PNM1(I)) IF(I.GE.IFRONT) B(I) =-2.-PHI*VISCR*EM*DX*DX/(K*DT*PNM1(I)) IF(I.GE.IFRONT) C(I) = 1. +EM*DX*DPDX/(2.*PNM1(I)) IF(I.GE.IFRONT) W(I) =-(PHI*VISCR*EM*DX*DX/(K*DT)) CONTINUE A(1) = 99. B(1) = 1. C(1) = 0. W(1) = PLEFT A(IMAX) = 0. B(IMAX) = 1. C(IMAX) = 99. W(IMAX) = PRIGHT IF(VISCIN.EQ.VISCDP) GO TO 240 A(IFRONT) = 1./VISCL B(IFRONT) = -1./VISCL -1./VISCR C(IFRONT) = 1./VISCR W(IFRONT) = 0. CALL TRIDI(A,B,C,VECTOR,W,IMAX)

394 Supercharge, Invasion and Mudcake Growth 250

260

DO 250 I=1,IMAX P(I) = VECTOR(I) CONTINUE PGRAD = (P(IFRONT)-P(IFRONT-1))/DX XFRONT = XFRONT - (K*DT/(PHI*VISCL))*PGRAD IFRONT = XFRONT/DX +1 DO 260 I=1,IMAX PNM1(I) = P(I) CONTINUE .

Figure 5.8a. Fortran source code (Example 5-6).

INPUT PARAMETER SUMMARY: Rock core permeability (darcies): Rock core porosity (decimal nbr): Viscosity, invading liquid (cp): Viscosity of displaced gas (cp): Compr .. invading liquid (1/psi): Muskat m exponent of gas (real#): Pressure at left boundary (psi): Pressure at right boundary (psi): Pressure, initial time t=0 (psi): Length of rock core sample (ft): Initial "xfront" position (feet): Integration space step size (ft): Integration time step size (sec): Maximum allowed number of steps: Number spatial DX grids selected: Time (sec) .000E+00 .600E+02 .120E+03 .180E+03 .240E+03 .300E+03 .360E+03 .420E+03 .480E+03 .540E+03 .600E+03 .660E+03 .720E+03 .780E+03 .840E+03 .900E+03 .960E+03 .102E+04 . . .192E+04 .198E+04 .204E+04 .210E+04 .216E+04 .222E+04 .228E+04 .234E+04 .240E+04 .246E+04 .252E+04 .258E+04 .264E+04 .270E+04

Position (ft)

.100E-02 .100E+00 .100E+01 .200E-01 .300E-05 .700E+00 .200E+03 .100E+03 .100E+03 .100E+01 .200E+00 .100E-01 .100E+01 .100E+05 .101E+03

.200E+00 .221E+00 .239E+00 .256E+00 .272E+00 .287E+00 .302E+00 .316E+00 .329E+00 .342E+00 .354E+00 .366E+00 .378E+00 .389E+00 .400E+00 .411E+00 .421E+00 .431E+00

______________________________ | * | * | * | * | * | * | * | * | * | * | * | * | * | * | * | * | * | *

.563E+00 .571E+00 .578E+00 .586E+00 .593E+00 .601E+00 .608E+00 .615E+00 .622E+00 .629E+00 .636E+00 .643E+00 .650E+00 .656E+00

| | | | | | | | | | | | | |

* * * * * * * * * * * * * *

Figure 5.8b. Numerical results (Example 5-6).

Numerical Supercharge, Pressure and Multiphase Methods 395

One Fortran subtlety deserves elaboration. The 200 do-loop defines two separate difference equations for the flows left and right of the front, but the pressure updating in the 260 do-loop refers to a single pressure. So long as the front does not move more than one mesh in a time step, errors due to copying liquid pressure as gas pressure, or conversely, do not exist, assuming small capillary pressures. Pressure continuity assures that both blocks will contain identical pressures. INPUT PARAMETER SUMMARY: Rock core permeability (darcies): Rock core porosity (decimal nbr): Viscosity, invading liquid (cp): Viscosity of displaced gas (cp): Compr .. invading liquid (1/psi): Muskat m exponent of gas (real#): Pressure at left boundary (psi): Pressure at right boundary (psi): Pressure, initial time t=0 (psi): Length of rock core sample (ft): Initial "xfront" position (feet): Integration space step size (ft): Integration time step size (sec): Maximum allowed number of steps: Number spatial DX grids selected: Time (sec) .000E+00 .600E+02 .120E+03 .180E+03 .240E+03 .300E+03 .360E+03 .420E+03 .480E+03 .540E+03 .600E+03 .660E+03 .720E+03 . . .204E+04 .210E+04 .216E+04 .222E+04 .228E+04 .234E+04 .240E+04 .246E+04 .252E+04 .258E+04 .264E+04 .270E+04 .276E+04 .282E+04 .288E+04 .294E+04 .300E+04

Position (ft)

.100E-02 .500E-01 .100E+01 .200E-01 .300E-05 .700E+00 .200E+03 .100E+03 .100E+03 .100E+01 .200E+00 .100E-01 .100E+01 .100E+05 .101E+03

.200E+00 .239E+00 .272E+00 .302E+00 .329E+00 .354E+00 .378E+00 .400E+00 .421E+00 .441E+00 .461E+00 .479E+00 .497E+00

______________________________ | * | * | * | * | * | * | * | * | * | * | * | * | *

.796E+00 .807E+00 .818E+00 .829E+00 .839E+00 .850E+00 .860E+00 .870E+00 .880E+00 .890E+00 .900E+00 .910E+00 .920E+00 .929E+00 .939E+00 .948E+00 .957E+00

| | | | | | | | | | | | | | | | |

*

* * *

* * *

* * *

* * *

* * *

*

Figure 5.8c. Numerical results (Example 5-6).

396 Supercharge, Invasion and Mudcake Growth

Example 5-7. Simultaneous mudcake buildup and displacement front motion for incompressible liquid flows.

In this last exercise, we reconsider the problem of dynamically coupled invasion front motion and mudcake growth in lineal flow; this was studied analytically before, where it was solved in closed form, but we will approach its solution numerically. This is pursued for several reasons. First, we wish to demonstrate how problems with moving boundaries and disparate space scales (characterizing mudcake and rock) are formulated and solved with finite differences. Second, computational methods are ultimately needed because they are more convenient when cake compaction, time-dependent applied pressures, and formation heterogeneities are required. Because the present problem can be described analytically, we at least possess a tool with which to evaluate the quality of more approximate solution methods. In the foregoing examples, we emphasized how the effects of transients due to fluid compressibility, and the nonlinear effects of gas displacement by liquids, can be numerically modeled. For the present, we return to simple incompressible flows of liquids to illustrate the main ideas, so that we need not address the complicating, but nonetheless straightforward, effects. Here, we have instead two moving boundaries: the displacement front within the rock and the surface of the mudcake, which moves in such a way as to increase cake thickness with time. Thus, analytical and computational changes to our schemes are required. In addition, as we have noted, disparate space scales enter the numerical formulation in a subtle way: mudcakes are thin relative to the distance that the filtrate penetrates the formation. The problem domain is shown in Figure 5.9a. Flow Front

Cake

Rock

Pmud

Pres

x = -xc < 0 i=1

0 iwall

xf if

imax

Figure 5.9a. Three-layer lineal flow problem.

Numerical Supercharge, Pressure and Multiphase Methods 397

For simplicity, we assume that in the cake and rock, the permeabilities kc and kr are constant, although they can be different constants. Therefore, whether we start with d(kc dp/dx)/dx = 0 or d(kr dp/dx)/dx = 0, the permeabilities factor out, leaving d2p(x)/dx2 = 0 (5-10) in either case. Now, we can approximate Equation 5-10 with the central difference formula used earlier, namely, d2p(xi )/dx2 = {pi-1 - 2pi + pi+1}/( x)2 + O( x)2 = 0 (5-14) Our combined mudcake-growth and displacement-front-movement problem, with its clearly disparate length scales, is not unlike boundary layer or shock layer type flows in classical fluid mechanics. That is, the cake is extremely thin, while the scale of the front motion is orders of magnitude larger: any attempt to characterize both flows using the same physical measures of length is likely to result in inaccuracy. Therefore, we would like to select x, as usual, for the rock, but xc for the mudcake, with xc 0 p(L) = Pres

(5-54) (5-55)

398 Supercharge, Invasion and Mudcake Growth

for the mud and the farfield reservoir. That is, we assume that p1 = Pmud and pimax = Pres where L is the core length. This leads to the coupled equations p1 p1 -2p2 + p3 p2 - 2p3 + p4 p3 - 2p4 + p5

i = 2: i = 3: i = 4: . i = iwall . i = ifront or if : . i = imax-2: i = imax-1:

or

| | | | | | | | | |

1 1 ... ... ... ...

0 -2 1

Pmud 0 0 0

(5-56)

= 0 pimax-3 - 2pimax-2 + pimax-1 pimax-2 -2pimax-1+ pimax = 0 pimax = Pres

1 -2

1

= = = =

1

-2 1

1 -2 0

| | | | | | | | 1 | 1 |

| | | | | | | | | |

p1 p2 p3 . . . . pimax-2 pimax-1 pimax

| | | | | | | | | |

= = = = = = = = = =

| | | | | | | | | |

Pmud 0 0 .. .. .. .. 0 0 Pres

| | | | | (5-57) | | | | |

just as we had obtained for simple liquid flows. The crucial differences, however, arise from the matching conditions that need to be enforced at the mud-to-mudcake and displacement front interfaces. Let the subscripts c and r denote cake and rock properties, while mf and o denote mud filtrate and formation oil or displaced fluid. At the displacement front interface separating invading from displaced fluids, velocity continuity requires that -(kr/ mf) dpr(xf -)/dx to the left of the front equal the velocity -(kr/ o) dpr(xf +)/dx just to the right.

Numerical Supercharge, Pressure and Multiphase Methods 399

Matching conditions at displacement front. Since rock permeability cancels, we have (1/ mf) dp(xf -)/dx = (1/ o) dp(xf +)/dx. Now we will denote by if- and if+ the spatial locations infinitesimally close to the left and right of the front x = xf, which is itself indexed by i = i f. Then, we can approximate the pressure gradient dp(xf -)/dx using backward differences, whereas the gradient dp(xf +)/dx can be modeled using forward differences. (Again, differentiation through the interface itself is forbidden since the pressure gradient in general changes suddenly.) This process leads to (1/ mf) (pif- - pif-1)/ x = (1/ o) (pif+1 - pif+)/ x, or (1/ mf) (pif- - pif-1) = (1/ o) (pif+1 - pif+). Now, since surface tension is unimportant, pressure continuity requires that pif- = pif+ or simply pif. Thus, at the interface, the matching condition

(1/ mf) pif-1 - (1/ mf + 1/ o)pif + (1/ o) pif+1 = 0

(5-58)

applies, and straightforward changes are made to Equations 5-56 and 557 corresponding to the row defined by i = ifront. Unlike the central difference approximation, which is second-order accurate, our use of backward and forward differences in deriving Equation 5-58 renders it only O( x) accurate. Matching conditions at the cake-to-rock interface. It is tempting to invoke similar arguments at the index i = iwall representing the cake-to-rock interface, modifying Equation 5-58 in the obvious manner to account for differences between mudcake and rock permeabilities. This would lead to kc piwall-1 - (kc + kr) piwall + kr piwall+1 = 0 (5-59) In Equation 5-59, viscosity drops out identically, since the same filtrate flows through the mudcake as through the flushed zone in the rock. However, any attempt to use Equation 5-59 would produce gross numerical error and poor physical resolution in the mudcake, since identical grid sizes x are implicit in its derivation. Also, the fact that Equation 5-59 is not as numerically stable as pi-1 - 2pi + pi+1 = 0, say, would lead to inaccuracies if our algorithm were extended to transient compressible flows. Thus, we need to return to basics and consider the more general statement - (kc/ mf) dp(xwall -)/dx = - (kr/ mf) dp(xwall+)/dx (5-60)

400 Supercharge, Invasion and Mudcake Growth

Since physical length scales in the mudcake are much smaller than those characterizing the rock, we wish to use the mesh xs in the cake, and the usual x in the rock, such that xs xc,equil). We have selected a cross-section of examples, ranging from constant density, two-fluid flows without mudcake to flows with transients due to compressibility to problems with mudcake thickening with time. Naturally, other combinations of problems with lineal, radial, and spherical geometries, single or multiple fluids in formations, compressible mudcake, general transient effects, and so on, can be modeled by combining appropriate pieces of theory and source code. Finally, this author warns prospective users of canned computational fluid mechanics software of likely formulation errors. In an environment driven by high-resolution graphics and user-friendly screen interfaces, it is important to understand precisely which equations are solved and the methodology employed. The highly specialized problems typical of formation invasion applications are unlikely to be pre-programmed in commercial solvers; users should direct technical questions to development and not sales staff.

5.2 Forward and Inverse Multiphase Flow Modeling. In this final section, we present new ideas on immiscible and miscible flow modeling with respect to formation invasion and time lapse analysis. In particular, we first study forward simulation methods, where the evolution of an initial state dynamically in time is considered. Then, we focus on inverse time lapse analysis applications that attempt to uncover formation evaluation information from data collected by well logging instruments. Whereas our earlier models assume piston-like flows associated with discontinuous step changes in fluid properties, the

406 Supercharge, Invasion and Mudcake Growth

forward and inverse, miscible and immiscible flow models here are generally smeared by diffusion, stretched by geometric spreading, and characterized by steep saturation shock fronts. We pose, and importantly solve, what we call the resistivity migration problem, wherein the distinct fronts from which prescribed smeared profiles evolve are recovered by reverse diffusion using methods similar to the parabolized wave methods in seismic migration. We also show how the sharp saturation discontinuities obtained in immiscible water-oil flows can be unshocked in order to recover the original smooth saturation distributions for further information processing. The work in this chapter is not a tutorial on multiphase flow, although all derivations do proceed from first principles. This book assumes some exposure to reservoir flow analysis, for example, to concepts such as Darcy’s law, miscible flow, relative permeability, and capillary pressure, but it does not require any exposure to the research literature or any experience in numerical simulation. Problem hierarchies. We first discuss immiscible two-phase flows in the Buckley-Leverett limit of zero capillary pressure, and we provide exact, analytical, closed-form solutions for early-time, near-well invasion problems, which can be modeled by a planar flow. Also, since mud filtration rates are at their largest, the effects of capillary pressure can be ignored. For the problem in which saturation shocks form, shockfitting is used to obtain the correct physical solution. Then, we turn our attention to miscible flows, where the competing effects of convection and diffusion are important. (This model can be used to refine the waterphase description obtained in the immiscible discussion.) Here new closed form solutions are given, and numerical models are developed. Using these models, the basic ideas behind resistivity migration and undiffusion are introduced using lineal and radial flow examples. With these discussions completed, we proceed to two more difficult problems. First, we consider deep, late-time invasion, when filtration rates are likely to be the smallest; here, the effects of radial geometric divergence and capillary pressure cannot be ignored. A two-phase flow model is formulated which assumes that a highly impermeable mudcake controls the filtration rate into the flushed zone. This is solved numerically for a range of parameters that alter the ratio of inertial to capillary forces from very low to very high, in the latter case, showing how shock formation as suggested by the Buckley-Leverett limit of zero capillary pressure is recovered. We show how nearly discontinuous saturation solutions can be reversed or unshocked using a high-order

Numerical Supercharge, Pressure and Multiphase Methods 407

accurate numerical finite difference model. Second, the immiscible flow problem where mud filtrate invades a radial core is considered, but the usual assumption that a highly impermeable mudcake controls the flow rate into the core is not invoked. Thus, the model applies to mud filtrate invasion into very low permeability hydrocarbon zones with mixed water and oil. This combined analytical and computational model is developed using ideas obtained by integrating our two-phase flow formulation with the numerical mudcake growth model designed earlier. Finally, because much of two-phase flow modeling, by virtue of its inherent mathematical difficulties, is necessarily numerical, we refer the reader to the discussion on artificial viscosity, numerical diffusion, and convergence to correct solutions given in Chin (2017). 5.2.1 Immiscible Buckley-Leverett lineal flows without capillary pressure. In this section, we will study the immiscible, constant density flow through a homogeneous lineal core where the effects of capillary pressure are insignificant. In particular, we will derive exact, analytical, closed form solutions for the forward modeling problem for a single core. These solutions include those for saturation, pressure and shock front velocity, for arbitrary relative permeability and fractional flow functions. We will determine what formations properties can be inferred, assuming the existence of a propagating front, when the front velocity is known. The Darcy velocities are qw = -(kw/ w) Pw/ x (5-74)

qnw = -(knw/ nw) Pnw/ x (5-75) where w and nw are viscosities, and kw and knw are relative permeabilities, the subscripts w and nw here denoting wetting and nonwetting phases. For mathematical simplicity, we assume zero capillary pressures Pc, so that Pnw - Pw = Pc = 0 (5-76) For water injection problems, this assumes that the displacement is fast (or, inertia dominated), so that surface tension can be neglected; however, when water breakthrough occurs, the assumption breaks down locally. In formation invasion, this zero capillary pressure assumption may be valid during the early periods of invasion near the well, when high filtrate influx rates are possible, as the resistance offered by

408 Supercharge, Invasion and Mudcake Growth

mudcakes is minimal. For slow flows, capillary pressure is important; but generally, fast and slow must be characterized dimensionlessly in the context of the model. Since Pnw = Pw holds, the pressure gradient terms in Equations 5-74 and 5-75 are identical. If we divide Equation 5-75 by Equation 5-74, these cancel and we obtain qnw = (knw w/kw nw)qw (5-77) At this point, we invoke mass conservation, and assume for simplicity a constant density, incompressible flow. Then, it follows that qw/ x = - Sw/ t (5-78) qnw/ x =-

Snw/ t

(5-79)

where Sw and Snw are the wetting and non-wetting saturations. Since the fluid is incompressible, these saturations must sum to unity; that is, Sw + Snw = 1 (5-80) Then, upon adding Equations 5-78 and 5-79, and simplifying with Equation 5-80, it follows that (qw + qnw)/ x = 0 (5-81) Thus, we conclude that a one-dimensional, lineal, constant density flow without capillary pressure admits the general total velocity integral qw + qnw = q(t) (5-82) where an arbitrary functional dependence on time is permitted. We have not yet stated what q(t) is, or how it is to be determined; this crucial issue is discussed in detail later. It is convenient to define the fractional flow function f w for the wetting phase by the quotient fw = qw/q (5-83) Then, for the nonwetting phase, we obtain fnw = qnw/q = (q - qw )/q = 1 - fw

(5-84)

where we used Equation 5-82. Equations 5-83 and 5-84 can be rewritten as q w = q fw (5-85) qnw = q (1 - fw)

(5-86)

Substituting into Equation 5-77, the function q(t) drops out, so that 1 - fw = (knw w/kw nw) fw (5-87) fw(Sw, w/ nw) = 1/{1 + (knw w/kw nw)}

(5-88)

Numerical Supercharge, Pressure and Multiphase Methods 409

The function fw(Sw, w/ nw) in Equation 5-88, we emphasize, is a function of the constant viscosity ratio w/ nw and the saturation function Sw itself. According to Equation 5-85, qw must likewise be a function of Sw. Thus, we can rewrite Equation 5-78 with the more informed nomenclature Sw/ t =- -1 qw/ x =- -1q fw(Sw, w/ nw)/ x =- -1q dfw(Sw, w/ nw)/Sw Sw/ x (5-89) or Sw/ t + {q(t)/ } dfw(Sw, w/ nw)/dSw Sw/ x = 0 (5-90) Equation 5-90 is a first-order nonlinear partial differential equation for the saturation Sw(x,t). Its general solution can be easily constructed using concepts from elementary calculus. The total differential dSw for the function Sw(x,t) can be written in the form dSw = Sw/ t dt + Sw/ x dx If we divide Equation 5-91 by dt, we find that dSw/dt = Sw/ t + dx/dt Sw

(5-91) (5-92)

Comparison with Equation 5-90 shows that we can certainly set dSw/dt = 0 (5-93) provided dx/dt = {q(t)/ } dfw(Sw, w/ nw)/dSw (5-94) Equation 5-93 states that the saturation Sw is constant along a trajectory whose speed is defined by Equation 5-94. (This constant may vary from trajectory to trajectory.) In two-phase immiscible flows, we conclude that it is the characteristic velocity dx/dt = {q(t)/ } dfw(Sw, w/ nw)/dSw that is important, and not the simple dx/dt = q(t)/ obtained for single-phase flow. But when shocks form, it turns out that Equation 5-112 applies. Example boundary value problems. If the filtration rate q(t) is a constant, say qo, Equation 5-94 takes the form dx/dt = {qo/ } dfw(Sw, w/ nw)/dSw (5-95)

410 Supercharge, Invasion and Mudcake Growth

Since the derivative dfw(Sw, w/ nw)/dSw is also constant along trajectories (as a result of Equation 5-93), depending only on the arguments Sw and w/ nw, it follows that Equation 5-95 can be integrated in the form x - {(qo/ ) dfw(Sw, w/ nw)/dSw} t = constant (5-96) That Sw is constant when x - { ...} t is constant can be expressed as Sw(x,t) = G(x - {(qo/ ) dfw(Sw, w/ nw)/dSw}t) (5-97) where G is a general function. Note that the method by which we arrived at Equation 5-97 is known as the method of characteristics (Hildebrand, 1948). General initial value problem. We now explore the meaning of Equation 5-97. Let us set t = 0 in Equation 5-97. Then, we obtain Sw(x,0) = G(x) (5-98) In other words, the general saturation solution to Equation 5-90 for constant q(t) = qo satisfying the initial condition Sw(x,0) = G(x), where G is a prescribed initial function, is exactly given by Equation 5-97! Thus, it is clear that the finite difference numerical solutions offered by some authors are not really necessary because problems without capillary pressure can be solved analytically. Actually, such computational solutions are more damaging than useful because the artificial viscosity and numerical diffusion introduced by truncation and round-off error smear certain singularities (or, infinities) that appear as exact consequences of Equation 5-90. Such numerical diffusion, we emphasize, appears as a result of finite difference and finite element schemes only, and can be completely avoided using the more laborintensive method of characteristics. For a review of these ideas, refer to Chin (2017). As we will show later, capillary pressure effects become important when singularities appear; modeling these correctly is crucial to correct strength and shock position prediction. To examine how these singularities arise in the solution of Equation 5-90, take partial derivatives of Equation 5-97 with respect to x, so that Sw(x,t)/ x = {G’}{1 - t (qo/ ) d2fw/dSw2 Sw(x,t)/ x} (5-99) Solving for Sw(x,t)/ x, we obtain Sw(x,t)/ x = G’/{1 + t (qo/ ) (G') d2fw/dSw2} (5-100)

Numerical Supercharge, Pressure and Multiphase Methods 411

Now, the fractional flow function f w(Sw, w/ nw) is usually obtained from laboratory measurement and is to be considered as prescribed for the purposes of analysis. Let us focus our attention on the denominator of Equation 5-100. If it remains positive, then the spatial derivative Sw(x,t)/ x is well-behaved for all time. If, however, (qo/ ) (G’) d2fw/dSw2 < 0, then it follows that the denominator vanishes in the finite breakthrough time given by tbreakthrough = - /{qo G’ d2fw/Sw2} (5-101) at which point the spatial derivative of saturation Sw(x,t)/ x becomes singular, approaching infinity, increasing without bound. In reservoir engineering, this is known by various terms including water breakthrough, shocks, or saturation discontinuities. Since Sw undergoes rapid change, it is also said to be multivalued, or doublevalued. Whether or not this discontinuity exists in reality cannot be determined within the scope of our zero capillary pressure analysis. When saturation gradients become large, the capillary forces that we have neglected may become important, and cannot be excluded a priori in any analysis. When infinite saturation gradients form, as they have formed here, low-order theory breaks down, and recourse to a model that offers finer physical resolution is required. General boundary value problem for infinite core. Note that the argument of G{ } appearing in the solution of Equation 5-97 takes the general form x - {(qo/ ) dfw(Sw, w/ nw)/dSw} t. There is nothing sacred about this expression, and we could have multiplied it by two, five, or - /(qo dfw/dSw ). With the last choice, we can rewrite Equation 5-97 as Sw(x,t) = H{t - x /(qo dfw/dSw )} (5-102) If we set x = 0 throughout in Equation 5-102, we find that Sw(0,t) = H(t). Thus, the saturation solution to Equation 5-90 satisfying the boundary condition Sw(0,t) = H(t), where H is a prescribed function, is given by Equation 5-102. Variable q(t). If the filtration rate q(t) is a general function of time, we return to Equation 5-94 and rewrite it in the differential form dx = {q(t)/ } dfw(Sw, w/ nw)/dSw dt (5-103)

412 Supercharge, Invasion and Mudcake Growth

Since Equation 5-93 states that Sw is still constant along a trajectory, the term dfw(Sw, w/ nw)/dSw is likewise constant. Thus, the integral of Equation 5-103 is simply x - -1dfw(Sw, w/ nw)/dSw q(t) dt = constant (5-104) where q(t) dt denotes the indefinite integral (e.g., qo dt = qot is obtained for our constant rate problem). Following reasoning similar to that leading to Equation 5-97, since Sw is constant whenever the left side of Equation 5-104 is constant, we have the equivalent statement Sw(x,t) = G(x - -1dfw(Sw, w/ nw)/dSw q(t) dt) (5-105) Equation 5-105 is the general saturation solution for time-dependent q(t). If the function q(t) dt vanishes for t = 0, this solution satisfies the initial condition specified by Equation 5-98. If the function does not vanish, some algebraic manipulation is needed to obtain the correct format. Mudcake-dominated invasion. So far, we have not stated how the velocity q(t), possibly transient, is determined. If we assume that the flow at the inlet to our lineal core is controlled by mudcake, as is often the case, the fluid dynamics within the core will be unimportant in determining q(t). (This assumption is removed in our last example.) Then, the general mudcake model in for single-phase filtrate flows provides the required q(t). In fact, xf (t) = eff-1 {2k1(1- c)(1-fs)(pm-pr)t/( ffs)} (5-106) when the effect of spurt and the presence of the formation are neglected. The fluid influx rate q(t) through the mudcake is therefore given by q(t) = eff dxf (t)/dt = ½ t-½ {2k1(1- c)(1-fs)(pm-pr)/( ffs)}

(5-107)

which can be substituted in the nonlinear saturation equation Sw/ t + {q(t)/ } dfw(Sw, w/ nw)/dSw Sw/ x = 0

(5-108)

This can be integrated straightforwardly using the method of characteristics. So long as singularities and saturation fronts do not form, saturations obtained as a function of space and time will be smooth, and shocks will not appear. Shock velocity. We will consider the problem that arises when saturation shocks do form. (Problems with smooth but rapidly varying properties are addressed in our capillary pressure analysis.) In order to discuss saturation discontinuities and steep gradients, we must complete the formulation by specifying initial and boundary conditions. We

Numerical Supercharge, Pressure and Multiphase Methods 413

assume that at t = 0, our core is held at the constant water saturation S wi throughout, where the italicized i denotes initial conditions. At the left boundary x = 0, where fluid influx occurs, we assume that the water saturation is fixed at a constant value Swl where the italicized l denotes left. (Normally, this value is unity for water filtrates, but it may differ for certain water-oil muds.) That is, we take Sw(x,0) = Swi (5-109) Sw(0,t) = Swl (5-110) As discussed, we can expect shockwaves and steep saturation discontinuities to form in time, depending on the exact form and values of our fractional flow functions and initial conditions. We will assume that the particular functions do lead to piston-like shock formation very close to the borehole. The shock boundary value problem just stated can be solved in closed form, and, in fact, is the petroleum engineering analogue of the classic nonlinear signaling problem ( t + c( ) x = 0,

= o for x > 0, t = 0, and = g(t) for t > 0, x = 0) discussed in the wave mechanics book of Whitham (1974). We will not rederive the mathematics, but will draw on Whitham’s results only. For brevity, define for convenience the function Q(Sw) = {q(t)/ } dfw(Sw, w/ nw)/dSw (5-111) where q(t) is given in Equation 5-107. It turns out that the shock propagates with a shock speed equal to Vshock = {Qw(Swl) - Qw(Swi)}/(Swl - Swi) (5-112) If the injection rate q(t), the core porosity , and the speed of the front Vshock separating saturations Swl from Swi are known, then since Swl is available at the inlet of the core, Equations 5-111 and 5-112 yield information relating the initial formation saturation Swi to the fractional flow derivative dfw(Sw, w/ nw)/dSw. Equation 5-88 shows that the fractional flow function satisfies fw(Sw, w/ nw) = 1/{1 + (knw w/kw nw)}. Thus, if additional lithology information is available about the form of the relative permeability functions, the viscosity ratio w/ nw can be extracted, thus yielding nw. We emphasize that this

414 Supercharge, Invasion and Mudcake Growth

solution for the nonlinear saturation problem does not apply to the linear single-phase flow where red water displaces blue water. Pressure solution. Now we derive the solution for the corresponding transient pressure field. Let us substitute Darcy’s laws qw = - (kw/ w) Pw/ x and qnw = - (knw/ nw) Pnw/ x) into Equation 5-82 (or qw + qnw = q(t)). Also, from Equation 21-3, we find that Pnw = Pw. Thus, we obtain the governing pressure equation {(kw(Sw )/ w) + (knw(Sw )/ nw)} Pw/ x = - q(t) (5-113) so that the pressure gradient satisfies Pw/ x = - q(t)/{(kw(Sw )/ w) + (knw(Sw )/ nw)} (5-114) Since the saturation function Sw(x,t), following Whitham’s solution to the signaling problem is a simple step function in the x direction whose hump moves at the shock velocity, we conclude that the pressure gradient in Equation 5-114 takes on either of two constant values, depending on whether Sw equals Swi or Swl locally. Thus, on either side of the shock front, we have different but linear pressure variations with space, when time is held fixed. This situation is shown in Figure 5.10. At the shock front itself, the requirement that pressure be continuous and single-valued, a consequence of our zero capillary pressure assumption, is itself sufficient to uniquely define the time-varying pressure distribution across the entire core. Now we give the computational procedure. At the left of the core, the saturation specification Swl completely determines the value of the linear variation Pw (Swl)/ x, following the arguments of the preceding paragraph. Since the exact value of pressure P l is assumed to be known at x = 0 (that is, the interface between the rock core and the mudcake), knowledge of the constant rate of change of pressure throughout completely defines the pressure variation starting at x = 0. Unlike reservoir engineering problems, we are not posing a pressure problem for the core in order to calculate flow rate; our flow rate is completely prescribed by the mudcake. In this problem, saturation constraints fix both pressure gradients, which in turn fix the right-side pressure. The radial flow extension of this procedure leads to an estimate for reservoir pore pressure.

Numerical Supercharge, Pressure and Multiphase Methods 415 P

Shock front x

Figure 5.10. Pressure in lineal core. In finite length core flows without mudcakes, it is appropriate to specify both the left and right pressures Pl and Pr, and determine the corresponding q(t). Since q(t) is now unknown, the shock velocity cannot be written a priori, so that the manner in which the step solution for saturation propagates is uncertain. Strong nonlinear coupling between the pressure and saturation equations is obtained, and iterative numerical solutions are required, which will be discussed later. Before embarking on radial flows with capillary pressure, we turn to multiphase flows of miscible fluids, where diffusive processes predominate. 5.2.2 Molecular diffusion in fluid flows.

Fluid flows need not be purely homogeneous, as in single-phase flows, nor need they be definable by clearly discernible differences in properties, as in multiphase immiscible flows. For simplicity, let us consider mixtures having two components only; the composition of the mixture is described by the concentration C, defined as the ratio of the mass of one component to the total mass of the fluid in a given volume element. With the passage of time, this concentration changes in two ways. When there is macroscopic motion of the fluid, mechanical mixing of the flow results; if we ignore thermal conduction and internal friction, this change is thermodynamically reversible and does not result in energy dissipation. But a change in composition will also occur by the molecular transfer of the components from one part of the fluid to another. The equalization of concentration by this direct change of composition is called diffusion. Diffusion is an irreversible process; like thermal conduction and internal viscous friction, it is one of the sources of energy dissipation in fluid mixtures. If we denote by the total density of the fluid, the equation of mass continuity for the total mass of the fluid is, as before,

416 Supercharge, Invasion and Mudcake Growth

/ t+ ( q) = 0 (5-115) where q is the velocity vector and denotes the gradient operator from vector calculus. The corresponding momentum, or Darcy equations, remain unchanged. In the absence of diffusion, the composition of any given fluid element would remain unchanged as it moved about. That is, the total derivative dC/dt would be zero, so that dC/dt = C/ t + q C = 0. This can be written, using Equation 5-115 for mass continuity, in the form ( C)/ t + ( qC) = 0 (5-116) as a continuity equation for a component of the mixture. But when diffusion occurs, besides the flux qC of the component under investigation, there is another flux which results in the transfer of the components even when the fluid mass as a whole is at rest. The general concentration equation describing both mass transport and diffusion takes the form (Peaceman, 1977) 2C C/ t + q C = (5-117) where is the diffusivity coefficient. In radial cylindrical coordinates, Equation 5-117 can be written as C/ t + v(r) C/ r = { 2C/ r2 + (1/r) C/ r} (5-118) where v(r) is the underlying radial Darcy velocity, for example, as obtained in Chin (2017) or Chin and Proett (2005), or 2C/ r2 C/ t + (v(r) - /r) C/ r = (5-119) What might be a typical value of ? Peaceman and Rachford (1962), for example, assumed a value of = 10-3 sq cm/sec. This corresponded to an experimental situation where oil was flooded by solvent of equal density, from a thin rectangular channel in Lucite packed with uniform Ottawa sand. We will discuss Equation 5-119 in more detail later, but for now, it is useful to consider lineal flows for which motivating exact analytical solutions are available. Exact lineal flow solutions. For one-dimensional lineal flows, the convective-diffusion equation for a constant velocity U takes the form 2C/ x2 C/ t + U C/ x = (5-120) Let us assume that at t = 0, the concentration varies linearly with x in the form C0 + x, whereas at the inlet boundary x = 0, the concentration is imposed in the form C1 + t. While the linear variations appear

Numerical Supercharge, Pressure and Multiphase Methods 417

somewhat limiting, they can be generally interpreted as first-order Taylor series representations to more general initial and boundary conditions. In mathematical form, C(x > 0,0) C(0,t > 0)

= C0 + = C1 +

x t

(5-121) (5-122)

The exact solution to this initial-boundary value problem is easily obtained using Laplace transforms and can be shown to be C(x,t) = C0 + (x-Ut/ ) (5-123) + ½ (C1- C0){erfc ½(x-Ut/ )/( t)½ + eUx/ erfc ½(x+Ut/ )/( t)½} + {( + U/ )/(2U/ )}{(x+Ut/ )eUx/ erfc ½(x+Ut/ )/( t)½ - (x-Ut/ ) erfc ½(x-Ut/ )/( t)½}

where erfc denotes the complementary error function. These solutions show that, in a coordinate system moving with the speed U, the width of the transition zone increases and smears with time (Marle, 1981). Several limits of Equation 5-123 come to mind. If = = 0, C(x,t) = C0 +½ (C1 -C0){erfc ½(x-Ut/ )/( t)½+eUx/ erfc ½(x+Ut/ )/( t)½} (5-124)

If, in addition, U = 0, C(x,t) = C0 + ½ (C1 - C0){erfc ½x/( t)½ + erfc ½x/( t)½}

(5-125) This solution, at least in lineal flows, describes the large-time behavior in problems with thick mudcakes that effectively shut off the influx of filtrate. Numerical analysis. The numerical formulation for the heat-like 2C/ x2 given in Equation 5-120 equation C/ t + U C/ x = proceeds in the same manner as that for Equation 5-19, or 2p(x,t)/ x2 = ( c/k) p/ t, since the former can be written as 2C/ x2 = / C/ t (5-126) in the U = 0 limit. In this limit, Equations 5-22 and 5-23 apply without change. If we replace P in Equation 5-23 by C, and c/k by / , we have Ci-1,n - [2 + { ( x)2/( t)}] Ci,n + Ci+1,n (5-127) = - { ( x)2/(

t)}Ci,n-1

418 Supercharge, Invasion and Mudcake Growth

Then, the algorithm and Fortran implementation developed for compressible transient flows applies without change. In the limit when U does not vanish, we write the governing PDE in the form 2C/ x2 = / C/ t + U/ C/ x, or 2C/ x2 - U/

C/ x = /

C/ t

(5-128)

Applying central differences to all spatial derivatives and backward differences to the first-order time derivative, we have (Ci-1,n -2Ci,n +Ci+1,n)/ x2 - (U/ ) (Ci+1,n - Ci-1,n)/(2 x) =

or

/ (Ci,n - Ci,n-1)/ t

(5-129)

Ci-1,n -2Ci,n + Ci+1,n - (U x2/ ) (Ci+1,n - Ci-1,n)/(2 x) (5-130) = {( x2)/( t)}(Ci,n - Ci,n-1)

Thus, we again have the familiar tridiagonal difference equation [1 + U x/2 ] Ci-1,n (5-131) - [2 + ( x2)/( t)] Ci,n + [1 - U x/2 ] Ci+1,n = - ( x2)/( t)} Ci,n-1 which bears superficial resemblance to our pressure equation for radial flows. Peaceman and Rachford (1962) discuss this model in their investigation of miscible reservoir flow modeling. Also, Lantz (1971) offers very enlightening discussions on numerical diffusion, and in particular examines the types of numerical diffusion and truncation error that arise in different kinds of discretization schemes. For example, instead of the central differencing used in Equation 5-129 for the first derivative, we might have assumed C/ x (U/ ) (Ci+1,n - Ci,n)/ x (5-132a)

or

C/ x

(U/ ) (Ci,n - Ci-1,n)/ x

(5-132b)

C/ x

(U/ ) (Ci+1,n-1 - Ci,n-1)/ x

(5-132c)

C/ x (U/ ) (Ci,n-1 - Ci-1,n-1)/ x (5-132d) We caution that issues beyond accuracy are involved. As noted earlier, the computed diffusivity is not the physical diffusivity , but a combination of that plus numerical diffusion due to truncation errors.

Numerical Supercharge, Pressure and Multiphase Methods 419

Diffusion in cake-dominated flows. Close to the well, immiscible flows containing propagating saturation discontinuities may exist. But very often, flows are obtained that do not contain shocks. These include immiscible flows with and without capillary pressure, and miscible flows governed by highly diffusive processes, where discontinuities never form. Flow Front Cake

Rock core

"1"

"2"

Figure 5.11. Diffusive front motion. For purely diffusive flows, sharp (fresh versus saline water resistivity) discontinuities always smear in time. The dynamics of such flows are very important in log interpretation. For this class of problems, the speed of the fresh-to-saline water interface slows appreciably once the mudcake establishes itself at the borehole walls, as we have demonstrated earlier. This is especially true in the case of radial flows, where geometric spreading significantly slows the front. For such problems, the speed of the underlying flow U can be neglected after some time, when diffusion predominates. The problem is shown in Figure 5.11. Resistivity migration. Let us suppose that the ultimate electromagnetic wave resistivity tool were available and capable of determining the exact, continuous, or even discontinuous variation of electrical properties in the formation as a function of the radial coordinate r in a concentric problem. (Resistivity and concentration are used interchangeably, since they are related through logging tool measurements.) In order to use the piston-like displacement results assumed previously for time lapse analysis, a front having a distinct constant radius would have to be inferred from a generally continuous distribution of resistivities. Typically, this is done in any of several ways: by eye, by arithmetic, geometric, or harmonic averaging, or by using the improved method of Chin et al. (1986) as discussed earlier, all of which are ad hoc. Actually, a simple and exact solution to this

420 Supercharge, Invasion and Mudcake Growth

problem is possible. What we wish to do, at any particular instant in time, given a smeared concentration profile that will generally vary with radial position, is to extrapolate that profile back to time t = 0 when the front is truly discontinuous. This problem formulation appears incredible, since diffusion is physically irreversible. For example, in heat transfer, the effect of an instantaneous point heat source is a diffusion width that grows with time; the diffused temperature distribution never evolves backward to become a point source. However, while physical diffusion is irreversible, the computational process isn’t. It turns out that we can undiffuse a smeared front using reverse diffusion and recover original sharp transitions by marching backward in time using a host diffusion equation. Of course, the initial profile must be sufficiently transient, since a steady-state profile is obviously devoid of historical content. Such migration methods are used in seismic imaging and geophysics. In particular, wave-equation-based methods, introduced by Claerbout (1985a,b) at M.I.T. and Stanford, and formalized by the multiple scale analyses in Chin (1994), lead to a parabolized wave equation which is just the heat equation in disguise. By applying these methods to our smeared concentrations, we can recover any sharp discontinuities, if they in fact existed. In doing so, we obtain the location of the radial front for use in the plug-flow time lapse analysis equations developed and used in earlier discussions. In addition to this front position, we can uncover the time scale of the reverse diffusion process as a byproduct of the reverse time integration. The key idea is simple: differential equations of evolution do exist, and their application to deconvolution is not at all unusual. There are some problems, however. Since the end starting conditions are likely to be complicated functions of space, determined at discrete points, the reverse diffusion must be accomplished numerically in time. But finite difference methods produce truncation and round-off errors that are associated with their own thermodynamic irreversibility and entropy production. Thus, the scheme has to be designed so that it is perfectly reversible in order to be usable for time lapse analysis purposes. This is accomplished by retaining the next highest order finite difference contributions neglected in Chapter 20. Lineal diffusion and “un-diffusion” examples. For simplicity, consider the fresh-to-saline water invasion problem, where mudcake forms and grows at the inlet entrance. At first, mud filtrate motions are extremely rapid, and fluid movements dominate the convection-diffusion

Numerical Supercharge, Pressure and Multiphase Methods 421

process. However, as mudcake forms, the influx of filtrate decreases rapidly with time, and eventually, diffusion dominates the dynamics. For simplicity, we first study lineal flows where the effects of radial geometric spreading are unimportant. In our examples, because fluid 2C/ x2 = convection is negligible, we consider C/ t. For numerical purposes, we fix the left-side (x = 1) concentration at C = 10%, while the right (x = 11) is held at C = 90%. For visual clarity, all concentrations to the left side of x = 6 are initially 10%, while those values to the right are 90%. There are several objectives for pursuing the test cases described here. For one, if the initial value problem when time reversal starts has progressed to steady-state, straight-line conditions, it is clear that all transient information will have been lost and that no amount of reverse diffusion will return the steady-state system to its initial step profile. (The steady-state solution is obtained by solving d2C/dx2 = 0, taking the straight line joining C values at the left and right boundaries.) The degree of smear and its percentage approach to steady state are therefore important research questions. Second, we need to determine if the method is applicable to radial flows, if it proves successful for lineal ones. This objective is important because any spatial distribution of concentration obtained radially is a consequence of both diffusion and geometric spreading. Geometric spreading worsens the undiffusion process because diffusion effects are less clear. The method must account for both mechanisms if the initial step profile is to be recovered properly. In the following results, we de-emphasize the values of the numerical inputs themselves; note that ten one-foot grid blocks were selected, with 500 time steps taken forward, then followed by 500 taken backwards. The real parameters of computational significance, of course, are the dimensionless ones that affect truncation errors. Solutions are both tabulated and plotted using a simple ASCII text plotter; the wiggles in our plotter are due to character spacing and font control issues and not instability, as tabulated results clearly show. Observe the strong initial discontinuity in the C(x,t) profile used. The bottom solution in Figure 5.12a represents the final spatial profile obtained before we reverse integrate in time. The profile is smeared, almost to the point where a straight-line steady solution is obtained. Carefully study the reverse diffusion results in Figure 5.12b.

422 Supercharge, Invasion and Mudcake Growth Concentration vs distance @ time .5000E+00 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .100E+02 | * .300E+01 .100E+02 | * .400E+01 .100E+02 | * .500E+01 .103E+02 | * .600E+01 .897E+02 | * .700E+01 .900E+02 | * .800E+01 .900E+02 | * .900E+01 .900E+02 | * .100E+02 .900E+02 | * .110E+02 .900E+02 | * Concentration vs distance @ time .1000E+02 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .100E+02 | * .300E+01 .100E+02 | * .400E+01 .102E+02 | * .500E+01 .153E+02 | * .600E+01 .847E+02 | * .700E+01 .898E+02 | * .800E+01 .900E+02 | * .900E+01 .900E+02 | * .100E+02 .900E+02 | * .110E+02 .900E+02 | * Concentration vs distance @ time .1000E+03 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .103E+02 | * .300E+01 .117E+02 | * .400E+01 .176E+02 | * .500E+01 .351E+02 | * .600E+01 .649E+02 | * .700E+01 .824E+02 | * .800E+01 .883E+02 | * .900E+01 .897E+02 | * .100E+02 .900E+02 | * .110E+02 .900E+02 | * Concentration vs distance @ time .2495E+03 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .124E+02 | * .300E+01 .172E+02 | * .400E+01 .265E+02 | * .500E+01 .413E+02 | * .600E+01 .587E+02 | * .700E+01 .735E+02 | * .800E+01 .828E+02 | * .900E+01 .874E+02 | * .100E+02 .892E+02 | * .110E+02 .900E+02 | *

Figure 5.12a. A diffusing lineal flow. Despite truncation errors after 1,000 time steps, the last tabulationplot in Figure 5.12b shows that we have recaptured the step initial condition in three ways: we (1) obtained the exact left-to-right concentration values of 10% and 90%, (2) correctly imaged the transition boundary between the x = 5 to 6 ft nodes, and (3) extracted the two solutions just quoted using exactly the same number of backward time steps as we did forward time steps. In time lapse analysis, the front position obtained in the last plot might be used as input to the pistondisplacement formulas derived earlier. Similar results for radial flows can be obtained.

Numerical Supercharge, Pressure and Multiphase Methods 423 Concentration vs distance @ time .2000E+03 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .116E+02 | * .300E+01 .154E+02 | * .400E+01 .243E+02 | * .500E+01 .401E+02 | * .600E+01 .599E+02 | * .700E+01 .757E+02 | * .800E+01 .846E+02 | * .900E+01 .883E+02 | * .100E+02 .896E+02 | * .110E+02 .900E+02 | * Concentration vs distance @ time .1000E+03 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .103E+02 | * .300E+01 .117E+02 | * .400E+01 .176E+02 | * .500E+01 .350E+02 | * .600E+01 .650E+02 | * .700E+01 .824E+02 | * .800E+01 .883E+02 | * .900E+01 .897E+02 | * .100E+02 .899E+02 | * .110E+02 .900E+02 | * Concentration vs distance @ time .8000E+01 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .999E+01 | * .300E+01 .100E+02 | * .400E+01 .105E+02 | * .500E+01 .130E+02 | * .600E+01 .870E+02 | * .700E+01 .895E+02 | * .800E+01 .899E+02 | * .900E+01 .900E+02 | * .100E+02 .900E+02 | * .110E+02 .900E+02 | * Concentration vs distance @ time .0000E+00 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .999E+01 | * .300E+01 .100E+02 | * .400E+01 .106E+02 | * .500E+01 .833E+01 |* .600E+01 .917E+02 | * .700E+01 .894E+02 | * .800E+01 .900E+02 | * .900E+01 .900E+02 | * .100E+02 .900E+02 | * .110E+02 .900E+02 | *

Figure 5.12b. An “un-diffusing” lineal flow. Radial diffusion and “un-diffusion” examples. We repeat the prior problem with the same inputs, except that the equation 2C/ x2 = C/ t is replaced by its cylindrical radial counterpart, namely, ( 2C/ r2 + 1/r C/ r) = C/ t. We introduce strongly divergent radial effects by assuming a small borehole radius of 0.25 ft relative to our one-foot grid blocks. Again, the difference scheme is integrated 500 time steps, at which point the smeared and geometrically distorted concentration profile is undiffused in time for 500 time steps. Once more, our computed results suggest that smeared resistivity profiles can be deconvolved to produce the original sharp front. The last display

424 Supercharge, Invasion and Mudcake Growth

below represents the final profile obtained before reverse integration begins. The time-reversed computations are shown in Figure 5.13b. Concentration vs distance @ time .5000E+00 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .100E+02 | * .300E+01 .100E+02 | * .400E+01 .100E+02 | * .500E+01 .103E+02 | * .600E+01 .897E+02 | * .700E+01 .900E+02 | * .800E+01 .900E+02 | * .900E+01 .900E+02 | * .100E+02 .900E+02 | * .110E+02 .900E+02 | * Concentration vs distance @ time .1000E+01 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .100E+02 | * .300E+01 .100E+02 | * .400E+01 .100E+02 | * .500E+01 .107E+02 | * .600E+01 .895E+02 | * .700E+01 .900E+02 | * .800E+01 .900E+02 | * .900E+01 .900E+02 | * .100E+02 .900E+02 | * .110E+02 .900E+02 | * Concentration vs distance @ time .1000E+02 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .100E+02 | * .300E+01 .100E+02 | * .400E+01 .103E+02 | * .500E+01 .159E+02 | * .600E+01 .853E+02 | * .700E+01 .898E+02 | * .800E+01 .900E+02 | * .900E+01 .900E+02 | * .100E+02 .900E+02 | * .110E+02 .900E+02 | * Concentration vs distance @ time .1000E+03 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .107E+02 | * .300E+01 .128E+02 | * .400E+01 .202E+02 | * .500E+01 .393E+02 | * .600E+01 .685E+02 | * .700E+01 .839E+02 | * .800E+01 .887E+02 | * .900E+01 .898E+02 | * .100E+02 .900E+02 | * .110E+02 .900E+02 | * Concentration vs distance @ time .2000E+03 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .141E+02 | * .300E+01 .194E+02 | * .400E+01 .299E+02 | * .500E+01 .466E+02 | * .600E+01 .652E+02 | * .700E+01 .789E+02 | * .800E+01 .860E+02 | * .900E+01 .888E+02 | * .100E+02 .897E+02 | * .110E+02 .900E+02 | * Concentration vs distance @ time .2495E+03 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .163E+02 | * .300E+01 .226E+02 | * .400E+01 .334E+02 | * .500E+01 .486E+02 | * .600E+01 .648E+02 | * .700E+01 .774E+02 | * .800E+01 .848E+02 | * .900E+01 .882E+02 | * .100E+02 .895E+02 | * .110E+02 .900E+02 | *

Figure 5.13a. A diffusing radial flow.

Numerical Supercharge, Pressure and Multiphase Methods 425 Concentration vs distance @ time .2000E+03 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .141E+02 | * .300E+01 .194E+02 | * .400E+01 .299E+02 | * .500E+01 .466E+02 | * .600E+01 .653E+02 | * .700E+01 .789E+02 | * .800E+01 .860E+02 | * .900E+01 .888E+02 | * .100E+02 .897E+02 | * .110E+02 .900E+02 | * Concentration vs distance @ time .5000E+02 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .101E+02 | * .300E+01 .107E+02 | * .400E+01 .141E+02 | * .500E+01 .303E+02 | * .600E+01 .745E+02 | * .700E+01 .875E+02 | * .800E+01 .897E+02 | * .900E+01 .900E+02 | * .100E+02 .900E+02 | * .110E+02 .900E+02 | * Concentration vs distance @ time .4000E+01 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .100E+02 | * .300E+01 .999E+01 | * .400E+01 .107E+02 | * .500E+01 .109E+02 | * .600E+01 .894E+02 | * .700E+01 .896E+02 | * .800E+01 .900E+02 | * .900E+01 .900E+02 | * .100E+02 .900E+02 | * .110E+02 .900E+02 | * Concentration vs distance @ time .0000E+00 sec. Position (ft) C% ______________________________ .100E+01 .100E+02 | * .200E+01 .100E+02 | * .300E+01 .996E+01 | * .400E+01 .108E+02 | * .500E+01 .815E+01 |* .600E+01 .916E+02 | * .700E+01 .895E+02 | * .800E+01 .900E+02 | * .900E+01 .900E+02 | * .100E+02 .900E+02 | * .110E+02 .900E+02 | *

Figure 5.13b. An “undiffusing” radial flow. As before, we recaptured the initial step concentration profile, to include concentration values, discontinuity location, and total time to undiffuse. While the starting radial concentration profile is substantially smeared, and significantly different from the lineal flow solution obtained at this point in Figure 5.12a, we again successfully undiffuse our starting flow. 5.2.3 Immiscible radial flows with capillary pressure and prescribed mudcake growth.

We will consider immiscible radial flows with capillary pressure and prescribed mudcake growth. In particular, we will derive the relevant governing equations, develop the numerical finite difference algorithm and the Fortran implementation, and proceed to demonstrate the computational model in both forward and inverse modes.

426 Supercharge, Invasion and Mudcake Growth

Governing saturation equation. Let us now repeat the lineal flow derivation given earlier but include the effects of radial geometric spreading and non-vanishing capillary pressure. Again, analogous Darcy laws apply, namely, qw =- (kw/ w) Pw/ r (5-133)

qnw =- (knw/ nw) Pnw/ r (5-134) Unlike flows in rectangular systems, the mass continuity equations in cylindrical radial coordinates take the form qw/ r + qw/r = Sw/ t (5-135) qnw/ r + qnw/r = -

Snw/ t

(5-136)

If we add Equations 5-135 and 5-136, and observe that Sw + Snw = 1

(5-137)

is constant for incompressible flows, it follows that r (qw+qnw)/ r + (qw+qnw) = 0

(5-138)

or, equivalently, {r (qw+qnw)}r = 0, so that r (qw+qnw) = Q(t)

(5-139)

Here the function Q(t), having dimensions of length squared per unit time (not to be confused with volume flow rate), is determined by its value at the wellbore sandface. In particular, since only mud filtrate is obtained there, we have Q(t) = Rwell q(t) (5-140) where Rwell is the radius of the borehole and q(t) is the velocity through the mudcake obtained on a lineal flow basis, given by the expression derived earlier in this chapter, namely, q(t) = eff dxf (t)/dt = ½ t-½ {2k1(1- c)(1-fs)(pm-pr)/( ffs)}

(5-107)

A means for handling the square root singularity at t = 0 is given later. Note that another choice of q(t), for thick mudcakes, is found in the radial cake growth formula derived in Chapter 19 of Chin (2017). At this point, it is convenient to introduce the capillary pressure function Pc and write it as a function of the water saturation Sw, taking Pc(Sw) = Pnw - Pw (5-141)

Numerical Supercharge, Pressure and Multiphase Methods 427

Then, the nonwetting velocity in Equation 5-134 can be written in the form qnw = - (knw/ nw) Pnw/ r = - (knw/ nw) (Pc + Pw)/ r. If we substitute this and Equation 5-133 into Equation 5-139, we obtain r (kw/ w+ knw/ nw) Pw/ r + r (knw/ nw) Pc/ r = - Q(t) (5-142) or, more precisely, r (kw/ w+ knw/ nw) Pw/ r + r (knw/ nw) Pc'(Sw) Sw/ r = - Q(t)

(5-143)

This yields Pw/ r = - {Q(t) + r (knw/ nw) Pc’(Sw) Sw/ r}/{r (kw/ w+ knw/ nw)}

(5-144)

so that Equation 5-133 becomes qw = (kw/ w){Q(t)+r (knw/ nw)Pc’(Sw) Sw/ r}/{r (kw/ w+knw/ nw)}

(5-145)

If we combine Equations 5-145 and 5-135, that is, qw/ r + qw/r = Sw/ t, we have -

Sw/ t = ( / r + 1/r)

(5-146)

(kw/ w){Q(t)+r (knw/ nw)Pc’(Sw) Sw/ r}/{r (kw/ w+knw/ nw)}

where it is understood that the relative permeabilities kw and knw are both prescribed functions of Sw. This is the nonlinear governing equation for water saturation. Once Sw is known, the oil saturation Snw can be obtained using Equation 5-137 as Snw = 1- Sw. In order to simplify notation, let us reintroduce the fractional flow function first used in Equation 5-88, namely, F(Sw) = 1/{1 + wknw/ nwkw} (5-147) and, in addition, the function G(Sw) = {knw/ nw}F(Sw)Pc’(Sw)

(5-148)

Then, Equation 5-146 can be expressed succinctly in the form - Sw/ t - {Q(t)F'(Sw) + G(Sw)}/r Sw/ r

(5-149)

= G’(Sw)( Sw/ r)2 + G(Sw) 2Sw/ r2 Numerical analysis. Equation 5-149 is conveniently solved, again using finite difference time-marching schemes. We always central difference our first and second-order space derivatives, while backward differencing in time, with respect to the nodal point (ri,tn). Furthermore, we will evaluate all nonlinear saturation-dependent coefficients at their previous values in time. This leads to

428 Supercharge, Invasion and Mudcake Growth [1 - (QnF'i,n-1+Gi,n-1) r/(2Gi,n-1ri) - G'i,n-1 ( Sw/ r)i,n-1 r/(2Gi,n-1)] SWi-1,n + [-2 + ( r)2/(Gi,n-1 t)] SWi,n + [1 + (QnF'i,n-1+Gi,n-1) r/(2Gi,n-1ri) + G'i,n-1( Sw/ r)i,n-1 r/(2Gi,n-1)] SWi+1,n = + ( r)2SWi,n-1/(Gi,n-1 t)

(5-150)

which importantly assumes tridiagonal form for rapid matrix inversion while maintaining O( x)2 accuracy in space. Note that ri = Rwell + (i1) r. Straightforward von Neumann analysis shows that the timedependent scheme implied by Equation 5-150 is conditionally stable, with the exact time step limitations depending on the form of the relative permeability and capillary pressure functions. Following the rules established in Chapter 20 of Chin (2017), we write Equation 5-150 for the internal nodes i = 2, 3, ..., imax-1, and augment the resulting system of linearized equations with the mud filtrate boundary condition SW1,n = SWl = 1 (for 100% water saturation) and the saturation SWimax,n = SWr < 1 at a distant effective radius. To start the time-marching calculations, the right side of Equation 5-150 is assumed as SWi,n-1 = SWr < 1 for the very first value of the time index n. In this discussion, SWr also represents the initial uniform water saturation in the reservoir. Once the left side of Equation 5-150 is inverted using the tridiagonal matrix solver TRIDI, SWi,n is copied into SWi,n-1 on the right side, and the calculations are continued recursively. C

. START RECURSIVE TIME INTEGRATION DO 300 N=1,NMAX T = T+DT THOURS = T/3600. DO 200 I=2,IMAXM1 RI = WELRAD+(I-1)*DR SW = SNM1(I) DSDR =(SNM1(I+1)-SNM1(I-1))/(2.*DR) TERM1=((Q(T)*FP(SW)+G(SW))*DR)/(2.*G(SW)*RI) TERM2= DR*DR*PHI/(G(SW)*DT) TERM3= (GP(SW)*DR/G(SW))*DSDR/2. A(I) = 1.- TERM1-TERM3 B(I) = -2.+ TERM2 C(I) = 1.+ TERM1+TERM3 W(I) = TERM2*SNM1(I)

Numerical Supercharge, Pressure and Multiphase Methods 429 200

250 260 300

CONTINUE A(1) = 99. B(1) = 1. C(1) = 0. W(1) = SL A(IMAX) = 0. B(IMAX) = 1. C(IMAX) = 99. W(IMAX) = SR CALL TRIDI(A,B,C,VECTOR,W,IMAX) DO 250 I=1,IMAX S(I) = VECTOR(I) CONTINUE DO 260 I=1,IMAX SNM1(I) = S(I) CONTINUE CALL GRFIX(S,XPLOT,IMAX) CONTINUE .

Figure 5.14. Nonlinear saturation solver.

Fortran implementation. Equation 5-150 is easily programmed in Fortran. Because the implicit scheme is second-order accurate in space, thus rigidly enforcing the diffusive character of the capillary pressure effects assumed in this formulation, we do not obtain the oscillations at saturation shocks or the saturation overshoots having S w > 1 often cited. The exact Fortran producing the results shown later is displayed in Figure 5.14 and in several function statements given later. For convenience, the saturation derivatives F’(Sw) and G’(Sw) are denoted FP and GP (P indicates prime for derivatives). Typical calculations. In this section, we will perform a suite of validation runs designed to demonstrate the stability and physical correctness of the two-phase flow algorithm. In the calculations, a borehole radius of 0.2 ft and an effective reservoir radius of 2 ft are assumed. The water saturation at the borehole sandface is assumed to be unity, since it consists entirely of water-base mud filtrate. At the farfield boundary or effective radius, the water saturation is taken as 0.10. (This is also assumed to be the initial reservoir water saturation.) In addition, we discretize the radial coordinate using 0.1 ft grids, assume time steps of 0.001 sec, and take the porosity of the rock as 20%. Note that for the twenty grid block mesh used, 1,000 time steps requires approximately one second on typical Pentium class personal computers. Multiphase flow properties are conveniently defined in Fortran function statements. In our calculations, the relative permeability curves and fractional flow functions are specified in code fragment

430 Supercharge, Invasion and Mudcake Growth FUNCTION F(SW) REAL KDARCY,KABS,KW,KNW KDARCY = 0.001 KABS = KDARCY*0.00000001/(12.*12.*2.54*2.54) KW = KABS * SW**2. KNW = KABS*(SW-1.)**2. VISCIN = 1. VISCDP = 2. VISCL = 0.0000211*VISCIN VISCR = 0.0000211*VISCDP F = 1. +VISCL*KNW/(VISCR*KW) F = 1./F RETURN END

In the preceding calculations, an absolute permeability of 0.001 Darcies is assumed for the formation, and the wetting and nonwetting relative permeability functions, defined in terms of the water saturation Sw, are taken in the form kw = Sw2 and knw = (Sw-1) 2 for simplicity. Our water and oil viscosities are taken as 1 and 2 cp, respectively. The fractional flow function just defined is independent of the absolute permeability, of course, and depends only on the viscosity ratio. The function G(Sw) is similarly defined by FUNCTION G(SW) REAL KDARCY,KABS,KNW KDARCY = 0.001 KABS = KDARCY*0.00000001/(12.*12.*2.54*2.54) KNW = KABS*(SW-1.)**2. VISCDP = 2. VISCR = 0.0000211*VISCDP G = KNW*F(SW)*PCP(SW)/VISCR RETURN END

while the capillary pressure function is defined by Pc = 35 (1-Sw) psi, again for simplicity, through the function block FUNCTION PC(SW) PC = 1.-SW PC = 144.*35.*PC RETURN END

Derivatives of Pc, F, and G with respect to water saturation can be easily taken by introducing function statements that define the differentiation process. We now discuss typical calculations, designed to test the

Numerical Supercharge, Pressure and Multiphase Methods 431

properties of the scheme, such as saturation overshoots, unstable oscillations, and so on. We will find that the algorithm given is physically consistent. For example, it will not yield water saturations that exceed unity or fall below zero; thus, oil will not be created or destroyed, at least not in an obvious manner. The fully implicit scheme, unlike the explicit schemes used in many commercial IMPES models (to be discussed), does not produce numerical oscillations at the head of the shock. But instabilities do arise when the saturation shock reflects back upstream from the fictitious i = imax effective radius boundary; these instabilities, however, are irrelevant to our simulations. Finally, when mud filtrate is completely shut off, the water-oil saturation front never moves and must remain stationary – a trait not shared by several commercial simulators because of numerical round-off. Let us now discuss specific calculations. In this very first example, we set our mud filtration invasion rate identically to zero, using the function statement FUNCTION Q(T) Q = 0. RETURN END

The partial results in Figure 5.15a for the near-well nodes indicate that the water front correctly stays absolutely static, with the remainder of the flow domain remaining unperturbed, despite the 1,000 steps taken. Water saturation at time (hrs): .167E-04 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .100E+00 | * .300E+01 .100E+00 | * .400E+01 .100E+00 | * .500E+01 .100E+00 | * .600E+01 .100E+00 | * .700E+01 .100E+00 | * Water saturation at time (hrs): .150E-03 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .100E+00 | * .300E+01 .100E+00 | * .400E+01 .100E+00 | * .500E+01 .100E+00 | * .600E+01 .100E+00 | * .700E+01 .100E+00 | * Water saturation at time (hrs): .267E-03 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .100E+00 | * .300E+01 .100E+00 | * .400E+01 .100E+00 | * .500E+01 .100E+00 | * .600E+01 .100E+00 | * .700E+01 .100E+00 | *

Figure 5.15a. Zero mud filtrate influx.

432 Supercharge, Invasion and Mudcake Growth

Again, we note that Q is not the volume flow rate, but the product of well radius and radial Darcy velocity at the sandface. For the assumed radius of 0.2 ft, a typical velocity may be assumed as 0.1 ft/hr, so that Q = (0.2 ft)(0.1 ft/hr) = 0.0000055 ft2/sec in the units used. We determine if the calculated invasion rates are physically reasonable, and in the process, we examine the stability of the algorithm. In Figure 5.15b below, 50,000 time steps of 0.001 sec are taken, requiring one minute of Pentium PC computations, and sample early and late time results are given. Truncation error is negligible in this stable scheme. Water saturation at time (hrs): .167E-04 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .100E+00 | * .300E+01 .100E+00 | * .400E+01 .100E+00 | * .500E+01 .100E+00 | * . . .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | * Water saturation at time (hrs): .903E-02 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .110E+00 | * .300E+01 .100E+00 | * .400E+01 .100E+00 | * .500E+01 .100E+00 | * . . .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | * Water saturation at time (hrs): .126E-01 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .114E+00 | * .300E+01 .100E+00 | * .400E+01 .100E+00 | * .500E+01 .100E+00 | * .600E+01 .100E+00 | * .700E+01 .100E+00 | * .800E+01 .100E+00 | * .900E+01 .100E+00 | * .100E+02 .100E+00 | * .110E+02 .100E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | *

Figure 5.15b. Very slow constant injection rate. The question of large versus small Q are only meaningful dimensionlessly, when the effects of inertia are measured against those due to capillary pressure. Since the functional form of these changes from problem to problem, because relative permeability and capillary

Numerical Supercharge, Pressure and Multiphase Methods 433

pressure curves will often vary substantially, a parameter as simple or as elegant as the Reynolds number in elementary Newtonian fluid mechanics is not in general available. But, fortunately, we can understand the stability properties of the numerical scheme by examining different parameter limits of the present problem. It is clear from the two foregoing runs that inertia is not important, since little of the fluid is actually moving. In this next example, we assume the comparatively large constant value of Q = 1, to simulate water breakthrough known to reservoir engineers. Large Qs model rapid influxes of injected water and should result in sharp saturation discontinuities; for such problems, there is little smearing at the shock due to capillary pressure. This is not to say that capillary pressure is unimportant: it is, because of the singular role it plays in defining the correct saturation discontinuity. (The shock-fitting used in the Buckley-Leverett solution process is unnecessary in the present high-order formulation.) For the Q = 1, 2, and 3 calculations shown in the following figures, 3,000 time steps of 0.001 sec each were used. In Figures 5.15c,d,e, note how the effects of radial geometrical spreading are captured in the gently sloping curve, while steep saturation gradients are computed as sudden changes. Also note that the calculations shown are extremely stable and that no numerical oscillations appear in the results. Moreover, we never obtain any water saturations that exceed unity in our O( x2) accurate implicit scheme. However, we have found that instabilities will arise after the shock reaches the farfield computational boundary and reflects. By this time, the calculations have no physical meaning, so that the existence of this instability is not germane to our applications. Water saturation at time (hrs): .167E-04 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .938E+00 | * .300E+01 .877E+00 | * .400E+01 .827E+00 | * .500E+01 .789E+00 | * .600E+01 .758E+00 | * .700E+01 .664E+00 | * .800E+01 .278E+00 | * .900E+01 .116E+00 | * .100E+02 .101E+00 | * .110E+02 .100E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | *

434 Supercharge, Invasion and Mudcake Growth Water saturation at time (hrs): .500E-04 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .975E+00 | * .300E+01 .945E+00 | * .400E+01 .913E+00 | * .500E+01 .881E+00 | * .600E+01 .849E+00 | * .700E+01 .819E+00 | * .800E+01 .794E+00 | * .900E+01 .774E+00 | * .100E+02 .760E+00 | * .110E+02 .751E+00 | * .120E+02 .739E+00 | * .130E+02 .682E+00 | * .140E+02 .372E+00 | * .150E+02 .130E+00 | * .160E+02 .102E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | * Water saturation at time (hrs): .667E-04 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .981E+00 | * .300E+01 .957E+00 | * .400E+01 .931E+00 | * .500E+01 .903E+00 | * .600E+01 .875E+00 | * .700E+01 .848E+00 | * .800E+01 .822E+00 | * .900E+01 .798E+00 | * .100E+02 .779E+00 | * .110E+02 .764E+00 | * .120E+02 .754E+00 | * .130E+02 .747E+00 | * .140E+02 .739E+00 | * .150E+02 .708E+00 | * .160E+02 .507E+00 | * .170E+02 .164E+00 | * .180E+02 .100E+00 | *

Figure 5.15c. Q = 1, constant rate, high inertia flow. Water saturation at time (hrs): .167E-04 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .964E+00 | * .300E+01 .925E+00 | * .400E+01 .884E+00 | * .500E+01 .846E+00 | * .600E+01 .811E+00 | * .700E+01 .784E+00 | * .800E+01 .764E+00 | * .900E+01 .749E+00 | * .100E+02 .716E+00 | * .110E+02 .510E+00 | * .120E+02 .165E+00 | * .130E+02 .106E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | * Water saturation at time (hrs): .333E-04 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .981E+00 | * .300E+01 .957E+00 | * .400E+01 .931E+00 | * .500E+01 .903E+00 | * .600E+01 .876E+00 | * .700E+01 .848E+00 | * .800E+01 .822E+00 | * .900E+01 .798E+00 | * .100E+02 .778E+00 | * .110E+02 .762E+00 | * .120E+02 .752E+00 | * .130E+02 .745E+00 | * .140E+02 .737E+00 | * .150E+02 .709E+00 | * .160E+02 .520E+00 | * .170E+02 .172E+00 | * .180E+02 .100E+00 | *

Figure 5.15d. Q = 2, constant rate, high inertia flow.

Numerical Supercharge, Pressure and Multiphase Methods 435 Water saturation at time (hrs): .167E-04 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .976E+00 | * .300E+01 .947E+00 | * .400E+01 .916E+00 | * .500E+01 .883E+00 | * .600E+01 .851E+00 | * .700E+01 .821E+00 | * .800E+01 .794E+00 | * .900E+01 .772E+00 | * .100E+02 .756E+00 | * .110E+02 .746E+00 | * .120E+02 .735E+00 | * .130E+02 .688E+00 | * .140E+02 .396E+00 | * .150E+02 .138E+00 | * .160E+02 .103E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | *

Figure 5.15e. Q = 3, constant rate, high inertia flow. Mudcake dominated flows. Now we consider time-dependent influx flows of the kind created by real mudcakes. Earlier we found that the invasion speed at t = 0 was infinite, behaving like t-1/2. Such singularities, if implemented exactly, would cause instabilities in finite difference schemes. Fortunately, we can circumvent this difficulty without introducing any artificial devices, by considering the effects of nonzero mud spurt. From an earlier result, the filtration thickness x(t) in a lineal flow varies like dx/dt = /x, where is a constant. If x(0) = xspurt, it follows that x(t) = (2 t + xspurt2); then, the speeds dx/dt = / (2 t + xspurt2) and q(t) = dx/dt = / (2 t + xspurt2) are never infinite. This lineal model is used because the controlling mudcake forms lineally; a radial model can, of course, be substituted in slimhole applications. The spurt model is implemented by the Fortran function definition C

FUNCTION Q(T) MUDCAKE MODEL, ALPHA = 1. PHI = 0.2 WELRAD = 0.2 SPURT =0.1 SPURT2 = SPURT**2 ALPHA = 1. Q = WELRAD*ALPHA*PHI/SQRT(SPURT2+2.*ALPHA*T) RETURN END

Figure 5.16. Mudcake-dominated invasion. In the sequence of snapshots in Figures 5-81a,b,c, the formation and movement of the saturation shocks are shown for high, very high, and very slow invasion rates, all using 0.001 sec time steps. Again, complete stability is obtained, without numerical saturation oscillations.

436 Supercharge, Invasion and Mudcake Growth Water saturation at time (hrs): .167E-04 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .810E+00 | * .300E+01 .475E+00 | * .400E+01 .132E+00 | * .500E+01 .101E+00 | * .600E+01 .100E+00 | * .700E+01 .100E+00 | * .800E+01 .100E+00 | * .900E+01 .100E+00 | * .100E+02 .100E+00 | * .110E+02 .100E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | * Water saturation at time (hrs): .117E-03 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .905E+00 | * .300E+01 .831E+00 | * .400E+01 .776E+00 | * .500E+01 .639E+00 | * .600E+01 .222E+00 | * .700E+01 .108E+00 | * .800E+01 .100E+00 | * .900E+01 .100E+00 | * .100E+02 .100E+00 | * .110E+02 .100E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | * Water saturation at time (hrs): .383E-03 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .939E+00 | * .300E+01 .879E+00 | * .400E+01 .829E+00 | * .500E+01 .793E+00 | * .600E+01 .764E+00 | * .700E+01 .688E+00 | * .800E+01 .340E+00 | * .900E+01 .121E+00 | * .100E+02 .101E+00 | * .110E+02 .100E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | * Water saturation at time (hrs): .833E-03 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .956E+00 | * .300E+01 .908E+00 | * .400E+01 .863E+00 | * .500E+01 .823E+00 | * .600E+01 .792E+00 | * .700E+01 .771E+00 | * .800E+01 .750E+00 | * .900E+01 .671E+00 | * .100E+02 .312E+00 | * .110E+02 .119E+00 | * .120E+02 .101E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | *

Figure 5.17a. High filtration rate mudcake model ( = 1).

Numerical Supercharge, Pressure and Multiphase Methods 437 Water saturation at time (hrs): .167E-04 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .891E+00 | * .300E+01 .813E+00 | * .400E+01 .726E+00 | * .500E+01 .385E+00 | * .600E+01 .125E+00 | * .700E+01 .101E+00 | * .800E+01 .100E+00 | * .900E+01 .100E+00 | * .100E+02 .100E+00 | * .110E+02 .100E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | * Water saturation at time (hrs): .150E-03 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .953E+00 | * .300E+01 .904E+00 | * .400E+01 .857E+00 | * .500E+01 .817E+00 | * .600E+01 .788E+00 | * .700E+01 .767E+00 | * .800E+01 .738E+00 | * .900E+01 .595E+00 | * .100E+02 .202E+00 | * .110E+02 .108E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | *

Water saturation at time (hrs): .833E-03 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .978E+00 | * .300E+01 .952E+00 | * .400E+01 .924E+00 | * .500E+01 .894E+00 | * .600E+01 .864E+00 | * .700E+01 .836E+00 | * .800E+01 .810E+00 | * .900E+01 .788E+00 | * .100E+02 .772E+00 | * .110E+02 .760E+00 | * .120E+02 .752E+00 | * .130E+02 .743E+00 | * .140E+02 .711E+00 | * .150E+02 .511E+00 | * .160E+02 .162E+00 | * .170E+02 .105E+00 | * .180E+02 .100E+00 | *

Figure 5.17b. Very high filtration rate mudcake model ( = 5).

438 Supercharge, Invasion and Mudcake Growth Water saturation at time (hrs): .167E-04 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .101E+00 | * .300E+01 .100E+00 | * .400E+01 .100E+00 | * .500E+01 .100E+00 | * .600E+01 .100E+00 | * .700E+01 .100E+00 | * .800E+01 .100E+00 | * .900E+01 .100E+00 | * .100E+02 .100E+00 | * .110E+02 .100E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | * Water saturation at time (hrs): .120E-02 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .186E+00 | * .300E+01 .102E+00 | * .400E+01 .100E+00 | * .500E+01 .100E+00 | * .600E+01 .100E+00 | * .700E+01 .100E+00 | * .800E+01 .100E+00 | * .900E+01 .100E+00 | * .100E+02 .100E+00 | * .110E+02 .100E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | * Water saturation at time (hrs): .167E-02 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .231E+00 | * .300E+01 .104E+00 | * .400E+01 .100E+00 | * .500E+01 .100E+00 | * .600E+01 .100E+00 | * .700E+01 .100E+00 | * .800E+01 .100E+00 | * .900E+01 .100E+00 | * .100E+02 .100E+00 | * .110E+02 .100E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | *

Figure 5.17c. Very slow filtration rate model ( = 0.001). “Un-shocking” a saturation discontinuity. In time lapse analysis, we may detect a moving saturation front, and may wish to look within or unscramble the steeply formed flow for additional fluiddynamical information. Here, resistivity migration means unsteepening the shock, carefully untracing its history, being dynamically consistent with the effects of capillary pressure and nonlinear relative permeability functions. Unlike the miscible flow problem, where the dominant physical process involved the unsmearing of a diffused front, several complications enter the present problem. First, radial spreading again exists. But the high-order derivative term, now related to capillary pressure instead of molecular diffusion, serves dual purposes: it smears

Numerical Supercharge, Pressure and Multiphase Methods 439

the flow throughout the entire flow domain, and it is instrumental in controlling the shock formation. (Shocks, remember, do not exist in the miscible flow problem.) . C

200 300

START TIME INTEGRATION DO 300 N=1,NMAX IF(N.LT.2000) T = T+DT IF(N.GE.2000) T = T-DT THOURS = T/3600. DO 200 I=2,IMAXM1 RI = WELRAD+(I-1)*DR SW = SNM1(I) DSDR =(SNM1(I+1)-SNM1(I-1))/(2.*DR) IF(N.LT.2000) TERM1=((Q(T)*FP(SW)+G(SW))*DR)/(2.*G(SW)*RI) IF(N.GE.2000) TERM1=((-Q(T)*FP(SW)+G(SW))*DR)/(2.*G(SW)*RI) TERM2= DR*DR*PHI/(G(SW)*DT) TERM3= (GP(SW)*DR/G(SW))*DSDR/2. A(I) = 1.- TERM1-TERM3 B(I) = -2.+ TERM2 C(I) = 1.+ TERM1+TERM3 W(I) = TERM2*SNM1(I) CONTINUE . CONTINUE

Figure 5.18. “Un-shocking” a steep gradient. Can we undo all of these two-phase flow effects? The answer appears to be a definitive, “Yes.” To evaluate this numerical reversibility, we execute the program for 2,000 time steps, assuming = 1, and then reverse the direction of time as well as the direction of filtrate movement, as shown in Figure 5.18 by the bold print modifications to our earlier source code. Forward simulation results are given in Figure 5.19a, while successfully migrated, or unshocked results, are shown in Figure 5.19b. The potential applications of this important capability are vast indeed and are under investigation. Water saturation at time (hrs): .167E-04 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .810E+00 | * .300E+01 .475E+00 | * .400E+01 .132E+00 | * .500E+01 .101E+00 | * .600E+01 .100E+00 | * .700E+01 .100E+00 | * .800E+01 .100E+00 | * .900E+01 .100E+00 | * .100E+02 .100E+00 | * .110E+02 .100E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | *

440 Supercharge, Invasion and Mudcake Growth Water saturation at time (hrs): .117E-03 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .905E+00 | * .300E+01 .831E+00 | * .400E+01 .776E+00 | * .500E+01 .639E+00 | * .600E+01 .222E+00 | * .700E+01 .108E+00 | * .800E+01 .100E+00 | * .900E+01 .100E+00 | * .100E+02 .100E+00 | * .110E+02 .100E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * Water saturation at time (hrs): .350E-03 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .937E+00 | * .300E+01 .876E+00 | * .400E+01 .825E+00 | * .500E+01 .790E+00 | * .600E+01 .758E+00 | * .700E+01 .651E+00 | * .800E+01 .257E+00 | * .900E+01 .112E+00 | * .100E+02 .101E+00 | * .110E+02 .100E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | *

Figure 5.19a. Forward shock formation. Water saturation at time (hrs): .550E-03 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .947E+00 | * .300E+01 .893E+00 | * .400E+01 .845E+00 | * .500E+01 .806E+00 | * .600E+01 .779E+00 | * .700E+01 .753E+00 | * .800E+01 .658E+00 | * .900E+01 .277E+00 | * .100E+02 .115E+00 | * .110E+02 .101E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | * Water saturation at time (hrs): .211E-03 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .923E+00 | * .300E+01 .855E+00 | * .400E+01 .806E+00 | * .500E+01 .765E+00 | * .600E+01 .645E+00 | * .700E+01 .240E+00 | * .800E+01 .110E+00 | * .900E+01 .101E+00 | * .100E+02 .100E+00 | * .110E+02 .100E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | *

Numerical Supercharge, Pressure and Multiphase Methods 441 Water saturation at time (hrs): .106E-04 Position node Water Sat ______________________________ .100E+01 .100E+01 | * .200E+01 .747E+00 | * .300E+01 .255E+00 | * .400E+01 .108E+00 | * .500E+01 .101E+00 | * .600E+01 .100E+00 | * .700E+01 .100E+00 | * .800E+01 .100E+00 | * .900E+01 .100E+00 | * .100E+02 .100E+00 | * .110E+02 .100E+00 | * .120E+02 .100E+00 | * .130E+02 .100E+00 | * .140E+02 .100E+00 | * .150E+02 .100E+00 | * .160E+02 .100E+00 | * .170E+02 .100E+00 | * .180E+02 .100E+00 | *

Figure 5.19b. Backward shock migration. 5.2.4 Immiscible flows with capillary pressure and dynamically coupled mudcake growth.

In the foregoing formulation, we assumed that q(t) was available from a knowledge of mudcake properties, and we solved for the resulting two-phase flow in the rock. Of course, this is not generally the case. Consider the limit in which no mudcake forms on the rock: here the time-dependent flow through the rock is determined by the saturations and pressures at the inlet and outlet boundaries of the problem. For the cake-free problem just described, the PDEs governing saturation and pressure are nonlinearly coupled, and the time-dependent flow rate through the core must be determined iteratively. This is also the case when the mobility in the mudcake is comparable to that of the formation. But there is a complication. For such problems, this cake growth must be additionally determined as part of the solution; it does not alone dictate the filtrate influx but depends strongly on two-phase flow details in the rock. Flows without mudcakes. In order to solve the latter, it is instructive to formulate and discuss the former one without mudcake first. In doing so, we derive the complete set of two-phase flow equations required later, and we demonstrate some essential ideas. Let us recall that we had determined r (kw/ w+ knw/ nw) Pw/ r + r (knw/ nw) Pc'(Sw) Sw/ r = - Q(t) (5-143) In the previous section, Q(t) was assumed to be known; this being the case, the derived saturation equation could be solved independently of the pressure equation, so that a pressure differential equation was not required. Now, we expect that any derived governing pressure equation must reduce to an anticipated 2Pw/ r2 + (1/r) Pw/ r = 0 in the single-

442 Supercharge, Invasion and Mudcake Growth

phase flow limit. This can be accomplished by differentiating Equation 5-143 with respect to the radial coordinate; this differentiation eliminates the explicit appearance of Q(t) which is, again, unknown. Since kw= kw(Sw) and knw= knw(Sw), straightforward manipulations show that we can write the desired equation as 2P / r2 + [1/r +{(k ’/ w w

w+knw’/ nw)/

= - [( 2Sw/ r2 + 1/r Sw/ r) ( + {(

wknw’/ nwkw)Pc’(Sw)+

[1 + (

wknw/ nwkw)]

(kw/

w+knw/ nw)}

wknw/ nwkw)

(

Sw/ r] Pw/ r

Pc’(Sw)

wknw/ nwkw)Pc”(Sw)}(

Sw/ r)2]/

(5-151)

Now, there exist two dependent variables in the present problem, namely, pressure and saturation. Pressure is governed by Equation 5151, while saturation satisfies Equation 5-146, - Sw/ t = ( / r + 1/r) (5-146) (kw/ w){Q(t)+r (knw/ nw)Pc’(Sw) Sw/ r}/{r (kw/ w+knw/ nw)}

where Q(t), now not a prescribed function, merely stands for the functional combination Q(t) = - r (kw/ w+ knw/ nw) Pw/ r - r (knw/ nw) Pc’(Sw) Sw/ r (5-152) as is clear from Equation 5-143. If the initial spatial distributions for Pw and Sw are prescribed, a reasonable numerical solution process might solve Equations 5-151 and 5-146 sequentially for a time step, before proceeding to the next. We will, in fact, adopt this procedure. The solution procedure for saturation has been discussed and was implemented using Equation 5-150. We will retain that procedure for the present problem. For the pressure solution, in order to simplify our nomenclature, we recast Equation 5-151 in the form 2P / r2 + COEF P / r = RHS w w

(5-153)

where COEF and RHS denote the coefficient and right-hand-side terms. Then, adopting the central difference approximation (PWi-1 - 2 PWi + PWi+1)/ r2 + COEFi (PWi+1 - PWi-1)/(2 r) = RHSi, we rewrite Equation 5-153 as (1 - COEFi r/2) PWi-1- 2 PWi + (1 + COEFi r/2) PWi+1 = RHSi r2 (5-154) Insofar as the tridiagonal solver TRIDI is concerned, the coefficients A, B, C, and W take the form Ai = (1 - COEFi r/2), Bi = -2, Ci = (1 +

Numerical Supercharge, Pressure and Multiphase Methods 443

COEFi r/2), and Wi = RHSi r2 for the internal nodes i = 2, 3, ..., imax1. In addition, A(1) = 99, B(1) = 1, C(1) = 0, W(1) = Pleft, and A(IMAX) = 0, B(IMAX) = 1, C(IMAX) = 99, W(IMAX) = Pright, where Pleft and Pright denote the prescribed pressures at the inlet and outlet boundaries. Observe that COEF and RHS will always be evaluated by secondorder accurate central differences in space at the previous time step. Also, a starting initial pressure distribution is required that is analogous to our initial condition for saturation. Selected portions of the Fortran source code engine required to implement this algorithm are given in Figure 5.20. They are obtained by simple modification of our earlier program designed to solve two-phase flows when the flow rate is a prescribed function of time. Finally, observe that we do not use the outlet saturation boundary condition of Collins (1961) because our outlet is a fictitious computational boundary that is internal to the reservoir. Note that “Multiple Factors That Influence Wireline Formation Tester Pressure Measurements and Fluid Contact Estimates,” by M.A. Proett, W.C. Chin, M. Manohar, R. Sigal, and J. Wu, SPE Paper 71566, presented at the 2001 SPE Annual Technical Conference and Exhibition in New Orleans, Louisiana, September 30 – October 3, 2001, extends the work in this chapter to higher order, ensuring that mass is accurately conserved at strong saturation discontinuities.

444 Supercharge, Invasion and Mudcake Growth C

100 C C

C C

. . INITIALIZATION T = 0. DO 100 I=1,IMAX SNM1(I) = SZERO XPLOT(I) = WELRAD+(I-1)*DR P(I) = PINIT CONTINUE START TIME INTEGRATION DO 300 N=1,NMAX T = T+DT THOURS = T/3600.

PRESSURE EQUATION DO 150 I=2,IMAXM1 RI = WELRAD+(I-1)*DR SW = SNM1(I) DSDR = (SNM1(I+1)-SNM1(I-1))/(2.*DR) DSDR2 = DSDR**2. D2SDR2 = (SNM1(I-1)-2.*SNM1(I)+SNM1(I+1))/(DR*DR) DEL2S = D2SDR2+(1./RI)*DSDR COEF = 1./RI 1 +((KWP(SW)/VISCL+KNWP(SW)/VISCR)/ 2 (KW(SW) /VISCL+KNW(SW) /VISCR))*DSDR RHS = PCP(SW)*DEL2S*(VISCL*KNW(SW)/(VISCR*KW(SW))) 1 +DSDR2*VISCL*KNWP(SW)*PCP(SW)/(VISCR*KW(SW)) 2 +DSDR2*PCPP(SW)*(VISCL*KNW(SW)/(VISCR*KW(SW))) RHS = -RHS*F(SW) A(I) = 1.-COEF*DR/2. B(I) = -2. C(I) = 1.+COEF*DR/2. W(I) = RHS*DR*DR 150 CONTINUE A(1) = 99. B(1) = 1. C(1) = 0. W(1) = PLEFT A(IMAX) = 0. B(IMAX) = 1. C(IMAX) = 99. W(IMAX) = PRIGHT CALL TRIDI(A,B,C,VECTOR,W,IMAX) DO 160 I=1,IMAX P(I) = VECTOR(I) 160 CONTINUE C C SATURATION EQUATION DO 200 I=2,IMAXM1 RI = WELRAD+(I-1)*DR SW = SNM1(I) DSDR = (SNM1(I+1)-SNM1(I-1))/(2.*DR) DPDR = (P(I+1)-P(I-1))/(2.*DR) Q = -RI*(KW(SW)/VISCL+KNW(SW)/VISCR)*DPDR 1 -RI*(KNW(SW)/VISCR)*PCP(SW)*DSDR TERM1=((Q*FP(SW)+G(SW))*DR)/(2.*G(SW)*RI) TERM2= DR*DR*PHI/(G(SW)*DT) TERM3= (GP(SW)*DR/G(SW))*DSDR/2. A(I) = 1.- TERM1-TERM3 B(I) = -2.+ TERM2 C(I) = 1.+ TERM1+TERM3 W(I) = TERM2*SNM1(I) 200 CONTINUE

Numerical Supercharge, Pressure and Multiphase Methods 445

250 260

280

281 300

C

A(1) = 99. B(1) = 1. C(1) = 0. W(1) = SL A(IMAX) = 0. B(IMAX) = 1. C(IMAX) = 99. W(IMAX) = SR CALL TRIDI(A,B,C,VECTOR,W,IMAX) DO 250 I=1,IMAX S(I) = VECTOR(I) CONTINUE DO 260 I=1,IMAX SNM1(I) = S(I) CONTINUE IF(MOD(N,60).NE.0) GO TO 300 WRITE(*,10) WRITE(4,10) WRITE(*,280) THOURS WRITE(4,280) THOURS FORMAT(' Water saturation at time (hrs):' E9.3) CALL GRFIX(S,XPLOT,IMAX,1) WRITE(*,281) THOURS WRITE(4,281) THOURS FORMAT(' Pressure versus r @ time (hrs):' E9.3) CALL GRFIX(P,XPLOT,IMAX,2) CONTINUE . . STOP END FUNCTION F(SW) REAL KDARCY,KABS,KW,KNW KDARCY = 0.001 KABS = KDARCY*0.00000001/(12.*12.*2.54*2.54) KW = KABS * SW**2. KNW = KABS*(SW-1.)**2. VISCIN = 1. VISCDP = 2. VISCL = 0.0000211*VISCIN VISCR = 0.0000211*VISCDP F = 1. +VISCL*KNW/(VISCR*KW) F = 1./F RETURN END

Figure 5.20. Implicit pressure – implicit saturation solver. In the following calculations, two tabulations are shown per frozen instant in time, the first for spatial saturation distribution and the second for the corresponding pressure. (The pressure and time units shown are not germane to our discussion, since they were chosen to replicate an entire range of weak to strong inertia-to-capillary force effects.) The tabulated solution sets appear in Figures 5.21a,b,c.

446 Supercharge, Invasion and Mudcake Growth Water saturation at time (hrs): .167E-03 Position (ft) Water Sat ______________________________ .200E+00 .100E+01 | * .300E+00 .807E+00 | * .400E+00 .388E+00 | * .500E+00 .120E+00 | * .600E+00 .101E+00 | * .700E+00 .100E+00 | * .800E+00 .100E+00 | * .900E+00 .100E+00 | * .100E+01 .100E+00 | * .110E+01 .100E+00 | * .120E+01 .100E+00 | * .130E+01 .100E+00 | * .140E+01 .100E+00 | * .150E+01 .100E+00 | * .160E+01 .100E+00 | * .170E+01 .100E+00 | * .180E+01 .100E+00 | * .190E+01 .100E+00 | * Pressure versus r @ time (hrs): .167E-03 Position (ft) Pressure ______________________________ .200E+00 .144E+09 | * .300E+00 .129E+09 | * .400E+00 .108E+09 | * .500E+00 .888E+08 | * .600E+00 .765E+08 | * .700E+00 .663E+08 | * .800E+00 .574E+08 | * .900E+00 .496E+08 | * .100E+01 .426E+08 | * .110E+01 .363E+08 | * .120E+01 .305E+08 | * .130E+01 .252E+08 | * .140E+01 .203E+08 | * .150E+01 .157E+08 | * .160E+01 .114E+08 |* .170E+01 .738E+07 * .180E+01 .359E+07 | .190E+01 .000E+00 |

Figure 5.21a. Early time saturation and pressure. The early time saturation solution in Figure 5.21a indicates that inertia effects are not yet strong. This is clear, since reference to our source code shows that we have initialized our pressure field to a constant value throughout, so that the flow is initially stagnant. At t = 0+, a sudden applied pressure differential is introduced (that is, PLEFT PRIGHT > 0), and fluid movement commences. However, the saturation shock has not formed, and the flow is controlled by capillary pressure. Note how the computed pressure shows a mild slope discontinuity.

Numerical Supercharge, Pressure and Multiphase Methods 447 Water saturation at time (hrs): .667E-03 Position (ft) Water Sat ______________________________ .200E+00 .100E+01 | * .300E+00 .919E+00 | * .400E+00 .854E+00 | * .500E+00 .811E+00 | * .600E+00 .756E+00 | * .700E+00 .491E+00 | * .800E+00 .149E+00 | * .900E+00 .103E+00 | * .100E+01 .100E+00 | * .110E+01 .100E+00 | * .120E+01 .100E+00 | * .130E+01 .100E+00 | * .140E+01 .100E+00 | * .150E+01 .100E+00 | * .160E+01 .100E+00 | * .170E+01 .100E+00 | * .180E+01 .100E+00 | * .190E+01 .100E+00 | * Pressure versus r @ time (hrs): .667E-03 Position (ft) Pressure ______________________________ .200E+00 .144E+09 | * .300E+00 .129E+09 | * .400E+00 .117E+09 | * .500E+00 .106E+09 | * .600E+00 .962E+08 | * .700E+00 .845E+08 | * .800E+00 .697E+08 | * .900E+00 .599E+08 | * .100E+01 .514E+08 | * .110E+01 .438E+08 | * .120E+01 .368E+08 | * .130E+01 .304E+08 | * .140E+01 .245E+08 | * .150E+01 .189E+08 | * .160E+01 .138E+08 |* .170E+01 .891E+07 * .180E+01 .433E+07 | .190E+01 .000E+00 |

Figure 5.21b. Intermediate time saturation and pressure. Figure 5.21b illustrates the start of saturation shock formation, an event not unlike the piston-like displacements assumed early in this book. It is interesting to observe that immiscible two-phase flow theory will predict piston-like fronts when they exist, but when they do not, will produce smooth flows. Thus, immiscible flow theory is more general and more powerful. However, it suffers from several practical disadvantages. Calculations are almost always numerical and produce little intuitive insight; also, the relative permeability and capillary pressure functions that are required may not be known accurately. Water saturation at time (hrs): .283E-02 Position (ft) Water Sat .200E+00 .300E+00 .400E+00 .500E+00 .600E+00 .700E+00 .800E+00 .900E+00 .100E+01 .110E+01 .120E+01 .130E+01 .140E+01 .150E+01 .160E+01 .170E+01 .180E+01 .190E+01

.100E+01 .978E+00 .951E+00 .921E+00 .891E+00 .861E+00 .834E+00 .811E+00 .793E+00 .780E+00 .772E+00 .765E+00 .755E+00 .691E+00 .359E+00 .127E+00 .102E+00 .100E+00

______________________________ | * | * | * | * | * | * | * | * | * | * | * | * | * | * | * | * | * | *

448 Supercharge, Invasion and Mudcake Growth Pressure versus r @ time (hrs): .283E-02 Position (ft) Pressure ______________________________ .200E+00 .144E+09 | * .300E+00 .125E+09 | * .400E+00 .111E+09 | * .500E+00 .100E+09 | * .600E+00 .899E+08 | * .700E+00 .809E+08 | * .800E+00 .726E+08 | * .900E+00 .648E+08 | * .100E+01 .576E+08 | * .110E+01 .508E+08 | * .120E+01 .445E+08 | * .130E+01 .386E+08 | * .140E+01 .331E+08 | * .150E+01 .275E+08 | * .160E+01 .196E+08 | * .170E+01 .117E+08 |* .180E+01 .562E+07 * .190E+01 .000E+00 |

Figure 5.21c. Late time saturation and pressure. Finally, note that while the saturation profiles in Figures 5.21a to 5.21c have evolved significantly from the beginning to the end of the calculations, our pressure profiles have remained more or less invariant with time. This indicates the existence of two global time scales in the problem, one governing pressure and the other governing saturation. Also, while the pressure gradient profile is mildly discontinuous, the saturation profile is strongly discontinuous. The invariance of the pressure solution with time is not unexpected, although it is not always obtained. Since-steady state all-water and steady-state all-oil pressure distributions are identical for a fixed pressure differential, one might expect that all of the intervening mixed fluid pressure states will not deviate far from the profile obtained on a singlephase flow basis. The latter solution can therefore be used to initialize the pressure solver for rapid convergence. The converged solution would contain the propagating slope discontinuities required at the water-oil interface. Intuitive arguments such as this, when plausible, can motivate more efficient numerical schemes for research purposes. It is important to observe that the transient saturation equation may be either parabolic or hyberbolic in nature (e.g., see Hildebrand (1948)), depending on the importance of the capillary pressure term relative to the convection term. The form of the equation given in Equation 5-149 illustrates this distinction very clearly. When capillary pressure is important, the G(Sw) Sw/ r term must be retained, so that -

Sw/ t - {Q(t)F’(Sw) + G(Sw)}/r Sw/ r = = G’(Sw)( Sw/ r)2 + G(Sw) 2Sw/ r2

(5-149)

is heat-like. The equation Sw/ t Sw/ r is clearly diffusive, and it is not unlike the pressure diffusion equation used in transient

Numerical Supercharge, Pressure and Multiphase Methods 449

compressible well test simulation. But when inertia is more important, the second derivative term Sw/ r can be neglected, at least until shocks form. With this term neglected, Equation 5-149 reduces to the first-order wave equation -

Sw/ t - {Q(t)F'(Sw) + G(Sw)}/r Sw/ r = G'(Sw)( Sw/ r)2 (5-155)

which is the radial form of the Buckley-Leverett equation studied earlier for lineal flows. Whether or not the saturation equation is parabolic or hyperbolic, the pressure equation 2P / r2 + COEF P / r = RHS w w

(5-153)

is always elliptic-like and time-independent, at least to the extent that the variables COEF and RHS are evaluated at the previous time step. In any event, both governing equations, Equations 5-149 and 5-153, contain second-order spatial derivative terms and are associated with welldefined boundary value problems and boundary conditions. These formulations were solved using second-order accurate implicit schemes in the work just presented; that is, our approach was implicit pressure, implicit saturation. This is in contrast to the popular implicit pressure, explicit saturation codes used in the industry, which are only conditionally stable. (The von Neumann stability of both implicit and explicit schemes was considered earlier.) This so-called IMPES scheme, in addition to its stability problems, yields undesirable saturation oscillations and overshoots that are often fixed by upstream (that is, backward) differencing of spatial derivatives. But this solution actually introduces more problems than it fixes. As Lantz (1971) demonstrates, this stabilizes the numerical problem, at the expense of adding artificial viscosity by way of the truncation terms. Thus, the physically meaningful diffusion coefficient G in the G(Sw) 2Sw/ r2 term of Equation 5-149 is no longer the only diffusion in the problem: a numerical diffusion comparable in size to G is introduced that contaminates the computed solution. This has the effect of misplacing the position of the saturation shock and miscalculating the magnitude of the saturation discontinuity. These problems are well known and solved in the aerospace industry, where they arise in high-speed wing design. Mathematical problems should be addressed within the context of the equation itself. However, the basic issues (Chin, 1993a) are still overlooked by petroleum investigators overly concerned with field agreement.

450 Supercharge, Invasion and Mudcake Growth

Modeling mudcake coupling. Now that we understand immiscible two-phase flow formulations, both analytically and numerically, we address the problem where an additional mudcake Darcy flow appears at the inlet to our radial geometry. This flow satisfies its own pressure differential equation and is characterized by a moving mudto-mudcake boundary and a fixed mudcake-to-rock interface. The problem is shown in Figure 5.22, where x applies to both lineal and radial flows. In order to solve this coupled problem, the algorithms developed in prior Example 5-7 and the immiscible flow problem just completed must be coupled. Cake

Immiscible two-phase flow in rock

Pmud i=1 x = -xc < 0

Pres iwall x=0

imax x=L

Figure 5.22. Two-layer mudcake-rock, immiscible flow model. Let us first review the mudcake formulation developed earlier. Again, the flow in the mudcake is assumed to be single-phase, and because compressibility is neglected, the pressure distribution P(x,t) satisfies d(kc dP/dx)/dx = 0 (5-156) where the mudcake absolute permeability kc may be prescribed as a function of x, or given as a function of P, or taken as a constant for simplicity. We choose the latter for convenience, so that the simple ordinary differential equation d2P/dx2 = 0 (5-157) applies. Note that P(x,t) will depend parametrically on time, even though there are no time-dependent derivatives in Equation 5-157, because a moving boundary will be allowed. We also assume that the mud filtrate and the formation water are identical, so that only two fluids need to be modeled. Other formulations are possible but will not be treated here. If oil base muds are used, three separate fluids must be accounted for, namely, the oil filtrate, the formation hydrocarbon, and the connate water. If two different waters (e.g., fresh versus saline) are present, gravity effects may have to be accounted for. And if combined water-oil

Numerical Supercharge, Pressure and Multiphase Methods 451

muds are considered, the mudcake flow formulation is necessarily twophase as it is in the formation. These formulations add to numerical complexity and do not introduce new ideas. Now how do we couple Equation 5-157 for the single-phase flow in the growing mudcake to Equations 5-151, 5-146, and 5-152 describing the two-phase immiscible flow in the rock? It is clear that the grid expansion method used in Example 5-7 and suggested in Figure 5.22 cannot be used: the number of nodes increases with time as the cake thickens, but the saturation solution Si,n requires information at earlier nonexistent spatial nodes. An elementary solution to the problem, fortunately, is available, and requires us first to transform the boundary value problem for the cake into a boundary condition for the rock flow. Since Equation 5-157 applies, where x actually refers to the radial coordinate, the exact solution P = Ar + B applies. Then, the simple solution P = A(r - Rcake) + Pleft satisfies P = Pleft at r = Rcake. (Here, Pleft is the borehole mud pressure, acting on the exposed face of the mud cake located at r = Rcake.) The pressure at the mudcake-to-rock interface is given by the expression P = A(Rwell - Rcake) + Pleft, where r = Rwell is the wellbore radius without cake. The fluid velocity at the cake-torock interface is kcake dP/dr or kcakeA. This must be equal to the Darcy velocity krock(PW2,n-PW1,n)/ r evaluated from the two-phase flow solution. Setting the two equal, that is, kcake A = krock (PW2,nPW1,n)/ r, and noting that pressure continuity requires that PW1,n = A(Rwell - Rcake) + Pleft lead to the fact that [kcake r + krock (Rwell - Rcake)] PW1,n

(5-158)

- krock(Rwell - Rcake) PW2,n = kcakePleft r where we have eliminated the constant A, and PW is the wetting phase pressure. Since the mudcake-to-rock interface is completely saturated with water, the permeability k rock is exactly the absolute permeability. Unchanging mudcake thickness. In dynamic filtration, the mudcake ceases to grow once equilibrium conditions are achieved in the borehole. This invasion is modeled by a cake thickness that is a prescribed constant, which does not vary with time. Then, the only algorithmic change to the Fortran code in the foregoing section requires us to replace

452 Supercharge, Invasion and Mudcake Growth

by

A(1) B(1) C(1) W(1)

= = = =

99. 1. 0. PLEFT

KCAKE = 0.001 KC = KCAKE*0.00000001/(12.*12.*2.54*2.54) RCAKE = 0.01/12. . . A(1) = 99. B(1) = KC*DR + K*(WELRAD-RCAKE) C(1) = -K*(WELRAD-RCAKE) W(1) = KC*PLEFT*DR

Typical saturation and pressures in Figures 5.23a,b,c for early, intermediate, and late times illustrate shock formation and propagation. The parameters were selected to cover the entire range of inertial-tocapillary force ratios. Water saturation at time (hrs): .167E-03 Position (ft) Water Sat ______________________________ .200E+00 .100E+01 | * .300E+00 .715E+00 | * .400E+00 .213E+00 | * .500E+00 .105E+00 | * .600E+00 .100E+00 | * .700E+00 .100E+00 | * .800E+00 .100E+00 | * .900E+00 .100E+00 | * .100E+01 .100E+00 | * .110E+01 .100E+00 | * .120E+01 .100E+00 | * .130E+01 .100E+00 | * .140E+01 .100E+00 | * .150E+01 .100E+00 | * Pressure versus r @ time (hrs): .167E-03 Position (ft) Pressure ______________________________ .200E+00 .117E+09 | * .300E+00 .103E+09 | * .400E+00 .807E+08 | * .500E+00 .682E+08 | * .600E+00 .589E+08 | * .700E+00 .510E+08 | * .800E+00 .442E+08 | * .900E+00 .382E+08 | * .100E+01 .328E+08 | * .110E+01 .279E+08 | * .120E+01 .235E+08 | * .130E+01 .194E+08 | * .140E+01 .156E+08 | * .150E+01 .121E+08 | * .160E+01 .878E+07 |* .170E+01 .568E+07 * .180E+01 .276E+07 | .190E+01 .000E+00 |

Figure 5.23a. Early time solution. Water saturation at time (hrs): .100E-02 Position (ft) Water Sat ______________________________ .200E+00 .100E+01 | * .300E+00 .929E+00 | * .400E+00 .867E+00 | * .500E+00 .823E+00 | * .600E+00 .792E+00 | * .700E+00 .720E+00 | * .800E+00 .374E+00 | * .900E+00 .125E+00 | * .100E+01 .101E+00 | * .110E+01 .100E+00 | * .120E+01 .100E+00 | * .130E+01 .100E+00 | * .140E+01 .100E+00 | * .150E+01 .100E+00 | * .160E+01 .100E+00 | * .170E+01 .100E+00 | * .180E+01 .100E+00 | * .190E+01 .100E+00 | *

Numerical Supercharge, Pressure and Multiphase Methods 453 Pressure versus r @ time (hrs): .100E-02 Position (ft) Pressure ______________________________ .200E+00 .119E+09 | * .300E+00 .106E+09 | * .400E+00 .954E+08 | * .500E+00 .863E+08 | * .600E+00 .782E+08 | * .700E+00 .706E+08 | * .800E+00 .603E+08 | * .900E+00 .503E+08 | * .100E+01 .431E+08 | * .110E+01 .367E+08 | * .120E+01 .308E+08 | * .130E+01 .255E+08 | * .140E+01 .205E+08 | * .150E+01 .159E+08 | * .160E+01 .115E+08 |* .170E+01 .747E+07 * .180E+01 .363E+07 | .190E+01 .000E+00 |

Figure 5.23b. Intermediate time solution. Water saturation at time (hrs): .267E-02 Position (ft) Water Sat ______________________________ .200E+00 .100E+01 | * .300E+00 .971E+00 | * .400E+00 .936E+00 | * .500E+00 .900E+00 | * .600E+00 .866E+00 | * .700E+00 .835E+00 | * .800E+00 .809E+00 | * .900E+00 .791E+00 | * .100E+01 .779E+00 | * .110E+01 .770E+00 | * .120E+01 .752E+00 | * .130E+01 .634E+00 | * .140E+01 .249E+00 | * .150E+01 .112E+00 | * .160E+01 .101E+00 | * .170E+01 .100E+00 | * .180E+01 .100E+00 | * .190E+01 .100E+00 | * Pressure versus r @ time (hrs): .267E-02 Position (ft) Pressure ______________________________ .200E+00 .116E+09 | * .300E+00 .102E+09 | * .400E+00 .907E+08 | * .500E+00 .816E+08 | * .600E+00 .736E+08 | * .700E+00 .664E+08 | * .800E+00 .597E+08 | * .900E+00 .536E+08 | * .100E+01 .479E+08 | * .110E+01 .426E+08 | * .120E+01 .377E+08 | * .130E+01 .324E+08 | * .140E+01 .244E+08 | * .150E+01 .183E+08 | * .160E+01 .133E+08 | * .170E+01 .861E+07 |* .180E+01 .418E+07 * .190E+01 .000E+00 |

Figure 5.23c. Late time solution. Transient mudcake growth. When transient mudcake growth is allowed, for example, as in static filtration or nonequilibrium dynamic filtration, conceptual but simple coding changes are required. For thin mudcake-to-borehole radii ratios, the lineal cake growth model dxc/dt = - {fs/{(1-fs)(1- c)}}|vn| < 0 (5-63)

applies, where |vn| is proportional to the Darcy velocity (kc/ mf) dp(xc)/dx at the cake surface. Note that Equation 5-62 describing displacement fronts in the rock is not used here, since saturation

454 Supercharge, Invasion and Mudcake Growth

discontinuities are allowed to form naturally in immiscible flows, if they exist. Equation 5-63 is approximated by xc,new = xc,old + {fs/{(1-fs)(1- c)}}{kc t/( mf x)}(p2 - p1)old (5-64) where (p2 - p1)old/ x represents the pressure gradient in the cake. But our mudcake pressure solution P = Ar + B shows that dP/dr = A does not depend on position, and that at any instant, it is a constant that does not change through the cake. This being the case, its value can be extrapolated from the velocity matching interfacial condition kcake dP/dr = krock(PW2,n-PW1,n)/ r, that is, dP/dr = (krock/kcake)(PW2,n-PW1,n)/ r

(5-159)

Thus, the only required addition to the Fortran immediately preceding is the following update logic in boldface type. C

C

INITIAL SETUP RCAKE = WELRAD KCAKE = 0.001 KC = KCAKE*0.00000001/(12.*12.*2.54*2.54) FS=0.2 PHIMUD=0.2 . . Update cake position immediately after pressure integration. RATIO = K/KC PGRADC = RATIO*(P(2)-P(1))/DR RCAKE = RCAKE+(FS/((1.-PHIMUD)*(1.-FS)))*(KC/VISCL)*PGRADC*DT . . A(1) = 99. B(1) = KC*DR + K*(WELRAD-RCAKE) C(1) = -K*(WELRAD-RCAKE) W(1) = KC*PLEFT*DR

The uppermost line represents the mudcake initial condition; that is, at time t = 0, the surface of the infinitesimally thin cake coincides with the borehole radius. In Figures 5.24a,b,c, the computational parameters are identical to those in Figures 5.23a,b,c, except that the cake grows from zero thickness, as opposed to being fixed at 0.01 in. for all time. Since the mudcake considered in Figure 5.24 is typically thinner than that in Figure 5.23 for any instant in time, we expect greater relative invasion. In fact, we do observe increased water saturation and deeper penetration of the saturation shock into the rock.

Numerical Supercharge, Pressure and Multiphase Methods 455 Water saturation at time (hrs): .167E-03 Position (ft) Water Sat ______________________________ .200E+00 .100E+01 | * .300E+00 .803E+00 | * .400E+00 .373E+00 | * .500E+00 .118E+00 | * .600E+00 .101E+00 | * .700E+00 .100E+00 | * .800E+00 .100E+00 | * .900E+00 .100E+00 | * .100E+01 .100E+00 | * .110E+01 .100E+00 | * .120E+01 .100E+00 | * .130E+01 .100E+00 | * .140E+01 .100E+00 | * .150E+01 .100E+00 | * .160E+01 .100E+00 | * .170E+01 .100E+00 | * .180E+01 .100E+00 | * .190E+01 .100E+00 | * Pressure versus r @ time (hrs): .167E-03 Position (ft) Pressure ______________________________ .200E+00 .141E+09 | * .300E+00 .125E+09 | * .400E+00 .104E+09 | * .500E+00 .862E+08 | * .600E+00 .743E+08 | * .700E+00 .644E+08 | * .800E+00 .558E+08 | * .900E+00 .482E+08 | * .100E+01 .414E+08 | * .110E+01 .352E+08 | * .120E+01 .296E+08 | * .130E+01 .245E+08 | * .140E+01 .197E+08 | * .150E+01 .152E+08 | * .160E+01 .111E+08 |* .170E+01 .717E+07 * .180E+01 .349E+07 | .190E+01 .000E+00 |

Figure 5.24a. Early time solution. Another interesting observation concerns pressure drops computed at different points in the radial core sample. In the normalized units selected, Pleft = 0.144 x 109 was assumed at the borehole edge of the mudcake, while Pright = 0 was taken at the far right effective radius. Figure 5.24c shows that a pressure of 0.113 x 109 was obtained at the mudcake-to-rock interface. In this calculation, the rock and not the mudcake supports the greatest portion of the total pressure drop. The computations pursued here, in loose terms, model invasion in very tight zones and in problems having highly permeable cakes. Water saturation at time (hrs): .100E-02 Position (ft) Water Sat ______________________________ .200E+00 .100E+01 | * .300E+00 .937E+00 | * .400E+00 .879E+00 | * .500E+00 .833E+00 | * .600E+00 .803E+00 | * .700E+00 .774E+00 | * .800E+00 .650E+00 | * .900E+00 .249E+00 | * .100E+01 .111E+00 | * .110E+01 .101E+00 | * .120E+01 .100E+00 | * .130E+01 .100E+00 | * .140E+01 .100E+00 | * .150E+01 .100E+00 | * .160E+01 .100E+00 | * .170E+01 .100E+00 | * .180E+01 .100E+00 | * .190E+01 .100E+00 | *

456 Supercharge, Invasion and Mudcake Growth Pressure versus r @ time (hrs): .100E-02 Position (ft) Pressure ______________________________ .200E+00 .131E+09 | * .300E+00 .117E+09 | * .400E+00 .105E+09 | * .500E+00 .947E+08 | * .600E+00 .857E+08 | * .700E+00 .776E+08 | * .800E+00 .693E+08 | * .900E+00 .570E+08 | * .100E+01 .481E+08 | * .110E+01 .409E+08 | * .120E+01 .344E+08 | * .130E+01 .284E+08 | * .140E+01 .228E+08 | * .150E+01 .177E+08 | * .160E+01 .129E+08 |* .170E+01 .832E+07 * .180E+01 .404E+07 | .190E+01 .000E+00 |

Figure 5.24b. Intermediate time solution. We emphasize that we have obtained stable numerical results, without saturation overshoots and local oscillations, all using secondorder accurate spatial central differencing without having to introduce special upwind operators. The methods are stable and require minimal computing since they are based on tridiagonal equations. Several subtle aspects of numerical simulation as they affect miscible diffusion and immiscible saturation shock formation are discussed in Chin (2017). Water saturation at time (hrs): .267E-02 Position (ft) Water Sat ______________________________ .200E+00 .100E+01 | * .300E+00 .973E+00 | * .400E+00 .941E+00 | * .500E+00 .907E+00 | * .600E+00 .874E+00 | * .700E+00 .843E+00 | * .800E+00 .816E+00 | * .900E+00 .796E+00 | * .100E+01 .783E+00 | * .110E+01 .773E+00 | * .120E+01 .764E+00 | * .130E+01 .734E+00 | * .140E+01 .525E+00 | * .150E+01 .170E+00 | * .160E+01 .105E+00 | * .170E+01 .100E+00 | * .180E+01 .100E+00 | * .190E+01 .100E+00 | * Pressure versus r @ time (hrs): .267E-02 Position (ft) Pressure ______________________________ .200E+00 .113E+09 | * .300E+00 .987E+08 | * .400E+00 .882E+08 | * .500E+00 .794E+08 | * .600E+00 .717E+08 | * .700E+00 .647E+08 | * .800E+00 .583E+08 | * .900E+00 .523E+08 | * .100E+01 .468E+08 | * .110E+01 .417E+08 | * .120E+01 .369E+08 | * .130E+01 .323E+08 | * .140E+01 .269E+08 | * .150E+01 .192E+08 | * .160E+01 .137E+08 | * .170E+01 .885E+07 |* .180E+01 .430E+07 * .190E+01 .000E+00 |

Figure 5.24c. Late time solution.

Numerical Supercharge, Pressure and Multiphase Methods 457

General immiscible flow model. Earlier we showed how a firstorder nonlinear equation arises in immiscible two-phase flow. We derived Equation 5-90, that is, Sw/ t + {q(t)/ } dfw(Sw, w/ nw)/dSw Sw/ x = 0 for saturation in one-dimensional systems, and indicated that it applied to high-rate invasion problems where capillary pressure could be ignored. This equation was accurate at least until the appearance of saturation shocks and steep flow gradients. Then, the low-order description breaks down locally, but it could still be used provided we introduce a shock that satisfies certain externally imposed constraints that fall outside the scope of the simple formulation. In fact, mass conservation requires us to take the shock velocity in the form given by Equation 5-112, namely, Vshock = {Qw(Swl) - Qw(Swi)}/(Swl - Swi). But the patched solution is incomplete, since the structure and thickness of the shock cannot be resurrected. In order to recover the complete details of the flow, recourse to the high-order partial differential equation with capillary pressure is necessary. In radial flow, the required Equation 5-149 shows that the more detailed physical model is - Sw/ t - {Q(t) F’(Sw) + G(Sw)}/r Sw/ r = G’(Sw)( Sw/ r)2 + G(Sw) 2Sw/ r2. The G(Sw) 2Sw/ r2 term is all-important, as we have seen (e.g., refer to Figure 5.15c) because it produces the shock structure naturally; also, it will affect the propagation speed somewhat, and the shock speed so obtained will differ from the Vshock given here. In addition, this second-order derivative completely determines the particular flux that is conserved across shocks and implicitly contains the entropy condition that dictates the manner in which shocks form. The key idea, we emphasize, is the crucial role that the high-order derivative term plays: it may be negligible for a while, but it must be correctly accounted for at the shock because it is large. This being the case, it is imperative that the correct high-order terms be modeled and that the included terms remain free of undesirable numerical diffusion. In this chapter, the coupling of dynamic mudcake growth to immiscible fluid flow was studied as a purely radial problem. Idealizations were undertaken in order to extend the diffusion ideas presented early on to broader problems involving two-phase flow. In practical applications, many physical mechanisms are simultaneously at work in the formation, for example, reservoir heterogeneities, axial

458 Supercharge, Invasion and Mudcake Growth

variations, and miscible mixing. At the same time, the auxiliary conditions that apply to well logging tools are far from simple. Consider formation-testing-while-drilling. During drilling, two-phase flow invasion takes place while the mudcake builds; this establishes initial conditions that apply once fluid sampling commences. When pretest samples are taken, mudcake is first removed at the piston, and complicated three-dimensional boundary conditions that model skin and flow line storage effects must be used. This operational procedure can be simulated in detail using the building blocks described in this book. A comprehensive numerical model has been developed for tester applications. It can predict (1) the pumping times required to purge the near-well formation of mud filtrate before uncontaminated petroleum fluids are accessible, (2) the tool power requirements associated with such pumping processes, and (3) continually refined formation evaluation parameters based on compressible and incompressible fluid flow pressure transients. This modeling effort is reported in “Sample Quality Prediction with Integrated Oil and Water-Based Mud Invasion Modeling,” SPE Paper No. 77964, SPE Asia Pacific Oil & Gas Conference and Exhibition (APOGCE), Oct. 2002, Melbourne, Australia by M. Proett, D. Belanger, M. Manohar and W. Chin. 5.3 Closing Remarks. This section brings our formation testing volume, the latest addition to Wiley-Scrivener’s Petroleum Engineering Handbook Series to a close. While “handbooks” normally refer to summaries of decades-old technologies, the present edition is timely because numerous advances have been made in pressure transient analysis, forward and inverse modeling, supercharge, mudcake growth and invasion formulations, and contamination and cleaning multiphase methods – and all during the past fifteen years by the present authors. In addition, there has been rapid and complementary growth in hardware design, at both COSL and at other leading oilfield manufacturers – to include the newer “triple probe” formation testers discussed in our companion 2021 book publication. All of this will overwhelm petroleum practitioners and researchers alike with “information overload.” To lighten this burden, we have developed means for custom and individualized delivery of content to engineers, researchers, interpretation specialists, and so on. The FTSim menus for our system, shown on the following pages, are self-explanatory, and individuals may update menu items with their own content as needed.

Numerical Supercharge, Pressure and Multiphase Methods 459

Figure 5.25a. COSL formation testing software platform.

460 Supercharge, Invasion and Mudcake Growth

Figure 5.25b. COSL formation testing software platform.

Numerical Supercharge, Pressure and Multiphase Methods 461

Figure 5.25c. COSL formation testing software platform.

462 Supercharge, Invasion and Mudcake Growth

Figure 5.25d. COSL formation testing software platform.

Numerical Supercharge, Pressure and Multiphase Methods 463

Figure 5.25e. COSL formation testing software platform. Again, with the exception of the finite difference models in this chapter, the present volume focuses on “source” (or “sink”) models for forward and inverse applications, with and without supercharge. Our companion 2021 volume Formation Testing – Multiprobe Design and Pressure Analysis, by Tao Lu, Minggao Zhou, Yongren Feng, Yuqing Yang and Wilson Chin focuses on formation testers with azimuthally displaced probes, for example, as shown in the photographs of Figures 1.20 and 1.21, which require completely different three-dimensional models. In addition, the forward models are entire numerical, while inverse methods draw upon statistical approaches in “big data” simulation. The methods discussed in that book are broad, as they are also applicable to conventional azimuthal tools with one active sink probe and a passive horizontal observation probe in the cross-plane of the tool.

464 Supercharge, Invasion and Mudcake Growth

5.4 References. Carnahan, B., Luther, H.A., and Wilkes, J.O., Applied Numerical Methods, John Wiley & Sons, New York, 1969. Chin, W.C., Borehole Flow Modeling in Horizontal, Deviated and Vertical Wells, Gulf Publishing, Houston, 1992. Chin, W.C., Petrocalc 14: Horizontal and Vertical Annular Flow Modeling, Petroleum Engineering Software for the IBM PC and Compatibles, Gulf Publishing, Houston, 1992. Chin, W.C., Modern Reservoir Flow and Well Transient Analysis, Gulf Publishing, Houston, 1993. Chin, W.C., 3D/SIM: 3D Petroleum Reservoir Simulation for Vertical, Horizontal, and Deviated Wells, Petroleum Engineering Software for the IBM PC and Compatibles, Gulf Publishing, Houston, 1993. Chin, W.C., Wave Propagation in Petroleum Engineering, with Applications to Drillstring Vibrations, Measurement-While-Drilling, Swab-Surge and Geophysics, Gulf Publishing, Houston, 1994. Chin, W.C., Formation Invasion, with Applications to MeasurementWhile-Drilling, Time Lapse Analysis and Formation Damage, Gulf Publishing, Houston, 1995. Chin, W.C., Computational Rheology for Pipeline and Annular Flow, Butterworth-Heinemann, Boston, MA, 2001. Chin, W.C., RheoSim 2.0: Advanced Rheological Flow Simulator, Butterworth-Heinemann, Boston, MA, 2001. Chin, W.C., Electromagnetic Well Logging: Models for MWD/LWD Interpretation and Tool Design, John Wiley & Sons, Hoboken, New Jersey, 2014. Chin, W.C., Wave Propagation in Drilling, Well Logging and Reservoir Applications, John Wiley & Sons, Hoboken, New Jersey, 2014. Chin, W.C., Quantitative Methods in Reservoir Engineering, Second Edition – with New Topics in Formation Testing and Multilateral Well Flow Analysis, Elsevier Science, Woburn, MA, 2017.

Numerical Supercharge, Pressure and Multiphase Methods 465

Chin, W.C., Suresh, A., Holbrook, P., Affleck, L., and Robertson, H., “Formation Evaluation Using Repeated MWD Logging Measurements,” Paper No. U, SPWLA 27th Annual Logging Symposium, Houston, TX, June 9-13, 1986. Claerbout, J.F., Fundamentals of Geophysical Data Processing, Blackwell Scientific Publishers, Oxford, 1985. Claerbout, J.F., Imaging the Earth’s Interior, Blackwell Scientific Publishers, Oxford, 1985. Collins, R.E., Flow of Fluids Through Porous Materials, Reinhold Publishing, New York, 1961. Dewan, J.T., and Chenevert, M.E., “Mudcake Buildup and Invasion in Low Permeability Formations: Application to Permeability Determination by Measurement While Drilling,” Paper NN, SPWLA 34th Annual Logging Symposium, June 13-16, 1993. Dewan, J.T., and Holditch, S.A., “Radial Response Functions for Borehole Logging Tools,” Topical Report, Contract No. 5089-2601861, Gas Research Institute, Jan. 1992. Holditch, S.A., and Dewan, J.T., “The Evaluation of Formation Permeability Using Time Lapse Logging Measurements During and After Drilling,” Annual Report, Contract No. 5089-260-1861, Gas Research Institute, December 1991. Lane, H.S., “Numerical Simulation of Mud Filtrate Invasion and Dissipation,” Paper D, SPWLA 34th Annual Logging Symposium, June 13-16, 1993. Lantz, R.B., “Quantitative Evaluation of Numerical Diffusion (Truncation Error),” Society of Petroleum Engineers Journal, Sept. 1971, pp. 315-320. Lu, T., Zhou, M., Feng, Y., Yang, Y. and Chin, W.C., Multiprobe Pressure Analysis and Interpretation, John Wiley & Sons, Hoboken, New Jersey, 2021. Marle, C.M., Multiphase Flow in Porous Media, Gulf Publishing, Houston, 1981.

466 Supercharge, Invasion and Mudcake Growth

Muskat, M., Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York, 1937. Muskat, M., Physical Principles of Oil Production, McGraw-Hill, New York, 1949. Outmans, H.D., “Mechanics of Static and Dynamic Filtration in the Borehole,” Society of Petroleum Engineers Journal, Sept. 1963, pp. 236-244. Peaceman, D.W., and Rachford, H.H., “Numerical Calculation of Multidimensional Miscible Displacement,” Society of Petroleum Engineers Journal, Dec. 1962, pp. 327-339. Proett, M.A., Belanger, D., Manohar, M., and Chin, W.C., “Sample Quality Prediction with Integrated Oil and Water-Based Mud Invasion Modeling,” SPE Paper No. 77964, SPE Asia Pacific Oil & Gas Conference and Exhibition (APOGCE), Oct. 2002, Melbourne, Australia. Streeter, V.L., Handbook of Fluid Dynamics, McGraw-Hill, New York, 1961. Whitham, G.B., Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974.

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Index Barrier, 144, 237, 273, 280 Barrier detection, 237, 280 Barriers, 8, 139, 221, 273, 280, 299 Basic Reservoir Characteristic Tester, 13 BASIC-RCT, 13 Batch, 115-116, 237, 281-284 Batch mode, 115-116, 282, 284 Bedding plane effects, 39, 49 Boundary condition, 24, 30, 143, 391, 400, 411, 428, 443, 451 Boundary conditions, 36, 49, 82, 302-303, 306, 308, 311, 313, 316, 322, 324, 328, 331, 343, 366, 370, 374, 397, 412, 417, 449, 458 Boundary conforming, 32, 376 Boundary layer, 397 Boundary value, 24, 28, 36, 113, 290, 300, 302-303, 305, 332, 335, 363, 374, 409, 411, 413, 417, 449, 451 Boundary value problem, 28, 36, 113, 300, 305, 332, 335, 374, 411, 413, 417, 451 Boundary value problems, 24, 290, 302-303, 363, 409, 449 Buckley-Leverett, 406-407, 433, 449 Buildup, 2, 9, 23, 25-26, 28, 37, 55-56, 58, 61, 63-64, 74-75, 81, 85-86, 110, 112-113, 122-125, 127-129, 132-133, 136, 144, 173, 177, 180, 184, 186, 188, 193-194, 216, 226, 238-239, 241-243, 249, 252, 255-256, 259-260, 262-263, 277, 282, 292, 302, 314, 318-321,

A Absolute permeability, 430, 450451 ADI, 50 Adiabatic, 118-119, 302, 315, 390 Aerodynamics, 367 Amplification, 371, 389 Amplitude attenuation, 81, 86 Anisotropic, 25, 27-28, 31-32, 34, 36, 38, 49, 52-53, 64, 81-82, 84, 89, 97-99, 103, 105-107, 117, 223, 265, 267, 273, 280, 302, 317 Anisotropy, 1, 8, 10, 23, 28, 32, 37-38, 40, 47, 52, 86, 111, 113, 119, 224, 264, 271, 273, 361 Annular, 304, 310, 336-337, 360, 364, 464 Annulus, 290 Artificial viscosity, 49, 407, 410, 449 Asymptotic, 26, 37, 260 Attenuation, 81, 86, 90 Axial, 2, 18, 38-39, 49, 231, 267, 346, 457 Axially displaced, 18, 24, 26, 38, 50, 86, 264, 273 Axisymmetric, 31, 221, 223, 231, 328 Axisymmetry, 221 Azimuthal, 32, 38-39, 43, 51-52, 231, 463 Azimuthally displaced, 50, 463

B Backward differences, 365, 367, 369, 374, 399, 418 481

482 Supercharge, Invasion and Mudcake Growth

323, 327, 337, 341-345, 351, 357, 371, 396, 465

C Cake, 139, 217, 289-292, 301, 306, 312, 314, 317-321, 323-330, 334-346, 349, 352-353, 355-358, 380, 383, 396-403, 419, 426, 441, 450-451, 453-454 Calibration, 7, 34-35, 38, 62, 65 Capillary pressure, 303, 305, 406408, 410-412, 414-415, 419, 425426, 428-430, 432-433, 438, 441, 446-448, 457 Cartesian, 306 Catscan, 217-220, 293, 297-298, 312, 337, 339 Central difference, 365, 369, 389, 397, 399, 403, 427, 442 Chemical, 1, 304 China Oilfield Services, 22 Circular coordinates, 221 Clean-up, 113, 133, 217, 221, 224 Coefficient matrix, 351, 367 Color, 33, 36, 100, 133, 137, 228 Compaction, 306, 317, 319, 337, 363, 373, 396, 404 Complex complementary error function, 43, 113, 226 Complex conjugate, 38 Complex variables, 114 Compressibilities, 37, 143, 227, 296, 314, 324, 328, 373, 386-387, 392 Compressibility, 1, 8, 23-24, 30, 47, 61, 82, 89-91, 94, 96-98, 100, 111-113, 123-125, 127, 132, 139, 144-145, 148-159, 161, 163, 166169, 171-177, 179-185, 187-188,

190-193, 200, 204, 238-239, 243244, 249, 255-256, 260, 262, 271, 281, 295, 300, 304-306, 311, 321322, 331-334, 363, 385-386, 390, 392-393, 396, 404-405, 450 Computational rheology, 360, 464 Concentration, 137, 222-226, 228, 230-232, 234, 359, 415-416, 419-425 Concentration profile, 224, 359, 420, 423, 425 Constant flow rate, 123, 238-239 Constant rate pumping, 40, 44, 239 Contamination, 3, 20-21, 23, 32, 38, 61, 113, 133, 137, 217, 221222, 226, 234-236, 289-290, 292, 294, 318, 458 Convergence, 69, 158, 272, 318, 371, 389, 407, 448 COSL, 4, 6-9, 13, 17-19, 22, 24, 29, 35, 458-463 Cubic equation, 38 Curvature, 22, 26, 62, 142, 221, 257, 296 Cylindrical, 32, 34, 37, 61, 113, 136, 139-140, 217, 222-223, 228, 230, 232, 257, 290, 292, 296, 303, 306, 313-315, 327, 341-343, 347, 371, 373, 380, 383-384, 388389, 405, 416, 423, 426 Cylindrical coordinates, 61, 416 Cylindrical flow, 139

D

Darcy, 1, 24, 27, 29-30, 36, 38, 50, 52, 55, 61, 81-82, 117, 122, 142-143, 220-221, 256, 258, 289,

Index 483

292, 296, 302-306, 310-311, 315316, 319-322, 325, 336, 341, 343, 367, 373-374, 400, 406-407, 414, 416, 426, 432, 450-451, 453 Darcy flow, 29, 52, 61, 142, 220221, 256, 258, 296, 304, 306, 319, 321, 336, 341, 343, 367, 450 Data integrity, 46 Deconvolution, 420 Depth of investigation, 82, 90, 117, 227, 231, 235, 276 Deviated well, 46, 84-85 Deviated well interpretation, 85 Diagonally dominant, 372, 374 Diffuse, 304 Diffusion, 18, 26-27, 40-41, 47, 49, 52, 55, 104, 117, 226-227, 234, 280, 298, 300, 303-304, 309, 331, 361, 406-407, 410, 415-416, 418-421, 423, 438, 448-449, 456457, 465 Diffusion coefficient, 449 Diffusion equation, 416, 420, 448 Diffusive, 26, 223, 372, 415, 419, 429, 448 Dimensionless, 145, 151, 155, 159, 161, 163, 169, 174, 177, 181, 185, 188, 272, 299, 314-315, 317, 326, 341, 421 Dip, 22, 27-28, 41-46, 81-82, 8485, 88, 96-100, 106-107, 112, 117-118, 235, 237, 264-265, 268, 272-273 Dip angle, 22, 27, 41-45, 82, 8485, 88, 97-100, 106, 112, 117, 235, 237, 264-265, 268, 272-273 Direct gas solver, 272 Dirichlet, 306

Discontinuities, 359, 406, 411413, 419-420, 433, 443, 448, 454 Discontinuity, 309, 359, 411, 421, 425, 433, 438, 446, 449 Discontinuous, 405-406, 419-420, 448 Dispersive, 372 Dissolved gas, 7, 17, 22 Distant probe, 46 Do-loop, 237, 387, 395 DOI, 116-117, 237, 273, 276, 280 Double drawdown, 62-63, 132 Double-drawdown, 62, 64, 132 Drawdown, 2, 9, 15, 23, 25-28, 33, 37, 47, 55-56, 58, 61-68, 7478, 80-81, 85-86, 110, 112-113, 122-129, 132, 144-145, 147, 149, 151-153, 155, 157-159, 161, 163, 165-168, 171-174, 177, 180-181, 184-186, 188, 193-194, 202, 206, 210, 214, 216, 226, 237-239, 241242, 247, 252, 256, 259-264, 267, 270, 276-277, 282, 302, 314, 371 Drawdown buildup, 144 Drawdown only, 2, 55, 58, 122, 126, 128, 144, 147, 149, 152-153, 157, 165-167, 171-173, 237, 259260, 276 Drawdown-buildup, 2, 23, 25-26, 28, 37, 55, 61, 81, 85-86, 110, 112-113, 122-125, 127-128, 132, 173, 177, 180, 184, 186, 188, 193-194, 216, 239, 259, 262-263, 277, 282 Drawdown-only, 144-145, 151, 155, 158-159, 161, 163, 168, 260, 262, 267 Drawdowns, 17, 61, 63-64, 67, 74, 110, 132

484 Supercharge, Invasion and Mudcake Growth

Drilling mud, 217 DST, 13, 227, 231 Dual packer, 6 Dynamic filtration, 310, 331, 336-337, 341, 405, 451, 453, 466 Dynamically coupled, 221, 289, 291, 301, 318, 351, 355, 396, 441

E

EFDT, 9-10, 16, 20, 298, 361 Effective permeability, 107, 264 Effective porosity, 309, 325, 346, 353 Effective properties, 308 Effective radius, 145, 148-149, 151, 153-155, 157-159, 161, 163, 166-169, 171-172, 174-177, 179181, 183-185, 187-188, 191-192, 244, 249, 255, 271, 315, 328, 347, 349, 355-358, 382-385, 428429, 431, 455 Electromagnetic, 2, 21, 26-27, 52, 81-82, 86, 112, 117, 339, 348, 360, 419, 464 Electromagnetic logging, 2, 21, 26-27, 52, 81-82, 86, 112, 117 Elementary solution, 451 ELIS, 10 Ellipsoidal, 26, 30-32, 36-38, 49, 61, 79, 82-84, 88, 139-141, 226, 289 Ellipsoidal flow, 31, 139-140, 289 Ellipsoidal source, 36, 82, 84 Elliptic, 300, 449 Elongated pad, 86 Enhanced Formation Dynamic Tester, 9 Entropy, 420, 457 Equilibrate, 46

Equilibration, 40-41, 61, 267 Erosion, 299, 317, 331, 338 Error function, 43, 113-114, 226, 417 Eulerian, 303-306, 308, 312-313 Excitation, 90-92, 94, 100 Experiment, 318 Experimental results, 217, 336 Explicit, 311, 370-371, 389, 392, 431, 442, 449 Exponential, 26, 37, 55, 64, 8384, 90, 111-112, 114, 123 Exponential model, 114

F

Farfield, 37, 50, 143, 220, 223, 257, 296, 346, 380-381, 383-384, 398, 429, 433 Field operations, 22 Filter paper, 299, 318, 320, 327, 341-344 Filtrate, 224-226, 228, 231, 289293, 301, 309, 314, 318-321, 323328, 337-338, 341-342, 346, 349, 351-352, 355-356, 358, 361, 390, 396, 398-399, 405, 407, 412, 417, 420-421, 426, 428-429, 431, 439, 441, 450, 458, 465 Filtration, 221, 299, 310, 319, 321, 330-331, 334, 336-338, 341342, 345, 401, 405-406, 409, 411, 431, 435-438, 451, 453, 466 Finite difference, 7, 32, 34, 37, 39, 49-50, 79, 91, 99-100, 117, 119, 310, 318, 323, 363-366, 370, 380, 385-386, 392, 400, 407, 410, 420, 425, 427, 435, 463 Finite difference equations, 392 Finite differences, 396 Finite differencing, 392

Index 485

Finite element, 25, 32, 37, 49, 117, 142, 220-221, 257, 296, 364, 379, 410 Fishing, 15-16 Flowline storage, 24-25, 30, 36, 43, 55, 65, 69, 79, 111, 113, 120, 122-123, 132, 143, 145, 147, 149, 151-153, 155, 157, 159, 161, 163, 165-168, 171-172, 200, 221, 258 Flowline volume, 8, 21, 23, 27, 30, 47, 56, 67-71, 73-74, 76-80, 110, 112, 120-122, 132, 145, 148, 150-151, 154-155, 159, 161, 163, 168-169, 173-174, 177, 181, 184185, 187-188, 193, 216, 243, 264, 281 Fluid contact, 10, 15, 139, 361, 443 Fluid contacts, 168 Fluid sampling, 5, 10, 458 Formation evaluation, 1, 5, 13, 27, 81, 142, 217, 221, 236, 294, 336, 338, 361, 405, 458, 465 Formation rate analysis, 25 Formation tester, 1, 5, 7, 13, 1618, 21-22, 25, 27-29, 32, 34, 37, 49, 54, 61-62, 81, 85, 87, 90, 107, 110-113, 118, 136, 139, 141-142, 144-145, 151, 155, 159, 161, 163, 168, 174, 177, 181, 185, 188, 217, 220-221, 228, 237, 257, 262, 270, 273-274, 298, 361, 443 Formation testing, 1-6, 8-10, 12, 14, 16-18, 20-26, 28, 30, 32-36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116-118,

120, 122-124, 126, 128, 130, 132134, 136, 138-142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 232, 234-238, 240, 242, 244, 246, 248, 250, 252, 254, 256, 258, 260, 262, 264, 266, 268, 270, 272, 274, 276, 278, 280, 282, 284, 286, 288-290, 292, 294-298, 300, 302, 304, 306, 308, 310, 312, 314, 316, 318, 320, 322, 324, 326, 328, 330, 332, 334, 336, 338, 340, 342, 344, 346, 348, 350, 352, 354, 356, 358, 360-362, 364, 366, 368, 370, 372, 374, 376, 378, 380, 382, 384, 386, 388, 390, 392, 394, 396, 398, 400, 402, 404, 406, 408, 410, 412, 414, 416, 418, 420, 422, 424, 426, 428, 430, 432, 434, 436, 438, 440, 442, 444, 446, 448, 450, 452, 454, 456, 458-466 Formation-testing-while-drilling, 224, 458 Fortran, 100, 144, 299, 363, 368369, 372, 376-377, 379, 381-382, 386-388, 393-395, 401-402, 404405, 418, 425, 429, 435, 443, 451, 454 Forward analysis, 65, 68, 70, 7273, 152, 166, 173, 177, 184, 187188, 190 Forward difference, 365

486 Supercharge, Invasion and Mudcake Growth

Forward model, 39, 144, 150, 152, 154, 193, 200, 258 Fourier, 94, 332-333, 371, 389 FRA, 25 Fractional flow, 407-408, 411, 413, 427, 429-430 Fracture, 317, 351 FT-00, 36-39, 43, 46-47, 49, 52, 55-57, 59, 65, 67-68, 70-74, 76, 79, 94, 110, 113-119, 129, 139, 141-143, 193, 200-201, 205, 209, 213, 237-239, 243, 245, 250, 256, 258, 264-265, 269, 271, 273, 275276, 281-284, 296 FT-01, 37-38, 41-43, 46-47, 110, 117-118, 122, 139, 141-143, 257, 264, 268, 271, 275, 280, 296 FT-03, 38, 49-52, 110 FT-04, 52 FT-06, 59-60, 79, 91, 110, 119121, 237 FT-07, 79, 91, 119-121, 237 FT-PTA, 55, 58, 110, 122-123, 130, 132, 141-142, 145, 147-148, 150, 152-154, 157, 159, 163, 165168, 171-173, 175-180, 183-184, 187-188, 192-193, 255-257, 264, 296 FT-PTA-DD, 256, 264 FT-PTA-DDBU, 55, 58, 110, 122-123, 130, 132, 141-142, 193, 255-257, 264, 296 FTWD, 20, 26, 38, 50, 55, 227 Function statement, 431

G

Gas, 7, 17, 22, 28, 32, 37, 59, 61, 111, 113, 118-120, 133, 142, 271272, 300, 302, 309, 315-316, 327,

333, 361, 363, 373, 390-391, 393396, 458, 465-466 Gas displacement, 327, 363, 390391, 393, 396 Gas exponent, 272 Gas flow, 315, 391 Gas pumping, 59, 61 Gas solver, 272 Gaussian elimination, 367 Geometric factor, 30, 65-67, 69, 71-72, 74-78, 80, 142, 145, 147151, 153-155, 157-159, 161, 163, 165-169, 171-181, 183-185, 187188, 191-192, 244, 249, 254-257, 271, 296 Geometric spreading, 83, 86, 90, 104, 306, 317, 383-384, 406, 419, 421, 426 GeoTapTM, 21, 26, 112, 245 Gradient, 55, 139, 168, 312-314, 316, 341, 374, 399, 408, 414, 416, 439, 448, 454

H

Harmonic, 32, 419 Heat equation, 300, 304, 369, 371, 385, 420 Heat transfer, 300, 302, 332, 420 Heterogeneities, 22, 86, 273, 300, 396, 403, 457 Heterogeneity, 280 History matching, 38, 47, 49, 52, 61, 110, 116, 237, 281, 283, 304 Homogeneous, 36, 49-50, 81-82, 100, 280, 300, 309, 311, 313-315, 317, 319, 327, 331-332, 343, 361, 366, 369, 373, 384, 407, 415, 466 Homogeneous media, 49, 81-82, 100

Index 487

Horizontal permeability, 33, 8283, 269, 283 Horizontal well, 38, 83, 97 Horizontal well limit, 83, 97 Horner, 23, 37, 74 Hydrate, 113 Hydraulic, 13, 16, 86-87, 298 Hyperbolic, 304, 449

I

Immiscible, 32, 220, 223, 292, 299-300, 303, 305, 308-310, 336, 347, 359, 363, 390, 405-407, 409, 415, 419, 425, 441, 447, 450-451, 454, 456-457 Immiscible flow, 347, 359, 390, 406-407, 447, 450-451, 457 IMPES, 431, 449 Implicit, 83, 344, 370-371, 388389, 392, 399, 429, 431, 433, 445, 449 Inertial effects, 217, 220 Infill drilling, 143, 257, 289, 296, 300 Initial conditions, 24, 220, 302303, 329, 332, 335, 385, 413, 458 Initial value, 317, 375, 410, 421 Initial value problem, 375, 410, 421 Injection, 37, 39, 269, 304, 407, 413, 432 Interface, 12, 36, 59, 79, 100-101, 107, 122, 127-129, 137, 223, 290, 300, 305-307, 309, 319, 322, 324, 328, 335, 342-343, 374-375, 398399, 401, 414, 419, 448, 450-451, 455 Interference, 10, 47 Internal nodes, 370, 397, 428, 443

Interpretation, 4, 7, 18, 22-23, 28, 34, 38-40, 46-47, 52, 55, 58, 62, 65, 82-83, 85-86, 99, 111-113, 117, 120, 132, 140, 217, 224, 237, 274, 290, 302, 309, 321, 342, 354, 359-360, 383, 419, 458, 464 Invasion, 20, 30, 32, 34, 37, 55, 61, 110, 113, 133, 136, 141, 217219, 221, 223-224, 226, 228-231, 234, 237, 257, 289-293, 295-315, 317-319, 321-323, 325-331, 333, 335-341, 343, 345-361, 363-364, 373, 375-376, 383, 396, 403, 405407, 412, 420, 431-432, 435, 451, 454-455, 457-458, 464-466 Inverse, 20-21, 35-38, 40-42, 46, 49-50, 55, 57, 61, 63, 65, 67-70, 74, 76, 79-80, 110-111, 113, 117120, 122-123, 125, 127, 129-132, 138-149, 151-153, 155-158, 163169, 171-187, 189-194, 200, 204, 208, 212, 216, 235, 237-239, 241245, 248, 250, 252, 254-258, 260, 262-264, 267-273, 289, 295-296, 327, 334, 336, 339-340, 342, 345, 349, 351, 357-359, 405-406, 425, 458, 463 Inverse analysis, 20, 65, 113, 130, 184, 186, 264, 327, 334, 336, 345 Inverse model, 41, 57, 68, 74, 80, 118, 122-123, 125, 132, 147, 149, 152-153, 156-157, 165-167, 171173, 175-176, 178-180, 183, 187, 191-193, 200, 204, 208, 212, 216 Inverse problem, 42, 248, 269 Isothermal, 118-119, 302, 315316, 390

488 Supercharge, Invasion and Mudcake Growth

Isotropic, 25-28, 30-31, 33, 36, 38, 48, 50, 63, 65, 79, 81-82, 8990, 96-99, 103-105, 107-108, 112, 139, 142, 145, 147, 149, 151-153, 155, 157, 159, 161, 163, 165-168, 171-172, 235, 256, 264, 296, 300, 310, 317, 327, 343, 390 Iterative, 74, 354, 372, 415

J

Job planning, 4, 32, 50, 117, 217, 221, 224, 226-227, 235, 271, 273, 275, 298-299

L

Laboratory, 35, 63, 217, 318, 321, 327, 339-342, 411 Lagrangian, 303-305, 308, 310, 312-313, 317, 330 Layer, 32, 85, 99-101, 105-110, 137, 228, 230-231, 289, 324-325, 327-328, 335, 343, 345-347, 396397, 450 Layered media, 51-52, 81, 221, 235 Layering, 104, 107 Layers, 32, 85, 99, 107, 110, 221, 228, 322, 324-325, 328-329, 335, 339 Line plots, 100, 116, 119, 133, 137 Linear, 37, 65, 74, 118-119, 217220, 292, 297, 299, 302, 304, 306, 311, 315-316, 318, 321, 323, 327, 333, 339, 348, 350-351, 363, 366-367, 369, 397, 414, 416, 466 Liquid, 36-37, 59, 62, 90, 114120, 133, 145, 150-151, 154-155, 159, 161, 163, 168-169, 173-174, 177, 180-181, 184-185, 188, 268,

272, 292, 300, 311-316, 320-321, 323-329, 331, 333-334, 341, 351352, 366, 369, 373, 380, 383, 385, 390-392, 394-396, 398 Liquid displacement, 323, 373, 380, 383, 385 Liquids, 61, 82, 113, 118, 122, 129, 142, 256, 296, 300, 302, 315-319, 321, 324, 328, 331-333, 363, 373, 390, 392-393, 396 Logarithmic, 315 Low mobility, 3, 18, 20-21, 2326, 28, 40, 55, 63-64, 67, 81, 83, 85, 88-89, 110, 117, 122-123, 132, 145, 147, 149, 151-153, 155, 157, 159, 161, 163, 165-168, 171172, 193, 200, 236, 258, 273, 283, 294 LWD, 81, 360, 464

M

Mass conservation, 302, 306, 316, 330, 408, 457 Maxwell's equations, 27, 52, 81, 86 MDDBU, 129, 193, 255 Measurement While Drilling, 1, 26, 36, 465 Measurement-While-Drilling, 360, 464 Mesh generation, 32 Migration, 359, 406, 419-420, 438, 441 Mini-DST, 227, 231 Miscible, 32, 133, 136, 220, 223, 234, 292, 299-300, 303, 310, 336, 363, 405-406, 415, 418-419, 438439, 456, 458, 466 Miscible flow, 136, 405-406, 438-439

Index 489

Mobility, 1, 3, 6, 8, 18, 20-21, 2328, 34-35, 40, 43, 55, 57, 61-64, 66-68, 71, 73-74, 76, 80-81, 83, 85, 88-89, 91, 94, 110-113, 117, 120, 122-125, 127, 132, 139, 144145, 147-159, 161, 163, 165-168, 171-173, 175-176, 179-180, 182184, 187-188, 190-193, 200, 204, 208, 212, 216-217, 236, 238-239, 243-244, 249, 255-256, 258, 260, 262, 271, 273, 275, 283, 294-295, 298, 312, 326, 334, 340, 357, 375, 392, 441 Modeling hierarchies, 23, 28 Molecular diffusion, 331, 415, 438 Monophase, 9-10 Movie, 137, 226 Moving boundaries, 289, 318, 335, 363, 396 Moving boundary, 217, 290, 319, 335, 363, 450 Moving front, 306, 374, 392 Moving interfaces, 322 MPST, 10 Mud, 16, 32, 61, 65, 133, 136137, 217, 223-224, 229, 231, 233234, 289-291, 293, 298-299, 301, 305, 309, 314, 318-319, 323-324, 326-328, 335, 337-339, 341-343, 345, 347-349, 351-352, 355-358, 361, 379, 398, 400-403, 405-407, 420, 426, 428-429, 431, 435, 450451, 458, 465-466 Mud cake, 314, 355-358, 401403, 451 Mud filtrate, 224, 231, 289, 291, 293, 301, 309, 318, 323-324, 327, 337, 349, 351-352, 355-356, 358,

361, 398, 407, 420, 426, 428-429, 431, 450, 458, 465 Mudcake, 16, 20, 34, 133, 136, 139, 141, 217, 219, 221, 223, 234-235, 237, 289-293, 295, 297299, 301, 303, 305, 307, 309-311, 313-315, 317-321, 323-331, 333347, 349, 351-353, 355-359, 361, 363, 372-373, 375, 385, 390, 396402, 404-407, 412, 414, 419-421, 425-426, 435-437, 441, 450-451, 453-455, 457-458, 465 Mudcake buildup, 133, 136, 292, 314, 318-321, 323, 327, 343-344, 351, 396, 465 Mudcake growth, 20, 34, 217, 235, 289, 292-293, 295, 297-299, 301, 303, 305, 307, 309, 311, 313, 315, 317-321, 323, 325-327, 329, 331, 333, 335-337, 339, 341343, 345, 347, 349, 351, 353, 355, 357, 359, 361, 396, 400, 405, 407, 425, 441, 453, 457-458 Multilateral, 361, 464 Multilayer, 339, 348, 359 Multiphase, 32, 61, 113, 133, 138, 217, 220, 292, 299-300, 310, 336, 363, 365, 367, 369, 371, 373, 375, 377, 379, 381, 383, 385, 387, 389, 391, 393, 395, 397, 399, 401, 403, 405-407, 409, 411, 413, 415, 417, 419, 421, 423, 425, 427, 429, 431, 433, 435, 437, 439, 441, 443, 445, 447, 449, 451, 453, 455, 457-459, 461, 463, 465 Multiphase flow, 61, 217, 336, 363, 405-406, 429, 465 Multiphase invasion, 61, 133, 299

490 Supercharge, Invasion and Mudcake Growth

Multiple drawdown, 55, 58, 113, 122-123, 129, 132, 193, 216, 238239 Multiple drawdown and buildup, 58, 113, 122-123, 129, 132, 238 Multiple drawdown-buildup, 113, 216, 239 Multiprobe, 1, 18, 20, 35-36, 237, 264, 273, 463, 465 Multirate, 39, 129-130, 235, 255 Multirate pumping, 130 MWD, 1, 5, 26, 81, 113, 217, 236, 294, 334, 336, 339, 360-361, 364, 464-465

N

Navier-Stokes, 302, 306 Nearfield, 36, 137, 221 Neumann, 306, 371, 388, 390, 428, 449 Newton-Raphson, 272 Newtonian, 310, 433 Non-constant flow rate, 238 Non-Newtonian, 310 Non-wetting, 408 Nonlinear, 37, 97-98, 118-119, 271, 300, 302, 308, 315, 329, 332-333, 343, 347, 392, 396, 404, 409, 412-415, 427, 429, 438, 457, 466 Nonlinear gas, 37, 118-119, 271 Nonlinearity, 302 Nonuniform initial pressure, 256 Nozzle, 3, 7, 18, 22, 26, 30, 50, 141-142, 187, 221, 223-224, 226228, 231, 257, 289, 296, 299, 314-315 Numerical diffusion, 49, 361, 407, 410, 418, 449, 457, 465

O

Observation probe, 6, 18, 22, 27, 40-41, 44-47, 52-53, 59, 81, 8387, 91, 93, 99-100, 104, 106-110, 112, 115, 117, 119, 227-228, 233, 235, 241, 247, 252, 264, 266-269, 273, 280, 283, 286-288, 463 Observation probes, 46, 52, 66, 81, 86-88, 94, 99, 104, 112, 114115, 226-227, 237, 264, 271, 273275 ODE, 404 ODEs, 317 Oil saturation, 223, 309, 427, 431 Ordinary differential equation, 303, 311-312, 320, 343, 347, 450 Oval, 7, 50 Oval pad, 50 Overbalance, 113, 123-127, 132133, 142, 144-159, 161-188, 190192, 256, 259-260, 262, 289-290, 298 Overbalanced, 55, 136, 139-144, 191, 237, 257, 259, 296, 310, 315

P

Packer, 6, 9, 21, 39, 50, 62, 223, 231-232, 299 Packers, 62 Pad nozzle, 50, 299 Parabolic, 300, 324, 328, 404, 448-449 Partial differential equation, 50, 143, 300, 311, 313, 324, 332, 369, 372, 385, 409, 457 Pathline, 317 PDE, 318, 389, 418 PDEs, 367, 441 Periodicity, 83, 88

Index 491

Permeabilities, 1, 23, 25, 28, 3435, 37-38, 42, 46, 50-51, 61, 6365, 67, 81, 84-86, 90, 98, 100, 103, 117, 228, 230, 235, 268-270, 283, 302, 309, 324, 328, 335, 339-340, 347, 397, 399, 403, 407, 427 Permeability, 1, 8-10, 13, 17, 20, 23-28, 32-33, 35, 37-38, 40, 4244, 46-47, 52, 61-62, 65, 67, 8186, 88-91, 94, 96-99, 104-107, 111-113, 118-119, 133, 139, 144145, 150-151, 154-155, 159, 161, 163, 168-169, 173-174, 177, 180181, 184-185, 188, 217, 223-224, 227, 231, 234-235, 238, 255, 264, 267, 269, 272, 275, 280-281, 283, 289, 291, 295, 298, 300-301, 303, 305-306, 309, 311, 320-321, 324330, 332, 334, 336, 338, 340-342, 345, 347-350, 352-359, 361, 373, 378, 382-383, 385, 388, 391, 394395, 399, 401-402, 404, 406-407, 413, 428-430, 432, 438, 447, 450451, 465 Phase delay, 26-28, 40, 52, 54, 81-91, 94, 96-100, 102-103, 110, 112, 264, 274 Phase wrapping, 91 Piecewise constant, 37, 119, 129, 193, 264 Piston, 27, 30, 52, 81, 89-90, 112, 173, 180, 186, 190, 202, 238, 245, 250, 305, 308, 310, 321, 331, 354, 359, 363, 373, 376, 390, 405, 413, 419, 447, 458 Plane fracture, 317 Playback, 137

Point source, 32, 34, 36, 49, 221, 314, 420 Pore pressure, 1, 8, 20-21, 23, 2528, 42, 55, 57, 61, 63, 67-68, 71, 73-74, 76, 80, 86, 111-113, 122127, 132, 139, 143-145, 148-159, 161, 163, 166-169, 171-177, 179185, 187-188, 190-193, 200, 204, 208, 212, 216, 229, 237-239, 243244, 249, 255-257, 259-260, 262, 269, 271-272, 289, 295-296, 298, 315, 328, 338, 345, 347-348, 350, 352-355, 357-359, 386, 414 Porosity, 23, 47, 82, 89-91, 94, 96-98, 104, 133, 228, 258, 281, 289, 291, 300-301, 306, 308-309, 311-312, 320-322, 325, 327, 332, 334, 336-340, 345-346, 348-349, 353-358, 378, 382-383, 385, 388, 393-395, 401-402, 404, 413, 429 Pressure drop, 7, 19, 46-47, 111, 118, 238, 243, 249, 268, 319-320, 347, 367, 455 Pressure transient, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19-21, 23, 28, 34, 41, 47, 53, 56-57, 67, 79, 85, 91, 111, 123, 129, 139, 144-145, 147148, 150-151, 155, 158-159, 161, 163, 165, 168, 170, 173-174, 177178, 180-182, 185-186, 188, 190, 200, 217, 224, 226-228, 235-238, 265, 275-276, 280, 283, 294, 458 Pressure transient analysis, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19-21, 23, 56, 139, 217, 227, 235-236, 294, 458 Probe, 1-2, 5-7, 9-10, 16, 18-19, 21-22, 24, 26-29, 32-33, 38-42, 44-47, 50, 52-53, 57, 59, 62, 66-

492 Supercharge, Invasion and Mudcake Growth

69, 71-72, 74-81, 83-87, 89-91, 93-94, 96-100, 104, 106-113, 115-119, 122-123, 125, 137, 140, 144-145, 147-151, 153-155, 157159, 161-163, 165-169, 171-181, 183-185, 187-188, 191-192, 200, 202, 206, 210, 214, 222-223, 225228, 230-231, 233-235, 237, 240241, 244, 246-247, 249, 252, 254257, 260-261, 264, 266-277, 280, 283, 286-288, 298, 361, 458, 463 Probes, 2, 5-6, 17-19, 22-24, 26, 33, 39, 46, 50, 52, 66, 81, 86-89, 94, 99, 104, 110, 112, 114-115, 140, 221, 223, 226-227, 233, 237, 264, 271, 273-275, 463 PTA, 1, 4, 7-8, 55, 58, 68, 74, 80, 110, 122-123, 130, 132, 141-142, 145, 147-148, 150, 152-154, 157, 159, 163, 165-168, 171-173, 175180, 183-184, 187-188, 192-193, 200, 217, 221, 244, 249, 254-257, 264, 271, 296 Pulse interaction, 40, 46-47, 49, 52 Pulse interference, 47 Pump out, 9 Pumpout, 2, 40, 136, 202, 206, 210, 214, 238, 240-241, 246, 251, 264-265, 281 PVT, 9-10

R

Radial flow, 217-218, 223, 292293, 299, 303, 306, 313, 327, 334-335, 337, 339-340, 343, 345, 351, 355, 359, 380, 406, 414, 424-425, 457 Rational polynomial, 64, 123 Rectangular wave excitation, 94

Relative permeability, 289, 303, 305, 324-325, 329, 347, 406-407, 413, 428-430, 432, 438, 447 Reservoir characterization, 111, 113 Reservoir engineering, 112, 220, 227, 235, 319, 342, 361, 411, 414, 464 Residual oil, 309, 324 Resistivity, 26-27, 52, 81-82, 86, 117, 217, 264, 290, 309, 312, 330, 339-340, 354, 359, 383, 406, 419, 423, 438 Resistivity logging, 27, 264, 330 Resistivity migration, 359, 406, 419, 438 Reynolds number, 302, 306, 433 RFT, 62, 68, 80 RFTTM, 61-62 Rheology, 112, 360, 464 Ring source, 32, 61, 221 Round nozzle, 18 Round-off, 24, 35, 37, 117, 372, 410, 420, 431 Round-off error, 37, 117, 410

S

Sample quality, 234, 458, 466 Saturation, 222-223, 234, 309, 336, 359, 406-407, 409-415, 419, 426-449, 451-457 Saturation discontinuity, 433, 438, 449 Saturation equation, 412, 426, 441, 444, 448-449 Saturation profile, 448 Schlumberger, 22, 61-67, 132 Sealing, 7, 227, 235, 237, 289 Separation of variables, 332 Shock front, 407, 414-415

Index 493

Shock position, 410 Shock velocity, 412, 414-415, 457 Signaling, 87, 413-414 Simulation, 33, 35, 39, 42-43, 46, 50, 56-57, 61, 66, 79, 91, 120, 133-135, 137, 142, 148, 217, 220, 224, 226-228, 234-235, 256, 264265, 276-277, 280-281, 283, 285288, 296, 302, 336, 349, 355-361, 363-364, 369, 371, 373, 376, 381, 405-406, 439, 449, 456, 463-465 Single probe, 1-2, 29, 38-40, 223, 228, 257, 264, 267, 270 Single-phase flow, 36, 221, 300, 305, 310, 325, 409, 414, 451 Single-valued, 322, 414 Singular, 36, 411, 433 Singularity, 426 Sink, 1, 3, 5-8, 18-19, 22, 30, 49, 61, 110, 221, 267, 273, 463 Sink probe, 18-19, 110, 273, 463 Sinusoidal excitation, 91 Skin, 25, 30, 32, 36-37, 42-43, 46, 49, 65, 139, 142, 256, 268, 272, 296, 342, 458 Skin damage, 139 Skin effect, 36 Slimhole, 289, 383, 435 Slimholes, 292, 343, 345 Slot, 7, 17-18 Slotted, 7 Smear, 410, 419, 421 Smearing, 40, 117, 433 Software, 4, 6, 13, 35-36, 39, 4243, 63, 68, 74, 91, 94, 96, 100, 112, 116, 123-125, 127, 133, 138, 144-145, 147-148, 150, 152-154, 157-159, 161, 163, 165-168, 171-

173, 175-180, 183-184, 187-188, 192-193, 200, 217, 223, 227, 229, 242, 259, 269, 272, 360, 405, 459-464 Software reference, 68, 74, 145, 147-148, 150, 152-154, 157-159, 161, 163, 165-168, 171-173, 175180, 183-184, 187-188, 192, 242 Source, 1, 3, 5, 7-8, 21-23, 25-27, 29-41, 43-47, 49, 51-53, 55, 5759, 61, 63, 65-69, 71, 73, 75-77, 79-87, 89-91, 93-95, 97, 99-113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139-143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187, 189, 191, 193, 195, 197, 199, 201-203, 205-207, 209-211, 213-215, 217, 219, 221, 223, 225-227, 229, 231, 233-235, 240-241, 243, 246-247, 249, 251-252, 257, 264, 266-273, 275, 277, 280, 283, 286-288, 296, 314, 318, 320, 377, 381-382, 384, 386, 388, 393-394, 401-402, 405, 420, 439, 443, 446, 463 Source code, 58, 100, 377, 381382, 384, 386, 388, 393-394, 401402, 405, 439, 443, 446 Source model, 49, 52, 140, 226, 264 Source probe, 33, 40, 45-47, 5253, 57, 66-69, 76, 79-81, 83-85, 91, 100, 107, 115, 125, 163, 202, 206, 210, 214, 226-227, 234-235, 240-241, 246, 249, 264, 268-269, 272-273, 277, 280, 286-288

494 Supercharge, Invasion and Mudcake Growth

SPE, 6, 28, 32-33, 111, 113, 117, 142, 273, 361, 443, 458, 466 Specific heat, 302 Spherical, 1, 21-23, 25-35, 37-39, 41, 43-47, 49, 51, 53, 55, 57, 59, 61-63, 65, 67, 69, 71, 73, 75, 77, 79, 81-83, 85, 87-89, 91, 93, 95, 97-99, 101, 103, 105, 107, 109, 111-113, 115, 117, 119, 121-123, 125, 127, 129, 131, 133, 135, 137, 139-143, 145, 147-149, 151, 153-155, 157-159, 161, 163, 165169, 171-173, 175-177, 179-181, 183-185, 187, 189, 191-193, 195, 197, 199-201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225-227, 229, 231, 233, 235, 244, 249, 255-257, 267, 271, 289, 296, 302-303, 306, 314-315, 341, 371, 373, 383-384, 388-389, 405 Spherical flow, 31, 61, 139-140, 143, 303, 315, 341, 373, 384, 389 Spherical mobility, 25, 122-123, 148-149, 153-154, 157-158, 166168, 171-172, 175-176, 179-180, 183-184, 187, 191-192, 200, 244, 249, 255, 271 Spherical permeabilities, 23, 28, 81 Spherical permeability, 1, 43-44, 85, 88, 97-98, 111-112, 119, 267 Spherical source, 21, 23, 25-27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 8183, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135,

137, 139, 141-143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187, 189, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 233, 235, 257, 296 Spurt, 291, 301, 323, 326-327, 330, 338-339, 341, 352, 412, 435 Spurt loss, 338-339 SPWLA, 28, 142, 217, 220, 236, 256, 289, 294, 361-362, 465 Stability, 50, 221, 318, 354, 371372, 386, 388-390, 392, 429, 432433, 435, 449 Stagnant, 446 Static filtration, 336-337, 345, 453 Steady, 21, 23, 30, 34, 37, 40-43, 46-47, 61-63, 65-69, 88, 111-113, 117-119, 122-123, 130, 132, 143, 200, 241, 257, 259, 264, 267-269, 271-273, 275-276, 280, 304-305, 308, 314, 332, 363-364, 366, 420421, 448 Steady state, 21, 30, 68, 132, 273, 280, 314, 421, 448 Steady-state, 23, 34, 37, 41-43, 47, 61-62, 65-67, 88, 111, 113, 117-118, 130, 200, 241, 259, 264, 267-268, 271, 275, 305, 308, 332, 363, 366, 420-421, 448 Storage, 24-25, 30, 36, 43, 55, 65, 69, 79, 111, 113, 120, 122-123, 132, 143, 145, 147, 149, 151-153, 155, 157, 159, 161, 163, 165-168, 171-172, 200, 221, 258, 346, 458

Index 495

Straddle packer, 9, 21, 223, 231232, 299 Streamline, 283, 346 Streamlines, 223, 308 Stuck, 15-16, 20, 24, 61, 217, 224, 237, 289, 298, 330-331, 345, 361 Stuck pipe, 16, 217, 237, 289, 298, 330-331, 345 Stuck release, 16, 298 Stuck release arms, 16, 298 Stuck tool, 16, 224, 298 Stuck tools, 15, 24, 61 Subroutine, 368, 370 Subroutine TRIDI, 368 Subsidence, 319 Supercharge, 3, 16, 20-21, 23, 30, 55, 61, 113, 123-125, 127-128, 132, 139-142, 144, 160, 166-167, 171, 173, 193, 224, 226, 236, 255-264, 289, 292, 294-297, 299, 301, 303, 305, 307, 309, 311, 313, 315, 317, 319, 321, 323, 325, 327, 329, 331, 333, 335, 337, 339, 341, 343, 345, 347, 349, 351, 353, 355, 357, 359, 361, 363, 365, 367, 369, 371, 373, 375, 377, 379, 381, 383, 385, 387, 389, 391, 393, 395, 397, 399, 401, 403, 405, 407, 409, 411, 413, 415, 417, 419, 421, 423, 425, 427, 429, 431, 433, 435, 437, 439, 441, 443, 445, 447, 449, 451, 453, 455, 457-459, 461, 463, 465 Supercharged, 20, 125, 145, 147, 149, 151-153, 155, 157, 159, 161, 163, 165-168, 171-172

Supercharging, 32, 34-35, 110, 113, 133, 139, 141, 143-144, 148, 152, 156, 160, 173, 179, 217, 221-223, 225, 228-231, 235, 237, 256-257, 262, 289, 293, 295-296, 298-299 Superposition, 37, 65, 118-119, 129, 302, 333 Supersonic, 367 Sweep efficiency, 304

T

Temperature, 10, 304, 332, 342, 420 Thermal, 38, 415 Thermodynamic, 118-119, 271, 315-316, 390, 420 Thermodynamics, 302, 333 Three-dimensional, 22-23, 30, 32, 34-36, 38, 49-52, 99, 110, 221, 306, 458, 463 Tight, 25, 46, 50, 111-112, 142, 191, 390, 455 Tight zone, 25 Time delay, 27, 83-84, 87, 89, 91, 94, 96-98 Time integration, 377, 381, 387, 390, 392-393, 401, 420, 428, 439, 444 Time lapse, 217, 312, 316, 327, 334, 336, 340-342, 345-347, 349352, 354, 356-360, 405, 419-420, 422, 438, 464-465 Time lapse logging, 336, 465 Time to plug, 345 Tool sticking, 43, 46-47 Total differential, 307, 323, 409 Tracer, 312-313 Transient, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19-21, 23, 25, 28, 30, 32, 34,

496 Supercharge, Invasion and Mudcake Growth

36-37, 40-41, 43-44, 47, 50, 5253, 55-57, 61, 64-65, 67-68, 71, 79-80, 85, 90-91, 111, 114-115, 117-119, 123, 125, 129, 131-133, 139, 144-145, 147-148, 150-151, 155, 158-159, 161-163, 165, 168170, 173-174, 177-178, 180-182, 185-188, 190, 193, 200, 217, 221, 224, 226-228, 235-238, 259-261, 265, 273, 275-277, 280, 283, 289, 291-292, 294, 301, 304-305, 311, 313-314, 316, 319, 321, 324, 328, 331-333, 342, 359-360, 364, 369, 372-375, 386, 388, 390, 392, 397, 399, 405, 412, 414, 418, 420-421, 448, 453, 458, 464 Transversely isotropic, 26, 30-31, 36, 38, 50, 65, 79, 82, 89, 97, 99, 139, 264 TRIDI, 368, 372, 377, 382, 387, 393, 402, 428-429, 442, 444-445 Tridiagonal, 367-368, 370, 372, 386, 389, 418, 428, 442, 456 Tridiagonal equations, 386, 456 Tridiagonal matrix, 367-368, 389, 428 Triple probe, 6, 18-19, 21-22, 50, 274, 458 Truncation, 24, 37, 49, 117, 318, 359, 361, 364, 372, 389-390, 410, 418, 420-422, 432, 449, 465 Truncation error, 318, 359, 361, 364, 372, 418, 432, 465 Two-phase, 303, 305, 309, 324325, 329, 347, 359, 390, 406-407, 409, 429, 439, 441, 443, 447, 450-451, 457-458

Two-phase flow, 305, 309, 324325, 329, 406-407, 429, 439, 441, 447, 450-451, 457-458

U

Underbalanced, 55, 133, 139-141, 143, 257, 296, 310, 315, 351 Underbalanced drilling, 55, 139141, 143, 351 Undercharge, 55 Undiffusion, 406, 421 Uniform media, 81 Unsteady, 36, 43, 64, 113, 132, 200, 299, 302, 304-305, 308, 336, 339, 374-375 Unwrapped, 346 User interface, 36, 122, 127-128, 223 Variable meshes, 225 Vertical permeability, 38, 82-83, 86, 97, 104, 118, 227, 269, 275, 280 Vertical probe, 18 Vertical well, 83 Vertical well limit, 83 Viscosity, 8, 24, 35, 47, 49, 62, 65, 67, 82, 89-91, 94, 96-98, 100, 122, 137, 139, 144-145, 150-151, 154-155, 159, 161, 163, 168-169, 173-174, 177, 180-181, 184-185, 188, 217, 233-235, 268, 272, 291, 300-301, 305-306, 311-312, 320321, 323, 325-327, 330, 332, 338, 340-342, 345, 347-350, 352-359, 376, 378-379, 382-383, 385-386, 388, 394-395, 399, 402, 405, 407, 409-410, 413, 430, 449 Volume flow rate, 40, 47, 62, 69, 71-72, 74, 80, 115, 119, 129, 143, 145, 147-151, 153-155, 157-159,

Index 497

161, 163, 165-169, 171-181, 183185, 187-188, 191-192, 202, 206, 210, 214, 241, 244, 247, 249, 252, 254-255, 260, 267-268, 270272, 319, 426, 432 von Neumann, 371, 388, 390, 428, 449 von Neumann stability, 371, 388, 390, 449 Vortex, 306

W

Water breakthrough, 407, 411, 433 Water saturation, 413, 426-441, 445-447, 452-456 Waveform, 47, 81, 273 Well test, 302, 316, 333, 449 Wetting, 407-408, 430, 451 Windows, 36, 55, 101, 115, 125, 128, 133, 137, 379 Wireline, 1, 5, 9, 16, 25, 33, 36, 50, 81, 113, 132, 142, 256, 289, 294, 298, 361-362, 443 Withdraw, 193, 231 Withdrawal, 22, 123, 186, 202, 223, 259, 265, 269 End of Index.

About the Authors

Tao Lu

Xiaofei Qin

Yongren Feng Yanmin Zhou

Wilson Chin

Tao Lu, Ph.D., Vice President, China Oilfield Services Limited, leads the company’s logging and directional well R&D activities, also heading its formation testing research, applications and marketing efforts. Mr. Lu is recipient of numerous awards, including the National Technology Development Medal, National Engineering Talent and State Council Awards, and several COSL technology innovation prizes. Xiaofei Qin graduated from Huazhong University of Science and Technology with a M.Sc. in Mechanical Science and Engineering. At China Oilfield Services Limited, he is engaged in the research and development of petroleum logging instruments and their applications. Mr. Qin has published twelve scientific papers and obtained twenty patents. Yongren Feng is a Professor Level Senior Engineer and Chief Engineer at the Oilfield Technology Research Institute of China Oilfield Services Limited. He has been engaged in the research and development of offshore oil logging instruments for three decades, mainly responsible for wireline formation testing technology, electric core sampling technology and formation testing while drilling technology. Yanmin Zhou received her Ph.D. in Geological Resources Engineering from the University of Petroleum, Beijing and serves as Geophysics Engineer at COSL. She participated in the company’s Drilling and Reservoir Testing Instrument Development Program, its National Science and Technology Special Project, and acts as R&D engineer for national formation testing activities. Wilson Chin earned his Ph.D. from M.I.T. and his M.Sc. Caltech. He has authored over twenty books with John Wiley & Sons and Elsevier Scientific Publishing, more than four dozen domestic and international patents, and over one hundred journal articles, in the areas of reservoir engineering, formation testing, well logging, Measurement While Drilling, and drilling and cementing rheology. Inquiries: [email protected].

498

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