Reserves Estimation for Geopressured Gas Reservoirs 0323950884, 9780323950886

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Reserves Estimation for Geopressured Gas Reservoirs

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Reserves Estimation for Geopressured Gas Reservoirs

TONGWEN JIANG PetroChina Exploration & Production Company, Beijing, China

HEDONG SUN PetroChina Research Institute of Petroleum Exploration and Development, Beijing, China

HONGFENG WANG PetroChina Tarim Oilfield Company, Korla, China

XIANGJIAO XIAO PetroChina Tarim Oilfield Company, Korla, China

Gulf Professional Publishing is an imprint of Elsevier 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, United Kingdom Copyright © 2023 Petroleum Industry Press. Published by Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-323-95088-6 For Information on all Gulf Professional Publishing publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Glyn Jones Editorial Project Manager: Naomi Robertson Production Project Manager: Surya Narayanan Jayachandran Cover Designer: Greg Harris Typeset by MPS Limited, Chennai, India

Contents About the authors Preface Acknowledgments

ix xi xiii

1. Introduction

1

1.1 Geopressured natural gas resources and development in China 1.1.1 Overview of geopressured natural gas resources 1.1.2 Gas reservoir characteristics 1.1.3 Development status 1.1.4 Development strategies 1.2 Classification of geopressured gas reservoirs 1.2.1 Classification by depth 1.2.2 Classification by pressure and pressure coefficient 1.3 Reserves terms 1.3.1 Standards in China 1.3.2 Terms related to Securities and Exchange Commission reserves 1.4 Dynamic reserves 1.4.1 Definition 1.4.2 Calculation methods 1.4.3 Challenges for dynamic reserve estimation

2. Pressure monitoring of geopressured gas wells 2.1 Downhole temperature and pressure monitoring of high pressure & high temperature (HPHT) gas wells 2.1.1 Role and function of dynamic monitoring 2.1.2 Content and means of dynamic monitoring 2.1.3 Downhole temperature and pressure monitoring technology for HPHT gas wells 2.2 Static pressure conversion for gas wells 2.2.1 Gas column density conversion method 2.2.2 Static pressure gradient conversion method 2.2.3 Wellhead static pressure conversion method 2.3 Calculation of average gas reservoir pressure 2.3.1 Arithmetic average method 2.3.2 Weighted average method

1 2 2 3 6 8 9 10 11 11 15 16 16 20 23

27 27 27 28 32 41 42 42 43 50 50 51

v

vi

Contents

3. Physical properties of natural gas and formation water 3.1 Composition and properties of natural gas 3.1.1 Composition of natural gas 3.1.2 Equation of state of an ideal gas 3.2 Behavior of real gas 3.2.1 Natural gas deviation factor 3.2.2 Compressibility factors for natural gases 3.2.3 Gas formation volume factor 3.2.4 Natural gas viscosity 3.3 Deviation factor of ultra-high-pressure gas 3.3.1 DPR or DAK extrapolation method 3.3.2 LXF-RMP (Li et al., 2010) fitting method 3.4 Properties of formation water 3.4.1 Formation water volume factor 3.4.2 Formation water viscosity 3.4.3 Natural gas solubility in water 3.4.4 Isothermal compressibility factor of formation water

4. Material balance equation of a gas reservoir 4.1 Material balance equation for homogeneous gas reservoirs 4.1.1 Volumetric gas reservoir 4.1.2 Closed gas reservoir 4.1.3 Water-drive gas reservoir 4.1.4 Water-drive gas reservoir with water-soluble gas 4.1.5 Linear form of pressure depletion curve 4.2 Material balance equation for compartmented gas reservoirs 4.2.1 Payne method 4.2.2 Hagoort
Hoogstra method 4.2.3 Gao method 4.2.4 Sun method 4.3 Drive index of gas reservoir 4.4 Apparent initial gas in place of gas reservoir 4.5 Sensitivity analysis of key parameters 4.5.1 Rock compressibility factor 4.5.2 Size of water body 4.5.3 Pressure depletion 4.5.4 Apparent reservoir pressure 4.5.5 Influence of water-soluble gas

53 53 53 54 58 59 71 75 75 80 80 82 87 88 91 92 94

97 98 99 104 109 116 118 119 119 123 125 131 137 140 142 142 152 152 152 155

Contents

5. Original gas in place estimations for geopressured gas reservoirs 5.1 Classical two-segment method 5.1.1 Hammerlindl method 5.1.2 Chen method 5.1.3 Gan
Blasingame method 5.1.4 Discussion on the time of inflection point of p/Z curve 5.2 Linear regression method 5.2.1 Ramagost
Farshad method 5.2.2 Roach method 5.2.3 Poston
Chen
Akhtar method 5.2.4 Becerra-Arteaga method 5.2.5 Havlena
Odeh method 5.2.6 Sun method 5.3 Nonlinear regression method 5.3.1 Binary regression method 5.3.2 Nonlinear regression method 5.3.3 Starting point of the nonlinear regression method 5.4 Type curve matching analysis method 5.4.1 Ambastha method 5.4.2 Fetkovich method 5.4.3 Gonzalez method 5.4.4 Sun method 5.4.5 Multiwell production decline analysis method 5.5 Trial-and-error analysis method 5.6 Original gas in place estimation procedure of geopressured gas reservoirs 5.6.1 Summary of calculation methods 5.6.2 Recommended methods 5.6.3 Recommended procedure 5.6.4 Basic data preparation 5.6.5 Comparative analysis of results Appendices Bibliography Index

vii

157 158 158 161 164 171 172 173 175 178 190 195 199 206 206 209 215 223 223 227 231 235 240 246 255 255 255 258 258 260 265 301 315

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About the authors Tongwen Jiang, PhD, SPE&AAPG member, born in 1968, professor, earned his PhD degree in oil and gas field development engineering from Southwest Petroleum Institute in 1996. He has been working for Tarim Oilfield Company since 1996 and has rich experience in reservoir study and management. He has published over 40 papers on complicated oil and gas reservoir studies such as condensate gas reservoirs and fractured reservoirs. He has authored six books published by Petroleum Industry Press and Elsevier, including Dynamic Description Technology of Fractured Vuggy Carbonate Gas Reservoir and Abnormal High Pressure Gas Field Development. Hedong Sun, PhD, SPE member, born in 1973, professor, earned his PhD degree from Xi’an Jiaotong University in 2004. Since 2004, he has been working as a research engineer at Research Institute of Petroleum Exploration and Development of PetroChina. Prof. Sun has about 20 years of reservoir engineering experience with a focus on well test analysis and production decline analysis. He has published over 60 papers in peerreviewed journals and SPE conferences. He has authored four books published by Elsevier, including Advanced Production Decline Analysis and Application and Well Test Analysis for Multilayered Reservoir with Formation Crossflow. Hongfeng Wang, born in 1978, senior engineer, earned his PhD degree in oil and gas field development engineering from Southwest Petroleum University in 2022. He has been working for Tarim Oilfield Company since 2004 and has rich experience in gas reservoir study and management. He has published 3 books and more than 10 papers on highpressure and high-temperature gas reservoirs and condensate gas reservoirs study and management. ix

x

About the authors

Xiangjiao Xiao, PhD, SPE member, born in 1968, professor, earned her PhD degree in oil and gas field development engineering from China University of Petroleum in 2012. She has been working for Tarim Oilfield Company since 1992 and has rich experience in geopressured gas reservoir and condensate gas reservoir study. She has published more than 40 papers about geopressured gas reservoirs. She has authored six books published by Petroleum Industry Press.

Preface Since the beginning of the 21st century, geopressured gas reservoirs have been discovered throughout the world. In China, a series of major breakthroughs have been made in the field of high-pressure and hightemperature (HPHT) gas reservoirs, where a number of HPHT gas fields in the Tarim Basin and the Sichuan Basin were successively discovered and developed, such as KL2 gas field, DN2 gas field, and AnYue gas field, with proven reserves of more than 100 billion cubic meters. Accurate estimation of the reserves of HPHT gas reservoirs is the key to scientifically formulating development technology policies. Owing to the limitation of data conditions and the low degree of understanding of development rules, the reserves estimated by different methods vary greatly. Therefore how to scientifically and accurately estimate reserves is extremely challenging for gas reservoir engineers. This book aims to introduce the principles and methods for calculating reserves of geopressured gas reservoirs with the material balance method and to present the advantages, disadvantages, and applicable conditions of various methods. This book, based on manual analysis, explains the analysis methods and calculation steps with examples of more than 30 gas reservoirs. It is expected to help gas reservoir engineers to learn the basic principles and calculation methods and to familiarize themselves with the content of the software black box, which in turn can help improve the level of gas field performance analysis as well as the level of gas field development. A gap in a petroleum engineer’s library for a comprehensive book covering the state of the art in original gas in place (OGIP) estimation for geopressured gas reservoir has existed for some time now. Practicing petroleum engineers need a single up-to-date reference for OGIP estimation. Reserves Estimation for Geopressured Gas Reservoirs has been written to fill that gap. This book has five chapters, which are summarized as follows: In recent years, newly discovered gas fields around the world have gradually moved toward deeper layers, where geological conditions have become more and more complex, thus making development more difficult. The use of production data to evaluate OGIP is the key and difficult point for the efficient development of such gas reservoirs. Chapter 1 first xi

xii

Preface

introduces China’s deep-seated natural gas resources and development overview and then the classification of deep-seated high-pressure gas reservoirs and some basic concepts related to the reserves. Chapter 2 focuses on the bottom-hole pressure monitoring technology and bottom-hole static pressure conversion method for ultradeep (more than 7000 m) and ultrahigh-pressure (tubing pressure more than 100 MPa) gas wells and aims to provide reliable pressure data for OGIP calculation using the material balance method. Chapter 3 introduces the physical properties of natural gas and formation water, commonly used empirical relations, and their applicable conditions. Chapter 4 focuses on introducing the material balance equation of homogeneous gas reservoirs, the compartmental material balance equation for heterogeneous gas reservoirs, and the sensitivity analysis of key parameters. Chapter 5 focuses on OGIP calculation methods based on material balance in combination with gas field examples. There are 5 types (22 methods), namely classical slope two-segment analysis method, linear regression analysis method, nonlinear regression analysis method, type curve matching, and trial-and-error analysis method; finally, the process and recommendations for OGIP estimation of high-pressure gas reservoirs are introduced. This book is a summary and refinement of the authors’ research over the years, which reflects the combination, promotion, and redevelopment of the gas reservoir engineering theory and field practice. Many of the results have been applied in the China’s geopressured gas fields, improving the scientific, predictable, and economic benefits of gas field development. In short, the book has been written to reflect current practices and technology and is more reader friendly, with introductions to term and concepts as well as examples using Microsoft Excel as the computational tool. We hope that this book acts as a useful tool in the development of geopressured gas reservoirs throughout the world. Owing to the limited level of the authors’ knowledge and experience, there may be erroneous statements in this book. Your comments and criticism are thereby warmly welcomed.

Acknowledgments We would like to thank the editorial and production staff of Elsevier for their work and professionalism, most notably Naomi Robertson, Surya Narayanan Jayachandran, and Greg Harris. Thanks also go to Songbai Zhu and Wen Cao for their assistance and help during writing and proofreading of the book. We also acknowledge financial support from China National Petroleum Corporation Major Science and Technology Project “Research and Application of Key Technologies for Deep-Ultra-Deep Gas Field Development in Kuqa Depression” (No. 2018E-1803).

xiii

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CHAPTER 1

Introduction Contents 1.1 Geopressured natural gas resources and development in China 1.1.1 Overview of geopressured natural gas resources 1.1.2 Gas reservoir characteristics 1.1.3 Development status 1.1.4 Development strategies 1.2 Classification of geopressured gas reservoirs 1.2.1 Classification by depth 1.2.2 Classification by pressure and pressure coefficient 1.3 Reserves terms 1.3.1 Standards in China 1.3.2 Terms related to Securities and Exchange Commission reserves 1.4 Dynamic reserves 1.4.1 Definition 1.4.2 Calculation methods 1.4.3 Challenges for dynamic reserve estimation

1 2 2 3 6 8 9 10 11 11 15 16 16 20 23

In recent years, global gas exploration has been increasingly targeting deep zones or deep-water zones, where the relatively complex geological conditions bring greater challenges in resource development. To develop such kinds of reservoirs efficiently, their reserves must be evaluated. In this chapter, deep natural gas resources and development in China are introduced. In addition, the classification of deep high-pressure gas reservoirs and some basic concepts related to reserves are explained.

1.1 Geopressured natural gas resources and development in China Developing deep hydrocarbon resources is an important strategic prospect of China and a pragmatic option for enhancing domestic petroleum exploration and development. Such thermally evolved resources mainly occur as natural gas. This section provides a brief introduction to the deep natural gas resources in China and the development characteristics and strategies. Reserves Estimation for Geopressured Gas Reservoirs © 2023 Petroleum Industry Press. DOI: https://doi.org/10.1016/B978-0-323-95088-6.00001-8 Published by Elsevier Inc. All rights reserved.

1

2

Reserves Estimation for Geopressured Gas Reservoirs

1.1.1 Overview of geopressured natural gas resources According to China’s fourth petroleum resource evaluation, the quantity of onshore conventional gas resources in China is 41 3 1012 m3, including 70.3% in deep strata with proved reserves of 4.02 3 1012 m3, mainly in seven petroliferous basins (i.e., Sichuan, Tarim, Songliao, Ordos, Qaidam, Junggar, and Bohai Bay, China) (Fig. 1
1). The Sichuan Basin and Tarim Basin are the major development targets with the most abundant deep natural gas resources (Yu et al., 2018). Deep natural gas made up 34.8% of the additionally proved gas initiallyin-place (GIIP) in 2008
17 and took a proportion in the total proved GIIP, increasing from 13% in 2008 to 38% in 2017. By the end of 2018, the total proved GIIP of producing deep gas fields reached 3.32 3 1012 m3. Deep natural gas production was 428 3 108 m3 in 2018, accounting for 30.2% of the total natural gas production in China (Li et al., 2019).

1.1.2 Gas reservoir characteristics Deep gas reservoirs are characterized by high in situ stress, strong heterogeneity, high temperature, and high pressure. For example, the deep gas reservoirs in the Tarim Basin are complex in surface landform (e.g., high-steep mountains, blade mountains, and cliffs) and subsurface structures (e.g., highsteep overthrust nappe and thrust) as a result of multiphase tectonic 100,000

Resources，108m3

80,000

60,000

80965

54709

40,000

20,000

20118

19062 7946

7670

6622

0

Sichuan Tarim Songliao Ordos Qaidam Junggar Bohai Bay Figure 1
1 Histogram of deep natural gas resources in major onshore petroliferous basins in China.

Introduction

3

p/Z

Normal gas reservoir Geopressured gas reservoir

0.0

0.2

0.4

0.6

0.8

1.0

Gp/G Figure 1
2 p/Z curve of deep high-pressure gas reservoirs.

movements. These reservoirs are very deep (up to 8271 or 6850 m averagely) and exhibit low matrix porosity (6% averagely), low matrix permeability (avg. 0.05 3 1023 μm2), high reservoir temperature (up to 190°C or 143°C averagely), high pressure (up to 144 or 112 MPa averagely), high in situ stress (up to 180 or 130 MPa averagely, with a stress difference up to 60 MPa), strong fracture heterogeneity, and strong water activity. Compared with normal-pressure gas reservoirs, deep gas reservoirs contain much larger elastic energy and also very different drive energy composition—expansion of rock particles and bound water, stratigraphic compaction, water invasion of shale connected to the reservoirs, and precipitation of dissolved gas in immobile water and aquifer, in addition to the gas expansion and net water encroachment in normal-pressure gas reservoirs. The relation curve between apparent formation pressure and cumulative production during development generally presents an upward-convex behavior (Fig. 1
2).

1.1.3 Development status A few factors have been reported to influence the effects and economics of deep natural gas development, including efficient placement of well, rational development technology and strategy determination, safe and rapid drilling/completion, effective stimulation, and safe and clean production. Since the beginning of the 21st century, China has made several significant breakthroughs in deep natural gas exploration, with the successful discovery of several large deep high-pressure gas fields, mainly in the Tarim Basin and the Sichuan Basin (Table 1
1). These reservoirs are

Table 1
1 Geological and reservoir properties of major deep gas fields in China. Basin

Gas field

Depth /m

Development layer series

Lithology

Reservoir type

Average porosity /%

Permeability /1023μm2

Formation pressure /MPa

Formation temperature /°C

Proved reserves /108m3

Sichuan

Chuanxi

5500
6300

Triassic Leikoupo Fm.

Dolomite, limestone

5.30

1.27

63
68

141
152

1140

Xinchang Xu-2 Yuanba

4500
5300

Triassic Xujiahe Fm.

Sandstone

3.75

0.07

72
81

127
132

1250

6300
7200

Permian Changxing Fm.

5.67

0.47

118
120

147
153

2195

Puguang

4800
5500

Porous

7.3
8.1

0.01
3334

56

120
135

4121

Longgang

5400
6200

Limestone, dolomite

Fractured porous

6.50

1.00

61

130
150

720

Shuangyushi Anyue

7000
8000 4500
6000

Dolomite Dolomite

Porous, cave Fractured vuggy

4.00 3.8
4.3

2.26 0.51
0.96

95
100 56
78

155
160 140
160

811 10570

Diana-1

4800
5600

Sandstone

0.99

106

136

1659

6000
7800

6.20

0.06

103
136

150
184

6320

Dabei

5500
7300

7.30

0.08

89
119

130
165

1749

Tazhong-I

4500
7000

Fractured porous Fractured porous Fractured porous Fractured vuggy, cave

8.80

Keshen

Triassic Feixianguan Fm., Permian Changxing Fm. Triassic Feixianguan Fm., Permian Changxing Fm. Permian Qixia Fm. Cambrian Longwangmiao Fm., Sinian Dengying Fm. Paleogene, Kumugeliemu Group Cretaceous Bashijiqike Fm. Cretaceous Bashijiqike Fm. Ordovician Lianglitag Fm., Yijianfang Fm. and Yingshan Fm.

Dolomite, limestone Dolomite

Porous, fracturedporous Fracturedporous Porous

2.30

0.01
452

45
72

125
145

4133

Tarim

Sandstone Sandstone Limestone, dolomite

Introduction

5

complex and diverse, including sandstone reservoirs, carbonate reservoirs, lithologic reservoirs, and structural reservoirs. With the deep natural gas targets becoming more complex, the development focus has gradually shifted from efficient production to long-term stable production, and the deep natural gas development technologies are also being improved. The available technologies include deep seismic imaging and reservoir prediction technology, deep, complex gas reservoir development optimization technology, deep high pressure & high temperature (HPHT) gas well drilling and completion technology, deep, complex reservoir precision stimulation technology, and deep special fluid gas production technology, which have effectively supported the leap growth of deep natural gas production. 1.1.3.1 Sichuan Basin The Sichuan Basin has the most abundant deep natural gas resources in China. Since 2000, several large deep gas fields, such as Puguang, Longgang, Yuanba, Anyue, and Chuanxi, have been discovered in the basin, with proven GIIP of more than 2.0 3 1012 m3, and the annual production capacity of deep natural gas over 3 3 1010 m3. In China, the Anyue gas field is the largest monolithic carbonate gas reservoir, with an annual production capacity of 1.5 3 1010 m3, while the Puguang gas field is the largest and most abundant marine high-sulfur gas field, with an annual production capacity of 6.3 3 109 m3. The Yuanba gas field is a rare ultradeep and high-sulfur reef gas field globally, with an annual production capacity of 4 3 109 m3. At present, the Sichuan Basin is still at the peak of deep gas discoveries and a rapid reserves growth period (Li et al., 2020b). Except for marine carbonate rocks, the tight sandstone gas reservoir of Triassic Xujiahe Formation wide-spreading in the Western Sichuan depression holds 3P reserves close to 1.0 3 1012 m3, which, however, is hard to be commercially developed under the current economic and technological conditions owing to its characteristics of deep burial, tight lithology, and complex gas
water relationship. However, it will still be a critical replacement for deep natural gas development in the Sichuan Basin. In addition, a significant breakthrough has been made in the exploration of deep Permian volcanic rocks in the western Sichuan Basin, making it a new target for increasing reserves and production of deep natural gas in the basin. 1.1.3.2 Tarim Basin The Tarim Basin has many source rocks and hydrocarbon plays in the Cambrian
Mesozoic. Deep natural gas resources are mainly endowed in

6

Reserves Estimation for Geopressured Gas Reservoirs

the Cretaceous
Paleogene clastic rocks in the Kuqa sag and the Cambrian
Ordovician carbonate rocks in platform
basin. The pay zones are generally deeper than 6000 m. The proven GIIP of deep natural gas exceeds 10,000 3 108 m3, and the annual production capacity is nearly 200 3 108 m3. Several large deep gas fields have been successfully developed, such as Dina-2, Keshen, and Dabei. In China, the Dina-2 gas field is the largest deep high-pressure condensate field, with an annual production capacity of 45 3 108 m3, while the Keshen gas field is the largest ultra-deep and ultra-high-pressure gas field, with an annual production capacity of 105 3 108 m3. The Dabei gas field is currently the main block for increasing reserves and production of deep natural gas in the Tarim Basin, with an annual production capacity of 35 3 108 m3. The TazhongI gas field in the Tazhong uplift is a rare fractured vuggy carbonate condensate field in China, with an annual production capacity of 13 3 108 m3. Moreover, significant breakthroughs have been made in exploring Cambrian subsalt natural gas in the Qiulitage structural belt of Kuqa sag and platform
basin. These two gas-rich zones are expected to become the main replacement for deep natural gas development in the Tarim Basin.

1.1.4 Development strategies Deep gas reservoirs are scientifically developed following the theories of practice and contradiction. Specifically, the objective knowledge and development laws of gas reservoirs are summarized from early development practices and then used to guide the subsequent development activities. In each stage of this process, principal contradictions and principal aspects of a contradiction in gas reservoir development are recognized and analyzed carefully. 1.1.4.1 High-precision seismic survey Deep gas reservoirs with complex geological conditions are usually developed at a high cost, with high uncertainty and risk. First, the highprecision seismic survey must be performed to provide reliable threedimensional seismic data, with which structural morphology and reservoir distribution can be inferred accurately to support well placement. The three-dimensional seismic data should be available before deploying development wells and should accurately reflect the geological characteristics of gas reservoirs. In Kuqa, Northwestern Sichuan, and other foreland thrust belts, the seismic data often have large offset errors due to the high-steep

Introduction

7

strata and complex structures. Therefore, such seismic data must be repeatedly processed by prestack depth migration, in combination with drilling data, and secondary three-dimensional seismic acquisition and processing should be performed when necessary. 1.1.4.2 Pilot test and production test The mode of exploration and production integration has emerged to become a popular and effective means to improve the exploration and development efficiency and maximize the return on investment. However, for deep gas reservoirs whose geological characteristics and development laws need to be recognized in a long period, the accelerated development by pure exploration and production integration may be risky. Instead, the combination of the pilot test and production test can help accurately understand the basic characteristics of gas reservoirs so that an appropriate development strategy can be developed to reduce the uncertainty and risks. 1.1.4.3 Determining technical policies based on geological features An appropriate development strategy allows a gas reservoir to be exploited efficiently and economically. The development strategy will be made for deep gas reservoirs after their complex geological characteristics are analyzed and clarified. In the Keshen gas field of the Tarim Basin, the gas reservoir matrix is tight. It was initially considered that such a reservoir could be developed by copying the overseas practice for tight gas, horizontal well, and large-scale sand fracturing stimulation. However, the horizontal well test failed in Keshen. Although the large-scale sand fracturing technique enabled a huge increase in single-well productivity in the initial stage, it did work in just a short period and induced serious wellbore blockage. The technique does not apply to the gas reservoirs in Keshen. Through in-depth study, it is found that the Keshen gas field has the presence of faults and fractures, good reservoir connectivity, active edge and bottom water, and natural fractures controlling productivity. Accordingly, deploying wells at high positions along the axis is proposed in addition to making appropriate stimulation treatment to communicate natural fractures. New wells are placed to obtain the maximum natural productivity, and specific engineering measures represented by fracture network acid fracturing are taken. As a result, the development effect has been greatly improved. The drilling success rate has increased from 50% (2012) to

8

Reserves Estimation for Geopressured Gas Reservoirs

100%, (2016) and the ratio of production in the second year to the design productivity has risen from 64% (2012) to 100% (2016). 1.1.4.4 Geology
engineering integration with geomechanics as a bridge Geomechanics plays an important role in many aspects of petroleum exploration and development, such as formation pressure prediction, casing program design, directional well trajectory design optimization, wellbore stability analysis, reservoir fractured performance evaluation, fracture effectiveness evaluation, induced fracture network prediction, stimulation scheme optimization, perforation interval selection, sand production mechanism analysis, early warning of casing failure, fault activity evaluation, productivity prediction, well placement optimization, geological modeling of fractures, and fluid
solid coupling numerical simulation. In recent years, with the exploration and development of fractured gas reservoirs, it has been realized that in situ stress field (especially present in situ stress field) is also a key factor affecting the permeability and fluid flow of fractured reservoirs. Therefore, geomechanics is a discipline bridging petroleum geology and petroleum engineering. It can translate complex and abstract geological information into data that can be directly used in engineering design. Geomechanics is of great significance for deep natural gas development through geology
engineering integration. 1.1.4.5 Technological innovation and integrated application To develop deep gas reservoirs, we need to strengthen the technological innovation and integrated applications to address key bottlenecks. First, R&D should be made throughout exploration and development to improve the equipment, technology, and service capabilities. Second, a problem-oriented research system is required to replace the current projectbased research system. Third, an integrated technology R&D and production/manufacturing system need to be built. Fourth, a multidisciplinary and multisector collaborative research system should be created so that new technologies and methods can be quickly promoted to practical application. Furthermore, technological innovation and integrated application should take technologies’ applicability and economics into consideration.

1.2 Classification of geopressured gas reservoirs Hydrocarbon reservoirs can be broadly divided into oil reservoirs and gas reservoirs. The latter can be subdivided by single and combined factors

9

Introduction

(GB/T 26979
2011). When a single factor is not enough to reflect the main development characteristics of a gas reservoir, two or more factors can be used to classify the same.

1.2.1 Classification by depth Depth affects the development difficulty and investment of gas fields. At present, “deep” strata are not defined uniformly in any international standard. It is generally accepted that formations with a depth of higher than 4500 m are deep strata (Zhang et al., 2015). In China, the drilling engineering industry determines 4500 and 6000 m as the depth limits for deep strata and ultra-deep strata, respectively (GB/T 28911
2012). According to the Classification of Gas Reservoirs (GB/T 26979
2011) issued by the National Standardization Committee, gas reservoirs are divided into five categories according to depth (Table 1
2), in which the strata with the depth of 3500 to 4500 m and higher than 4500 m are deep and ultra-deep, respectively. On this basis, the Ministry of Natural Resources issued the Regulations of Petroleum Reserves Estimation (DZ/T 0217
2020) and the Regulations of Shale Gas Resources/Reserves Estimation (DZ/T 0254
2020) issued by adopting the following standard for classification. When determining the industrial gas flow, the depth is subdivided into six intervals (Table 1
3). Table 1
2 Classification of gas reservoirs by depth. Class

Shallow

Midshallow

Mid-deep

Deep

Ultradeep

Mid-depth of gas reservoir (m)

, 500

500
2000

2000
3500

3500
4500

$ 4500

Table 1
3 Criteria for reserve estimation. Depth of gas reservoir (m)

# 500

500
1000

1000
2000

2000
3000

3000
4000

$ 4000

Lower limit of single-well production (104 m3/d)

0.05

0.10

0.30

0.50

1.00

2.00

10

Reserves Estimation for Geopressured Gas Reservoirs

In practice, gas reservoirs are further divided according to the geothermal field and hydrocarbon accumulation characteristics. Those with depth 3500 to 4500 m and 4500 to 6000 m are defined as deep reservoirs in eastern and western China, respectively; those with depth $ 4500 m and $ 6000 m are defined as ultra-deep reservoirs in eastern and western China, respectively (Zhang et al., 2015; Li et al. 2020b). Reserves are estimated accordingly.

1.2.2 Classification by pressure and pressure coefficient Formation pressure is generally expressed by the pressure coefficient, which is defined as the ratio of the original formation pressure to the hydrostatic column pressure at the same depth: pi (1.1) α5 CD where α—formation pressure coefficient, dimensionless; pi—original pressure at the mid-depth of the gas reservoir, MPa; D—mid-depth of the gas reservoir, m; C—hydrostatic column pressure gradient, 0.980665 MPa/ 100 m. The classification of gas reservoirs by formation pressure coefficient is given in Table 1
4. In addition, gas production engineering and gathering designs usually incorporate specific engineering technologies and construction materials depending on reservoir pressures, which are commonly bounded at 35 MPa (intermediate pressure), 70 MPa (high pressure), and 105 MPa (ultra-high pressure). The classification of high-pressure gas reservoirs based on both reservoir pressure and pressure coefficient is given in Table 1
5.

Table 1
4 Classification of gas reservoirs by formation pressure coefficient. Class

Low pressure

Normal pressure

High pressure

Ultra-high pressure

Formation pressure coefficient

,0.9

0.9
1.3

1.3
1.8

$ 1.8

Introduction

11

Table 1
5 Classification of gas reservoirs by reservoir pressure and pressure coefficient. Reservoir pressure /MPa

Intermediate pressure High pressure Ultra-high pressure

Pressure coefficient

35
70 70
105 $ 105

High pressure

Ultra-high pressure

1.3
1.8

$ 1.8

① ③ ⑤

② ④ ⑥

1.3 Reserves terms Hydrocarbon reserves are the material basis for development and operational decision-making. For many oil and gas companies, hydrocarbon reserves are the core assets and a key indicator of corporate market value and development potential. Since its IPO in the United States in 2000, PetroChina has always been following the Securities and Exchange Commission (SEC) standards in its estimation and disclosure of proved hydrocarbon reserves. This section explains the terms relating to reserves defined following standards defined by China and SEC.

1.3.1 Standards in China In 2020, the State Market Supervision Administration of the China and the State Standardization Administration approved the issuance and implementation of the Classifications for Petroleum Resources and Reserves (GB/ T19492
2020), which defines the terms of resources/reserves. 1.3.1.1 Definitions Total petroleum initially-in-place is the available natural accumulation of oil and gas formed by geological processes in the earth’s crust. It is characterized by quantity, quality, and spatial distribution. Its quantity is expressed in surface conditions converted to 20°C and 0.101325 MPa and can be further divided into undiscovered petroleum initially-in-place and discovered petroleum initially-in-place. Undiscovered petroleum initially-in-place is the quantity of petroleum discovered, not verified by drilling, and estimated depending on comprehensive

12

Reserves Estimation for Geopressured Gas Reservoirs

geological conditions, geological law research, and geological survey of petroleum. Discovered petroleum initially-in-place is the petroleum quantity estimated according to seismic, drilling, logging, testing, and other data after petroleum is found in drilling. It includes possible petroleum initially-in-place, probable petroleum initially-in-place, and proved petroleum initially-inplace, which are ranked in ascending order of the exploration and development degree and the geological knowledge. Possible petroleum initially-in-place is the estimated petroleum quantity in the reservoirs that have revealed oil/gas flows by drilling or are interpreted comprehensively to contain oil and gas layers and are believed to be worthy of exploration. Its certainty is low. Probable petroleum initially-in-place is the estimated petroleum quantity in the reservoirs that have revealed commercial oil/gas flows by drilling and are believed worthy of exploitation through preliminary drilling appraisal. Its certainty is moderate. Proved petroleum initially-in-place is the estimated petroleum quantity in the reservoirs that have revealed commercial oil/gas flows and are confirmed worthy of exploitation through drilling appraisal. Its certainty is high. Technically recoverable reserves refer to the ultimately recoverable part of discovered petroleum initially-in-place estimated according to certain technical conditions. This term is unique to China and corresponds to no benchmark in international standards (Chen, 2009). Probable technically recoverable reserves are ultimately recoverable parts of probable petroleum initially-in-place estimated according to the presupposed technical conditions. Proved technically recoverable reserves are the ultimately recoverable part of proved petroleum initially-in-place estimated according to the technical conditions that will be implemented. Commercially recoverable reserves can be defined as the recoverable commercial part of technically recoverable reserves estimated according to the economic conditions. Probable commercially recoverable reserves are the recoverable commercial part of probable technically recoverable reserves estimated according to the reasonably predicted economic conditions (e.g., price, production allocation, and cost). The remaining probable commercially recoverable reserves refer to the probable commercially recoverable reserves minus the cumulative petroleum production.

Introduction

13

Proved commercially recoverable reserves are the recoverable commercial part of proved technically recoverable reserves estimated according to the reasonably predicted economic conditions (e.g., price, production allocation, and cost). The remaining proved commercially recoverable reserves are the proved commercially recoverable reserves minus the cumulative petroleum production. 1.3.1.2 Classification of resources/reserves Petroleum resources/reserves are classified according to the geological reliability of hydrocarbon reservoirs and the technical and economic conditions. The classes of reserves are shown in the dashed line box in Fig. 1
3. The framework of Classifications for Petroleum Resources and Reserves is shown in Fig. 1
4. Compared with the Classifications for Petroleum Resources and Reserves (GB/T 19492
2004), the 2020 version does not subdivide the undiscovered petroleum initially-in-place. The discovered petroleum initially-inplace is classified into three levels: possible petroleum initially-in-place, probable petroleum initially-in-place, and proved petroleum initially-inplace. When using and disclosing the data of discovered petroleum initially-in-place, possible petroleum initially-in-place, probable petroleum initially-in-place, and proved petroleum initially-in-place shall be listed separately and shall not be added together. The possible petroleum initially-in-place is estimated under the preconditions that the structural form and reservoir conditions are preliminarily ascertained, oil/gas flows have been obtained, oil/gas zones have been

Figure 1
3 Classes of petroleum resources/reserves and estimation.

14

Reserves Estimation for Geopressured Gas Reservoirs

Figure 1
4 Framework of Classifications for Petroleum Resources and Reserves.

encountered, and the reservoir is immediately adjacent to the probable or proved petroleum initially-in-place area, is predicted to contain oil/gas zones, and is believed to be worthy of exploration after comprehensive analysis. It corresponds to low geological reliability. The probable petroleum initially-in-place is estimated under the preconditions that the structural form, reservoir change, oil/gas zone distribution, reservoir type, fluid property, and productivity are basically ascertained and the reservoir is immediately adjacent to the proved petroleum initially-in-place area. It corresponds to moderate geological reliability. It can be used as the basis for reservoir appraisal and development of conceptual design. The proved petroleum initially-in-place is estimated under the preconditions that the structural form, oil/gas zone distribution, reservoir space type, reservoir type, drive type, fluid property, and productivity are ascertained, the fluid boundary or the lowest known hydrocarbons is confirmed by drilling, logging, testing, or pressure data, and there is reasonable drilling control extent, or primary development well-pattern scheme is available. It corresponds to high geological reliability.

Introduction

15

When estimating the technically recoverable reserves, the probable technically recoverable reserves in the probable petroleum initially-inplace are estimated according to the technical conditions, and the proved technically recoverable reserves in the proved petroleum initially-in-place are estimated according to the technical conditions. When estimating the commercially recoverable reserves, the probable commercially recoverable reserves in the probable technically recoverable reserves are estimated according to the economic viability, and the proved commercially recoverable reserves in the proved technically recoverable reserves are estimated according to the economic viability.

1.3.2 Terms related to Securities and Exchange Commission reserves In November 2011, the Society of Petroleum Engineers (SPE), the American Association of Petroleum Geologists, the World Petroleum Council, the Society of Petroleum Evaluation Engineers, and the Society of Exploration Geophysicists jointly updated and issued the Guidelines for Application of the Petroleum Resources Management System (PRMS). PRMS introduces the background and basis of the preparation of the guidelines, expounds the definitions, classification, and categorization of petroleum resources/reserves, and explains the principle and application of petroleum resources/reserves estimation procedures and methods with example cases. It also discusses theoretical and practical issues related to petroleum resource management, asset evaluation, and information disclosure. This informative and pragmatic document has also been adopted by SEC as an important support for its “latest rules for petroleum disclosure” (Liu et al., 2019). SEC divides reserves into three classes: Proved Reserves (P1), Probable Reserves (P2), and Possible Reserves (P3). The classification by development status applies to all grades of reserves, which are Proved Developed Reserves (P1D), Proved Undeveloped Reserves (P1UD), Probable Developed Reserves (P2D), Probable Undeveloped Reserves (P2UD), Possible Developed Reserves (P3D), and Possible Undeveloped Reserves (P3UD). PRMS further divides the developed reserves into Developed Producing Reserves and Developed Non-Producing Reserves according to the production status, which applies to all types of reserves.

16

Reserves Estimation for Geopressured Gas Reservoirs

1.4 Dynamic reserves As their name suggests, dynamic reserves refer to the reserve calculated using dynamic methods, which is an important prerequisite for correctly evaluating the development effect of the gas reservoir, accurately predicting the development performance of gas reservoirs, and making a plan for gas reservoir development. The most commonly used methods for calculating gas reservoir reserves include analogy method, volumetric method, material balance method, production decline method, well test method, and statistical simulation method (Yang et al., 1998). This section introduces the fundamentals of dynamic reserves and summarizes the methods for their calculation.

1.4.1 Definition The term “dynamic reserves” is unique to China and is not a relevant standard. Dynamic reservers are also known as movable reserves and performance-based reserves. The details are shown in Table 1
6. To sum up, dynamic reserves have the following characteristics: 1. They refer to the gas reservoir or single well. 2. They are theoretically movable and usually less than the reserves by the volumetric method. 3. They are obtained from dynamic data. 4. They contain recoverable reserves and nonrecoverable reserves. 5. They are related to the current technology level and well pattern. 6. They are time-efficient, especially for fractured vuggy carbonate reservoirs and low-permeability reservoirs. 7. According to the Classifications for Petroleum Resources and Reserves (GB/T 19492
2020), the term is a range between proved petroleum initially-in-place and technically recoverable reserves. The term technically recoverable reserves is unique to China, but it is not clearly defined and is unpredictable; thus, it is difficult to determine the same in practical application. Accordingly, a dynamic reserve is defined as the cumulative gas production calculated using dynamic methods, when the formation pressure drops to 1 standard atmospheric pressure within the sweeping range based on the dynamic data (e.g., the production and pressure of a single well or gas reservoir), under the conditions that the existing technologies and well pattern are not unchanged. Its quantity is the limit value of technically recoverable reserves.

Table 1
6 Definition of dynamic reserves. No.

Year

Author

Source

Description

1

Li (1994)

Natural gas industry

2

Zhang (1996)

Natural gas exploration and development

3

Hao and Bian (1999)

Well testing and production technology

About dynamic reserves of gas fields

4

Feng et al. (2002)

Natural gas industry

Discussion on calculating dynamic reserves in the early stage of heterogeneous gas reservoir development

Discussion on the dynamic reserve of low permeability gas reservoir calculated by pressure buildup data The concept of movable reserves and the method of determining economically recoverable reserves

The term “dynamic reserves” is created for the first time, but its definition is not given precisely. Under the condition that the existing technologies and the existing well pattern remain unchanged, the total amount of natural gas that can flow out of the gas reservoir in the developed geological reserves when all wells are put into production and the formation pressure within the affected range drops to zero. The portion of the geological gas reserves that can participate in percolation or flow during development. Under the conditions of existing production technologies, the total gas volume can flow effectively after being converted under standard conditions in the connected pore volume of a gas reservoir. (Continued)

Table 1
6 (Continued) No.

Year

Author

5

Ministry of Land and Resources (2005)

Geological and mineral industry standard of China “Regulations of Petroleum Reserves Estimation” (DZ/ T 0217
2005)

Regulations of petroleum reserves estimation

Li et al. (2008)

Journal of Chongqing University of Science and Technology Special oil and gas reservoirs

Comparative study of gas well dynamic reserves determination methods An overview of prediction methods for dynamic reserves of gas reservoirs A new method to calculate dynamic reserves of lowpermeability gas reservoir Dynamic reserve evaluation technologies

Chen et al. (2009)

6

Source

Shen et al. (2010)

Science technology and engineering

Sun (2012)

Modern well test analysis and productivity evaluation of complex gas reservoirs

Description

When geological reserves are difficult to be calculated using volumetric method, dynamic methods (material balance method, elastic two-phase method, etc.) should be used to calculate. Based on the reliability of production and pressure data, they can be divided into proved geological reserves and probable geological reserves. Reserves are calculated by dynamic methods such as material balance, transient well test, production decline, and production accumulation, among others.

Based on the dynamic production data (such as production and pressure) of a single well or gas reservoir, the cumulative gas (Continued)

Table 1
6 (Continued) No.

7

Year

Author

Li et al. (2018)

Source

Description

Petroleum exploration and development

production of “when the production of a gas well drops to zero, and the formation pressure drops to 1 standard atmospheric pressure in the sweeping range” is calculated by the gas reservoir engineering method under the condition of the existing technologies and the production pattern are unchanged. The natural gas reserves involved in the flowing within the sweeping scope of pressure drop in the production process. It is the key parameter to evaluate the utilization degree of geological reserves of gas reservoirs. It is also the basis for the design of the development plan, the determination of infill drilling potential, and the calculation of gas reservoir recovery degree.

Correlation between per-well average dynamic reserves and initial absolute open flow potential for large gas fields in China and its application

20

Reserves Estimation for Geopressured Gas Reservoirs

1.4.2 Calculation methods 1.4.2.1 Conventional methods Conventional methods for calculating dynamic reserves include material balance method, advanced production decline analysis method, well test method, etc., as given in Table 1
7. In practice, dynamic reserves can be estimated using multiple methods depending on the gas reservoirs. The material balance method calculates the discovered petroleum initially-in-place of a gas reservoir based on the shut-in pressure data of the whole gas reservoir in different periods. It is based on the principle of matter conservation. The material balance method can be used efficiently only when (1) appropriate calculation method is selected depending on the gas reservoir type (e.g., volumetric gas reservoir, closed gas reservoir, and waterdriven gas reservoir), (2) the gas reservoir has been recovered by a certain percent, which is generally greater than 10% for conventional gas reservoirs and greater than 20% for gas reservoirs with complex lithologic trap, multifracture systems, low-permeability, high tightness, or high heterogeneity, and (3) a certain amount of shut-in pressure data (acquired at least through three to five measurements) of the whole gas reservoir is available. The advanced production decline analysis method can quantitatively analyze the dynamic reserves and related formation parameters of the single well (or connected well group) by using the production data of gas wells and the type curve matching. Gas well production data must include the transient flow in the early stage to boundary-dominated flows in the middle to late stages. Such method is represented by classical Fetkovich method and modern Blasingame, Agarwal
Gardner, and normalized pressure integral (NPI) methods. The Fetkovich method only needs production data under production at constant pressure and in the boundarydominated stage. The Blasingame, Agarwal
Gardner, and NPI methods require production to reach the boundary-dominated stage (or in pseudosteady-state flow status under production at a constant rate). These methods are derived from the theory of single-phase gas flow. When there is water invasion or retrograde condensation in strata, the applicability of the methods should be considered depending on the actual situation. Since bottom-hole pressure is not continuously monitored in most gas wells during production, it is often converted from wellhead pressure. If a large error can occur when the dynamic gas column method is used for conversion under the condition of two-phase flow in wellbore or high rate, the flowing pressure gradient test data should be used for constraint.

Table 1
7 Methods for calculating dynamic reserves. Method

Applicable conditions

Calculation method

Application range

Material balance method

1. With a certain recovery degree; 2. Gas reservoirs or gas wells with shut-in pressure test data.

Material balance equation of gas reservoirs with constant volume Balance equation for water drive gas reservoirs Balance equation for highpressure gas reservoirs Balance equation for condensate gas reservoir Fetkovich method

Gas reservoirs with constant volume

Advanced production decline analysis method

1. With certain production data; there is no need to shut in wells for pressure measurement; 2. Gas wells with natural gas seepage achieving quasistable flow state.

Elastic twophase method

1. Small gas reservoirs with constant volume; 2. Have single well test data.

Well test method

1. Buildup test; 2. Have certain production data.

Blasingame method Agarwal
Gardner method normalized pressure integral (NPI) method Flow material balance method Elastic two-phase method

Volumetric method

Water drive index .0.1 Pressure coefficient .1.3 Condensate oil content .50 g/m3 All kinds of gas reservoirs with constant-pressure production conditions and declining production. Variable production and pressure, in boundary flow, dominated state.

Small gas reservoirs with constant volume; with pressure drop test data under stable production conditions. Pressure propagates to the boundary

22

Reserves Estimation for Geopressured Gas Reservoirs

The pseudo-steady state flow may be achieved for small volumetric gas reservoirs or small fault-block gas reservoirs, or relatively small singlewell drainage areas. With the pressure drop test data obtained under stable production conditions, the elastic two-phase method can determine the discovered petroleum initially-in-place (Petroleum and Natural Gas Industry Standard of the People's Republic of China, SY/T 6098, 2010). A well test analysis method using “entire history match” can greatly increase the utilization ratio of data and decrease the ambiguity of interpretation results. Besides, it can be used to estimate the single-well dynamic reserves. The essence of this method is material balance, that is, to estimate the drainage area of the gas well by combining the short-term well test with the long-term online production data and using the volumetric method to calculate the dynamic reserves of a single well. The single-well drainage area can be determined using the following three methods. First, for a log
log plot with boundary response, the volumetric method can be directly used to calculate the reserves. Second, for a log
log plot without boundary response, the infinite approximating method can be used to match the long-term online production history to determine the boundary range, and then the dynamic reserves within the equivalent scope spread by the current pressure wave are derived. Third, for a well producing for a long time and having past pressure buildup test data, the deconvolution method can be used to determine the boundary (Jiang et al., 2018). 1.4.2.2 Securities and Exchange Commission methods Deterministic or probabilistic methods are used for SEC reserves estimation, while deterministic methods are widely used in China. The specific methods used for an estimate can be chosen in accordance with various factors such as the data acquisition, the stage of development and production, the complexity of oil and gas reservoirs, and the drive mechanism. In general, for estimating proved reserves, after the appraisal stage, the volumetric method is used before or at the initial development stage, and a dynamic method is used in the later stages of development and production. Analogous reservoirs and reliable technologies shall be rationally applied in assessment to ensure that the estimation results meet the reasonable certainty requirement of proved reserves (Wang et al., 2016). The volumetric method is an indirect estimation method and cannot directly determine the reserves. It is suitable for the early stage of reservoirs. In reserve assessment, the volumetric method is first used to estimate

Introduction

23

Table 1
8 Dynamic methods for SEC proved reserve estimation. Dynamic method

Decline curve analysis method

Application conditions

Exponential decline Hyperbolic decline Harmonic decline

Material balance method

Numerical simulation method Other production performance trend analysis

Elastic two-phase method

There is a clear decline trend, with sufficient monthly data points. Recovery percent .10%; Formation pressure changes; Sufficient pressure points. History matches by all available production performance data. There are pressure drop test data under stable production conditions; There are enough data points.

the petroleum initially-in-place (PIIP); the analogous method is used to estimate the recovery efficiency (RE), and then the estimated ultimate recovery is derived from the product of independent PIIP and RE; the proved reserve is derived by subtracting the cumulative production from the estimated ultimate recovery. Dynamics methods use the real data of hydrocarbon reservoir to predict the forthcoming development tendency and thus obtain the proved reserves in a combination of the gathering and transportation conditions and economic limit. They are relatively accurate in reserves estimation. It mainly includes decline curve analysis method, material balance method, reservoir numerical simulation method, and other auxiliary methods (including the water cut versus cumulative production method, oil cut versus cumulative production method, water/oil ratio versus cumulative production method, water drive type curve method, injection
production relation method, and other production performance trend analysis methods, for oil; and the elastic two-phase method, for natural gas). The estimation methods and their application conditions are shown in Table 1
8.

1.4.3 Challenges for dynamic reserve estimation For deep gas reservoirs, which are generally characterized by high or ultra-high pressure, tight matrix, and fracture presence, there is a high

24

5000

100

Reserve, 108m3

Proved reserve Dynamic reserve

4000

80

3000

60

2000

40

1000

20

0

0

KS1 KS2

KS8

KS9

KS6

KS5 KS24

DB

LWM

DN2

Ratio of reserve to proved reseve, %

Reserves Estimation for Geopressured Gas Reservoirs

Figure 1
5 Ratio of dynamic reserves to static reserves for typical deep gas reservoirs.

uncertainty in the estimation of dynamic reserves. Dynamic reserves are very different from static reserves (where the ratio of dynamic reserves to static reserves is between 37% and 94%), as shown in Fig. 1
5. Reservoir engineers find accurate estimation of dynamic reserves of deep high-pressure gas reservoirs challenging due to the difficulties in the acquisition of dynamic data, the recognition of development laws, and the determination of reservoir engineering parameters. 1.4.3.1 Difficulty in performance surveillance Deep gas reservoirs are usually characterized by high temperature, high pressure, and complex well conditions. For example, in the Krasu gas field of the Tarim Basin, the reservoir is deeper than 8000 m, with the formation pressure . 144 MPa, formation temperature . 180°C, and maximum wellhead pressure of 100 MPa, which can result in high risks in well control. Therefore, wellhead high-precision pressure gage testing, capillary pressure measurement, and permanent temperature and pressure monitoring technologies are insufficient for dynamic reservoir description. 1.4.3.2 Difficulty in understanding development laws By drive type, gas reservoirs can be divided into volumetric, closed, and water-drive gas reservoirs. As described in Section 1.1, compared with normal-pressure gas reservoirs, deep gas reservoirs contain very different drive energy compositions, including expansion of rock particles and

Introduction

25

p/Z，MPa

45

40 Extrapolated 61.69×108m3

35 Modified 31.96×108m3

Extrapolated 33.09×108m3

30 0

2

4

6

8

10

12

Gp, 108m3 Figure 1
6 Material balance pressure depletion curve of NS2B gas reservoir. Modified from Hammerlindl, D.J., 1971. Predicting gas reserves in abnormally pressure reservoirs. SPE 3479-MS, Permission to publish by the SPE, Copyright SPE.

bound water, stratigraphic compaction, water invasion of shale connected to the reservoirs, and precipitation of dissolved gas in immobile water and aquifer, in addition to the gas expansion and net water encroachment that are seen in normal-pressure gas reservoirs. For a long time in the past, researchers always assigned the gas reservoirs with high or ultra-high pressure into another category and summarized the development characteristics completely different from normal-pressure gas reservoirs (Li and Lin, 1985; Deng et al., 2002), that is, the pressure depletion curve of a normal-pressure gas reservoir is a straight line, while that of a highpressure gas reservoir is a broken line (Hammerlindl, 1971; Chen, 1983), as shown in Fig. 1
6. The p/Z curve of a deep high-pressure gas reservoir is smooth (Li and Lin, 1985), and the broken line is only in its approximate form. In the development process, the relation curve between apparent formation pressure and cumulative production is usually upward-convex. For tight fractured gas reservoirs, the p/Z curve becomes more complicated with edge/bottom water, so it is difficult to accurately determine the dynamic reserves from the p/Z curve. 1.4.3.3 Difficulty in determining gas reservoir parameters The material balance method is a traditional method for calculating the dynamic reserves of high-pressure gas reservoirs. The material balance

26

Reserves Estimation for Geopressured Gas Reservoirs

equation of high-pressure water-drive gas reservoirs (see Chapter 4 for details) can be simplified as:
 Gp p pi ð1 2 Ce Δp 2 ωÞ 5 12 (1.2) Z Zi G where p—average pressure of the gas reservoir, MPa; Δp—average pressure drop of the gas reservoir, MPa; Ce—effective compressibility, defined as Ce 5 Cw1S2wi S1wiCf ,1/MPa; Cw—compressibility of formation water, 1/MPa; Cf—compressibility of rock, 1/MPa; Swi—immobile water saturation, %; ω—volume coefficient of water storage volume in the gas reservoir (difference between water influx and water production), W 2W B ω 5 e GBgip w , dimensionless; Bw—volume coefficient of formation water, 1/MPa; Bgi—volume coefficient of natural gas under original conditions, 1/MPa; We—water influx volume of the gas reservoir, 108 m3; Wp—cumulative water production of the gas reservoir, 108 m3; Gp—cumulative gas production of the gas reservoir, 108 m3; G—GIIP of gas reservoir, 108 m3; Z—gas deviation factor, dimensionless; i—initial conditions. The accurate application of the material balance method depends on the accuracy of several key parameters in Eqs. (1.1) and (1.2), such as the average pressure of the gas reservoir (p), the gas deviation factor (Z), the compressibility of rock (Cf), etc. It is closely related to the recovery percent (or decline of apparent formation pressure), water size and other factors. For example, for a gas reservoir with high or ultra-high pressure in the early stage of production, even if the test production time is up to one year and the pressure drop reaches 3% (or above) of the original formation pressure, the deviation from the starting point of the early straight section would not occur. The misapplication of the classical two-segment method will lead to an overestimation of reserves. The following chapters of this book will present the determination of these key parameters, the methods for dynamic reserves estimation, and the respective conditions for their applications.

CHAPTER 2

Pressure monitoring of geopressured gas wells Contents 2.1 Downhole temperature and pressure monitoring of high pressure & high temperature (HPHT) gas wells 27 2.1.1 Role and function of dynamic monitoring 27 2.1.2 Content and means of dynamic monitoring 28 2.1.3 Downhole temperature and pressure monitoring technology for HPHT gas wells 32 2.2 Static pressure conversion for gas wells 41 2.2.1 Gas column density conversion method 42 2.2.2 Static pressure gradient conversion method 42 2.2.3 Wellhead static pressure conversion method 43 2.3 Calculation of average gas reservoir pressure 50 2.3.1 Arithmetic average method 50 2.3.2 Weighted average method 51

The average pressure of a gas reservoir is one of the critical parameters in the original gas-in-place (OGIP) estimation by the material balance method. It is usually calculated by the weighted average of single-well bottom-hole pressure data. There are two methods for calculating the bottom-hole pressure of a gas well: one is from bottom-hole pressure data monitoring, while the other is based on the conversion by the static wellhead pressure. This chapter focuses on introducing the bottom-hole pressure monitoring technology and bottom-hole static pressure conversion method for ultra-deep (i.e., more than 7000 m) and ultra-high pressure (i.e., tubing pressure more than 100 MPa) gas wells. It aims to provide reliable pressure data for the calculating OGIP by the material balance method.

2.1 Downhole temperature and pressure monitoring of high pressure & high temperature (HPHT) gas wells 2.1.1 Role and function of dynamic monitoring Dynamic monitoring and performance analysis run through the process of gas field development. They are essential techniques to thoroughly Reserves Estimation for Geopressured Gas Reservoirs © 2023 Petroleum Industry Press. DOI: https://doi.org/10.1016/B978-0-323-95088-6.00002-X Published by Elsevier Inc. All rights reserved.

27

28

Reserves Estimation for Geopressured Gas Reservoirs

understand the characteristics and performance of gas reservoirs/wells, optimize the development strategy, ensure the regular operation of the production system, evaluate the effects of stimulation treatment and development plan, and provide the basis for development adjustment and potential tapping. The quality of dynamic monitoring and performance analysis directly affects the level and effect of gas field development. The objectives and tasks of active monitoring and performance analysis vary with development stages (Table 2
1). The dynamic monitoring and performance analysis of gas reservoir engineering are complementary, but their emphases are different. Dynamic monitoring aims to collect and briefly analyze the basic data such as single well pressure, temperature, output, and fluid properties in different exploitation periods and special test stages to learn the basic performance of gas reservoirs/wells and determine the anomalies in time. Performance analysis uses dynamic monitoring data to thoroughly and comprehensively understand the gas reservoir/well characteristics and the production performance. It puts forward the technical requirements for dynamic monitoring data acquisition based on gas reservoir development and the gas reservoir engineering theory.

2.1.2 Content and means of dynamic monitoring The development of dynamic monitoring of gas field involves gas reservoir engineering, gas production engineering, and surface gathering engineering. The dynamic monitoring of gas reservoir engineering helps identify the characteristics and development laws of gas reservoirs and the productivity of gas wells and provide the basis for formulating an appropriate development strategy. The dynamic monitoring of gas production engineering aims to evaluate the wellbore quality and the adaptability and safety of related technologies or to carry out new technology tests to provide a basis for the optimization of gas production technologies. The dynamic monitoring of surface gathering engineering aims to understand the operation of gathering pipelines/facilities and provide the basis for ensuring the safe and stable operation of the production system and conducting real-time optimization and adjustment. The specific purpose and content of dynamic monitoring of gas reservoir engineering are listed in Table 2
2. Gas well monitoring modes are divided into downhole monitoring and wellhead monitoring. Though downhole monitoring is robust to

Table 2
1 Objectives and tasks of dynamic monitoring and performance analysis at development stages (Li et al., 2016). Stage

Objectives

Tasks

Appraisal

Accomplish the conceptual design of gas reservoir development and submit proved reserves.

Continuously further understand the geological characteristics and development laws of the gas reservoir

Accomplish the preparation of gas reservoir development plan

Key research contents

Track and analyze gas well productivity

Track, analyze, and evaluate gas reservoir reserves

Build and perfect gas reservoir description model

Put forward static and dynamic data acquisition requirements and deploy appraisal wells.

Conduct production tests, optimize development modes, divide development strata and development blocks, and determine the well spacing pattern, gas well proration, and gas recovery rate of the gas reservoir.

Initial formation pressure of gas reservoir; pressure, temperature, and fluid distribution; flow characteristics of the reservoir; open flow potential of the gas well; contamination or improvement status of a gas well. Formation pressure, permeability, and other flow characteristic parameters; gas well productivity equation; interwell and interlayer connectivity; recoverability of reserves; contamination or improvement status of a gas well. (Continued)

Table 2
1 (Continued) Stage

Objectives

Productivity construction

Achieve production scale designed in the development plan

Stable production

Improve rate maintenance capability of the gas reservoir and prolong production plateau

Tasks

Key research contents

Put forward additional data acquisition requirements; optimize gas well proration, location of development well to be drilled, and drill sequence. Maintain normal production of the gas reservoir, optimize production system of gas wells, conduct pertinent treatment such as production enhancement stimulation for abnormal conditions; control water in water drive gas reservoir, and drill new development wells in case of need.

Difference between actual and predicted productivities; factors affecting productivity; and rational output of gas wells.

Pressure, productivity, flow characteristics, connectivity, contamination or improvement status, water invasion performance, well controlled reserves, remaining reserve distribution, and corresponding variation laws.

(Continued)

Table 2
1 (Continued) Stage

Objectives

Production decline

Slow down the decline of gas reservoir production

Low yield

Improve ultimate recovery of gas reservoir

Tasks

Key research contents

In case of need, deploy development adjustment wells and infill wells to improve reserve producing level, control water in water drive gas reservoir, and conduct pertinent treatment such as production enhancement stimulation for abnormal conditions. Reduce the abandonment pressure of the gas reservoir, prolong the production time of gas wells, and tap the development potential of gas reservoir as far as possible.

Pressure, productivity, flow characteristics, connectivity, contamination or improvement status, water invasion performance, wellcontrolled reserves, remaining reserve distribution, corresponding variation laws, and production decline law.

Productivity, remaining reserve distribution, recovery factor, and so on.

32

Reserves Estimation for Geopressured Gas Reservoirs

Table 2
2 Purpose and content of dynamic monitoring of gas reservoir engineering (Li et al., 2016).

Purpose

Provide necessary basic data for production management and performance analysis

Content

Routine monitoring

Special monitoring

Evaluate production status and stability of gas wells Review and evaluate gas reserves Master flow characteristics of the reservoir Describe formation pressure and fluid distribution Judge interwell and interlayer connectivity Identify water invasion Analyze decline trend Diagnose contamination or improvement status of gas wells Evaluate the effect of the development plan Determine the abandonment conditions and predict the recovery factor Wellhead pressure and temperature monitoring Flow rate or injection rate monitoring Produced fluid component monitoring Salinity and main ion content monitoring of produced water Wellbore pressure, temperature gradient, bottom-hole pressure, and temperature monitoring Well test Production logging Produced fluid sampling for pressurevolume-temperature (PVT) analysis

abnormal factors and can provide relatively accurate results, it is costly and challenging to implement. Wellhead monitoring, on the other hand, is feasible and straightforward; however, it is not precise in complex conditions, and its application is limited. For example, for a gas well producing high water flow, the wellhead monitoring data cannot correctly reflect the bottom-hole pressure variation, thus failing to support the quantitative analysis on gas reservoir development and gas rate characteristics.

2.1.3 Downhole temperature and pressure monitoring technology for HPHT gas wells The static reservoir pressure of a single well is a direct reflection of formation energy in the drainage area. Therefore, it is of great significance for

Pressure monitoring of geopressured gas wells

33

estimating dynamic reserves, verifying well productivity, and predicting development effect to timely and accurately master the formation pressure variation of a single well. For this purpose, production wells and observation wells are explicitly tested regularly. Static pressure test before production of a new well, shut-in pressure buildup of production well, and pressure monitoring of observation well are the direct methods to determine the formation pressure of gas wells. Production management, technical conditions, and economic factors may restrict the shut-in test time of production wells. When the shut-in pressure is challenging to be kept stable, the well test analysis (WTA) method is used to calculate the formation pressure. 2.1.3.1 Challenges As more and more high or ultra-high-pressure gas wells are deployed, various well complexities occur, such as (ultra-)high pressure, high H2S (CO2), (ultra-)high temperature, and high yield. In the Krasu gas field of the Tarim Basin, for example, the reservoir is deeper than 8000 m, with the formation pressure . 144 MPa, formation temperature . 180°C, and maximum wellhead pressure of 100 MPa, which suggest high well control risks. Therefore, the downhole temperature and pressure monitoring are incredibly challenging. For ultra-deep and ultra-high-pressure gas wells, the downhole temperature and pressure data can be acquired by (Zhuang et al., 2020): 1. Wellhead high-precision pressure gage test. This technology has several advantages such as simple operation, low cost, and safe wellhead. However, the bottom-hole pressure converted from wellhead pressure exhibits a significant error, making it ineffective to truly and effectively reflect the behavior of formation pressure (which thus cannot be used for WTA). Before introducing safer and more accurate data acquisition technology, this technology is the most common means for data acquisition in ultra-deep and ultra-high-pressure gas wells. 2. Capillary pressure measurement. Capillary is used for transmitting the pressure changes at the measuring point to the ground through inert gas, and then the data at the measuring point are converted. This technology is advantageous for safe wellheads but inferior for high cost, complex maintenance, and low accuracy. It has been tested in a few wells only.

34

Reserves Estimation for Geopressured Gas Reservoirs

3. In-pipe cable-hanging test. This technology has the advantages such as simple operation and low cost. However, during the medium-/long-term hanging test, the wellhead is exposed to high pressure, and the cable is in the medium flow channel at all times, so that well control risk may occur and the cable may fall in the hole. 4. Permanent temperature and pressure monitoring. The optical fiber pressure gauge or electronic pressure gauge is run in a hole in the completion process. Then, the data acquired at the test point are transmitted to the ground through the optical fiber fixed outside tubing or steel pipe cable. This technology has the advantages such as safe wellhead and high accuracy, but it is costly and not justified in ultra-deep and ultra-high-pressure wells. To secure personal and work safety, it is urgent to effectively avoid accidents such as wellhead leakage, downhole paraffin scraper blocking, and wireline/cable corrosion. Tarim Oilfield Company has developed a downhole dropping-fishing test technology for highpressure gas wells (Fig. 2
1), characterized by a high success rate and low cost.

50

48 43

Number of tests

40 32

30 27

20

19

12

10 7

0 2014

2015

2016

2017

2018

2019

2020

Year Figure 2
1 Number of downhole dropping-fishing tests in Krasu gas field, 2014
20.

Pressure monitoring of geopressured gas wells

35

2.1.3.2 Wireline-conveyed downhole temperature and pressure monitoring technology The pressure gauge is suspended or seated mechanically via wireline in the production string near the producing formation to realize long-term monitoring on downhole pressure and temperature. After landing, the conveying tool and wireline are pulled out, and the wellhead blowout prevention device for wireline operation is dismantled. When the data have been collected, the pressure gauge is retrieved via wireline, and the stored data are read back. This way, the time-varying dynamic monitoring data of formation and downhole pressure and temperature are obtained. The tool landing is illustrated in Fig. 2
2. The pressure gauge is landed in two modes: seating and hanging. In the seating mode, a special unique reducing nipple is run in a hole together with the completion string. Then, the pressure gauge is landed onto the reducing nipple to monitor the pressure and temperature during acidizing or fracturing or permanently, and after the test, the pressure gauge is retrieved to provide the monitoring data. The hanging mode is suitable for wells where the reducing nipple is not run in a hole in advance. The downhole test tool cannot be seated. The pressure gauge string is connected to a unique tubing hanger and then together landed to the desired depth, while the slip of tubing hanger opens and sticks on the inner wall of tubing along with the speed variation of draw work. After the landing tool is released from the pressure measuring tool, it is pulled out, while the pressure measuring tool is left for downhole pressure and

Figure 2
2 Landing of the downhole wireline-conveyed test tool string.

36

Reserves Estimation for Geopressured Gas Reservoirs

temperature monitoring. After the test, the pressure measuring tool string is fished out to provide the monitoring data. Compared with downhole cable-conveyed test technology, the wirelineconveyed downhole temperature and pressure monitoring technology are superior in antivibration, safety, economics, and blockage prevention. 2.1.3.3 Safety control technology for HPHT gas well testing 1. Three-stage blowout preventer (BOP) at the wellhead The wellhead seal is the greatest difficulty in downhole temperature and pressure monitoring for ultra-high-pressure gas wells. The wirelineconveyed downhole temperature and pressure monitoring technology adopts a three-stage blowout prevention assembly consisting of the pack-off seal, choke tube seal, and BOP at the wellhead for sealing (Fig. 2
3). The first stage is the pack off, which realizes sealing by ground pressurization when the primary seal is leaking. The second stage is the choke tube, which replaces the wireline seal by injecting the sealing grease. The third stage is the BOP. If leakage occurs above the BOP, the upper double ram is closed first. If the leakage cannot be controlled, the lower double ram is closed. 2. Wellhead grease injection sealing system The wellhead blowout prevention system plays a vital role in securing test safety. It should be constructed to ensure good sealing when

Figure 2
3 Three-stage blowout prevention assembly at the wellhead and its field structure. (A) Three-stage blowout prevention assembly at surface; (B) Working yard.

Pressure monitoring of geopressured gas wells

37

Figure 2
4 Wireline-conveyed test wellhead grease injection sealing system for ultra-high-pressure gas wells. 1. Choke tube; 2. Lubricator; 3. stuffing box; 4. Trave; 5. Grease injection component; 6. Recovery component; 7. Hold-up vessel; 8. Hydraulic pump; 9. Filter basket;10. Filter screen; 11. Grease return line; 12. Grease injection port; 13. Grease return port; 14. Grease injection line.

the wireline is at rest and good dynamic sealing when lifted and lowered. The grease injection sealing system is used as a standard for lubricating wireline and sealing wellhead in wireline-conveyed test operation. In a conventional wireline-conveyed process, the wellhead is sealed with a stuffing box, which does not work effectively for high-pressure gas wells. When the choke tube grease injection sealing system in cable test (Fig. 2
4) is combined with the pack off seal in wireline test, a wireline-conveyed test wellhead grease injection sealing system is formed (Fig. 2
5). This system can effectively ensure the wellhead sealing in the process of downhole temperature and pressure monitoring in ultra-deep and ultra-high-pressure gas wells. The sealing grease must maintain appropriate viscosity during use. If the viscosity is too high, the flow resistance of the grease will be increased to

38

Reserves Estimation for Geopressured Gas Reservoirs

(A) Grease injection control head

(B) Hydraulic control skid of grease injection

Figure 2
5 Photos of grease injection control head and hydraulic control skid of grease injection.

tamper with the operation of the grease injection pump. If the viscosity is too low, the wellhead sealing will not be good. The choke tube sealing grease should be selected depending on the season. In summer, the sealing grease with high viscosity (8
16 mPa  s) is used; in winter, the sealing grease with low viscosity (4
5 mPa  s) is used. 3. Optimization of downhole monitoring tools A set of countermeasures concerning wellhead sealing and temperature/pressure resistance of pressure gauge have been developed for ultra-deep and ultra-high-pressure gas wells (Table 2
3). 2.1.3.4 Application In 2014, the wireline-conveyed downhole temperature and pressure monitoring technology was successfully pioneered in the Tarim Oilfield to acquire the pressure and temperature data in an ultra-deep and ultrahigh-pressure gas well with wellhead pressure higher than 100 MPa. Up to now, the technology has recorded the test success rate of 100% and the data accuracy rate of 100%. It has been applied to the deepest monitoring depth of 8038 m, the highest downhole pressure of 136 MPa, and the highest downhole temperature of 187°C (Figs. 2
6
2
8). The wireline-conveyed downhole temperature and pressure monitoring technology for ultra-deep and ultra-high-pressure gas wells lays a foundation for the dynamic description of the Krasu gas field. It provides reliable pressure data for dynamic reserve estimation.

Table 2
3 Optimization of wireline-conveyed test tools for ultra-deep and ultra-high-pressure gas wells. Problem

Countermeasures

Effect

Due to high CO2 content, wireline may be eroded and broken

A set of wellhead blowout control parameter standards is formulated, and the sulfur- and CO2resisting wireline is used.

The wireline corrosion is detected in the whole process of the test.

At depth . 8000 m, the wireline bears large tension and thus may be broken.

A set of wireline inspection and abandonment standards is formulated. For the 3.8 mm wireline, it is sampled for tensile test before it is run in hole. 140 MPa blowout preventer (BOP) is adopted.

The test has been made in 90 wells without corrosioninduced wireline breakage. The test has been made in 90 wells without wireline breakage.

Wellhead risk has been minimized.

Third-party performance verification, including flaw detection and thickness check, is performed. PPS2800 pressure gauge

Wellhead pressure exceeds 100 MPa.

Under high formation temperature and pressure, the pressure gauge works at a high load, making its monitoring time shorter.

The selection standard for differential pressure gauges is formulated, and the battery service life is prolonged.

The longest monitoring time exceeds 60 days.

Photo

Remarks

The wireline wear is checked in the whole process. The wireline can be run in a hole only 20 times.

40

Reserves Estimation for Geopressured Gas Reservoirs

Figure 2
6 Maximum well depth of downhole test for high-pressure gas wells.

Figure 2
7 Maximum wellhead pressure of downhole test for high-pressure gas wells.

Pressure monitoring of geopressured gas wells

41

Figure 2
8 Maximum bottom-hole temperature of downhole test for high-pressure gas wells.

2.2 Static pressure conversion for gas wells The measured average formation pressure refers to the bottom-hole static pressure measured by long-term shut-in in the production area. If the shut-in time is long enough, the pressure tends to be balanced in the limited area controlled by the well, and the measured static pressure can represent the average formation pressure of the block. This method is direct and reliable, but it requires high test quality and may have a significant impact on production. During the production of gas wells, it is almost impossible to land the pressure gauge to the bottom-hole for measuring the formation pressure. When the pressure gauge cannot be anchored to the mid-depth of the reservoir, the static pressure at middepth of the reservoir in a single well can be determined using the gas column density conversion method, the static pressure gradient conversion method, and the wellhead static pressure conversion method based on the temperature and pressure data at the measuring point. The pressure conversion method does not require any basic data with high quality; however, it is not as accurate as the direct method. An appropriate method should be selected depending on its reliability and applicability for actual conditions.

42

Reserves Estimation for Geopressured Gas Reservoirs

2.2.1 Gas column density conversion method Based on the temperature and pressure data at the measuring point, the state equation is used to determine the gas density. Then, the gas column pressure between the measuring point and the mid-depth is used to convert the pressure at mid-depth of the reservoir. Example 1: In a gas well, the mid-depth of the reservoir is 7500 m, the setting depth of the pressure gauge is 6800 m, the gas gravity is 0.6, and the pressure and temperature at the measuring point are 120 MPa and 423K, respectively. Calculation of the pressure at mid-depth of the reservoir is described below. Solution: 1. Based on the temperature and pressure at the measuring point, the DAK (Dranchuk-Abu-Kassem, 1975) extrapolation is used to calculate the gas deviation factor (see Chapter 3, Physical Properties of Natural Gas and Formation Water for details) as Z 5 1.8751. According to the equation of state, we have ρ5

pM g 120 3 28:97 3 0:60 5 5 316:3 kg=m3 ZRT 1:8751 3 8:3143 3 10-3 3 423

2. Calculation of the gas column pressure between the measuring point and the mid-depth of the reservoir. Δp 5 ρgΔH 5 316:3 3 9:8 3 ð7500 2 6800Þ 5 2:17 MPa 3. Calculation of the pressure at mid-depth of the reservoir. p 5 pmeasure point 1 Δp 5 120 1 2:17 5 122:17 MPa

2.2.2 Static pressure gradient conversion method If there are measured data of static pressure gradient in the lowering or lifting process of pressure gauge, the gradient trend can be used to get the pressure at mid-depth of the reservoir. Example 2: In a gas well, the static wellhead pressure is 69.28 MPa, the wellhead temperature is 20.73°C, the mid-depth of the reservoir is 6100 m, the setting depth of pressure gauge is 5362 m, the pressure and temperature at the measuring point are 89.4 MPa and 112.57°C, respectively, and the static pressure gradient data are shown in

Pressure monitoring of geopressured gas wells

43

Figure 2
9 Static pressure gradient of a high-pressure gas well.

Fig. 2
9. Calculation of the pressure at mid-depth of the reservoir is described below. Solution: 1. According to the wellbore static pressure gradient test data, the gradient is calculated as 0.37 MPa/100 m. 2. Calculation of the gas column pressure between the measuring point and the mid-depth of the reservoir. Δp 5 Gradient 3 ΔH 5

0:37 3 ð6100 2 5362Þ 5 2:73 MPa 100

3. Calculation of the pressure at mid-depth of the reservoir. p 5 pmeasure point 1 Δp 5 89:4 1 2:73 5 92:13 MPa

2.2.3 Wellhead static pressure conversion method If only wellhead static pressure data are available, the average temperature and average deviation factor method or the Cullender
Smith method can be used for bottom-hole static pressure conversion (Yang, 1992). 2.2.3.1 Average temperature and average deviation factor method The bottom-hole flowing pressure equals the sum of wellhead flowing pressure, the pressure exerted on bottom-hole by gas column mass, kinetic

44

Reserves Estimation for Geopressured Gas Reservoirs

energy change, and energy loss resulting from friction. If the kinetic energy variation under steady flow is ignored, the energy equation of single-phase gas flowing in a vertical pipe is dz 1

1000 fv 2 dp 1 dL 5 0 ρg g 2gD

(2.1)

where dz—change in vertical distance, m; dL—change in the distance along wellbore trajectory, m; dp—change in pressure, MPa; ρg—gas density, g/cm3; f—friction coefficient, dimensionless; v—gas flow velocity, m/s; D—tubing ID (considering flow in tubing), m; g—gravity acceleration, m/s2. Substituting the gas density into Eq. (2.1), after integration, we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1:324 3 10210 fq2 T Z 2S ðe 2 1Þ (2.2) pwf 5 p2wh e2S 1 D5 where pwf—bottom-hole flowing pressure, MPa; pwh—wellhead flowing pressure, MPa; q—gas rate (under standard condition), 104 m3/d; T —the average temperature of the wellbore, K; Z—gas deviation factor under average pressure and temperature of the wellbore, dimensionless; 0:03416γ H S—intermediate parameter, S 5 T Z g ; H—actual vertical depth in the middle of pay zone, m; γg —gas gravity, dimensionless. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1:324 3 10210 fq2 T Z 2S pwf 5 p2wh e2S 1 ðe 2 1Þ (2.3) ðDc 2Dt Þ3 ðDc 1Dt Þ2 where Dc—casing ID, m; Dt—tubing OD, m. If the production satisfies q 5 0, the equation of bottom-hole pressure calculated by static gas column is obtained as pwf 5 pwh e

0:03416γg H TZ

(2.4)

The calculation procedure is as follows. 1. Estimate the initial value of bottom-hole pressure pwf, which affects the number of iterations but not the final calculation results. The initial value of bottom-hole pressure can be estimated by the equation of static gas column pressure of ideal gas p0wf 5 pwh 1 9:80665 3 1023 ρg H. For highpressure gas wells, the wellbore gas density is mostly 0.25
0.35 g/cm3, and the gas density can be approximately estimated according to the actual situation of the gas well. 2. Based on wellhead and bottom-hole temperatures, calculate the average temperature of the wellbore using the average arithmetic method.

Pressure monitoring of geopressured gas wells

3. 4. 5. 6.

7.

45

When the bottom-hole temperature is unknown, it can be estimated using the geothermal gradient and the mid-depth of pay zone in the gas wells of the same area. Calculate the average pressure of the wellbore p 5 pwh 12 pwf . Calculate the gas deviation factor Z under the average pressure and temperature of the wellbore, and calculate the S value. Calculate the bottom-hole pressure according to Eq. (2.4). Compare the bottom-hole pressure obtained in step (5) with that in the previous calculation (estimated value for the first time). If the difference is significant, return to step (3) to conduct an iterative analysis based on the calculation results in step (5). Repeat steps (3) to (6) until the calculation results meet the acceptable error.

Example 3: In a gas well, the mid-depth of the reservoir is 6100 m, the static wellhead pressure is 69.28 MPa, the wellhead temperature is 20.73° C, and the gas gravity is 0.7436. Calculation of the pressure at mid-depth of the reservoir is given below. Solution: 1. Estimate the average temperature of the wellbore as 350K. 2. Based on the wellhead static pressure data, calculate the gas deviation factor using the DAK extrapolation method, Z 5 1.4266. According to the state equation, we have ρ5

pM g 69:4 3 28:97 3 0:7436 5 5 360:0 kg=m3 ZRT 1:4266 3 8:3143 3 10-3 3 350

pwf 5 pwh 1 9:80665 3 1023 ρg H 5 69:4 1 9:80665 3 1023 3 360 3 6100=103 5 90:9 MPa 3. Calculate the average pressure of the wellbore. p5

pwh 1 pwf 69:4 1 90:9 5 80:2 MPa 5 2 2

4. Knowing that the gas deviation factor under average pressure and temperature of the wellbore is 1.5581, calculate the S value. S5

0:03416γ g H TZ

5

0:03416 3 0:7436 3 6100 5 0:2841 350 3 1:5581

46

Reserves Estimation for Geopressured Gas Reservoirs

5. Calculate the bottom-hole pressure according to Eq. (2.4). pwf 5 pwh e

0:03416γ g H TZ

5 69:4 3 e0:2841 5 92:2 MPa

6. The difference between the calculation result in step (5) and the initial value is 92.2 2 90.9 5 1.3 MPa. Let pwf 5 92:2 MPa, and repeat steps (3) to (5). We have p5

pwh 1 pwf 69:4 1 92:2 5 80:8 MPa 5 2 2

Z 5 1:5660 S5

0:03416γ g H TZ

5

0:03416 3 0:7436 3 6100 5 0:2827 350 3 1:5660

0:03416γg H

pwf 5 pwh e

TZ

5 69:4 3 e0:2827 5 92:07 MPa

7. The difference between the calculation result in step (6) and the initial value is 92.07 2 92.2 5 20.13 MPa. Let pwf 5 92:07 MPa, and repeat steps (3) to (5). We have p5

pwh 1 pwf 69:4 1 92:07 5 80:735 MPa 5 2 2

Z 5 1:5670 S5

0:03416γg H TZ

5

0:03416 3 0:7436 3 6100 5 0:2825 350 3 1:5670

0:03416γg H

pwf 5 pwh e

TZ

5 69:4 3 e0:2825 5 92:06 MPa

8. The difference between step (7) calculation result and the initial value is 92.06 2 92.07 5 20.01 MPa, which meets the accuracy requirements. Therefore, this method’s bottom-hole static pressure is 92.06 MPa. 2.2.3.2 Cullender
Smith method This method uses the numerical integration method. For the typical tubing production, we have ð pwf p TZ (2.5)  p 2 1:324 3 10210 fq2 dp 5 0:03416γg H pwh TZ 1 D5

Pressure monitoring of geopressured gas wells

47

The bottom-hole pressure can be calculated using the piecewise integral numerical calculation method according to the above equation. Multiple calculation position points are set from the wellhead to the bottom-hole. Then, the pressure of each position point is calculated from top to bottom based on the wellhead pressure, and finally, the bottomhole pressure is figured out. For Eq. (2.5), the trapezoidal method is used to obtain the product, and the pressure equations are obtained as 0:03416γ g H    p p TZ TZ 1 p 2 1:324 3 10210 fq2 210 2 2 ðTZp Þ 11:324 3D105 fq 1 ðTZÞ 1 D5 m

pwfm 5 pwf 1 1 

0:03416γg H    p p TZ TZ 1 p 2 1:324 3 10210 fq2 210 2 2 ðTZp Þ 11:324 3D105 fq m ðTZÞ 1 D5 2

pwf2 5 pwfm 1 

(2.6)

(2.7)

where subscript 1 represents the upper position point of the adjacent position points, subscript 2 represents the lower position point, and subscript m represents the middle position point between them. If the production q 5 0, the equation of bottom-hole pressure calculated by static gas column is obtained as follows: 0:03416γ g H

pwfm 5 pwf 1 1

TZ TZ 1 p p 1

(2.8) m

0:03416γg H

pwf2 5 pwfm 1

TZ TZ 1 p p m

(2.9) 2

The specific calculation procedure is as follows. 1. Calculate the initial value of pressure in the middle of the wellbore, when subscript 1 numerical term in the denominator of Eq. (2.8) is taken under the wellhead pressure and temperature conditions, and it is assumed that the numerical term of the midpoint is equal to that of subscript 1. 2. Based on the temperature in the middle of the wellbore and the above-calculated pressure value, calculate subscript m numerical term in the denominator of Eq. (2.8), and then, calculate the pressure in the middle of the wellbore according to Eq. (2.8) once more.

48

Reserves Estimation for Geopressured Gas Reservoirs

3. Compare the bottom-hole pressure obtained in step (2) with that in the previous calculation (estimated value for the first time). If the difference is significant, repeat step (2) to conduct an iterative analysis until the calculation results are acceptable. 4. After the pressure in the middle of the wellbore is determined, repeat the above steps to calculate the bottom-hole pressure further iteratively. Example 4: In a gas well, the mid-depth of the reservoir is 6100 m, the static wellhead pressure is 69.28 MPa, the wellhead temperature is 20.73° C, the mid-depth temperature is 125.74°C, and the gas gravity is 0.7436. Calculation of the pressure at mid-depth of the reservoir is given below. Solution: Assumptions include that: (1) there is only one point in the middle of the wellbore; (2) the wellbore is divided into two sections; and (3) the temperature at such point is 346.4K. (1) Based on wellhead pressure and temperature data, p

69:4 5 5 0:1503 TZ 1 293:88 3 1:5711 According to Eq. (2.8), 0:03416γ g H

pwfm 5 pwf 1 1

TZ TZ 1 p p 1

5 69:4 1

m

0:03416 3 0:7436 3 6100 5 81:05 MPa 2=0:1503

When pwfm 5 81:05 MPa, Tm 5 346:4 K and Zm 5 1:6482. Thus, p

81:05 5 5 0:1420 TZ m 346:4 3 1:6482 0:03416γ g H

pwfm 5 pwf 1 1

TZ TZ 1 p p 1

m

0:03416 3 0:7436 3 6100    5 80:712 MPa 5 69:4 1  1=0:1503 1 1=0:1420

Pressure monitoring of geopressured gas wells

49

When pwfm 5 80:712 MPa, Tm 5 346:4 K and Zm 5 1:6433. Thus, p

81:712 5 5 0:1418 TZ m 346:4 3 1:6433 0:03416γ g H

pwfm 5 pwf 1 1

TZ TZ 1 p p 1

m

0:03416 3 0:7436 3 6100    5 80:705 MPa 5 69:4 1  1=0:1503 1 1=0:1418 (2) pwfm 5 80:705 MPa is the pressure in the middle of the wellbore. Then, calculate the bottom-hole pressure according to Eq. (2.8), 0:03416γg H

pwf2 5 pwfm 1

TZ TZ 1 p p m

5 80:705 1

2

0:03416 3 0:7436 3 6100 5 91:69 MPa 2=0:1418

When p2 5 91:69 MPa, T2 5 398:89 K and Z2 5 1:7016. Thus, we have p

91:69 5 0:1351 5 TZ 2 398:89 3 1:7016 0:03416γg H

pwf2 5 pwfm 1

TZ TZ 1 p p m

2

0:03416 3 0:7436 3 6100    5 91:43 MPa 5 91:69 1  1=0:1418 1 1=0:1351 When p2 5 91:43 MPa, T 2 5 398:89 K, and Z 2 5 1:6984. Thus, we have p

91:43 5 0:1350 5 TZ 2 398:89 3 1:6984 0:03416γg H

pwf 2 5 pwfm 1

TZ TZ 1 p p m

2

50

Reserves Estimation for Geopressured Gas Reservoirs

0:03416 3 0:7436 3 6100    5 91:42 MPa 5 91:69 1  1=0:1418 1 1=0:1350 After two iterations, the pressure at the mid-depth of the reservoir is determined as 91.42 MPa. Assuming that two points are taken in the middle of the wellbore, the wellbore is divided into three sections, and the temperatures at these two points are 328.88 and 363.88K, respectively, according to the above steps, the pressure at the mid-depth of the reservoir is calculated as 91.64 MPa. The above method is mainly applicable to net gas wells in dry gas reservoirs. For condensate gas wells or water production wells, this method exhibits a significant error. In this case, the static pressure gradient test should be carried out, and the gas column density conversion method or static pressure gradient conversion method at the measuring point should be used.

2.3 Calculation of average gas reservoir pressure The average gas reservoir pressure is a crucial parameter necessary for the estimation of dynamic reserves, water invasion, and productivity. During the development of most gas reservoirs, the pressures in blocks are hard to be ultimately the same. Therefore, it is necessary to determine the average pressure representing the overall energy of the gas reservoir. Once the static pressure of a single well is obtained, the arithmetic average and weighted average methods can be used to calculate the average gas reservoir pressure.

2.3.1 Arithmetic average method The arithmetic average method, expressed as Eq. (2.10), is suitable for gas reservoirs exploited relatively uniformly. n P

p5

j51

n

pj (2.10)

Where n—number of wells, p—formation pressure, MPa, p
average formation pressure, MPa.

Pressure monitoring of geopressured gas wells

51

2.3.2 Weighted average method Depending on the geological and production conditions, the methods such as thickness weighting, area weighting, volume weighting, output weighting, and cumulative production weighting can be used to calculate the average pressure, with the equations as follows: ! n n X X p5 pj hj = hj (2.11) j51 n X

p5

j51

! pj Aj =

j51 n X

p5

p5

! Aj

(2.12)

j51

! pj Aj hj =

j51 n X

n X

n X

! Aj hj

(2.13)

j51

! pj Aj hj φj =

j51

n X

! Aj hj φj

(2.14)

j51

When the formation pressure drop rate for all wells is the same, the cumulative production of the gas well is almost proportional to the pore volume effectively controlled by a single well. Suppose the wells are put into production roughly at the same time. In such a case, the output is almost proportional to the pore volume effectively controlled by a single well. The weighted average method of output or cumulative production is thus derived. ! ! n n X X (2.15) p5 pj qj = qj j51

p5

n X j51

j51

! Gpj pj =

n X

! Gpj

(2.16)

j51

where A—drainage area of the gas well, m2, h—net-pay thickness, m, φ
porosity, decimal, q—gas rate, 104 m3/d, Gp—cumulative production of a single well, 104 m3, n—number of wells, dimensionless.

52

Reserves Estimation for Geopressured Gas Reservoirs

All the above methods use the formation pressure data of a single well. When the well pattern covers the gas reservoir to a great extent, the trend surface interpolation method can calculate the formation pressure at any position of the gas reservoir based on the formation pressure of the gas well. On this basis, the discrete grid method can be used for interpolation calculation to get the formation pressure at all grid points, and then, the arithmetic average for all grid points is obtained to determine the average gas reservoir pressure, or the multiple integral numerical calculation method is used to calculate the average gas reservoir pressure.

CHAPTER 3

Physical properties of natural gas and formation water Contents 3.1 Composition and properties of natural gas 3.1.1 Composition of natural gas 3.1.2 Equation of state of an ideal gas 3.2 Behavior of real gas 3.2.1 Natural gas deviation factor 3.2.2 Compressibility factors for natural gases 3.2.3 Gas formation volume factor 3.2.4 Natural gas viscosity 3.3 Deviation factor of ultra-high-pressure gas 3.3.1 DPR or DAK extrapolation method 3.3.2 LXF-RMP (Li et al., 2010) fitting method 3.4 Properties of formation water 3.4.1 Formation water volume factor 3.4.2 Formation water viscosity 3.4.3 Natural gas solubility in water 3.4.4 Isothermal compressibility factor of formation water

53 53 54 58 59 71 75 75 80 80 82 87 88 91 92 94

The high-pressure physical properties of natural gas and formation water are essential parameters in calculating reserves using the material balance equation. This chapter discusses the physical properties of natural gas and formation water, commonly used empirical relations, and the conditions for their applications.

3.1 Composition and properties of natural gas 3.1.1 Composition of natural gas Natural gas is a mixture of hydrocarbon and nonhydrocarbon gases. Hydrocarbon gases are mainly methane, ethane, propane, butane, pentane, and a small amount of hexane and heavy hydrocarbons. Nonhydrocarbon gases (i.e., impurities) include carbon dioxide, hydrogen Reserves Estimation for Geopressured Gas Reservoirs © 2023 Petroleum Industry Press. DOI: https://doi.org/10.1016/B978-0-323-95088-6.00003-1 Published by Elsevier Inc. All rights reserved.

53

54

Reserves Estimation for Geopressured Gas Reservoirs

sulfide, and nitrogen. Sometimes, trace amounts of rare gases, such as helium and argon, are also found in natural gas. The main physical and chemical properties of standard components in natural gas are listed in Table 3
1, which can be used for general calculation.

3.1.2 Equation of state of an ideal gas According to the kinetic theory of gas molecules, the equation of state of an ideal gas can be expressed as pV 5 nRT

(3.1)

where p: gas pressure, MPa, V: gas volume, m3, T: absolute temperature, K, n: the amount of gaseous substance, kmol, R: universal gas constant, which is 8.3143 3 1023 MPa  m3/(kmol  K). The amount of gaseous substance n is defined as the ratio of the gas mass to the molecular weight. m n5 (3.2) M where m: gas mass, kg, M: molecular weight of the gas, kg/kmol. Combining Eq. (3.1) and Eq. (3.2), m pV 5 RT M

(3.3)

According to the definition of density, Eq. (3.3) can be expressed as ρg 5

m pM 5 V RT

(3.4)

where ρg: density of the gas, kg/m3. Example 1: Put 1.3608 kg of n-butane in a container with a pressure of 0.4137 MPa and a temperature of 322.2K. Calculate its volume and density in an ideal gas state (the example cases in Sections 3.1 and 3.2 are all cited from the Reservoir Engineering Handbook, which is written by Tarek Ahmed and translated by Sun et al. (2021b). Solution 1. From Table 3
1, the molecular weight of n-butane is 58.123.

Table 3
1 Physical property constants of hydrocarbons and nonhydrocarbon gases in natural gas. No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Compound

Methane Ethane Propane Isobutane N-butane Isopentane N-pentane N-hexane N-heptane N-octane Isooctane N-nonane N-decane Cyclopentane Cyclohexane Ethylene Acrylic Isobutylene Isoprene Acetylene Benzene Toluene

Formula

CH4 C2H6 C3H8 C4H10 C4H10 C5H12 C5H12 C6H14 C7H16 C8H18 C8H18 C9H20 C10H22 C5H10 C6H12 C2H4 C3H6 C4H8 C5H8 C2H2 C6H6 C7H8

Molecular weight

Boiling point at atmospheric pressure, °C

The freezing point at atmospheric pressure, °C

16.043 30.070 44.097 58.123 58.123 72.150 72.150 86.177 100.204 114.231 114.231 128.258 142.285 70.134 84.161 28.054 42.081 56.108 68.119 26.038 78.114 92.141

2 161.52 2 88.61 2 42.08 2 11.79 2 0.51 27.84 36.07 68.73 98.42 125.67 99.24 150.82 174.16 49.25 80.72 2 103.74 2 47.69 2 6.89 34.06 2 84.72 80.10 110.63

2 182.47 2 182.80 2 187.63 2 159.60 2 138.36 2 159.90 2 129.73 2 95.32 2 90.58 2 56.77 2 107.37 2 53.49 2 29.64 2 93.84 6.54 2 169.15 2 185.25 2 140.36 2 145.96 2 81.39 5.53 2 95.00

Critical parameter Pressure, MPa

Temperature, K

4.5947 4.8711 4.2472 3.6397 3.7962 3.3812 3.3688 3.0123 2.7358 2.4869 2.5676 2.2877 2.1043 4.5078 4.0734 5.0401 4.6098 4.0003 3.8473 6.1391 4.8980 4.1058

190.56 305.33 369.85 407.85 425.16 460.43 469.71 507.37 540.21 568.83 543.96 594.64 617.59 511.59 553.48 282.34 364.91 417.90 484.26 308.34 562.16 591.80

Relative density under standard conditions

0.3000 0.3562 0.5070 0.5629 0.5840 0.6247 0.6311 0.6638 0.6882 0.7070 0.6962 0.7219 0.7342 0.7505 0.7835 — 0.5209 0.6004 0.6862 0.8990 0.8845 0.8719 (Continued)

Table 3
1 (Continued) No.

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

Compound

Phenyl Ethane Styrene Carbon monoxide Carbon dioxide Hydrogen sulfide Sulfur dioxide Ammonia Air Hydrogen Oxygen Nitrogen Chlorine Water Helium Hydrochloric acid

Formula

Molecular weight

Boiling point at atmospheric pressure, °C

The freezing point at atmospheric pressure, °C

C8H10

106.167

136.20

C8H8 CO

104.152 28.010

CO2

Critical parameter

Relative density under standard conditions

Pressure, MPa

Temperature, K

2 94.98

3.6060

617.20

0.8717

145.14 2 191.49

2 30.61 2 205.00

4.0527 3.4991

645.93 132.91

0.9111 0.7894

44.010

2 78.48

2 56.57

7.3843

304.21

0.8180

H2S

34.080

2 60.28

2 85.49

8.9632

373.40

0.8014

SO2

64.060

2 9.94

2 75.48

7.8807

430.82

1.3974

NH3 N2 1 O 2 H2 O2 N2 Cl2 H2O He HCl

17.031 28.963 2.016 31.999 28.013 70.906 18.015 4.003 36.461

2 195.81 2 194.33 2 252.75 2 182.96 2 195.81 2 33.96 100.00 2 268.94 2 146.82

2 77.71 — 2 259.59 2 218.79 2 210.00 2 100.96 0.00 — 2 114.18

11.3487 3.7707 1.2969 5.0428 3.3998 7.9772 22.0549 0.2275 8.3082

405.48 132.42 33.21 154.58 126.20 416.90 647.13 5.20 324.69

0.6183 0.8748 0.0710 1.1421 0.8094 1.4244 1.0000 0.1251 0.8513

Source: Modified from Tarek, A., 2019. Reservoir Engineering Handbook, fifth ed. Elsevier.

57

Physical properties of natural gas and formation water

2. According to Eq. (3.3), we have V5

m 1:3608 3 8:3143 3 1023 3 322:2 RT 5 5 0:1516 m3 pM 0:4137 3 58:123

3. According to Eq. (3.4), we have ρg 5

pM 0:4137 3 58:123 5 5 8:9760 kg=m3 RT 8:3143 3 1023 3 322:2

3.1.2.1 Apparent molecular weight If yi is used to represent the mole fraction of the ith component in a gas mixture, the apparent molecular weight is expressed as X yi M i (3.5) Ma 5 where Ma: apparent molecular weight of natural gas, Mi: molecular weight of the ith component in natural gas, yi: mole fraction of the ith component in natural gas. 3.1.2.2 Standard volume Standard volume refers to the volume of gas occupied by 1 kmol gas under standard conditions (in this book, i.e., 0.101325 MPa and 293.15K). Applying the standard conditions to Eq. (3.1), Vsc 5

RTsc 8:3143 3 1023 3 293:15 5 5 24:05 m3 0:101325 psc

(3.6)

where Vsc: standard volume, m3. Under the conditions of 0.101325 MPa and 273.15K, the gas volume occupied by 1 kmol gas is 22.41 m3. 3.1.2.3 Density The density of an ideal gas mixture is calculated by replacing the molecular weight of the pure component in Eq. (3.4) with the apparent molecular weight of the gas mixture to give ρg 5

pMa RT

where ρg: density of the ideal gas mixture, kg/m3.

(3.7)

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Reserves Estimation for Geopressured Gas Reservoirs

3.1.2.4 Gas gravity Under normal circumstances, gas gravity is defined as the ratio of the density of gas to the density of air under standard conditions, which can be expressed as γ g 5 ρg =ρair

(3.8)

Or ð3:9Þ

Example 2: In a gas well, the proportions of CO2, C1, C2, and C3 are 0.05, 0.90, 0.03, and 0.02, respectively. Calculate the apparent molecular weight, gas gravity, and density under the pressure of 13.79 MPa and temperature of 338.89K. Solution According to Table 3
1, the molecular weights of CO2, C1, C2, and C3 are 44.01, 16.04, 30.07, and 44.10, respectively. According to Eq. (3.5), the apparent molecular weight is determined as X yi Mi 5 18:421 Ma 5 According to Eq. (3.9), the gas gravity is determined as γg 5

Ma 18:421 5 0:636 5 28:96 28:96

According to Eq. (3.7), the gas density is determined as ρg 5

pM 13:79 3 18:421 5 5 90:156 kg=m3 RT 8:3143 3 1023 3 338:89

3.2 Behavior of real gas Natural gas is a multicomponent real gas. When considering the influence of the volume occupied by gas molecules and the interaction between molecules, the deviation between the real gas and the ideal gas under the same conditions increases with the increase of pressure and temperature and varies with the composition of gases.

59

Physical properties of natural gas and formation water

3.2.1 Natural gas deviation factor For a real gas, the equation of state is expressed as pV 5 ZnRT

(3.10)

Where Z is the gas deviation factor, which is a dimensionless quantity and is defined as the ratio of the volume occupied by the real gas to the volume occupied by the same amount of ideal gas at the same temperature and pressure. The gas deviation factor is a function of pressure, temperature, and gas composition and is usually determined by the Standing
Katz (1942) chart, as shown in Fig. 3
1. This chart is also suitable for natural gas with a minor nonhydrocarbons. In Fig. 3
1, Tpr and ppr are dimensionless pseudo-reduced temperature and pseudo-reduced pressure, respectively. These dimensionless terms are defined as

2.0

ppr 5 p=ppc

(3.11)

Tpr 5 T =Tpc

(3.12)

Tpr 1.05 1.40 1.80 2.40

1.6

1.10 1.50 1.90 2.60

1.20 1.60 2.00 2.80

1.30 1.70 2.20 3.00

Z

1.2

0.8

0.4

0.0 0

3

6

ppr

9

12

15

Figure 3
1 Standing and Katz deviation factors. Modified from Tarek, A., 2019. Reservoir Engineering Handbook, fifth ed. Elsevier.

60

Reserves Estimation for Geopressured Gas Reservoirs

ppc 5

X

yi pci

(3.13)

yi Tci

(3.14)

i51

Tpc 5

X i51

where p: system pressure, MPa, ppc: pseudo-critical pressure, MPa, ppr: pseudoreduced pressure, dimensionless, T: system temperature, K, Tpc: pseudo-critical temperature, K, Tpr: pseudo-reduced temperature, dimensionless. Example 3: The initial pressure and temperature of a reservoir are 20.68 MPa and 355.56K, respectively. In natural gas, the proportions of CO2, N2, C1, C2, C3, i-C4, and n-C4 are 0.02, 0.01, 0.85, 0.04, 0.03, 0.03, and 0.02, respectively. Calculate the deviation factor under initial reservoir conditions. Solution According to Table 3
1, the Tci and pci of each component can be found. According to Eqs. (3.13) and (3.14), Component

yi

Tci, K

yiTci, K

pci, MPa

yipci, MPa

CO2 N2 C1 C2 C3 i-C4 n-C4 Sum

0.02 0.01 0.85 0.04 0.03 0.03 0.02

304.39 126.38 190.74 305.51 370.03 408.03 425.34

6.09 1.26 162.13 12.22 11.10 12.24 8.51 213.55

7.384 3.400 4.595 4.871 4.250 3.640 3.796

0.148 0.034 3.905 0.195 0.127 0.109 0.076 4.595

ppc 5

X

yi pci 5 4:595 MPa Tpc 5

X

yi Tci 5 213:55K

According to Eq. (3.11) and Eq. (3.12), ppr 5

T pr 5

p 20:68 5 4:502 5 ppc 4:595 T 355:56 5 5 1:665 Tpc 213:55

Physical properties of natural gas and formation water

61

According to Fig. 3
1, the deviation factor Z is determined to be 0.850. If the composition of natural gas is unknown, the gas gravity of natural gas can be used to predict the pseudo-critical pressure and pseudo-critical temperature, as shown in Fig. 3
2 (Brown, 1948). Standing (1977) expressed this graphical correlation with mathematical forms. For dry gas, 168 1 325γg 2 12:5γ2g

Tpc 5

1:8

ppc 5

677 1 15γg 2 37:5γ 2g 145:038

(3.15)

(3.16)

For condensate gas, T pc 5

ppc 5

187 1 330γg 2 71:5γ2g 1:8 706 2 51:7γg 2 11:1γ2g 145:038

(3.17)

(3.18)

Figure 3
2 Pseudo-critical properties of natural gas. Modified from Tarek, A., 2019. Reservoir Engineering Handbook, fifth ed. Elsevier.

62

Reserves Estimation for Geopressured Gas Reservoirs

Example 4: Using Eq. (3.15) and Eq. (3.16), as well as the data in Example 3, calculate the pseudo-critical properties of the gas. Solution According to Eq. (3.9), γg 5

Ma 20:23 5 0:6986 5 28:96 28:96

According to Eq. (3.15), Tpc 5

168 1 325 3 0:6986 2 12:5 3 0:69862 5 216:08K 1:8

According to Eq. (3.16), ppc 5

677 1 15 3 0:6986 2 37:5 3 0:69862 5 4:6138 MPa 145:038

According to Eq. (3.11), ppr 5

p 20:68 5 5 4:4822 ppc 4:6138

According to Eq. (3.12), Tpr 5

T 355:56 5 5 1:646 Tpc 216:08

According to Fig. 3
1, Z can be determined as 0.845. According to the equation of state, the gas density is obtained as ρg 5

pMa 20:68 3 20:23 5 5 167:48 kg=m3 ZRT 0:845 3 8:3143 3 1023 3 355:56

3.2.1.1 Correction for nonhydrocarbon components Hydrocarbon gases are classified as sweet gas or sour gas depending on the content of H2S. Both sweet gas and sour gas may contain N2 or CO2, or both. Hydrocarbon gas is a sour gas if the H2S content is higher than 0.02311 g/m3. Concentrations of up to 5% of nonhydrocarbon components will not affect the calculation accuracy. When the content of nonhydrocarbon components in the gas mixture is higher, the error in the deviation factor calculation may be as high as 10%. 1. The Wichert
Aziz correction method

Physical properties of natural gas and formation water

63

When natural gas contains H2S and/or CO2, the calculation method of Wichert and Aziz (1972) is 0

Tpc 5 Tpc 2 ε

(3.19)

0

p0pc

5

ppc Tpc

(3.20)

Tpc 1 Bð1 2 BÞε 0

where p0pc : corrected pseudo-critical pressure, MPa, Tpc : corrected pseudo-critical temperature, K, B: mole fraction of H2S in the gas mixture, ε: pseudo-critical temperature adjustment factor, which is expressed as     120 A0:9 2 A1:6 1 15 B0:5 2 B4:0 ε5 (3.21) 1:8 A 5 yH2 S 1 yCO2 Example 5: Sour gas has a gas gravity of 0.70 and contains CO2 (5%) and H2S (10%). Calculate the gas density at pressure 24.13 MPa and temperature 344.44K. Solution According to Eq. (3.15), T pc 5

168 1 325 3 0:7 2 12:5 3 0:72 5 216:32K 1:8

According to Eq. (3.16), 677 1 15 3 0:7 2 37:5 3 0:72 5 4:6134 MPa 145:038

ppc 5

According to Eq. (3.21),     120 3 0:150:9 2 0:151:6 1 15 3 0:100:5 2 0:104:0 ε5 5 11:52K 1:8 According to Eq. (3.19), 0

Tpc 5 Tpc 2 ε 5 216:32 2 11:52 5 204:80K

64

Reserves Estimation for Geopressured Gas Reservoirs

According to Eq. (3.20), 0

p0pc 5

ppc Tpc Tpc 1 Bð1 2 BÞε

5

4:613 3 204:80 5 4:347 MPa 216:32 1 0:10 3 ð1 2 0:10Þ 3 11:52

According to Eq. (3.11), ppr 5

24:13 5 5:55 4:347

According to Eq. (3.12), Tpr 5

344:44 5 1:68 204:80

According to Fig. 3
1, Z can be determined as 0.890. According to Eq. (3.9), we have Ma 5 28:96 3 0:70 5 20:27 The gas density can be calculated as, ρg 5

pMa 24:13 3 20:27 5 191:92 kg=m3 5 ZRT 0:890 3 8:3143 3 1023 3 344:44

2. The Carr-Kobayashi-Burrows correction method As per the Carr
Kobayashi
Burrows (1954) method, if the gas gravity is known, Eq. (3.15) and Eq. (3.16) are used to calculate the pseudocritical temperature and pseudo-critical pressure, respectively. The estimated pseudo-critical properties are corrected using the following equations. 0

130yH2 S 2 80yCO2 2 250yN2 1:8

(3.22)

600yH2 S 1 440yCO2 2 170yN2 145:038

(3.23)

Tpc 5 Tpc 1 p0pc 5 ppc 1

3.2.1.2 Correction for high-molecular-weight gases For high-molecular-weight gases (γ g . 0.75) containing more components with higher molecular weight than heptane, the method proposed by Stewart et al. (1959) is used for correction, which can reduce the calculation error. The correction steps are as follows.

65

Physical properties of natural gas and formation water

Step 1: Calculate Stewart-related parameters. " "
#
0:5 #2 1 X T ci 2 X T ci J5 0:01241yi 0:1114yi 1 3 i 3 i pci pci

K5

X i

! T ci 0:14946yi pffiffiffiffiffi pci

(3.24)

(3.25)

where pci: critical pressure of component i, MPa, Tci: critical temperature of component i, K, Step 2: Calculate the correction parameters. " 

0:5 #2 1 Tc 2 Tc FJ 5 0:01241y 0:1114y 1 (3.26) 3 pc C71 3 pc C71

EJ 5 0:6081FJ 1 1:1325FJ 2 2 14:004FJ yC71 1 64:434FJ y2C71 ! Tc Ek 5 0:14946 pffiffiffiffi pc



(3.27)

0:3129yC71 2 4:8156y2C71 1 27:3751y3C71



C71

(3.28) Step 3: Correct the parameters obtained in Step 1. 0

J 5 J 2 EJ

(3.29)

0

K 5 K 2 Ek

(3.30)

Step 4: Calculate the pseudo-critical parameters.  0 2 K 0 Tpc 5 1:8J 0

(3.31)

0

p0pc

5 0:01241

Tpc J0

Step 5: Get the deviation factor according to Fig. 3
1.

(3.32)

66

Reserves Estimation for Geopressured Gas Reservoirs

Example 6: For a natural gas, the proportions of C1, C2, C3, n-C4, n-C5, C6, and C71 are 0.83, 0.06, 0.03, 0.02, 0.02, 0.01, and 0.03, respectively. The C71 fraction is characterized by a molecular weight and gas gravity are dimensionless. Calculate the gas density at pressure 13.79 MPa and temperature 338.89K using the Stewart method. Solution Riazi and Daubert (1987) established an equation for predicting the physical properties of a mixture of pure components and undefined hydrocarbons. The equation takes the molecular weight M and gas gravity γ of undefined components as parameters expressed as: θ 5 aM b γc exp ðdM 1 eγÞ where θ: any physical property, a-e: constant parameters of each property shown in the table below, Θ

Tc, K pc, MPa

a

B

302.4444 311.6632

c

0.2998 2 0.8063

d

1.0555 1.6015

e 24

2 1.3478 3 10 2 1.8078 3 1023

2 0.61641 2 0.3084

γ: gas gravity of the component, M : molecular weight, Tc : critical temperature, K, pc : critical pressure, MPa. According to the equation proposed by Riazi and Daubert (1987), we have ðT c ÞC71 5 659:53K   pc C71 5 2:152 MPa First, prepare the data table of relevant parameters, which is shown as follows. Calculation data table of Example 6. Component

yi

Mi

Tci, K

pci, MPa

yiMi

yi(Tci/pci)

yi(Tci/pci)0.5

yiTci/pci0.5

C1 C2 C3 n-C4 n-C5 C6 C71 Sum

0.83 0.06 0.03 0.02 0.02 0.01 0.03

16.0 30.1 44.1 58.1 72.2 84.0 161.0

190.74 305.51 370.03 425.34 469.78 512.78 659.53

4.595 4.871 4.250 3.796 3.369 3.330 2.152

13.28 1.81 1.32 1.16 1.44 0.84 4.83 24.69

34.456 3.763 2.612 2.241 2.789 1.540 9.194 56.594

5.348 0.475 0.280 0.212 0.236 0.124 0.525 7.200

73.857 8.305 5.385 4.366 5.119 2.810 13.487 113.329

Physical properties of natural gas and formation water

According to Eqs. (3.24) and (3.25), we have 1 2 J 5 ð0:01241 3 56:594Þ 1 ð0:1114 3 7:200Þ2 5 0:663 3 3 K 5 0:14946 3 113:329 5 16:938 According to Eqs. (3.26), (3.27), and (3.28), we have 1 2 FJ 5 ð0:01241 3 9:194Þ 1 ð0:1114 3 0:525Þ2 5 0:0403 3 3 EJ 5 0:6081 3 0:0403 1 1:1325 3 0:04032 2 14:004 3 0:0403 3 0:03 1 64:434 3 0:0403 3 0:032 5 0:012

 ffiffiffiffiffiffiffiffi 0:3129 3 0:03 2 4:8156 3 0:032 Ek 5 0:14946 3 p659:53 2:152 1 27:3751 3 0:033 Þ 5 0:389 According to Eqs. (3.29) and (3.30), we have 0

J 5 0:663 2 0:012 5 0:651 0

K 5 K 2 Ek 5 16:938 2 0:389 5 16:549 According to Eqs. (3.31) and (3.32), we have 0

Tpc 5

16:5492 5 233:72K 1:8 3 0:651

p0pc 5 0:01241 3

233:72 5 4:455 MPa 0:651

According to Eqs. (3.11) and (3.12), we have ppr 5

13:79 5 3:09 4:455

Tpr 5

338:89 5 1:45 233:72

67

68

Reserves Estimation for Geopressured Gas Reservoirs

According to Fig. 3
1, Z is determined as 0.745. The gas density can be calculated as ρg 5

pMa 13:79 3 24:69 5 162:20 kg=m3 5 ZRT 0:745 3 8:3143 3 1023 3 338:89

3.2.1.3 Direct calculation of deviation factor In the 1970s, Hall and Yarborough (1973), Dranchuk, Abu and Kassem (1975), and Dranchuk, Purvis and Robinson (1973) used different empirical formulas to mathematically describe the Standing
Katz deviation factor chart (Fig. 3
1) and established corresponding correlations. 1. The Hall
Yarborough method Based on the Starling
Carnahan equation of state, Hall and Yarborough (1973) digitized the Standing
Katz deviation factor chart by the following expression.
 0:06125ppr t 21:2ð12tÞ2 Z5 (3.33) e Y where t: reciprocal of the pseudo-reduced temperature, 1/Tpr, Y: reduced density, which can be obtained by the following equation. F ð Y Þ 5 X1 1

Y 1 Y2 1 Y3 2 Y4 2 X 2 Y 2 1 X 3 Y X4 ð12Y Þ3

(3.34)

where X1 5 2 0:06125ppr te21:2ð12tÞ ; X2 5 14:76t 2 9:76t 2 1 4:58t 3 ; 2 3 X3 5 90:7t 2 242:2t 1 42:4t ; X 4 5 2:18 1 2:82t° Eq. (3.34) is a nonlinear equation and can be solved using the Newton
Raphson iterative method, according to the steps as follows. Step 1: Assign an initial value to the unknown parameter Yk by the following equation. k is the iteration counter. 2

Y k 5 0:06125ppr te21:2ð12t Þ

2

Step 2: Substitute the initial value into Eq. (3.34) to calculate F ðY Þ. Step 3: According to Eq. (3.35), calculate Yk11, which is obtained as Y k11 5 Y k 2 F 0 ðY Þ 5

FðY k Þ F 0 ðY k Þ

1 1 4Y 1 4Y 2 2 4Y 3 1 Y 4 2 2X2 Y 1 X3 X4 Y X4 21 ð12Y Þ4

(3.35) (3.36)

Physical properties of natural gas and formation water

69

Step 4: Iterate repeatedly until the error condition is met. Step 5: Substitute the Y value into Eq. (3.33) to calculate the deviation factor. This method is suitable for Tpr . 1. 2. The Dranchuk
Abu-Kassem (DAK) method Dranchuk and Abu-Kassem (1975) introduced the reduced gas density to calculate the gas deviation factor. The reduced gas density is defined as the ratio of the gas density at a specific pressure and temperature to the gas density at the critical pressure and temperature. It is expressed as ρr 5

ρ p=ZT  5 ρc p=ZT c

The critical gas deviation factor Zc is approximately 0.27, and the reduced density expression can be simplified as ρr 5

0:27ppr ZTpr

(3.37)

The equation used to calculate the reduced gas density is     R2 2 1 R3 ρ2r 2 R4 ρ5r 1 R5 1 1 A11 ρ2r ρ2r e2A11 ρr 1 1 5 0 f ρr 5 R1 ρr 2 ρr (3.38) where ppr A2 A3 A4 A5 1 3 1 4 1 5 R2 5 0:27 Tpr Tpr Tpr Tpr Tpr

1 TA28 R4 5 A9 TApr7 1 TA28 R5 5 AT103

R1 5 A1 1 R3 5 A6 1

A7 Tpr

pr

pr

pr

After performing nonlinear regression fitting on the 1500 data points in the Standing
Katz chart, we get A1 5 0.3265, A2 5 21.0700, A3 5 20.5339, A4 5 20.01569, A5 5 20.05165, A6 5 0.5475, A7 5 20.7361, A8 5 0.1844, A9 5 0.1056, A10 5 0.6134, A11 5 0.7210. Eq. (3.38) is a nonlinear equation, which can be solved using the Newton
Raphson iterative method according to the steps as follows. Step 1: Assign an initial value to the unknown parameter ρkr by the following equation. k is the iteration counter.

70

Reserves Estimation for Geopressured Gas Reservoirs

ρr 5

0:27ppr ZTpr

2: Substitute the initial value into Eq. (3.38) to calculate  Step  f ρr . Step 3: Calculate ρrk11 , which is determined as   f ρkr k11 k ρr 5 ρr 2 0  k  (3.39) f ρr    R2 2  f 0 ρr 5 R1 1 2 1 2R3 ρr 2 5R4 ρ4r 1 2R5 ρr e2A11 ρr 1 1 2A11 ρ3r ρr   2 A11 ρ2r 1 1 A11 ρ2r Step 4: Iterate repeatedly until the error condition is met. Step 5: Substitute the ρr value into Eq. (3.37) to calculate the deviation factor. This method is applicable under conditions that 1:0 , T pr , 3:0 and 0:2 , ppr , 15. 3. The Dranchuk
Purvis
Robinson method (DPR) Dranchuk et al. (1973) established the following reduced density equation based on the Benedict
Webb
Rubin-type equation of state    T5 2 1 1 T 1 ρr 1 T 2 ρ2r 1 T 3 ρ5r 1 T 4 ρ2r 1 1 A8 ρ2r e2A8 ρr 2 5 0 (3.40) ρr where T1 5 A1 1

ppr A2 A3 A5 A5 A6 A7 1 3 T2 5 A4 1 T3 5 T4 5 3 T5 5 0:27 Tpr Tpr Tpr Tpr Tpr Tpr

A1 5 0.31506237; A2 5 21.0467099; A3 5 20.57832720; A4 5 0.53530771; A5 5 20.61232032; A6 5 20.10488813; A7 5 0.68157001; A8 5 0.68446549 The solution procedure of Eq. (3.40) is similar to that of Dranchuk
Abu-Kassem. The method is applicable under conditions that 1:05 , T pr , 3:0 and 0:2 , ppr , 3:0.

Physical properties of natural gas and formation water

71

3.2.1.4 Comparison of methods The Standing
Katz natural gas deviation factor chart has been widely used since the last 60 years, when it was first introduced. Two main types of empirical formulas are used to calculate the deviation factor. The first is to try to fit directly by the analytical curve method, and the second is to calculate by using the equation of state. Khaled (1995) compared eight methods with their recommended applications, as shown in Table 3
2. The data points of the Standing
Katz experiment may be in error within the pseudo-reduced temperature range of 1.05
1.1 and the pseudo-reduced pressure range of 0.2
5.0. Research by scholars in China has also shown that when Tpr # 1.10, the calculation accuracy of ppr in the range of 1.24
5.78 is poor; the smaller the Tpr, the greater the calculation error (Yang et al., 2017).

3.2.2 Compressibility factors for natural gases The isothermal compressibility factor of natural gas refers to the change in volume per unit volume for a unit change in pressure, which is expressed as
 1 @V Cg 5 2 (3.41) V @p T where Cg—isothermal compressibility factor, 1/MPa. According to Eq. (3.10), we have V5

ZnRT p

Differentiating the above equation concerning pressure, we have 
 
 @V 1 @Z Z 5 nRT 2 2 @p T p @p p Substituting the above equation into Eq. (3.41), we have
 1 1 @Z Cg 5 2 p Z @p T

(3.42)

For an ideal gas, Z 5 1, we have Cg 5 1=p

(3.43)

72

Reserves Estimation for Geopressured Gas Reservoirs

Table 3
2 Recommended applications of gas deviation factor calculation methods. Method

Pseudo-reduced temperature

Pseudo-reduced pressure

Dranchuk and Abu-Kassem (1975) Dranchuk et al. (1973)

[1.05, [1.10, [1.05, [1.10, [1.05, [1.10, [1.05, [1.10, [1.15, 1.05 [1.60, [1.05, 1.60 [1.70,

[5.0, 15] [0.20, 15] [5.0, 15] [0.20, 15] [5.0, 15] [0.55,15] [5.0, 15] [0.55, 15] [0.20, 15] [1.0, 1.5] Out of range [0.20, 5.0] [3.0, 6.0] (0.20, 13] [0.20, 15.0]

Dranchuk et al. (1973) Hall and Yarborough (1973) Brill and Beggs (1991) Gopal (1977) Papay (1968) Leung (1965)

1.1] 3.0] 1.10] 3.0] 1.10] 3.0] 1.10] 3.0] 2.40] 3.0] 1.5] 3.0]

Source: Modified from Khaled, A.A.-el F., 1995. Analysis shows magnitude of Z-factor error. Oil Gas J 93 (48), 65
69.

The order of magnitude of the compressibility factor can be determined by Eq. (3.43). Given the pseudo-reduced pressure and temperature, Eq. (3.42) is rewritten as
 1 1 @Z Cpr 5 2 (3.44) ppr Z @ppr Tpr Cpr 5 Cg ppc

(3.45)

pseudo-reduced compressibility factor, dimensionWhere,  Cpr: isothermal  less, @Z=@ppr Tpr can be calculated from the slope of the Tpr isotherm on the Standing
Katz chart. Example 7: A hydrocarbon gas mixture has a gravity of 0.72. Calculate the isothermal compressibility factor under pressure 13.79 MPa and temperature 333.33K. Solution Assuming that the hydrocarbon mixture is an ideal gas, according to Eq. (3.43), we have

Physical properties of natural gas and formation water

Cg 5

73

1 1 5 5 7:252 3 1022 MPa21 p 13:79

Assuming that the hydrocarbon mixture is a real gas, according to Eq. (3.15), we have T pc 5

168 1 325 3 0:72 2 12:5 3 0:722 5 219:73K 1:8

According to Eq. (3.16), ppc 5

677 1 15 3 0:72 2 37:5 3 0:722 5 4:608 MPa 145:038

According to Eq. (3.11), ppr 5

p 13:79 5 2:99 5 ppc 4:608

According to Eq. (3.12), T pr 5

T 333:33 5 5 1:52 T pc 219:73

According to Fig. 3
1, z 5 0.78. Then, ! @Z 5 2 0:022 @ppr T pr 51:52

According to Eq. (3.44), C pr 5

1 1 2 ð 20:022Þ 5 0:3627 2:99 0:78

According to Eq. (3.45), Cg 5

0:3627 5 7:870 3 1022 MPa21 4:608

Trube (1957) plotted the charts of the isothermal pseudo-reduced gas compressibility factor versus the pseudo-reduced pressure and temperature, as shown in Fig. 3
3 and Fig. 3
4, from which the isothermal gas compressibility factor can be obtained. Mattar et al. (1975) proposed an analytical method to calculate the isothermal gas compressibility. Eq. (3.37) is differentiated with respect to ppr to give

74

Reserves Estimation for Geopressured Gas Reservoirs

Figure 3
3 Pseudo-reduced gas compressibility chart (1). Modified from Tarek, A., 2019. Reservoir Engineering Handbook, fifth ed. Elsevier.

Figure 3
4 Pseudo-reduced gas compressibility chart (2). Modified from Tarek, A., 2019. Reservoir Engineering Handbook, fifth ed. Elsevier.

 "  # @Z=@ρr T pr @Z 0:27   5 @ppr ZT pr 1 1 ρZr @Z=@ρr T pr Substituting Eq. (3.46) into Eq. (3.44),  "  # @Z=@ρ r 1 0:27 T pr   2 2 C pr 5 ppr Z T pr 1 1 ρZr @Z=@ρr T pr

(3.46)

(3.47)

75

Physical properties of natural gas and formation water

Differentiating Eq. (3.40),
   @Z 2 5 T1 1 2T2 ρr 1 5T3 ρ4r 1 2T4 ρr 1 1 A8 ρ2r 2 A28 ρ4r e2A8 ρr @ρr Tpr (3.48) where the coefficients are the same as the definition of Eq. (3.40).

3.2.3 Gas formation volume factor The volume of natural gas is determined under surface standard conditions, but gas reservoir engineering calculations often require the volume under formation pressure and temperature. The conversion factor between surface and subsurface volumes is referred to as the gas formation volume factor. The gas formation volume factor is defined as the ratio of the volume of gas under the formation pressure and temperature conditions to the volume of gas under the surface standard conditions (0.101325 MPa, 293.15K). Bg 5 Vr =Vsc

(3.49)

where Bg: gas formation volume factor, m3/m3, Vr: the volume of gas under reservoir conditions, m3, Vsc: the volume of gas under standard conditions, m3. According to the real gas equation of state (10), Bg 5

psc ZT ZT 5 3:4564 3 1024 Tsc p p

(3.50)

where Tsc: temperature under standard conditions, 293.15K, psc: pressure under standard conditions, 0.101325 MPa.

3.2.4 Natural gas viscosity Natural gas viscosity is one of the essential parameters in reservoir engineering. It measures internal friction resistance to fluid flows and is a function of pressure, temperature, and composition. The gas viscosity increases with pressure, either in low- or high-pressure conditions. Generally, the gas viscosity is not commonly measured through experiments because it can be estimated precisely from empirical formulas. 3.2.4.1 The Carr
Kobayashi
Burrows method The method proposed by Carr et al. (1954) is realized in the steps described as follows.

76

Reserves Estimation for Geopressured Gas Reservoirs

Figure 3
5 Natural gas viscosity chart. (A) gas viscosity correlation; (B) nonhydrocarbon (N2, CO2, H2S) correction. Modified from Tarek, A., 2019. Reservoir Engineering Handbook, fifth ed. Elsevier.

Step 1: Calculate the pseudo-critical pressure, temperature, and apparent molecular weight based on the gas gravity or composition. Corrections to these pseudo-critical parameters should be made if the content of nonhydrocarbon gases (CO2, N2, and H2S) is greater than 5%. Step 2: Obtain the viscosity of natural gas at atmospheric pressure according to Fig. 3
5 at a given temperature and perform the nonhydrocarbon correction. Nonhydrocarbon components tend to increase the viscosity of the gas phase. The influence of nonhydrocarbon components on the viscosity of natural gas can be expressed by Eq. (3.51),   μ1 5 μ1 uncorrected 1 ðΔμÞN2 1 ðΔμÞCO2 1 ðΔμÞH2 S (3.51) where μ1 : gas viscosity after correction under the conditions of formation temperature and atmospheric pressure, mPa  s, ðΔμÞN2 ; ðΔμÞCO2 ; ðΔμÞH2 S : viscosity corrections in the presence of different nonhydrocarbon gases,  mPa  s, μ1 uncorrected : uncorrected gas viscosity, mPa  s. Step 3: Calculate the pseudo-reduced pressure and temperature. Step 4: According to the pseudo-reduced pressure and temperature, obtain μg =μ1 according to Fig. 3
6. The term μg represents the gas viscosity under the specified conditions. Example 8: A gas reservoir has a formation pressure of 13.79 MPa, a temperature of 322.22K, and a gas gravity of 0.72. Calculate the gas viscosity.

Physical properties of natural gas and formation water

77

Figure 3
6 Natural gas viscosity ratio chart. Modified from Tarek, A., 2019. Reservoir Engineering Handbook, fifth ed. Elsevier.

Solution According to Eq. (3.9), we get M a 5 28:96 3 0:72 5 20:85. According to Fig. 3
5, we obtain the value of μ1 5 0.0113 mPa  s. According to Eq. (3.15), T pc 5

168 1 325 3 0:72 2 12:5 3 0:722 5 219:73K 1:8

According to Eq. (3.16), ppc 5

677 1 15 3 0:72 2 37:5 3 0:722 5 4:608 MPa 145:038

According to Eq. (3.11), ppr 5

p 13:79 5 5 2:99 ppc 4:608

According to Eq. (3.12), Tpr 5

T 322:22 5 1:47 5 Tpc 219:73

According to Fig. 3
6, μg =μ1 5 1:5. The viscosity at a pressure of 13.79 MPa and a temperature of 322.22K is, μg 5 1:5 3 0:0113 5 0:01695 mPa  s

78

Reserves Estimation for Geopressured Gas Reservoirs

3.2.4.2 The Standing method Standing (1977) proposed a mathematical expression for calculating,   μ1 5 μ1 uncorrected 1 ðΔμÞN2 1 ðΔμÞCO2 1 ðΔμÞH2 S

  μ1 uncorrected 5 1:709 3 1025 2 2:062 3 1026 γg ð1:8T 2 460Þ 1 8:118 3 1023 2 6:15 3 1023 lgγ g

ðΔμÞCO2 5 yCO2 9:08 3 1023 lgγg 1 6:24 3 1023

(3.53)



ðΔμÞN2 5 yN2 8:48 3 1023 lgγ g 1 9:59 3 1023

(3.54)



ðΔμÞH2 S 5 yH2 S 8:49 3 1023 lgγ g 1 3:73 3 1023

(3.55)

where T: reservoir temperature, K. 3.2.4.3 The Dempsey method Dempsey (1965) proposed a method for calculating μg =μ1 , " !#

μg ln Tpr 5 a0 1a1 ppr 1 a2 p2pr 1 a3 p3pr 1 Tpr a4 1 a5 ppr 1 a6 p2pr 1 a7 p3pr μ1



1 Tpr2 a8 1 a9 ppr 1 a10 p2pr 1 a11 p3pr 1 Tpr3 a12 1 a13 ppr 1 a14 p2pr 1 a15 p3pr (3.56) where a0 5 22.46211820 a1 5 2.970547414; a2 5 22.86264054 3 1021; a3 5 8.05420522 3 1023; a4 5 2.80860949; a5 5 23.49803305; a6 5 3.60373020 3 1021; a7 5 21.044324 3 1022;

a8 5 27.93385648 3 1021 a9 5 1.39643306; a10 5 21.49144925 3 1021; a11 5 4.41015512 3 1023; a12 5 8.39387178 3 1022; a13 5 21.86408848 3 1021; a14 5 2.03367881 3 1022; and a15 5 26.09579263 3 1024.

3.2.4.4 The Lee
Gonzalez
Eakin method Lee, Gonzalez, and Eakin (1966) presented an empirical relationship for calculating the viscosity of natural gases.

Physical properties of natural gas and formation water

79

  ρg Y μg 5 10 K exp X 1000

(3.57)

ð9:4 1 0:02 Ma Þð1:8T Þ1:5 209 1 19 Ma 1 1:8T

(3.58)

24

K0 5

0

X 5 3:5 1

986 1 0:01 Ma 1:8T

Y 5 2:4 2 0:2X

(3.59) (3.60)

The proposed correlation can predict viscosity values with a standard deviation of 2.7% and a maximum deviation of 8.99%. The correlation becomes less precise for gases with higher gas gravity. The method cannot be used for sour gases. Example 9: A gas reservoir has a formation pressure of 13.79 MPa, a temperature of 322.22K, and a gas gravity of 0.72. Calculate the gas viscosity by using the Lee
Gonzalez
Eakin method. Solution According to Eq. (3.9), Ma 5 28:96 3 0:72 5 20:85. The gas density can be calculated by ρg 5

13:79 3 20:85 5 137:60 kg=m3 8:3143 3 10-3 3 322:22 3 0:78

According to Eqs. (3.58), (3.59), and (3.60), K0 5

ð9:4 1 0:02 3 20:85Þ 3 ð1:8 3 322:22Þ1:5 5 115:70 209 1 19 3 20:85 1 1:8 3 322:22

X 5 3:5 1

986 1 0:01 3 20:85 5 5:4085 1:8 3 322:22

Y 5 2:4 2 0:2 3 5:4085 5 1:318

80

Reserves Estimation for Geopressured Gas Reservoirs

Figure 3
7 The Standing and Katz deviation factor chart. Modified from Katz, D.L., 1959. Handbook of Natural Gas Engineering. McGraw-Hill Publishing Co., New York City.

According to Eq. (3.57), μg 5 10-4 K 0 exp

" #
   ρg Y 137:61 1:318 24 5 10 3 115:70 3 exp 5:408 3 X 1000 1000

5 0:0172 mPaUs

3.3 Deviation factor of ultra-high-pressure gas In reserve estimation, production calculation, production performance analysis, and vertical flowing calculation of gas reservoirs, the state equation used generally contains deviation factors. However, the currently widely used Standing
Katz chart (Fig. 3
1) needs to meet a condition for its application, that is, 0.2 # ppr # 15 and 1.05 # Tpr # 3.0, which is not suitable for the determination of the deviation factor of ultra-high-pressure gas. Katz (1959) gave a deviation factor chart for gases with a pressure range of 68.9
137.9 MPa in his work, as shown in Fig. 3
7.

3.3.1 DPR or DAK extrapolation method As shown in Figs. 3
1 and 3
7, when ppr $ 7, the gas deviation factor curve at a given Tpr is a linear function of the pseudo-reduced pressure. The straight line in Fig. 3
1 is extrapolated to ppr 5 30 according to the original law, and the extrapolated value is in good agreement with the

Physical properties of natural gas and formation water

81

results of DAK or DPR (Yang et al., 2017). Fig. 3
1 to Fig. 3
7 show the deviation factor chart in Fig. 3
8. Example 10: A gas well has a composition shown in Table 3
3. The original formation pressure is 110 MPa, and the deviation factor data measured by the PVT is shown in Table 3
4. Calculate the natural gas deviation factors at 404.55K, 373.15K, 343.15K, and 313.15K using the DPR extrapolation method and compare the results. Solution First, calculate the pseudo-critical physical property parameters through gas composition. Composition

Carbon dioxide Nitrogen Methane Ethane Propane Isobutane n-butane Isopentane n-pentane Hexane Heptane Sum

Mole percentage

Critical pressure, MPa

Critical temperature, K

Weighted product

yi

pci

Tci

yipci

yiTci

0.38

7.3843

304.21

2.8060

115.60

0.91 87.14 7.29 2.82 0.58 0.58 0.1 0.06 0.11 0.03

3.3998 4.5947 4.8711 4.2472 3.6397 3.7962 3.3812 3.3688 3.0123 2.7358

126.2 190.56 305.33 369.85 407.85 425.16 460.43 469.71 507.37 540.21

3.0938 400.3822 35.5103 11.9771 2.1110 2.2018 0.3381 0.2021 0.3314 0.0821 4.5904

114.84 16605.40 2225.86 1042.98 236.55 246.59 46.04 28.18 55.81 16.21 207.34

According to Eq. (3.13) and Eq. (3.14), X yi pci 5 4:5904 ppc 5 i51

Tpc 5

X

yi Tci 5 207:34

i51

The calculation results of the DPR extrapolation method are shown in Fig. 3
9, which are entirely consistent with the experimental data. In this example case, the four temperature values correspond to the

82

Reserves Estimation for Geopressured Gas Reservoirs

Figure 3
8 The complete Standing
Katz deviation factor chart. Table 3
3 Gas composition of Well DN201. Carbon dioxide

0.38

Nitrogen Methane Ethane Propane Isobutane nIsopentane nHexane Heptane Butane pentane

0.91

87.14

7.29

2.82

0.58

0.58

0.1

0.06

0.11

0.03

quasicritical temperature interval of 1.40 # Tpr # 2.0. Therefore, the DPR or DAK extrapolation method can be used to calculate the deviation factor when 15 # ppr # 30 and 1.40 # Tpr # 3.0. However, for the interval of 1.05 # Tpr # 1.40, the method’s reliability needs to be further verified by experimental data.

3.3.2 LXF-RMP (Li et al., 2010) fitting method Li (2001) established an analytical calculation model for the deviation factor of high-pressure natural gas based on Figs. 3
1 and 3
7. The model is composed of two parts, medium-to-high-pressure part (8 # ppr # 15) and ultra-high-pressure part (15 # ppr # 30). In 2010, Li et al. reoptimized the curve matching method based on the original work and matched natural gas’s deviation factor in the entire temperature and pressure ranges (six intervals) and established a high-precision analytical calculation model of natural gas deviation factor. In the high-pressure area (9 # ppr # 15 and 1.05 # Tpr # 3.0), the deviation factor has a linear relationship with the pseudo-reduced pressure

Table 3
4 Measured data of deviation factor for Well DN201. 404.55K

373.15K

343.15K

313.15K

p，MPa

Z

p，MPa

Z

p，MPa

Z

p，MPa

Z

110 108 105.92 104 102 100 98 96 94 92 90 88 86 84 82 80 78 76 74 72 70 67 64 60 57

1.8639 1.8410 1.8175 1.7956 1.7728 1.7502 1.7275 1.7059 1.6826 1.6606 1.6379 1.6154 1.5921 1.5687 1.5453 1.5240 1.5012 1.4794 1.4568 1.4338 1.4116 1.3740 1.3372 1.2911 1.2584

110 108 105.92 104 102 100 98 96 93 90 87 84 81 78 75 72 69 66 63 60 57 54 51 48 45.39

1.9451 1.9195 1.8924 1.8689 1.8458 1.8206 1.7958 1.7710 1.7314 1.6917 1.6549 1.6167 1.5796 1.5430 1.5061 1.4682 1.4299 1.3915 1.3523 1.3138 1.2718 1.2322 1.1964 1.1596 1.1288

110 108 105.92 103 100 97 94 91 88 85 82 79 76 73 70 67 64 61 58 55 52 49 46.55 43 42

2.0129 1.9887 1.9612 1.9234 1.8804 1.8390 1.7994 1.7596 1.7186 1.6757 1.6355 1.5941 1.5526 1.5107 1.4694 1.4263 1.3837 1.3428 1.2986 1.2585 1.2163 1.1772 1.1446 1.0915 1.0793

110 108 105.92 103 100 97 94 91 88 85 82 79 76 73 70 67 64 61 58 55 52 49 47.05 46 43

2.1036 2.0776 2.0480 2.0050 1.9609 1.9179 1.8726 1.8282 1.7841 1.7386 1.6936 1.6482 1.6027 1.5569 1.5108 1.4646 1.4180 1.3716 1.3256 1.2814 1.2359 1.1922 1.1611 1.1413 1.0949 (Continued)

Table 3
4 (Continued) 404.55K

373.15K

343.15K

313.15K

p，MPa

Z

p，MPa

Z

p，MPa

Z

p，MPa

Z

53 50 47 44 43.46 43 42 41 40 39 38 37 36 35 33 31 29 26 23 20 17

1.2126 1.1782 1.1438 1.1102 1.1055 1.1016 1.0919 1.0839 1.0748 1.0673 1.0597 1.0522 1.0446 1.0363 1.0202 1.0055 0.9919 0.9754 0.9597 0.9471 0.9401

43 42 41 39 37 35 32 29 26 23 20 17 14

1.1007 1.0877 1.0744 1.0515 1.0305 1.0095 0.9833 0.9605 0.9369 0.9175 0.9064 0.9005 0.9043

40 38 35 32 29 26 23 20 17 14

1.0476 1.0206 0.9807 0.9448 0.9120 0.8825 0.8629 0.8465 0.8415 0.8470

40 38 36 34 32 29 26 23 20 16 12

1.0455 1.0135 0.9824 0.9496 0.9227 0.8793 0.8412 0.8057 0.7783 0.7574 0.7598

85

Physical properties of natural gas and formation water

Figure 3
9 Comparison of measured deviation factors and DPR calculation results for Well DN201.

and a fourth power relationship with the pseudo-reduced temperature, Z 5 x1 ppr 1 x2

(3.61)

where x1 5 2 0:002225T 4pr 1 0:0108T 3pr 1 0:015225T 2pr 2 0:153225T pr 1 0:241575

(3.62) x2 5 0:1045T 4pr 2 0:8602T 3pr 1 2:3695T 2pr 2 2:1065T pr 1 0:6299 (3.63) In the ultra-high-pressure area (15 # ppr # 30 and 1.05 # Tpr # 3.0), the deviation factor also has a linear relationship with the pseudo-reduced pressure and a fourth power relationship with the pseudo-reduced temperature, Z 5 x1 ppr 1 x2

(3.64)

where x1 5 0:0155T 4pr 2 0:145836Tpr3 1 0:5153091Tpr2 2 0:8322091T pr 1 0:5711 (3.65) x2 5 2 0:1416T 4pr 1 1:34712T 3pr 2 4:77535T 2pr 1 7:72285T pr 2 4:2068 (3.66)

The comparison between the calculation results of the LXF-RMP model and the DAK method is shown in Table 3
5. In the ultra-high-

Table 3
5 Comparison between the calculation results of the LXF-RMP model and the DAK method. Pseudo-reduced pressure

30 28 26 24 22 20 18 16 14 12

Deviation factors from the chart under different pseudo-reduced temperatures

Relative error of deviation factor calculated by the DAK model, %

Relative error of deviation factor calculated using the LXF-RMP model, %

1.4

2.0

2.6

1.4

2.0

2.6

1.4

2.0

2.6

2.660 2.510 2.359 2.207 2.055 1.905 1.752 1.600 1.455 1.300

2.123 2.027 1.93 1.83 1.742 1.635 1.54 1.44 1.344 1.246

1.876 1.805 1.735 1.662 1.59 1.52 1.449 1.377 1.307 1.238

22.50 20.67 18.19 26.86 29.71 26.69 20.25 28.86 15.63 18.31

1.44 1.30 1.15 1.12 0.36 0.66 0.19 0.04 0.37 0.51

2.44 2.23 1.92 1.77 1.53 1.14 0.81 0.56 0.23 0.07

0.017 0.049 0.042 0.011 0.072 0.037 0.168 0.261 0.201 0.149

0.130 0.015 0.060 0.021 0.578 0.094 0.324 0.240 0.234 0.096

0.011 0.060 0.172 0.113 0.111 0.240 0.314 0.322 0.007 0.032

Physical properties of natural gas and formation water

87

Table 3
6 Results of the LXF-RMP method. Pseudo-reduced pressure

25 24 23 22 21 20 19 18 17 16 15

Pressure

Deviation factor under different temperatures

MPa

405.55K

373.15K

343.15K

313.15K

114.25 109.68 105.11 100.54 95.97 91.40 86.83 82.26 77.69 73.12 68.55

1.9116 1.8602 1.8088 1.7574 1.7061 1.6547 1.6033 1.5519 1.5006 1.4492 1.3978

1.9836 1.9275 1.8713 1.8152 1.7591 1.7030 1.6468 1.5907 1.5346 1.4784 1.4223

2.0725 2.0106 1.9487 1.8868 1.8249 1.7630 1.7011 1.6392 1.5773 1.5154 1.4535

2.1938 2.1241 2.0544 1.9847 1.9150 1.8453 1.7757 1.7060 1.6363 1.5666 1.4969

pressure area (15 # ppr # 30 and 1.05 # Tpr # 3.0), the error of the LXF-RMP result is basically within 0.5%. The error of the DAK method gradually increases with the increase of the pseudo-reduced pressure. When the pseudo-reduced temperature is 1.4, the error of the DAK method is extensive. Example 11: Recalculate the deviation factor of natural gas with data shown in Example 8 using the LXF-RMP method. Solution The quasicritical pressure is 4.57 MPa, and the quasicritical temperature is 211.45K. According to Eqs. (3.64), (3.65), and (3.66), the calculation results are shown in Table 3
6. The comparison with the actual data is shown in Fig. 3
10.

3.4 Properties of formation water When calculating the reserves of ultra-high-pressure gas reservoirs through the material balance method, high-pressure physical property parameters of formation water, such as formation water volume factor, gas solubility in water, and formation water compressibility, are sometimes used. Formation water exists in bound water or free water in the edge and bottom water parts under reservoir conditions. Values of physical parameters

88

Reserves Estimation for Geopressured Gas Reservoirs

Figure 3
10 Comparison of measured deviation factors and LXF-RMP calculation results for Well DN201.

mainly depend on the formation pressure, formation temperature, gas solubility in formation water, and the salinity of formation water. Generally, bound water and free water have the same properties. Downhole sampling and PVT analysis can be used to obtain these parameters accurately, and empirical formulas are usually used for calculation.

3.4.1 Formation water volume factor The volume factors of pure water and water saturated with natural gas at different temperatures and pressures are shown in Fig. 3
11. When the pressure is constant, the volume factor and the temperature increase. When the temperature is constant, the volume factor decreases with pressure. Experiments reflect that the gas solubility in formation water decreases with formation water salinity. The formation water volume factor can be calculated using the following equation (Hewlett-Packard, 1982). Bw 5 A1 1 A2 ð145:038pÞ 1 A3 ð145:038pÞ2

(3.67)

where Ai 5 a1 1a2 ð1:8T 2 460Þ1a3 ð1:8T 2460Þ , p: pressure, MPa, T: temperature, K. 2

89

Physical properties of natural gas and formation water

Figure 3
11 Formation water volume factor.

For pure water, Ai

a1

a2

a3

A1 A2 A3

0.9947 2 4.228 3 1026 1.30 3 10210

5.80 3 1026 1.8376 3 1028 2 1.3855 3 10212

1.02 3 1026 2 6.77 3 10211 4.29 3 10215

For gas-saturated water, Ai

a1

a2

a3

A1 A2 A3

0.9911 2 1.093 3 1026 2 5.0 3 10211

6.35 3 1025 2 3.497 3 1029 6.43 3 10213

8.50 3 1027 4.57 3 10212 2 1.43 3 10215

The brine volume factor (Yang and Liu, 1994) is Bwb 5 Bw ðSC Þ Where

 SC 5 b1 ð145:038pÞ 1 b2 1 b3 ð145:038pÞð1:8T 2 520Þ  1 b4 1 b5 ð145:038pÞð1:8T 2520Þ2 gS 1 1:0

(3.68)

90

Reserves Estimation for Geopressured Gas Reservoirs

b1 5 5:1 3 1028 b4 5 2 3:23 3 1028

b2 5 5:47 3 1028 b5 5 8:5 3 10213

b3 5 2 1:95 3 10210

Where S represents the water salinity, which is expressed by the weight percentage of NaCl, %. The formation water volume factor can also be calculated using the following equation (McCain, 1988),   Bw 5 ð1 1 ΔVwt Þ 1 1 ΔVwp (3.69) Where 2 27 ΔVwt 5 2 0:01 3 1024 ð1:8T 2 460Þ 1 5:50654 3 10  1 1:33391210 2 ð1:8T 2460Þ 213 ΔVwp 5 2 2:25341 3 10 1 1:72834 3 10 ð1:8T 2 460 Þ p 2 3:58922 3 1027 1 1:95301 3 1029 ð1:8T 2 460Þ p

The error between Eq. (3.69) and the actual data is within 2%. This equation does not consider the influence of salinity, but McCain pointed out that when the salinity changes, there is compensation between ΔVwt (slightly increases) and ΔVwt (slightly decreases). This compensation makes the error of the formula within the engineering accuracy range, so there is no need to consider the influence of salinity. Example 12: It is known that the pressure is 20.685 MPa, the temperature is 93.33°C, and the water salinity is 10%. Calculate the volume factor of water. Solution Applying the Hewlett
Packard method, we have A1 5 a1 1a2 ð1:8T 2 460Þ1a3 ð1:8T 2460Þ2 5 1:3067 A2 5 a1 1a2 ð1:8T 2 460Þ1a3 ð1:8T 2460Þ2 5 2 3:2579 3 1026 A3 5 a1 1a2 ð1:8T 2 460Þ1a3 ð1:8T 2460Þ2 5 2:4190 3 10211 Bw 5 A1 1 A2 ð145:038pÞ 1 A3 ð145:038pÞ2 5 1:0267 If the influence of salinity is considered,

91

Physical properties of natural gas and formation water

Figure 3
12 Water viscosity at different salinities and temperatures. (A) water viscosity at different salinities and temperatures; (B) pressure correction factor at different temperatures. Modified from Earlougher, Jr R.C. 1977. Advances in Well Test Analysis. Monograph. Society of Petroleum Engineers of AIME, vol. 5. Millet the Printer, Dallas, TX.

 SC 5 b1 ð145:038pÞ 1 b2 1 b3 ð145:038pÞð1:8T 2 520Þ  1 b4 1 b5 ð145:038pÞð1:8T 2520Þ2 gS 1 1:0 5 1:00213 Bwb 5 Bw ðSCÞ 5 1:0267 3 1:0021 5 1:0289 If the McCain method is applied, 2 27 ΔVwt 5 2 0:01 3 1024 ð1:8T 2 460Þ 1 5:50654 3 10  1 1:33391210 2 ð1:8T 2460Þ 5 0:03859 213 ΔVwp 5 2 2:25341 3 10 1 1:72834 3 10 ð1:8T 2 460 Þ p 2 3:58922 3 1027 1 1:95301 3 1029 ð1:8T 2 460Þ p 5 2 1:5379 3 1025

  Bw 5 ð1 1 ΔVwt Þ 1 1 ΔVwp 5 1:03857

3.4.2 Formation water viscosity The formation water viscosity is mainly sensitive to formation temperature, formation water salinity, and solubility of natural gas and less sensitive to formation pressure, as shown in Fig. 3
12. Meehan (1980) proposed an empirical formula for calculating the formation water viscosity after considering the influence of pressure and salinity,   μwT 5 109:574 2 8:40564ws 1 0:313314ws2 1 8:72213 3 1023 ws3 ð1:8T 2460Þ2D

(3.70)

92

Reserves Estimation for Geopressured Gas Reservoirs

where D 5 1:12166 2 0:0263951ws 1 6:79461 3 1024 ws2 1 5:47119 3 1025 ws3 2 1:55586 3 1026 ws4 μwT : brine viscosity at one atmosphere and reservoir temperature T, 10 23 μm2, T: reservoir temperature, K, ws : salinity of brine (weight percent), %. The equation is suitable for water with the temperature of 311K , T , 477:8K less than 26% salinity. If the influence of pressure on the formation water viscosity is considered, we have,  μw 5 μwT 0:9994 1 4:0295 3 1025 ð145:038pÞ 1 3:1062 3 1029 ð145:038pÞ2 It is applicable for 303K , T , 348K. When p , 68:9 MPa, the error is within 4%. When 68:9 , p , 96:5 MPa, the error is within 7%. When only the influence of reservoir temperature is considered, the empirical equation for formation water viscosity (Brill and Beggs, 1991) can be expressed as 22

μw 5 e1:00321:479 3 10

ð1:8T 2460Þ11:982 3 1025 ð1:8T 2460Þ2

(3.71)

3.4.3 Natural gas solubility in water The natural gas solubility in water is defined as the amount of natural gas dissolved per unit volume of formation water under formation conditions, as shown in Fig. 3
13. The natural gas solubility in water can be calculated using Eq. (3.75) (McCain, 1990), Rsw 5

A1 1 A2 ð145:038pÞ 1 A3 ð145:038pÞ2 5:615

(3.72)

where A1 5 2:12 1 3:45 3 1023 ð1:8T 2 460Þ 2 3:59 3 1025 ð1:8T 2460Þ2 A2 5 0:0107 2 5:26 3 1025 ð1:8T 2 460Þ 1 1:48 3 1027 ð1:8T 2460Þ2 A3 5 2 8:75 3 1027 1 3:90 3 1029 ð1:8T 2 460Þ 2 1:02 3 10211 ð1:8T 2460Þ2

In mineralized water (Yang, 1994), Rsb 5 Rsw ðSC Þ0

(3.73)

Physical properties of natural gas and formation water

93

Figure 3
13 Methane solubility in water. Modified from Dodson, C.R., Standing, M.B., 1944. Pressure-volume-temperature and solubility relations for natural-gas-water mixtures. API 44-173.

where ðSC Þ0 5 1 2 ½0:0753 2 0:000173ð1:8T 2 460ÞS S: water salinity, which is expressed in terms of the weight percentage of NaCl, %, T: reservoir temperature, K, Rsw —natural gas solubility in water, m3/m3. The above method is suitable for the formation of water with 344:4K , T , 513:3K less than 3% salinity. ln Rsw 5 2 3:3544 2 0:002277ð1:8T 2 460Þ 1 6:278 3 1026 ð1:8T 2460Þ2 2 0:004042S  2 1 0:9904 ln ð145:038pÞ 2 0:0311 ln ð145:038pÞ 1 3:204 3 1024 ð1:8T 2 460Þ ln ð145:038pÞ

(3.74) ln Rsw 5 2 1:4053 2 0:002332ð1:8T 2 460Þ 1 6:30 3 1026 ð1:8T 2460Þ2 2 0:004038S 2 7:579 3 1026 ð145:038pÞ 1 0:5013 ln ð145:038pÞ 1 3:235 3 1024 ð1:8T 2 460Þ ln ð145:038pÞ

(3.75) where S: water salinity, g/L, T: reservoir temperature, K, p: reservoir pressure, MPa, Rsw : natural gas solubility in water, m3/m3. The solubility calculated according to Eq. (3.74) is shown in (Fig. 3
14).

94

Reserves Estimation for Geopressured Gas Reservoirs

Figure 3
14 Methane solubility in brine calculated by Eq. (3.77). (A) Methane solubility in brine under reservoir temperature of 373k; (B) Methane solubility in brine under reservoir temperature of 443k.

Example 13: It is known that the pressure is 20.685 MPa, the temperature is 93.33°C, and the water salinity is 10%. Calculate the natural gas solubility in water. Solution A1 5 2:12 1 3:45 3 1023 ð1:8T 2 460Þ 2 3:59 3 1025 ð1:8T 2460Þ2 5 1:3776 A2 5 0:0107 2 5:26 3 1025 ð1:8T 2 460Þ 1 1:48 3 1027 ð1:8T 2460Þ2 5 0:0061 A3 5 2 8:75 3 1027 1 3:90 3 1029 ð1:8T 2 460Þ 2 1:02 3 10211 ð1:8T 2460Þ2

5 2 5:0294 3 1027 Rsw 5

A1 1 A2 ð145:038pÞ 1 A3 ð145:038pÞ2 5 2:6962 m3 =m3 5:615

If the influence of salinity is considered, ðSC Þ0 5 1 2 ½0:0753 2 0:000173ð1:8T 2 460ÞS 5 0:5924 Rsb 5 Rsw ðSC Þ0 5 2:6972 3 0:5924 5 1:5979 m3 =m3

3.4.4 Isothermal compressibility factor of formation water The compressibility factor of formation water depends on temperature, pressure, natural gas solubility in water, and the salinity of formation water, as shown in Fig. 3
15. Ignoring the influence of dissolved gas, the isothermal compressibility factor of formation water can be calculated using Eq. (3.76) (Meehan, 1980),

Physical properties of natural gas and formation water

95

Figure 3
15 Isothermal compressibility factor of formation water. Modified from McCain, W.D. 1991. Reservoir-fluid property correlations-state of the art (includes associated papers 23583 and 23594). SPE Reserv. Eng. 6 (2), 266
272, Permission to publish by the SPE, Copyright SPE.

 C w 5 1:45038 3 1024 C 1 1 C 2 ð1:8T 2 460Þ 1 C 3 ð1:8T 2460Þ2 (3.76) Where C 1 5 3:8546 2 0:000134ð145:038pÞ C 2 5 2 0:01052 1 4:77 3 1027 ð145:038pÞ C 3 5 3:9267 3 1025 2 8:8 3 10210 ð145:038pÞ T: reservoir temperature, K, p: reservoir pressure, MPa, Cw: compressibility factor of formation water, MPa21. For gas-saturated water,   C wg 5 C w 1 1 5 3 1022 Rsw (3.77) If the influence of salinity is considered (Numbere et al., 1977), C wb 5 C wg ðSC Þ

(3.78)

 SC 5 20:052 1 2:7 3 1024 ð1:8T 2 460Þ 2 1:14 3 1026 ð1:8T 2460Þ2 1 1:121 3 1029 ð1:8T 2460Þ3 S 0:7 1 1:0

96

Reserves Estimation for Geopressured Gas Reservoirs

The above method is suitable for the formation water with 300K , T , 394:4K, 6:89 MPa , p , 41:37 MPa, and salinity of less than 25%. Example 14: It is known that the pressure is 20.685 MPa, the temperature is 93.33°C, the water salinity is 10%, and the natural gas solubility is 2.8139 m3/m3. Calculate the isothermal compressibility of pure water, natural gas
saturated water, and brine. Solution First, calculate the isothermal compressibility factor of pure water. C 1 5 3:8546 2 0:000134ð145:038pÞ 5 3:4526 C 2 5 2 0:01052 1 4:77 3 1027 ð145:038pÞ 5 2 9:0889 3 1023 C 3 5 3:9267 3 1025 2 8:8 3 10210 ð145:038pÞ 5 3:6627 3 1025  Cw 5 1:45038 3 1024 C1 1 C2 ð1:8T 2 460Þ 1 C3 ð1:8T 2460Þ2 5 4:4933 3 1024 MPa21 Calculate the isothermal compressibility factor of gas-saturated water.   Cwg 5 Cw 1 1 5 3 1022 Rsw 5 5:1255 3 1024 MPa21 Calculate the isothermal compressibility factor of brine.  SC 5 20:052 1 2:7 3 1024 ð1:8T 2 460Þ 2 1:14 3 1026 ð1:8T 2460Þ2 1 1:121 3 1029 ð1:8T 2460Þ3 S0:7 1 1:0 5 0:8265 Cwb 5 Cwg ðSC Þ 5 5:1255 3 1024 3 0:8265 5 4:2363 3 1024 MPa21

CHAPTER 4

Material balance equation of a gas reservoir Contents 4.1 Material balance equation for homogeneous gas reservoirs 4.1.1 Volumetric gas reservoir 4.1.2 Closed gas reservoir 4.1.3 Water-drive gas reservoir 4.1.4 Water-drive gas reservoir with water-soluble gas 4.1.5 Linear form of pressure depletion curve 4.2 Material balance equation for compartmented gas reservoirs 4.2.1 Payne method 4.2.2 Hagoort
Hoogstra method 4.2.3 Gao method 4.2.4 Sun method 4.3 Drive index of gas reservoir 4.4 Apparent initial gas in place of gas reservoir 4.5 Sensitivity analysis of key parameters 4.5.1 Rock compressibility factor 4.5.2 Size of water body 4.5.3 Pressure depletion 4.5.4 Apparent reservoir pressure 4.5.5 Influence of water-soluble gas

98 99 104 109 116 118 119 119 123 125 131 137 140 142 142 152 152 152 155

For a long time, the material balance equation has been recognized as one of the essential tools used by reservoir engineers for predicting the development performance of the reservoir. It can be used to estimate the original reserves of oil and gas (original oil-in-place (OOIP) and original gas-in-place (OGIP)), reservoir performance, and the ultimate recovery factor. The concept of material balance was first proposed by Schilthuis in 1936. The simplest form of the equation is expressed in terms of volume, namely, original volume 5 remaining volume 1 removal volume. To apply the material balance equation, two necessary conditions must be satisfied. The first condition is that there is sufficient production history and PVT data, and the second is that the average formation pressure expressed by time or cumulative production must have a depletion trend. Reserves Estimation for Geopressured Gas Reservoirs © 2023 Petroleum Industry Press. DOI: https://doi.org/10.1016/B978-0-323-95088-6.00004-3 Published by Elsevier Inc. All rights reserved.

97

98

Reserves Estimation for Geopressured Gas Reservoirs

This chapter introduces the material balance equation of homogeneous gas reservoirs, the compartmental material balance equation for heterogeneous gas reservoirs, and the sensitivity analysis of critical parameters.

4.1 Material balance equation for homogeneous gas reservoirs There are two basic assumptions for material balance: (1) the reservoir maintains thermodynamic equilibrium during the entire development process, that is, the formation temperature remains constant, and (2) all parts of the reservoir have the same pressure in equilibrium in the same period (Tarek, 2019). Based on these assumptions, an actual gas reservoir can be simplified to one or more closed or unclosed underground containers (with natural water influx) storing gas. In each container, with the reservoir exploitation, the volume change of gas and water obeys the principle of conservation of mass. The equation established is referred to as the material balance equation for homogeneous gas reservoirs or compartmented gas reservoirs. This equation is characterized by underground balance, cumulative balance, and volumetric balance. According to the drive mechanism, homogeneous gas reservoirs can be divided into gas-drive reservoirs and water-drive reservoirs. Gas-drive reservoirs can be further divided into volumetric gas reservoirs and closed gas reservoirs (considering the influence of rock and irreducible water expansion). Water-drive reservoirs can also be subdivided according to the influence of water-soluble gas, as shown in Table 4
1. Table 4
1 Classification of homogeneous gas reservoirs. Class

Gas-drive gas reservoir Water-drive gas reservoir

Elastic gas drive

Volumetric gas reservoir Closed gas reservoir Without considering water-soluble gas Considering watersoluble gas

Rock and irreducible water expansion

Water drive

Solution gas drive

O O O

O O

O

O

O

O

O

Material balance equation of a gas reservoir

99

4.1.1 Volumetric gas reservoir For volumetric gas reservoirs, the volume of the gas reservoirs does not change during the production process. According to the volumetric balance relationship,   GBgi 5 G 2 Gp Bg (4.1)
 Bgi Gp 5 12 Bg G

(4.2)

where Bgi: gas formation volume factor at initial pressure pi, Bg: gas formation volume factor at current pressure p, Gp: cumulative production of natural gas, 108 m3, G: gas initially in place, 108 m3, Gp =G: recovery efficiency, decimal. Substituting Eq. (3.50) into Eq. (4.2), we have
 Gp p pi 12 5 (4.3) Z Zi G Eq. (4.3) shows a linear p=ZBGp relationship, as shown in Fig. 4
1. For all volumetric gas reservoirs, regardless of ultra-high-pressure or normal-pressure, high-pressure, or low-pressure, the p/Z curve is a straight line (Li, 2017). The slope of the straight-line segment is pi Slope 5 2 (4.4) Zi G

p/Z

pi/Zi

pa/Za

(Gp)a

0

Gp

Figure 4
1 The p/Z curve for volumetric gas reservoir.

G

100

Reserves Estimation for Geopressured Gas Reservoirs

According to the slope, the volume under initial conditions can be determined, and then, the area of the gas reservoir can be determined according to Eq. (4.5). G 5 Ahφð1 2 Swi Þ

(4.5)

where G: gas initially in place, 108 m3, A: area of the gas reservoir, m2, h: thickness of the gas reservoir, m, φ: porosity of the gas reservoir, decimal. According to the intersection between the p/Z curve and the vertical axis in Fig. 4
1, the recoverable reserves and the recovery efficiency at any apparent reservoir pressure can also be determined according to the straight line. When the p=ZBGp curve deviates upward from the linear relationship, it indicates that water invasion occurs (Fig. 4
2). There are various methods for detecting water invasion based on the material balance equation, one of which is the energy graph method. Transforming Eq. (4.3), 12

Gp p pi = 5 Z Zi G

(4.6)

Taking logarithm on both sides, we have
 p pi lg 1 2 = (4.7) 5 lgGp 2 lgG Z Zi

On log
log coordinates, the plot lg 1 2 Zp = Zpii BlgGp will yield a straight line with a slope of 1.0 when p 5 0. The intersection between the constant volume strong water drive moderate water drive weak water drive

p/Z

pi/Zi

0

Gp

Figure 4
2 p=ZBGp with water invasion (Tarek, 2019).

Material balance equation of a gas reservoir

101

lg[1-Zip/(Zpi)]

constant volume

water drive

45°

0 lgGp Figure 4
3 Schematic diagram of energy graph. Modified from Tarek, A. 2019. Reservoir Engineering Handbook fifth ed. Elsevier.

straight line and the horizontal axis is G, as shown in Fig. 4
3, which can be used to detect early water invasion. If there is water invasion, the slope of the plot will be less than 1.0 and will also decrease over time. If the gas escapes from the pay zone to other zones or the data are highly noisy, the slope may be greater than 1.0. Example 1: For a gas reservoir, the gas-bearing area is 4.290 3 106 m2, the thickness is 16.46 m, the porosity is 0.13, the irreducible water saturation is 0.5, and the formation temperature is 346.7K. The cumulative production and historical pressure data are shown in Table 4
2. Calculate the gas initially-in-place (GIIP) using the material balance method. Solution According to Eq. (3.50), the gas formation volume factor is Bg 5 3:4564 3 1024

ZT 0:869 3 346:7 5 3:4564 3 1024 3 5 0:008398 p 12:40

According to Eq. (4.5), we have G5

Ahφð1 2 Swi Þ 4:29 3 106 3 16:46 3 0:13 3 ð1 2 0:52Þ 5 0:008398 Bgi 8 3 5 5:25 3 10 m

102

Reserves Estimation for Geopressured Gas Reservoirs

Table 4
2 Production data of volumetric gas reservoir. t, a

p, MPa

Z

Gp, 108 m3

0.0 0.5 1.0 1.5 2.0

12.40 11.58 10.62 9.85 9.20

0.869 0.870 0.880 0.890 0.900

0.00 0.27 0.60 0.91 1.11

Modified from Tarek, A. 2019. Reservoir Engineering Handbook fifth ed. Elsevier.

15

p/Z, MPa

12

9

6

3 4.02

0 0

1

2

3

4

5

Gp , 10 8m3 Figure 4
4 The p/Z curve of a volumetric gas reservoir.

Prepare the p=ZBGp plot with the material balance method, as shown in Fig. 4
4. The intersection between the straight line and the horizontal axis is the GIIP, which is 4.02 3 108 m3. The gas production per unit pressure drop is defined as the cumulative gas production when the average formation pressure drops by one unit. The gas production of cumulative unit pressure drop is defined as the ratio of the cumulative gas production to the cumulative pressure drop from the initial reservoir pressure to the current formation pressure. Transforming Eq. (4.3), we have Gp p=Z 512 pi =Zi G

(4.8)

Material balance equation of a gas reservoir

103

The cumulative gas production when the formation pressure drops from the initial pressure pi to pj is  
 p=Z j Gp  512  (4.9) G j p=Z i The cumulative gas production when the formation pressure drops from the initial pressure pi to pj11 is  
 p=Z j11 Gp  512  (4.10) G j11 p=Z i The gas production per unit pressure drop (assuming pj 2 pj11 5 1) is    

 p=Z j p=Z j11 Gp Gp  2   2 5 (4.11) G j11 G j p=Z i p=Z i The dimensionless gas production of cumulative unit pressure drop is       Gp =G j 1 2 p=Z j = p=Z i 5 (4.12) pi 2 pj pi 2 pj The gas production per unit pressure drop and the cumulative unit pressure drop production are shown in Fig. 4
5. The formation pressure

Figure 4
5 Gas production per unit pressure drop and dimensionless gas production of cumulative unit pressure drop.

104

Reserves Estimation for Geopressured Gas Reservoirs

is 40 MPa, the formation temperature is 373.15K, and the gas gravity is 0.6. The theoretical calculation results show that due to the influence of the deviation factor, the gas production per unit pressure drop first increases and then decreases with the depletion of formation pressure, and the dimensionless gas production of cumulative unit pressure drop gradually increases with the depletion of formation pressure. When the formation pressure drops to atmospheric pressure, the value of the dimensionless gas production of cumulative unit pressure drop is the reciprocal of the initial reservoir pressure (0.025 in this case).

4.1.2 Closed gas reservoir A closed gas reservoir refers to a gas reservoir that is not connected to water bodies. The volume of the gas reservoir changes during the exploitation. Considering the expansion of rock and irreducible water, according to the volumetric balance relationship, we have
   Cw Swi 1 Cf GBgi 5 G 2 Gp Bg 1 GBgi Δp (4.13) 1 2 Swi 
 
 Bgi Gp Cw Swi 1 Cf 12 Δp 5 1 2 Bg 1 2 Swi G Substituting Eq. (3.50) into Eq. (4.14), we have
 Gp p pi ð1 2 Ce ΔpÞ 5 12 Z Zi G

(4.14)

(4.15)

where Ce : effective compressibility factor, Ce 5 Cw1S2wi S1wiCf , MPa21.   Eq. (4.15) shows that the plot p=Z ð1 2 Ce ΔpÞBGp yields a linear relationship (Fig. 4
6). Similar to the case of the volumetric gas reservoir, the slope of the straight line is pi Slope 5 2 (4.16) Zi G The intercept of the straight line is pi =Zi . Then, the GIIP of the gas reservoir can be determined according to the slope and intercept. Example 2: For a gas reservoir, the initial reservoir pressure is 74 MPa, the temperature is 377.15K, the gas gravity is 0.568, the initial water saturation is 0.32, the initial rock compressibility factor is 2.5 3 1023/MPa,

Material balance equation of a gas reservoir

105

  Figure 4
6 p=Z ð1 2 Ce ΔpÞ curve for closed gas reservoir. 1.5 Z=0.99796-0.0093p+3.64632×10-4p2-2.17327×10-6p3 (full pressure range)

1.4

Z

1.3 1.2 1.1 1.0 Z=0.66629+0.01025p (p>30MPa)

0.9 0

20

40

60

80

p, MPa Figure 4
7 Variation of gas deviation factor for Example 2.

and the formation water compressibility factor is 5.6 3 1024/MPa. Draw the p/Z curve and p=Z ð1 2 Ce ΔpÞ curve. Solution According to the temperature and gas gravity data, the variation law of the gas deviation factor is calculated using the DAK (Drankchuk and Abu-Kassem, 1975) extrapolation method, as shown in Fig. 4
7. In the high-pressure area, the plot is linear, and the regression formula is Z 5 0:66629 1 0:01025p

(4.17)

106

Reserves Estimation for Geopressured Gas Reservoirs

Table 4
3 Calculation results for Example 2. p, MPa

Δp, MPa

Z

p/Z, MPa

1 2 CeΔp

(p/Z)(CeΔp), MPa

Gp/G

74 73 72 71 70 ... 33 32 31 ... 1 0.1

0 1 2 3 4

1.4258 1.4167 1.4074 1.3979 1.3882

51.90 51.53 51.16 50.79 50.42

1.0000 0.9961 0.9921 0.9882 0.9842

0.0000 0.2030 0.4031 0.6003 0.7947

0.0000 0.0111 0.0221 0.0330 0.0437

41 42 43

1.0100 1.0025 0.9953

32.67 31.92 31.15

0.8385 0.8345 0.8306

5.2778 5.2820 5.2767

0.4722 0.4868 0.5016

73 73.9

0.9890 0.9970

1.01 0.10

0.7124 0.7088

0.2908 0.0292

0.9861 0.9986

Figure 4
8 Comparison of pressure depletion curves. (A) Dimensioned pressure depletion curve; (B) Dimensionless pressure depletion curve.

The calculation results using Eq. (4.15) are shown in Table 4
3. The two types of curves are shown in Fig. 4
8, where the maximum value   p=Z ðCe ΔpÞ appears at about 32 MPa. If the ordinate is expressed in a dimensionless form, as shown in Fig. 4
8B, the dimensionless pressure drop is about 0.615.   Fig. 4
8 shows that the difference between p/Z and p=Z ð1 2 Ce ΔpÞ     is p=Z ðCe ΔpÞ. Supposing that pe 5 p=Z ðCe ΔpÞ, Z 5 f ðpÞ, we have pe 5 5

pCe Δp f ðpÞ

(4.18)

107

Material balance equation of a gas reservoir

Differentiating Eq. (4.18), the maximum point is Ce ðpi 2 2pÞf ðpÞ 2 f 0 ðpÞCe pðpi 2 pÞ 50 f 2 ðpÞ According to Eq. (4.17), p0e 5

Z 5 f ðpÞ 5 a 1 bp a 5 0:66629 b 5 0:01025

(4.19)

(4.20)

Substituting them into Eq. (4.19), bp2 1 2ap 2 api 5 0

(4.21)

Solving the equation, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a 1 aða 1 bpi Þ 2 a 1 a2 1 abpi 2 a 1 aZi 5 5 p5 b b b According to Eq. (4.20), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a 1 aZ i 2 0:66629 1 0:66629 3 1:4258 5 5 30:1MPa p5 b 0:01025 Substituting p 5 30:1MPa into Eq. (4.18), we have ðpe Þmax 5

pCe Δp 30:1 3 0:1734 30:1 3 0:1734 5 5 5 5:280 f ðpÞ f ð30:1Þ 0:9885

This result is close to the data in Table 4
2. The slight pressure difference is due to the difference in the deviation factor equation. If the deviation factor f ðpÞ 5 0:99796 2 0:0093p 1 3:64632 3 1024 p2 2 2:17327 3 1026 p3 is substituted into Eq. (4.19), we get p 5 32:055MPa, Δp 5 41:945MPa, Ce Δp 5 0:1653 ðpe Þmax 5

pCe Δp 32:055 3 0:1734 32:055 3 0:1653 5 5 5 5:2820 f ðpÞ f ð32:055Þ 0:9980

The pressure at ðpe Þmax is p 5 32:055 5 0:43pi , which is close to what Li (2007) proposed. Then, ðpe Þmax 5

0:2451p2i Ce f ðpÞ

(4.22)

108

Reserves Estimation for Geopressured Gas Reservoirs

  Fig. 4
9 shows the dimensionless p=Z ð1 2 Ce ΔpÞ curve after the value of the effective compressibility factor is changed. As seen, all curves are pseudo-parabolic. As the effective compressibility factor increases, the ðpe Þmax points are the same. However, the cumulative production corresponding to different effective compressibility factors gradually increases. The greater the effective compressibility factor, the larger the error in evaluating dynamic reserves using early data, as shown in Fig. 4
10, where the green line indicates that the rock compressibility factor is 5 3 1024/MPa, and other

Figure 4
9 Comparison of pressure depletion curves under different effective compressibility factors.

Figure 4
10 Extrapolation comparison of pressure depletion curves under different effective compressibility factors.

Material balance equation of a gas reservoir

109

parameters remain unchanged. If rock and formation water compressibility is ignored, an error of about 10% may occur.

4.1.3 Water-drive gas reservoir A water-drive gas reservoir refers to a gas reservoir connected to water bodies. During the production process, water invasion may occur, and the gas reservoir volume will change. The water-drive gas reservoir can be divided into three classes, namely, active water drive, subactive water drive, and inactive water drive, which are described in terms of formation water activity, water influx replacement coefficient (ratio of net water influx to cumulative gas production under formation conditions), recovery range, and production characteristics, as shown in Table 4
4. Considering the expansion of rock and irreducible water, according to the volumetric balance relationship, we have
     Cw Swi 1 Cf GBgi 5 G 2 Gp Bg 1 GBgi Δp 1 We 2 Wp Bw (4.23) 1 2 Swi   
 We 2 Wp Bw Bgi Gp Cw Swi 1 Cf (4.24) 12 Δp 2 512 GBgi Bg 1 2 Swi G Substituting Eqs. (4.3)
(4.50) into Eq. (4.24),
 Gp p pi ð1 2 Ce Δp 2 ωÞ 5 12 Z Zi G

(4.25)

where We : water influx of the gas reservoir, 108 m3, Wp : cumulative gas production of the gas reservoir, 108 m3, ω: water storage (water influx— W 2W B water production) volume factor, ω 5 e GBgip w .   Eq. (4.25) shows that the plot p=Z ð1 2 Ce Δp 2 ωÞBGp is linear (Fig. 4
11). Similar to the volumetric gas reservoir, the slope of the straight line is pi Slope 5 2 (4.26) Zi G The material balance equation can be expressed in fluid withdrawal, fluid expansion, pore compression, and water invasion (Havlena and Odeh, 1963). Transforming Eq. (4.23), we have
   Cw Swi 1 Cf Δp 1 We (4.27) Gp Bg 1 Wp Bw 5 G Bg 2 Bgi 1 GBgi 1 2 Swi

Table 4
4 Classification of water-driven gas reservoirs. Class

Water drive

Formation water activity

Water influx replacement coefficient

Dimensionless abandoned apparent reservoir pressure

Recovery range

Active

$ 0.4

$ 0.5

0.4
0.6

Subactive

0.15
0.4

$ 0.25

0.6
0.8

Inactive

0
0.15

$ 0.05

0.7
0.9

Description of production characteristics

The movable edge/bottom water size is large. Generally, some gas wells begin to produce water in massive quantities or are flooded in the early stage of development (with the recovery percent ,20%). The stable production period of the gas reservoir is short, and the water invasion characteristic curve rises linearly. The large water body is locally connected to the gas reservoir, and the energy is relatively weak. Generally, local water channeling occurs in the middle and late stages of development, causing water production in some gas wells. Mostly closed reservoirs. In the midlate stage of development, water is occasionally produced in certain wells, or the gas reservoir does not produce water at all, and the water invasion energy is feeble. Behaviors of the elastic gas-drive reservoir are found during the production. (Continued)

Table 4
4 (Continued) Class

Gas drive

Formation water activity

Water influx replacement coefficient

0

Dimensionless abandoned apparent reservoir pressure

$ 0.05

Recovery range

0.7
0.9

Description of production characteristics

There is no edge/bottom water and mostly closed multifracture system, fault-compartment, sand, or ultrahigh-pressure gas reservoirs. There is no water invasion in the whole development process, with the characteristics of the elastic gas drive.

112

Reserves Estimation for Geopressured Gas Reservoirs

  Figure 4
11 p=Z ð1 2 Ce Δp 2 ωÞ curve for water-drive gas reservoir.

The underground fluid withdrawal F is defined as, F 5 Gp Bg 1 Wp Bw

(4.28)

The gas expansion Eg is defined as, Eg 5 Bg -Bgi

(4.29)

The water and rock expansion Ef, w is defined as,
 Cw Swi 1 Cf Ef ; w 5 Bgi Δp 1-Swi Eq. (4.27) can be simplified,  F 5 G Eg 1 Ef ; F Eg 1 Ef ;

5G1 w

 w

1 We

We Eg 1 Ef ;

(4.30)

(4.31)

(4.32) w

If the water and rock expansion Ef, w can be ignored relative to the gas expansion Eg, then Eq. (4.32) can be simplified as, F We 5G1 Eg Eg

(4.33)

Material balance equation of a gas reservoir

113

Figure 4
12 Influence of aquifer strength on the calculation of GIIP. Modified from Tarek, A. 2019. Reservoir Engineering Handbook fifth ed. Elsevier.

The shape of the F=Eg BGp curve is shown in Fig. 4
12. If the gas reservoir is volumetric, then We 5 0. The value is constant, and the intersection with the vertical axis is the GIIP. If the production of the gas reservoir is affected by natural water invasion, the F=Eg curve is usually a concave arc, and its exact shape depends on the size and strength of the aquifer and the gas production rate. The intersection between the F=Eg trend line and the ordinate is the GIIP (We 5 0). The F=Eg BGp curve is superior to other methods in regard to its high sensitivity in determining whether a gas reservoir is affected by water invasion. Even if the production data, pressure, temperature, and gas gravity are known for water-drive gas reservoirs, there are still two unknowns in the material balance equation, namely, GIIP and cumulative water influx. In order to calculate the GIIP using the material balance equation, some independent methods must be developed to estimate the cumulative water influx We. To date, a mature method for calculating the water influx of reservoirs has been established. The commonly used models include tank model, Schilthuis steady flow model (1936), Hurst modified steady flow model (1943), van Everdingen
Hurst unsteady flow model (edge water drive and bottom water drive) (1949), Carter
Tracy unsteady flow model (1960), Fetkovich method (radial aquifer and linear aquifer) Fetkovich (1971). For professional details, see Chapter 10 in the Reservoir Engineering Handbook (Tarek, 2019; Sun et al., 2021b), Chapter 5, OGIP Estimations for Geopressured Gas Reservoir in the Modern Reservoir Engineering (Chen, 2020), the Principles of Reservoir Engineering (Li, 2017), Chapter 12 in the

114

Reserves Estimation for Geopressured Gas Reservoirs

Practical Reservoir Engineering (Smith, 1992), and the Gas Reservoir Engineering (Ikoku, 1984; Lee and Wattenbarger, 1996). Because the gas is trapped by the invaded water (water-sealed gas), the recovery rate of water-drive gas reservoirs may be much lower than that of volumetric gas reservoirs. The material balance equation considering water-sealed gas can be expressed as Eq. (4.34) (Tarek, 2019). "  #   Vp wiz =Bg Gp (4.34) 5 G 1 Sgi 2 Sgrw 1 2 Bgi =Bg 1 2 Bgi =Bg   where Vp wiz : pore volume of the water-invaded zone, 108 m3, Sgrw: residual gas saturation to water displacement, decimal. The general material balance equation above shows that if the ðVp Þwiz =Bg Gp 1 2 Bgi =Bg B 1 2 Bgi =Bg plot is drawn, the intercept is the GIIP G. If expressed in p/Z form, there is

" #     Vp wiz =Bg Gp  5 G 1 Sgi 2 Sgrw        1 2 p=Z = pi =Zi 1 2 p=Z = pi =Zi

(4.35)

Example 3: For a gas reservoir, the initial reservoir pressure is 74 MPa. The temperature is 377.15K, the gas gravity is 0.568, the initial water saturation is 0.32, the initial rock compressibility factor is 2.5 3 1023/MPa, the formation water compressibility factor is 5.6 3 1024/MPa, the water volume ratio is 2.0, and  the cumulative water production   Wp 5 0. Try to draw the p/Z curve, p=Z ð1 2 Ce ΔpÞ curve, and p=Z ð1 2 Ce Δp 2 ωÞ curve. Solution The deviation factor is shown in Fig. 4
7, and the regression formula is f ðpÞ 5 0:99796 2 0:0093p 1 3:64632 3 1024 p2 2 2:17327 3 1026 p3 (4.36) Calculating the water influx according to the tank model (Tarek, 2019), we have We 5 ðCf 1 Cw ÞWi ðpi 2 pÞ

(4.37)

where Wi : water volume, 108 m3. The calculation results using Eq. (4.25) are shown in Table 4
5. There are three types of pressure depletion curves, as shown in Fig. 4
13, and the

Table 4
5 Calculation results for Example 3. p, MPa

Δp, MPa

Z

ω

p/Z, MPa

1 2 CeΔp

1 2 CeΔp-ω

(p/Z) (1 2 CeΔp), MPa

(p/Z) (1 2 CeΔp-ω), MPa

Gp/G

74 73 72 71 70 69 ... 7 5 4 3 2 1 0.1

0 1 2 3 4 5

1.4258 1.4167 1.4074 1.3979 1.3882 1.3783

0.0000 0.0061 0.0122 0.0184 0.0245 0.0306

51.90 51.53 51.16 50.79 50.42 50.06

1.0000 0.9961 0.9921 0.9882 0.9842 0.9803

1.0000 0.9899 0.9799 0.9698 0.9598 0.9497

51.8998 51.3235 50.7535 50.1890 49.6294 49.0742

51.8998 51.0082 50.1273 49.2565 48.3950 47.5424

0.0000 0.0111 0.0221 0.0330 0.0437 0.0544

67 69 70 71 72 73 73.9

0.9500 0.9603 0.9665 0.9733 0.9808 0.9890 0.9970

0.4100 0.4223 0.4284 0.4345 0.4406 0.4468 0.4523

7.37 5.21 4.14 3.08 2.04 1.01 0.10

0.7360 0.7281 0.7242 0.7203 0.7163 0.7124 0.7088

0.3260 0.3059 0.2958 0.2857 0.2757 0.2656 0.2566

5.4234 3.7912 2.9973 2.2201 1.4607 0.7203 0.0711

2.4020 1.5925 1.2243 0.8808 0.5622 0.2686 0.0257

0.8955 0.9270 0.9422 0.9572 0.9719 0.9861 0.9986

116

Reserves Estimation for Geopressured Gas Reservoirs

Figure 4
13 Comparison of the three types of pressure depletion curves.

Figure 4
14 Extrapolation comparison of the three types of pressure depletion curves.

extrapolation is shown in Fig. 4
14. Obviously, in water-drive gas reservoirs, the dynamic reserves will be significantly overestimated if the influence of water influx is ignored.

4.1.4 Water-drive gas reservoir with water-soluble gas For high-temperature and high-pressure gas reservoirs, the influence of solution gas in formation water sometimes cannot be underestimated. For a gas reservoir with irreducible water saturation of Swi and water volume ratio of M, assuming that the water and irreducible water are saturated

Material balance equation of a gas reservoir

117

with solution gas, and the injected gas is ignored, according to the volumetric balance, the material balance equation (Fetkovich et al., 1998) can be expressed as Fluid volume in reservoir VpR 1 Fluid volume in aquifer VpA 5 ðGas and water volume in the reservoir VgR 1 VwR Þ (4.38) 1 ðGas and water volume in aquifer VgA 1 VwA Þ where





 GBgi GBgi C f ðpi 2 pÞ Swi 2 VpR 5 GBgi 1 1 2 Swi 1 2 Swi
VpA 5


 GBgi GBgi M2 MC f ðpi 2 pÞ 1 2 Swi 1 2 Swi

(4.39)

   VgR 5 G 2 Gp 2 Wp Rsw Bg 1




(4.40)

 GBgi Swi ðRswi 2 Rsw ÞBg (4.41) 1 2 Swi Bwi




 GBgi 1 ðRswi 2 Rsw ÞBg VgA 5 M 1 2 Swi Bwi  GBgi Swi Bwi 2 Wp Bw 1 We VwR 5 1 2 Swi Bwi

(4.42)




 GBgi M Bw VwA 5 1 2 Swi Bwi

(4.43)




(4.44)

Substituting Eqs. (4.39)
(4.44) into Eq. (4.38), we have     G Bg 2 Bgi 1 GBgi C e Δp 1 We 5 Gp Bg 1 Wp Bw 2 Bg Rsw

(4.45)

Eq. (4.45) can also be expressed as,


 pi p Q 12 1 2 C e ðpÞðpi 2 pÞ 5 Z G Zi


Wp Bw 2 We Q 5 Gp 2 Wp Rsw 1 Bg

(4.46)

 (4.47)

118

Reserves Estimation for Geopressured Gas Reservoirs

C e ðpÞ 5

  1  Swi C tw 1 C f 1 M C tw 1 C f 1 2 Swi

1 Btw ðpÞ 2 Btw ðpi Þ Btw ðpi Þ pi 2 p  Btw ðpÞ 5 Bw ðpÞ 1 Rswi 2 Rsw ðpÞ Bg ðpÞ C tw 5

(4.48) (4.49) (4.50)

If the influence of the water body is not considered, that is, M 5 0, Eq. (4.48) can be simplified as C e ðpÞ 5

Swi C tw 1 C f 1 2 Swi

(4.51)

If water influx and water production are ignored, Eq. (4.46) can be simplified as,
 pi Gp p 1 2 C e ðpÞðpi 2 pÞ 5 12 (4.52) Z Zi G where p: average formation pressure, MPa, Z: gas deviation factor, dimensionless, Btw : two-phase formation water volume factor, dimensionless, Bg : gas formation volume factor, dimensionless, Rsw : solubility of natural gas in formation water, dimensionless, C e ðpÞ: cumulative effective compressibility factor, MPa, C tw : cumulative water compressibility factor, MPa, C f : cumulative rock compressibility factor, MPa, M: water volume ratio, G, Gp: dynamic reserves and cumulative production, 108 m3, Swi: initial water saturation, The subscript ii: an initial condition. Eq. (4.51) is consistent with Eq. (4.13) in representation, and the shape of the curve is also similar to that in Fig. 4
13. The difference lies in the expression of the compressibility terms; one is a variable, while the other is a constant. Under low-pressure conditions, water-soluble gas will make a particular contribution to production. When 1 2 C e ðpÞðpi 2 pÞ 5 0,  extrapolating the Zp 1 2 C e ðpÞðpi 2 pÞ BGp plot can get the gas reserves G in the reservoir. When p 5 0:101325MPa extrapolating, the plot can get the total gas reserves in the gas zone and solution gas.

4.1.5 Linear form of pressure depletion curve For volumetric gas reservoirs, the p/ZBGp curve is a straight line. Considering the influence of rock and irreducible water expansion for closed gas reservoirs, the p/ZBGp curve is a downward curve, and the p=Z ð1 2 Ce ΔpÞBGp curve is a straight line. Considering the influence

Material balance equation of a gas reservoir

119

of rock and irreducible water expansion andwater influx for water-invaded gas reservoirs, the p/ZBG p=Z ð1 2 Ce ΔpÞBGp curves are  p and  downward curves, and the p=Z ð1 2 Ce Δp 2 ωÞBGp curve is a straight line. Considering the influence of a water-soluble gas, the p=Z 1 2 C e Δp BQ curve is a straight line. High-pressure and ultrahigh-pressure gas reservoirs are not essentially different from normalpressure gas reservoirs, except that the pressure is slightly higher (Li, 2007).

4.2 Material balance equation for compartmented gas reservoirs The conventional material balance method cannot consider the widespread reservoir heterogeneity and its impact on flow and production. To address this shortcoming, many scholars (Gao, 1993; Gao et al., 1997, 2006; Payne, 1996; Hoogstra, 1999; Sun, 2011) have proposed the multicompartment material balance equation and used the same to predict the development index of volumetric gas reservoirs. A compartmented gas reservoir is defined as a reservoir that consists of two or more distinct regions that are allowed to communicate. Each compartment is described by its material balance, coupled with the neighboring compartments’ material balance through influx or efflux across the common boundaries.

4.2.1 Payne method The gas reservoir is divided into multiple small compartments, which can communicate. Such compartments can either be depleted directly by wells or indirectly through other compartments. Flow rate between compartments is set proportionally to either the difference in the squares of compartment pressure or the difference in pseudo-pressures. To illustrate the concept, consider a reservoir that consists of two compartments, 1 and 2, as shown schematically in Fig. 4
15. Before starting production, the two compartments are in equilibrium, with the same initial reservoir pressure. Gas production can be produced from either one or both compartments. Adopting the convention that influx is positive if gas flows from compartment 1 into compartment 2, the linear gas flow between the two compartments in terms of gas pseudo-pressure is given by q12 5

123:129KA  mðp1 Þ 2 mðp2 Þ TL

(4.53)

120

Reserves Estimation for Geopressured Gas Reservoirs

q1

q2

q12

G2

G1

Figure 4
15 Schematic diagram of two-compartment material balance model. Modified from Tarek, A. 2019. Reservoir Engineering Handbook fifth ed. Elsevier.

where q12 : flow rate between the two compartments, m3/d, mðp1 Þ: gas pseudo-pressure in compartment 1, MPa2/(mPa  s), mðp2 Þ: gas pseudopressure in compartment 2, MPa2/(mPa  s), A: cross-sectional area, m2, T : temperature, K, L: the distance between the center of the two compartments, m. This equation can be expressed in a more compacted form by including a “communication factor,” C12, between the two compartments.  q12 5 C12 mðp1 Þ 2 mðp2 Þ (4.54) For compartment 1, the communication factor is C1 5

123:129K1 A1 TL1

For compartment 2, the communication factor is C2 5

123:129K2 A2 TL2

The communication factor between the two compartments, C12, is given by the following harmonic average technique, C12 5

2C1 C2 C1 1 C2

where C: communication factor between two compartments, m3/d/ [MPa2/(mPa  s)], L: length of compartment, m, A: cross-sectional area of the compartment, m2, T : temperature, K.

Material balance equation of a gas reservoir

121

The cumulative gas in influx, Gp12, from compartment 1 to compartment 2 is given by the integration of flow rate over time t as ðt t X Gp12 5 q12 dt 5 ðΔq12 ÞΔt (4.55) 0

0

The individual compartment pressures are determined by assuming a straight-line relationship of p/Z versus Gpt, with the apparent total gas production, Gpt, from an individual compartment as defined by the following expression, Gpt 5 Gp 1 G12 where Gp is the cumulative gas production from wells in the compartment and G12 is the cumulative gas efflux/influx between the connected compartments. Assuming a positive flow from compartment 1 to compartment 2, the material balance equations for the two compartments are as follows,

 Gp1 1 Gp12 pi p1 5 Z1 12 (4.56) Zi G1

 Gp2 2 Gp12 pi 12 p2 5 Z2 Zi G2

(4.57)

with G1 5

ðAhφÞ1 ð1 2 Swi Þ Bgi

(4.58)

G2 5

ðAhφÞ2 ð1 2 Swi Þ Bgi

(4.59)

where subscripts 1 and 2 denote compartments 1 and 2, respectively, while the subscript i refers to an initial condition. The required input data for the Payne method are as follows. 1. initial gas in place in each compartment, that is, compartment dimensions, porosity, and saturation, 2. intercompartment communication factors, C12, 3. the initial pressure in each compartment, 4. production data profiles from the individual compartments.

122

Reserves Estimation for Geopressured Gas Reservoirs

5. The Payne method is performed fully explicit in time. At each time step, the pressures in various compartments are calculated, yielding a pressure profile that can be matched to the actual pressure decline. The specific steps of this iterative method are summarized as follows (Tarek, 2019; Sun et al., 2021b). Step 1. Prepare the available gas properties data versus pressure in tabulated and graphical forms, including Z, μg, 2p/(μgZ), and m(p). Step 2. Divide the gas reservoir into two compartments and determine the length L, thickness H, width W, and cross-sectional area A of each compartment. Step 3. Calculate the initial gas in place G1 and G2 from Eqs. (4.58) and (4.59). Step 4. For each compartment, make a plot of p/Z versusGp, which  can be constructed by simply drawing a straight line between p=Z i and G1 or G2. Step 5. Calculate the communication factors for compartments C1 and C2 and between compartments C12. Step 6. Select a small time step Δt, and determine the corresponding actual cumulative gas production, Gp, from each compartment. Assign Gp 5 0 if the compartment does not include a well. Step 7. Assume the pressure distribution throughput to be the selected compartmental system and determine the gas deviation factor Z at each pressure. For a two-compartment system, let the initial values be denoted by pk1 and pk2 . Step 8. Using the assumed values of pressure, pk1 and pk2 , determine the corresponding mðp1 Þ and mðp2 Þ from the data of Step 1. Calculate the gas influx rate q12 and cumulative gas influx Gp12 by applying Eqs. (4.54) and (4.55), respectively. Step 9. Substitute the Gp12, Z, Gp1, and Gp2 into Eqs. (4.56) and (4.57) k11 to calculate the pressure in each compartment, denoted by pk11 1  and p2 . k k11 k k11 Compare the assumed and calculated values, p1 2 p1  and p2 2 p2 . If a satisfactory match is achieved within a tolerance of 0.03
0.07 MPa, repeat Steps 3 through 7. If the match is not satisfactory, repeat the iterative cycle of Steps 4 through 7 and set pk1 5 p1k11 and pk2 5 p2k11 . Step 10. Repeat Steps 6 through 9 to produce a pressure-decline profile for each compartment that can be compared with the actual pressure profile for each compartment. Performing a material balance history match consists of varying the number of compartments required, the dimension of the compartments, and the communication factors until an acceptable match of the pressure

Material balance equation of a gas reservoir

(A) History matching by two-compartment material balance model

123

(B) History matching by traditional material balance model

Figure 4
16 Schematic diagram of the two-compartment material balance model. (A) History matching by two-compartment material balance model, results are well; (B) History matching by traditional material balance model, results are bad. Payne D. A. 1996. Material balance calculations in tight gas reservoirs: the pitfalls of p/Z plots and a more accurate technique. SPE Reservoir Engineering, 11 (4), 260
267. Permission to publish by the SPE, Copyright SPE.

decline is obtained. The improved accuracy in estimating the initial gas in place, resulting from determining the optimum number and size of compartments, stems from the proposed method’s ability to incorporate reservoir pressure gradients. Three gas reservoir example cases in the Waterton gas field show that the model results are consistent with the numerical simulation results, matching well with the actual pressure (Fig. 4
16).

4.2.2 Hagoort
Hoogstra method Based on the Payne method, Hagoort and Hoogstra (1999) developed a numerical method to solve the material balance of compartmented gas reservoirs that employs an implicit, iterative procedure. The iterative technique relies on adjusting the size of the compartments and the transmissibility values to match the historical pressure data for each compartment as a function of time. As shown in Fig. 4
15, it is assumed that there is a thin permeable layer with a transmissibility of Γ12 , which separates the two compartments. The instantaneous gas influx crossing through the thin permeable layer is expressed in terms of Darcy’s equation, which is given by q12 5

 123:129Γ12  2 p1 2 p22 TL

(4.60)

Γ1 Γ2 ðL1 2 L2 Þ L1 Γ2 1 L2 Γ1

(4.61)

Γ12 5

124

Reserves Estimation for Geopressured Gas Reservoirs

! KA Γ1 5 Zμg

(4.62) !

KA Γ2 5 Zμg

1

(4.63) 2

where q: influx gas rate, m3/d,Γ: the transmissibility between compartments, 1023 μm2  m2/(mPa  s), L: the distance between the centers of compartments, m, A: cross-sectional area, m2. Similar to the Payne method, the multicompartment material balance equation can be expressed as

 Gp1 1 Gp12 pi p1 5 Z1 12 (4.64) Zi G1

 Gp2 2 Gp12 pi p2 5 Z2 12 (4.65) Zi G2 To solve for the two unknowns p1 and p2 , the two expressions can be arranged to equate to zero, as follows,

 Gp1 1 Gp12 pi F1 ðp1 ; p2 Þ 5 p1 2 Z1 12 50 (4.66) Zi G1

 Gp2 2 Gp12 pi F2 ðp1 ; p2 Þ 5 p2 2 Z2 12 50 (4.67) Zi G2 The general methodology of applying the method is very similar to that of the Payne method and involves the following specific steps (Tarek, 2019; Sun et al., 2021b). Step 1. Prepare the available data on gas properties versus pressure in tabulated and graphical forms, including Z and μg . Step 2. Divide the gas reservoir into two compartments and determine the length L, height H, width W, and cross-sectional area A of each compartment. Step 3. Calculate the initial gas in place G1 and G2 of the two compartments from Eqs. (4.58) and (4.59). Step 4. For each compartment, make a plot of p/Z versus Gpt, which  can be constructed by simply drawing a straight line between p=Z i and G1 or G2.

Material balance equation of a gas reservoir

125

Step 5. Calculate the transmissibility Γ12 by applying Eq. (4.61). Step 6. Select a time step Δt and determine the corresponding cumulative gas production Gp. Step 7. Calculate the gas influx rate q12 and cumulative gas influx Gp12 by applying Eqs. (4.60) and (4.55), respectively. Step 8. Assume the initial pressures to be pk1 and pk2 . Using Newton’s iterative scheme, calculate p1k11 and p2k11 by solving the following linear equations as expressed in a matrix form. 2 @F ðpk ;pk Þ @F ðpk ;pk Þ 321 1 1 2 1 1 2    k11   k   @p1 @p2 2 F1 pk1 ; pk2  p1 p1 7 6 5 k 2 4 @F pk ;pk 5 @F2 ðpk1 ;pk2 Þ 2ð 1 2Þ 2 F2 pk1 ; pk2 pk11 p2 2 @p1

@p2

k11 Denote the results as pk11 1 and p2  . Then  compare  the assumed and k k11 . The iteration is calculated values, that is, p1 2 p1  and pk2 2 pk11 2 stopped when all differences fall within the tolerance of 0.03
0.07 MPa. Step 9. Generate the pressure profile for each compartment and compare the same with the actual pressure profile by repeating Steps 6 through 9. Step 10. Adjust the number and size of compartments and repeat Steps 2 through 9 until a match is achieved.

4.2.3 Gao method Early reservoir performance is usually predicted using traditional material balance and numerical simulation methods. The traditional material balance method is oversimplified. It treats the gas reservoir as a large homogeneous container and completely ignores the influence of the reservoir heterogeneity. Its indicators predicted consistently tend to be optimistic. The numerical simulation method is too complicated and requires the detailed distribution of formation parameters, which is difficult to obtain in the early development stage of the gas field. Furthermore, the numerical simulation method requires a large amount of time and workload, limiting its application. Gao et al. (2006) developed a multicompartment material balance method. According to the regional heterogeneity, the gas reservoir is divided into n interconnected relatively homogeneous compartments. It is assumed that every compartment meets the requirements of the traditional material balance method, and each compartment has its average formation

126

Reserves Estimation for Geopressured Gas Reservoirs

parameters, pressures, well numbers, and single-well production. Natural gas and groundwater flow from high-pressure compartments to lowpressure compartments, with the flow rate depending on the flow equation. It is assumed that the influx is proportional to the squared pressure difference between the two compartments. The influx is described with a communication factor. According to the principle of material balance, the following equation can be obtained for the underground volume of compartment j,     Bgj Gpj 5 Gj Bgj 2 Bgi 1 Gcj Bgj 2 Wpj Bw 2 Wcj 2 Wej (4.68) The equation can be further arranged to,   Wpj Bw 2 Wcj 2 Wej Gj Bgi Gj 2 Gpj 1 Gcj 5 Gj 1 Bgj Bgj " #   1 2 Gpj 2 Gcj =Gj pj pi   5 Zj Zi 1 1 Wpj Bw 2 Wcj 2 Wej =Gj Bgi

(4.69)

(4.70)

Wpj 5 WGRj Gpj

(4.71)

Wcj 5 WGRn Gcj Bw

(4.72)

Assuming that the flow rate is proportional to the squared pressure difference between the two compartments, the flow equation is n

X qcj 5 (4.73) djk p2k 2 p2j k51

The average gas production equation of a single well in a compartment is p2j 2 p2wj 5 aj qgj 1 bj q2gj

(4.74)

The total gas production equation of a compartment is qtj 5 Nj qgj

(4.75)

For the case with active edge water, the water influx equation of a compartment is   qwj 5 αwj pi 2 pj (4.76)

Material balance equation of a gas reservoir

127

For the case with limited edge water volume, they are Wet 5 Vw Ct ðpi 2 pe Þ   qwj 5 αwj pe 2 pj

(4.77) (4.78)

where Bg: gas formation volume factor, dimensionless, Bgi: gas formation volume factor at initial pressure, dimensionless, Gp: cumulative gas production, 108 m3, Gcj: cumulative gas influx from the neighboring compartment j, 108 m3, G: initial gas in place, 108 m3, Wcj: quantity of formation water in the cumulative gas influx from the neighboring compartment j, 108 m3, Wej: cumulative water influx from edge water to compartment j, 108 m3, Wpj: cumulative water production in compartment j, 108 m3, Bw: formation water volume factor, dimensionless, pi: initial reservoir pressure, MPa, Z: gas deviation factor, dimensionless, WGR: water
gas ratio, m3 /m3, qc: gas flow between two compartments, 104 m3/d, d: communication factor, 104 m3/(MPa2  d), pw: bottom hole pressure of gas well, MPa, qg: average single-well gas production, 104 m3/d, a, b: average binomial gas production equation coefficients of the gas well, qt: total gas production, 104 m3/d,N: number of wells, qwj: water influx of aquifer to compartment j, 104 m3/d, αw: water invasion coefficient, 104 m3/(MPa  d), Wet: cumulative water influx of edge water, 108 m3, Vw: edge water volume, 108 m3, pe: formation pressure in the edge water area, MPa. We can get the pressure, production, and gas influx changes over time by solving the above equations in each compartment. The compartments with relatively good physical properties are preferentially selected, while the remaining large low-permeable zones are ranked as the last compartment, contributing gas influx to adjacent compartments. There are usually only a small number of wells produced in the low-permeable compartment. In order to simulate the performance of the low-permeable compartment effectively, a production area can be delineated around each production well in the compartment, the production areas of wells can be virtually merged into a near-well production area, and the areas outside the near-well production area can be called the far-well recharge area. Thus, the low-permeable compartment is divided into two areas, namely, the near-well production area and the far-well recharge area, yielding one more compartment than the original scheme. This modified model is referred to as multicompartment material balance model. Note that under this modified model, the sizes of the near-well production area and the far-well recharge area will vary with the number of wells, but the sum of their sizes is a constant.

128

Reserves Estimation for Geopressured Gas Reservoirs

Example 4: Given a closed gas reservoir, the initial reservoir pressure is 31.38 MPa, the lowest flowing pressure is 3.0 MPa, the limit daily singlewell gas rate is 1.0 3 104 m3, the gas gravity is 0.59, the formation temperature is 378K, the formation water volume factor is 1.0, and the ratio of available reserves to initial gas in place is 0.62. According to the contour line of Kh 5 20.0 3 1023 μm2  m, eight compartments (compartments 1 to 8) with high yield are delineated, while the other low-yield areas are categorized as compartment 9 (the near-well production area) and compartment 10 (the far-well recharge area). The detailed data are shown in Table 4
6. Predict the production performance of the gas reservoir with an annual gas production of 15.56 3 108 m3 and a stable production for 12 years. Solution The optimized results are shown in Table 4
7. According to the well positions and single-well production given in the preliminary development plan, the number of wells and the average single-well production in the compartment before optimization are calculated, and the stable production period is calculated by the multicompartment material balance method. In Table 4
5, except for compartment 6, the stable production period of each compartment in the original preliminary development plan is relatively long, and the production is relatively low. Therefore, to maximize the productivity of each compartment, the production rate and the number of wells in the high-yield compartments can be increased, and the Table 4
6 Parameters for Example 4. Compartment

Compartment area, km2

Initial gas in place, 108 m3

aj

bj

Communication factor

1 2 3 4 5 6 7 8 9 10 Sum

14.05 31.95 14.88 27.30 46.80 82.73 34.20 29.08 213.63 544.40 1039.02

6.74 24.92 10.86 12.83 21.06 38.47 20.14 14.78 76.47 194.87 421.14

22.524 17.397 9.608 1.092 11.263 7.361 12.657 11.689 33.488

0.117 0.143 0.925 0.196 0.309 0.333 1.107 0.444 7.175

0.1405 0.1709 0.1466 0.0934 0.1553 0.1336 0.1700 0.1795 2.9960 0

Table 4
7 Well distribution and stable production status in compartments before and after optimization for Example 4. Compartment

1 2 3 4 5 6 7 8 9 10

Before optimization 4

3

After optimization

Number of wells

qg,10 m /d

Stable production period, a

Number of wells

qg, 104 m3/d

Stable production period, a

2 5 2 5 6 13 6 5 68 0

4.65 4.80 5.60 4.26 3.40 7.17 6.06 5.90 3.33 0.00

16.5 17.0 18.0 18.0 19.5 11.5 14.5 16.0 13.0 0.0

5 10 6 5 13 8 8 9 11

5.58 5.76 6.53 8.52 4.71 10.75 6.82 6.56 3.95 0.00

12.5 12.0 12.0 12.0 12.0 11.5 12.0 12.0 12.5

130

Reserves Estimation for Geopressured Gas Reservoirs

number of wells with low production in compartment 9 is decreased, so that the natural gas in some low-yield compartments can be recovered from the high-yield compartments by intercompartment communication, thereby improving development benefit. After well spacing optimization in compartments 1 and 10, the production performance for 30a is illustrated in Figs. 4
17 and 4
18. In Fig. 4
17, the

Figure 4
17 Performance profile of compartment 1. Modified from Gao, C, Lu, T, Gao, W, et al. (2006). Multi-region material balance method and its use in the performance prediction and optimal well pattern design of edge water reservoir. Petroleum Exploration and Development, 33(1), 103
106.

Figure 4
18 Performance profile of compartment 10. Modified from Gao, C, Lu, T, Gao, W, et al. (2006). Multi-region material balance method and its use in the performance prediction and optimal well pattern design of edge water reservoir. Petroleum Exploration and Development, 33(1), 103
106.

Material balance equation of a gas reservoir

131

formation pressure of the high-yield compartment (compartment 1) is decreasing with time. The bottom-hole pressure first decreases with time and remains unchanged after reaching the limit at about 12a. The average single-well production does not change in the beginning, but it decreases after about 12a. The average single-well gas flow between two compartments first increases rapidly with time and then slowly decreases. It accounts for a large proportion of single-well production, which increases with time in the later stage of development. Other compartments exhibit a similar performance profile to compartment 1. The production in the later stage of development mainly comes from intercompartment communication. To develop the gas field reasonably, it is necessary to correctly understand and estimate the change of intercompartment communication and allow it to give a full play.

4.2.4 Sun method It is assumed that a gas reservoir is divided into two interconnected and relatively homogeneous compartments, which are separated by a lowpermeability zone. The two compartments are the production zone and recharge zone, and the low-permeability zone is the transition zone (Fig. 4
19). It is conceivable that the resistance of the plane flow is concentrated on the contact surface between the compartments. Therefore, the flow resistance in the compartment is zero, and the pressure in the compartment is only related to time. This kind of surface can also be called a semipermeable wall (Gao and Sun, 2017). The pressure jumps when the fluid passes through the semipermeable wall, and the flow obstruction of the semipermeable wall can be calculated using the principle of equivalent flow resistance. qg

Figure 4
19 Two-compartment material balance model with recharge.

132

Reserves Estimation for Geopressured Gas Reservoirs

For the convenience of conducting research, the semipermeable wall model is used instead of actual formation, and the assumptions are as follows. 1. For the production zone, recharge zone, and transition zone, the lengths are L1, L2, and W, m, the widths are L, m, the formation thickness is h, m. 2. The stratum in each compartment is homogeneous with the same thickness. The permeability of the production zone and the recharge zone are K1, K2, 1023 μm2, respectively, and the permeability of the transition zone is K, 1023 μm2. 3. The initial gas in place in the production and recharge zones is G1 and G2, respectively, 108 m3, while the initial gas in place in the transition zone is ignored. 4. Single-well production is qg, 104 m3/d. 5. The flow of natural gas in the stratum satisfies Darcy’s law. It flows from the high-pressure compartment to the low-pressure compartment. The gas flow between two compartments is proportional to the pressure difference between the two compartments, and the flow distance is W 1 (L1 1 L2)/2. 6. The influence of water is ignored. The semipermeable wall model can ensure the pressure balance in each compartment at any time so that each compartment fulfills the conditions of traditional material balance. According to the principle of material balance, the flow in the production zone can be expressed as G1

dρ1 K  Lh @p ρ 5 2 ρsc qg 1 μ @x dt

(4.79)

The flow in the recharge zone can be expressed as G2

dρ2 K  Lh @p ρ 52 μ @x dt

(4.80)

where μ: gas viscosity, mPa  s, ρ: average density, kg/m3, ρsc: density under standard conditions, kg/m3, Subscripts 1 and 2: the production zone and the recharge zone, respectively, K  : average permeability within the flow distance, 1023 μm2. According to the principle of equivalent flow resistance, the resistance to flow from the high-pressure compartment to the low-pressure compartment is composed of three parts, that is, the resistance from the highpressure compartment to the transition zone, the resistance within the transition zone, and the resistance to the flow in the low-pressure

Material balance equation of a gas reservoir

133

compartment. Assuming that the fluid viscosity in the two compartments is the same, K  can be expressed as K 5

W 1 L1 12 L2 L1 L2 W 2K1 1 K 1 2K2

(4.81)

The initial condition is ρ1 ðt 5 0Þ 5 ρ2 ðt 5 0Þ 5 ρ0

(4.82)

where ρ0 is the gas density under initial condition, kg/m3. Eqs. (4.79) through (4.81) can be transformed into a second-order homogeneous linear differential equation with constant coefficients, given as d2 ρ1 dρ 1 β ð1 1 αÞ 1 2 αβγ 5 0 2 dt dt

(4.83)

where the parameter groups α, β, and γ are expressed as α5

ρ qg G1 K  Lh ; β5 ; γ 5 2 sc G1 μCg W G2 G1

Cg is the gas compressibility factor, MPa21. Solving Eq. (4.83), we have  p

p   qg t qg 2β ð11αÞt 5 12 2 12e (4.84) Z 1 Z i G1 1 G2 G1 β ð11αÞ2 p

Z

2

5

p  Z

i

  qg t qg 2β ð11αÞt 12 1 12e G1 1 G2 G2 β ð11αÞ2

(4.85)

where The subscript i: an initial condition, t: production time, a, p: average formation pressure at time t, MPa. According to Duhamel’s principle, in the production decline stage after the end of the stable production period, the average formation pressure in the production zone and the recharge zone are as follows. 8 9 j X > >   1 > > > > > > 12 ðqk 2 qk21 Þ 3 tj 2 tk21 > > = p

p < G1 1 G2 k51 5 j i> X > Z 1; j Z i> 1 ðqk 2 qk21 Þ h 2β k ð11αÞðtj 2tk Þ > > > 2 1 2 e > > > > : G1 ð11αÞ2 k51 ; βk (4.86)

134

Reserves Estimation for Geopressured Gas Reservoirs

p

Z

2;

8 9 j X > >   1 > > > > >12 > ðqk 2 qk21 Þ 3 tj 2 tk21 > > = p < G1 1 G2 k51 5 j i> X Z i> j 1 ðqk 2 qk21 Þ h > 2β k ð11αÞðtj 2tk Þ > > > 1 2 e 1 > > > > : G ð11αÞ2 ; β 2

k

k51

(4.87) Subscript j represents a specific moment. Assuming that there is only one well in the compartment, the binomial productivity equation is as follows after the productivity test. Δp2 5 p21 2 p2wf 5 Aqg 1 Bq2g

(4.88)

where pwf : bottom hole pressure, MPa, A, B: binomial productivity equation coefficients. The gas flow between two compartments qc is proportional to the pressure difference between the two compartments, 104 m3/d, which is expressed as qc 5 μ h

p22 2 p21 L1 =2 K1

1

L2 =2 K2

1

W K



(4.89)

Combining Eq. (4.84) and (4.89) can predict the long-term production performance of gas wells. Example 5: For a gas reservoir, the thickness is 10 m, the porosity is 0.1, the formation temperature is 80°C, the initial reservoir pressure is 30 MPa, the gas gravity is 0.6, the permeability in the production zone is 10 3 1023 μm2, the permeability in the recharge zone is 5 3 1023 μm2, and the permeability in the transition zone is 0.1 3 1023 μm2. A well is located in the middle of the production zone (Fig. 4
20). The absolute open flow potential is 59 3 104 m3/d, and the gas production equation is as follows Δp2 5 7:74344qg 1 0:127014q2g Predict the reservoir performance and analyze the characteristics of pressure decline.

Material balance equation of a gas reservoir

135

qg

Figure 4
20 Schematic diagram of parameters of an actual gas reservoir.

Figure 4
21 p/Z curve with gas recharge.

Solution We are assuming that the well is producing at a rate of 10.0 3 104 m3/d. Fig. 4
21 shows the relationship between apparent reservoir pressure (p/Z)1 and production Gp in the production zone. We can see that the plot shows an obvious two-segment trend. According to Eq. (4.85), due to the low-permeability zone and resistance, the (p/Z)1 curve for compartments yields reserves ΔG smaller than the standard p/Z curve.
 μCg W G22 qg (4.90) ΔG 5  K Lh G1 1 G2

136

Reserves Estimation for Geopressured Gas Reservoirs

Figure 4
22 Comparison of traditional material balance method and the proposed method.

According to the above equation, given certain compartment reserves, ΔG not only depends on the characteristic parameters of the gas reservoir but also relates to the initial production allocation in the production zone (Hower and Collins, 1989), which is shown in two aspects. 1. The more significant the permeability of the gas reservoir and the contact area between two compartments and the smaller the zone width between the two compartments, the smaller the residual reserves. 2. The more significant the initial production allocation, the larger the residual reserves, consistent with the experimental results. Assuming that the well is producing at 10.0 3 104 m3/d, the comparison between the traditional material balance method and the proposed method is illustrated in Fig. 4
22. The stable production period is 0.85 years for the proposed method and 1.32 years for the traditional method. The stable period is calculated to be 0.9 years through numerical simulation. The traditional material balance method yields an optimistic prediction since it treats the drainage area of a single well as a large homogeneous container and completely ignores the changes in reservoir heterogeneity, which leads to the pressure balance drop during actual production. Due to the existence of the low-permeability zone, the pressure between the production zone and the recharge zone is quite different, resulting in the occurrence of unbalanced production. The proposed method considers factors such as reservoir heterogeneity and flow resistance, which is close to the actual situation, as confirmed by numerical simulation.

Material balance equation of a gas reservoir

137

Figure 4
23 Performance profile by the two-compartment material balance model with recharge.

Assuming that the well is producing at 10.0 3 104 m3/d, the average formation pressure and production changes in the production zone and recharge zone are illustrated in Fig. 4
23. As the production continues, the recharge zone gradually plays a role, especially after entering the decline stage, and the proportion of the recharge zone’s contribution to the total production is even more critical.

4.3 Drive index of gas reservoir When two or more drive energies act on the reservoir simultaneously, the action degree of each energy can be expressed in terms of the drive index (Pirson, 1977). Transforming Eq. (4.27), we have   G Bg 2 Bgi GBgi Ce Δp We 1 1 5 1:0 (4.91) Gp Bg 1 Wp Bw Gp Bg 1 Wp Bw Gp Bg 1 Wp Bw The three terms on the left side of Eq. (4.91) represent the elastic energy of natural gas, the elastic energy of rock and irreducible water, and the drive index of the edge/bottom water energy, respectively, and they are defined as follows. For depletion drive index,   G Bg 2 Bgi EDI2G 5 (4.92) Gp Bg 1 Wp Bw

138

Reserves Estimation for Geopressured Gas Reservoirs

For expansion (rock and irreducible water) drive index, EDI2C 5

GBgi Ce Δp Gp Bg 1 Wp Bw

(4.93)

We Gp Bg 1 Wp Bw

(4.94)

For water-drive index, EDI2W 5

Then Eq. (4.91) can be expressed as EDI2G 1 EDI2C 1 EDI-W 5 1:0

(4.95)

Eq. (4.95) is referred to as the drive index equation. The above three indexes are constantly changing in the development process of waterdrive gas reservoirs. Usually, the depletion drive index predominates the development process of gas reservoirs. If the influence of water-soluble gas is considered, according to Eq. (4.45), we have   G Bg 2 Bgi GBgi C e Δp  1   Gp Bg 1 Wp Bw 2 Bg Rsw Gp Bg 1 Wp Bw 2 Bg Rsw (4.96) We   51 1 Gp Bg 1 Wp Bw 2 Bg Rsw where   1  Swi C tw 1 C f 1 M C tw 1 C f 1 2 Swi   1 Btw ðpÞ 2 Btw ðpi Þ C tw 5 Btw ðpi Þ pi 2 p  Btw ðpÞ 5 Bw ðpÞ 1 Rswi 2 Rsw ðpÞ Bg ðpÞ

C e ðpÞ 5

(4.97) (4.98) (4.99)

For depletion-drive index,

  G Bg 2 Bgi   EDI-G 5 Gp Bg 1 Wp Bw 2 Bg Rsw

(4.100)

For expansion (rock, irreducible water, formation water, and solution gas)-drive index, EDI-C 5

GBgi C e Δp   Gp Bg 1 Wp Bw 2 Bg Rsw

(4.101)

Material balance equation of a gas reservoir

139

For water-drive index, EDI2W 5

We   Gp Bg 1 Wp Bw 2 Bg Rsw

(4.102)

The expansion (rock, irreducible water, formation water, and solution gas)-drive index can be further divided into, EDI2C 5 EDI2Cwater 1 EDI2Csolution gas 1 EDI2Crock GBgi C w ΔpðSwi 1 M Þ   EDI2Cwater 5  Gp Bg 1 Wp Bw 2 Bg Rsw ð1 2 Swi Þ

(4.104)

GBgi ðSwi 1 M ÞðRswi 1 Rsw ÞBg   Gp Bg 1 Wp Bw 2 Bg Rsw ð1 2 Swi ÞBwi

(4.105)

GBgi C f Δpð1 1 M Þ   Gp Bg 1 Wp Bw 2 Bg Rsw ð1 2 Swi Þ

(4.106)

EDI2Csolution gas 5  EDI2Crock 5 

(4.103)

Example 6: For a gas reservoir, the initial pressure is 74 MPa, the temperature is 377.15K, the gas gravity is 0.568, the initial water saturation is 0.32, the initial rock compressibility factor is 2.5 3 1023/MPa, the formation water compressibility factor is 5.6 3 1024/MPa, the water volume ratio is 2.0, and the cumulative water production Wp 5 0. Analyze the change law of drive indexes in the development process of the gas reservoir. Solution The gas deviation factor is shown in Fig. 4
7, and the regression formula is f ðpÞ 5 0:99796 2 0:0093p 1 3:64632 3 1024 p2 2 2:17327 3 1026 p3 According to the tank model, calculate the water influx with Eq. (4.37) (Tarek, 2019), We 5 ðCf 1 Cw ÞWi ðpi 2 pÞ Calculate the volume factor according to Eqs. (4.3)
(4.50), Bg 5 3:4564 3 1024

ZT p

According to Eqs. (4.92), (4.93), and (4.94), three drive index curves are shown in Fig. 4
24.

140

Reserves Estimation for Geopressured Gas Reservoirs

Figure 4
24 Drive index curves for Example 6.

4.4 Apparent initial gas in place of gas reservoir Transforming Eq. (4.27) for water-drive gas reservoirs, we have Ga 5

Gp Bg 1 Wp Bw We 5G1 Bg 2 Bgi ð1 2 Ce ΔpÞ Bg 2 Bgi ð1 2 Ce ΔpÞ

(4.107)

where gas in place, that is, the sum of G and  Ga is apparent initial We = Bg 2 Bgi ð1 2 Ce ΔpÞ . Ignoring the elastic changes of rock and irreducible water, Eq. (4.107) can be simplified as Ga 5

Gp Bg 1 Wp Bw We 5G1 Bg 2 Bgi Bg 2 Bgi

(4.108)

Eq. (4.108) is the linear equation of apparent initial gas in place for water-drive gas reservoirs (Havlena and Odeh, 1963). For volumetric gas reservoirs, the initial gas in place has nothing to do with the cumulative production, so the plot of Ga versus G is a horizontal line. For water-drive gas reservoirs, the water influx will gradually increase as the production progresses. At this time, the plot of Ga versus G is an upward curved line. This plot can determine the initial gas in place and the water influx, as shown in Fig. 4
12. Example 7: For a gas reservoir, the initial reservoir pressure is 74 MPa, the temperature is 377.15K, the gas gravity is 0.568, the initial water saturation

Material balance equation of a gas reservoir

141

is 0.32, the initial rock compressibility factor is 2.5 3 1023/MPa, the formation water compressibility factor is 5.6 3 1024/MPa, the water volume ratio is 2.0, and the cumulative water production Wp 5 0. Analyze the change law of apparent initial gas in place during the development of the gas reservoir. Solution The gas deviation factor is shown in Fig. 4
7, and the regression formula is f ðpÞ 5 0:99796 2 0:0093p 1 3:64632 3 1024 p2 2 2:17327 3 1026 p3 Calculate the water influx according to the tank model Eq. (4.37) (Tarek, 2019), We 5 ðCf 1 Cw ÞWi ðpi 2 pÞ Calculate the volume factor according to Eqs. (4.3)
(4.50), Bg 5 3:4564 3 1024

ZT p

Ga/G

Calculate the apparent initial gas in place according to Eq. (4.87), as shown in Fig. 4
25.

Figure 4
25 Apparent initial gas in place for the water-drive gas reservoir in Example 7.

142

Reserves Estimation for Geopressured Gas Reservoirs

4.5 Sensitivity analysis of key parameters Eq. (4.46) shows the key parameters, including the average reservoir pressure (p), gas deviation factor (Z), and cumulative effective compressibility factor (C e a comprehensive reflection of rock compressibility factor, irreducible water saturation, and water body size). In addition, the material balance calculation is also highly sensitive to recovery efficiency. This section first introduces the concept of compressibility factor and then conducts sensitivity analysis on critical parameters.

4.5.1 Rock compressibility factor Reservoir rocks are affected by the internal stress exerted by the fluid in the pores and by the external stress exerted by the overlying rock. With the production of fluid, the pore pressure decreases and the effective stress increases, resulting in the compression of the rock skeleton, leading to changes in rock particles, pores, and total volume. 4.5.1.1 Rock compressibility factor Rock volume includes skeleton volume, pore volume, and appearance volume. Calculations in reservoir engineering are only sensitive to pore volume change, that is, the compressibility of pore volume to pore pressure, which Cp represents. In order to correspond to the fluid compressibility factor, it is usually referred to as the rock compressibility factor in reservoir engineering, which is usually represented by Cf (Petroleum and Natural Gas Industry Standard of the People's Republic of China, SY/T 6580, 2004) and is defined as Cf 5

dVp Vp dp

(4.109)

where Cf : rock (pore) compressibility factor, MPa21, Vp: pore volume, m3, p: pressure, MPa. Eq. (4.109) shows that the rock compressibility factor is a function of pressure. If the stress sensitivity of the pore volume is ignored, Cf is generally considered to be a constant. There are experimental analysis methods (Petroleum and Natural Gas Industry Standard of the People's Republic of China, SY/T 5815, 2016) and empirical formula methods to obtain the compressibility factor. The Hall chart is shown

Material balance equation of a gas reservoir

143

Figure 4
26 Hall empirical chart for rock compressibility factor. Modified from Hall, H.N. 1953. Compressibility of reservoir rocks. Journal of Petroleum Technology, 5(1), 17
19. Permission to publish by the SPE, Copyright SPE.

in Fig. 4
26 (12 testing gas reservoirs), and the empirical formula (Hall, 1953) is 2:587 3 10-4 (4.110) Cf 5 φ0:4358 where Cf: rock compressibility factor, MPa, φ: porosity, decimal. Based on 79 core samples with porosity ranging from 2% to 23%, Newman (1973) obtained the empirical formula for matching the compressibility factor of cemented sandstone, as follows. Cf 5

0:014115 ð1155:8721φÞ1:428586

(4.111)

Based on core samples of limestone with porosity ranging from 2% to 33%, Newman (1973) obtained the empirical formula for matching the compressibility factor of limestone, as follows. 123:7899 Cf 5  0:93023 112:36715 3 106 φ

(4.112)

Horne (1990) established the following three empirical formulas based on Newman’s experimental data. For cemented limestone, there is   C f 5 1:45038 3 1024 exp 4:026 2 23:07φ 1 44:28φ2 (4.113)

144

Reserves Estimation for Geopressured Gas Reservoirs

Figure 4
27 Data points and empirical formula of Newman’s rock compressibility factor. Modified from Newman, G.H. 1973. Pore-volume compressibility of consolidated, friable, and unconsolidated reservoir rocks under hydrostatic loading. Journal of Petroleum Technology, 25(2), 129
134. Permission to publish by the SPE, Copyright SPE.

For cemented sandstone, there is   Cf 5 1:45038 3 1024 exp 5:118 2 36:26φ 1 63:98φ2

(4.114)

For loose sandstone, there is Cf 5 1:45038 3 1024 exp ð34:012φ 2 6:8024Þ

(4.115)

Newman’s experimental data points and the empirical formula lines of Horne (1990) and Hall (1953) are shown in Fig. 4
27. It can be seen that the correlation between rock compressibility and porosity is poor, and the correlation can only give a preliminary estimate of the order of magnitude. Therefore, experimental determinations should be made to the actual reservoirs. For standard pressure systems, the rock compressibility factor usually ranges from 4.35 3 1024 to 8.70 3 1024/MPa. It is generally believed that the rock compressibility factor of high-pressure and ultra-highpressure gas reservoirs is very high—of the order of 1023/MPa, which is one order of magnitude higher than that of conventional gas reservoirs (Harville and Hawkins, 1969; Duggan, 1972; Ramagost and Farshad, 1981; Elsharkawy, 1995), ranging from 20 3 1024 to 30 3 1024/MPa, as shown in Table 4
8 and Table 4
9. The compressibility factor of the

Table 4
8 Rock compressibility factors of three typical ultra-high-pressure gas reservoirs. Gas reservoir

NS2B, North Ossun, Louisana

Anderson L

Offshore, Louisana

Literature

Harville and Hawkins (1969) 3810 0.24 200 61.51 1.64 393 0.34 43.51 (Initial pressure) 4.35

Duggan (1972) 3404 0.24 0.7
791 65.55 1.91 403 0.35 21.76 (Constant) 4.35

Ramagost and Farshad (1981) 4054 0.24 200 78.90 1.95 402 0.22 28.28 (Constant) 4.41

68.17 32.28

35.82 19.54

37.50 133.00

Buried depth, m Porosity, decimal Permeability, 1023 μm2 Initial pressure, MPa Pressure coefficient Formation temperature, K Irreducible water saturation, decimal Rock compressibility factor, 1024/MPa Formation water compressibility factor, 1024/MPa Effective compressibility factor, 1024/MPa Volumetric reserves, 108 m3

Table 4
9 Rock compressibility factor of Louisana ultra-high-pressure gas reservoir (Elsharkawy, 1995). Reservoir no.

162

269

164

183

33

268

70

195

Formation temperature, K Porosity, decimal Irreducible water saturation, decimal Initial pressure, MPa Buried depth, m Pressure coefficient Volumetric reserves, 108 m3

432 0.25 0.35 91.20 4433 2.06 5.0 4.35 36.26 22.05

403 0.25 0.26 93.08 4867 1.92 3.9 4.35 40.61 28.92

416 0.22 0.26 73.77 4676 1.58 4.1 4.35 39.16 27.85

407 0.24 0.28 51.90 4176 1.24 13.4 4.35 33.36 22.80

417 0.22 0.23 79.63 4341 1.83 56.6 4.35 34.81 25.80

390 0.27 0.26 62.74 4145 1.52 7.1 4.35 33.36 23.55

408 0.28 0.3 69.84 4572 1.54 3.5 4.35 37.71 25.09

407 0.24 0.28 75.06 4176 1.79 10.3 4.35 33.36 22.80

Effective compressibility factor, 1024/MPa Rock compressibility factor, 1024/MPa

Material balance equation of a gas reservoir

147

NS2B loose sandstone ultra-high-pressure gas reservoir is as high as 43.51 3 1024/MPa. If the porosity is 0.25, the rock compressibility factor is only 4.82 3 1024/MPa calculated using Hall’s empirical formula Eq. (4.110), and 5.65 3 1024/MPa calculated using Eq. (4.115). In the Kela 2, Dina 2, and Keshen 2 deep high-pressure gas reservoirs in the Tarim Basin, as the buried depth of the reservoir increases, the porosity becomes smaller, the permeability becomes lower, and the rock compressibility factor under the initial effective stress condition also gradually decreases (Table 4
10 and Fig. 4
28). The experimental results show that under the initial stress conditions, the rock compressibility factor is a function of porosity and permeability (Fig. 4
29). However, factors such as lithology and shale content are also influential (Xie et al., 2005). Therefore, using the univariate Hall or Newman empirical formula for material balance calculations is not recommended. 4.5.1.2 Effective compressibility factor In the calculation of the material balance equation, the formation water compressibility factor Cw and rock compressibility factor Cf appear in the form of effective compressibility factor, as follows. Ce 5

Cf 1 Swi Cw 1 2 Swi

(4.116)

where Ce: effective compressibility factor, MPa, Cw: formation water compressibility factor, MPa, Swi: irreducible water saturation, decimal. 4.5.1.3 Cumulative rock compressibility factor The cumulative compressibility factor C f of rock is defined (Fetkovich et al., 1998) as
 1 Vpi 2 Vp Cf 5 (4.117) Vpi pi 2 p Eq. (4.117) shows that the cumulative compressibility factor of rock is a function of pressure and a function of initial pressure. Although the changes can be measured under certain experimental conditions, as shown in Fig. 4
30, the experimental results are only for reference due to the influence of the number and representativeness of the tested samples.

Table 4
10 Rock compressibility factors of three high-pressure gas reservoirs in the Tarim Basin. Reservoir

Kela 2

Dina 2

Keshen 2

Buried depth, m Number of samples Porosity of samples, % Air permeability of samples, 1023 μm2 Initial reservoir pressure, MPa Pressure coefficient Formation temperature, K Irreducible water saturation, decimal Rock compressibility factor in original condition, 1024/MPa Formation water compressibility factor, 1024/MPa Effective compressibility factor, 1024/MPa Volumetric reserves, 108 m3

3750 27 6.4
20, avg. 13.90 0.1
722, avg. 107.8 74.35 2.02 373 0.32 26.33

5050 7 9
14, avg. 12 0.1
2.2, avg. 1.0 106.20 2.12 409 0.34 17.30

6500 10 1.53
9.00, avg. 6.0 0.01
0.12, avg. 0.05 116.78 1.83 440 0.35 7.64

5.65 41.38 2840.00

4.35 (empirical value) 28.45 1659.03

4.35 (empirical value) 14.10 1542.93

Material balance equation of a gas reservoir

149

102 KL2 DN2 KS2

Cfi , 10 -4 MPa-1

Avg. 26.33 Avg. 17.30

101 Avg. 7.64

100 60

70

80

90

100

110

120

pi , MPa Figure 4
28 Rock compressibility factors of three high-pressure gas reservoirs in the Tarim Basin in the initial condition.

Figure 4
29 Relationship between the rock compressibility factor in the initial condition and the permeability and porosity.

150

Reserves Estimation for Geopressured Gas Reservoirs

Figure 4
30 Schematic diagram of Cf versus C f for the sandstone gas reservoirs in Mexico Gulf. Modified from Fetkovich, M.J, Reese, D.E., Whitson, C.H. 1998. Application of a general material balance for high-pressure gas reservoirs (includes associated paper 51360). SPE Journal, 3(1), 3-13. Permission to publish by the SPE, Copyright SPE.

According to Eqs (4.116) and (4.117), C f equals Cf under the initial conditions. The relationship between C f and Cf is as follows. C f ðpi 2 pn Þ 5

n X

ðCf Þj ðΔpÞj

(4.118)

j51

4.5.1.4 Cumulative effective compressibility factor of gas reservoirs The cumulative effective compressibility factor of gas reservoirs C e ðpÞ is defined as Eq. (4.119) (Fetkovich et al., 1998), a function of the initial pressure and pressure, and its reliability depends on the core representativeness and experimental conditions. C e ðpÞ 5

  1  Swi C w 1 C f 1 M C w 1 C f 1 2 Swi

(4.119)

where M: water volume ratio, dimensionless, Swi: initial water saturation, decimal. If the influence of the water body is ignored, M 5 0, then Eq. (4.119) can be simplified as C e ðpÞ 5

Swi C w 1 C f 1 2 Swi

(4.120)

Material balance equation of a gas reservoir

151

Therefore, theoretically, when applying the ultra-high-pressure material balance equation, it is not advisable to confuse the rock compressibility factor Cf with the rock cumulative compressibility factor C f or simply replace C e ðpÞ with Ce . Example 8: For the Keshen 2 gas reservoir, the initial reservoir pressure is 116.7 MPa, the temperature is 440K, the gas gravity is 0.65, the initial water saturation is 0.35, the formation water compressibility factor is 4.35 3 1024/MPa, the rock compressibility factor is 7.64 3 1024/MPa, and the production data is shown in Table 4
11. Analyze the influence of rock effective compressibility factor on the estimation of dynamic reserves. Solution Assuming that the compressibility factors of formation water and rock are constant, calculate the effective compressibility factor according to Eq. (4.120). Ce ðpÞ 5

Swi Cw 1 Cf ð0:35 3 4:35 1 7:64Þ 3 1024 5 14:1 3 1024 MPa21 5 1 2 Swi 1 2 0:35

If Ce takes the values 0, 14.1 3 1024/MPa, and 28.2 3 1024/MPa, respectively, the pressure decline curves calculated by the pressure correction method (see Chapter 5, OGIP Estimations for Geopressured Gas Reservoir for details) are shown in Fig. 4
31. The reserves are estimated to be 1000, 750, and 593 3 108 m3, respectively. The ratio of the maximum to the minimum calculation results is 169%. The compressibility term has an essential influence on the calculation results of dynamic reserves. Table 4
11 Production data of Keshen 2 gas reservoir. Gp, 108 m3

p, MPa

Δp, MPa

0.04 33.33 53.55 59.78 69.55 83.74 95.76 108.46

116.22 104.17 96.56 95.43 93.46 91.39 88.44 87.32

12.05 19.66 20.79 22.76 24.83 27.78 28.90

Z

1.8465 1.7232 1.6464 1.6349 1.6150 1.5941 1.5643 1.5530

p/Z, MPa

pD [p/Z/(p/Z)i]

62.94 60.45 58.65 58.37 57.87 57.33 56.53 56.23

1.00 0.96 0.93 0.93 0.92 0.91 0.90 0.89

152

Reserves Estimation for Geopressured Gas Reservoirs

Figure 4
31 Influence of effective compressibility factor on the reserve estimation for Keshen 2 gas reservoir.

4.5.2 Size of water body As the formation pressure declines for water-drive gas reservoirs, the bottom water will continue to invade. The dynamic reserves will be overestimated using the data from the initial development stage. As shown in Fig. 4
14, when the water volume ratio is 2, the calculation result from the early data is 2.5 times the actual reserves.

4.5.3 Pressure depletion When using the classic two-segment method and nonlinear regression method (see Chapter 5, OGIP Estimations for Geopressured Gas Reservoir for details), the decline of apparent reservoir pressure is significant, as shown in Fig. 4
32. The parameters, such as the time when the inflection point appears and the starting point of the nonlinear regression method, will be discussed in Chapter 5, OGIP Estimations for Geopressured Gas Reservoir.

4.5.4 Apparent reservoir pressure Apparent reservoir pressure (p/Z) is an essential parameter in the material balance equation. When the recovery efficiency is low, the apparent reservoir pressure error significantly impacts the material balance calculation. The influence of the gas deviation factor is implicit in the apparent reservoir pressure. If there is no PVT data, the deviation factor can be

Material balance equation of a gas reservoir

153

Figure 4
32 Sensitivity analysis on depletion of apparent reservoir pressure.

calculated by the method proposed in Chapter 3, Physical Properties of Natural Gas and Formation Water. The apparent reservoir pressure is mainly affected by pressure buildup time and reservoir heterogeneity. Here, Eq. (4.3) is taken for sensitivity analysis of volumetric gas reservoirs. Making Eq. (4.3) dimensionless, we have pD 1 GpD 5 1

(4.121)

where the dimensionless cumulative production GpD 5 Gp =G and the . From Eq. (4.121), if dimensionless apparent reservoir pressure pD 5 pp=Z i =Zi the error of pD is δ, then the error of GpD is 2δ, as follows.   ðpD 1 δÞ 1 GpD 2 δ 5 1 (4.122) From Eq. (4.122), the relative error ΔG=G of reserves caused by the dimensionless apparent reservoir pressure error is ΔG=G 5 1 1 δ

(4.123)

If the dimensionless formation pressure increase δ, the reserves will be increase δ, otherwise, the reserves will be decrease δ, as shown in Fig. 4
33. Assuming that the apparent initial reservoir pressure is accurate, the reserve error caused by the average formation pressure error is shown in Fig. 4
34. The smaller the recovery percent within a certain range, the greater the pressure error and the greater the reserve error. When the recovery percent is more significant than 18.75% (dimensionless apparent reservoir pressure is 0.8125), the calculated reserve error in the two cases

154

Reserves Estimation for Geopressured Gas Reservoirs

Figure 4
33 Reserve error caused by apparent reservoir pressure error.

Figure 4
34 Influence of the apparent reservoir pressure error on the estimation of reserves (at accurate initial pressure). (A) When pD error is 6 1%, ΔG/G varies with Gp/G; (B) When pD error is 6 2%, ΔG/G varies with Gp/G.

is approximately equal to the pressure error, that is, 6 1.0% and 6 2.0%. When the recovery percent is equal to 10%, the calculated reserve error in the two cases is within 6 3% and 6 5%, respectively. When the recovery percent is equal to 5%, the calculated reserve error in the two cases is within 6 10% and 6 20% to 6 30%, respectively. Therefore, under normal circumstances, the recovery percent should be greater than 10% (Petroleum and Natural Gas Industry Standard of the People's Republic of China, SY/T 6098, 2010) when evaluating reserves with the material balance method. For low-permeability gas reservoirs or fractured tight gas reservoirs, the measured average shut-in pressure of all gas reservoirs may be lower,

Material balance equation of a gas reservoir

155

resulting in an underestimate of dynamic reserves. If the flow meets the boundary-dominated flow, then the flow material balance method or multicompartment material balance method can be used to estimate the reserves.

4.5.5 Influence of water-soluble gas The gas dissolved in irreducible water and formation water in original conditions can provide energy for reservoir development and be produced in the late stage of development (at pressure ,10 MPa). Generally, solution gas is about 2% to 10% of free gas reserves (Fetkovich et al., 1998), depending on the size of the aquifer and the original water
gas ratio. The solution gas reserves will be higher for gas reservoirs with a higher CO2 content (see Chapter 5 for case analysis).

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CHAPTER 5

Original gas in place estimations for geopressured gas reservoirs Contents 5.1 Classical two-segment method 5.1.1 Hammerlindl method 5.1.2 Chen method 5.1.3 Gan
Blasingame method 5.1.4 Discussion on the time of inflection point of p/Z curve 5.2 Linear regression method 5.2.1 Ramagost
Farshad method 5.2.2 Roach method 5.2.3 Poston
Chen
Akhtar method 5.2.4 Becerra-Arteaga method 5.2.5 Havlena
Odeh method 5.2.6 Sun method 5.3 Nonlinear regression method 5.3.1 Binary regression method 5.3.2 Nonlinear regression method 5.3.3 Starting point of the nonlinear regression method 5.4 Type curve matching analysis method 5.4.1 Ambastha method 5.4.2 Fetkovich method 5.4.3 Gonzalez method 5.4.4 Sun method 5.4.5 Multiwell production decline analysis method 5.5 Trial-and-error analysis method 5.6 Original gas in place estimation procedure of geopressured gas reservoirs 5.6.1 Summary of calculation methods 5.6.2 Recommended methods 5.6.3 Recommended procedure 5.6.4 Basic data preparation 5.6.5 Comparative analysis of results

158 158 161 164 171 172 173 175 178 190 195 199 206 206 209 215 223 223 227 231 235 240 246 255 255 255 258 258 260

Reserves Estimation for Geopressured Gas Reservoirs © 2023 Petroleum Industry Press. DOI: https://doi.org/10.1016/B978-0-323-95088-6.00005-5 Published by Elsevier Inc. All rights reserved.

157

158

Reserves Estimation for Geopressured Gas Reservoirs

Estimating the original gas in place (OGIP) of high-pressure, ultra-high pressure, and fractured stress-sensitive gas reservoirs has always been challenging. This chapter focuses on 22 different methods for OGIP calculation based on the material balance equation combined with gas field examples, which generally fall into five categories, namely, classical slope two-segment analysis method, linear regression analysis method, nonlinear regression analysis method, type curve matching, and trial-and-error analysis method. In addition, the process and recommendations for OGIP estimation of high-pressure gas reservoirs are introduced.

5.1 Classical two-segment method The classical analysis method assumes that the p/Z curve of high-pressure and ultrahigh-pressure gas reservoirs has two straight-line segments, and the second straight line shows a steeper slope than the first straight line (Harville and Hawkins, 1969). The early production data are used to plot the p/Z curve, and different methods, such as the Hammerlindl method (1971), Chen method (1983), and Gan
Blasingame method (2001), are adopted to correct the apparent original gas in place (OGIP) to actual OGIP.

5.1.1 Hammerlindl method Hammerlindl (1971) introduced two calculation methods, namely, the average compressibility method and the corrected reservoir volume method, to correct apparent original gas in place (OGIP) obtained from an extrapolation of the early straight line on the p/Z curve geopressured gas reservoirs. 5.1.1.1 Average compressibility method The calculation procedure is as follows, 1. Determine the rock compressibility Cfi at the point under the initial condition and Cf at the point under the condition of pressure coefficient higher than 1.13 MPa/100 m (i.e., 0.5 psi/ft). 2. Determine Cgi’s and Cg’s gas compressibility at the above two points. 3. Calculate the effective compressibility at the above two points. Cei 5

Cgi Sgi 1 Cwi Swi 1 Cf Sgi

(5.1)

Ce 5

Cg Sgi 1 Cwi Swi 1 Cf Sgi

(5.2)

Original gas in place estimations for geopressured gas reservoirs

159

  4. Find the Ce =Cg avg , which is approximately equal to       Ce =Cg avg 5 0:5 Ce =Cg i 1 Ce =Cg

(5.3)

5. Plot the p/Z curve and determine the apparent OGIP Ga. 6. Calculate the actual OGIP.   G 5 Ga = Ce =Cg avg

(5.4)

5.1.1.2 Corrected reservoir volume method The calculation equation is as follows, G 5 αGa

(5.5)

where α is the correction factor α5 

1. 2.

3. 4.



Egi 2 Eg

Egi 2 Eg 1

Eg ðCf 1 Cw Swi Þðpi 2 pÞ 1 2 Swi

Egi 2 Eg  5 ; Eg 5 1=Bg Egi 2 Eg 1 Eg ΔpCe

The calculation procedure is as follows. Plot the p/Z curve and determine the apparent OGIP Ga. Determine the gas formation volume factor Bgi at the point under the initial condition and Bg at the point under the condition of pressure coefficient higher than 1.13 MPa/100 m (i.e., 0.5 psi/ft). Calculate the effective compressibility Ce and the correction factor α. Calculate the actual OGIP using Eq. (5.5).

Example 1: The NS2B reservoir has a pressure coefficient of 1.64, with the static and dynamic data shown in Appendix 1 (Harville and Hawkins, 1969). Use the Hammerlindl method to calculate the OGIP. Solution The calculation result of the average compressibility method is shown in Table 5
1. Based on the data in Appendix Table 1
2, the apparent OGIP determined by the p/Z curve is Ga 5 62.3 3 108 m3 (Fig. 5
1), and as Eq. (5.4), the OGIP is   G 5 Ga = Ce =Cg avg 5 62:3=1:937 5 32:2 3 108 m3

160

Reserves Estimation for Geopressured Gas Reservoirs

Table 5
1 Calculation results of the average compressibility method. Pressure condition

Cf, MPa21

Cg, MPa21

Ce, MPa21

Ce/Cg

(Ce 1 Cg)avg

Original pressure Dew point pressure

3.18 3 1023

4.35 3 1023

9.40 3 1023

2.16

1.937

7.06 3 1023

1.21 3 1022

1.71

50

p/Z, MPa

40

30

20

10 36.8

62.3

0 0

10

20

30

40

50

60

70

Gp , 108m3 Figure 5
1 p/Z versus Gp for NS2B reservoir. Modified from Hammerlindl, D.J., 1971. Predicting gas reserves in abnormally pressure reservoirs. SPE 3479-MS. Permission to publish by the SPE, Copyright SPE.

Based on the corrected reservoir volume method and the data in Appendix Table 1
1 and Appendix Table 1
3, we have Egi 2 Eg Eg ðCf 1 Cw Swi Þðpi 2 pÞ Egi 2 Eg 1 1 2 Swi 302:5 2 281:5   5 5 0:518 281:5 3 3:18 3 1023 1 4:35 3 1024 3 0:34 3 ð61:51 2 47:71Þ 302:5 2 281:5 1 1 2 0:34

α5

Based on Eq. (5.5), the OGIP is G 5 αG a 5 0:518 3 62:3 5 32:2 3 108 m3

Original gas in place estimations for geopressured gas reservoirs

161

The OGIP is determined to be 36.8 3 108 m3 by directly extrapolating the middle-late time data in Fig. 5
1, slightly higher than that obtained by using the Hammerlindl method, which is mainly due to water influx in the late stage of development (Hammerlindl, 1971). 5.1.1.3 Relationship between the two methods The correction factor α in Eq. (5.5) can be expressed as (Ambastha, 1993)
 Zi Ce pΔp 21 (5.6) α 5 11 Zpi 2Zi p Eq. (5.3) can be further expressed as      Ce =Cg avg 5 0:5 1 1 Ce 1=Cg 1 1=Cgi

(5.7)

Based on Eq. (3.42), the gas compressibility can be expressed as Cg 5

1 1 @Z 2 p Z @p

@Z Zi 2 Z 5 @p pi 2 p

(5.8)

(5.9)

Substituting Eqs. (5.8) and (5.9) into Eq. (5.7),   Ce ΔpZi p Ce =Cg avg 5 1 1 x Zpi 2 Zi p

(5.10)


 pi Z 1 x 5 0:5 Zi p

(5.11)

For a high-pressure gas reservoir, pi =p . 1 and Z=Zi , 1. If x 5 1, the calculation   results of Eqs. (5.6) and (5.10) are consistent. If x . 1, then Ce =Cg avg . α, and the calculation result of the average compressibility method is smaller than that of the corrected reservoir volume method. Otherwise, the calculation result of the average compressibility method is greater than that of the corrected reservoir volume method.

5.1.2 Chen method As described in Section 4.1, a closed high-pressure gas reservoir’s p/Z curve is quasiparabolic and can be approximated to a broken line

162

Reserves Estimation for Geopressured Gas Reservoirs

(Hammerlindl, 1971; Chen, 1983). The first straight line represents the stage when the high-pressure gas reservoir is subject to recompaction, and the OGIP obtained from its extrapolation to p/Z 5 0 is the apparent OGIP Ga. The second straight line indicates the stage when the recompaction disappears, and the reservoir exhibits normal pressure behavior, and the OGIP obtained from its extrapolation to p/Z 5 0 is the actual OGIP, that is, G. The equation for G by Ga is G5

Ga C ðp 2 pws Þ 11 e i pi =Zi 2 1 pws =Zws

(5.12)

Where G: OGIP, 108 m3, Ga: apparent OGIP, 108 m3, pws: hydrostatic pressure of the gas reservoir, MPa, Zws: gas deviation factor under the condition of pws and gas reservoir temperature, dimensionless. The intercept a1 and slope b1 of the first straight line can be used to calculate the OGIP using Eq. (5.13) G5

a1 =b1 C ðp 2 pws Þ 11 e i pi =Zi 2 1 pws =Zws

(5.13)

Example 2: For the offshore high-pressure gas reservoir in Louisiana, the United States, with the production data shown in Appendix Table 2
1 (Ramagost and Farshad, 1981), the buried depth is 4055 m, the initial formation pressure is 78.90 MPa, the gas reservoir temperature is 128.4°C, the gas relative density is 0.6, the initial water saturation is 0.22, the formation water compressibility is 4.41 3 1024/MPa, and the rock compressibility is 3.325 3 1023/MPa. Use the Chen method to calculate the OGIP. Solution Calculate the effective compressibility based on Eq. (4.15). We have Ce 5

Cw Swi 1 Cf 4:41 3 1024 3 0:22 1 33:25 3 1024 5 4:387 3 1023 MPa21 5 1 2 Swi 1 2 0:22

Plot the p/Z curve based on the data in Appendix Table 2
1, as shown in Fig. 5
2.

163

Original gas in place estimations for geopressured gas reservoirs

60 p/Z=52.34489-0.28255Gp

50

p/Z , MPa

40 30 20 10 132.07

185.26

0 0

50

100

150

200

Gp , 108m3 Figure 5
2 p/Z curve of offshore gas reservoir.

The intercept and slope of the first straight line are 52.34489 and 20.28255, respectively. The apparent OGIP of the gas reservoir is Ga 5

a1 52:34489 5 185:26 3 108 m3 5 0:28255 b1

The buried depth is 4055 m, the corresponding hydrostatic pressure is 40.55 MPa, the gas deviation factor is 1.09, and the apparent hydrostatic pressure is 37.202 MPa. Substitute these parameters into Eq. (5.13). We have G5

Ga 185:26 5 5 132:07 3 108 m3 23 C e ðpi 2 pws Þ 4:387 3 10 3 ð78:90 2 40:55Þ 11 pi =Zi 2 1 1 1 52:743 2 1 37:202 pws =Zws

The parameter pws in Eq. (5.12) is the hydrostatic pressure of the gas reservoir, which means that only when the pressure coefficient is 1.0, the highpressure characteristics will be ended. Hammerlindl (1971) defined the pressure coefficient lower boundary as 1.13; correspondingly, the hydrostatic pressure is 45.82 MPa, the gas deviation factor is 1.15, the apparent hydrostatic pressure is 39.85 MPa, and the OGIP is 127.9 3 108 m3. If the pressure coefficient boundary is set to 1.2 (Cheng et al., 2016), the hydrostatic pressure is 48.66 MPa, the gas deviation factor

164

Reserves Estimation for Geopressured Gas Reservoirs

is 1.18, the apparent hydrostatic pressure is 41.27 MPa, and the OGIP is 125.4 3 108 m3. If the pressure coefficient lower boundary is set to 1.3 according to the Classification of Gas Reservoirs (National Standard of the People's Republic of China, GB/T 26979, 2011), the hydrostatic pressure is 52.72 MPa, the gas deviation coefficient is 1.22, the apparent hydrostatic pressure is 43.18 MPa, and the OGIP is 122.0 3 108 m3. Fig. 5
2 shows that all the data points with pressure coefficients greater than 1.3 (pressure greater than 52.72 MPa) fall on the regression line.

5.1.3 Gan
Blasingame method 5.1.3.1 Analysis method As described in Section 4.1, for a gas reservoir with the connate water saturation Swi and the water volume ratio (i.e., the ratio of the aquifer to reservoir volume) M, neglecting the water influx and gas injection terms, the material balance equation (Fetkovich et al., 1998) is
 pi Gp p 12 1 2 C e ðpÞðpi 2 pÞ 5 (5.14) Z Zi G Transforming Eq. (5.14),


 Gp pi =Zi C e ðpÞðpi 2 pÞ 5 1 2 12 p=Z G

(5.15)

Assuming that the p/Z curve has two segments inflecting at point A, as shown in Fig. 5
3, the two slope lines are extended to intersect with the coordinate axis. G is the actual OGIP, while Ga is the apparent OGIP. For any point C in the second straight line, according to Eq. (5.15),
  GpC pi =Zi  12 C e ðpÞðpi 2pÞ C 5 1 2  (5.16) G p=Z C According to the triangle similarity principle (Becerra-Arteaga, 1993),   p=Z C GpC  (5.17) 12 5  G p=Z 1   Substituting Eq. (5.17) into Eq. (5.16) to eliminate p=Z C ,  pi =Zi  C e ðpÞðpi 2pÞ C 5 1 2  p=Z 1

(5.18)

Original gas in place estimations for geopressured gas reservoirs

165

(p/Z)1

p/Z

pi/Zi

B A

(p/Z)B (p/Z)A

Gp C

(p/Z)C GpB

0

GpA

GpC

Ga

G

Gp

Figure 5
3 Schematic plot of p/Z curve for a geopressured gas reservoir. Modified from Gan, R.G., 2001. A new p/z technique for the analysis of abnormally pressured gas reservoirs. Master dissertation, Texas A&M University, College Station, Texas. Permission to publish by the SPE, Copyright SPE.

Eq. (5.18) shows that C e ðpÞðpi 2 pÞ is a constant at any point in the second straight line. Similarly, for inflection point A,   p=Z A GpA  12 (5.19) 5 G p=Z 1   Substituting Eq. (5.19) into Eq. (5.18) to eliminate p=Z 1 ,
  G pi =Zi  1 2 pA C e ðpÞðpi 2pÞ A 5 1 2  (5.20) G p=Z A Eq. (5.20) shows that C e ðpÞðpi 2 pÞ at any point in the second straight line is a constant, but its value is related to the location of point A. Similarly, for any point B in the first straight line,
  GpB pi =Zi  12 C e ðpÞðpi 2pÞ B 5 1 2  (5.21) G p=Z B According to the triangle similarity principle,   p=Z B GpB  12 5 Ga pi =Zi

(5.22)

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Reserves Estimation for Geopressured Gas Reservoirs

Substituting Eq. (5.22) into Eq. (5.21) to eliminate GpB ,  pi =Zi p =Z G G  1  i i a 2 a C e ðpÞðpi 2pÞ B 5 1 2  G p=Z B p=Z B G Thus, for any point in the first straight line,

 1 1 12 C e ðpÞðpi 2 pÞ 5 1 2 pD GD

(5.23)

(5.24)

Where pD : dimensionless apparent formation pressure, pD 5 pp=Z , GD : i =Zi ratio of actual OGIP to apparent OGIP, GD 5 G=Ga . Eq. (5.24) shows that C e ðpÞðpi 2 pÞ at any point in the first straight line is a variable and is a function of GD . Based on Eq. (5.24), we can plot the C e ðpÞΔpBpD dimensionless type curve, as shown in Fig. 5
4. Transforming Eq. (5.15), we have pD 5

1 2 GpD 1 2 C e ðpÞðpi 2 pÞ

(5.25)

Where GpD : the ratio of cumulative production to the OGIP defined as GpD 5 Gp =G. 100

Ce ' p

10-1

10-2

Inflection point

G 0.5 0.6 0.7 0.8 0.9

10-3 0.0

0.2

0.4

0.6

0.8

1.0

pD Figure 5
4 C e ðpÞðpi 2 pÞ versus pD for geopressured gas reservoir. Modified from Gan, R.G., 2001. A new p/z technique for the analysis of abnormally pressured gas reservoirs. Master dissertation, Texas A&M University, College Station, Texas. Permission to publish by the SPE, Copyright SPE.

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Original gas in place estimations for geopressured gas reservoirs

Substituting Eq. (5.24) into Eq. (5.25), we can obtain the relationship of pD BGpD , the first straight line, pD 5 1 2

Gp 5 1 2 GpD GD Ga

(5.26)

Substituting Eq. (5.20) into Eq. (5.25), we can obtain the relationship of pD BGpD , the second straight line, pD 5

 pDA  1 2 GpD 1 2 GpAD

(5.27)

Where ðp=Z Þ pDA : dimensionless apparent formation pressure at point A, pDA 5 pi =Zi A , G

GpAD : dimensionless cumulative production at point A, GpAD 5 GpA . Based on Eqs. (5.26) and (5.27), we can plot the pD BGpD curve, as shown in Fig. 5
5. The curve shape depends only on the location of the inflection point and is independent of factors such as rock compressibility.

1.0

GD 1.0

0.8

Inflection point

0.6 0.4

pD

0.6

0.4

0.2

0.0 0.0

0.2

0.4

GpD

0.6

0.8

1.0

Figure 5
5 Schematic plot of pD versus GpD for geopressured gas reservoir. Modified from Gan, R.G., 2001. A new p/z technique for the analysis of abnormally pressured gas reservoirs. Master dissertation, Texas A&M University, College Station, Texas. Permission to publish by the SPE, Copyright SPE.

168

Reserves Estimation for Geopressured Gas Reservoirs

5.1.3.2 Analysis procedure Prepare the C e ðpÞðpi 2 pÞBpD semilog plot, and then, prepare the pD BGpD plot. After that, conduct nonlinear regression analysis to determine the location of the inflection point and OGIP. The C e ðpÞðpi 2 pÞBpD semilog plot (Fig. 5
4) is prepared as follows. 1. Prepare a conventional p=ZBGp plot using production data and estimates Ga . 2. Assume a value of G and an estimate of the location of the inflection point A. We can use a pressure coefficient of 1.13
1.30 for point A and 0.95 Ga as an estimate for G. 3. Calculate C e ðpÞðpi 2 pÞ using Eq. (5.15) and plot the calculation results as the C e ðpÞðpi 2 pÞBpD semilog plot. On the same graph, plot the calculation results using Eq. (5.24) as the C e ðpÞðpi 2 pÞBpD theoretical curve, and change G and the location of point A to conduct curve matching. The pD BGpD plot (Fig. 5
5) is prepared as follows. 1. Prepare a conventional p=ZBGp plot using production data and estimates Ga . 2. Assume a value of G and an estimate of the location of the inflection point A. We can use a pressure coefficient of 1.13
1.30 for point A and 0.95 Ga as an estimate for G. 3. Prepare a pDBGpD plot using production data. On the same graph, plot the calculation results of Eqs. (5.26) and (5.27) as the pDBGpD theoretical curve, change G and the location of point A to conduct curve matching. 4. Based on the above hand-preliminary matching results, develop a program to conduct computer automatic matching (Gan, 2001). Example 3: The Anderson L high-pressure gas reservoir in the United States corresponds to the production data shown in Appendix Table 3
2 (Duggan, 1972). Use the Gan
Blasingame method to calculate the OGIP. Solution Prepare the p/ZBGp plot using the data in Appendix Table 3
2, as shown in Fig. 5
6, and use the early production data to determine Ga as 31.8 3 108 m3. Assuming that G 5 21.5 3 108 m3 and pD 5 0:80 at the inflection point, calculate the C e ðpÞðpi 2 pÞ according to Eq. (5.15), with the results

169

Original gas in place estimations for geopressured gas reservoirs

50 Early matching point Non-matching point Matching curve, p/Z=45.455-1.42883Gp

p/Z, MPa

40

30

20

10 31.8

0 0

5

10

15

20

25

30

35

Gp , 10 m 8

3

Figure 5
6 p/Z
Gp of Anderson L gas reservoir. Table 5
2 Calculation results for Example 3. p, MPa

Z

Gp, 108 m3

p/Z, MPa

GpD

pD

C e ðpÞðpi 2 pÞ

65.55 64.07 61.85 59.26 57.45 55.22 52.42 51.06 48.28 46.34 45.06 39.74 32.86 29.61 25.86 22.39

1.440 1.418 1.387 1.344 1.316 1.282 1.239 1.218 1.176 1.147 1.127 1.048 0.977 0.928 0.891 0.854

0.000 0.118 0.492 0.966 1.276 1.647 2.257 2.620 3.146 3.519 3.827 5.163 6.836 8.389 9.689 10.931

45.52 45.18 44.59 44.09 43.65 43.07 42.31 41.92 41.05 40.40 39.98 37.92 33.63 31.91 29.02 26.21

0.0000 0.0055 0.0229 0.0449 0.0593 0.0766 0.1050 0.1219 0.1463 0.1637 0.1780 0.2401 0.3179 0.3902 0.4507 0.5084

1.0000 0.9925 0.9796 0.9686 0.9590 0.9463 0.9295 0.9210 0.9018 0.8875 0.8783 0.8331 0.7389 0.7010 0.6375 0.5759

0.0000 2 0.0020 0.0025 0.0140 0.0191 0.0242 0.0371 0.0465 0.0534 0.0577 0.0641 0.0879 0.0769 0.1301 0.1383 0.1464

shown in Table 5
2. The C e ðpÞðpi 2 pÞBpD semilog curve is shown in Fig. 5
7, and GD 5 0.67. Plot the pDBGpD curve, pD 5 0:79 at the inflection point, as shown in Fig. 5
8. The final results are shown in Fig. 5
9, and the OGIP is 21.5 3 108 m3.

170

Reserves Estimation for Geopressured Gas Reservoirs

100 GD=0.67

Ce ' p

10-1

10-2

10-3 0.0

0.2

0.4

pD

0.6

0.8

1.0

Figure 5
7 C e ðpÞðpi 2 pÞ versus pD matching curve for Example 4. Modified from Gan, R.G., Blasingame, T.A. 2001. A semianalytical p/Z technique for the analysis of reservoir performance from abnormally pressured gas reservoirs [C]. SPE 71514-MS. Permission to publish by the SPE, Copyright SPE.

1.0

Inflection point

0.8

pD

0.6

0.4

0.2

0.0 0.0

0.2

0.4

GpD

0.6

0.8

1.0

Figure 5
8 pD versus GpD curve for Example 4. Modified from Gan, R.G., Blasingame, T.A. 2001. A semianalytical p/Z technique for the analysis of reservoir performance from abnormally pressured gas reservoirs [C]. SPE 71514-MS. Permission to publish by the SPE, Copyright SPE.

171

Original gas in place estimations for geopressured gas reservoirs

50 Early matching point Non matching point Matching curve，p/Z=45.455-1.42883Gp

p/Z, MPa

40

Inflection point

30

20

10 22.0

31.8

0 0

5

10

15

20 8

25

30

35

3

Gp , 10 m

Figure 5
9 Results of Gan
Blasingame method. Modified from Gan, R.G., Blasingame, T.A. 2001. A semianalytical p/Z technique for the analysis of reservoir performance from abnormally pressured gas reservoirs [C]. SPE 71514-MS. Permission to publish by the SPE, Copyright SPE.

5.1.4 Discussion on the time of inflection point of p/Z curve The above methods involve the recovery percent of reserves: How much is the pressure depletion degree? Can the two-segment inflection point occur? The analysis on 20 high-pressure and ultrahigh-pressure gas reservoirs shows that the ratio of OGIP to apparent OGIP is 0.43
0.77, with an average of 0.58 (Fig. 5
10), and the apparent formation pressure depletion degree at the inflection point is 0.14
0.38, with an average value of 0.22 (Fig. 5
11). The detailed data are given in Appendix 15
1. The empirical equation for the time of occurrence of the inflection point is as follows.
0:997076 p

p 5 0:674663 i ðGD Þ-0:272519 (5.28) Z A Zi The calculation method of OGIP based on the two-segment assumption of the p/Z curve can be used to roughly evaluate the OGIP as the pressure at the inflection point predicted with the pressure coefficient of 1.3 when the inflection point has not occurred. The method based on rock compressibility may generate uncertain results.

172

Reserves Estimation for Geopressured Gas Reservoirs

GpD at inflection point

1.0

0.8 Avg. 0.58

0.6

0.4

0.2

0.0 1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

Reservoir No. Figure 5
10 Ratio of OGIP to apparent OGIP.

pD at inflection point

1.0

Avg. 0.78

0.8

0.6

0.4

0.2

0.0 1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

Reservoir No. Figure 5
11 Time of occurrence of inflection point.

5.2 Linear regression method .

This method expresses the material balance equation in different linear forms and then uses the intersection of the linear regression line and the coordinate axis to determine OGIP. It mainly includes the Ramagost
Farshad pressure

Original gas in place estimations for geopressured gas reservoirs

173

correction method (1981), the Roach linear regression method (1981), the Poston
Chen
Akhtar improved Roach method (1994), the Becerra
Arteaga method (1993), the Havlena
Odeh (Elsharkawy, 1996a, b) method, and the Sun method (Sun et al., 2021a).

5.2.1 Ramagost
Farshad method As described in Section 4.1, the material balance equation for closed gas reservoirs (Ramagost and Farshad, 1981) can be expressed as
 Gp p pi ð1 2 Ce ΔpÞ 5 12 (5.29) Z Zi G Where Ce : effective compressibility, MPa21, defined as Ce 5

Cw Swi 1 Cf 1 2 Swi

(5.30)

  Eq. (5.29) shows that p=Z ð1 2 Ce ΔpÞBGp have a linear relationi ship, with the slope and intercept of the straight line being 2 Zpi G and pi =Zi , respectively. The method requires prior knowledge of rock compressibilities and formation water compressibilities. Example 4: The Anderson L high-pressure gas reservoir in the United States corresponds to the production data in Appendix Table 3
2 (Duggan, 1972), with the pressure coefficient of 1.91, the initial formation pressure of 65.55 MPa, the gas reservoir temperature of 130.0°C, the initial water saturation of 0.35, the effective thickness of 22.86 m, the formation water compressibility of 4.351 3 1024/MPa, the rock compressibility of 2.828 3 1023/MPa, and the volumetric OGIP of 19.68 3 108 m3. Use the Ramagost
Farshad method to calculate the OGIP. Solution First, calculate the effective compressibility. According to Eq. (5.29), we have Ce 5

Cw Swi 1 Cf 4:351 3 1024 3 0:35 1 2:828 3 1023 5 4:585 3 1023 MPa-1 5 1 2 Swi 1 2 0:35

  Assuming y 5 p=Z ð1 2 Ce ΔpÞ, calculate y using the data in Appendix Table 3
2. We get data shown in Table 5
3.

174

Reserves Estimation for Geopressured Gas Reservoirs

Table 5
3 Calculation results for Example 4. No.

Gp 108 m3

p, MPa

Z

p/Z, MPa

Δp, MPa

y, MPa

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.000 0.118 0.492 0.966 1.276 1.647 2.257 2.620 3.146 3.519 3.827 5.163 6.836 8.389 9.689 10.931

65.55 64.07 61.85 59.26 57.45 55.22 52.42 51.06 48.28 46.34 45.06 39.74 32.86 29.61 25.86 22.39

1.440 1.418 1.387 1.344 1.316 1.282 1.239 1.218 1.176 1.147 1.127 1.048 0.977 0.928 0.891 0.854

45.52 45.18 44.59 44.09 43.65 43.07 42.31 41.92 41.05 40.40 39.98 37.92 33.63 31.91 29.02 26.21

0.00 1.48 3.70 6.29 8.10 10.33 13.13 14.49 17.27 19.21 20.49 25.81 32.69 35.94 39.69 43.16

45.52 44.87 43.83 42.82 42.03 41.03 39.76 39.14 37.80 36.84 36.22 33.43 28.59 26.65 23.74 21.03

p/Z , p/Z(1-Ce' p) , MPa

50

p/Z p/Z(1-Ce' p)

40

30

20

10 31.8

20.25

0 0

5

10

15

20

25

30

35

Gp , 108m3 Figure 5
12 Comparison of calculation results for Example 4.

  Plot the p=Z ð1 2 Ce ΔpÞBGp and p=ZBGp curves respectively, as shown in Fig. 5
12. Ga is determined as 31.8 3 108 m3 using early production data and as 20.25 3 108 m3 by the Ramagost
Farshad method.

Original gas in place estimations for geopressured gas reservoirs

175

5.2.2 Roach method 5.2.2.1 Analysis method Transforming Eq. (5.29), we have
 1 pi =Zi pi =Zi Gp 21 5 2 Ce ðΔpÞp=Z G Δp p=Z

(5.31)

Let x5

Gp pi =Zi ðΔpÞp=Z


 1 pi =Zi 21 y5 Δp p=Z

(5.32)

(5.33)

Eq. (5.31) is converted into y5

x 2 Ce G

(5.34)

Eq. (5.34) shows that y and x have a linear relationship, with the slope and intercept of 1=G and Ce (Roach, 1981). This method can be applied only when the initial formation pressure, cumulative pressure drop, and cumulative production data are known. Example 5: Use the Roach method to calculate the OGIP of the Anderson L gas reservoir, with the basic parameters shown in Appendix Table 3
2 (Duggan, 1972). Solution Calculate x and y using the given data, with the results shown in Table 5
4. Plot the yBx curve, as shown in Fig. 5
13. Finally, based on the linear regression slope and intercept, the effective compressibility is obtained as 2.67 3 1023/MPa, based on the slope, the OGIP is estimated to be 22.36 3 108 m3. 5.2.2.2 Discussion on Roach analysis method This simple and convenient method can be used to calculate the OGIP and calculate the effective compressibility. However, it is susceptible to the initial apparent formation pressure (Ambastha, 1993). Here, a normally pressured gas reservoir is taken as an example to explain how the initial

176

Reserves Estimation for Geopressured Gas Reservoirs

Table 5
4 Calculation results for Example 5. No.

Gp, 108 m3

p, MPa

Z

p/Z, MPa

Δp, MPa

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.000 0.118 0.492 0.966 1.276 1.647 2.257 2.620 3.146 3.519 3.827 5.163 6.836 8.389 9.689 10.931

65.55 64.07 61.85 59.26 57.45 55.22 52.42 51.06 48.28 46.34 45.06 39.74 32.86 29.61 25.86 22.39

1.440 1.418 1.387 1.344 1.316 1.282 1.239 1.218 1.176 1.147 1.127 1.048 0.977 0.928 0.891 0.854

45.52 45.18 44.59 44.09 43.65 43.07 42.31 41.92 41.05 40.40 39.98 37.92 33.63 31.91 29.02 26.21

0.00 1.48 3.70 6.29 8.10 10.33 13.13 14.49 17.27 19.21 20.49 25.81 32.69 35.94 39.69 43.16

x, 108 m3/ MPa

y, MPa21

0.0801 0.1356 0.1586 0.1642 0.1686 0.1850 0.1964 0.2020 0.2064 0.2127 0.2401 0.2830 0.3330 0.3829 0.4398

0.0051 0.0056 0.0051 0.0053 0.0055 0.0058 0.0059 0.0063 0.0066 0.0068 0.0078 0.0108 0.0119 0.0143 0.0171

0.020 y=0.04471x-0.00267

y, MPa -1

0.015

0.010 G=22.36×108m3

0.005

0.000 Ce=0.00267 MPa-1

-0.005 0.0

0.1

0.2

0.3

0.4

0.5

x,10 m /MPa 8

3

Figure 5
13 Calculation results for Example 5 using Roach method. Modified from Roach, R.H., 1981. Analyzing geopressured reservoirs—A material balance technique. SPE 9968-Unsolicated. Permission to publish by the SPE, Copyright SPE.

apparent formation pressure affects the calculation using the Roach method. Example 6: Use the data in Table 5
5 to analyze how the initial pressure affects the results by the Roach method.

177

Original gas in place estimations for geopressured gas reservoirs

Table 5
5 Calculation results for Example 6 (Ambastha, 1993). No.

Gp, 108 m3

p, MPa

Z

p/Z, MPa

Δp, MPa

1 2 3 4 5 6

0.0 1.9 4.0 6.7 8.8 10.3

14.30 13.00 11.20 8.30 6.10 4.50

0.759 0.767 0.787 0.828 0.866 0.900

18.84 16.95 14.23 10.02 7.04 5.00

0.00 1.30 3.10 6.00 8.20 9.80

x, 108 m3/MPa

y, MPa21

1.6246 1.7082 2.0988 2.8705 3.9604

0.0858 0.1045 0.1466 0.2042 0.2825

20 p/Z=19.80-1.44Gp 19.3

p/Z, MPa

15

10

5

13.7

0 0

3

6

9

12

15

Gp , 108m3 Figure 5
14 p/Z versus Gp for Example 6. Modified from Ambastha, A.K., van Kruysdijk., 1993. Effects of input data errors on material balance analysis for volumetric, gas and gas-condensate reservoirs. PETSOC-93-04.

Solution First, plot the p/ZBGp curve, as shown in Fig. 5
14. Except for the point with the initial apparent formation pressure pi/Zi 5 18.84 MPa, all other points are in a straight line. When the linear regression intercept pi/Zi 5 19.80 MPa, the OGIP is 13.7 3 108 m3. The calculation results of the Roach method at different initial apparent formation pressures are shown in Fig. 5
15, indicating that the initial formation pressure has a significant impact on the early data points. Fig. 5
15 shows that if the early data points of the Roach method drop, the initial formation pressure value may be lower. Otherwise, the initial formation pressure value may

178

Reserves Estimation for Geopressured Gas Reservoirs

0.35

18.84MPa 19.80MPa y=0.7217x

0.30

y, MPa-1

0.25 0.20 0.15

G=13.86×108m3

0.10 0.05 0.00 0

1

2

3

4

5

x,10 m /MPa 8

3

Figure 5
15 Effect of initial formation pressure on Roach data points. Modified from Ambastha, A.K., van Kruysdijk., 1993. Effects of input data errors on material balance analysis for volumetric, gas and gas-condensate reservoirs. PETSOC-93-04.

be higher. However, Poston and Chen (1987, 1989) speculated that this phenomenon might result from compressibility change.

5.2.3 Poston
Chen
Akhtar method 5.2.3.1 Analysis method As described in Chapter 4, the material balance equation for water-drive gas reservoirs can be expressed as
 Gp p pi ð1 2 Ce Δp 2 ωÞ 5 12 (5.35) Z Zi G ω5

We 2 Wp Bw GBgi

(5.36)

According to the Roach method, Eq. (5.35) can be expressed as
 1 pi =Zi pi =Zi Gp ω (5.37) 21 5 2 Ce 2 Δp p=Z Δp ðΔpÞp=Z G

Original gas in place estimations for geopressured gas reservoirs

179

Let x5

pi =Zi Gp ðΔpÞp=Z

(5.38)


 1 pi =Zi 21 y5 Δp p=Z

(5.39)

Eq. (5.37) is converted into


 x ω y 5 2 Ce 1 G Δp

(5.40)

Eq. (5.40) shows that y and x are linearly related. G can be calculated from the slope of the straight line, and the integrated value of effective compressibility and water influx Ce 1 ω=Δp can be calculated from the intercept of the straight line. This method does not require compressibility and water influx data. Fig. 5
16 is a schematic curve plot of pressure depletion. In Fig. 5
16, the data points are convex-upward in the early time and become a straight line in the late time. When the elastic drive is the

G=1/Slope

y

Critical Slope line

0 Time increase Ce+ω /' p

0

x

Figure 5
16 Pressure depletion by Poston method. Modified from Poston, S.W., Chen, H.Y., Akhtar, M.J., 1994. Differentiating formation compressibility and water-influx effects in overpressured gas reservoirs. SPE Reserv. Eng. 9 (3), 183
187. Permission to publish by the SPE, Copyright SPE.

180

Reserves Estimation for Geopressured Gas Reservoirs

dominant mechanism, the data points are always a straight line. The dotted line crossing the origin represents the critical slope line. Eq. (5.39) shows that the intercept must be negative. Experience indicates that the positive intercept of the line is usually reflects a substantial water influx. Fig. 5
17 is a typical curve of water influx. In segment t1-t2, water influx is significant, and the curve shifts rightward. The straight lines with the same slope crossing points t1 and t2 have different intercepts with the y axis, and the intercept of the dotted line reflects the net water influx. The curve may shift rightward either in the early time or in the late time. If the rock compressibility is assumed to be a constant, at t1 and t2, the intercepts are yintercept t1 5 C e

(5.41)

yintercept t 2 5 C e 1 ω=Δp

(5.42)

Combined with Eq. (5.36), we have   We 2 Wp Bw 5 yintercept t1 2 yntercept t2 ΔpGBgi

t1

(5.43)

t2

t0

y

G=1/Slope

0 Ce

Ce+ ω/' p

0

Time increase

x

Figure 5
17 Water influx effect by Poston method. Modified from Poston, S.W., Chen, H.Y., Akhtar, M.J., 1994. Differentiating formation compressibility and water-influx effects in overpressured gas reservoirs. SPE Reserv. Eng. 9 (3), 183
187. Permission to publish by the SPE, Copyright SPE.

Original gas in place estimations for geopressured gas reservoirs

181

Eq. (5.43) can be used to calculate the water influx in any two time intervals. Fig. 5
18 is a typical curve when the effective compressibility is a constant. Segment t1-t2 is linear, indicating that the system is in a stable state. Rock compressibility can be calculated from intercept according to Eq. (5.44) Cf 5 yintercept ð1 2 Swi Þ 2 Cw Swi

(5.44)

Experience indicates that the rock compressibility ranges from 8.7 3 1024 to 29 3 1024/MPa. A calculated value beyond this range suggests that local water influx has occurred (Poston et al., 1994). Fig. 5
19 is a typical curve when the effective compressibility is a variable. In Case a, the p/Z curve is concave when the compressibility increases. In Case b, when the compressibility decreases, the p/Z curve shows an early concave decline, followed by an upward concave shape (Fetkovich et al., 1998). Fig. 5
20 is a typical curve in the case of water-drive gas reservoir. In any stage of gas reservoir development life cycle, water influx may occur, and the curve would shift rightward. Significant early water influx may form a straight line at the beginning, and the calculated rock compressibility may be greater than 36 3 1024/MPa, indicating that water influx has occurred.

t2

y

G=1/Slope t1

0 Time increase

Ce

0

x

Figure 5
18 Case with effective compressibility as a constant. Modified from Poston, S.W., Chen, H.Y., Akhtar, M.J., 1994. Differentiating formation compressibility and waterinflux effects in overpressured gas reservoirs. SPE Reserv. Eng. 9 (3), 183
187. Permission to publish by the SPE, Copyright SPE.

182

Reserves Estimation for Geopressured Gas Reservoirs

t3

G=1/Slope

y

t1

t2

Case b

Case a

t1

0 Time increase

Ce

0

x

Figure 5
19 Effective compressibility as a variable. Modified from Poston, S.W., Chen, H.Y., Akhtar, M.J., 1994. Differentiating formation compressibility and water-influx effects in overpressured gas reservoirs. SPE Reserv. Eng. 9 (3), 183
187. Permission to publish by the SPE, Copyright SPE.

t3 late

y

t2 t1

midway

early

0

Time increase

0

x

Figure 5
20 Water-drive case type curves. Modified from Poston, S.W., Chen, H.Y., Akhtar, M.J., 1994. Differentiating formation compressibility and water-influx effects in overpressured gas reservoirs. SPE Reserv. Eng. 9 (3), 183
187. Permission to publish by the SPE, Copyright SPE.

Original gas in place estimations for geopressured gas reservoirs

183

5.2.3.2 Analysis procedure The analysis procedure of the Poston
Chen
Akhtar method is as follows. 1. Calculate x and y with Eqs. (5.31) and (5.32), respectively, using the production data. 2. Plot the x versus y curve and analyze the driving mechanism. Draw the critical slope line crossing the origin, and then, select the data points above the critical slope line for linear regression analysis. The reciprocal of the slope is the OGIP. 3. Calculate the rock compressibility with Eq. (5.44). If the rock compressibility is greater than 29 3 1024/MPa, the reservoir has an energy supplement. 4. Draw lines parallel to the first regression line, which in turn determines the cumulative water influx at any time. Example 7: For the Anderson L reservoir, as described in Example 5, Duggan (1972) believed that the water influx in mudstone affects the production performance of the gas reservoir. The initial water saturation is 0.35, and the formation water compressibility is 4.351 3 1024/MPa. Use the Poston
Chen
Akhtar method to estimate OGIP. Solution Calculate x and y using the given data, with the results shown in Table 5
4. Plot the x versus y curve, as shown in Fig. 5
21. Draw the critical slope line crossing the origin, and use the six data points above the critical slope line for linear regression. Based on the slope and intercept, the reserves and effective compressibility are determined to be 22.40 3 108 m3 and 2.64 3 1023/MPa, respectively. Based on Eq. (5.45), the rock compressibility is determined as Cf 5 yintercept ð1 2 Swi Þ 2 Cw Swi 5 2:64 3 1023 3 ð1 2 0:35Þ 2 4:351 3 1024 3 0:35 5 15:6 3 1024 MPa-1 Example 8: For the NS2B reservoir as described in Example 1, the OGIP calculated by the Hammerlindl method is 32.2 3 108 m3. Use the Poston
Chen
Akhtar method to estimate OGIP. Solution Calculate x and y using the given data, with the results shown in Table 5
6. Plot the x versus y curve, as shown in Fig. 5
22. Draw the critical slope line crossing the origin, and use the four data points above

184

Reserves Estimation for Geopressured Gas Reservoirs

0.020 y=0.04465x-0.00264

y, MPa -1

0.015 G=22.40×108m3

0.010

Critical slope line

0.005

0.000 Ce=0.00264 MPa-1

-0.005 0.0

0.1

0.2

0.3

0.4

0.5

x,10 m /MPa 8

3

Figure 5
21 Analysis results for Anderson L reservoir by Poston
Chen
Akhtar method. Modified from Poston, S.W., Chen, H.Y., Akhtar, M.J., 1994. Differentiating formation compressibility and water-influx effects in overpressured gas reservoirs. SPE Reserv. Eng. 9 (3), 183
187. Permission to publish by the SPE, Copyright SPE. Table 5
6 Calculation results for Example 8. No.

Gp, 108 m3

p, MPa

Z

p/Z, MPa

Δp, MPa

1 2 3 4 5 6 7 8 9

0.00 0.19 0.93 2.94 4.41 6.79 7.89 9.54 11.36

61.51 60.98 57.38 51.14 47.44 41.81 37.85 32.96 28.30

1.473 1.465 1.4 1.288 1.219 1.13 1.075 0.967 0.887

41.76 41.63 40.98 39.70 38.91 37.00 35.21 34.09 31.90

0.0000 0.5240 4.1300 10.3697 14.0722 19.6983 23.6559 28.5442 33.2120

x,108 m3/MPa

y, MPa21

0.3578 0.2305 0.2987 0.3366 0.3889 0.3953 0.4095 0.4475

0.0059 0.0046 0.0050 0.0052 0.0065 0.0079 0.0079 0.0093

the critical slope line for linear regression. Based on the slope and intercept, the reserves and effective compressibility are determined to be 25.15 3 108 m3 and 8.42 3 1023/MPa, respectively. Based on Eq. (5.45), the rock compressibility is determined as Cf 5 yintercept ð1 2 Swi Þ 2 Cw Swi 5 8:64 3 1023 3 ð1 2 0:34Þ 2 4:351 3 1024 3 0:34 5 55:54 3 1024 MPa-1

185

Original gas in place estimations for geopressured gas reservoirs

0.015 y=0.03976x-0.00842

0.010

y, MPa-1

Critical slope line

0.005 G=25.15×108m3

0.000

-0.005 Ce=0.00842 MPa-1

-0.010 0.0

0.1

0.2

0.3

0.4

0.5

x,10 m /MPa 8

3

Figure 5
22 Analysis results for NS2B reservoir by Poston
Chen
Akhtar method. Modified from Poston, S.W., Chen, H.Y., Akhtar, M.J., 1994. Differentiating formation compressibility and water-influx effects in overpressured gas reservoirs. SPE Reserv. Eng. 9 (3), 183
187. Permission to publish by the SPE, Copyright SPE.

The curve does not show an apparent right shift, but the rock compressibility is higher, indicating that water influx has occurred. The OGIP is slightly smaller than those calculated using the Gan
Blasingame method (Appendix Table 15). Example 9: Use the Poston
Chen
Akhtar method to estimate the OGIP of the Gulf of Mexico reservoir. Solution Calculate x and y using the given data, with the results shown in Table 5
7. Plot the x versus y curve, as shown in Fig. 5
23. Draw the critical slope line crossing the origin, and use the three data points above the critical slope line for linear regression. Based on the slope and intercept, the reserves and effective compressibility are determined to be 38.34 3 108 m3 and 26.08 3 1023/MPa, respectively. Based on Eq. (5.45), the rock compressibility is determined as C f 5 yintercept ð1 2 Swi Þ 2 C w S wi 5 26:08 3 1023 3 ð1 2 0:34Þ 2 4:351 3 1024 3 0:34 5 170:6 3 1024 MPa-1

186

Reserves Estimation for Geopressured Gas Reservoirs

Table 5
7 Calculation results for Example 9. No.

Gp, 108 m3

p, MPa

Z

p/Z, MPa

Δp, MPa

1 2 3 4 5 6 7 8 9

0.00 0.19 0.93 2.94 4.41 6.79 7.89 9.54 11.36

61.51 60.98 57.38 51.14 47.44 41.81 37.85 32.96 28.30

1.473 1.465 1.4 1.288 1.219 1.13 1.075 0.967 0.887

41.76 41.63 40.98 39.70 38.91 37.00 35.21 34.09 31.90

0.0000 0.5240 4.1300 10.3697 14.0722 19.6983 23.6559 28.5442 33.2120

x, 108 m3/MPa

y, MPa21

0.3578 0.2305 0.2987 0.3366 0.3889 0.3953 0.4095 0.4475

0.0059 0.0046 0.0050 0.0052 0.0065 0.0079 0.0079 0.0093

0.06 y=0.02608x-0.02607

y, MPa -1

0.04

Critical slope line

0.02

G=38.34×108m3

0.00

-0.02 Ce=0.02608MPa-1

0

1

2

3

4

x,10 m /MPa 8

3

Figure 5
23 Analysis results for Gulf of Mexico reservoir. Modified from Poston, S.W., Chen, H.Y., Akhtar, M.J., 1994. Differentiating formation compressibility and water-influx effects in overpressured gas reservoirs. SPE Reserv. Eng. 9 (3), 183
187. Permission to publish by the SPE, Copyright SPE.

Although the curve does not show an apparent right shift, the rock compressibility is seriously high, indicating a strong effect of water drive. The OGIP is slightly smaller than those calculated using the Gan
Blasingame method (Appendix Table 15
1). Example 10: For the M1 reservoir, the initial formation pressure is 74.35 MPa, the formation water compressibility is 5.645 3 1024/MPa,

187

Original gas in place estimations for geopressured gas reservoirs

the rock compressibility is 25.01 3 1024 /MPa, the initial water saturation is 0.32, the volumetric OGIP is 2833 3 108 m3, and the production data are shown in Appendix Table 16
1. Use the Poston
Chen
Akhtar method to estimate OGIP. Solution Calculate x and y using the given data, with the results shown in Table 5
8. Plot the x versus y curve, as shown in Fig. 5
24. Draw the critical slope line crossing the origin, and use the three data points above Table 5
8 Calculation results for Example 10. No.

Gp, 108 m3

p, MPa

Z

p/Z, MPa

Δp, MPa

1 2 3 4 5 6 7 8 9 10

0.00 58.78 150.89 258.20 365.80 473.11 687.73 902.64 1117.26 1332.18

74.35 72.73 70.67 68.09 65.32 62.47 56.70 50.61 44.82 39.16

1.44 1.42 1.40 1.37 1.34 1.31 1.25 1.19 1.12 1.06

51.56 51.05 50.40 49.54 48.58 47.55 45.30 42.68 39.90 36.86

0.0000 1.6200 3.6800 6.2600 9.0300 11.8800 17.6500 23.7400 29.5300 35.1900

x, 108 m3/MPa

y, MPa21

36.6425 41.9500 42.9284 42.9940 43.1842 44.3463 45.9349 48.8945 52.9501

0.0061 0.0063 0.0065 0.0068 0.0071 0.0078 0.0088 0.0099 0.0113

0.020 y=3.64635×10-4x-0.00796

y, MPa -1

0.015 0.010 Critical slope line

0.005 G=2742×108m3

0.000 -0.005 Ce=0.00796MPa-1

-0.010 0

10

20

30

40

50

60

x,10 m /MPa 8

3

Figure 5
24 Analysis results for M1 reservoir by Poston
Chen
Akhtar method.

188

Reserves Estimation for Geopressured Gas Reservoirs

the critical slope line for linear regression. Based on the slope and intercept, the reserves and effective compressibility are determined to be 2742 3 108 m3 and 7.96 3 1023/MPa, respectively. Based on Eq. (5.45), the rock compressibility is determined as C f 5 yintercept ð1 2 S wi Þ 2 C w S wi 5 7:96 3 1023 3 ð1 2 0:32Þ 2 5:645 3 1024 3 0:32 5 52:32 3 1024 MPa-1 The curve does not show an apparent right shift, but the rock compressibility is seriously higher, indicating that the water drive has yielded a particular effect. The OGIP is close to the volumetric reserves. Example 11: For the M2 reservoir, the initial formation pressure is 74.22 MPa, the formation water compressibility is 5.6 3 1024/MPa, the formation water volume factor is 1.0, the initial water saturation is 0.32, the gas reservoir temperature is 100°C, the volumetric OGIP is 2400 3 108 m3, and the production data are shown in Appendix Table 17
1. Use the Poston
Chen
Akhtar method to estimate OGIP. Solution Calculate x and y using the given data, with the results shown in Table 5
9. Plot the x versus y curve, as shown in Fig. 5
25. Draw the critical slope line crossing the origin, and use the five data points above the critical slope line for linear regression. Based on the slope and intercept, the reserves and effective compressibility are determined to be 2100 3 108 m3 and 2.0 3 1023/MPa, respectively. The curve shifts rightward from point 7, showing a water-driven characteristic. Starting from point 8, draw lines parallel to the critical slope line crossing these points respectively, and then, determine ω/Δp depending on the intersection with the y axis, and finally, calculate WeBWpBw and We using Eq. (5.36), with the results shown in Table 5
9. The OGIP is close to the volumetric reserves. Owing to the lack of data points above the critical slope line, the OGIP represents the lower limit, slightly conservative. Example 12: The M3 reservoir is a typical radial substantial water drive gas reservoir (Wang and Teasdale, 1987), with the initial formation pressure of 29.25 MPa, the formation water compressibility of 4.5 3 1024/MPa, the initial water saturation of 0.32, the gas reservoir temperature of 94°C, and the volumetric OGIP of 230.8 3 108 m3. The production data are shown in Appendix Table 18
1. Use the Poston
Chen
Akhtar method to estimate OGIP.

Table 5
9 Calculation results for Example 11. Gp, 108 m3

Wp,104 m3

p/Z, MPa

Δp, MPa

x, 108 m3/MPa

y, MPa21

ω/Δp

We
WpBw,106 m3

We,106 m3

0.00 2.14 34.61 118.72 228.32 345.40 457.91 553.39 614.31 689.23 769.08 839.48 901.98 958.96 1029.23 1095.80 1157.65

0.00 0.00 0.37 2.11 4.65 7.50 10.93 16.61 18.40 21.11 23.60 26.42 29.52 35.67 39.10 46.26 57.22

51.27 51.00 50.43 48.52 45.76 42.76 40.28 38.56 37.48 36.11 34.65 33.31 32.13 31.08 29.81 28.64 27.57

0.0000 0.2400 1.3000 4.2500 8.4200 12.9000 16.6200 19.2300 20.9200 23.0100 25.2200 27.1600 28.8500 30.3600 32.1700 33.8300 35.3500

& 8.9646 27.0693 29.5187 30.3845 32.1049 35.0718 38.2638 40.1693 42.5269 45.1207 47.5850 49.8899 52.1022 55.0257 57.9915 60.9101

& 0.0224 0.0129 0.0133 0.0143 0.0154 0.0164 0.0171 0.0176 0.0182 0.0190 0.0199 0.0206 0.0214 0.0224 0.0234 0.0243

& & & & & & & 0.0009 0.0013 0.0018 0.0023 0.0026 0.0029 0.0032 0.0036 0.0041 0.0045

& & & & & & & 8.9454 14.8330 22.0122 30.2740 37.2904 44.3832 51.6776 61.6989 72.5505 83.7610

& & & & & & & 8.9471 14.8348 22.0143 30.2764 37.2930 44.3861 51.6812 61.7028 72.5551 83.7667

190

Reserves Estimation for Geopressured Gas Reservoirs

0.030 y=4.76184×10-4x-0.0002

0.025 G=2100×108m3

y, MPa -1

0.020 0.015

Critical slope line

0.010 Ce=0.0002MPa-1

0.005 0.000

Ce + ω/' p=0.0047MPa-1

-0.005 0

10

20

30

40

50

60

70

x,10 m /MPa 8

3

Figure 5
25 Analysis results for M2 reservoir by Poston
Chen
Akhtar method.

Solution Calculate x and y using the given data, with the results shown in Table 5
10. Plot the x versus y curve, as shown in Fig. 5
26, suggesting a strong water drive characteristic. Case b in Fig. 5
19 shows that the critical slope line crossing the origin cannot be drawn. Thus, the origin is connected with the third point. Based on the slope, the OGIP is 282.3 3 108 m3, which represents the upper limit. With the Everdingen
Hurst radial water model, the OGIP is estimated to be 244.3 3 108 m3 (Wang and Teasdale, 1987). On this basis, draw the critical slope line. Starting from point 4, draw lines parallel to the critical slope line crossing these points respectively, determine ω/Δp depending on the intersection with the vertical axis, and finally, calculate WeBWpBw and We using Eq. (5.36), with the results shown in Table 5
10. This case does not belong to the high-pressure gas reservoir and is only used to explain how the Poston
Chen
Akhtar method is applied to estimate the OGIP in substantial water-drive gas reservoirs.

5.2.4 Becerra-Arteaga method As described in Chapter 3, in Figs. 3
1, when ppr $ 7, the gas deviation factor curve at a given Tpr is a linear function of pseudo-reduced pressure,

Table 5
10 Calculation results for Example 12. Gp, 108 m3

Wp, 104 m3

p/Z, MPa

Δp, MPa

x, 108 m3/MPa

y, MPa21

ω/Δp

We
WpBw, 106 m3

We,106 m3

0.00 6.64 8.30 16.24 28.97 47.00 53.84 64.05 74.54 84.36 92.77 99.91 104.83 109.65 118.10 120.31 122.72 127.38 131.78 147.80 155.55 163.04 164.55

0.00 0.00 0.01 0.15 0.44 0.91 1.00 1.24 1.48 1.88 2.76 3.20 3.30 3.83 4.88 5.04 5.46 6.98 7.28 10.61 13.70 18.35 19.80

30.10 29.68 29.35 28.36 27.33 25.96 25.30 25.14 24.64 23.62 23.35 23.10 22.78 22.95 21.95 21.83 21.69 21.54 21.06 18.82 17.76 17.34 17.28

0.0000 0.5378 0.9308 2.1236 3.3302 4.8608 5.5847 5.7571 6.2880 7.3429 7.6187 7.8807 8.1979 8.0324 9.0321 9.1424 9.2803 9.4320 9.9009 12.0451 13.0173 13.3965 13.4585

& 12.5248 9.1507 8.1159 9.5835 11.2109 11.4718 13.3249 14.4849 14.6392 15.6956 16.5226 16.8950 17.9082 17.9340 18.1425 18.3499 18.8731 19.0260 19.6311 20.2534 21.1238 21.3040

& 0.0266 0.0277 0.0289 0.0305 0.0328 0.0340 0.0343 0.0353 0.0373 0.0379 0.0385 0.0392 0.0388 0.0411 0.0414 0.0418 0.0421 0.0434 0.0498 0.0534 0.0549 0.0552

& 0.0246 0.0098 0.0043 0.0087 0.0131 0.0129 0.0202 0.0240 0.0226 0.0263 0.0292 0.0300 0.0345 0.0323 0.0328 0.0333 0.0351 0.0345 0.0306 0.0295 0.0316 0.0320

& 1.3948 0.9606 0.9625 3.0661 6.7025 7.6038 12.2563 15.8930 17.4528 21.1129 24.1936 25.8666 29.1645 30.6858 31.6150 32.5843 34.8726 35.9709 38.7525 40.4617 44.5096 45.4018

& 1.3955 0.9614 0.9641 3.0689 6.7072 7.6092 12.2627 15.9004 17.4613 21.1221 24.2036 25.8771 29.1755 30.6976 31.6270 32.5965 34.8853 35.9841 38.7673 40.4772 44.5259 45.4182

192

Reserves Estimation for Geopressured Gas Reservoirs

0.06 0.05

y, MPa -1

0.04 G=244.3×108m3

0.03

Critical slope line Parallel to the critical slope line

Point 1

0.02 G=282.3×108m3

0.01 0.00 0

5

10

15

20

25

x,10 m /MPa 8

3

Figure 5
26 Analysis results for M3 reservoir by Poston
Chen
Akhtar method.

that is, the pressure and the gas deviation factor have a linear relationship. Thus, @Z ðpi 2 pÞ @p

(5.45)


 1 1 @Z Cgi 5 2 pi Zi @p

(5.46)

Z 5 Zi 2 Based on Eqs. (5.3)
(5.42),

Substituting Eq. (5.9) into Eq. (5.46), 
  1 Z 5 Zi pi Cgi 1 2 Cgi p pi Substituting Eq. (5.47) into Eqs. (5.4)
(5.15),   ðpi 2 pÞ pi Cgi 1 Ce p Gp 5 pi Cgi ðpi 2 pÞ 1 p G

(5.47)

(5.48)

Eq. (5.48) shows that the equation is valid as long as the gas deviation factor is linearly related to the pressure. Fig. 5
27 illustrates the relations between pressure and cumulative production and between apparent pressure and cumulative production.

193

Original gas in place estimations for geopressured gas reservoirs

80 p p/Z

pi

p, p/Z, MPa

60 p pi/Zi

40 p/Z

20

Gp

0 0

Gu

Go

G

1

2

3

4

5

Gp ,10 m 8

3

Figure 5
27 p/Z versus Gp and p versus Gp. Modified from Becerra-Arteaga. 1993. Analysis of Abnormally Pressured Gas Reservoirs. Texas A&M University.

Extrapolating the early p/Z curve would yield an overestimated reserve, denoted as Go. Extrapolating the early p curve would yield an underestimated reserve, denoted as Gu. According to the principle of triangle similarity, the equation expressing the underestimate in the extrapolation of early pressure data is Gu 2 Gp p 5 pi Gu Rearranging Eq. (5.49),




Gu 5 Gp

p 11 pi 2 p

(5.49)  (5.50)

Similarly, the equation expressing the overestimate in the extrapolation of early apparent pressure data is Gp pi =Zi 2 p=Z 5 pi =Zi Go Rearranging Eq. (5.51),


Go 5 Gp 1 1

p=Z pi =Zi 2 p=Z

(5.51)  (5.52)

194

Reserves Estimation for Geopressured Gas Reservoirs

The correct reserves G should be between Go and Gu, and the linear interpolation is G 5 Gp 1 γ ðGo 2 Gu Þ

(5.53)

where γ is the interpolation ratio. The apparent formation pressure can be approximately expressed as p pi 2 Cgi pi ðpi 2 pÞ (5.54) 5 Z Zi Substituting Eq. (5.53) into Eq. (5.52), 
 p γ G 5 Gp 1 1 12γ1 pi 2 p ZCgi pi

(5.55)

If α is defined as the mean value in the linear range of the deviation factor and the pressure, we have

Ð pi γ 1 2 γ 1 pmin ZCgi pi dp α5 (5.56) pi 2 pmin  
 Gp pi αp pi 12 5 Zi αp 1 ðpi 2 pÞ Zi G

(5.57)

i Eq. (5.57) shows a linear relation with the slope 2 Zpi G and intercept pi . Parameter α is the function of pressure coefficient pid and initial apparZi ent formation pressure, and its empirical relation is pi α 5 2:229 1 2:256355 pid 2 0:099306 (5.58) Zi

Eq. (5.58) is transformed into a type curve, as shown in Fig. 5
28. Eq. (5.57) can also be expressed as   αp G 5 Gp 1 1 (5.59) ðpi 2 pÞ Calculate the G value of each point, and the optimal mean value is the required value. This method does not require rock compressibility. Example 13: The Anderson L high-pressure gas reservoir in the United States corresponds to the production data in Appendix Table 3
2 (Duggan, 1972). Use the Becerra-Arteaga method to calculate the OGIP.

195

Original gas in place estimations for geopressured gas reservoirs

2.2 pressure coefficient 1.47 1.58 1.70 1.81 1.93

2.0

α

1.8 1.6 1.4 1.2 1.0 35

40

45

50

55

60

p/Z, MPa Figure 5
28 α versus p/Z. Modified from Becerra-Arteaga. 1993. Analysis of Abnormally Pressured Gas Reservoirs. Texas A&M University.

Solution From the above conditions, we have pid 5 1.91 and pi/Zi 5 45.52 MPa. Then, according to Eq. (5.58), the coefficient α is calculated as p α 5 2:229 1 2:256355 pid 2 0:099306 i 5 2:229 1 2:256355 Zi 3 1:91 2 0:099306 3 45:52 5 2:04 h i According to Eq. (5.57), set y 5

pi αp Zi αp 1 ðpi 2 pÞ

. According to

Eq. (5.59), calculate G, with the results shown in Table 5
11. Plot the y versus Gp curve, as shown in Fig. 5
29. According to the linear regression intercept, the OGIP is estimated to be 22.5 3 108 m3. Plot the G versus Gp curve, as shown in Fig. 5
30. According to the linear regression intercept, the OGIP is estimated to be (21.0
22.5) 3 108 m3. The latter results are higher, which may be caused by a weak water influx.

5.2.5 Havlena
Odeh method Assume that water influx is expressed as We 5 αΔp

(5.60)

196

Reserves Estimation for Geopressured Gas Reservoirs

Table 5
11 Calculation results for Example 13. No.

Gp, 108 m3

p, MPa

Z

p/Z, MPa

y, MPa

G, 108 m3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.000 0.118 0.492 0.966 1.276 1.647 2.257 2.620 3.146 3.519 3.827 5.163 6.836 8.389 9.689 10.931

65.55 64.07 61.85 59.26 57.45 55.22 52.42 51.06 48.28 46.34 45.06 39.74 32.86 29.61 25.86 22.39

1.440 1.418 1.387 1.344 1.316 1.282 1.239 1.218 1.176 1.147 1.127 1.048 0.977 0.928 0.891 0.854

45.52 45.18 44.59 44.09 43.65 43.07 42.31 41.92 41.05 40.40 39.98 37.92 33.63 31.91 29.02 26.21

45.52 45.01 44.22 43.27 42.58 41.70 40.54 39.96 38.73 37.83 37.22 34.53 30.60 28.54 25.97 23.40

0.00 10.50 17.25 19.54 19.73 19.61 20.64 21.46 21.08 20.84 20.99 21.38 20.85 22.49 22.56 22.50

50 y=45.122-2.01Gp

y, MPa

40

30

20

10

0 0

5

10

15

20

25

Gp , 108m3 Figure 5
29 y versus Gp. Modified from Becerra-Arteaga. 1993. Analysis of Abnormally Pressured Gas Reservoirs. Texas A&M University.

The material balance equation for a closed gas reservoir can be expressed as 
    Cw Swi 1 Cf 1 α Δp (5.61) Gp Bg 1 Wp Bw 5 G Bg 2 Bgi 1 GBgi 1 2 Swi

Original gas in place estimations for geopressured gas reservoirs

197

30 25

22.5 21.0

G, 108m3

20 15 10 5 0 0

2

4

6

8

10

12

Gp, 10 m 8

3

Figure 5
30 G versus Gp. Modified from Becerra-Arteaga. 1993. Analysis of Abnormally Pressured Gas Reservoirs. Texas A&M University.

Rearranging Eq. (5.61), we have 
  Gp Bg 1 Wp Bw Cw Swi 1 Cf Δp   5 G 1 GBgi  1α  1 2 Swi Bg 2 Bgi Bg 2 Bgi 
  Cw Swi 1 Cf y 5 G 1 GBgi 1α x 1 2 Swi

(5.62)

(5.63)

where y5

Gp Bg 1 Wp Bw   Bg 2 Bgi

x5 

Δp  Bg 2 Bgi

Plot the x versus y curve, and the intercept is G (Havlena, 1963, 1964). The Elsharkawy (1996a, b) method is similar to the Havlena
Odeh method, except that the slope term contains a different element, that is, water influx in mudstone is added compared with Eq. (5.63). Example 14: The Anderson L high-pressure gas reservoir in the United States corresponds to the production data in Appendix Table 3
2 (Duggan, 1972). Use the Havlena
Odeh method to calculate the OGIP.

198

Reserves Estimation for Geopressured Gas Reservoirs

Solution Calculate x and y using the given data, with the results shown in Table 5
12. Then, plot the x versus y curve, as shown in Fig. 5
31. The OGIP is estimated to be 20.5 3 108 m3. Table 5
12 Calculation results for Example 14. Gp, 108 m3

p, MPa

Z

p/Z, MPa

Δp, MPa

Bg

x, MPa

y, 108 m3

0.00 0.12 0.49 0.97 1.28 1.65 2.26 2.62 3.15 3.52 3.83 5.16 6.84 8.39 9.69 10.93

65.55 64.07 61.85 59.26 57.45 55.22 52.42 51.06 48.28 46.34 45.06 39.74 32.86 29.61 25.86 22.39

1.440 1.418 1.387 1.344 1.316 1.282 1.239 1.218 1.176 1.147 1.127 1.048 0.977 0.928 0.891 0.854

45.52 45.18 44.59 44.09 43.65 43.07 42.31 41.92 41.05 40.40 39.98 37.92 33.63 31.91 29.02 26.21

0.0000 1.4824 3.7025 6.2881 8.1014 10.3284 13.1277 14.4860 17.2715 19.2089 20.4913 25.8072 32.6882 35.9357 39.6934 43.1614

0.003060 0.003083 0.003124 0.003159 0.003191 0.003234 0.003292 0.003323 0.003393 0.003448 0.003484 0.003673 0.004141 0.004365 0.004800 0.005314

64531.25 58010.91 63485.74 61903.20 59428.48 56530.50 55182.05 51860.77 49542.42 48325.41 42088.23 30228.30 27535.81 22810.77 19153.01

15.81 24.07 30.82 31.11 30.65 32.00 33.16 32.05 31.29 31.45 30.93 26.18 28.06 26.73 25.78

35

y, 108m3

30

25 G=20.5×108m3

20

15 0

10,000

20,000

30,000

40,000

50,000

60,000

x, MPa Figure 5
31 Analysis results for Anderson L reservoir by Havlena
Odeh method.

Original gas in place estimations for geopressured gas reservoirs

199

This method was initially used to calculate the OGIP of water-drive gas reservoirs. Here, it is imported to the high-pressure gas reservoir. This method is advantageous in that it does not require prior knowledge of water size and rock compressibility. However, it should be noted that the calculation results are sensitive to initial formation pressure and early data. For a closed gas reservoir, this method can calculate both OGIP and rock compressibility.

5.2.6 Sun method 5.2.6.1 Gas production of cumulative unit pressure drop As described in Chapter 4, the gas production of cumulative unit pressure drop is defined as the ratio of cumulative gas production to the cumulative drawdown of formation pressure from the initial formation pressure condition to the current formation pressure condition.
 1 dG C52 (5.64) G dp T When separation variables are integrated, we have ð pi   Gp 5 G Cdp 5 GC ðpi 2 pÞ

(5.65)

p

where GC is the gas production of cumulative unit pressure drop, 108 m3/MPa. When the formation pressure drops to 0.101325 MPa, we have   Gp 5 G 5 GC pi (5.66) That is, when the formation pressure drops to normal atmospheric pressure, the dimensionless gas production of cumulative unit pressure drop is the reciprocal of the initial formation pressure. As shown in Figs. 4
5, the relation curve of gas production of cumulative unit pressure drop and cumulative pressure drop is linear at early and middle times and gradually flattens out in the late stage, with the flattened amplitude sensitive to the deviation factor. Example 15: Suppose that the formation pressure is 40 MPa and the gas relative gravity is 0.6. Analyze the change of gas production of cumulative unit pressure drop with formation pressure drop at the temperature of constant volume gas reservoir being 333.15K, 373.15K, and 413.15K.

200

Reserves Estimation for Geopressured Gas Reservoirs

Solution According to Eqs. (4.12), calculate the gas production of cumulative unit pressure drop of the closed gas reservoir, with the results shown in Fig. 5
32. It can be seen that the relation curve of the gas production of cumulative unit pressure drop and the cumulative pressure drop is linear in the early and middle time. When the linear relation is prolonged to intersect with the vertical axis, the gas production ratio of cumulative unit pressure drop to intersection value is approximately equal to the minimum deviation factor (Fig. 5
33). In Fig. 5
32A, the temperature is 333.15K, a 5 0.025/0.0284 5 0.88  0.85, and the relative error is 3.6%. In Fig. 5
32B, the temperature is 373.15K, a 5 0.025/0.0264 5 0.95  0.94, and the relative error is 1.1%. In Fig. 5
32C, the temperature is 413.15K, a 5 0.025/0.0267 5 0.94  0.96, and the relative error is 2.1%. The minimum deviation factor of natural gas depends on the pseudo-critical property, as shown in Fig. 5
33. When the pseudo-reduced temperature is more than 1.9, the minimum deviation factor of natural gas is above 0.90. Example 16: For a closed gas reservoir, the initial pressure is 74 MPa, the temperature is 377.15K, the gas gravity is 0.568, the initial water saturation is 0.32, the initial rock compressibility is 2.5 3 1023/MPa, and the formation water compressibility is 5.6 3 1024/MPa (Example 2). Plot the gas production of the gas reservoir’s cumulative unit pressure drop curve and analyze its characteristics. Solution Plot the gas production of the gas reservoir’s cumulative unit pressure drop curve using the data in Table 4
3, as shown in Fig. 5
34. Its characteristics are the same as those of constant-volume gas reservoirs, a 5 0.0135/0.0141 5 0.957  0.935, and the relative error is 2.3%. 5.2.6.2 Analysis procedure The analysis procedure of the method is as follows. 1. Prepare the Cartesian plot of gas production of Gp/ΔpBΔp, conduct linear regression, extend the straight line to intersect with the vertical axis, and record the intersection coordinates. 2. Determine the minimum deviation factor based on natural gas temperature and physical parameters and refer to Fig. 5
34. 3. Multiply the initial formation pressure, intersection value, and minimum deviation factor to get the product, which is the OGIP.

∆ p , MPa 0

5

10

15

20

25

30

35

40 0.030

1.10

0.0284

Z Gp/(G∆p)

1.05

0.027

1.00

0.024

0.95

0.021

0.90

0.018

0.85

Gp/(G ∆ p), MPa-1

Z

0.025

0.015 0.852

0.80 40

0.012 35

30

25

20

15

10

5

0

30

35

40

p, MPa

(A) 333.15 K 0

5

10

15

∆p, MPa 20

25

1.10 0.0264

0.025

0.024

0.022

1.00

0.020 0.95

0.94

Gp/(G ∆ p), MPa-1

Z

1.05

0.026

0.018

Z Gp/(G∆p)

0.016

0.90 40

35

30

25

20

15

10

5

0

p, MPa

(B) 373.15 K 5

10

15

∆ p, MPa 20

25

30

35

1.10

Z Gp/(G∆p)

40 0.028 0.0267

0.026 0.025

1.05

Z

0.024 0.022

1.00 0.964

0.020

0.95

Gp/(G ∆ p), MPa-1

0

0.018 0.90 40

0.016 35

30

25

20

15

10

5

0

p, MPa

(C) 413.15K

Figure 5
32 Effect of temperature on gas production of cumulative unit pressure drop. (A) gas production of cumulative unit pressure drop when temperature is 333.15K; (B) gas production of cumulative unit pressure drop when temperature is 373.15K; (C) gas production of cumulative unit pressure drop when temperature is 413.15K.

202

Reserves Estimation for Geopressured Gas Reservoirs

1.2

1.0

Zmin

0.8

0.6

0.4

0.2 1.0

1.5

2.0

Tpr

2.5

3.0

3.5

Figure 5
33 Minimum deviation factor at different pseudo-reduced temperatures.

' p, MPa 15

30

45

60

75 0.0145

1.5

Z Gp/(G'pR)

1.4

0.0140

0.0140

0.0135 0.0135

1.3

0.0130

Z

0.0125 1.2 0.0120 1.1

0.0115

Gp/(G'p), MPa-1

0

0.0110 1.0 0.0105

0.935

0.9 75

0.0100 60

45

30

15

0

p, MPa Figure 5
34 Gas production of cumulative unit pressure drop curve of a closed gas reservoir.

Example 17: The Anderson L high-pressure gas reservoir in the United States corresponds to the production data in Appendix Table 3
2 (Duggan, 1972). Use the gas production of the cumulative unit pressure drop method to calculate the OGIP.

Original gas in place estimations for geopressured gas reservoirs

203

Solution Calculate the gas production of cumulative unit pressure drop using the given data, with the results shown in Table 5
13. Next, plot the Gp/ΔpBΔp curve. Under atmospheric pressure, the gas production of cumulative unit pressure drop is 0.31 3 108 m3/MPa, as shown in Fig. 5
35. Moreover, finally, given the gas gravity of 0.665, the minimum deviation factor is calculated as 0.934. Thus, the OGIP is 18.96 3 108 m3. Without correction, the value is 20.30 3 108 m3. If the OGIP is calculated directly by the p/Z curve method, the value is 26.6 3 108 m3, as shown in Fig. 5
36, which is much higher than that calculated by the volumetric method. Example 18: For the M2 reservoir, the initial formation pressure is 74.22 MPa, and the formation water compressibility is 5.6 3 1024/MPa. It is assumed that the formation water volume factor is 1.0, the initial water saturation is 0.32, and the gas reservoir temperature is 100°C, and the volumetric OGIP is 2400 3 108 m3. The production data are shown in Appendix Table 17
1. Use the gas production of the cumulative unit pressure drop method to calculate the OGIP. Solution According to the analysis results for Example 1, the M2 reservoir is a water-drive gas reservoir. First, calculate the gas production of cumulative Table 5
13 Calculation results for Example 17. No.

Gp, 108 m3

p, MPa

Δp, MPa

Z

p/Z, MPa

Gp/Δp, 108 m3/MPa

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.000 0.118 0.492 0.966 1.276 1.647 2.257 2.620 3.146 3.519 3.827 5.163 6.836 8.389 9.689 10.931

65.55 64.07 61.85 59.26 57.45 55.22 52.42 51.06 48.28 46.34 45.06 39.74 32.86 29.61 25.86 22.39

0.00 1.48 3.70 6.29 8.10 10.33 13.13 14.49 17.27 19.21 20.49 25.81 32.69 35.94 39.69 43.16

1.440 1.418 1.387 1.344 1.316 1.282 1.239 1.218 1.176 1.147 1.127 1.048 0.977 0.928 0.891 0.854

45.52 45.18 44.59 44.09 43.65 43.07 42.31 41.92 41.05 40.40 39.98 37.92 33.63 31.91 29.02 26.21

0.079 0.133 0.154 0.157 0.160 0.172 0.181 0.182 0.183 0.187 0.200 0.209 0.233 0.244 0.253

204

Reserves Estimation for Geopressured Gas Reservoirs

0.35 Gp/' p=0.0027' p+0.1326

Gp/' p, 108m3/MPa

(65.548,0.31)

R2 =0.9745

0.30 0.25 0.20 0.15 0.10 0.05 0.00 0

10

20

30

40

50

60

' p, MPa Figure 5
35 Gas production of cumulative unit pressure drop curve for Example 17.

50 p/Z=45.99084-1.73144Gp R2＝0.99

p/Z, MPa

40

30

20

10 26.6

0 0

5

10

15

20

25

30

Gp , 10 m 8

3

Figure 5
36 p/Z curve for Example 17.

unit pressure drop using the given data, with the results shown in Table 5
14. Next, plot the Gp/ΔpBΔp curve, as shown in Fig. 5
37. Starting from point 6, the curve is upward, indicating that water influx has occurred. According to the high-pressure physical properties of natural gas,

205

Original gas in place estimations for geopressured gas reservoirs

Table 5
14 Calculation results for Example 18. No.

Gp, 108 m3

Wp, 104 m3

p, MPa

Z

p/Z, MPa

Δp, MPa

Gp/Δp, 108 m3/MPa

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0.00 2.14 34.61 118.72 228.32 345.40 457.91 553.39 614.31 689.23 769.08 839.48 901.98 958.96 1029.23 1095.80 1157.65

0.00 0.00 0.37 2.11 4.65 7.50 10.93 16.61 18.40 21.11 23.60 26.42 29.52 35.67 39.10 46.26 57.22

74.22 73.98 72.92 69.97 65.80 61.32 57.60 54.99 53.30 51.21 49.00 47.06 45.37 43.86 42.05 40.39 38.87

1.450 1.450 1.446 1.442 1.438 1.434 1.430 1.426 1.422 1.418 1.414 1.413 1.412 1.411 1.411 1.410 1.410

51.27 51.00 50.43 48.52 45.76 42.76 40.28 38.56 37.48 36.11 34.65 33.31 32.13 31.08 29.81 28.64 27.57

0.0000 0.2400 1.3000 4.2500 8.4200 12.9000 16.6200 19.2300 20.9200 23.0100 25.2200 27.1600 28.8500 30.3600 32.1700 33.8300 35.3500

8.9167 26.6231 27.9348 27.1158 26.7749 27.5516 28.7776 29.3647 29.9533 30.4948 30.9088 31.2643 31.5864 31.9935 32.3914 32.7482

45

Gp/' p, 10 8m3/MPa

42.0

40 G=74.22×42×0.93=2899×108m3

35

30

28.0

25

G=74.22×28×0.93=1933×108m3

20 0

15

30

45

60

75

' p, MPa Figure 5
37 Gas production of cumulative unit pressure drop curve for Example 18.

the minimum deviation factor is 0.93. The OGIP is 1933 3 108 m3 based on the data of the first straight-line segment and 2899 3 108 m3 based on the data of the second straight-line segment. If calculated by the p/Z pressure drop method, OGIP is 2468 3 108 m3, as shown in Fig. 5
38.

206

Reserves Estimation for Geopressured Gas Reservoirs

60 p/Z=50.597-0.0205Gp R2 =0.99

p/Z, MPa

50 40 30 20 10 G=2468

0 0

500

1000

1500

2000

2500

Gp, 10 m 8

3

Figure 5
38 p/Z curve for Example 18.

This method is advantageous because it requires only production and pressure data and is relatively accurate for constant-volume and closed gas reservoirs. However, this method should be combined with other analysis methods, otherwise, the calculation results may deviate largely.

5.3 Nonlinear regression method The nonlinear regression method expresses the material balance equation as different nonlinear forms and uses multivariate or nonlinear regression methods to estimate the OGIP. It mainly includes the binary regression method (Chen and Hu, 1993) and the nonlinear regression method (Gonzalez, 2008; Zheng and Liu, 2011; Jiao, 2017; Sun et al., 2019).

5.3.1 Binary regression method As described in Section 4.1, the material balance equation for closed gas reservoirs (Ramagost and Farshad, 1981) can be expressed as
 Gp p pi ð1 2 Ce ΔpÞ 5 12 (5.67) Z Zi G

Original gas in place estimations for geopressured gas reservoirs

207

Rearranging Eq. (5.67) (Li and Lin, 1985; Chen and Hu, 1993), we have
 G ð1 2 Ce pi Þ p GCe p2 Gp 5 G 2 2 (5.68) pi =Zi Z pi =Zi Z y 5 a0 1 a1 x1 1 a2 x2

(5.69)

Where p2 Z G ð 1 2 Ce p i Þ a1 5 2 pi =Zi

y 5 Gp x 1 5 a0 5 G

p Z

x2 5

a2 5 2

GCe pi =Zi

Based on pressure, cumulative production, and gas deviation factor, the binary regression method determines the coefficients (see Appendix 27 for analysis principle), where a0 is the OGIP. Then, based on a1 and a2, the effective compressibility can be calculated by Eq. (5.70) a2 Ce 5 (5.70) a1 1 a2 pi Example 19: The offshore high-pressure gas reservoir in Louisiana, United States, corresponds to the production data in Appendix Table 2
1 (Ramagost and Farshad, 1981). The initial water saturation is 0.22, the formation water compressibility is 4.41 3 1024/MPa, and the rock compressibility is 3.325 3 1023/MPa. Use the binary regression method to calculate the OGIP. Solution Use Eq. (5.69) to calculate x1 and x2, with the results shown in Table 5
15. Next, use Microsoft Excel software to conduct a binary regression analysis, where x1 and x2 are independent variables, and Gp is a dependent variable, with the matching results shown in Fig. 5
39, a0 5 129.99, a1 5 21.7467, a2 5 29.69 3 1023, that is, OGIP is 130.0 3 108 m3. And finally, use Eq. (5.70) to calculate the effective compressibility as follows Ce 5

a2 9:69 3 1023 5 38:58 3 1024 MPa21 5 a1 1 a2 pi 1:7467 1 9:69 3 1023 3 78:90

208

Reserves Estimation for Geopressured Gas Reservoirs

Table 5
15 Calculation results for Example 19. Gp, 108 m3

p, MPa

Z

p/Z, MPa

p2/Z, MPa2

0.00 2.81 8.11 15.18 22.00 28.72 34.09 41.07 45.49 51.64 56.00 61.08 66.76 69.64

78.90 73.60 69.85 63.80 59.12 54.51 50.88 47.21 44.04 40.18 37.29 34.47 31.03 28.75

1.496 1.438 1.397 1.330 1.280 1.230 1.192 1.154 1.122 1.084 1.057 1.033 1.005 0.988

52.74 51.18 50.00 47.97 46.18 44.32 42.69 40.91 39.25 37.06 35.28 33.37 30.87 29.10

4161.66 3766.50 3492.60 3060.19 2730.24 2415.72 2172.05 1931.19 1728.94 1489.03 1315.84 1150.49 957.82 836.66

Figure 5
39 Results by the binary regression method for Example 19.

Original gas in place estimations for geopressured gas reservoirs

209

5.3.2 Nonlinear regression method The material balance equation for closed gas reservoirs with compressibility as a variable (Fetkovich et al., 1998) can be expressed as
 pi Gp p 12 1 2 C e ðpÞðpi 2 pÞ 5 (5.71) Z Zi G where C e ðpÞ is the effective cumulative compressibility and is defined as i 1 h C e ðpÞ 5 Swi C wi 1 C f 1 M ðC w 1 C f Þ (5.72) 1 2 Swi

5.3.2.1 C e ðpÞðpi 2 pÞBGp linear relation The relation is critical in the material balance equation for high-pressure and ultrahigh-pressure gas reservoirs. Gonzalez (2008) proposed the following approximate linear relation. C e ðpÞðpi 2 pÞ  λGp

(5.73)

Substituting Eq. (5.73) into Eq. (5.71), we have pD 5

1 2 Gp =G 1 2 aGp p=Z 5 5 pi =Zi 1 2 λGp 1 2 bGp

(5.74)

where a and b (a 5 1=G, b 5 λ) in Eq. (5.74) can be obtained by nonlinear regression (Jiao, 2017). If bGp ,, 1, According to the Taylor series expansion, we have 1 5 1 1 bGp 1 b2 Gp2 1 . . . 1 2 bGp

(5.75)

Substituting Eq. (5.75) into Eq. (5.74), we obtain the parabolic (Gonzalez, 2008) and cubic (Zheng and Liu, 2011) relations as follows. pD 5 1 1 ðb 2 aÞGp 2 abGp2

(5.76)

pD 5 1 1 ðb 2 aÞGp 1 bðb 2 aÞGp2 2 ab2 Gp3

(5.77)

Eq. (5.74) is the limiting form of parabolic and cubic relations. When the condition bGp ,, 1 does not hold, the results calculated by Eqs. (5.76) and (5.77) can be overestimated, while the nonlinear regression results of Eq. (5.74) are more accurate.

210

Reserves Estimation for Geopressured Gas Reservoirs

Example 20: For a closed gas reservoir, the initial pressure is 74 MPa, the gas reservoir temperature is 377.15K, the gas gravity is 0.568, the initial water saturation is 0.32, the initial rock compressibility is 2.5 3 1023/ MPa, and the formation water compressibility is 5.6 3 1024/MPa (Example 4
2). Plot the Gp/GBλ curve. Solution According to Eq. (5.71) and Eq. (5.73), we have 
 Gp B pi =Zi  gi  λ5 12 12 p=Z G Gp =G

(5.78)

Plot the Gp/GBλ curve using the data in Table 4
2, as shown in Fig. 5
40. The Gp/GBλ relation can be approximately expressed as a linear function. Therefore, it is reasonable to set λ as a linear function of Gp/G. Plot the gas production of the gas reservoir’s cumulative unit pressure drop curve using the results calculated by Eq. (5.78), as shown in Fig. 5
34. Its characteristics are the same as those of constant-volume gas reservoirs. A 5 0.0135/0.0140 5 0.957  0.935, and the relative error is 2.3%.

0.00100 0.00095

λ

0.00090 0.00085 linear fit

0.00080 0.00075 0.00070 0.0

0.2

0.4

0.6

0.8

1.0

Gp/G Figure 5
40 Gp/GBλ plot for closed gas reservoir. Modified from Gonzalez, F.E., Ilk, D., Blasingame, T.A., 2008. A quadratic cumulative production model for the material balance of an abnormally pressured gas reservoir. SPE 114044-MS. Permission to publish by the SPE, Copyright SPE.

211

Original gas in place estimations for geopressured gas reservoirs

Example 21: The Anderson L high-pressure gas reservoir in the United States corresponds to the production data in Appendix Table 3
2 (Duggan, 1972), and the gas reservoir temperature is 130°C. Use the binomial and nonlinear regression methods to calculate the OGIP and the λ value. Solution Conduct regression according to Eqs. (5.74) and Eq. (5.76), with the comparison results shown in Fig. 5
41. Calculate λ according to Eq. (5.78), and λ decreases with the increase of OGIP, as shown in Fig. 5
42. According to the nonlinear regression results, λ 5 0.02025. Use the binomial regression method to calculate the OGIP as 21.0 3 108 m3, with the regression equation as follows. pD 5 1 2 0:02939 Gp 2 8:52869 3 1024 Gp2 Use the nonlinear regression method to calculate the OGIP as 19.8 3 108 m3, with the regression equation as follows. pD 5

1 2 0:05047 Gp 1 2 0:02025 Gp

(5.79)

The difference between reserves calculated by these two methods is 6%. 1.0 actual point Binomial regression Nonlinear regression

0.8

pD

0.6

0.4

0.2

0.0 0

5

10

15

Gp , 10 m 8

3

Figure 5
41 Nonlinear regression results for Example 4.

20

25

212

Reserves Estimation for Geopressured Gas Reservoirs

0.025 λ =0.02025

0.020 λ =0.016

λ

0.015

0.010

0.005 Binomial Nonlinear

0.000 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Gp/G Figure 5
42 λ value for Example 4.

Example 22: The M4 reservoir is a typical water drive gas reservoir (Jiao, 2017), with an initial formation pressure of 74.48 MPa, the gas reservoir temperature of 100°C, and the volumetric OGIP of 2091.5 3 108 m3. The production data are shown in Appendix Table 19
1. Use the binomial and nonlinear regression methods to calculate the OGIP. Solution Conduct regression according to Eqs. (5.74) and (5.76), with the comparison results shown in Fig. 5
43. Use the binomial regression method to calculate the OGIP as 2760 3 108 m3, which has a 25.9% difference from the volumetric reserves, with the regression equation as follows. pD 5 1 2 7:95488 3 1025 Gp 2 1:01514 3 1027 Gp2 Use the nonlinear regression method to calculate the OGIP as 2158.3 3 108 m3, which has a 1.5% difference from the volumetric reserves, with the regression equation as follows. pD 5

1 2 4:63328 3 1024 Gp 1 2 3:4874 3 1024 Gp

(5.80)

The difference between reserves calculated by these two methods is 27.9%. Therefore, the nonlinear regression method is recommended for analysis.

213

Original gas in place estimations for geopressured gas reservoirs

1.0 Actual point Binomial regression Nonlinear regression

0.8

pD

0.6

0.4

0.2 2158

2760

0.0 0

500

1000

1500

2000

2500

3000

Gp , 10 m 8

3

Figure 5
43 Nonlinear regression results for Example 22.

5.3.2.2 C e ðpÞðpi 2 pÞBGp power function relation Assume that C e ðpÞðpi 2 pÞ  bGpc

(5.81)

Substituting Eq. (5.81) into Eq. (5.71), we have pD 5

1 2 aGp 1 2 bGpc

(5.82)

According to Eq. (5.74), use the nonlinear regression method to determine the OGIP of 20 gas reservoirs in Appendix 1
14. Next, plot the C e ðpÞðpi 2 pÞBGp =G log
log plot, as shown in Fig. 5
44, and the empirical value of c is 1.02847, that is, pD 5

1 2 aGp 1 2 bGp1:02847

(5.83)

Example 23: Use Eq. (5.83) to calculate the OGIP of the M4 reservoir. Solution Conduct nonlinear regression according to Eq. (5.83), with the results shown in Fig. 5
45. The OGIP is 2120.0 3 108 m3, which has a 1.4%

214

Reserves Estimation for Geopressured Gas Reservoirs

100 Ce'p=-0.29242+1.02847(Gp/G) R2=0.81027

Ce(pi-p)

10-1

10-2 19 gas reservoir Anderson L linear regression

10-3 10-3

10-2

10-1

Gp/G

100

Figure 5
44 C e ðpÞðpi 2 pÞBGp =G log
log plot for 20 gas reservoirs.

1.0 Actual point Binomial regression Nonlinear regression Power function regression

0.8

pD

0.6

0.4

0.2 2120

2158

2760

0.0 0

500

1000

1500

2000

Gp , 10 m 8

3

Figure 5
45 Nonlinear regression results for Example 23.

2500

3000

Original gas in place estimations for geopressured gas reservoirs

215

difference from the volumetric reserves, with the regression equation as follows. pD 5

1 2 4:71656 3 1024 G p 1 2 2:9436 3 1024 G 1:02847 p

(5.84)

5.3.3 Starting point of the nonlinear regression method Eq. (5.83) was used to calculate the OGIP of 20 developed high-pressure and ultrahigh-pressure gas reservoirs, as shown in Table 5
16. Except for No. 6, 12, 14, and 17 reservoirs, the OGIP in other reservoirs calculated using Eq. (5.83) is the same as other methods but slightly smaller than those calculated using the binomial nonlinear regression method (Gonzalez et al., 2008). The calculation results by various methods for No. 6, 12, 14, and 17 gas reservoirs are quite different, which is speculated to be attributable to the apparent pressure depletion. Statistical analysis results show that for the two-segment straight-line method represented by the Gan and Blasingame (2001) method, the apparent pressure depletion degree with respect to the time point at which the inflection point appears is 0.14
0.38, or 0.23 averagely, as shown in Fig. 5
10, the apparent pressure depletion degree with respect to the time point at which the reserves can be calculated from the second straight line is 0.23
0.50, or 0.33 averagely, as shown in Fig. 5
46, the corresponding degree of reserve recovery is 0.33
0.65, or 0.45 averagely, as shown in Fig. 5
47. Statistical analysis of the nonlinear regression method using Eq. (5.83) shows that if the 10% difference between the calculated OGIP and those at all points is taken as the standard, the corresponding apparent pressure depletion degree is 0.16
0.62, or 0.33 averagely, as shown in Fig. 5
48, and the corresponding degree of reserve recovery is 0.28
0.62, or 0.48 averagely, as shown in Fig. 5
49. For fractured, stress-sensitive gas reservoirs, because the fracture compressibility is challenging to be determined, the natural gas reserves can also be estimated by the nonlinear regression method. Example 24: For a gas reservoir, the connate water saturation of matrix is 0.25, the water saturation of fracture is 0, the water phase compressibility is 4.35 3 1024/MPa, the matrix compressibility is 2.90 3 1023/MPa, the fracture porosity is 0.01, and the fracture storativity ratio is 0.5.

Table 5
16 Calculation results by various methods for 20 developed high-pressure and ultrahigh-pressure gas reservoirs. No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Gas reservoir

Reservoir 197 SE Texas Reservoir 70 Reservoir 41 Reservoir 33 Reservoir 268 Reservoir 195 Reservoir 117 ROB 43
1 Cajun Louisiana GOM Field 38 Example 4 GOM Case2 Stafford North Ossun South LA Offshore LA Anderson L

Data source

Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix

4 13 4 5 6 7 8 9 11 14 12 5 9 8 10 6 1 7 2 3

Apparent pressure depletion degree 1 2 pD

0.70 0.57 0.76 0.39 0.84 0.27 0.79 0.46 0.50 0.82 0.43 0.27 0.49 0.16 0.63 0.51 0.24 0.38 0.45 0.42

Volumetric method 108 m3

61.73

28.32

32.28

19.68

Literature 108 m3

Extrapolation Guehria Extrapolation Hammerlindl Hammerlindl Hammerlindl Hammerlindl Hammerlindl Roach Roach Guehria Ramagost Roach Havlena-Odeh Roach Ramagost Bourgoyne Bourgoyne Ramagost Ramagost

4.11 59.75 3.11 11.61 61.45 8.64 15.04 159.28 28.6 62.3 30.87 6.34 22.68 13.62 40.21 7.08 33.41 4.53 133.09 20.39

Gan

Gonzalez

Eq. (5.83)

108 m3

108 m3

108 m3

4.13 75.89 3.43 13.76 60.91 9.2 14.64 130.63 30.47 60.6 36.39 4.36 19.82 10.28 46.3 6.46 25.37 3.91 140.88 21.38

3.88 74.36 3.34 13.96 59.15 8.55 14.22 142.55 33.13 58.79 38.79 4.28 19.48 15.23 51.2 6.65 24.49 3.77 125.05 20.81

4.17 64.22 3.32 10.22 57.74 5.32 14.06 114.42 26.83 57.94 32.12 3.48 19.71 9.42 46.82 5.5 24.58 2.91 122.4 19.37

217

Original gas in place estimations for geopressured gas reservoirs

Apparent formation pressure depletion degree 1-pD

0.75

0.60 Mean value 0.33

0.45

0.30

0.15

0.00 5

10

15

20

Reservoir No. Figure 5
46 Time inflection point for Gan method.

Degree of reserve recovery Gp/G

0.75 Mean value 0.45

0.60

0.45

0.30

0.15

0.00 5

10

15

20

Reservoir No. Figure 5
47 Degree of reserve recovery at time point with the error of OGIP less than 10%.

218

Reserves Estimation for Geopressured Gas Reservoirs

Apparent formation pressure depletion degree 1-pD

0.75

0.60 Mean value 0.33

0.45

0.30

0.15

0.00 5

10

15

20

Reservoir No. Figure 5
48 Time point with the error of OGIP less than 10%.

Degree of reserve recovery Gp/G

0.75 Mean value 0.48

0.60

0.45

0.30

0.15

0.00 5

10

15

20

Reservoir No. Figure 5
49 Degree of reserve recovery at time point with the error of OGIP less than 10%.

Original gas in place estimations for geopressured gas reservoirs

219

The production data are shown in Appendix Table 20
1. Use different methods to calculate the OGIP. Solution Use Eqs. (5.74), (5.76), and (5.83) to conduct nonlinear regression, and obtain the reserves as 46.7 3 108 m3, 45.6 3 108 m3, and 45.3 3 108 m3, respectively, which are close to the initial calculation results of 48.14 3 108 m3 (Aguilera, 2008). Neglecting the influence of fracture compressibility, conduct linear regression and extrapolation for the first four data points, and obtain the gas reserves as 58.3 3 108 m3, which are overestimated, as shown in Fig. 5
50. Fig. 5
51 shows that when the apparent formation pressure depletion degree is higher than 0.35, the reserves calculated by Eq. (5.83) are closer, and the critical point is the same as the above statistical result (0.33). Suppose that C e ðpÞðpi 2 pÞ and Gp in the equation are linearly correlated. After the dimensionless apparent formation pressure pD, dimensionless degree of reserve recovery GpD, and dimensionless linear coefficient λD have been introduced, Eq. (5.74) can be expressed as 1 2 Gp =G 1 2 GpD p=Z  5 1 2 λGp pi =Zi 1 2 λD GpD where GpD 5 Gp =G, λD 5 λG. pD 5

(5.85)

1.0

0.8

pD

0.6

0.38

0.4

real data Binomial, G=46.7×108m3 Nonlinear, G=45.6×108m3 Power function, G=45.3×108m3 Extrapolation, G=58.3×108m3

0.2

0.0 0

10

20

30

Gp , 108m3 Figure 5
50 pD
Gp regression curve for Example 24.

40

50

60

220

Reserves Estimation for Geopressured Gas Reservoirs

1.0

0.8

pD

0.65

0.6

0.54

0.38

0.4

Real data Eq.(5-83)(all point) ,G=45.3×108m3 Eq.(5-83)(first 5 points),G=46.9×108m3 Eq.(5-83)(first 4 points),G=55.0×108m3 Extrapolation (first 5 points),G=58.3×108m3

0.2

0.0 0

10

20

30

40

50

60

Gp, 10 m 8

3

Figure 5
51 Sensitivity analysis of apparent formation pressure depletion degree.

Since there is no analytic solution to the starting point of the pD
Gp curve deviating from the early straight line, the starting point cannot be obtained by theoretical calculation. According to Eq. (5.85), plot the pD
GpD curve (Fig. 5
52) and use the graphical method to obtain this point. Conduct linear regression on the early data points of the pD
GpD curve, determine the slope and intercept of the linear regression curve, and determine the location of the inflection point depending on two discriminant conditions: (1) The relative error between the intercept value of the pD
GpD linear regression curve and 1.0 is less than 0.25%, and (2) the relative error between the fitted pD linear regression value corresponding to the inflection point abscissa GpD and the actual inflection point pD value is less than 0.50%. With different values of λD, the starting points of the pD
GpD curve deviating from the early straight line are significantly different, and the corresponding apparent formation pressure depletion degrees (1 2 pD) are between 0.06 and 0.38. The statistics on the 22 developed high-pressure and ultrahigh-pressure gas reservoirs show that the λD is between 0.20 and 0.75, and 1 2 pD corresponding to the starting point deviating from the early straight line is between 0.14 and 0.38, consistent with the calculation results of the graphical method. Fig. 5
52 shows that for high-pressure and ultrahigh-pressure large gas reservoirs in the evaluation stage of early production test, even if the production test has lasted for one year and the drawdown amplitude has

Original gas in place estimations for geopressured gas reservoirs

221

1.0

0.8 λD

pD

0.6

0.4

0.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Starting point theoretically deviating from the straight line Starting point deviating from the straight line of 22 developed gas reservoirs

0.0 0.0

0.2

0.4

0.6

0.8

1.0

GpD Figure 5
52 Schematic diagram for starting points of pD
GpD curve deviating from the early straight line.

reached 3%
5% or even higher of initial formation pressure, the starting point deviating from the early straight line does not occur, which is far from reaching the starting condition for the material balance method to calculate the reserves. If the linear regression of these data points is used to calculate the reserves, the calculation results will inevitably be too large to determine the productivity under the development plan accurately and thus mislead the development decision. Based on the starting points deviating from the early straight line in different λD situations (Fig. 5
52), we determine the corresponding G/Ga values and plot the λD versus G/Ga curves, as shown in Fig. 5
53A. If linear regression is adopted, the empirical equation is G=Ga 5 1:03242 2 0:96207λD

(5.86)

If nonlinear regression is adopted, the empirical equation is G 1:0 2 0:97602λD 5 (5.87) Ga 1:0 2 0:18793λD After statistical analysis on the production data of 20 high-pressure and ultrahigh-pressure gas reservoirs in China and abroad, we obtain the relation between λD and G=Ga , as shown in Fig. 5
53B. G=Ga is not a constant but a function of λD , which is consistent with the changing trend of the theoretical curve.

222

Reserves Estimation for Geopressured Gas Reservoirs

1.0

G/Ga

0.8

0.6

0.4

0.2

Data point Linear G/Ga＝1.03242−0.96207 λD Nonlinear G/Ga＝(1.00−0.97602 λ D)/(1-0.18793 λ D)

0.0 0.0

0.2

0.4

0.6

0.8

1.0

λD (A) Theoretical curve result analysis 1.0

G/Ga

0.8

0.6

0.4

0.2 20 gas reservoirs in Appendix 1-14 Eq.(5-87)

0.0 0.0

0.2

0.4

0.6

0.8

1.0

λD (B) Data distribution of developed gas reservoirs Figure 5
53 λD BG=Ga curve corresponding to starting point of pD BGpD curve deviating from an early straight line. (A) Theoretical curve result analysis: linear regression and nonlinear regression; (B) Starting point deviating from an early straight line of 20 developed gas reservoirs.

Original gas in place estimations for geopressured gas reservoirs

223

To sum up, when the nonlinear regression method is used to calculate the reserves, it avoids uncertain parameters such as effective rock compressibility, aquifer volume, and water influx, and it is superior for its simple calculation process and good practicability. Therefore, the nonlinear regression method that does not require compressibility is recommended for estimating the reserves in high-pressure and ultrahighpressure gas reservoirs. The theoretical method cannot obtain the starting point for the nonlinear regression method to calculate the reserves, and the graphical method’s statistical results show that the (1 2 pD) value corresponding to the starting point in different λD situations lies between 0.06 and 0.38.

5.4 Type curve matching analysis method Agarwal introduced the type curve matching method into the petroleum industry Agarwal et al. (1970). This tool has been widely used in well test analysis and reserve estimation using the material balance method. It uses the material balance equation to establish the semilog or log
log type curve, which is matched to obtain the parameters such as reserves and effective compressibility. It includes the Ambastha method (Ambastha, 1991), Fetkovich method (Fetkovich et al., 1998), multiwell advanced production decline analysis method (Marhaendrajana and Blasingame, 2001), Gonzalez method (2008), and Sun method (Sun et al., 2020).

5.4.1 Ambastha method As described in Section 5.3, the material balance equation for closed gas reservoirs can be expressed as follows (Bourgoyne, 1972):
 G ð1 2 Ce pi Þ p GCe p2 Gp 5 G 2 2 (5.68) Z pi =Zi Z pi =Zi After dimensionless conversion (Ambastha, 1991), we have GpD 5 1 2 ð1 2 CD ÞpD 2 CD ZD p2D where

    GpD 5 Gp =G pD 5 p=Z = pi =Zi CD 5 Ce pi ZD 5 Z=Zi

(5.88)

224

Reserves Estimation for Geopressured Gas Reservoirs

If the initial pressure, temperature, and gas properties are given, the ZD can be calculated, and the type curve can be established. Taking the Cajun ultrahigh-pressure gas reservoir as an example, the initial formation pressure is 79.0 MPa, the gas reservoir temperature is 401.3K, the gas gravity is 0.6, and the deviation factor under initial conditions for Eq. (A22
1) in Appendix 22 is 1.4795. Thus, the type curve of the gas reservoir is shown in Fig. 5
54. Ambastha (1991) presented a type curve matching procedure based on a Cartesian plot according to Fig. 5
54A. However, the procedure requires further processing of dimensionless data, possibly leading to significant uncertainty in the calculation. Nonetheless, it works well for CD 5 0
0.6. If logarithm is taken for the horizontal axis, the type curve is shown in Fig. 5
54B. At this time, the semilog type curve matching method can be used for analysis with the following procedure. 1. Draw a series of data points (Gp, pD) on the pD versus Gp semilog plot based on the production data, and then superpose the plot on the pD versus GpD semilog type curve plot (Fig. 5
54B). 2. Move the data points up and down to align the vertical axis, and then, move the data points leftward and rightward to match the curve corresponding to a CD in the pD versus GpD semilog type curve. To determine CD, take any point to read its coordinate values (Gp, pD) and (GpD, pD) on the pD versus Gp and pD versus GpD semilog plots, respectively, and then, use Eq. (5.88) to determine G and Ce. Example 25: Use the Ambastha semilog type curve to calculate the reserves of the Cajun gas reservoir, which corresponds to the production data in Appendix Table 22
1. Solution Prepare the pD versus Gp plot, as shown in Fig. 5
55, and then, superpose Fig. 5
55 on Fig. 5
54B for matching, as shown in Fig. 5
56, and finally, read the match point data. The match point corresponds to the coordinate (0.3, 0.7) on the theoretical type curve and (4.0, 0.7) on an actual curve, and the reserves are estimated to be 13.3 3 108 m3. According to the CD match value of 0.3, the effective compressibility is determined as 3.8 3 1023/MPa.

Original gas in place estimations for geopressured gas reservoirs

225

1.0

0.8

pD

0.6

0.4

CD 0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.2

0.0 0.0

0.2

0.4

GpD

0.6

0.8

1.0

(A) Cartesian plot 1.0

0.8

pD

0.6

0.4

0.2

0.0 10-2

CD 0.0 0.1 0.2 0.3 0.4 0.5 0.6

10-1

GpD

100

(B) Semi-log plot Figure 5
54 Ambastha type curve for Cajun gas reservoir. (A) cartesian plot of Ambastha type curve for Cajun gas reservoir; (B) semi-log plot of Ambastha type curve for Cajun gas reservoir. Modified from Ambastha, A K. 1991. A type curve matching procedure for material balance analysis of production data from geopressured gas reservoirs. Journal of Canadian Petroleum Technology, 30(5), 61-65.

226

Reserves Estimation for Geopressured Gas Reservoirs

1.0

pD

0.9

0.8

0.7

0.6 10-1

100

101

Gp,10 m 8

3

Figure 5
55 Actual curve for Example 25.

1.0

1.0

pD

0.9

0.8

0.8

0.7

0.6 10-1

100

101

0.6

0.4

CD 0.0 0.1 0.2 0.3 0.4 0.5 0.6

10-2

pD

Gp,108m3

0.2

10-1

GpD

0.0 100

Figure 5
56 Type curve matching for Example 25.

The type curve is a function of the gas deviation factor. When this method is used to analyze the data of a gas reservoir, the theoretical type curve of the gas reservoir should be established. While matching, due to the similar curve shape, there is significant uncertainty in the result, especially when the apparent formation pressure decreases very little.

Original gas in place estimations for geopressured gas reservoirs

227

5.4.2 Fetkovich method As described in Chapter 4, for closed ultrahigh-pressure gas reservoirs with the connate water saturation Swi and the water volume ratio M, neglecting the water influx and gas injection terms, the material balance equation considering water-soluble gas (Fetkovich et al., 1998) is
 pi Gp p 1 2 C e ðpÞðpi 2 pÞ 5 12 (5.89) Z Zi G C e ðpÞ 5

  1  Swi C w 1 C f 1 M C w 1 C f 1 2 Swi

(5.90)

where the effective cumulative compressibility of formation water is expressed as C w ðpÞ 5

Btw 2 Btwi Btwi Δp

(5.91)

This equation considers the escape of dissolved gas, and the water includes connate water, water within nonpay reservoir rock, and any limited aquifer volume. Btw 5 Bw 1 ðRswi 2 Rsw ÞBg

(5.92)

Eqs. (3
67) and (3
72) can be used for calculating the formation water volume factor and the gas solubility in formation water. The nonpay water volume ratio M represents the ratio of the sum of nonnet-pay (NNP) volume and limited aquifer (AQ) volume to the pore volume (PV) in pay, and it is expressed by Eq. (5.93). M5

MNNP 1 MAQ MPV

(5.93)

This method can be used to analyze the reserves of the gas reservoir, cumulative effective compressibility, and nonpay water volume ratio, with the procedure as follows. 1. Plot the p=ZBGp and pBGp curves on the same graph to determine the maximum and minimum values of G. 2. Assuming a series of G values, back-calculate the cumulative effective compressibility according to Eq. (5.94) and plot the C e ðpÞBp curve. 
 Gp pi =Zi 1 C e ðpÞ 5 1 2 12 (5.94) p=Z G ðpi 2 pÞ

228

Reserves Estimation for Geopressured Gas Reservoirs

3. Use the empirical equation to determine the formation water volume factor and the gas solubility. Assuming a series of M values, calculate the cumulative effective compressibility according to Eq. (5.90) and plot the C e ðpÞBp curve on the same graph. 4. Match the curves of the two methods, and finally, determine the M and G values. Example 26: The Anderson L gas reservoir corresponds to the production data in Appendix Table 3
2 (Duggan, 1972), with the initial formation pressure of 65.55 MPa, the gas reservoir temperature of 130.0°C, the initial water saturation of 0.35, the formation water compressibility of 4.64 3 1024/MPa, and the rock compressibility of 4.64 3 1024/MPa (Fetkovich et al., 1998). Use the Fetkovich method to calculate the OGIP. Solution 1. Plot the p/ZBGp and pBGp curves, as shown in Fig. 5
57. The G value is roughly estimated as (18
25) 3 108 m3. 2. Suppose that the reserves are 18 3 108 m3, 20.4 3 108 m3, and 25 3 108 m3, respectively, back-calculate the cumulative effective compressibility according to Eq. (5.94), and plot the C e ðpÞBp curve, as shown in Fig. 5
58. The solid line represents the C e ðpÞBp curve

75 p p/Z

p, p/Z, MPa

60

45

30

15 18

0 0

5

10

15

25

20

25

Gp , 10 m 8

3

Figure 5
57 Reserve preliminary estimation for Example 26.

30

35

229

Original gas in place estimations for geopressured gas reservoirs

0.008

G= 18.0×108m3 20.4×108m3 25.0×108m3

0.007

Ce , MPa-1

0.006 0.005 0.004 M=2.25

0.003 0.002 0.001 0.000 0

10

20

30

40

50

60

70

p, MPa Figure 5
58 Back-calculated cumulative effective compressibility at various reserves. Modified from Fetkovich, M. J., Reese, D. E., Whitson, C. H. 1998. Application of a general material balance for high-pressure gas reservoirs (includes associated paper 51360). SPE Journal, 3(1), 3
13. Permission to publish by the SPE, Copyright SPE.

from the reservoir property parameters. Therefore, the nonpay water volume ratio is 2.25, and the reserves are 20.4 3 108 m3. Based on these parameters, the reserves predicted by the Ramagost
Fashad method are 21.52 3 108 m3, containing 1.12 3 108 m3 dissolved gas, as shown in Fig. 5
59. This method requires the parameters such as gas solubility in water and formation water volume factor, which can be calculated by Eqs. (3
67) and (3
72) provided in Chapter 3, with the results shown in Figs. 5
60 and 5
61. When the pressure is higher than 10 MPa, C w ðpÞ changes slowly, but when the pressure is lower than 6.89 MPa, C w ðpÞ changes basically according to the 1=p. The results for Example 26 show that for high-pressure gas reservoirs, the escape of natural gas dissolved in water in nonpay is also essential driving energy, and its reserve may account for 2%
10% of the pure gas area, depending on the nonpay water volume ratio M and the gas solubility in water. For gas reservoirs with high CO2 content, the proportion may be higher. For example, in the Ellenburger reservoir (in the natural gas component, CO2 mole content accounts for 28%), the proportion of dissolved gas reserve is more than 15% (Fetkovich et al., 1998).

230

Reserves Estimation for Geopressured Gas Reservoirs

1.0

0.8

pD

0.6

0.4

0.2 21.52

0.0 0

5

10

15

20

25

Gp , 10 m 8

3

Figure 5
59 p/Z
Gp curve.

0.006

Ce , MPa-1

0.005 0.004 0.003 0.002 0.001 0.000 0

10

20

30

40

50

60

70

p, MPa Figure 5
60 Cumulative effective compressibility of formation water. Modified from Fetkovich, M. J., Reese, D. E., Whitson, C. H. 1998. Application of a general material balance for high-pressure gas reservoirs (includes associated paper 51360). SPE Journal, 3 (1), 3
13. Permission to publish by the SPE, Copyright SPE.

Original gas in place estimations for geopressured gas reservoirs

1.5

231

7.5 Btw

1.3

4.5

1.2

3.0

Btw

6.0

1.1

1.5

Bw

1.0 0

Rsw , m3/m3

Rsw

1.4

10

20

30

40

50

60

0.0 70

p, MPa Figure 5
61 Rsw and Bw. Modified from Fetkovich, M. J., Reese, D. E., Whitson, C. H. 1998. Application of a general material balance for high-pressure gas reservoirs (includes associated paper 51360). SPE Journal, 3(1), 3
13. Permission to publish by the SPE, Copyright SPE.

5.4.3 Gonzalez method The binomial expansion of the material balance equation shown in Eq. (5.74) is 2 pD 5 1 2 ð1 2 λD ÞGpD 2 λD GpD

The pressure integral function pDi is defined as ð 1 GpD pDi 5 ð1 2 pD ÞdGpD GpD 0

(5.95)

(5.96)

Substituting Eq. (5.95) into Eq. (5.96), 2 λD GpD ð1 2 λD ÞGpD 2 2 3 Substituting Eq. (5.74) into Eq. (5.96) and integrating,   λD GpD 1 ð1 2 λD Þln 1 2 λD GpD pDi 5 1 2 λ2D GpD   Expanding ln 1 2 λD GpD and taking the first three terms, ! 2 λ2D GpD     λD GpD 1 ln 1 2 λD GpD  λD GpD 1 1 2 3

pDi 5 1 2

(5.97)

(5.98)

(5.99)

232

Reserves Estimation for Geopressured Gas Reservoirs

Substituting Eq. (5.99) into Eq. (5.98), 2 2 2 λD GpD λ2D GpD λD GpD ð1 2 λD ÞGpD ð1 2 λD ÞGpD 2 2  12 2 pDi 5 1 2 2 3 3 2 3

(5.100) 2 =3Þ terms are ignored, Eq. (5.100) is consistent with If the ðλ2D GpD Eq. (5.97). The Gonzalez
Ilk
Blasingame log
log type curve is shown in Fig. 5
62, and it can be used for log
log matching or verification of other methods. Eqs. (5.74) and (5.98) are used for plotting. The type curve is shown in Fig. 5
63. The type curves of these two methods are shown in Fig. 5
64. According to Eq. (5.95), when λD . 0.43, a significant deviation ( . 10%) starts to occur in the binomial approximation method, as shown in Fig. 5
65. The Gonzalez
Ilk
Blasingame log
log type curve can verify the results of the aforesaid methods such as nonlinear regression, or further verified by combining with the Δ(p/Z)BGp and Δ(p/Z)/GpBGp type curves.

100

pDi

1-pD

λ D=

λD=

1-pD, pDi

10-1

0.00 0.25 0.50 0.75 1.00

0.00 0.25 0.50 0.75 1.00

10-2

10-3 10-3

10-2

GpD

10-1

100

Figure 5
62 Gonzalez log

log type curve. Modified from Gonzalez, F.E., Ilk, D., Blasingame, T.A., 2008. A quadratic cumulative production model for the material balance of an abnormally pressured gas reservoir. SPE 114044-MS. Permission to publish by the SPE, Copyright SPE.

Original gas in place estimations for geopressured gas reservoirs

100

pDi

1-pD

λD=

λ D= 0.05 0.25 0.50 0.75 0.95

0.05 0.25 0.50 0.75 0.95

1-pD, pDi

10-1

233

10-2

10-3 10-3

10-2

GpD

10-1

100

Figure 5
63 Gonzalez limiting form log
log type curve. 100

1-pD

10-1

Eq.（5-95） Eq.（5-74) λ D= λ D= 0.00 0.00 0.25 0.25 0.50 0.50 0.75 0.75 1.00 1.00

10-2

10-3 10-3

10-2

GpD

10-1

100

Figure 5
64 Comparison of Gonzalez binomial and limiting form log
log type curves.

Example 27: Use the Gonzalez
Ilk
Blasingame type curve to verify the calculation results for the Anderson L reservoir in Example 21. Solution In Example 21, reserves are 19.8 3 108 m3, λ 5 0.02025, λD 5 0.02025/0.05047 5 0.40, and the difference between binomial and

234

Reserves Estimation for Geopressured Gas Reservoirs

Gbinomial/Glimit , %

103

102

101

100 0.0

0.2

0.4

λD

0.6

0.8

1.0

Figure 5
65 Comparison of binomial and limiting form results.

100

pDi

1-pD

λ D=

λ D=

1-pD, pDi

10-1

0.05 0.25 0.40 0.75 0.95

0.05 0.25 0.40 0.75 0.95 Actual data point 1-pD pDi

10-2

10-3 10-3

10-2

GpD

10-1

100

Figure 5
66 Gonzalez limiting form log
log type curve.

limiting form regression results is only 6%. The data matching results of the Gonzales
Ilk
Blasingame log
log type curve are shown in Fig. 5
66, where the actual data points match well with the type curve, indicating that the calculated reserves are appropriate.

Original gas in place estimations for geopressured gas reservoirs

235

Example 28: Use the Gonzales
Ilk
Blasingame type curve to verify the calculation results for the M4 reservoir in Example 22. Solution In Example 22, the reserves calculated using the nonlinear regression method are 2158.3 3 108 m3, λ53.4874 3 1024, and λD 50.7527. Fig. 5
65 shows that binomial approximation is unsuitable for this case. The matching effect is poor if the binomial approximation type curve is adopted, as shown in Fig. 5
67. If the type curve plotted according to Eq. (5.74) is used for data matching, as shown in Fig. 5
68, the matching is well.

5.4.4 Sun method As described in Section 5.3, the material balance equation for a closed gas reservoir can be expressed as Eq. (5.85). If the horizontal axis in Fig. 5
52 is changed to logarithmic form, we have With the help of the log
log analysis principle of the well test (Appendix 21), the type curve matching method can be used for estimating the reserves, with the procedure as follows. 1. Prepare the pDBGp semilog plot using actual production data and superpose it on the theoretical type curve (Fig. 5
69). 2. Move the semilog curve up and down to align the pD data axis scale with the theoretical type curve, and then, move the semilog curve 100

pDi

1-pD

λD=

1-pD, pDi

10-1

λD= 0.05 0.25 0.40 0.75 0.95

0.05 0.25 0.40 0.75 0.95 Actual data point 1-pD pDi

10-2

10-3 10-3

10-2

GpD

10-1

Figure 5
67 Log
log type curve matching by binomial approximation.

100

236

Reserves Estimation for Geopressured Gas Reservoirs

100

pDi

1-pD

λD=

λ D=

1-pD, pDi

10-1

0.05 0.25 0.40 0.75 0.95

0.05 0.25 0.40 0.75 0.95 Actual data point 1-pD pDi

10-2

10-3 10-3

10-2

GpD

10-1

100

Figure 5
68 Log
log type curve matching by limiting form.

1.0 0.8 λD

pD

0.6

0.4

0.2

0.0 10-2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Starting point deviating from the straight line 10-1

100

GpD Figure 5
69 Semilog type curve.

leftward and rightward for matching. After fully matching, read the ½λD M value on the theoretical and  type curve,  finally, take any point and read the corresponding GpD ; pD M and Gp ; pD M values of the match points on the theoretical and semilog type curves respectively.

Original gas in place estimations for geopressured gas reservoirs

237

3. Determine the reserves G and ω according to the match point on the ½Gp  D M . horizontal axis, that is, G 5 ½G M and λ 5 ½λG pD M

4. If the production time is short, conduct linear regression for the pD BGp curve (the horizontal axis displays normal scale) to obtain the apparent OGIP. Determine G=Ga according to the ½λD M match value readout in Step (2) and by Eq. (5.86) or Eq. (5.87), and then, calculate G. Example 29: Use the semilog type curve matching method to calculate the OGIP of Anderson L reservoir. Solution The apparent formation pressure depletion degree (1 2 pD) of the Anderson L reservoir is 0.43, which meets the starting condition for the material balance method to calculate the reserves of ultrahigh-pressure gas reservoirs. The reserves obtained from the nonlinear regression method are 19.9 3 108 m3, as shown in Fig. 5
70. Use the semilog type curve matching method for matching, with the results shown in Fig. 5
71, where the λD match value is 0.4. Select point M (2.0 3 108, 0.8) on the pDBGp semilog plot, corresponding to the coordinate

1.0

0.8

pD

0.6 Linear regression

0.4 Nonlinear regression

0.2 pD=(1-0.05047Gp)/(1-0.02025Gp)

19.9

R2=0.99724

0.0 0

5

10

15

31.0 20

25

30

Gp, 10 m 8

3

Figure 5
70 Linear and nonlinear regression results for Anderson L reservoir.

35

Reserves Estimation for Geopressured Gas Reservoirs

1.0

1.0

0.8

0.8

0.6 10-1

100

Gp/108m3

0.6

101

pD

pD

238

0.4

λD

0.0 0.2 0.4 0.6 0.8 10-2

0.1 0.3 0.5 0.7 0.9

0.2

10-1

0.0 100

GpD Figure 5
71 Type curve matching results for Anderson L reservoir.

(0.1, 0.8) on the pDBGpD semilog plot. The reserves are calculated to be 20.0 3 108 m3, by G5

Gp 2:0 3 108 5 20:0 3 108 5 GpD 0:1

Substitute λD 5 0.4 into Eq. (5.87) to calculate G/Ga as 0.659, and then, conduct linear regression for the early data points on the pD versus Gp curve, deriving the Gapp as 31.0 3 108 m3 and the G as 20.4 3 108 m3. Example 30: For the M6 high-pressure gas reservoir, the initial formation pressure is 105.89 MPa, the gas reservoir temperature is 132°C, and the volumetric OGIP is 1704 3 108 m3, currently, the formation pressure is 79.04 MPa, and the cumulative gas production is 426 3 108 m3. The production data are given in Appendix 23. Use the semilog type curve matching method to calculate the OGIP. Solution The dimensionless apparent formation pressure of the M6 reservoir drops by 11.2%, which does not meet the starting condition for the material balance method to calculate reserves. The apparent OGIP is 3750 3 108 m3, as shown

239

Original gas in place estimations for geopressured gas reservoirs

in Fig. 5
72. The semilog type curve matching results are illustrated in Fig. 5
73. The theoretical match point is ½0:10; 0:80M , and the actual data match point is ½175; 0:8M ; the match line ½λD M is 0.60. The reserves are ½G p  8 3 calculated as, G 5 ½G M 5 175:0 0:1 5 1750:0 3 10 m . pD M

1.0

0.8

pD

0.6

0.4

0.2

0.0 0

1000

2000

3000

4000

Gp , 10 m 8

3

1.0

1.0

0.8

0.8

0.6

102 G

101

/108m3 p

0.6 103

0.4

λD

0.0 0.2 0.4 0.6 0.8 10-2

pD

pD

Figure 5
72 p/Z curve linear regression for M6 reservoir.

0.1 0.3 0.5 0.7 0.9

0.2

10-1

GpD Figure 5
73 Semilog type curve matching results for M6 reservoir.

0.0 100

240

Reserves Estimation for Geopressured Gas Reservoirs

According to the matching results of ½λD M , we have G 1:0 2 0:97602λD 1:0 2 0:97602 3 0:60 5 5 5 0:467 Ga 1:0 2 0:18793 3 0:60 1:0 2 0:18793λD According to the above proportional relation, G 5 0:467Ga 5 0:467 3 3750 5 1751:3 3 108 m3 . These values are consistent with the reserves obtained from the volumetric method.

5.4.5 Multiwell production decline analysis method It is assumed that a well in the center of a circular closed gas reservoir produces at a constant rate. When preparing the decline type curve, Palacio and Blasingame (1993) introduced the normalized pseudo pressure, normalized pseudo pressure normalized rate (q/Δpp), and material balance time tca to consider the production performance at variable bottom-hole flow pressure and the change of PVT properties of gas with formation pressure under constant rate conditions. To improve the accuracy of type curve matching, the pressure-normalized rate integral, pressure-normalized rate integral derivative, and β derivative type curves were derived from pressure normalized rate (Ilk et al., 2007; Mattar, 2011, 2015), as shown in Fig. 5
74. After having introduced the β derivative curve, the time to judge whether to enter the boundary-dominated flow can be advanced by 1/2 1

10

reD

qDd,qDdi,qDdid,βD

qDdi

10 20 50 2 10 3 10 4 β D 105 10 6 10 7 10

0

10

qDdid

-1

10

qDd -2

10

-3

10

-2

10

Figure 5
74 Blasingame type curve.

-1

10

0

tcDd

10

1

10

2

10

Original gas in place estimations for geopressured gas reservoirs

241

logarithmic cycle, which significantly reduces the ambiguity of interpretation results. The pressure-normalized rate curve is a cluster of different re/ rwa curves in the early transient flow stage, and the data converge into a harmonic decline curve in the boundary flow stage. The normalized pseudo pressure is defined as ð μgi Zi p p pp 5 dp (5.101) pi 0 μg Z where Z: gas deviation factor, Zi: gas deviation factor under initial conditions, μg: gas viscosity, mPa  s,μgi: gas viscosity under initial conditions, mPa  s. The normalized pseudo pressure
normalized rate (pressure-normalized rate for short) is defined as q q 5 (5.102) Δpp ppi 2 ppwf where Δpp 5 ppi 2 ppwf ppwf

μgi Zi 5 pi

ð pwf 0

p dp μg Z

Where ppi and ppwf are the normalized pseudo pressure at initial formation pressure and wellbore flowing pressure, respectively, q and ppwf are the rate and pressure, respectively, the function of time t. The pressure-normalized rate integral is defined as
 ð q 1 tca qðτ Þ 5 dτ (5.103) Δpp i tca 0 Δpp ðτ Þ The pressure-normalized rate integral derivative is defined as

q
 d Δpp i q 52 Δpp id dlntca The β derivative is defined as β5



(5.104)



q Δpp id

d Δpq p i

(5.105)

242

Reserves Estimation for Geopressured Gas Reservoirs

The material balance pseudo time is defined as ð μgi Cti t qðτ Þ tca 5 dτ qðt Þ 0 μg C t or, for the multiwell situation, defined as ð μgi Cti t qt ðτ Þ tca 5 dτ qðt Þ 0 μg C t

(5.106)

(5.107)

where qðt Þ: flow rate of gas well varying over time, 104 m3/d,qt ðt Þ: flow rate of gas well varying overtime for all wells in the entire connected unit, 104 m3/d, Cti : total rock and fluid compressibility under initial conditions, MPa21,t: production time, d. During depletion exploitation of gas reservoirs, the PVT properties of gas change significantly, and therefore, the change of gas viscosity and compressibility should be considered in the material balance equivalent time function. The dimensionless time is defined as tcaDd 5

0:0864 Ktca 1 1 2 1  2 φμi Cti rwa reD 2 1 2 lnreD 2 12

The dimensionless rate is defined as 
  q re 1 qDd 5 5 qD ln 2 qi 2 rwa qD 5

1:842 3 104 qBgi μ   Kh ppi 2 ppwf

The dimensionless normalized rate integral is defined as ð 1 tcaDd qDdi 5 qDd ðτÞdτ tcaDd 0

(5.108)

(5.109)

(5.110)

(5.111)

The dimensionless normalized rate integral derivative is defined as qDdid 5 2

dqDdi dlntcaDd

(5.112)

The dimensionless βD derivative is defined as βD 5

qDdid ðtDd Þ qDdi ðtDd Þ

(5.113)

Original gas in place estimations for geopressured gas reservoirs

243

In the Blasingame method, four sets of curves are matched simultaneously on the same plot. Set 1. normalized rate curve matching, that is, qDd BtcaDd curve matches with the Δpq p Btca curve. Set 2. normalized rate integral curve matching, that is, the qDdi BtcaDd curve matches with the ðΔpq p Þi Btca curve. Set 3. normalized rate integral derivative curve matching, that is, the qDdid BtcaDd curve matches with the ðΔpq p Þid Btca curve. Set 4. β derivative curve matching, that is, the β D BtcaDd curve matches with the βBtca curve. The material balance pseudo time tca is calculated by the procedure as follows. 1. Calculate the following relational data (Table 5
17) according to the gas properties. 2. Plot the p=ZBp and p=ðμZ ÞBp relation curves. 3. Assume an initial G value. G 4. Calculate pZ from pZ 5 Zpii ð1 2 Gp Þ using the Gp Bt data of each production stage, use the p=ZBp relation to determine the p of each production stage, and then, obtain the corresponding μ, C g and normalized pseudo pressure pp .   μ C Ð t t ðτ Þ dτ and ppi 2 pp for each production stage. 5. Calculate tca 5 qgiðt Þti 0 qμC g The calculation process and results are given in Table 5
18. ppi 2 pp 1. Plot the ð q ÞBtca curve and regress to obtain the slope. 2. Calculate G from the slope,G 5 Cti 31 slope, and Cti 5 1 2CSt wi . 3. Take G as the input value of Step (4), and repeat Steps (4)
(7) until convergence occurs. The tca and G obtained at this time are the Table 5
17 Relational data. P

μ

Z

p=ðμZ Þ

p=Z

Cg

pp

^

Table 5
18 Calculation results of Blasingame method. Given data

Intermediate variables

t

q

pwf

...

...

...

p

ppi 2 pp

Δpp

Calculation results

tca

q Δpp





q Δpp i





q Δpp id

β

244

Reserves Estimation for Geopressured Gas Reservoirs

required values. Calculate the time point, and then, draw the

q q q Δpp , ðΔpp Þi , and ðΔpp Þid values at each q q q Δpp Btca , ðΔpp Þi Btca , ðΔpp Þid Btca , βBtca

curves in the log
log plot. Perform curve matching to obtain reD and match point values ½Δpq p M , ½qDd M , ½tca M and ½tDd M , and then, determine the following relevant parameters.  
 1:842 3 104 μi Bgi q=Δpp 1 lnreD 2 K5 2 h qDd M sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   0:1728K t 2   ca rwa 5 1 φCti μi reD 2 1 lnreD 2 2 tDd M
 rw S 5 ln rwa re 5 reD 3 rwa         Bgi q=Δpp Bgi q=Δpp 1 tca tca

G5  Cti 1 2 21 qDd M tDd M Cti qDd M tDd M r eD

Example 31: The M7 reservoir is 2 km long and 1 km wide, with an effective thickness of 10 m, a porosity of 0.1, an initial formation pressure of 120 MPa, a gas reservoir temperature of 180°C, and volumetric OGIP of 8.3 3 108 m3. There are two production wells. The first well produces gas at 50 3 104 m3/d for 365 days, and the second well produces gas at the same rate from 101th day, with the production data shown in Appendix Table 24
1. Use the Blasingame multiwell advanced production decline analysis method to calculate the reserves of the M7 reservoir. Solution If only the pressure and rate data of the first well are used, the obtained normalized rate curve runs downward (due to the influence of offset well), as shown in Fig. 5
75. If the influence of offset well is considered, the data matching is well, as shown in Fig. 5
76, and the calculated reserves are 8.3 3 108 m3. In a gas reservoir with good connectivity, due to the influence of offset, the Blasingame production curve of a single well also goes inward

245

Original gas in place estimations for geopressured gas reservoirs

10

1

reD

qDd,qDdi,qDdid,βD

qDdi 10

10 20 50 2 10 3 10 4 βD 10 5 10 6 10 7 10

0

10

-1

10

-2

qDdid

qDd

Offset influence

10

-3

10

-2

10

-1

tcDd

10

0

10

1

2

10

Figure 5
75 Matching for a single well. 1

10

reD

qDd,qDdi,qDdid,βD

qDdi

10 20 50 2 10 3 10 4 βD 105 10 6 10 7 10

0

10

qDdid

-1

10

qDd -2

10

-3

10

-2

10

-1

10

0

tcDd

10

1

10

2

10

Figure 5
76 Matching for two wells.

in the boundary-dominated flow stage, showing the characteristics similar to the material balance curve of ultrahigh-pressure gas reservoirs (Fig. 5
76). The material balance curve of the whole reservoir is

246

Reserves Estimation for Geopressured Gas Reservoirs

featured as a constant-volume gas reservoir. In this case, the sum of single-well reserves (often too large) cannot be simply regarded as the reserves of the reservoir. The influence of multiwell interference should be considered, and just the system-based material balance time be redefined to conduct advanced production decline analysis of well clusters. This method requires total compressibility, and the complex calculation process can be aided with appropriate commercial software.

5.5 Trial-and-error analysis method For the material balance equation considering solution gas, the trial-anderror method (Walsh, 1998) can be used for the solution, and its material balance equation can be expressed as F 5 Gfgi Et 1 We   F 5 Gp Bg 1 Wp Bw 2 Rsw Bg     Bgi ðSwi 1 M Þ Bgi ð1 1 M Þ Et 5 Eg 1 Ew 1 Ef Bwi ð1 2 Swi Þ ð1 2 Swi Þ

(5.114) (5.115) (5.116)

Eg 5 Bg 2 Bgi

(5.117)

Ew 5 Btw 2 Btwi

(5.118)

Btw 5 Bw 1 Bg ðRswi 2 Rsw Þ

(5.119)

Ef 5 Cf ðpi 2 pÞ

(5.120)

W5

Gfgi Bgi ðSwi 1 M Þ Bwi ð1 2 Swi Þ

G 5 Gfgi 1 WRswi

(5.121) (5.122)

where Bg: gas volume factor, m3/m3, Bw: formation water volume factor, m3/m3, Ef: rock expansion, m3/m3, Eg: gas expansion, m3/m3, Ew: formation water expansion, m3/m3, Et: total expansion, m3/m3, F: fluid output, 108 m3, Gfgi: initial free gas volume, 108 m3, Gp: cumulative gas production, 108 m3, M: water volume ratio or the ratio of the aquifer to reservoir volume, dimensionless, Rsw: solubility of the gas in water, m3/m3, Swi: initial

Original gas in place estimations for geopressured gas reservoirs

247

M too small

F

Gfgi

M too large

0

Et

Figure 5
77 Effect of M on the shape of the curve. Modified from Walsh, M.P. 1998. Discussion of application of material balance for high pressure gas reservoirs. SPE Journal, 3 (1), 402-404. Permission to publish by the SPE, Copyright SPE.

water saturation, decimal, W: initial water phase volume, 108 m3, We: water influx, 108 m3, Wp: cumulative water production, 108 m3. If water influx is ignored, the FBEt curve crosses the origin, and its slope is Gfgi . Similar to the determination of the size of the gas cap by the Havlena method, M affects the shape of the curve. When M is too small, the curve bends upward; otherwise, the curve bends downward (Fig. 5
77). Eq. (5.114) can also be expressed in Eq. (5.123). However, since the expansion of the water phase is much smaller than that of the gas phase, the results may be distorted, so this method is not recommended. Eg 1 Bgi Ef F 5 Gfgi 1W Ew 1 Bwi Ef Ew 1 Bwi Ef

(5.123)

Example 32: Use the trial-and-error method to calculate the reserves of the Anderson L reservoir. The data are shown in Appendix 3. Solution Calculate the variables F and Et according to Appendix Table 3
2 and Appendix Table 3
3, with the results shown in Table 5
19. The final matching results (M 5 2.25) are shown in Fig. 5
78, Gfgi 5 20:7 3 108 m3 .

Table 5
19 Calculation results for Example 32. p, MPa

Bw

Rsw, m3/m3

Bg, m3/m3

Btw

F

Eg

Ew

Ef

Et (M 5 2.25)

65.55 64.07 61.85 59.26 57.45 55.22 52.42 51.06 48.28 46.34 45.06 39.74 32.86 29.61 25.86 22.39

1.0560 1.0564 1.0569 1.0576 1.0581 1.0586 1.0594 1.0597 1.0604 1.0609 1.0612 1.0625 1.0641 1.0649 1.0658 1.0666

5.6497 5.5958 5.5103 5.4035 5.3240 5.2212 5.0838 5.0138 4.8637 4.7539 4.6789 4.3476 3.8701 3.6257 3.3276 3.0380

0.00282 0.00287 0.00294 0.00300 0.00304 0.00308 0.00314 0.00317 0.00323 0.00328 0.00332 0.00354 0.00400 0.00432 0.00478 0.00530

1.0559 1.0566 1.0576 1.0587 1.0594 1.0602 1.0613 1.0618 1.0629 1.0637 1.0643 1.0668 1.0711 1.0736 1.0771 1.0808

0.0000 0.0004 0.0016 0.0033 0.0044 0.0057 0.0080 0.0093 0.0114 0.0129 0.0142 0.0202 0.0299 0.0391 0.0495 0.0614

0.00000 0.00005 0.00012 0.00018 0.00022 0.00026 0.00032 0.00035 0.00041 0.00046 0.00050 0.00072 0.00118 0.00149 0.00194 0.00247

0.0000 0.0007 0.0017 0.0028 0.0035 0.0043 0.0054 0.0059 0.0070 0.0078 0.0084 0.0109 0.0152 0.0178 0.0212 0.0249

0.0000 0.0007 0.0017 0.0029 0.0038 0.0048 0.0061 0.0067 0.0080 0.0089 0.0095 0.0120 0.0152 0.0167 0.0184 0.0200

0.0000 6.865 3 1024 1.598 3 1024 2.523 3 1024 3.111 3 1024 3.794 3 1024 4.639 3 1024 5.063 3 1024 5.999 3 1024 6.732 3 1024 7.266 3 1024 1.005 3 1023 1.559 3 1023 1.9217 3 1023 2.440 3 1023 3.027 3 1023

Original gas in place estimations for geopressured gas reservoirs

249

0.07 0.06

F, 108m3

0.05 0.04 0.03 Gfgi=20.7×108m3

0.02 0.01 0.00 0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

Et , m /m 3

3

Figure 5
78 Matching results for Example 32. Modified from Walsh, M.P. 1998. Discussion of application of material balance for high pressure gas reservoirs. SPE Journal, 3(1), 402
404 Permission to publish by the SPE, Copyright SPE.

According to Eq. (5.121), we have W5

Gfgi Bgi ðSwi 1 M Þ 20:7 3 0:00282 3 ð2:25 1 0:35Þ 5 0:22 3 108 m3 5 1:056 3 ð1 2 0:35Þ Bwi ð1 2 Swi Þ

According to Eq. (5.122), we have G 5 Gfgi 1 WRswi 5 20:7 1 0:22 3 5:6497 5 21:9 3 108 m3 The proportion of dissolved gas is 5.5%, which is consistent with the calculation results of the Fetkovich method (5.3%). Example 33: Use the trial-and-error method to calculate the reserves of the Ellenburger reservoir. The data are shown in Appendix 25. The initial formation pressure is 46.02 MPa, the gas reservoir temperature is 93.3°C, the porosity is 5%, the initial water saturation is 0.35, the CO2 mole content in gas is 28%, and the rock compressibility is 9.42 3 1024/MPa. Solution Calculate the variables F and Et according to Appendix Table 25
1 and Appendix Table 25
2, with the results shown in Table 5
20.

Table 5
20 Calculation results for Example 33. p, MPa

Bw

Rsw, m3/m3

Bg, m3/m3

Btw

F

Eg

Ew

Ef

Et (M 5 3.3)

46.00 35.60 30.84 29.89 28.75 27.98 26.81 24.62 23.87 23.04 21.89 19.44 18.55 16.80 15.71 13.53 13.38 12.28 11.26 10.22 9.66 9.58

1.0761 1.0768 1.0769 1.0769 1.0770 1.0770 1.0770 1.0769 1.0769 1.0769 1.0768 1.0766 1.0766 1.0763 1.0762 1.0758 1.0757 1.0755 1.0752 1.0749 1.0747 1.0746

12.0183 10.7482 10.0542 9.9057 9.7225 9.5955 9.3976 9.0053 8.8629 8.7023 8.4686 7.9341 7.7238 7.2852 6.9900 6.3505 6.3018 5.9480 5.5968 5.2205 5.0065 4.9752

0.00290 0.00336 0.00395 0.00405 0.00416 0.00423 0.00433 0.00453 0.00461 0.00471 0.00487 0.00537 0.00562 0.00627 0.00680 0.00822 0.00835 0.00930 0.01036 0.01160 0.01236 0.01247

1.0846 1.0864 1.0911 1.0917 1.0923 1.0926 1.0930 1.0937 1.0940 1.0944 1.0951 1.0982 1.1000 1.1051 1.1096 1.1224 1.1235 1.1325 1.1427 1.1549 1.1624 1.1635

0.0000 0.5490 1.0941 1.2456 1.3773 1.4652 1.6388 1.8917 2.1322 2.1496 2.4316 2.9153 3.2725 3.8796 4.4254 5.5920 5.9323 6.8432 7.8769 9.0145 9.8517 10.1193

0.00000 0.00046 0.00104 0.00113 0.00124 0.00131 0.00141 0.00161 0.00169 0.00178 0.00194 0.00244 0.00269 0.00332 0.00385 0.00525 0.00537 0.00631 0.00735 0.00858 0.00932 0.00944

0.0000 0.0017 0.0065 0.0071 0.0077 0.0080 0.0084 0.0091 0.0093 0.0097 0.0105 0.0135 0.0154 0.0205 0.0249 0.0377 0.0389 0.0479 0.0580 0.0702 0.0777 0.0789

0.0000 0.0098 0.0143 0.0152 0.0163 0.0170 0.0181 0.0202 0.0209 0.0216 0.0227 0.0250 0.0259 0.0275 0.0286 0.0306 0.0308 0.0318 0.0327 0.0337 0.0343 0.0343

0.000 6.784 3 1024 1.426 3 1023 1.550 3 1023 1.688 3 1023 1.776 3 1023 1.905 3 1023 2.156 3 1023 2.253 3 1023 2.372 3 1023 2.568 3 1023 3.156 3 1023 3.454 3 1023 4.211 3 1023 4.830 3 1023 6.482 3 1023 6.625 3 1023 7.738 3 1023 8.966 3 1023 1.041 3 1022 1.129 3 1022 1.143 3 1022

9.25 8.85 8.76 8.27 8.02 7.60 7.10 6.59

1.0745 1.0744 1.0743 1.0741 1.0740 1.0739 1.0736 1.0734

4.8482 4.6853 4.6521 4.4489 4.3445 4.1663 3.9450 3.7152

0.01294 0.01356 0.01369 0.01450 0.01492 0.01566 0.01662 0.01764

1.1682 1.1744 1.1757 1.1838 1.1881 1.1956 1.2054 1.2159

10.5916 11.2108 11.4304 12.2736 12.7920 13.5771 14.5401 15.6645

0.00990 0.01051 0.01064 0.01143 0.01185 0.01258 0.01352 0.01453

0.0835 0.0898 0.0910 0.0992 0.1035 0.1110 0.1207 0.1312

0.0346 0.0350 0.0351 0.0356 0.0358 0.0362 0.0367 0.0371

1.197 3 1022 1.269 3 1022 1.284 3 1022 1.378 3 1022 1.427 3 1022 1.514 3 1022 1.625 3 1022 1.744 3 1022

252

Reserves Estimation for Geopressured Gas Reservoirs

The final matching results (M 5 3.3) are shown in Fig. 5
79, Gfgi 5 905 3 108 m3. The reserves calculated by the nonlinear regression method are 1017 3 108 m3 (Fig. 5
80). 20

F, 108m3

16

12

8 Gfgi=892×108m3

4

0 0.000

0.005

0.010

0.015

0.020

Et , m /m 3

3

Figure 5
79 Trial-and-error matching results for Example 33.

1.0

0.8

pD

0.6

0.4

0.2 pD=(1-9.83527E-4Gp)/(1-2.7147E-4Gp) 1017

0.0 0

200

400

600

800

Gp , 108m3 Figure 5
80 Nonlinear regression matching results for Example 33.

1000

Original gas in place estimations for geopressured gas reservoirs

253

According to Eq. (5.121), we have W5

Gfgi Bgi ðSwi 1 M Þ 892 3 0:0029 3 ð3:3 1 0:35Þ 5 13:50 3 108 m3 5 Bwi ð1 2 Swi Þ 1:0761 3 ð1 2 0:35Þ

According to Eq. (5.122), we have G 5 Gfgi 1 WRswi 5 892 1 13:50 3 12:02 5 1054 3 108 m3 The total reserves of the gas reservoir are 1054 3 108 m3, which are close to the calculation results of the nonlinear regression method. Specifically, the proportion of dissolved gas is 15.4%, which is consistent with the calculation result of the Fetkovich method (14.9%). At low pressure, the elastic expansion of water-soluble gas (high CO2 content) is essential driving energy. In this case, due to the unknown water production, the calculated reserves are slightly different from the results given in the literature (1048 3 108 m3; Fetkovich et al., 1998). Example 34: Use the trial-and-error method to calculate the reserves of Duck Lake gas reservoir. The data are shown in Appendix 26. The initial formation pressure is 40.0 MPa, gas reservoir temperature is 115.7°C, porosity is 25%, initial water saturation is 0.18, the gas gravity is 0.65, and the rock compressibility is 4.93 3 1024/MPa. Solution The fluid property parameters are not available, so the empirical equation given in Chapter 3 is used to calculate them, with the results shown in Appendix Table 26
2. The calculation results of the nonlinear regression method are shown in Fig. 5
81, revealing the reserves of 189.8 3 108 m3. The calculation results of the trial-and-error method are 195.1 3 108 m3, close to the value obtained from the nonlinear regression method, as shown in Fig. 5
82. The proportion of dissolved gas is 9.5%. According to Eq. (5.121), we have W5

Gfgi Bgi ðSwi 1 M Þ 176:5 3 0:00355 3 ð6:5 1 0:18Þ 5 18:64 3 108 m3 5 Bwi ð1 2 Swi Þ 1:0463 3 ð1 2 0:18Þ

According to Eq. (5.122), we have G 5 Gfgi 1 WRswi 5 176:5 1 18:64 3 3:82 5 195:1 3 108 m3

254

Reserves Estimation for Geopressured Gas Reservoirs

1.0

0.8

pD

0.6

0.4

0.2 pD=(1-0.00527Gp)/(1-0.00156Gp) 189.8

0.0 0

50

100

150

200

Gp , 10 m 8

3

Figure 5
81 Nonlinear regression matching results for Example 33.

4

F, 108m3

3

2

Gfgi=176.5×108m3

1

0 0.000

0.005

0.010

0.015

Et , m /m 3

3

Figure 5
82 Trial-and-error matching results for Example 33.

0.020

0.025

Original gas in place estimations for geopressured gas reservoirs

255

5.6 Original gas in place estimation procedure of geopressured gas reservoirs 5.6.1 Summary of calculation methods In the above five sections, a total of 22 calculation methods in five categories (i.e., classical two-segment method, linear regression method, nonlinear regression method, type curve matching method, and trial-and-error method) are introduced (Table 5
21). These methods have their respective advantages and disadvantages, and most of them do not require compressibility. In addition to the above methods, there are also methods based on the change of rock compressibilities, such as those proposed by Begland and Whitehead (1989); Yale et al. (1993), and Guehria (1996), which are not reviewed in this book.

5.6.2 Recommended methods The reserve calculation methods for high-pressure gas reservoirs are selected with consideration of. 1. Compressibility. The compressibility is inversely proportional to the reserves and is highly sensitive. Moreover, it cannot be accurately determined by experiment and varies significantly at different locations of the gas reservoir, as shown in Fig. 4
28. Thus, the method considering compressibility is not recommended. 2. Initial formation pressure. The Roach method and Ambastha type curve and its improved analysis method are susceptible to the initial pressure, so they are not recommended. 3. Degree of reserve recovery. In case of a slight pressure drop, the methods such as the Gan
Blasingame method and the binary regression method are not recommended. 4. Nonlinear regression. Although binomial approximation and trinomial approximation methods are convenient and straightforward, they apply to specific conditions, and the derived curve shape is uncontrollable, possibly leading to significant error. Thus, these two methods are not recommended. 5. Water influx. The Poston
Chen
Akhtar improved Roach method is recommended. To sum up, we recommend the reserve calculation methods only relying on pressure data and production data, such as the gas production of cumulative unit pressure drop method (No. 10, applicable to all stages of

Table 5
21 Summary of reserve calculation methods for high-pressure gas reservoirs. No.

1

Calculation method

Classical twosegment

Hammerlindl

Average compressibility Corrected reservoir volume

2

Compressibility required or not

Inflection point sensitive or not

Assumptions, advantages, and disadvantages

O

╳

Pressure coefficient at data points . 1.13

O

╳

3

Chen

O

O

4

Gan
Blasingame

╳

O

Ramagost
Farshad pressure correction

O

O

6

Roach

╳

╳

7

Poston
Chen
Akhtar improved Roach

╳

O

8 9

Becerra-Arteaga Havlena
Odeh

╳ ╳

O ╳

10

Gas production of cumulative unit pressure drop

╳

╳

5

Linear regression

Constant-volume closed gas reservoir The pressure coefficient at the inflection point is 1.2
1.3 It does not require compressibility and can work whether the inflection point occurs. Closed gas reservoir It requires compressibility. It can back-calculate compressibility but is sensitive to initial pressure data. Water drive gas reservoir It calculates reserves before water influx, size, and effective compressibility. It requires a pressure coefficient. It is sensitive to initial pressure and early data for gas reservoirs with water influx. For a closed gas reservoir, it can calculate rock compressibility. It requires gas deviation factor variation but cannot exclude the influence of water energy.

11

╳

╳

╳

O

╳

O

╳

O

╳

O

Ambastha type curve matching

╳

O

Fetkovich type curve matching

╳

╳

Binomial form

╳

╳

Limiting form

╳

╳

Nonlinear regression

12

Binary regression Nonlinear regression

13 14 15 16

17

18

Binomial approximation Trinomial approximation Limiting form Power function form

Type curve matching

Gonzales type curve matching

19 20

Semilog type curve matching

╳

O

21

Multiwell advanced production decline

O

Entering boundarydominated flow

22

Trial-and-error analysis

O

╳

It requires gas deviation factor variation and can calculate reserves and compressibility. When λD , 0.4, binomial approximation works. The curve shape is uncontrolled, possibly leading to a significant error. The calculation is relatively accurate for the closed gas reservoir. The calculation is relatively accurate for the closed gas reservoir. Closed gas reservoir Plotting requires initial pressure and temperature Ambiguous solutions The effect of dissolved gas in nonpay Compressibility and water size to be backcalculated. Mainly for verifying the results of other methods The limiting form type curve is recommended. Closed gas reservoir It can calculate reserves and effective compressibility. It does not consider the effect of water influx and can only calculate the reserves of the connected well cluster. It separates the calculation of water-soluble gas reserves.

258

Reserves Estimation for Geopressured Gas Reservoirs

development), the nonlinear regression method (No. 14 and 15, applicable to middle-late stage of development), the semilog type curve matching method (No. 20, applicable to middle-late stage of development). The Poston
Chen
Akhtar improved Roach method (No.7, applicable to water-drive gas reservoirs). In practical applications, the methods should be cooperated and coordinated organically, coupling with static results, to reduce the uncertainty in reserve evaluation.

5.6.3 Recommended procedure The following analysis procedure is recommended. 1. Calculate average formation pressure and deviation factor, sort out pressure and cumulative production data, plot the p/ZBGp curve, and analyze whether the curve goes downward. 2. Retain at least 10% to 20% of historical production data to validate the reliability of the selected method. 3. Use the Poston
Chen
Akhtar improved Roach method and Havlena
Odeh method to judge whether there is a water influx behavior and determine the inflection point of the water influx effect. 4. If there is water influx behavior, use the Poston
Chen
Akhtar improved Roach method and the Fetkovich method, or use other methods with the data obtained before the occurrence of inflection point. If there is no water influx, use No. 10 or No. 14 and 15, depending on the available data. 5. When the nonlinear regression method is used to estimate the reserves, the p/ZBGp curve is required to go downward. Thus, use the limiting form (No. 14) or power function form (No.15) nonlinear regression method. 6. If the p/ZBGp curve does not go downward, use the gas production of cumulative unit pressure drop method (No. 10) to calculate the reserves, and use the Gonzalez limiting form matching method and the semilog type curve matching method to verify the calculation results. Fig. 5
83 illustrates the analysis procedure.

5.6.4 Basic data preparation The gas reservoir, wellbore completion, and daily production data are collected (Table 5
22). Table 5
23 lists the common issues in the data analysis of reserves, which should be considered carefully before production data analysis.

Original gas in place estimations for geopressured gas reservoirs

259

Plot p/Z~Gp Curve

Y(Early data)

Judge water influx By Havlena – Odeh method

Y

N N

Downward behavior

Y Gas production of cumulative unit pressure drop

Improved Roach method

Nonlinear regression N

Gonzales

Gonzalez

Method

Method

Verification

Semilog method

Fetkovich Method

Y Output

Figure 5
83 Recommended analysis procedure.

Table 5
22 Basic data for reserve analysis. Reservoir properties

Reservoir area, porosity, adequate thickness, rock compressibility, total compressibility, and initial water saturation

Gas properties

PVT report, volume factor, gas deviation factor, viscosity, compressibility, gas gravity Volume factor, the solubility of the gas in water, viscosity, compressibility Pressure at the measuring point, reservoir mid-depth, initial formation pressure, gas reservoir temperature, wellhead static temperature, wellhead static pressure Tubing and casing pressure data, oil, gas, and water flow rate data, flowing pressure gradient test data, static pressure gradient test data Cumulative gas production, cumulative water production, cumulative condensate oil production

Formation water properties Average pressurerelated data Daily production data Cumulative production data

260

Reserves Estimation for Geopressured Gas Reservoirs

Table 5
23 Common problems in production decline analysis (Ilk et al., 2007). Problem

Pressure

Production

General

Influence severity

No measured pressure data The incorrect calculated initial pressure value Incorrect pts
pws conversion Liquid accumulation in the wellbore (to affect pts
pws conversion) Incorrect pressure measurement position Inaccurate cumulative gas production metering Inaccurate cumulative water production metering Inaccurate compressibility Inaccurate deviation factor Poor synchronism of time, pressure, and production Poor correlation of time, pressure, and production

High High Moderate Moderate Very high Very high Moderate Very high Very high Moderate/High Very high

5.6.5 Comparative analysis of results Herein, the calculation results are analyzed and compared with Anderson L reservoir and M2 reservoir as examples. Example 35: Compare the reserves of Anderson L reservoir obtained from the Poston
Chen
Akhtar method (Example 7), the gas production of cumulative unit pressure drop method (Example 17), the nonlinear regression method (Example 21), and the semilog type curve matching method (Example 29). The reserves obtained from the volumetric method are 19.6 3 108 m3. Solution According to the recommended analysis procedure, use four methods to calculate the reserves of the Anderson L gas reservoir. 1. Poston
Chen
Akhtar method. By linear regression for six data points above the critical slope line, the reserves are obtained as 22.40 3 108 m3, as shown in Fig. 5
84A.

0.020 y=0.04465x-0.00264

y, MPa-1

0.015 G=22.40×108m3

0.010

Critical slope line

0.005

0.000 Ce=0.00264 MPa-1

-0.005 0.0

0.1

0.2

0.3

0.4

0.5

x,108m3/MPa (A) Poston-Chen-Akhtar method 0.35 Gp/' p=0.0027' p+0.1326

Gp/' p, 108m3/MPa

(65.548,0.31)

R2 = 0.9745

0.30 0.25 0.20 0.15 0.10 0.05 0.00 0

10

20

30

40

50

60

' p, MPa (B) Gas production of cumulative unit pressure drop method 1.0 Actual point Binomial regression Nonlinear regression

0.8

pD

0.6

0.4

0.2

0.0 0

5

10

15

20

25

Gp, 108m3 1.0

0.8

0.8

0.6 10-1

100

Gp/108m3

0.6

101 0.4

λD 0.0 0.2 0.4 0.6 0.8 10-2

pD

pD

(C) Nonlinear regression method 1.0

0.1 0.3 0.5 0.7 0.9

0.2

10-1

0.0 100

GpD

(D) Semilog type curve matching method

Figure 5
84 Reserves of Anderson L reservoir obtained by different methods. (A) Poston
Chen
Akhtar method analysis plot; (B) gas production of cumulative unit pressure drop method analysis plot; (C) nonlinear regression method analysis plot; (D) semilog type curve matching method analysis plot.

262

Reserves Estimation for Geopressured Gas Reservoirs

2. Gas production of cumulative unit pressure drop method. The reserves are obtained as 18.96 3 108 m3, as shown in Fig. 5
84B. 3. Limiting from nonlinear regression method. The reserves are obtained as 19.8 3 108 m3, as shown in Fig. 5
84C. 4. Semilog type curve matching method. The reserves are obtained as 20.0 3 108 m3, as shown in Fig. 5
84D. The calculation results of the four methods are similar. The Poston
Chen
Akhtar method reflects the most significant relative error, that is, 13.7%, while the other three methods have the relative error less than 5.0%. Example 36: Compare the reserves of the M2 reservoir obtained from the Poston
Chen
Akhtar method (Example 11), the gas production of cumulative unit pressure drop method (Example 18), the nonlinear regression method, and the semilog type curve matching method. The reserves obtained from the volumetric method are 2400 3 108 m3.

to 1.

2. 3.

4.

Solution According to the recommended analysis procedure, use four methods calculate the reserves of the M2 gas reservoir. Poston
Chen
Akhtar method. By linear regression with the five data points above the critical slope line, the reserves are obtained as 2100 3 108 m3, as shown in Fig. 5
85A. Gas production of cumulative unit pressure drop method. The reserves are obtained as 1933 3 108 m3, as shown in Fig. 5
85B. The curve does not show a downward trend, so the nonlinear regression and semilog type curve matching methods cannot be used. If the early production data are used for linear regression, the reserves are obtained as 2100 3 108 m3, as shown in Fig. 5
85C. The gas reservoir was affected by water in the early stage of development. The water/gas ratio (WGR) increases year by year, as shown in Fig. 5
85D. Thus, the reserves of 2100 3 108 m3 are the upper limit value. To sum up, the reserves of the M2 reservoir are taken as 1933 3 108 m3.

0.030 y=4.76184×10-4x-0.0002

0.025 G=2100×108m3

y, MPa-1

0.020 0.015

Critical slope line

0.010 Ce=0.0002MPa-1

0.005 0.000

Ce  ω/' p=0.0047MPa-1

-0.005 0

10

20

30

40

50

60

70

x,108m3/MPa (A) Poston-Chen-Akhtar method 45 42.0

Gp/' p, 108m3/MPa

40 G=74.22×42×0.93=2899×108m3

35

30

28.0

25

G=74.22×28×0.93=1933×108m3

20 0

15

30

45

60

75

' p, MPa (B) Gas production of cumulative unit pressure drop method 1.0

0.8

pD

0.6

0.4

0.2 2100

2600

0.0 0

500

1000

1500

2000

2500

3000

1000

1200

Gp , 108m3

(C) Nonlinear regression method

WGR, m3/106m3

20

Annual WGR Cumulative WGR

16

12

8

4

0 0

200

400

600

800

Gp , 108m3 (D) WGR curve

Figure 5
85 Reserves of M2 reservoir obtained by different methods. (A) Poston
Chen
Akhtar method analysis plot; (B) gas production of cumulative unit pressure drop method analysis plot; (C) nonlinear regression method analysis plot; (D) WGR curve of M2 reservoir.

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Appendices Appendix 1 Pertinent data of NS2B gas reservoir The NS2B gas reservoir is located in the North Ossun oil field in Lafayette County, Louisiana, Unites States. It is a closed gas reservoir with limited water aquifer. Table A1
1, Table A1
2 (Harville, 1969), and Table A1
3 present the relevant gas reservoir information, the pressureproduction history data, and the rock compressibility and volume factors under initial conditions and dew point conditions, respectively Table A1
1 Basic data of NS2B gas reservoir.

Depth, m Initial pressure, MPa Pressure coefficient, MPa/100m Temperature, K Gas/water contact, m Average net sand, m Porosity, fraction Initial water saturation, fraction Critical pressure, MPa Gas gravity, fraction

3810 61.51 1.64

Permeability, 1023μm2 Producing wells Dew point pressure, MPa

200 4 47.71

392.2 3810 30.48 0.235

Initial GOR, m3/m3 Condensate gravity, fraction Initial Z Initial gas in place (volumetric), 108m3 Initial gas compressibility, MPa21 Critical temperature, K

1113 0.7927 1.472 32.28

0.34 4.199 0.65

4.35 3 1023 263.89

Source: Modified from Harville, D.W., Hawkins, M.F., 1969. Rock compressibility and failure as reservoirs mechanisms in geopressured gas reservoirs. Journal of Petroleum Technology, 21 (12), 1528
1530.

Table A1
2 NS2B gas reservoir pressure-production history. p, MPa

Z

p/Z, MPa

Gp, 108m3

61.51 60.98 57.38 51.14 47.44 41.81 37.85 32.96 28.30

1.47 1.47 1.40 1.29 1.22 1.13 1.08 0.97 0.89

41.76 41.63 40.98 39.70 38.91 37.00 35.21 34.09 31.90

0.00 0.19 0.93 2.94 4.41 6.79 7.89 9.54 11.36

Source: Modified from Harville, D.W., Hawkins, M.F., 1969. Rock compressibility and failure as reservoirs mechanisms in geopressured gas reservoirs. Journal of Petroleum Technology, 21 (12), 1528
1530.

265

266

Appendices

Table A1
3 Rock and gas compressibility of NS2B gas reservoir. Cf, MPa21

Initial pressure Dew point pressure

3.18 3 10

Cg, MPa21 23

23

4.35 3 10 7.06 3 1023

Eg

302.5 281.5

Source: Modified from Harville, D.W., Hawkins, M.F., 1969. Rock compressibility and failure as reservoirs mechanisms in geopressured gas reservoirs. Journal of Petroleum Technology, 21 (12), 1528
1530.

Appendix 2 Pertinent data of offshore gas reservoir The pressure-production history data of the offshore high-pressure gas reservoir in Louisiana is shown in Table A2
1 (Ramagost, 1981). The buried depth of the gas reservoir is 4055 m, the initial pressure is 78.90 MPa, the gas reservoir temperature is 128.4°C, and the gas gravity is 0.6. The initial water saturation is 0.22, and the formation water compressibility coefficient is 4.41 3 1024 MPa21.

Table A2
1 Offshore gas reservoir pressure-production history. No.

Date

Gp, 108m3

p, MPa

Z

p/Z, MPa

0 1 2 3 4 5 6 7 8 9 10 11 12 13

1966.01.25 1967.02.01 1968.02.01 1969.06.01 1970.06.01 1971.06.01 1972.06.01 1973.09.01 1974.08.01 1975.08.01 1976.08.01 1977.08.01 1978.08.01 1979.08.01

0.00 2.81 8.11 15.18 22.00 28.72 34.09 41.07 45.49 51.64 56.00 61.08 66.76 69.64

78.90 73.60 69.85 63.80 59.12 54.51 50.88 47.21 44.04 40.18 37.29 34.47 31.03 28.75

1.496 1.438 1.397 1.330 1.280 1.230 1.192 1.154 1.122 1.084 1.057 1.033 1.005 0.988

52.74 51.18 50.00 47.97 46.18 44.32 42.69 40.91 39.25 37.06 35.28 33.37 30.87 29.10

Source: Modified from Ramagost, B.P., Farshad, F.F., 1981. P/Z abnormally pressured gas reservoirs. SPE 10125-MS.

Appendices

267

Appendix 3 Pertinent data of Anderson L gas reservoir The Mobil
David field was discovered by the Mobil Oil Corp. in May 1965, in an area of unrelated shallow production. The field is located on the Lower Gulf Coast in Nueces County about 12 miles southwest of Corpus Christi, Tex. The Anderson L reservoir was discovered in December 1965, by a confirmation well located 2.84 km north of the discovery well of another field. The Anderson L sandstone gas reservoir is a complex fault anticline structure. The structure axis is NNE-SSW. The structure is 8.45 km long and 5.63 km wide. The reservoir depth is 3810 m, the pressure coefficient is 1.91, and the initial formation pressure is 65.55 MPa. The gas reservoir temperature is 130°C, the initial water saturation is 0.35, the effective thickness is 22.86 m, the formation water compressibility coefficient is 4.351 3 1024 MPa21, the porosity is 0.24, the rock compressibility coefficient is 2.828 3 1023 MPa21, and the dew point pressure is 42.18 MPa, the gas gravity is 0.665, the condensate gravity is 0.7624, and the volumetric reserve is 19.68 3 108 m3. The composition of natural gas at the initial stage of development is shown in Table A3
1. The pressureproduction history data are shown in Table A3
2 (Duggan, 1972). Total water cumulative compressibility are shown in Table A3
3.

Table A3
1 Anderson L gas reservoir Wellstream analysis data. Wellstream, Mol percent

CO2

N2

C1

C2

C3

i-C4

n-C4

i-C5

n-C5

C6

C71

Total

yi

0.5

0.24

79.79

6.5

3.36

1.33

1.11

0.73

0.45

0.91

5.08

100

Source: Modified from Duggan, J.O., 1972. The Anderson “L” -An abnormally pressured gas reservoir in South Texas. Journal of Petroleum Technology, 24 (2), 132
138.

Appendices

269

Table A3
2 Anderson L gas reservoir pressure-production history. No.

Date

Gp, 108m3

Wp, 104m3

p, MPa

Z

p/Z, MPa

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1965.12.22 1966.3.1 1966.6.22 1966.9.29 1966.11.17 1966.12.30 1967.3.23 1967.5.15 1967.7.31 1967.9.14 1967.10.19 1968.3.5 1968.9.4 1969.3.19 1969.9.29 1970.3.31

0.000 0.118 0.492 0.966 1.276 1.647 2.257 2.620 3.146 3.519 3.827 5.163 6.836 8.389 9.689 10.931

0.00 0.48 1.96 3.83 5.04 6.47 8.92 10.35 12.35 13.74 14.95 19.48 25.69 30.42 33.96 36.70

65.55 64.07 61.85 59.26 57.45 55.22 52.42 51.06 48.28 46.34 45.06 39.74 32.86 29.61 25.86 22.39

1.440 1.418 1.387 1.344 1.316 1.282 1.239 1.218 1.176 1.147 1.127 1.048 0.977 0.928 0.891 0.854

45.52 45.18 44.59 44.09 43.65 43.07 42.31 41.92 41.05 40.40 39.98 37.92 33.63 31.91 29.02 26.21

Source: Modified from Duggan, J.O., 1972. The Anderson “L” -An abnormally pressured gas reservoir in South Texas. Journal of Petroleum Technology, 24 (2), 132
138.

Table A3
3 Calculation of total water cumulative compressibility for the Anderson L reservoir. p, MPa

Bw

Rsw, m3/m3

Z

Bg, m3/m3

Btw

Ctw, 1024MPa21

65.57 62.05 55.16 48.26 41.37 34.47 27.58 20.68 13.79 10.34 6.89 5.17 3.45 1.72 0.69 0.10

1.056 1.057 1.059 1.060 1.062 1.064 1.065 1.067 1.068 1.069 1.069 1.069 1.069 1.069 1.069 1.069

5.66 5.52 5.20 4.84 4.45 4.01 3.49 2.87 2.10 1.66 1.21 0.89 0.59 0.28 0.09 0.00

1.4401 1.3923 1.2991 1.2072 1.1176 1.0325 0.9562 0.8977 0.8744 0.8832 0.9078 0.9258 0.9472 0.9708 0.9835 1

0.00282 0.00288 0.00303 0.00322 0.00347 0.00385 0.00446 0.00558 0.00815 0.01098 0.01693 0.02302 0.03533 0.07242 0.18341 1.2686

1.056 1.057 1.060 1.063 1.066 1.070 1.075 1.083 1.097 1.113 1.145 1.179 1.249 1.459 2.092 8.254

3.4809 3.5244 3.6404 3.8435 4.0320 4.3221 4.7572 5.5984 7.5274 9.7029 14.4311 19.2898 29.3553 59.7548 151.1710 1041.1554

Source: Modified from Fetkovich, M.J., Reese, D.E., Whitson, C.H., 1991. Application of a general material balance for high pressure gas reservoir. SPE 22921-MS.

270

Appendices

Appendix 4 Pertinent data of Gulf Coast gas reservoir In the SPE16861, Prasad gave the pressure-production history data of seven high-pressure gas reservoirs in Gulf Coast, which was digitized by Gan (2001), which is now excerpted as follows (Tables A4
1 to A4
7).

Table A4
1 Reservoir-33 gas reservoir pressure-production history. p/Z, MPa

Gp, 108m3

p/Z, MPa

Gp, 108m3

p/Z, MPa

Gp, 108m3

40.33 37.47 37.47 36.65 35.46 34.03 31.05

0.00 6.03 6.09 9.31 12.53 15.97 20.05

25.41 22.45 17.32 15.59 11.03 10.89 9.11

28.83 32.96 39.17 41.24 44.93 47.01 48.86

9.28 7.62 6.27 6.75 6.25

49.51 50.82 52.07 52.78 53.43

Source: Modified from Gan, R.G., 2001. A new p/z technique for the analysis of abnormally pressured gas reservoirs. Master dissertation, Texas A&M University, College Station, Texas.

Table A4
2 Reservoir-41 gas reservoir pressure-production history. Gp, 108m3

p/Z, MPa

Gp, 108m3

p/Z, MPa

Gp, 108m3

p/Z, MPa

0.00 0.32 0.75 0.90 1.19 1.29 2.29 2.34

43.44 42.54 43.25 42.53 41.22 38.82 39.45 38.51

2.69 3.02 3.44 3.67 4.19 4.28 4.50 4.72

39.08 37.48 36.74 39.78 35.35 33.97 32.67 30.85

5.08 5.20 5.65 6.13 6.20 6.62

32.58 30.76 29.74 29.36 27.77 26.53

Source: Modified from Gan, R.G., 2001. A new p/z technique for the analysis of abnormally pressured gas reservoirs. Master dissertation, Texas A&M University, College Station, Texas.

Table A4
3 Reservoir-70 gas reservoir pressure-production history.

p/Z, MPa Gp, 108m3

47.78 0.00

44.28 0.20

39.07 0.78

34.39 1.06

26.77 1.65

20.42 2.04

Source: Modified from Gan, R.G., 2001. A new p/z technique for the analysis of abnormally pressured gas reservoirs. Master dissertation, Texas A&M University, College Station, Texas.

11.31 2.66

Appendices

271

Table A4
4 Reservoir-117 gas reservoir pressure-production history. p/Z, MPa

Gp, 108m3

p/Z, MPa

Gp, 108m3

p/Z, MPa

Gp, 108m3

47.33 47.25 46.59 46.16 44.56 44.85 46.66 43.91 44.34 43.54

0.00 0.40 1.99 6.16 7.13 10.50 13.16 15.82 18.65 19.01

42.45 43.32 42.38 41.87 40.85 40.71 40.12 39.76 37.51 38.96

19.89 21.84 24.59 27.43 27.51 31.06 38.41 42.23 46.12 49.58

35.76 36.71 33.29 32.49 29.96 30.46 27.85 27.78 25.46

53.57 56.58 65.97 66.77 71.46 73.15 74.83 76.60 81.56

Source: Modified from Gan, R.G., 2001. A new p/z technique for the analysis of abnormally pressured gas reservoirs. Master dissertation, Texas A&M University, College Station, Texas.

Table A4
5 Reservoir-195 gas reservoir pressure-production history.

p/Z, MPa Gp, 108m3

47.92 0.00

47.45 0.44

43.66 1.96

40.98 2.90

36.88 4.04

30.73 6.72

23.54 8.56

14.47 10.11

14.42 11.08

11.67 11.76

10.42 11.93

7.52 11.94

10.16 12.24

Source: Modified from Gan, R.G., 2001. A new p/z technique for the analysis of abnormally pressured gas reservoirs. Master dissertation, Texas A&M University, College Station, Texas.

273

Appendices

Table A4
6 Reservoir-197 gas reservoir pressure-production history.

p/Z, MPa Gp, 108m3

46.81 0.00

43.09 0.44

22.99 2.09

13.98 2.97

Source: Modified from Gan, R.G., 2001. A new p/z technique for the analysis of abnormally pressured gas reservoirs. Master dissertation, Texas A&M University, College Station, Texas.

Table A4
7 Reservoir-268 gas reservoir pressure-production history.

p/Z, MPa Gp, 108m3

46.82 0.00

46.46 0.15

45.65 0.65

43.04 1.32

39.56 2.42

36.59 2.94

34.05 3.36

Source: Modified from Gan, R.G., 2001. A new p/z technique for the analysis of abnormally pressured gas reservoirs. Master dissertation, Texas A&M University, College Station, Texas.

Appendix 5 Pertinent data of GOM gas reservoir Gan (2001) gave the pressure-production history data of the GOM gas reservoir in his master’s thesis, which is now excerpted as follows. The initial pressure is 84.46 MPa, the gas reservoir temperature is 130°C, the pressure coefficient is 2.13, the initial water saturation is 0.35, and the gas gravity is 0.875. The pressure-production history data are shown in Table A5
1.

Table A5
1 GOM gas reservoir pressure-production history.

p/Z, MPa

50.07

49.42

48.42

47.46

46.32

45.86

44.93

44.38

42.30

41.10

39.82

38.84

36.62

Gp, 108m3

0.00

0.11

0.24

0.38

0.52

0.65

0.77

0.93

1.08

1.23

1.38

1.51

1.70

Source: Modified from Gan, R.G., 2001. A new p/z technique for the analysis of abnormally pressured gas reservoirs. Master dissertation, Texas A&M University, College Station, Texas.

Appendices

275

Appendix 6 Pertinent data of Stafford gas reservoir Gan (2001) gave the pressure-production history data of Stafford gas reservoir in his master’s thesis, which is now excerpted as follows. The initial pressure of the gas reservoir is 49.64 MPa, the gas reservoir temperature is 121.1°C, the pressure coefficient is 1.80, and the gas gravity is 0.65. The pressure-production history is shown in Table A6
1.

Table A6
1 Stafford gas reservoir pressure-production history. p, MPa

Gp, 108m3

Z

p/Z, MPa

49.64 48.09 46.34 44.33 42.97 43.05 40.93 39.43 36.87 31.18 25.31 21.48 19.55 19.49 19.12 19.05

0.00 0.20 0.43 0.70 0.94 1.04 1.21 1.47 1.72 2.35 3.01 3.47 3.73 3.83 3.90 3.97

0.00 1.18 1.17 1.15 1.13 1.11 1.11 1.09 1.08 1.05 1.00 0.96 0.94 0.93 0.93 0.93

41.93 41.21 40.37 39.37 38.65 38.68 37.52 36.65 35.11 31.21 26.48 22.98 21.06 21.00 20.62 20.55

Source: Modified from Gan, R.G., 2001. A new p/z technique for the analysis of abnormally pressured gas reservoirs. Master dissertation, Texas A&M University, College Station, Texas.

Appendix 7 Pertinent data of South Louisiana gas reservoir Bourgoyne (1990) gave the pressure-production history of the South Louisiana gas reservoir, which is now excerpted as follows. The burial depth of the gas reservoir is 3962.4 m, the initial pressure is 75.73 MPa, the gas reservoir temperature is 120.2°C, the pressure coefficient is 1.91, the dew point pressure is 49.64 MPa, the gas gravity is 0.665, and condensate gravity is 0.7927. The pressure-production history is shown in Table A7
1.

276

Appendices

Table A7
1 South Louisiana gas reservoir pressure-production history. p, MPa

Gp, 108m3

Z

p/Z, MPa

75.73 57.32 48.70 43.09 33.96

0.00 0.82 1.55 1.77 2.12

1.49 1.27 1.17 1.17 1.08

50.86 45.06 41.49 36.83 31.50

Source: Modified from Bourgoyne Jr, A.T., 1990. Shale water as a pressure support mechanism in gas reservoirs having abnormal formation pressure. Journal of Petroleum Science & Engineering, 3 (4), 305
319.

Appendix 8 Pertinent data of Example-4 gas reservoir Wang (1999) gave the pressure-production history data of Example-4 gas reservoir as shown in Table A8
1. Table A8
1 Example-4 gas reservoir pressure-production history.

p/Z, MPa Gp, 108m3

50.98 0.00

50.48 0.12

48.40 1.06

47.17 1.67

46.62 1.93

44.66 2.77

42.83 3.20

Source: Modified from Wang, S.W., Stevenson, V.M., Ohaeri, C.U., et al. 1999. Analysis of overpressured reservoirs with a new material balance method. SPE 56690-MS.

Appendix 9 Pertinent data of Field-38 gas reservoir Poston (1989) gave the pressure-production history data of Field-38 gas reservoir as shown in Table A9
1. Table A9
1 Field-38 gas reservoir pressure-production history.

p/Z, MPa Gp, 108m3

44.98 0.00

32.14 6.88

30.61 8.32

26.48 10.05

24.68 10.71

23.16 11.31

Source: Modified from Poston, S.W., Chen, H.Y., 1989. Case history studies: abnormal pressured gas reservoirs. SPE 18857-MS.

Appendix 10 Pertinent data of Gulf of Mexico gas reservoir Poston (1989) gave the pressure-production history data of Gulf of Mexico gas reservoirs as shown in Table A10
1.

Table A10
1 Gulf of Mexico gas reservoir pressure-production history.

p/Z, MPa Gp, 108m3

39.30 0.00

34.60 9.58

32.71 12.59

25.81 22.11

22.70 26.88

23.30 27.64

21.62 28.95

Source: Modified from Poston, S.W., Chen, H.Y., 1989. Case history studies: abnormal pressured gas reservoirs. SPE 18857-MS.

19.93 31.52

16.29 34.26

14.68 35.07

278

Appendices

Appendix 11 Pertinent data of ROB43-1 gas reservoir Poston (1989) gave the ROB43-1 gas reservoir pressure-production history data, as shown in Table A11
1.

Table A11
1 ROB43-1 gas reservoir pressure-production history.

p/Z, MPa Gp, 108m3

55.16 0.00

45.20 10.58

39.27 14.53

35.02 16.67

29.83 19.45

27.41 20.00

Source: Modified from Poston, S.W., Chen, H.Y., 1989. Case history studies: abnormal pressured gas reservoirs. SPE 18857-MS.

Appendix 12 Pertinent data of Louisiana gas reservoir Guehria (1996) gave the pressure-production history data of Louisiana gas reservoirs, as shown in Table A12
1.

Table A12
1 Louisiana gas reservoir pressure-production history.

p/Z, MPa Gp, 108m3

51.02 0.00

49.50 1.92

47.34 4.68

46.06 8.08

44.82 9.62

42.06 11.13

40.64 13.01

39.30 13.94

37.71 16.07

Source: Modified from Guehria, F.M., 1996. A new approach to p/z analysis in abnormally pressured reservoirs. SPE 36703-MS.

34.96 17.43

33.58 19.17

31.06 20.63

28.96 21.76

280

Appendices

Appendix 13 Pertinent data of SE Texas gas reservoir Guehria (1996) gave the pressure-production history data of SE Texas gas reservoirs, as shown in Table A13
1. Table A13
1 SE Texas gas reservoir pressure-production history. p, MPa Gp, 108m3

97.22 0.00

89.98 9.41

84.46 17.12

77.91 27.96

71.02 30.81

61.16 39.36

52.26 44.21

41.71 50.48

Source: Modified from Guehria, F.M., 1996. A new approach to p/z analysis in abnormally pressured reservoirs. SPE 36703-MS.

Appendix 14 Pertinent data of Cajun gas reservoir Becerra-Arteaga (1993) gave the pressure-production history data of Cajun gas reservoir in his doctoral thesis, as shown in Table A14
1.

Table A14
1 Cajun gas reservoir pressure-production history. p/Z, MPa

Gp, 108m3

p/Z, MPa

Gp, 108m3

p/Z, MPa

Gp, 108m3

53.72 51.57 50.43 48.13 46.62 42.37

0.00 5.72 8.62 12.52 15.78 19.79

34.65 30.84 23.98 21.24 16.69 14.31

27.88 32.34 38.53 40.63 44.46 47.01

12.51 11.36 10.78 10.14 9.70

48.47 50.37 51.01 51.83 52.55

Source: Modified from Becerra-Arteaga, 1993. Analysis of Abnormally Pressured Gas Reservoirs. Texas A&M University.

Appendix 15 Inflection point statistics table of 20 developed gas reservoirs by Gan
Blasingame method In the article SPE71514, Gan and Blasingame (2001) used the method they established to evaluate the reserves of 20 developed gas reservoirs. The inflection point appeared time of p/Z curve is shown in Table A15
1, and the pressure-production history data are shown in the Appendix 1
14.

281

Appendices

Table A15
1 Gan
Blasingame method inflection point statistics table. No.

Reservoir

Ga, 108m3

G, 108m3

GD

pDA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Anderson L GOM reservoir North Ossun Offshore Louisiana South Louisiana Stafford Example-4/Wang Field-38 Gulf of Mexico Louisiana Cajun ROB43-1 Reservoir-117 Reservoir-195 Reservoir-197 Reservoir-268 Reservoir-33 Reservoir-41 Reservoir-70 SE Texas

33.33 7.87 43.61 185.79 8.35 12.80 21.32 25.34 76.34 81.78 141.05 58.59 250.80 22.00 5.55 16.68 104.77 25.71 4.42 140.11

21.17 4.32 28.63 123.37 3.87 6.39 10.18 19.63 45.84 36.03 60.00 30.17 129.34 14.50 3.87 9.11 60.31 13.63 3.39 75.14

0.64 0.55 0.66 0.66 0.46 0.50 0.48 0.77 0.60 0.44 0.43 0.51 0.52 0.66 0.70 0.55 0.58 0.53 0.77 0.54

0.79 0.80 0.86 0.70 0.81 0.86 0.85 0.62 0.65 0.81 0.89 0.76 0.76 0.73 0.69 0.86 0.79 0.86 0.78 0.78

Source: Modified from Gan, R.G., Blasingame, T.A., 2001. A semianalytical p/z technique for the analysis of reservoir performance from abnormally pressured gas reservoirs. SPE 71514-MS.

Appendix 16 Pertinent data of M1 gas reservoir The burial depth of M1 gas reservoir is 3750 m, the initial pressure is 74.35 MPa, the formation water compressibility coefficient is 5.645 3 1024 MPa21, the initial water saturation is 0.32, the gas gravity is 0.568, and the volumetric reserve is 2833 3 108 m3. The pressureproduction history data are shown in Table A16
1 (Xia, 2007). Table A16
1 M1 gas reservoir pressure-production history. p, MPa

Gp, 108m3

Z

p/Z, MPa

74.35 72.73 70.67 68.09 65.32 62.47 56.70 50.61 44.82 39.16

0.00 58.78 150.89 258.20 365.80 473.11 687.73 902.64 1117.26 1332.18

1.44 1.42 1.40 1.37 1.34 1.31 1.25 1.19 1.12 1.06

51.56 51.05 50.40 49.54 48.58 47.55 45.30 42.68 39.90 36.86

Source: Modified from Xia, J., Xie, X., Ji, G., et al., 2007. Derivation and application of material balance equation for over-pressured gas reservoir with aquifer. Acta Petrolei Sinica 28 (3), 96
99.

282

Appendices

Appendix 17 Pertinent data of M2 gas reservoir The initial pressure of the M2 gas reservoir is 74.22 MPa, the formation water compressibility coefficient is 5.6 3 1024 MPa21, the initial water saturation is 0.32, the gas reservoir temperature is 100°C, and the volumetric reserve is 2400 3 108 m3. The pressure-production history data are shown in Table A17
1.

Table A17
1 M2 gas reservoir pressure-production history. No.

Gp, 108m3

Wp, 104m3

p, MPa

Z

p/Z, MPa

Δp, MPa

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0.00 2.14 34.61 118.72 228.32 345.40 457.91 553.39 614.31 689.23 769.08 839.48 901.98 958.96 1029.23 1095.80 1157.65

0.00 0.00 0.37 2.11 4.65 7.50 10.93 16.61 18.40 21.11 23.60 26.42 29.52 35.67 39.10 46.26 57.22

74.22 73.98 72.92 69.97 65.80 61.32 57.60 54.99 53.30 51.21 49.00 47.06 45.37 43.86 42.05 40.39 38.87

1.450 1.450 1.446 1.442 1.438 1.434 1.430 1.426 1.422 1.418 1.414 1.413 1.412 1.411 1.411 1.410 1.410

51.27 51.00 50.43 48.52 45.76 42.76 40.28 38.56 37.48 36.11 34.65 33.31 32.13 31.08 29.81 28.64 27.57

0.0000 0.2400 1.3000 4.2500 8.4200 12.9000 16.6200 19.2300 20.9200 23.0100 25.2200 27.1600 28.8500 30.3600 32.1700 33.8300 35.3500

Appendix 18 Pertinent data of M3 gas reservoir The M3 gas reservoir is a typical strong water drive gas reservoir (Wang, 1987). The initial pressure is 29.25 MPa, the formation water compressibility coefficient is 4.5 3 1024 MPa21, the initial water saturation is 0.32, and the gas reservoir temperature is 94°C, the volumetric reserve is 230.8 3 108 m3. The pressure-production history data are shown in Table A18
1.

Appendices

283

Table A18
1 M3 gas reservoir pressure-production history. Time, d

Gp, 108m3

Wp, 104m3

p, MPa

Z

p/Z, MPa

Δp, MPa

0 283 452 799 1296 2172 2607 2966 3298 3663 4028 4390 4767 5156 5414 5504 5579 5809 5940 6293 6501 6807 6875

0.00 6.64 8.30 16.24 28.97 47.00 53.84 64.05 74.54 84.36 92.77 99.91 104.83 109.65 118.10 120.31 122.72 127.38 131.78 147.80 155.55 163.04 164.55

0.00 0.00 0.01 0.15 0.44 0.91 1.00 1.24 1.48 1.88 2.76 3.20 3.30 3.83 4.88 5.04 5.46 6.98 7.28 10.61 13.70 18.35 19.80

29.25 28.72 28.32 27.13 25.92 24.39 23.67 23.50 22.97 21.91 21.64 21.37 21.06 21.22 20.22 20.11 19.97 19.82 19.35 17.21 16.24 15.86 15.80

0.9718 0.9676 0.9651 0.9566 0.9486 0.9395 0.9357 0.9348 0.9322 0.9275 0.9264 0.9253 0.9242 0.9248 0.9214 0.9211 0.9207 0.9202 0.9190 0.9146 0.9142 0.9143 0.9143

30.10 29.68 29.35 28.36 27.33 25.96 25.30 25.14 24.64 23.62 23.35 23.10 22.78 22.95 21.95 21.83 21.69 21.54 21.06 18.82 17.76 17.34 17.28

0.0000 0.5378 0.9308 2.1236 3.3302 4.8608 5.5847 5.7571 6.2880 7.3429 7.6187 7.8807 8.1979 8.0324 9.0321 9.1424 9.2803 9.4320 9.9009 12.0451 13.0173 13.3965 13.4585

Source: Modified from Wang, B., Teasdale, T.S., 1987. Gaswat-Pc: a microcomputer program for gas material balance with water influx. SPE 16484-MS.

Appendix 19 Pertinent data of M4 gas reservoir The M4 gas reservoir is a typical water drive gas reservoir (Jiao, 2017). The initial pressure is 74.48 MPa, the gas reservoir temperature is 100°C, and the volumetric reserves are 2091.5 3 108 m3. The pressureproduction history data are shown in Table A19
1.

284

Appendices

Table A19
1 M4 gas reservoir pressure-production history. p, MPa

Z

Gp, 108m3

p/Z, MPa

pD

74.48 74.42 73.77 72.15 70.14 67.74 65.20 62.74 60.15 57.49 54.59 51.52 48.53 45.28 42.17 39.51 37.62 35.58

1.4037 1.4031 1.397 1.3814 1.3616 1.3372 1.3108 1.2848 1.2572 1.2287 1.1978 1.1656 1.1349 1.1027 1.0734 1.0498 1.0338 1.0176

0.0 2.6 35.0 119.7 226.5 338.6 451.0 555.3 659.5 764.0 868.2 970.4 1069.8 1169.6 1260.3 1341.2 1403.5 1462.5

53.06 53.04 52.81 52.23 51.52 50.66 49.74 48.83 47.85 46.79 45.57 44.20 42.76 41.06 39.29 37.64 36.39 34.96

1.0000 0.9996 0.9952 0.9843 0.9708 0.9547 0.9374 0.9203 0.9017 0.8818 0.8589 0.8330 0.8058 0.7738 0.7404 0.7093 0.6857 0.6589

Source: Modified from Jiao Yuwei, Xia Jing, Liu Pengcheng, et al,2017. New material balance analysis method for abnormally high-pressured gas-hydrocarbon reservoir with water influx. International Journal of Hydrogen Energy, 42 (29),18718
18727.

Appendix 20 Pertinent data of M5 gas reservoir The M5 gas reservoir is a typical fractured stress-sensitive gas reservoir (Aguilera, 2008). The initial pressure is 17.31 MPa, the matrix-irreducible water saturation is 0.25, the fracture water saturation is 0, and the water compressibility is 4.35 3 1024 MPa21, the matrix compressibility is 2.90 3 1023 MPa21, the fracture porosity is 0.01, and the fracture storage volume ratio is 0.5. The pressure-production history data are shown in Table A20
1. Table A20
1 M5 gas reservoir pressure-production history. p, MPa

p/Z, MPa

Gp, 108m3

Δp, MPa

17.31 14.27 13.18 11.27 9.59 7.00

19.75 16.25 14.96 12.67 10.66 7.60

0.00 10.48 14.16 20.81 25.15 31.71

0.00 3.03 4.12 6.03 7.72 10.31

Source: Modified from Aguilera, R., 2008. Effect of fracture compressibility on gas-in-place calculations of stresssensitive naturally fractured reservoirs. SPE Reservoir Evaluation & Engineering, 11 (2), 307
310.

Appendices

285

Appendix 21 Basic principles of type curve matching analysis method A type curve is a graphical representation of the theoretical solution of the flow equation, usually expressed in terms of dimensionless variables, such as dimensionless pressure pD , dimensionless time tD , dimensionless radius rD , dimensionless wellbore storage coefficient CD , etc., rather than actual variables (such as: Δp, t, r, C). Type curve analysis means that when the production or pressure changes, the graph of the actual measured data is superimposed on the type curve graph to find a theoretical typical curve that can “match” the actual response of the test well and the reservoir, and then, the reservoir and well parameters are calculated according to the matching result (Tarek, 2019). When any variable is multiplied by a set of constants with its reciprocal dimension, it can become dimensionless. The choice of this set of constants depends on the type of problem to be solved. For example, to create a dimensionless pressure pD , the actual pressure difference in MPa is multiplied by the parameter group A in MPa21, that is, pD 5 AΔp. The parameter group A, which makes the variable dimensionless, comes from the equation describing the fluid flow in the reservoir. The following takes the radial and incompressible fluid steady flow equation as an example to introduce this concept. The Darcy equation is q5

0:5428KhΔp μB½lnðre =rwa Þ 2 0:5

(A21
1)

where rwa is the effective radius, defined as rwa 5 rw e2S . By deforming the formula (A21
1), the parameter group A expression can be obtained, which is
   0:5428Kh ln re =rwa 2 0:5 5 Δp 5 AΔp qμB The left side of the equal sign of the above equation is dimensionless, so the right side is also dimensionless. Thus, pD is defined as
 0:5428Kh pD 5 Δp (A21
2) qμB The logarithm of both sides of the formula (A21
2) is
 0:5428Kh lg pD 5 lg 1 lg Δp 5 lg A 1 lg Δp qμB

(A21
3)

286

Appendices

Eq. (A21
3) shows that for the constant production situation, there is only a constant difference from the pD
Δp double logarithmic graph
 0:5428Kh lg 5 lg A qμB In the same way, dimensionless time can be defined, with
 3:6 3 1023 K Δt tD 5 φμCt rw2 Take the logarithm of both sides, there are
 3:6 3 1023 K 1 lg Δt 5 lg D 1 lg Δt lg tD 5 lg φμCt rw2

(A21
4)

Therefore, the lg ΔpBlg Δt curve and the lg pD Blg tD curve have the same shape, with a vertical distance of lg A, and a horizontal distance of lg D, as shown in Fig. A21
1. These two curves not only have the same shape, but if they move relative to each other until they coincide or match, the vertical and horizontal displacements required to achieve matching are related to these constants in Eqs. (A21
3 and A21
4). Once these constants are determined according to the vertical and

10-1

100

0

10

t，h 101

102

103 2 10

pD

pD

pD/'p

10-1

101 tD/t

10-2

101

102

103

104

'p，MPa

'p

100 105

tD Figure A21
1 The logarithmic matching principle. Modified from Tarek, A. 2019. Reservoir Engineering Handbook, fifth ed. Elsevier.

Appendices

287

horizontal displacements, the reservoir and well parameters, such as permeability and skin coefficient, can be estimated. The process of matching two curves through vertical and horizontal displacements and determining reservoir or well parameters is called type curve fitting.

Appendix 22 Pertinent data of Cajuna gas reservoir The Cajuna gas reservoir pressure-production history data are shown in Table A22
1. The initial pressure is 79.0 MPa, the gas reservoir temperature is 401.3 K, the gas gravity is 0.6 (Ambastha, 1991), and the natural gas deviation factor is Z 5 1:56871 3 1028 p4 2 4:1861 3 1026 p3 1 4:27663 3 1024 p2 2 0:00938p 1 1:00438

(A22
1)

Table A22
1 Cajuna gas reservoir pressure-production history. p, MPa

p/Z, MPa

Gp, 108m3

pD

79.00 73.70 70.00 63.90 59.20 54.60 51.00 47.30

53.40 51.79 50.57 48.39 46.54 44.56 42.89 41.02

0.00 0.28 0.81 1.57 2.20 2.87 3.41 4.10

1.0000 0.9699 0.9471 0.9062 0.8716 0.8346 0.8032 0.7683

Source: Modified from Ambastha, A.K., 1991. A type curve matching procedure for material balance analysis of production data from geopressured gas reservoirs. Journal of Canadian Petroleum Technology, 30 (5), 61
65.

Appendix 23 Pertinent data of M6 gas reservoir The initial pressure of the M6 gas reservoir is 105.89 MPa, the gas reservoir temperature is 132°C, and the volumetric reserves are 1704 3 108 m3; the current pressure is 79.04 MPa, and the cumulative gas production is 426 3 108 m3 (Table A23
1)

Table A23
1 M6 gas reservoir pressure-production history.

Gp, 108m3 pD

0 1

45.2 0.988

93.5 0.977

143.2 0.962

196.4 0.947

242.3 0.935

289.2 0.924

334.5 0.912

380.3 0.900

426.4 0.888

Appendices

289

Appendix 24 Pertinent data of M7 gas reservoir The initial pressure of the M7 gas reservoir is 120 MPa, the gas reservoir temperature is 180°C, and the volumetric reserves are 8.3 3 108 m3. The output of the two wells is 50 3 104 m3/d, respectively. The first well produced 365 days, and the second well started from the 101st day. The simulated pressure data is shown in the Table A24
1.

Table A24
1 M7 gas reservoir pressure-production history. t, d

pwf, MPa

t, d

pwf, MPa

t, d

pwf, MPa

t, d

pwf, MPa

t, d

pwf, MPa

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

119.1 118.7 118.4 118.0 117.7 117.3 117.0 116.6 116.3 116.0 115.6 115.3 114.9 114.6 114.3 114.0 113.6 113.3 113.0 112.7 112.3 112.0 111.7 111.4 111.1 110.8 110.4 110.1 109.8 109.5

73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 105 107 109 111 113 115 117 119 121 123 125 127 129 131 133

107.4 107.1 106.7 106.4 106.1 105.8 105.5 105.2 104.9 104.6 104.3 104.0 103.7 103.4 103.1 101.9 101.3 100.7 100.2 99.6 99.0 98.4 97.9 97.3 96.8 96.2 95.7 95.1 94.6 94.0

147 149 151 155 157 159 161 163 165 167 169 171 173 175 177 179 181 183 185 187 189 191 193 195 197 199 201 205 207 209

90.3 89.8 89.3 88.2 87.7 87.2 86.7 86.2 85.7 85.2 84.7 84.2 83.8 83.3 82.8 82.3 81.8 81.4 80.9 80.4 80.0 79.5 79.0 78.6 78.1 77.7 77.2 76.3 75.9 75.4

223 225 227 229 231 233 235 237 239 241 243 245 247 249 251 255 257 259 261 263 265 267 269 271 273 275 277 279 281 283

72.4 72.0 71.6 71.2 70.7 70.3 69.9 69.5 69.1 68.7 68.3 67.9 67.5 67.1 66.7 65.9 65.6 65.2 64.8 64.4 64.0 63.7 63.3 62.9 62.5 62.2 61.8 61.4 61.1 60.7

297 299 301 305 307 309 311 313 315 317 319 321 323 325 327 329 331 333 335 337 339 341 343 345 347 349 351 355 357 359

58.3 57.9 57.6 56.9 56.5 56.2 55.9 55.5 55.2 54.9 54.6 54.2 53.9 53.6 53.3 52.9 52.6 52.3 52.0 51.7 51.4 51.0 50.7 50.4 50.1 49.8 49.5 48.9 48.6 48.3

(Continued)

290

Appendices

Table A24
1 (Continued) t, d

pwf, MPa

t, d

pwf, MPa

t, d

pwf, MPa

t, d

pwf, MPa

t, d

pwf, MPa

61 63 65 67 69 71

109.2 108.9 108.6 108.3 108.0 107.7

135 137 139 141 143 145

93.5 92.9 92.4 91.9 91.3 90.8

211 213 215 217 219 221

75.0 74.6 74.1 73.7 73.3 72.8

285 287 289 291 293 295

60.4 60.0 59.7 59.3 58.9 58.6

361 363 365

48.0 47.7 47.4

Appendix 25 Pertinent data of Ellenburger gas reservoir In his paper SPE 22921-MS, Fetkovich cited an example of reserves calculation in the Ellenburger gas reservoir considering a water-soluble gas. The initial pressure of the gas reservoir is 46.02 MPa, the gas reservoir temperature is 93.3°C, the porosity is 5%, and the initial water saturation is 0.35. The reservoir has developed microfractures and has good connectivity. The molar content of CO2 in the natural gas component is 28%. The pressure-production history data are shown in Table A25
1; the rock compressibility is 9.42 3 10 24 MPa21, and the high-pressure physical properties of other fluids are shown in Table A25
2.

Table A25
1 Ellenburger gas reservoir pressure-production history. Gp, 108m3

p/Z, MPa

Gp, 108m3

p/Z, MPa

Gp, 108m3

p/Z, MPa

0.0 165.6 280.8 312.0 336.0 351.6 384.0 423.6 469.1 463.1

43.55 37.96 34.43 33.65 32.67 31.98 30.90 28.74 27.96 27.07

506.3 550.7 590.3 627.5 659.9 689.9 721.1 746.3 771.5 788.3

25.80 22.95 21.88 19.72 18.34 15.60 15.40 14.03 12.75 11.48

808.7 823.1 830.3 838.7 847.1 859.1 869.9 879.5 887.9 901.1

10.79 10.69 10.30 9.81 9.71 9.12 8.83 8.34 7.75 7.16

Source: Modified from Fetkovich, M.J., Reese, D.E., Whitson, C.H., 1991. Application of a general material balance for high pressure gas reservoir. SPE 22921-MS.

Appendices

291

Table A25
2 Calculation of total water cumulative compressibility for the Ellenburger reservoir. p, MPa

Bw

Rsw, m3/m3

Z

Bg, m3/m3

Btw

Ctw, 1024MPa21

46.02 41.37 34.47 27.58 20.68 17.24 13.79 12.07 10.34 8.62 6.89 3.45 5.17 1.72 0.69 0.10

1.0761 1.0765 1.0768 1.0770 1.0767 1.0764 1.0758 1.0754 1.0749 1.0743 1.0735 1.0716 1.0727 1.0704 1.0695 1.0689

12.02 11.49 10.60 9.53 8.21 7.39 6.43 5.88 5.27 4.59 3.85 2.08 3.01 1.03 0.34 0.00

1.0464 0.9962 0.9262 0.8732 0.8493 0.8513 0.8638 0.8742 0.8872 0.9028 0.9208 0.9621 0.9408 0.9833 0.9946 1

0.00292 0.0031 0.00345 0.00407 0.00528 0.00635 0.00805 0.00932 0.01103 0.01347 0.01717 0.03588 0.02339 0.07335 0.18548 1.2686

1.076 1.078 1.082 1.087 1.097 1.106 1.121 1.133 1.149 1.174 1.214 1.428 1.284 1.876 3.236 16.319

3.9885 4.1045 4.5251 5.5694 7.5999 9.5869 12.8937 15.4753 19.0722 24.4096 32.7201 76.8546 47.1802 167.8792 442.8384 3084.6256

Source: Modified from Fetkovich, M.J., Reese, D.E., Whitson, C.H., 1991. Application of a general material balance for high pressure gas reservoir. SPE 22921-MS.

Appendix 26 Pertinent data of Duck Lake gas reservoir In his paper SPE 22921-MS, Fetkovich cited examples of reserve calculations in Duck Lake gas reservoirs considering a water-soluble gas. The initial pressure of the gas reservoir is 40.0 MPa, the gas reservoir temperature is 115.7°C, the porosity is 25%, the initial water saturation is 0.18, the gas gravity is 0.65, and the rock compressibility is 4.93 3 1024 MPa21. The pressure-production history data of this gas field are shown in Table A26
1.

292

Appendices

Table A26
1 Duck Lake gas reservoir pressure-production history. Gp, 108m3

p/Z, MPa

Gp, 108m3

p/Z, MPa

Gp, 108m3

p/Z, MPa

0.25 4.04 9.09 13.39 18.44 24.25 32.33 45.21 48.50 52.54 58.35

36.41 35.76 35.11 34.54 33.81 33.00 31.78 29.75 29.18 28.36 27.63

61.38 66.18 68.70 71.73 77.79 82.59 85.12 92.44 95.73 105.33 111.89

26.58 26.50 26.09 25.52 24.54 23.57 22.92 21.62 21.54 19.75 18.37

116.19 127.05 137.91 143.72 149.78 153.57 158.37 162.16 164.93

17.31 15.36 12.84 11.30 10.08 9.35 7.96 6.83 6.18

Source: Modified from Fetkovich, M.J., Reese, D.E., Whitson, C.H., 1991. Application of a general material balance for high pressure gas reservoir. SPE 22921-MS.

This example does not provide the fluid property parameters. The empirical formula in Chapter 3 is used for calculation. The calculation results are shown in Table A26
2. Z is calculated according to the Standing method, Bw is calculated according to Eq. (3
70), and Rsw is calculated according to Eq. (3
75), Btw is calculated according to Eq. (5
119).

Table A26
2 Fluid property calculation for the Duck Lake reservoir. p, MPa

Bw

Rsw, m3/m3

Bg, m3/m3

Btw

F

Eg

Ew

Ef

Et (M 5 6.5)

40.00 37.51 36.53 35.57 34.75 33.72 32.61 31.01 28.48 27.80 26.85 26.02 24.86 24.77 24.33 23.73 22.71 21.73 21.08 19.82 19.75 18.07 16.81 15.87

1.0463 1.0468 1.0470 1.0472 1.0474 1.0476 1.0479 1.0482 1.0487 1.0489 1.0491 1.0493 1.0495 1.0496 1.0497 1.0498 1.0500 1.0502 1.0504 1.0507 1.0507 1.0510 1.0513 1.0515

3.82 3.78 3.75 3.73 3.70 3.66 3.62 3.55 3.41 3.37 3.31 3.26 3.18 3.17 3.14 3.09 3.01 2.93 2.87 2.76 2.75 2.59 2.47 2.37

0.00355 0.00369 0.00376 0.00383 0.00389 0.00398 0.00407 0.00423 0.00452 0.00461 0.00474 0.00486 0.00506 0.00507 0.00515 0.00527 0.00548 0.00570 0.00586 0.00622 0.00624 0.00681 0.00732 0.00776

1.0463 1.0470 1.0473 1.0476 1.0479 1.0483 1.0487 1.0494 1.0506 1.0510 1.0515 1.0520 1.0528 1.0529 1.0532 1.0536 1.0544 1.0553 1.0559 1.0572 1.0573 1.0594 1.0612 1.0628

0.0039 0.0047 0.0189 0.0384 0.0557 0.0768 0.1022 0.1400 0.2074 0.2264 0.2519 0.2867 0.3132 0.3385 0.3567 0.3805 0.4286 0.4735 0.5016 0.5770 0.5997 0.7189 0.8207 0.9040

0.00000 0.00014 0.00021 0.00028 0.00034 0.00043 0.00052 0.00068 0.00097 0.00106 0.00119 0.00131 0.00151 0.00152 0.00160 0.00172 0.00193 0.00215 0.00231 0.00267 0.00269 0.00326 0.00377 0.00421

0.0000 0.0007 0.0010 0.0013 0.0016 0.0019 0.0024 0.0030 0.0043 0.0046 0.0052 0.0057 0.0065 0.0065 0.0068 0.0073 0.0081 0.0090 0.0096 0.0109 0.0110 0.0131 0.0149 0.0165

0.0000 0.0012 0.0017 0.0022 0.0026 0.0031 0.0036 0.0044 0.0057 0.0060 0.0065 0.0069 0.0075 0.0075 0.0077 0.0080 0.0085 0.0090 0.0093 0.0099 0.0100 0.0108 0.0114 0.0119

0.000 2.006 3 1024 2.918 3 1024 3.855 3 1024 4.695 3 1024 5.806 3 1024 7.084 3 1024 9.095 3 1024 1.273 3 1023 1.383 3 1023 1.545 3 1023 1.698 3 1023 1.932 3 1023 1.951 3 1023 2.045 3 1023 2.183 3 1023 2.431 3 1023 2.698 3 1023 2.887 3 1023 3.296 3 1023 3.323 3 1023 3.973 3 1023 4.556 3 1023 5.063 3 1023 (Continued)

Table A26
2 (Continued) p, MPa

Bw

Rsw, m3/m3

Bg, m3/m3

Btw

F

Eg

Ew

Ef

Et (M 5 6.5)

14.15 11.95 10.60 9.53 8.89 7.65 6.61 6.01

1.0519 1.0524 1.0528 1.0530 1.0532 1.0534 1.0537 1.0538

2.17 1.91 1.74 1.60 1.52 1.35 1.20 1.11

0.00875 0.01047 0.01190 0.01334 0.01438 0.01688 0.01969 0.02176

1.0663 1.0724 1.0775 1.0826 1.0863 1.0952 1.1053 1.1128

1.1134 1.4449 1.7112 1.9988 2.2096 2.6735 3.1935 3.5899

0.00520 0.00692 0.00835 0.00979 0.01083 0.01333 0.01614 0.01821

0.0200 0.0261 0.0311 0.0362 0.0400 0.0489 0.0590 0.0665

0.0127 0.0138 0.0145 0.0150 0.0153 0.0160 0.0165 0.0168

6.173 3 1023 8.094 3 1023 9.686 3 1023 1.128 3 1022 1.244 3 1022 1.520 3 1022 1.831 3 1022 2.060 3 1022

Appendices

295

Appendix 27 Principles of multiple (two) meta-regression analysis When dealing with measurement data, it is often necessary to study the relationship between variables. The relationship between variables is generally divided into two types. One is a completely deterministic relationship, that is, a functional relationship; the other is a correlation relationship, that is, although there is a close relationship between variables, the value of another variable cannot be calculated from the value of one or more variables. Multiple regression refers to a regression model with one dependent variable and multiple independent variables. This kind of regression that includes two or more independent variables is called multiple regression. Applying this method can deepen the understanding of the conclusions of qualitative analysis and draw the quantitative dependence between various elements, thereby further revealing the internal laws of each element. Assume that x1 ; x2 ; . . .; xp is p variables that can be accurately measured or controlled. If the internal relationship between the variable y and x1 ; x2 ; . . .; xp is linear, then after n trials, n sets of data can be  obtained: yi ; xi1 ; xi2 ; . . .; xip , i 5 1; 2; . . .; n, the relationship between them can be expressed as: y1 5 b0 1 b1 x11 1 b2 x12 1 . . . 1 bp x1p 1 ε1 y2 5 b0 1 b1 x21 1 b2 x22 1 . . . 1 bp x2p 1 ε2 ... yn 5 b0 1 b1 xn1 1 b2 xn2 1 . . . 1 bp xnp 1 εn where b0 ; b1 ; b2 ; . . .bp is p 1 1 parameters to be estimated, εi represents the influence of random factors on yi in the i-th experiment. The above formula is the mathematical model of p-variable linear regression (Fengling Fu, 2003). It can be solved by the least square method and can be automatically regressed in Microsoft Excel software. The regression method will not be repeated.

Appendix 28 Nomenclature A B Bg, Bo, and Bw Bgi, Boi

area of gas reservoir, m2; formation volume factor, m3/m3; formation volume factors of gas, oil, and water, respectively, m3/m3; formation volume factor of gas and oil evaluated at pi, m3/m3;

296 C Ce Cg Cf Cpr Cs Cw Ct Cti dz dL dp D D Dc Dt f g G Ga h H K m M M m(p1) m(p2) n p p pD pi ppc p0pc pr ppr psc pt pw pwf pws Δp qsc q12 qw R

Appendices

hydrostatical pressure gradient 0.980665MPa/100m;

effective compressibility, Ce 5 Cw1S2wi S1wiCf , MPa21; isothermal gas compressibility, MPa21; rock compressibility, MPa21; isothermal pseudo-reduced compressibility, dimensionless; matrix compressibility, 1/MPa; water compressibility, MPa21; total compressibility, MPa21; total compressibility evaluated at pi, MPa21; vertical distance change, m; change in distance along the wellbore trajectory, m; pressure change, MPa; mid-deep gas reservoir, m; inner diameter of tubing (considering the flow in the tubing), m; inner diameter of casing, m; outer diameter of tubing, m; friction coefficient, dimensionless; gravity, m/s2; original gas in place, 108m3; apparent gas originally in place, 108m3; average reservoir thickness, m; vertical depth in the middle of the pay zone, m; permeability, 1023μm2; gas mass, kg; molecular weight, kg/kmol; volume ratio, dimensionless; pseudo-pressure of block 1, MPa2/(mPa  s); pseudo-pressure of block 2, MPa2/(mPa  s); number of moles of gas, kmol; current average formation pressure, MPa; average pressure, MPa; dimensionless pressure; initial pressure in the middle depth of the gas reservoir, MPa; pseudo-critical pressure, MPa; corrected pseudo-critical pressure, MPa; contrast pressure, dimensionless; pseudo-reduced pressure, dimensionless; standard pressure, psc 5 0.101325 MPa; tubing pressure, MPa; bottom hole pressure, MPa; bottom hole flow pressure, MPa; bottom shut-in pressure, MPa; pressure difference, MPa; gas well production under standard conditions, 104m3/d; flow rate between the two compartments, m3/d; daily water production, m3/d; universal gas constant, 8.3143 3 1023 (MPa  m3)/(kmol  K);

Appendices

Sg、So、Sw Sgi、Soi、Swi Sgrw t tca tcaDd tD T Tc Tf Tr Tpr Tpc T 0pc Tsc v V   V p wiz Vp Vr Vsc We Wi Wp Z Z Zi Zws Δpp q





Δpp q i Δpp q id q Δpp





q Δpp i q Δpp id

gas, oil, water saturation, decimal; initial gas, oil, water saturation, decimal; residual gas saturation of water drive, decimal; time, h; Ðt q gas well material balance pseudo time, tca 5 ðμCq t Þi 0 μC dt, d; t Blasingame dimensionless time of gas well, tcaDd 5 bma;pssa tca , dimensionless; 3 1023 Kt dimensionless time, tD 5 3:6μφC , dimensionless; 2 t rw temperature, K; critical temperature, K; gas reservoir temperature, K; contrast temperature, dimensionless; quasi-contrast temperature, dimensionless; pseudo-critical temperature, K; corrected pseudo-critical temperature, K; standard temperature, K, 293.15 K; gas flow rate, m/s; volume of gas, m3; pore volume of the water-invaded zone, 108m3; reservoir pore volume, m3; gas volume under formation conditions, m3; gas volume under standard conditions, m3; gas reservoir water influx, 108m3; water volume, 108m3; cumulative water production, 104m3; real gas deviation coefficient, dimensionless; average real gas deviation coefficient, dimensionless; natural gas deviation coefficient under initial conditions, dimensionless; gas deviation coefficient under pws pressure and TR temperature, dimensionless;   p 2p Δp normalized pressure for gas well, q p 5 pi q pwf ; MPa= m3 =d ; normalized pressure integral for gas well,

 3  Ð Δpp 1 tca Δpp 5 q i q dτ; MPa= m =d ; tca 0 normalized pressure Δp  integral derivative for gas well,

 3  d qp Δpp i q id 5 tca dtca ; MPa= m =d ; normalized rate for gas well, Δpq p 5 pp 2q pp ; m3 =d=MPa; i wf Ðt normalized rate integral for gas well, Δpq p 5 t1ca 0ca Δpq p dτ; m3 =d=MPa; i

normalized rate integral derivative for gas well, q ðΔpq p Þid 5 2 tca

dðΔpp Þi dtca

; m3 =d=MPa;

Greek alphabet: α β φ

297

formation pressure coefficient, dimensionless; derivative, dimensionless; porosity, decimal;

298

Appendices

γ

relative density (liquid relative to water, gas relative to air), dimensionless; gas specific gravity, dimensionless; oil specific gravity, dimensionless; viscosity, mPa  s; corrected gas viscosity at 1 atm and reservoir temperature, mPa  s; viscosity corrections due to the presence of N2/CO2/H2S, mPa  s; uncorrected gas viscosity, mPa  s; average viscosity, mPa  s; formation gas viscosity, mPa  s; formation water viscosity, mPa  s; density of gas, oil, water, kg/m3; density under standard conditions of gas and oil, kg/m3; W 2W B water invasion
related parameters, ω 5 e GBgip w , dimensionless; transmissibility between compartments, 1023μm2  m2/(mPa  s);

γg γo μ μ1 ðΔμÞN2 ; ðΔμÞCO2 ; ðΔμÞH2 S μ1 uncorrected μ μg μw ρg, ρo, ρw ρgsc ; ρosc ω Γ

Subscripts: D e f g i M t wf

dimensionless; external boundary; formation; gas; initial or the number of sequence; matched or type curve match point; total; wellbore and flowing.

Appendix 29 Conversion relationship between SI units and other units (1) Length Unit

km

m

cm

Mile

ft

in

1 km

1

103

105

39370.08

0.6214

3280.84

1m

10

1

102

6.214 3 1024

3.28084

39.37008

1 cm

1025

1022

1

6.214 3 1026

0.0328084

0.393701 63360

23

1 mile

1.60934

1609.34

1.60934 3 105

1

5280

1 ft

3.48 3 1024

0.3048

30.48

1.839 3 1024

1

12

1 in

2.54 3 1025

0.0254

2.54

1.5783 3 1025

0.08333

1

299

Appendices

(2) Area Unit

m2

cm2

ft2

in2

1m2 1cm2 1ft2 1in2

1 1024 0.092903 6.4516 3 1024

104 1 929.03 6.4516

10.7639 1.07639 3 1023 1 6.9444 3 1023

1550 0.155 144 1

(3) Volume m3

Unit 3

1m

1

10 26

3

1cm 1ft3 1bbl 1L

cm3

10 0.0283168 0.158988 1023

ft3

6

bbl

35.3147

1 2.83168 3 104 1.58988 3 105 103

L

103

6.28978 25

3.53147 3 10 1 5.6146 3.53147 3 1022

26

1023 28.3168 158.99 1

6.28978 3 10 0.17811 1 6.28978 3 1023

(4) Pressure Unit

MPa

kPa 3

atm

bar

kg/cm2

psi

145.038

1MPa

1

10

9.86923

10

10.1972

1kPa

1023

1

9.86923 3 1023

1022

0.0101972

0.145038

1atm

0.101325

101.325

1

1.01325

1.03323

14.6959

1021

102

0.986923

1

1.01972

14.5038

0.0980665

98.0665

0.967841

0.980665

1

14.2233

0.00689476

6.89476

0.068406

0.0689476

0.070307

1

1bar 1kg/cm 1psi

2

(5) Temperature Unit

°C

K

°F

°R

1°C 1K 1°F 1°R

t T 2 273.15 5(f 2 32)/9 5r/9 2 273.15

t 1 273.15 T 5(f 1 459.67)/9 5r/9

1.8t 1 32 1.8T 2 459.67 f r 2 459.67

1.8t 1 491.67 1.8T f 1 459.67 r

300

Appendices

(6) Oil Production m3/d

Unit 3

1m /d 1cm3/s 1bbl/d

cm3/s

bbl/d

4

1 0.0864 0.158988

10 /864 1 1.84014

6.28978 0.543437 1

(7) Gas production 104m3/d

Unit 4

3

110 m /d 1cm3/s 1103ft3/d 1106ft3/d

cm3/s 8

1 864 3 1028 2.83168 3 1023 2.83168

10 /864 1 327.741 327741

Mscf/d

MMscf/d

353.147 3.05119 3 1023 1 103

0.353147 3.05119 3 1026 1023 1

(8) Compressibility Unit

1/MPa

1/atm

1/(kg/cm2)

1/psi

1/MPa 1/atm 1/(kg/cm2) 1/psi

1 9.86923 10.1972 145.038

0.101325 1 1.03323 14.6959

0.0980665 0.967841 1 14.2233

0.00689476 0.068406 0.070307 1

(9) Permeability Unit

μm2

cm2

m2

mD

D

1μm2 1mD

1 0.98692 3 1023

1028 9.8692 3 10212

10212 9.8692 3 10216

1.0133 3 103 1

1.0133 1023

(10) Viscosity 1mPa  s 5 1023Pa  s 5 103μPa  s 5 1cP (11) Relative density of stock tank oil (γo) and °API (oil density is expressed in degrees of API gravity, a standard of the American Petroleum Institute) 141:5 2 131:5 γo   γo 5 141:5= 131:5 1 ° API °

API 5

(12) Gas oil ratio 1m3/ m3 5 5.615 scf/STB 1 scf/STB 5 0.1781 m3/ m3

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Further reading Adefidipe, O.A., Xu, Y.H., 2014. Estimating effective fracture volume from early-time production data: a material balance approach. SPE 171673-MS. Adrian, P.M., Cabrera, M.R., 2018. Application of Blasingame type curves to a multiwell gas-condensate reservoir, field case study. SPE 191214-MS. Agarwal, R.G., Al-Hussainy, R., Ramey Jr, H.J., 1965. The importance of water influx in gas reservoirs. Journal of Petroleum Technology, 17 (11), 1336
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1082. Andersen, M.A., 1997. Tips, tricks and traps of material balance calculations. Journal of Canadian Petroleum Technology, 36 (11), 34
48. Anderson, D.M., Stotts, G.W., Mattar, L., et al., 2010. Production data analysis challenges, pitfalls, diagnostics. SPE Reservoir Evaluation & Engineering, 13 (3), 538
552. Anisur Rahman, N.M., Mattar, L., Anderson, D.M., 2006. New, rigorous material balance equation for gas flow in a compressible formation with residual fluid saturation. SPE100563-MS. Azis, H., Febrianto, D.H., Wijayanti, E., et al., 2019. Flowing material balance analysis and production optimization in HPHT sour gas field. SPE 196360-MS. Baker, R.O., Regier, C., Sinclair, R., 2003. PVT error analysis for material balance calculations. PETSOC-2003-203. Bass, D.M., 1972. Analysis of abnormally pressured gas reservoirs with partial water influx. SPE 3850-MS. Bernard, W.J., 1985. Gulf coast geopressured gas reservoirs: drive mechanism and performance prediction. SPE 14362-MS.

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Index Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively.

A Ambastha method, 223
226 Anderson L gas reservoir, 211, 267 Apparent molecular weight, 57 Apparent reservoir pressure, 152
155 Arithmetic average method, 50 Average arithmetic method, 44
45 Average compressibility method, 160t Average gas reservoir pressure arithmetic average method, 50 weighted average method, 51
52

B Becerra
Arteaga method, 172
173, 190
195 Binomial regression method, 211 Brine volume factor, 89
90

C Cajun gas reservoir, 280 Cajuna gas reservoir, 287 Calculation methods conventional methods, 20
22 securities and exchange commission methods, 22
23 Capillary pressure measurement, 33 Carr
Kobayashi
Burrows method, 64, 75
77 Carter
Tracy unsteady flow model, 113
114 Chen method, 161
164 Classical two-segment method Gan
Blasingame method analysis method, 164
167 analysis procedure, 168
170 Hammerlindl method average compressibility method, 158
159 Chen method, 161
164 corrected reservoir volume method, 159
161

correction factor, 161 gas compressibility, 161 high-pressure gas reservoir, 161 time of inflection point of p/Z curve, 171 Classifications for Petroleum Resources and Reserves, 13, 16 Closed gas reservoir, 104
109, 200, 210 Commercially recoverable reserves, 12 Communication factor, 120 Compartmented gas reservoirs Gao method, 125
131 Hagoort
Hoogstra method, 123
125 Payne method, 119
123 Sun method, 131
137 Compressibility, 72, 74f, 94
96, 255 factors, 108
109 Conduct linear regression, 219 Conventional methods, 20
22 Corrected reservoir volume method, 159
161 Correction factor, 161 Cullender-Smith method, 46
50 Cumulative gas production, 103

D DAK extrapolation method, 80
82, 105
106 Darcy’s equation, 123
124 Dempsey method, 78 Density, 57 Depletion drive index, 137
138 Deviation factor, 59
71 Discovered petroleum initially-in-place, 12 Dranchuk
Abu-Kassem (DAK) method, 69
70 Dranchuk
Purvis
Robinson (DPR) method, 70 Drive index of gas reservoirs, 137
139 Duck Lake gas reservoir, 291
292 Duhamel’s principle, 133
134

315

316

Index

Dynamic monitoring, 28 Dynamic reserves defined, 16 estimation, 23
26

E Ellenburger gas reservoir, 290

F Far-well recharge area, 127
128 Fetkovich method, 227
230 Field-38 gas reservoir, 276 Formation water, 87
96 formation water viscosity, 91
92 formation water volume factor, 88
91 isothermal compressibility factor of, 94
96 natural gas solubility in water, 92
94 Fracture compressibility, 219

G Gan
Blasingame method, 186
187 inflection point statistics table, 280 Gao method, 125
131 Gas compressibility, 161 Gas field development, 27
28 Gas formation volume factor, 75 Gas gravity, 58 Gas initially-in-place (GIIP), 101 Gas production per unit pressure drop, 102
103 Gas reservoirs apparent initial gas in, 140
141 apparent reservoir pressure, 152
155 compartmented gas reservoirs Gao method, 125
131 Hagoort
Hoogstra method, 123
125 Payne method, 119
123 Sun method, 131
137 drive index of, 137
139 homogeneous gas reservoirs, 109
116 closed gas reservoir, 104
109 pressure depletion curve, 118
119 volumetric gas reservoir, 99
104 water-drive gas reservoir, 109
116 with water-soluble gas, 116
118 influence of water-soluble gas, 155

parameters, 25
26 pressure depletion, 152 sensitivity analysis cumulative rock compressibility factor, 150
151 effective compressibility factor, 147 rock compressibility factor, 142
147 size of water body, 152 Gas wells monitoring modes, 28
32 static pressure conversion gas column density conversion method, 42 static pressure gradient conversion method, 42
43 wellhead static pressure conversion method average temperature and average deviation factor method, 43
46 Cullender-Smith method, 46
50 Geopressured gas reservoirs basic data preparation, 258
259 calculation methods, 255 comparative analysis of results, 260
264 recommended methods, 255
258 recommended procedure, 258 Geopressured natural gas resources and development classification by depth, 9
10 by pressure and pressure coefficient, 10 development status Sichuan Basin, 5 Tarim Basin, 5
6 development strategies geology-engineering integration, 8 high-precision seismic survey, 6
7 pilot test and production test, 7 technical policies, 7
8 technological innovation and integrated application, 8 gas reservoir characteristics, 2
3 Gonzalez method, 231
235 Ground pressurization, 36 Gulf Coast gas reservoir, 270

Index

317

Gulf of Mexico gas reservoir, 273, 276
277

K

H

L

Hagoort
Hoogstra method, 123
125 Hammerlindl method, 183 average compressibility method, 158
159 Chen method, 161
164 corrected reservoir volume method, 159
161 correction factor, 161 gas compressibility, 161 high-pressure gas reservoir, 161 Havlena
Odeh method, 172
173, 195
199 Hewlett
Packard method, 90 High-pressure gas reservoir, 161 High pressure & high temperature (HPHT) gas wells downhole temperature and pressure monitoring technology for application, 38
40 challenges, 33
34 safety control technology for, 36
38 wireline-conveyed downhole temperature, 35
36 dynamic monitoring content and means of, 28
32 role and function of, 27
28 Homogeneous gas reservoirs, 109
116 closed gas reservoir, 104
109 pressure depletion curve, 118
119 volumetric gas reservoir, 99
104 water-drive gas reservoir, 109
116 with water-soluble gas, 116
118 Hurst modified steady flow model, 113
114 Hydrocarbon gas mixture, 72 Hydrocarbon reserves, 11

Lee-Gonzalez-Eakin method, 78
80 Linear regression, 183
185, 188
190 Linear regression method Poston
Chen
Akhtar method analysis method, 178
182 analysis procedure, 183
190 Becerra
Arteaga method, 190
195 Havlena
Odeh method, 195
199 Sun method, 199
206 Ramagost
Farshad method, 173
174 Roach method analysis method, 175 discussion on, 175
178 Louisiana gas reservoir, 278 LXF-RMP fitting method, 82
87

I Initial formation pressure, 255 In-pipe cable-hanging test, 34 Invaded water, 114 Isothermal compressibility factor, 72

Krasu gas field, 38

M M1 gas reservoir, 281 M2 gas reservoir, 282 M3 gas reservoir, 282 M4 gas reservoir, 283 M5 gas reservoir, 284 M6 gas reservoir, 287
288 M7 gas reservoir, 289 Material balance for compartmented gas reservoirs Gao method, 125
131 Hagoort
Hoogstra method, 123
125 Payne method, 119
123 Sun method, 131
137 for homogeneous gas reservoirs, 109
116 closed gas reservoir, 104
109 pressure depletion curve, 118
119 volumetric gas reservoir, 99
104 water-drive gas reservoir, 109
116 with water-soluble gas, 116
118 McCain method, 91 Multiple (two) meta-regression analysis, 295 Multiwell production decline analysis method, 240
246

318

Index

N Natural gas apparent molecular weight, 57 composition of, 53
54 density, 57 deviation factor, 59
71 equation of state of ideal gas, 54
58 gas gravity, 58 standard volume, 57 viscosity, 75
80 Carr
Kobayashi
Burrows method, 75
77 Dempsey method, 78 Lee-Gonzalez-Eakin method, 78
80 standing method, 78 Newman’s experimental data, 143
144 Newton
Raphson iterative method, 68
70 Nonhydrocarbon components, 62
64, 76 Nonlinear regression method, 255 binary regression method, 206
208 C e(p)(pi–p)BGp linear relation, 209
212 C e(p)(pi–p)BGp power function relation, 213
215 starting point of, 215
223 NS2B gas reservoir, 265

O Offshore gas reservoir, 266 Offshore high-pressure gas reservoir, 207

P Payne method, 119
124 Performance surveillance, 24 Permanent temperature, 34 Petroleum resources/reserves, 13 Possible petroleum initially-in-place, 12 Poston
Chen
Akhtar method, 172
173, 185 analysis method, 178
182 analysis procedure, 183
190 Becerra
Arteaga method, 190
195 Havlena
Odeh method, 195
199 Sun method, 199
206 Poston method, 180f Pressure coefficient boundary, 163
164

Pressure depletion, 152 Pressure monitoring, 34 Probable commercially recoverable reserves, 12 Probable petroleum initially-in-place, 12 Probable technically recoverable reserves, 12 Proved commercially recoverable reserves, 13 Proved petroleum initially-in-place, 12 Proved technically recoverable reserves, 12 Pseudo-critical parameters, 65 Pseudo-critical pressure, 61 Pseudo-critical temperature, 61 Pseudo-reduced gas compressibility chart, 74f p/Z curve method, 203

R Ramagost
Farshad method, 172
174 Real gas comparison of methods, 71 direct calculation of deviation factor, 68
70 gas formation volume factor, 75 high-molecular-weight gases, 64
68 natural gas deviation factor, 59
71 natural gas viscosity, 75
80 nonhydrocarbon components, 62
64 Remaining proved commercially recoverable reserves, 12
13 Roach linear regression method, 172
173 Roach method analysis method, 175 discussion on, 175
178 ROB43-1 gas reservoir, 278 Rock compressibility, 171, 180, 183
185

S Schilthuis steady flow model, 113
114 SE Texas gas reservoir, 280 Securities and Exchange Commission (SEC) standards, 11, 22
23 Semilog type curve matching method, 262 Sensitivity analysis cumulative rock compressibility factor, 150
151

Index

effective compressibility factor, 147 rock compressibility factor, 142
147 SI units and other units, conversion relationship between, 298
300 Society of Petroleum Engineers (SPE), 15 South Louisiana gas reservoir, 275 Stafford gas reservoir, 275 Standard volume, 57 Standing
Katz chart, 68
70 Standing method, 78 Static pressure test, 33 Stress-sensitive gas reservoirs, 215
219 Sun method, 131
137, 172
173, 199
206, 235
240

T Tank model, 113
114 Technically recoverable reserves, 12 Total petroleum initially-in-place, 11 Trial-and-error analysis method, 246
254 Type curve matching analysis method Ambastha method, 223
226 basic principles, 285
287 Fetkovich method, 227
230 Gonzalez method, 231
235 Sun method, 235
240

319

U Ultra-high-pressure gas DPR/DAK extrapolation method, 80
82 LXF-RMP fitting method, 82
87 Undiscovered petroleum initially-in-place, 11
12

V van Everdingen
Hurst unsteady flow model, 113
114 Volumetric gas reservoir, 99
104 Volumetric reserves, 188, 212

W Water-drive gas reservoirs, 109
116, 178, 181 Water/gas ratio (WGR), 262
264 Water influx, 255 Water phase compressibility, 215
219 Weighted average method, 51
52 Wellhead grease injection sealing system, 36
38 Wellhead high-precision pressure gage test, 33 Well test analysis (WTA) method, 33 Wichert
Aziz correction method, 62
64