Geomechanical Aspects of Operation of Underground Gas Storage (Springer Geology) 3031347641, 9783031347641

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Springer Geology

Vladimir Karev Yuri Kovalenko

Geomechanical Aspects of Operation of Underground Gas Storage

Springer Geology Series Editors Yuri Litvin, Institute of Experimental Mineralogy, Moscow, Russia Abigail Jiménez-Franco, Barcelona, Spain Tatiana Chaplina, Ishlinsky Institute for Problems in Mechanics, Moscow, Russia

The book series Springer Geology comprises a broad portfolio of scientific books, aiming at researchers, students, and everyone interested in geology. The series includes peer-reviewed monographs, edited volumes, textbooks, and conference proceedings. It covers the entire research area of geology including, but not limited to, economic geology, mineral resources, historical geology, quantitative geology, structural geology, geomorphology, paleontology, and sedimentology.

Vladimir Karev • Yuri Kovalenko

Geomechanical Aspects of Operation of Underground Gas Storage

Vladimir Karev Fontanars dels Alforins, Valencia, Spain

Yuri Kovalenko Nicosia, Cyprus

ISSN 2197-9545 ISSN 2197-9553 (electronic) Springer Geology ISBN 978-3-031-34764-1 ISBN 978-3-031-34765-8 (eBook) https://doi.org/10.1007/978-3-031-34765-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Introduction

Underground gas storage facilities (UGSF) is a complex of engineering structures in reservoir formations of geological structures; in mine workings; excavations-tanks created in rock salt deposits designed for gas injection, storage, and subsequent extraction including a subsurface area bounded by a mining allotment, a fund of wells of various purposes, gas collection and treatment systems, compressor shops. Russia currently has a well-developed underground gas storage system, which performs the following functions: • • • •

Regulating the seasonal irregularity of gas consumption. Storage of gas reserves in case of abnormally cold winters. Regulating the unevenness of gas export supplies. Ensuring gas supply in case of emergency situations in the Unified Gas Supply System (UGSS). • The creation of long-term gas reserves in case of force majeure circumstances of gas production or transportation. Underground gas storage facilities are an integral part of the Russian UGSS and are located in major gas consumption areas. There are 20 underground gas storage facilities within the Russian UGSS, 14 of which have been established in depleted fields. The efficiency of underground gas storage (UGS) operations is determined by the productivity of production wells in extraction cycles and their injectivity in gas injection cycles, as well as by the length of the time between workovers. The value of this period is mainly related to the intensity of bottomhole zone (BHZ) destruction which is caused by sand production, destruction, and abrasion of the subsurface equipment. Bottomhole zone destruction is facilitated by high pressure gradients on the wellbore wall during gas filtration, reservoir water breakthrough during unreasonable underbalance, changes in the direction of filtration flows during gas extraction and injection, uncontrolled sudden increase of well flow rate (for example, during well gas-dynamic studies). In addition to formation damage, the formation of local filtration channels of small size and high conductivity is possible. Local v

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Introduction

filtration channels can be identified both by geophysical methods and by the results of gas-dynamic surveys. Experience in the operation of underground gas storages has shown that in some cases the natural structural and mechanical properties of reservoir formations in UGSF cannot ensure the stability of rock in the bottomhole zone for a long time. Passive methods of sand control by reducing well productivity contradict the main purpose of UGS facilities—to ensure gas supply to consumers in the required volume; therefore, special measures are required to strengthen the bottomhole zone. At present, methods of sand control are mainly limited to the application of technologies aimed at preserving the BHZ by creating screens at the filtration surface with sand control filters and increasing the strength of the rock in the BHZ by strengthening it with various polymeric binders, resins, cements, etc. (Walaa et al. 2020; Song et al. 2022; Khamehchi et al. 2015). However, despite a wide arsenal of tools the field practice does not yet have a universal technology of bottomhole zone treatment, which provides high efficiency of sand production retention. The duration of effect of applied methods of reservoir stabilization is not sufficient and varies from several weeks to several months, and in rare cases it reaches 1 to 2 years. Therefore, increasing the duration of the strengthening effect is an important problem, the solution of which will expand the technological capabilities of wells and reservoirs in general, ensure high gas extraction rates, and reduce the volume of drilling. Thus, impacting on structural and mechanical properties of the near-wellbore rock has become an element of underground storage technology, largely determining the efficiency and quality of gas storage facilities. Despite many years of experience in operating wells with sand production, so far there is no sufficiently substantiated model of reservoir failure and mathematical description of the processes occurring in the reservoir-well system. As a result, there are no reliable criteria for establishing a rational mode of operation for an underground gas storage facility well under conditions of reservoir failure. The main hypothesis of sand production occurrence accepted by many researchers is related to the stress-strain state of the rock in the borehole zone (Gastelum 2021; Teatini et al. 2011; Zain ul et al. 2019; Jiang et al. 2019; ZainUl-Abedin and Henk 2020; Wanyan et al. 2023; Iwaszczuk et al. 2022; Liu et al. 2022; Karev 2018). Its failure occurs when the stresses in this area exceed the rock strength limit. In this regard, reduction of underbalance in boreholes is suggested as the main direction to prevent failure of the borehole zone. To improve UGS operation efficiency and reduce the number of complications during drilling and well operation it is necessary to have reliable data on geomechanical and filtration properties of rocks and their dependence on changes of stress-strain state in reservoirs during operations. The availability of such knowledge is important, given the large range of reservoir pressure changes during cyclic “extraction-injection” regimes. Particular attention should be paid to studying changes in the filtration characteristics of the host rock in order to assess the risks of deterioration of the insulating properties of the pool cap during the operation of the UGSF. For this purpose, stress-strain state of formation systems, dynamics of fracture growth, deformation processes, anthropogenic and natural changes of

Introduction

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properties of the near-wellbore zone should be studied. Inclusion of the obtained results into geomechanical and hydrodynamic models of underground gas storages will allow to study deformation and dynamic processes during cyclic operation of underground gas storages more fully and in detail. The mathematical models of elastic-plastic deformation of rocks in the bottomhole zone of UGS wells will provide calculations for specific wells and elaboration of recommendations on operation and completion of wells at UGS in conditions of sand production. The book provides an overview of current approaches and solutions to the research topic. A method of geomechanical modelling of deformation, fracture, and filtration processes in reservoirs of underground gas storages is presented in order to substantiate rational and safe operating regimes of wells. The method is based on the analysis of changes in the stress state in the reservoir occurring during the operation of an underground gas storage facility and testing of rock specimens on a true triaxial loading facility, in which real stresses occurring in the reservoir during gas injection and extraction are created. Geomechanical analysis and testing of rock specimens on a unique Triaxial Independent Load Test System (TILTS) developed at the Institute for Problems in Mechanics of the Russian Academy of Sciences has shown that changes in reservoir pressure play a key role in reservoir rock failure processes in general, and especially in the stage of maximum gas extraction. Changes in reservoir pressure result in changes in horizontal stress in the reservoir bed. That is, the effective stresses (acting on the soil skeleton of the formation) across the reservoir as a whole become unequal, even if the initial stresses in the reservoir were all-round uniform compression. Tests of the specimens based on analysis of changes in the stress-strain state in the BHZ during injection and gas extraction allow us to determine at what reservoir pressures creep strains occur in the rock ultimately leading to its failure and sand production. The above methodology has been applied to the conditions of a specific underground gas storage facility—UGS №5. Permissible intervals of reservoir pressure changes were determined, which do not cause rock destruction in the reservoir and do not lead to sand production without application of special filters and rock fixing in the bottomhole zone by binding agents. The book also deals with improving the productivity of underground gas storage wells, a decline in which is caused by a reduction in the permeability of rocks in the bottomhole zone of the reservoir during operation. Deterioration of filtration properties of rocks in the bottomhole formation zone occurs mainly at the stage of gas injection as a result of clogging of filtration channels by solid mechanical particles and compressor oil particles contained in the injected gas. As a result, a polluted screen is created around the well impairing hydrodynamic coupling with the reservoir. The urgency of the topic is caused by necessity of application of low-cost methods of well operation intensification with long-term effect due to restoration of rock permeability in bottomhole formation zone. To solve these problems the technologies based on geomechanical approach seem to be the most perspective due

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to their effectiveness, relatively low cost, and ecological safety. One of these technologies is the method of directed reservoir unload developed at the Institute for Problems in Mechanics of the Russian Academy of Sciences. The book examines the conditions for applying this method to wells in underground gas storage facilities taking into account the specifics of well operation—cyclic gas injection and extraction. It considers the question at what stage it is most effective to use the directed reservoir unload method—during gas injection or extraction. Physical modelling of directed reservoir unload method implementation for UGS №5 conditions was performed at the TILTS. The results suggest that there is a real possibility of improving the efficiency of UGS well operation by using the method of directed reservoir unload. In the book, the main emphasis is on solving the problem of rock fracture in the bottomhole reservoir zone based on physical modelling at the TILTS facility. Geomechanical modeling also includes mathematical modeling of processes occurring in a rock mass. These two approaches should develop together and complement each other. To solve the problem of stability of UGS wellbores on the basis of mathematical modeling, it is necessary to know the strain, strength, and filtration parameters of the medium included in the calculations. Moreover, these characteristics must be determined under the conditions of stresses actually acting in UGS reservoirs. This can be done on the basis of physical modeling of deformation processes at the TILTS facility. On the other hand, in order to construct physical modeling programs on the TILTS, it is necessary to be able to calculate the stresses that arise in the reservoir, taking into account the three-dimensionality of stress fields, plastic deformation of the rock, and its creep based on mathematical modeling. References Gastelum JIR (2021) Geomechanical analysis of injection and withdrawal on underground natural gas storage in Mondarra Field, Western Australia. In: Proceedings of the 2021 Asia Pacific Unconventional Resources Technology Conference Iwaszczuk N, Prytula M, Prytula N, Pyanylo Y, Iwaszczuk A (2022) Modeling of gas flows in underground gas storage facilities. Energies 15(19):7216 Jiang G, Qiao X, Wang X, Lu R, Liu L, Yang H, Su Y, Song L, Wang B, Wong T (2019) GPS observed horizontal ground extension at the Hutubi (China) underground gas storage facility and its application to geomechanical modeling for induced seismicity. Earth Planet Sci Lett 530(6318):115943 Karev V (2018) Geomechanical approach to improving the efficiency of the operation of underground gas storages. In: Physical and mathematical modeling of earth and environment processes, pp 150–158 Khamehchi E, Ameri O, Alizadeh A (2015) Choosing an optimum sand control method. Egypt J Pet 24:193–202 Liu T, Li Y, Ding G, Wang Z, Shi L, Liu Z, Tang X (2022) Simulation of pore space production law and capacity expansion mechanism of underground gas storage. Pet Explor Dev 49(6):1423–1429

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Song R, Zhang P, Tian X, Huang F, Li Z, Liu J (2022) Study on critical drawdown pressure of sanding for wellbore of underground gas storage in a depleted gas reservoir. Energies 15: 59134 Teatini P, Castelletto N, Ferronato M, Gambolati G, Janna C, Cairo E, Marzorati D, Colombo D, Ferretti A, Bagliani A, Bottazzi F (2011) Geomechanical response to seasonal gas storage in depleted reservoirs: a case study in the Po River basin, Italy. J Geophys Res Atmos 116:F02002 Walaa I, Haiduwa N, Magoti AJ, Dotto G (2020) Advanced technologies used in sand control completion. In: Proceedings of the International field exploration and development conference 2018, pp 1943–1964 Wanyan Q, Xu H, Song L, Zhu W, Pei G, Fan J, Zhao K, Liu J, Gao Y (2023) A novel performance evaluation method for gas reservoir-type underground natural gas storage. Energies 16(6):2640 Zain ul Abedin M, Henk A, Rudolph T (2019) Coupling of flow and geomechanical simulations for short-term underground gas storage - a case study from the Bavarian Molasse Basin. DGMK/ÖGEW-Frühjahrstagung, Fachbereich Aufsuchung und Gewinnung Zain-Ul-Abedin M, Henk A (2020) Building 1D and 3D Mechanical Earth models for underground gas storage—a case study from the Molasse Basin, Southern Germany. Energies 13(21):5722

Contents

1

2

3

Analysis of Existing Approaches to Solving the Problem of Sand Production During the Operation of UGS Wells . . . . . . . . . . . . . . . . 1.1 Features of Mechanical Behaviour of UGS Reservoir Rocks . . . . . 1.2 Factors Affecting Fracture of Reservoir Rocks in the Near-Wellbore Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Mathematical Modelling of the Reservoir Stress-Strain State . . . . . 1.4 Mathematical Models to Describe Rock Fracture and Sand Production in UGS Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Facilities for Studying the Deformation and Strength Properties of Rocks and Geomechanical Modelling . . . . . . . . . . . . . . 2.1 Existing Experimental Equipment for Studying the Deformation and Strength Properties of Rocks . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Installations Using Six Rigid Plates to Load the Specimen in Three Orthogonal Directions . . . . . . . . . . . . . . . . . . . . . 2.1.2 Installations with Six Flexible Load Plates . . . . . . . . . . . . . 2.1.3 Mixed Type Installations . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 True Triaxial Independent Load Test System (TILTS) . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methodology for Performing TILTS Experiments to Study the Deformation, Strength and Filtration Properties of Rocks . . . . . . . . 3.1 Experimental Procedure for Measuring Longitudinal Wave Velocities in the Test Specimens . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Methodology for Determining Strain and Strength Properties of Rocks (Triaxial Tests) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Methodology for Experiments to Simulate Sand Production Using Core Specimens with Holes in a Modified Hollow-Cylinder Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 6 7 8 9 9 10 11 11 12 15 17 17 18

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3.4

Determining the Dependence of Rock Specimen Permeability on the Type and Level of Applied Stresses . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

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Results of TILTS Testing of the Deformation, Strength and Filtration Characteristics of Reservoir Rocks at Underground Gas Storage Facilities Based on Triaxial Tests . . . . . . . . . . . . . . . . . 4.1 Test Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 UGS №4. Specimen KR-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Results of TILTS Tests of Deformation and Strength Properties of Rock Specimens Based on Triaxial Tests . . . . . . . . . . . . . . . . . 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geomechanical Modelling of the Stress-Strain State in UGS Reservoirs During Cyclic Changes in Reservoir Pressure and Downhole Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Main Factors Affecting the Stress-Strain State in the Vicinity of UGS Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Stress State in UGS Reservoirs Under Equal-Component Natural Stress State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Stress Changes in the Bottomhole Formation Zone When Underbalanced and Repressed at the Bottomhole . . . . . . . . 5.2.2 Stress-Strain State of the Reservoir Under Cyclic Reservoir Pressure Changes and Equal-Component Natural Stress State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Stresses Occurring in the Near-Wellbore Zone of the Reservoir During Changes in Reservoir Pressure in an Equal-Component Natural Stress State . . . . . . . . . . . . . . . 5.3 Stress State in the Near-Wellbore Zone of the Reservoir Under Unequal Natural Stress State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Effect of Reservoir Pressure Changes on the Stress State in UGS Reservoirs Under Unequal Natural Stress State . . . 5.3.2 Stresses Occurring in the Near-Wellbore Zone of the Reservoir During Changes in Reservoir Pressure in an Unequal Natural Stress State . . . . . . . . . . . . . . . . . . . . . . . 5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Modelling Programs of Rock Deformation and Fracture in the Near-Wellbore Zone of USG During Gas Injection and Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Equilibrium Natural Stress State in the Reservoir . . . . . . . . . . . . . 6.1.1 Specimen Load Programs for Physical Modelling of Drawdown and Repressions on Bottom Hole Without Regard to Reservoir Pressure Changes . . . . . . . . . . . . . . .

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6.1.2

6.2

6.3

6.4 7

Physical Modelling Programs at the TILTS of Deformation and Filtration Processes in the Bottomhole Zone Under the Action of Alternating Loads Arising from Cyclic Changes in Reservoir Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Modelling Programs for Unequal Natural Stress State . . . 6.2.1 Vertical Uncased Well . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Horizontal Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Modelling Programs at the TILTS of the Deformation and Filtration Processes Taking Place in the Bottomhole Zone of UGS Wells Under Cyclical Changes in Reservoir Pressure for Specific UGS Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 UGS №1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 UGS №2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 UGS №3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 UGS №4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 UGS №5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Results of Physical Modelling of Deformation and Filtration Processes in the Bottomhole Zone of UGS Wells During Gas Extraction and Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Results of Physical Modelling at the TILTS of Downhole Repressions and Underbalances Without Regard to Changes in Reservoir Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Test Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Test Results of KR-1 Specimen from UGS №4 . . . . . . . . . . . . . . . 7.3.1 Test 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Test 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Results of Physical Modeling of Deformation and Filtration Processes in the Bottomhole Zone of UGS Wells Taking into Account Cyclic Changes in Reservoir Pressure . . . . . . . . . . . . . . . 7.4.1 KS-5 Specimen Test Results from the UGS №2 . . . . . . . . . 7.4.2 Test Results of Rock Specimen AR-1.1 from UGS №5 . . . 7.4.3 Results of Physical Modeling of Deformation Processes in the Bottomhole Zone During Gas Injection and Extraction for the UGS №5 . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Rationale for the Possibility of Using the Directed Unload Reservoir Method (DUR Method) to Increase Rock Permeability in the Bottomhole Zone of UGS Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Test Methodology for Physical Modeling of Deformation Processes Using the TILTS on the Bottomhole of the Well During DUR Method Implementation . . . . . . . . . . . . . . . . . . . . . 8.2 Specimen Load Programs at the TILTS for Physical Modeling of the Deformation Processes in the Vicinity of a Horizontal Borehole During the Implementation of the DUR Method . . . . . . . 8.3 Results of TILTS Tests of Core Material from the UGS №5 to Determine whether the Permeability of Reservoir Rocks Can Be Increased by Implementing the DUR Method . . . . . . . . . . . . . . . . 8.3.1 Specimen AR-7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Specimen AR-10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Specimen AP-11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Specimen AP-12.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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135 136 137 139 139 141 141

Justification of Predictive Recommendations for Maximum and Minimum Allowable Drawdowns, Well Flow Rates and Prevention of Sand Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Chapter 1

Analysis of Existing Approaches to Solving the Problem of Sand Production During the Operation of UGS Wells

1.1

Features of Mechanical Behaviour of UGS Reservoir Rocks

The mechanical behaviour of reservoir rocks in UGS during their operation has a number of peculiarities in comparison with the behaviour of reservoir rocks in hydrocarbon fields during their development due to differences in both rock properties and well operation modes. The features of the collector’s properties include, first and foremost, the following. • The permeability of reservoir rocks is usually extremely high (up to several Darcies) and the porosity is also quite high. When choosing depleted gas fields for use as UGS facilities, this property is one of the determining ones; • The rock strength of UGS reservoirs is usually much lower than the average strength of hydrocarbon reservoirs. Features of the operating modes should include: • Significant reservoir pressure variations on a reservoir scale (up to tens of percent of average reservoir pressure) in the states of minimum and maximum saturation; • Cyclicality of injection and extraction stages determined by seasonality of gas consumption and other factors; • Very low pressure drawdown and gas injection overpressure (usually less than a few percent of the average reservoir pressure), which do not cause appreciable pore pressure gradients in near-wellbore areas; • Despite very low pressure drawdown and gas injection overpressure and consequently very low pressure gradients, the high permeability of the reservoir rocks results in high values of gas filtration rates.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Karev, Y. Kovalenko, Geomechanical Aspects of Operation of Underground Gas Storage, Springer Geology, https://doi.org/10.1007/978-3-031-34765-8_1

1

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1

Analysis of Existing Approaches to Solving the Problem of Sand. . .

In order to determine parameters when modelling gas extraction and injection processes, it is very important to have an understanding of the change in stress state in the reservoir rock during cyclic changes in reservoir pressure.

1.2

Factors Affecting Fracture of Reservoir Rocks in the Near-Wellbore Zone

Based on field data, the causes of reservoir failure have been discussed in numerous publications and monographs (Oñate and Rojek 2004; Fadeev 1987; Kaledin et al. 2002). Sand production is known to be one of the most common technogenic consequences of gas field operations. The works (Timurziev 2008; Morita 1994; Morita et al. 1989) summarizing the experience of combating sand production from the Cenomanian deposits of the Medvezhee and Urengoi fields are interesting in this respect. It should be borne in mind that the intensification of sand production in these wells coincided in time with the pulling of water and a decrease in reservoir pressure below the hydrostatic pressure. Similar circumstances are typical of UGS in aquifers. The conclusions of the authors (Timurziev 2008) are interesting, of which the following should be noted: 1. Bottomhole zone reservoir failure occurs as a result of the following set of factors: • An increase in gas moisture content with pressure drop in the reservoir conditions, • The presence of a static or dynamic water level in the filter interval, • An increase in effective rock pressure with the fall in reservoir pressure. 2. Reservoir stability depends on the total gas production of the well. 3. The least stable interlayers are inherently the most productive and stressed. 4. There is no direct relationship between reservoir failure and underbalance in the 0–0.6 MPa range (the accepted initial allowable underbalance is 0.5–0.6 MPa). 5. An appearance at the wellhead of mechanical impurities during short-term regime tests is not a direct consequence of a reservoir failure at a given time: it may be due to filtration rates and upward flow rates in the lift string sufficient to clear perforation channels and caverns of previous failure products, picking up and lifting impurities from the borehole face. 6. To determine the underbalance of reservoir failure, long-term observations of the dynamics of mechanical impurities appearance, long-term multi-cycle studies with gas deflation into the atmosphere are required. Given the acute facial and filtration variability of reservoir rocks and the ambiguity of their stress conditions (as a result of irregular drainage of reserves and watering of the reservoir), such studies must be carried out on practically every well. 7. Until now, there are no reliable and acceptable practical criteria for establishing the optimal well operation mode in conditions of violation of the stability of the reservoir bottomhole zone, and its destruction, the presence of a bottomhole

1.3

Mathematical Modelling of the Reservoir Stress-Strain State

3

fluidized or sandy plug, accumulation or export of fluid and mechanical impurities. Long-term field observations and an analysis of possible causes of sand production are also summarised in a review (Morita 1994). The authors reached the following conclusions: (a) Collapses of the caverns can be caused by too much pressure drop in the well; (b) Under certain conditions, sand production is caused by too high a flow rate in the well; (c) Water breakthrough in the borehole leads to an increase in sand production; (d) In some cases, a short-term increase in sand production is observed when the flow rate changes; (e) In some cases, the sand production is not stopped even when the flow rate decreases. In extreme cases, the well is completely filled with sand. Effects of cyclic load, tensile stresses, residual strains and capillary forces effect on adhesion are also of great importance.

1.3

Mathematical Modelling of the Reservoir Stress-Strain State

The failure of the reservoir is due to a breach of the strength conditions expressed in terms of the stress or strain tensor of the rock. The study (Morita et al. 1989) is based on analytical solutions for a poroelastoplastic strengthening material. As a result, two extreme situations or two types of reservoir failure are clarified: in the first case, the failure is mainly caused by filtration loads (this situation can be controlled by changing the flow rate); the second type of failure is mainly caused by high stresses in the reservoir. To prevent fracture in this case, a sufficiently high back pressure must be created in the well. The mathematical modelling of the mechanical behaviour of a massif under various kinds of influences is being realized by solving boundary value problems of elasticity, visco-elasticity, elastic-plasticity, etc. From the methodological point of view, elastic models are of a great importance; they are the easiest to analyze and, at the same time, they may serve as a basis for developing models which take into account inelastic properties. In conditions, where the constants determining rock deformation and strength are either not known or are approximated, elastic models provide at least a qualitative picture of the stress distribution, taking into account the heterogeneity of the rock mass. Mathematical methods for studying the stress-strain state (STS) parameters of rocks can be divided into two sections: exact analytical methods and approximate numerical methods. Because of great mathematical difficulties, exact analytical methods are applicable only for a limited range of simple models, in particularly, for areas in the vicinity of both perforation holes and open bottom of vertical and

4

1 Analysis of Existing Approaches to Solving the Problem of Sand. . .

horizontal wells. Approximate numerical calculations have great versatility, in addition, they can be performed with the accuracy provided by computational capabilities. An overview of both groups of mathematical methods is available in special literature (Oñate and Rojek 2004; Fadeev 1987; Kaledin et al. 2002). It is proved, that in the case of large number of interfaces (more than 3–4), their uneven surface, local heterogeneities and anisotropy of properties, the finite element method (FEM) is optimal in terms of accuracy and computational time. There is quite an extensive literature about the finite element method, well known to specialists, where features of FEM are described for various problems, the physical processes in which can be described by differential equations. The essence of the finite element method is as follows: a boundary value problem for the elastic potential of a rock mass, being under the action of its own weight with initial and boundary conditions, is considered as a Lagrange functional. The computational domain is divided into a large number of small elements: arbitrary hexagons, tetrahedrons or triangles (depending on the dimensionality of the domain), within which the potential varies linearly. Ultimately, the minimization of the functional reduces to solving a system of linear algebraic equations with a large number of unknowns—nodal potentials (displacements). When choosing the method for solving such a system, the ribbon nature of the matrix is taken into account. After solving the system and determining the displacement components according to Cauchy conditions, the deformations are calculated. Stresses at the centres of gravity or at the element nodes are calculated from the deformations based on the generalized Hooke’s law. The geometry of the model, the elastic properties and density of the rock, the kinematic relationships and the forces are the input data for the calculation (Timurziev 2008). An example of an approach to integrating mathematical models of filtration and geomechanics is demonstrated in Tortike and Farouk Ali (1993). In most filtration models, the response of the solid skeleton to changes in pore pressure is accounted for by introducing a porosity-pressure relationship. And this dependence is considered to be reversible in time. However, in the case of weak, poorly cemented sandstones with low adhesion, a linear (and even non-linear) elastic model does not adequately reflect the material behaviour. These rocks rather follow the laws of the mechanics of granular media (soils), reflecting the accumulation of irreversible plastic deformations. Moreover, depending on conditions, plastic deformation may be accompanied by dilatancy phenomenon. The author considers various rheological models: the step elastic-plastic model, a combination of the hypoelastic (step elasticnonlinear) model and the dilatancy relation. The hypoelastic model gives good results in some cases of monotonic load variations, but it is not applicable in the case of cyclic load. The elastic-plastic body model is most closely related to the real deformation processes but leads to more complicated computational problems compared to the hypoelastic model. A section devoted to analysis of the influence of inelastic behaviour of porous material on filtration is available in a well-known monograph (Barenblatt et al. 1972). Referring to K. Terzaghi, the authors consider “plastic” deformation of real sandstones as a consequence of partially irreversible disturbance of intergranular cement bonds. To account for plasticity, the porosity is represented as a function of

1.3

Mathematical Modelling of the Reservoir Stress-Strain State

5

pressure and the first invariant of the effective stress tensor. The load history is taken into account. A single load-unload cycle is considered. The effective stresses increase during the initial stage of oil field development and decrease during the second stage when the reservoir is flooded and pore pressure rises. To summarise the review of works on the subject under study, the main problems encountered in creating mathematical models of reservoir deformation under the influence of filtration loads can be highlighted: 1. Selection of defining relations to describe stress-strain relation for geomaterials comprising productive formations and host massifs of underground gas storages. The most realistic is the elastic-plastic body model, with the Mohr-Coulomb criterion as a condition for transition from the elastic to the plastic state. 2. Filtration and geomechanics model coupling. In the simplest case, when porosity and permeability are represented as unambiguous pressure functions, filtration and geomechanics problems are completely separated. Moreover, the former can be solved independently of the latter. Once pressures and saturations have been calculated, the geomechanics problem can be solved and the stress-strain dynamics simulated. In more complete models, permeability is represented by the effective stress history functional, which, in turn, is determined by changes in pressure distribution throughout the filtration domain. 3. Analytical and semi-analytical solutions are obtained for simple shape regions and for particular load cases (spherical or axial symmetry). All considered solutions refer to stationary case only. Exceptions are some problems of well investigations under elastic-plastic regime (see Barenblatt et al. 1972). Influence of cyclic load in considered analytical solutions is not taken into account. 4. Numerical models are based mostly on finite element method. Although some works use finite difference method and some propose combination of finite difference method for filtration equations and finite element method for stressstrain calculations. 5. Real accuracy of modern models of critical downhole pressure (or critical pressure drawdown) prediction, based on combined filtration and strength accounting, by estimates (Morita et al. 1989), can be brought to 5% of reservoir pressure, provided accuracy of initial data. Irreducible error is connected with approximation of representation of physics of processes, inaccuracy of experimental data, approximation of numerical solution. However, calibrating the model to actual reservoir fracture and sand production conditions in a particular area can improve accuracy. The method of calculating the stress state in the vicinity of a spherical cavity (perforation) and a cylindrical cavity (open-hole) using an elastic-plastic body model is promising for establishing the parameters of the technological mode of well operation and the permissible operating range without rock destruction during gas-hydrodynamic investigations. This methodology is described below. In order to complete the mathematical model, special laboratory studies of the mechanical properties of the rocks comprising the reservoir are required. The state of the work in other areas and proposals are presented in the section conclusions.

6

1.4

1

Analysis of Existing Approaches to Solving the Problem of Sand. . .

Mathematical Models to Describe Rock Fracture and Sand Production in UGS Reservoirs

The field data reviewed by the authors, as well as the results of physical and mathematical modelling, suggest several scenarios for rock failure and sand production. Combining them, a consolidated scenario is obtained, represented by the following sequence of events. Initially, the natural formation consists of a solid porous matrix, fluid saturating it and free fine rock particles on the pore walls. The equilibrium of the system around the wellbore is disturbed during the process of drilling the well and cementing the production casing. This results in mechanical destruction of the rock, change in stress state, and infiltration of drilling mud and cement slurry, as well as their suspended solids, into the formation. As a result, the strength and filtration properties of the porous matrix, the composition of saturating fluids and the concentration and composition of suspended solids change in some vicinity of the well. A so-called zone of impaired properties emerges. Additional changes in the system occur during perforation, among which is the formation of perforation channels surrounded by a layer of compacted rock of reduced permeability. The gas extraction is accompanied by changes in pore pressure and rock stresses in the formation. The filtration loads in the interwell space caused by fluid movement through the most productive interlayers are not great; the main influence is exerted by pressure changes in pores and pressure differences between high- and low-permeability interlayers. If, in this case, elastic limit is reached in individual interlayers, their constituent material starts to enter the plastic state with irreversible gradual accumulation of plastic deformations. The rock in the main interwell part of the formation will after a certain number of cycles, due to displacements limitation, change to a new elastic state, different from the initial one, having undergone hardening. The main changes occur in the vicinity of the wells. Here, the strongest changes in rock stress state are observed. And at the boundary of perforation holes rock displacements are not limited and under certain conditions its integrity can be broken and destroyed rock can be carried out of the reservoir. It is in the vicinity of perforation holes that the maximum filtration rates are observed and, consequently, the maximum probability of movement of suspended free solids, the concentration of which may increase due to detachment of particles from the matrix and their transition into the mobile state. This is facilitated by the weakening of the cement bonds between the grains and swelling of the clay particles due to enhance in moisture content. The accumulation of particles in certain areas of the pore space leads to a decrease in permeability and an increase in the filtration load on the matrix. The greatest deviations from the linear filtration law are achieved in the layers bordering the perforation hole, what contributes to stress concentration, particle detachment and the transition of rock into a plastic state. Particle removal leads to the growth of caverns and reduction of loads. However, under certain conditions the

1.5

Conclusions

7

vault of the cavern can become unstable and then a certain volume of sandstone will collapse. The resulting voids can be filled by the surrounding plastic rocks. Thus, to summarise the material considered, a number of limit conditions can be identified which determine the change in state of the system. Regarding the state of mathematical modelling, one can agree with the conclusions of (Morita et al. 1989): “None of the previously proposed methods can explain all the facts observed in the oil field, as each of them takes into account only one aspect of the problem”. It is also worth agreeing with the research program outlined in this chapter, which aims to improve the reliability of methods for predicting sand productions. The authors propose to base this work on three fundamental principles: • Revision of factors affecting the stability of caverns through parametric analysis; • Parameter sensitivity study using a simple analytical method; • Quantitative research using a realistic numerical model. We should mention (Susokolov 1984). The two-parameter, dilatantion model of V.N. Nikolayevsky was used in setting the problem. Numerical solution is obtained by finite element method. The local (near the wells) and global (on the scale of gas deposit diameter) changes of the stresses are modelled. The model of geomaterial behaviour is adapted from the results of the Japanese researchers’ experiments. The internal friction coefficient and dilatancy rate (positive and negative) are determined.

1.5

Conclusions

1. Methods for predicting and controlling rock failure and sand production based on mathematical modelling, field and laboratory tests have practical applications and work to improve them is ongoing. The theoretical basis of the methods is provided by models of rock deformation and fluid filtration. Deformation models, as a rule, take into account plastic, dilatant behaviour of strengthening material. However, there is no generally accepted model (defining relations). The importance of cyclic load (stopping and starting of a well) is noted. Computer models are based on application of finite element method. Analytical models are used for qualitative analysis. Experimental base provides, as a minimum, standard triaxial studies with reproduction of different load programs. 2. Based on the systematisation and analysis of field material and literature sources, the key processes leading to reservoir failure and sand production from field and UGSF wells are highlighted. 3. The mathematical models proposed by various authors for investigating the stress-strain state of the reservoir and determining filtration loads are analysed. It is also necessary to develop models of accompanying processes of particle transfer in pore space and physico-chemical influences leading to changes in filtration and reservoir strength properties.

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Analysis of Existing Approaches to Solving the Problem of Sand. . .

4. Under the considered conditions (availability of proven mathematical models) the decisive role belongs to the computational experiment as the cheapest, most flexible and fastest means of analysing the conditions of occurrence and development of sand production. 5. A special effort is advisable to collect experimental data on the strength properties of the rocks. 6. The review provides a concrete plan for prioritising the development of a methodology for predicting rock stress-strain state and sand production. • The primary use of experimental complexes is envisaged for predicting stressstrain, rock failure, sand production and remedies; • In terms of creating physical models, it seems necessary to use facilities that allow modelling of rock pressure, local stresses in the vicinity of the well, pore pressures, underbalances and flow rates, as well as reservoir heterogeneity.

References Barenblatt GI, Yentov VM, Ryzhik VM (1972) Theory of nonstationary filtration of liquid and gas (in Russian), p 472 Fadeev AB (1987) Finite element method in geomechanics (in Russian). Nedra, Moscow, p 221 Kaledin VO, Burnysheva TV, Lastovetskii VP (2002) Mathematical Modeling of stress fields for the problems of prospecting and mining the hydrocarbon deposits. J Min Sci 38:544–557 Morita N (1994) Field and laboratory verification of sand-production prediction models. SPE Drill Complet 9:227–235 Morita N, Whitfill DL, Fedde OP, Lovik TH (1989) Parametric study of sand-production prediction: analytical approach. SPE Prod Eng 4:25–33 Oñate E, Rojek J (2004) Combination of discrete element and finite element methods for dynamic analysis of geomechanics problems. Comput Methods Appl Mech Eng 193:3087–3128 Susokolov AI (1984) Development of calculation methods of stresses in reservoir of UGSF for justification of parameters of well bottomhole zone strengthening (in Russian). Ph.D. in technical sciences. Gubkin Russian State Oil and Gas University Timurziev AI (2008) Fracture prediction technology based on three-dimensional geomechanical and kinematic model of fractured reservoir (on the example of the Bely Tigre field) (in Russian). Geofizika 3:41–60 Tortike WS, Farouk Ali SM (1993) Reservoir simulation integrated with geomechanics (in Russian). J Can Pet Technol 35(5):28–37

Chapter 2

Experimental Facilities for Studying the Deformation and Strength Properties of Rocks and Geomechanical Modelling

In Chap. 1, it was noted that the key point of building geomechanical models for studying the stress-strain state of UGS reservoirs is to determine the deformation and strength characteristics of their constituent rocks. And it is necessary to study them at stresses which really arise in formation during operation of a deposit. The second important point is the possibility to carry out physical modelling of deformation and rock fracture processes in the vicinity of boreholes under conditions of real three-dimensional stress states in laboratory conditions using special experimental equipment. The use of facilities that allow modelling of rock pressure, local stresses in the vicinity of the well, pore pressures, underbalances and flow rates as well as reservoir heterogeneity seems necessary in this regard.

2.1

Existing Experimental Equipment for Studying the Deformation and Strength Properties of Rocks

The study of the mechanical properties of rocks under laboratory conditions involves a whole range of investigations in which tests to determine their basic deformation and strength properties occupy a special place. In recent decades, rock fracture testing methods have been developed. The simplest type of rock testing is uniaxial compression testing. These tests are carried out on compression testing machines (presses). The test records the deformation and strength of the specimen and determines the mechanical properties of the rock—modulus of elasticity, Poisson’s ratio and uniaxial compressive strength. The rocks in a rock massif are in a volumetric state of stress. Therefore, in order to study their mechanical properties to the fullest extent possible, they are tested in special laboratory apparatuses.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Karev, Y. Kovalenko, Geomechanical Aspects of Operation of Underground Gas Storage, Springer Geology, https://doi.org/10.1007/978-3-031-34765-8_2

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Experimental Facilities for Studying the Deformation and. . .

At present, experimental axisymmetric compression facilities, based on the Carman principle, are mainly used to determine the elastic and strength properties of rocks. The specimens to be tested are cylinders 70–80 mm long and 30–40 mm in diameter. The specimen is compressed along its axis by means of a press, while pressure along the lateral surface of the specimen is applied hydraulically by means of a hydraulic multiplier. Thus systems based on the Karman principle allow for creating conditionally triaxial stress states in the test specimen, which are characterized by the following relation of principal normal stresses σ 1 ≥ σ 2 = σ 3. It is possible to load specimens independently only in two axes in systems based on the Karman principle. This is sufficient to determine the deformation and strength properties of rocks assuming they are isotropic. It is difficult to determine these parameters using a Karman system for substantially anisotropic rocks; specimens must be cut at an angle to the bedding and the corresponding calculations must be carried out. However, it is the anisotropy of the elastic and strength properties of rocks that is the main cause of a number of serious problems encountered when drilling and operating deviated and horizontal wells. In addition, it is important to note that, since widespread installations using the Karman scheme carry out biaxial load of the specimen, these installations do not allow reproducing real stress-strain states occurring in the formation which are essentially three-dimensional. Meanwhile, it is obvious, that in order to solve geomechanical problems, it is necessary to be able to directly measure strain and strength properties of anisotropic rocks under true three-axis triaxial load. Attempts have been made for a long time to develop systems that allow to produce so-called ‘true triaxial loading’ on rock specimens. For triaxial testing cubic rock specimens with an edge of 40 mm or more are usually used. The main requirements for triaxial compression facilities are the ability to generate triaxial independent load of the specimen and to ensure homogeneity of the stress and strain fields in the specimen. Current installations for triaxial independent load of specimens can be divided into three types:

2.1.1

Installations Using Six Rigid Plates to Load the Specimen in Three Orthogonal Directions

The idea of using six sliding plates to load cubic specimens in three orthogonal directions (Hambly 1969; Pearce 1971; Wood 1974; Airey and Wood 1988) was realised in a number of installations. The same idea was later developed and used in other installations (Sture 1979; Arthur 1988; Ibsen and Praastrup 2002; Wood 2008; Alexeev et al. 2004). The main advantages of test systems using the principle of sliding load plates are.

2.1

Existing Experimental Equipment for Studying the Deformation and. . .

11

• Displacements caused by each of the load plates are the same across the specimen surface due to the stiffness of the plates; • There is no mutual interference of the specimen faces; • The three principal stresses can be changed arbitrarily and simultaneously during the experiment. The main disadvantage of this type of test machine is difficulty to eliminate friction between the contacting plates as well as between the plates and the specimen which introduces some inaccuracy into the measurement results. For this reason, six sliding load plates are hardly ever used in triaxial independent load setups today. Setups with three pairs of rigid load plates with gaps between plates in each pair of contiguous plates are more widely used (Matsuoka and Sun 1995; Matsuoka et al. 2002).

2.1.2

Installations with Six Flexible Load Plates

These machines use flexible diaphragms filled with fluid under pressure to load a cubic specimen in three orthogonal directions. This principle of a specimen loading was first proposed by Bell (1965). The point of loading specimens with liquidfilled rubber membranes is that the friction between the load surface, i.e. the rubber membrane and the surface of the specimen is relatively low. However, this type of installation has the significant disadvantage that the rubber load diaphragm covers only 60% of the specimen face. This makes it impossible to ensure homogeneous deformation inside the specimen during the test, especially in the vicinity of the edges of the cube. In addition, it is not possible to produce large deformations in the specimen during the experiment, because the end zones of the specimen begin to affect mutually to each other. Nevertheless, such installations and their modifications are used in a number of research centres (Sture and Desai 1979; Sivakugan et al. 1988; Reddy et al. 1992; Wawersik et al. 1997; Mandeville and Penumadu 2004; Prashant and Penemadu 2004; Choi et al. 2007).

2.1.3

Mixed Type Installations

In 1969, Green suggested the use of mixed installations in which the cube specimen is loaded in two directions by rigid plates and in the third direction by flexible rubber membranes (Green 1969). A gap is left between the rigid plates to prevent them from interacting. The idea was to try to combine the advantages of the two previous types of triaxial independent load machines, while at the same time getting rid of the inherent disadvantages. However, the main drawback remains in all these setups— the pressure elements do not cover the entire surface of the specimen which inevitably leads to non-uniformity of the strain and stress fields inside the specimen

12

2 Experimental Facilities for Studying the Deformation and. . .

during the experiment. Such installations have been created by a number of researchers (Green 1971; Hayno et al. 1999; AnhDan et al. 2006; Shapiro and Yamamuro 2003; Alshibli and Williams 2005). However, they have not become widespread because they have the same drawbacks, although to a lesser extent, and in manufacturing and operation they turned out to be much more complicated and unreliable. Thus, we can conclude that, at present, there are practically no installations in the world that allow independent triaxial rock specimens while ensuring homogeneity of strain and stress fields in them. The available installations have two main drawbacks—either the pressure elements do not cover the whole surface of the specimen which inevitably leads to non-uniformity of the strain and stress fields inside the specimen during the experiment, or it was difficult to get rid of friction between the contiguous plates, as well as between the plates and the specimen which makes it difficult to obtain reliable experimental data. These problems were largely solved with the creation of the Triaxial Independent Load Test System (TILTS) at the Institute for Problems in Mechanics of Russian Academy of Sciences.

2.2

True Triaxial Independent Load Test System (TILTS)

In order to justify rational well operation modes for specific conditions of the field in study, and reduce the risk of sand production, in addition to mathematical models of deformation and destruction of rocks of reservoirs, it is needed physical modelling of deformation and filtration processes in the reservoir bottomhole zone. The unique TILTS (Karev and Kovalenko 2013) of the Institute for Problems in Mechanics of the Russian Academy of Sciences, Fig. 2.1, is designed to study the deformation, strength and filtration characteristics of rocks in oil and gas, ore and coal fields. It allows to load rock specimens, which are cubes with 40 or 50 mm edges, independently along three axes and to measure rock permeability during load. The TILTS belongs to the class of electro-hydraulic test machines with an automated control system. The test system is a complex consisting of a true triaxial load power unit, an oil pumping station (OPS), an automatic control system, a measuring and information system, and an automated permeability measuring system. The power unit is a thick-walled cube (the wall thickness is 80 mm). Loading cylinders are mounted on the walls of the cube along each of its three axes. The three loading axes are mutually perpendicular and have a common point of intersection. Dynamometers are mounted on the inner rod of each double-rod hydraulic cylinder to measure the force exerted by the hydraulic cylinder. The pressure and support plates are mounted on precision-engineered flat roller bearings to ensure that the plates can move perpendicular to the power frame axis with low friction under relatively high load. Three pairs of plates form a load unit in which loads are applied over the entire surface of the cubic specimen. Load and supporting plates are fitted with sets of interchangeable tips to allow for testing of cube shaped specimens with

2.2

True Triaxial Independent Load Test System (TILTS)

13

Fig. 2.1 Triaxial Independent Load Test System (TILTS)

40 mm and 50 mm edges. Inductive displacement sensors are fitted to the tips of the pressure plates. The stroke of each pressure plate is 12 mm, the idle stroke (full extraction of the pressure plate) is 130 mm. The tips of the pressure plates form a working chamber in which the test rock specimen is placed. The peculiarity of the TILTS unit is that the pressure plate tip’s working surface (the surface that is in contact with the specimen) is larger than the size of the specimen face. The pressure plates are set so, that the active pressure plate, by moving axially and deforming the specimen, moves the two adjacent pressure plates (one active and one supporting plate) in the same direction. The kinematics of motion of the three pressure plate pairs is shown schematically in Figs. 2.2 and 2.3. Figure 2.2 shows schematically the position of the plates in the initial unloaded state and Fig. 2.3 shows the position of the plates when the specimen is deformed. This design ensures that loads are applied uniformly throughout the specimen during the entire deformation process, including the fracture stage, which greatly simplifies the analysis of experimental results, as there is no need to consider stress concentrations near the specimen edges. The TILTS hydraulic actuator (OPS) is designed to supply working fluid (oil) to the hydraulic cylinder control units. OPS provides a maximum travel speed of 3 mm/ s. The operating oil pressure can be set between 7 and 35 MPa. The maximum force of the hydraulic cylinder is 1000 kN. TIlTS uses a four-channel control system with

14

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Experimental Facilities for Studying the Deformation and. . .

Fig. 2.2 Starting position of the plates

Fig. 2.3 Position of the plates when the specimen is deformed

electrohydraulic transducers (servo valves). Three channels of the control system are used to control the hydraulic cylinders and one channel is redundant. Dynamometers installed in the hydraulic cylinder rods or travel sensors are used as feedback sensors in the control system. The ability to control the load process either by forces or displacements on each of the three channels allows practically any load (deformation) trajectory of the specimen, including the fracture process. An important feature of the TILTS is the possibility of investigating the dependence of rock permeability on the magnitude and type of acting stresses. It is known that rock permeability can both decrease and increase (including irreversibly) depending on the stresses acting in them. The type and level of these stresses are determined by the bottom hole structure (casing presence or absence, shape and density of perforation holes, etc.) and pressure created at the borehole bottom. The TILTS allows to simulate these conditions on rock specimens and continuously record changes in their permeability. The studies of the dependence of filtration properties of various rocks on the value of underbalance in the well allowed to develop a new technology to improve well productivity—the method of directed

References

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Fig. 2.4 Processing unit for producing test specimens

unloading reservoir which has been successfully tested on a number of oil fields in Western Siberia and the Kama region. The specimens to be tested at the TILTS are produced on a specially designed machining complex (Fig. 2.4), including two machines—a stone-cutting machine and a grinding machine. This complex allows the production of cubic specimens with high accuracy and with any orientation relative to the core axis. After finishing, the non-parallelism of the specimen faces and the deviation from perpendicularity at a base of 40 mm does not exceed 20 μm.

References Airey DW, Wood DM (1988) The Cambridge true triaxial apparatus. In: Advanced triaxial testing of soil and rock. ASTM STP 977. American Society for Testing and Materials, Philadelphia, PA, pp 796–805 Alexeev AD, Revva VN, Alyshev NA, Zhitlyonok DM (2004) True triaxial load apparatus and its application to coal outburst prediction. Int J Coal Geol 58:245–250. In Russian Alshibli KA, Williams HS (2005) A true triaxial apparatus for soil testing with mixed boundary conditions. Geotech Test J 28(6):534–543 AnhDan L, Koseki J, Sato T (2006) Evaluation of quasi-elastic properties of gravel using a largescale true triaxial apparatus. Geotech Test J 29(5):374–384 Arthur JRF (1988) Cubical devices: versatility and constraints. In: Advanced triaxial testing of soil and rock. STP 977. ASTM, Philadelphia, PA, pp 743–765

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Bell JM (1965) Stress-strain characteristics of cohesionless granular materials subjected to statically applied homogeneous loads in an open system. Ph.D.thesis. California Institute of Technology, Pasadena, CA Choi C, Arduino P, Harney MD (2007) Development of a true triaxial apparatus for sands and gravels. Geotech Test J 31(1):1–13 Green GE (1969) Strength and compressibility of granular materials under generalized strain conditions. Ph.D. thesis. University of London, London Green GE (1971) Strength and defjrmation of sand measured in an independent stress control cell. In: Stress-strain behaviour of soils, Proceedings of the Roscoe Memorial Symposium, Cambridge, UK, pp 285–323 Hambly EC (1969) A new triaxial apparatus. Geotechniqe 19(2):307–309 Hayno K, Koeski J, Sato T Totsuoka F (1999) Small strain characteristics of sedimentary soft mudstone from true triaxial tests. In: Prefailure characteristics of geomaterials, vol. 1, pp 191–198 Ibsen LB, Praastrup U (2002) The Danish rigid boundary true triaxial apparatus for soil testing. Geotech Test J 25(3):1–12 Karev VI, Kovalenko YF (2013) Triaxial load system as a tool for solving geotechnical problems of oil and gas production. In: True Triaxial testing of rocks. CRC Press, Leiden, pp 301–310 Mandeville D, Penumadu D (2004) True triaxial testing system for clay with proportional-integraldifferential control. Geotech Test J 27(2):1–11 Matsuoka H, Sun DA (1995) Extension of spatially mobilized plane (SMP) to frictional and cohesive materials and its application to cemented sands. Soils Found 35(4):63–72 Matsuoka H, Sun DA, Kogane A, Fukuzawa N, Ichihara W (2002) Stress-strain behaviour of unsaturated soil in true triaxial tests. Can Geotech J 39(3):608–619 Pearce JA (1971) A new true triaxial apparatus. In: Stress-strain behaviour of solids. Proceedings of the Roscoe Memorial Symposium, Cambridge, UK, pp 330–339 Prashant A, Penemadu D (2004) Effect of intermediate principal stress on overconsolidated kaolin clay. Geotech Test J 130(3):284–292 Reddy KR, Saxena SK, Budiman JS (1992) Development of a true triaxial testing apparatus. Geotech Test J 15(2):89–105 Shapiro S, Yamamuro JA (2003) Effects of silt on three-dimensional stress-strain behaviour of loose sand. J Geotech Geoenviron Eng 129(1):1–11 Sivakugan N, Chameau J-L, Holts RD, Altschaeffl AG (1988) Servo-controlled cuboidal shear device. Geotech Test J 11(2):119–124 Sture S (1979) Development of multiaxial cubical test device with pore water pressure monitoring facilities. Department of Civil Ehgineering, Virginia Polytechnic Institute and State University, Blaksburg, VA, Report №. VPI-E-79.18 Sture S, Desai CS (1979) Fluid cushion truly triaxial or multi-axial testing device. Geotech Test J 2(1):20–33 Wawersik WR, Carlson LW, Holcomb DJ, Williams RJ (1997) New method for true-triaxial rock testing. Int J Rock Mech Mini Sci 34(330):3–4 Wood DM (1974) Some aspects of the mechanical behaviour of kaolin under truly triaxial conditions of stress and strain. Ph.D.thesis. University of Cambridge, Cambridge, UK Wood DM (2008) Multiaxial testing at Boulder and elsewhere. In: Proceedings of the inaugural international conference of the Engineering Mechanics Institute, 19–21 May 2008, Minneapolis, MN. American Society of Civil Engineers, Reston, VA, p 31 [Abstr]

Chapter 3

Methodology for Performing TILTS Experiments to Study the Deformation, Strength and Filtration Properties of Rocks

Despite many years of experience in operating wells with sand production, so far there is no sufficiently substantiated model of reservoir failure and mathematical description of the processes occurring in the formation-well system. In addition to the need to fill mathematical models of deformation and fracture of reservoir rocks with constants, data on deformation, strength and filtration properties of rocks in the bottomhole zone are needed to establish technological regimes of well operation, to determine optimum methods of well completion, and to determine the possibility of improving operational performance by control the stress-strain state of rocks. The corresponding experiments were carried out at the TILTS under true triaxial stress conditions on reservoir rock specimens taken from reservoirs at a number of Russian underground gas storage. Cubic specimens with 40 mm edge were made of core material. It should be noted that it was very difficult to make specimens from UGS reservoirs rock due to the extremely low strength of weakly cemented sandstones. Specimens were marked as follows: axis 1 of the specimen coincided with the core axis, the orientation of axes 2 and 3 was arbitrary.

3.1

Experimental Procedure for Measuring Longitudinal Wave Velocities in the Test Specimens

In order to determine the type and degree of anisotropy of the elastic properties of the rocks under investigation, the velocities of longitudinal ultrasonic waves in three mutually orthogonal directions are measured in the specimens before testing with the TILTS. The velocities are measured in three directions: along axis 1, i.e. the core axis, and along axes 2 and 3 in two mutually perpendicular directions in the horizontal plane.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Karev, Y. Kovalenko, Geomechanical Aspects of Operation of Underground Gas Storage, Springer Geology, https://doi.org/10.1007/978-3-031-34765-8_3

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3

Methodology for Performing TILTS Experiments to Study the Deformation,. . .

Fig. 3.1 Schematic diagram of a longitudinal wave velocity measurement setup in the specimen: (1) transmitter, (2) receiver, (3) specimen, C9: oscilloscope, G5–63: pulse generator

2

F C9

3 1

G5-63

Figure 3.1 shows a schematic design of the installation. It consists of two sencorsgenerators of ultrasonic waves, between which the test specimen is placed. In order to visualize the measurement results, the electrical signals from both transducers are displayed on an oscilloscope. Waves with a length of 5 mm and a frequency of 1.25 MHz are passed between the sencors-generators and the transit time of the waves through the specimen and the attenuation of the vibration amplitude are determined. Based on the results of measurements of longitudinal wave velocities through the specimens, an elastic deformation model for the formation is being selected. If the velocities in all three specimen axes are approximately the same, an elastic isotropic model is selected. If velocities along axes 2 and 3 are close, and those along axis 1 (core axis) are noticeably lower (as a result of layering rock), a transversalisotropic model is chosen.

3.2

Methodology for Determining Strain and Strength Properties of Rocks (Triaxial Tests)

Reservoirs of UGSFs, as well as, in some cases, the reservoirs of hydrocarbon fields, in particular, the Cenomanian deposits in Western Siberia, are represented by weakly cemented sandstones. Under the influence of geomechanical and filtration loads, when the critical values of underbalance are exceeded, the bottomhole zone of such reservoirs can be destroyed in the course of operation with the subsequent sand production. This leads to destruction of subsurface and surface equipment and decrease of productivity, up to well shut-in. Mathematical models can be used for a variety of practical purposes, including justification of the operating mode of wells without destruction of the bottomhole zone and sand production, as well as justification of the well completion method—with or without a sand filter, with open or cased bottomhole. It is impossible to carry out mathematical modelling without knowing the whole set of data on mechanical properties of the reservoir rock, its filtration-capacitative properties, geomechanical parameters in the vicinity of the wellbore, etc. The most

3.2

Methodology for Determining Strain and Strength Properties of. . .

19

demanded and least studied of the rock mechanical properties in the near-wellbore zone, are the bulk strength values of weakly cemented sandstones, which determine rock failure near the wellbore. The bearing capacity of rocks is known to be conditioned mainly by shear and separation resistance. The critical value of shear resistance in a medium, which on average is considered homogeneous in all directions, at a given oriented site, i.e. local fracture criterion, is usually represented as Coulomb-More criterion (Tertsagi 1961). τ ≥ ½τ ½τ = C - σn tgρ where [τ] is the tensile strength, σ n is the normal stress at the site (compressive stresses are assumed to be negative), C is the coefficient of adhesion and ρ is the angle of internal friction—strength characteristics of the rock. If there are surfaces of weakening in the rock, which are essentially the layering planes, then failure will begin there first, as the shear strength [τ] is much lower there than in the other directions. In this case, C and ρ should be understood as the coefficient of adhesion and the friction angle of the layering planes. In order to determine the strength characteristics of C and ρ, special experiments are required to construct Mohr circles at different levels of load of the specimen. The difficulty lies in the fact that it is desirable to carry out tests at different reductions on the same specimen. In this case each load of the specimen (cycle) must be brought to a level where the specimen begins to deform plastically but still remains intact. This requires continuous monitoring of the specimen condition during the test, which can only be achieved by displaying the strain curves of the specimen online on a computer screen and simultaneously calculating the current strain characteristics of the rock. Moreover, load of the specimen should be controlled by displacements and not by load. Otherwise it would be practically impossible to stop the load of the specimen in the plastic region in time without destroying it. The TILTS makes it possible to carry out such tests. In order to construct Mohr circles corresponding to different load levels and to determine the volumetric strength constants in Mohr circles three load cycles must be performed on one specimen. At each of the cycles the specimens are preliminary subjected to all-round uniform compression up to one of the levels (S1 = S2 = S3 = 2 MPa, 10 MPa, 20 MPa), after that the additional load along the axis 1 of the specimen (vertical axis of the core) is realized under the constancy of stresses along the other two axes. The load of the specimen is brought to a level where the specimen begins to deform inelastically, but still remains intact. The deformation of the specimen in the direction of the additional load is continuously monitored during loading and the current tangential Young’s modulus in this direction is calculated. The specimen is loaded until the tangential Young’s modulus in the direction of the additional load decreases from the maximum recorded in the given test cycle by about 70–75%. In the first and second cycles, the specimen is then unloaded to the value of the current all-round compression, and in the third cycle, the specimen is loaded until it fractures.

20

3

Methodology for Performing TILTS Experiments to Study the Deformation,. . .

Fig. 3.2 Triaxial test program

Figure 3.2 shows the test program for the specimens. s1, s2, s3 are stresses acting along axes 1, 2, 3 of load unit of TILTS. An important feature of triaxial tests of cubic specimens on TILTS in comparison with triaxial tests of cylindrical specimens on Karman installations is the possibility of recording the strains of specimens in three directions. As an example, Figs. 3.3 and 3.4 show the strain curves of the specimen during the test and the dependence of the tangential modulus on the magnitude of the stress acting along the axis of the core, recorded during the second test cycle of the specimen of UGS №5. Based on the analysis of triaxial test results, elastic characteristics of rocks are calculated for each cycle of specimen loading—tangential Young’s modulus E and Poisson’s ratio ν. Elastic characteristics are calculated on rectilinear sections of the strain curves corresponding to the elastic deformation of specimens. In addition to elastic characteristics, the implemented specimen test program makes it possible to obtain three limit states at different compression stresses, construct Mohr circles and calculate rock strength characteristics—rock adhesion coefficient C and internal friction angle ρ (Tertsagi 1961).

Methodology for Experiments to Simulate Sand Production Using. . .

3.3

21

modulus E, 1000 Mpa

Fig. 3.3 Strain curves of the specimen 26 24 22 20 18 16 14 12 10 8 6 4 2 0

E

10

20

30

40

50

60

70

80

90 100 110 120 130 140 150

stress s1, MPa

Fig. 3.4 Change in tangential modulus during the experiment

3.3

Methodology for Experiments to Simulate Sand Production Using Core Specimens with Holes in a Modified Hollow-Cylinder Scheme

The TILTS allows direct modelling of the pressure drawdown lowering process. Such tests, in particular, are part of Total methodology for determining permissible underbalanced, the so-called “hollow-cylinder” test.

22

3

Methodology for Performing TILTS Experiments to Study the Deformation,. . .

Fig. 3.5 Schema of the modified “hollow” cylinder test

Two specially manufactured pressure plates with central channels are used in the TILTS load unit for the experiments “hollow cylinder”, Fig. 3.5. In the specimens to be tested according to the modified “hollow cylinder” scheme, a hole with a diameter of 10 mm is drilled in the centre of the specimen face in the direction of one of the axes. During the experiment air is blown through a channel in the tip of the upper active pressure plate coinciding with the hole in the specimen at a pressure of about 0.1 MPa. There is also an hole in the tip of the lower pressure plate, coinciding with the channel in the specimen. Through this opening, a special tube conveys the transported air with sand to an electronic scale connected to a computer. Compressive stresses are applied to the faces of the specimen which gradually increase over the course of the experiment. The specimen strains in the three directions are recorded and the weight of the sand carried out of the specimen by the airflow is measured. The accuracy of the electronic scales is 0.001 g and the recording is done every 2 s. The test is continued until the specimen fractures. The load program for the specimens was as follows. An the first stage, the specimen was loaded uniformly along the three axes at a constant speed. An the second stage, the loading of the specimen was stepwise. At each step, the stresses in all three or two axes perpendicular to the hole were evenly increased by 1 MPa, followed by a dwell time. Experiments using a modified “hollow-cylinder” scheme allow direct simulation of reservoir rock fracture processes in the vicinity of a wellbore during downhole pressure lowering. However, due to the scale effect, these experiments do not allow to quantify the magnitude of pressure drawdown leading to rock failure in the well bottomhole zone. Nevertheless, they provide an opportunity to see a picture of the deformation processes and determine the volume and characteristics of the broken rock particles.

3.3

Methodology for Experiments to Simulate Sand Production Using. . .

23

As an example, Figs. 3.6, 3.7, and 3.8 show test results using the described methodology and a photo of a rock specimen from the North Kamennomysskoye gas condensate field.

Fig. 3.6 Dependence of the carried out sand mass on specimen compression

Fig. 3.7 Strain curves of the specimen

24

3

Methodology for Performing TILTS Experiments to Study the Deformation,. . .

Fig. 3.8 Specimen with hole after test

The photo shows that the hole in the specimen was severely deformed and clogged with sand after the test. So, the TILTS has significant advantages over other installations used for the experiments described above. The purpose of classical “hollow cylinder” experiments is to record the stress at which the sand starts to escape from the hole. This stress is identified with the stress at which the rock starts to fracture on the walls of the borehole. The TILTS allows all three strain components to be measured along the three axes of a cubic specimen during the entire experiment. This provides important additional information. In particular, it turns out that the actual failure of the hole walls in the specimen starts at stresses lower than the stress of the beginning of the sand production from the specimen. As can be seen from the strain curves of the specimen shown in Fig. 3.7, as the total compressive stress increases, the deformation of the specimen first happens elastically (linear), which indicates the steady state of the hole in the specimen. When the compressive stress reaches a certain value, the specimen begins to deform nonlinearly, which is associated with the beginning of fracture of the hole walls. And sand production as such is not yet observed in the experiment. The difference between the beginning of non-linear deformation of the specimen and the beginning of sand production reached several tens of atmospheres. The above said gives grounds to state that testing of specimens according to the modified scheme “hollow cylinder” at TILTS allows to determine the moment of the beginning of fracture of hole walls with higher precision and reliability than in the classical experiments.

3.4

Determining the Dependence of Rock Specimen Permeability on the Type. . .

25

During the “hollow cylinder” tests of the reservoir rock specimens, the sand extracted from the holes was collected in special containers for particle sizes analysis. Granulometric analysis is the most important method of studying terrigenous rocks. Granulometric analysis is performed: 1. For classification purposes—to correctly identify and name rocks; 2. To assess rocks as reservoirs; 3. To identify genetic features needed for palaeogeographic reconstructions (mode and range of transport, debris migration routes, depositional conditions, etc.); 4. To determine the genetic type of the deposit; 5. Finally, as a preparatory stage for mineralogical analysis, because it isolates from rocks certain fractions which can then be studied by other methods.

3.4

Determining the Dependence of Rock Specimen Permeability on the Type and Level of Applied Stresses

The TILTS includes an Automatic Permeability Measurement System (APMS) which allows the permeability change to be monitored continuously during the tests of the specimen. The permeability measurement is carried out as follows. A gas (regular compressor air) is blown through the specimen during the test. For this purpose, one pair of pressure plates that form the load assembly is perforated. These plates have channels for feeding compressed gas into the specimen and outlet of the gas filtered through the specimen, as well as perforated inserts into plates tips for creation of even gas flow throw the specimen. The solution of the problem of permeability measurement under complex load required significant changes in the existing traditional techniques. First of all, it touched upon the issue of tightness of that part of permeability measuring system to which the assembly belongs: active pressure plate—specimen—supporting pressure plate. This issue is fundamentally important for the direct permeability measurement method, as the flow of gas filtered through the specimen is on average considered as one-dimensional and the permeability of the specimen is determined by the pressure drop across the specimen and the flow rate of the gas which filtered through the specimen. The traditional method of sealing lateral to the filtration direction faces of the specimen use of a thin rubber sheath in the load unit is not applicable due to the fact that dimensions of the specimen and the tip differ significantly and lateral movement of the tip is possible. Therefore, the specimen for testing under complex load with simultaneous permeability measurement along one of the specimen axes is prepared as follows. The axis of the specimen along which the gas is to be filtered is selected according to the objective of the test. A latex-based or aqueous polyvinyl acetate solution polymerising at room temperature is applied to the four faces of the specimen parallel to the filtration axis. The latex sheath is dried at room temperature for 4 to 6 h. The lateral cover thus formed are thin enough, not more than 50 μm, to not

26

3

Methodology for Performing TILTS Experiments to Study the Deformation,. . .

introduce a significant error in the strain measurement of the specimen. At the same time, such a shell has sufficient strength and elasticity, which ensures tightness even at relatively large deformations of the specimen up to the formation of main cracks. Before the experiment, a 2–3 mm wide strip of adhesive is applied along the contour of the faces through which filtration is carried out. The adhesive dries for 10 mins and then the specimen is placed in the load assemble, active pressure plates are move to the specimen and it is comprehensively loaded to a pressure of about 0.4 MPa. In this condition the specimen is allowed to stand for 20 mins to ensure that the specimen is sufficiently bonded to the tips of the active and supporting pressure plates along the filtration axis. In the final step of specimen preparation for testing, a check is made to ensure that the specimen is “glued” into the load assembly. Considering that the strength of the “gluing” is limited, the pressure drop at the inlet and outlet of the specimen must not exceed 0.5 MPa. The APMS is equipped with two flow meters that measure flow rate over a wide range from 0.5 ml/min to 5 l/min, and two digital pressure gauges that measure inlet and outlet pressure from the specimen. The signals from the flow meters and pressure gauges are transmitted to the Automated Control System controller, processed, displayed on the monitor screen and stored in the computer memory. The permeability coefficient is determined as follows. An interval of the inlet pressure p is preliminarily set, for which the dependence of the air flow rate Q on the difference of squares of the inlet and outlet pressure Q = Q p2 - p2а has a linear character (the outlet pressure is always atmospheric pа). The air pressure when testing the specimen is set in this interval, where the gas flow is described by Darcy’s law. The permeability coefficient is then defined by the formula k=

2lQμ Fpa ðp=2 - 1Þ

ð3:1Þ

where μ is the dynamic viscosity of the air, l is the specimen rib length, F is the crosssectional area of the specimen, p/ = p/pa. In general, μ depends on content in gas the water vapour, technical oil vapour and temperature. Since controlling the concentration of water vapour and oil vapour in the gas is costly and technically demanding, the APMS is equipped with filter driers that remove the water vapour and oil vapour from the filtering gas. Table data are used to determine the value of μ at a controlled temperature. Considering that the APMS is located in a laboratory room with a sufficiently stable temperature throughout the entire testing process of a single specimen, the gas supply to the APMS is carried out through relatively long copper pipes and the volume flow rate of the gas does not exceed 20 l/min, it can be assumed with a reasonable degree of accuracy that the gas temperature is constant and equal to the room air temperature.

Reference

27

The specimen testing process may also include a fracture stage, where particulate matter may be introduced into the gas stream, which may significantly affect the performance of the flow sensor. To avoid such a situation, a submicrofilter is installed at the inlet of the flow sensors.

Reference Tertsagi K (1961) Theory of soil mechanics (in Russian). Gosstroyizdat, Moscow

Chapter 4

Results of TILTS Testing of the Deformation, Strength and Filtration Characteristics of Reservoir Rocks at Underground Gas Storage Facilities Based on Triaxial Tests

Triaxial tests of core material from underground gas storage reservoirs at a number of UGS facilities were performed on a true triaxial load setup TILTS to determine deformation, strength and filtration characteristics of rocks under conditions of real three-axis stress states. The experiments were carried out according to the methodology described in Chap. 3.

4.1

Test Specimens

To determine deformation and strength properties 12 specimens were made: three specimens from UGS №1, two specimens from UGS №2, two specimens from UGS №3, one specimen from UGS №4 and four specimens from UGS №5. The details of the specimens tested are shown in Table 4.1. In order to determine the elastic properties and volumetric strength parameters of the investigated rocks, the rock specimens were tested at the TILTS using a triaxial test program which aimed to assess the stress required to fracture the rock specimen depending on the magnitude of the all-round uneven compression of the specimen. The specimen load programs are given in Sect. 3.2. The load was controlled by displacement. This method of load makes it possible to investigate the plastic behaviour of rocks under transcendent load modes. The elastic characteristics and the strength characteristics of the tested rocks were calculated for each load cycle of the specimens: Young’s modulus E and Poisson’s ratio ν, the adhesion modulus C and the angle of internal friction ρ. Note that the elastic characteristics using the above triaxial program is only correct for isotropic rocks. If the rock is anisotropic, the task becomes much more difficult. In particular, in the presence of layering, the rock becomes anisotropic, transversally isotropic, and its behavior is described by five elastic moduli instead of two, which require special experiments to be performed on the TILTS.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Karev, Y. Kovalenko, Geomechanical Aspects of Operation of Underground Gas Storage, Springer Geology, https://doi.org/10.1007/978-3-031-34765-8_4

29

30

4

Table 4.1 Specimens for determining deformation and strength characteristics

Results of TILTS Testing of the Deformation, Strength and. . . UGSF UGS №4 UGS №1 UGS №2 UGS №3 UGS №5 UGS №5 UGS №5 UGS №5

Depth, m 1195.60 1424.70 1401.85 1429.59 791.00 796.08 764.13 757.15 1249.43 1249.58 1201.15 1190.33

Specimen numbers KR-2 KSH-1 KSH-4 KS-8-2 KS-1 KS-4 У-1-2 У-5-2 AR-1-2 AR-2-1 AR-9-2 AR-13-1

To determine the type and degree of anisotropy of the investigated rocks, longitudinal ultrasonic waves propagation velocities were measured in the specimens in three mutually orthogonal directions before testing with the TILTS, see Sect. 3.1. Table 4.2 shows the measured longitudinal waves velocities in the test specimens in the three axes. Table 4.2 shows that the propagation velocities of longitudinal waves along the three axes in all specimens are close, which indicates that the elastic characteristics of the rock are isotropic. Therefore, in the experiments at the TILTS, the triaxial test programs described above were used to determine the strength and strain characteristics. During the triaxial tests, the permeability of specimens from all fields was measured at the APMS. The permeability measurement technique is given in Sect. 3.4.

4.2

UGS №4. Specimen KR-2

Figures 4.1, 4.2, 4.3, and 4.4 show as an example the results of triaxial tests for one specimen from UGS №4. For each test cycle, Figures (a) show diagrams of specimen loading and changes in its permeability at the parts of additional load, Figure (b) show strain curves of specimen in the course of the test. Figure 4.4 shows a picture of the specimen after the test.

4.3

Results of TILTS Tests of Deformation and Strength Properties of Rock. . .

Table 4.2 Data on ultrasonic sounding of the specimens

4.3

1

Specimen numbers KR-2

2

KSH-1

3

KSH-4

4

KSH-8-2

5

KS-1

6

KS-4

7

U-1-2

8

U-5-2

9

AR-1-2

10

AR-2-1

17

AR-9-2

22

AR-13-1

axes 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

31 а, m/s 2273 2150 2174 2632 2700 2535 1960 1960 1857 2702 2294 2303 1613 1639 1754 1492 1234 1265 1150 1156 1228 1618 1538 1538 1493 1429 1429 1483 1417 1417 2272 2325 2439 4253 4253 4253

Results of TILTS Tests of Deformation and Strength Properties of Rock Specimens Based on Triaxial Tests

Table 4.3 shows data of processing of results of specimens tests from all UGS—the values of elastic moduli of the studied rocks—Young’s modulus E and Poisson’s coefficient ν, and Coulomb-More parameters—adhesion coefficient С and internal

32

4

Results of TILTS Testing of the Deformation, Strength and. . .

Fig. 4.1 Test results of specimen KR-2. Cycle 1

Fig. 4.2 Test results of specimen KR-2. Cycle 2

Fig. 4.3 Test results of specimen KR-2. Cycle 3

friction angle ρ which calculated using rock strength limits S2 , on each of three cycles of loading (initial all-round compression 2 MPa, 10 MPa and 20 MPa respectively). The values of bulk strength coefficients were determined using

4.4

Conclusions

33

Fig. 4.4 Picture of specimen KR-2 after test

Mohr circles. In addition, for each load cycle the permeability of the specimen k at its beginning is given. When testing specimens AR-1-2 and AR-2-1 from UGS №5 it turned out that the rock was very weak. Therefore it was possible to perform two cycles of load in triaxial tests: for specimen AR-1-2 at 10 MPa and 20 MPa all-round compression and for specimen AR-2-1 at 2 MPa and 10 MPa all-round compression. Also only two cycles were possible for test specimen AR-13-1, but the reason here was technical.

4.4

Conclusions

Experimental studies of core material from gas storage reservoirs to determine deformation, strength and filtration characteristics of rocks under pseudo-three-axis stress conditions have been carried out. A distinctive feature of the experiments performed is that all three load cycles required to determine deformation-strength properties of each rock under study were performed on a one specimen loading was conducted with displacement control. This significantly improves the accuracy and reliability of the results obtained. Besides, in the course of experiments permeability of specimens in one of directions was measured. The result of the experiments carried out was: • Determination of elastic and plastic characteristics of reservoir rocks—gas storage facilities of UGS №1, UGS №2, UGS №3, UGS №4 and UGS №5; • Determination of the strength characteristics of the rocks of the UGS studied for the Coulomb-More criterion—moduli of adhesion and angles of internal friction; • The study of the dependence of the filtration characteristics of reservoir rocks— objects of gas storage—on the stresses created in them.

34

4

Results of TILTS Testing of the Deformation, Strength and. . .

Table 4.3 Results of triaxial tests UGS №4. KR-2 Cycle Compression, MPa 1 2 2 10 3 20 UGS №2. KC-1 Cycle Compression, MPa 1 2 2 10 3 20 UGS №2. KC-4 Cycle Compression, MPa 1 2 2 10 3 20 UGS №1. KSH-1 Cycle Compression, MPa 1 2 2 10 3 20 UGS №1 KSH-4 Cycle Compression, MPa 1 2 2 10 3 20 UGS №1. KSH-8-2 Cycle Compression, MPa 1 2 2 10 3 20 UGS №3. U-1-2 Cycle Compression, MPa 1 2 2 10 3 20 UGS №3 U-5-2 Cycle Compression, MPa 1 2 2 10 3 20 UGS №5 UGS. AR-1-2 Cycle Compression, MPa 1 10

Е*10-3, MPa 2.13 5.44 8.51

ν 0.30 0.18 0.06

S2 , MPa 21.0 51.0 86.0

С, MPa 3

ρ, degree 34

k, D 2.2 3.2 3.5

Е*10-3, MPa 5.0 7.25 7.7

ν 0.36 0.27 0.23

S2 , MPa 36.0 70.0 93.0

С, MPa 6.5

ρ, degree 34

k, D 4.9 4.3 3.2

Е*10-3, MPa 2.84 5.26 7.3

ν 0.24 0.16 0.19

S2 , MPa 23.0 48.0 76.0

С, MPa 5

ρ, degree 29

k, D 6.2 4.3 2.6

Е*10-3, MPa 4.8 6.0 6.5

ν 0.19 0.15 0.16

S2 , MPa 43.0 60.0 83.0

С, MPa 13.5

ρ, degree 20

k, D 1.5 1.1 0.9

Е*10-3, MPa 3.1 4.0 4.7

ν 0.27 0.24 0.20

S2 , MPa 25.0 39.0 60.0

С, MPa 7.5

ρ, degree 8

k, D 1.5 1.1 0.94

Е*10-3, MPa 6.94 9.1 10

ν 0.25 0.21 0.17

S2 , MPa 58.0 79.0 110.0

С, MPa 14.5

ρ, degree 29

k, D 0 0 0

Е*10-3, MPa 2.0 3.9 5.0

ν 0.13 0.06 0.01

S2 , MPa 17.5 36.0 57.0

С, MPa 4

ρ, degree 20

k, D 10.7 11.4 11.7

Е*10-3, MPa 2.6 4.6 6.5

ν 0.25 0.16 0.15

S2 , MPa 19.0 43.0 67.0

С, MPa 3.5

ρ, degree 28

k, D 6.1 4.4 3.3

Е*10-3, MPa 1.6

ν 0.01

S2 , MPa 29

С, MPa 1

ρ, degree 25

k, D 4.9

(continued)

4.4

Conclusions

35

Table 4.3 (continued) UGS №5 UGS. AR-1-2 Cycle Compression, MPa 2 20 UGS №5. AR-2-1 Cycle Compression, MPa 1 2 2 10 UGS №5. AR-9-2 Cycle Compression, MPa 1 2 2 10 3 20 UGS №5. AR-13-1 Cycle Compression, MPa 1 2 2 10

Е*10-3, MPa 0.6

ν 0.19

S2 , MPa 60

С, MPa

ρ, degree

k, D 3.9

Е*10-3, MPa 0.4 1.56

ν 0.28 0.06

S2 , MPa 7.3 34.6

С, MPa 1

ρ, degree 30

k, D 1.18 2.6

Е*10-3, MPa 3.8 7.9 9.7

ν 0.23 0.12 0.11

S2 , MPa 35 80 119

С, MPa 8

ρ, degree 38

k, D 12.9 11.8 9.4

Е*10-3, MPa 18.9 22.5

ν 0.31 0.33

S2 , MPa 105 140

С, MPa 22.5

ρ, degree 40

k, D 0 0

Table 4.3 shows that all the rocks studied are characterized by low values of elastic modulus (Young’s modulus), which fully agrees with the results of longitudinal wave velocity measurements in the specimens. The strength properties of the rocks are also low. The highest values of volumetric strength values are characterized by rocks of UGS №1, and the lowest ones—by rocks of UGS №5 and UGS №3 (Table 4.3). Specimens AR-1-2 and AR-2-1 turned out to be very weak with low adhesion modulus K and ultimate stress S2 , as well as low elastic Young’s modulus E. Specimen AR-13-1, on the contrary, turned out to be very strong with high strength and elastic characteristics. The elastic and strength characteristics of rock specimen AR-9-2 are intermediate between those mentioned above. The experiments showed that when the compressive and tangential stresses acting in the specimens increased in the course of the experiments, the permeability of the specimens decreased slightly. This fact requires further investigation.

Chapter 5

Geomechanical Modelling of the Stress-Strain State in UGS Reservoirs During Cyclic Changes in Reservoir Pressure and Downhole Pressure

5.1

Basic Equations

When selecting of basic equations of the mathematical model, attention should be paid to a number of specific features of UGS facilities and their operation mode which differ significantly from the mode of development of hydrocarbon fields. Such peculiarities should include: • The extremely high permeability of reservoir rocks (up to several Darcys), as well as a fairly high porosity; • The comparatively low strength properties of reservoir rocks; • Significant changes in reservoir pressure (up to tens of percent of average reservoir pressure) in the minimum and maximum filling states; • Very small working depressions and repressions (usually not exceeding a few percent of the average reservoir pressure); • Despite low working underbalances and repressions, high flows and flow rates due to the high permeability of the reservoir rock; • Cyclicality of injection and extraction stages determined by seasonality of gas consumption and other factors. However, it should be noted that in order to use the model as a tool for obtaining reliable predictions of the behaviour of UGS reservoirs, special experimental studies of the rocks of specific UGS reservoirs must be carried out: • Determination of elastic properties (Young’s modulus and Poisson’s ratio for isotropic media, five elastic constants in the case of transversally isotropic media); • Determination of strength properties (Coulomb-More or Drucker-Pragueur model parameters in the case of isotropy, generalised Hill model parameters in the case of anisotropy); • Determining the dependence of rock permeability on the stress state and its history;

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Karev, Y. Kovalenko, Geomechanical Aspects of Operation of Underground Gas Storage, Springer Geology, https://doi.org/10.1007/978-3-031-34765-8_5

37

38

5

Geomechanical Modelling of the Stress-Strain State in UGS. . .

• Special attention should be paid to identifying the influence of stress-strain state change cycling due to the cyclic nature of the gas injection and extraction processes. In order to determine parameters when modelling gas injection and extraction processes, it is important to have an understanding of the change in stress state in the reservoir rock during cyclic changes in reservoir pressure. At present, there is a serious problem associated with the lack of experimental data for mathematical models of the stress-strain state in the borehole zone. In order to fill in and refine the mathematical models, it is necessary to develop a mechanism for using the results of laboratory tests of core specimens in studies of the stressstrain state of the reservoir. This will make it possible to expand the possibilities of each of the studies taken separately and to determine the parameters of effective technological well operation mode. In the presence of one fluid (gas) component, it is assumed that the mechanical behaviour of the rock obeys the equations of poroelastic-plasticity (Biot 1935; Biot 1941; Zheltov and Khristianovich 1955; Tertsagi 1961). In absence of plastic deformations and assuming that filtration flow obeys Darcy’s law, this system can be written in Cartesian coordinates as follows σ ij,i þ f j = 0

ð5:1Þ

sij = σ ij þ ð1 - δÞpδij

ð5:2Þ

κij ðp, smn Þp,i

,j

= mμf

∂p ∂t

sij = Λijkl εElk εTij = εEij =

1 u þ uj,i 2 i,j

ð5:3Þ ð5:4Þ ð5:5Þ

Here σ ij, sij are components of total and effective (acting on the soil skeleton) stress tensors; fj—bulk forces, in particular, gravitational and inertial forces; εEij , εTij — components of elastic and total deformation tensors; ui—components of displacement vector; p—pore pressure; Λijkl—elasticity tensor; κ ij( p, smn)—permeability tensor generally depending on the magnitude of the acting pressure as well as the whole stress history; δij—unit tensor; m—porosity; μf—dynamic viscosity of the gas; δ—is the fraction of contact areas relative to the entire grain surface of the soil skeleton. It is usually assumed for sedimentary rocks δ = 0.2 (Lehnitsky 1977). Tensile stresses here are assumed to be positive, axis 3 of rectangular coordinate system is assumed to be vertical, indices after comma mean differentiation by corresponding coordinate. In the stationary case, the right-hand side of eq. (5.3) vanishes. Hereinafter, the index after the decimal point denotes the (covariant) differentiation by the corresponding coordinate, by the repeated indices the summation is implied.

5.1

Basic Equations

39

Equation (5.1) are equilibrium equations, eq. (5.2) expresses the relationship of total and effective stresses, eq. (5.3) is the filtration law, eq. (5.4) is generalised Hooke’s law written in terms of effective stresses, eq. (5.5) expresses the relationship between displacements and strains in the considered case of small deformations and rotations. In the case of a transversally isotropic medium, the equations of Hooke’s law take the form (5.6) with the total stresses replaced by the effective ones (Lehnitsky 1977). s11 = C11 εE11 þ C 12 εE22 þ C13 εE33 s22 = C12 εE11 þ C 12 εE22 þ C13 εE33 s33 = C13 εE11 þ C 13 εE22 þ C33 εE33 s12 = 2C66 εE12 s13 = 2C44 εE13

ð5:6Þ

s23 = 2C44 εE23 For an isotropic body sij = 2μεEij þ λεEkk δij

ð5:7Þ

Here, Cij are elasticity matrix coefficients for a transversally isotropic body; λ, μ are Lamé constants. It follows from generalization of a large number of experimental data (Karev and Kovalenko 2006; Karev et al. 2015) that permeability is a function of stress-strain history. In some cases a rather good approximation is to write the law of permeability change as a function of achieved intensity of shear stresses si, determined from experience. κ = κ ð si Þ si =

3 1 s - s δ 2 jk 3 ii jk

ð5:8Þ sjk -

1 s δ 3 ii jk

ð5:9Þ

This dependence must be determined experimentally either for all independent components of the permeability tensor or (at least) for the assumed direction of filtration flow. For UGS conditions, one would expect that the permeability value would also depend on the number of injection-extraction cycles carried out. Equations (5.1)–(5.5) together with boundary conditions for stresses (or displacements) and fluid pressure form a closed system. In the presence of plastic (inelastic) deformation, eq. (5.5) should be replaced by a similar one that takes into account the contribution of inelastic εPij deformation

40

5

Geomechanical Modelling of the Stress-Strain State in UGS. . .

εTij = εEij þ εPij =

1 u þ uj,i 2 i,j

ð5:10Þ

A variant of plastic flow theory with isotropic strengthening can be used to describe inelastic deformation.

5.1.1

Main Factors Affecting the Stress-Strain State in the Vicinity of UGS Wells

The stress-strain state in the vicinity of the UGS wells is largely determined by natural stresses from rock pressure acting in the reservoir as a whole. Two situations are possible here: • The natural stress state is close to a state of uniform all-round compression by rock pressure at a given depth. This can primarily be expected in the absence of pronounced geological disturbances and when the rocks comprising the reservoir are sufficiently plastic so that all stresses in the reservoir should have equalised over geological time; • The natural stress state is unequal. In the initial state, the formation is loaded with three main stresses from rock pressure, a vertical stress and two horizontal stresses. The issue of influence of non-uniform natural stress state in the reservoir on wellbore stability has recently gained particular relevance in connection with widespread introduction into practice of oil and gas production technology by drilling horizontal wells. But this factor is also very important for vertical wells, since non-uniform initial stress field has a significant impact on stress distribution on the contour of a vertical well, which results to sand production. The presence of non-uniform initial stress field in the formation leads to the fact that stresses in the well vicinity, and consequently stability of the wellbore will essentially depend on the direction of its drilling with respect to the principal stresses. During the operation of UGS, the magnitude of stresses acting in the vicinity of the well is significantly influenced by factors caused by gas injection and extraction. The first factor determining the stress state in the vicinity of the well is the periodic underbalanced (during gas extraction) and repression (during gas injection) created at the bottom of the well. The second factor is due to cyclical changes in reservoir pressure during gas injection/extraction. Both of these factors can be the cause of rock failure in the bottomhole formation zone and sand production.

5.2

Stress State in UGS Reservoirs Under Equal-Component Natural Stress State

5.2 5.2.1

41

Stress State in UGS Reservoirs Under Equal-Component Natural Stress State Stress Changes in the Bottomhole Formation Zone When Underbalanced and Repressed at the Bottomhole

Consider the change in stress state in the vicinity of a vertical uncased well, Fig. 5.1. Let it be that under the action of natural rock pressure, the formation is in a state of all-round equal-component compression stress q = - γH, where H—depth of occurrence of formation, γ—average specific gravity of overlying rocks, usually taking equal γ = 2.3 g/cm (Zheltov and Khristianovich 1955). Then, for a medium isotropic in terms of elastic properties, the distribution of total stresses due to the action of rock pressure in the vicinity of an open hole is determined by the wellknown solution of the Lamé problem (Leav 1935). σ r = - ðq þ pw ÞðRw =rÞ2 þ q σ θ = ðq þ pw ÞðRw =rÞ2 þ q σz = q

Fig. 5.1 Stresses acting in the vicinity of the vertical uncased well

ð5:11Þ

42

5

Geomechanical Modelling of the Stress-Strain State in UGS. . .

Where Rw is the radius of the borehole; r is the distance from the borehole axis, pw is the bottom hole pressure, σ i is the total stresses acting in the vicinity of the borehole (σ i ≤ 0, pw > 0). Tangential stresses τ = 1/2(σ r - σ θ) are equal to τ = - ðq þ pw ÞðRw =r Þ2

ð5:12Þ

The stresses acting on the ground skeleton are si = σ i þ pð1 - δÞ

ð5:13Þ

The values of the effective stresses acting in the soil skeleton in the vicinity of the borehole are then determined by the relations s r = - ð q þ pw Þ

Rw r

2

þ q þ pð r Þ ð 1 - δ Þ

Rw 2 þ q þ pð r Þ ð 1 - δ Þ r sz = q þ pð r Þ ð 1 - δ Þ s θ = ð q þ pw Þ

ð5:14Þ

From (5.14) and (5.12) it follows that at the borehole wall, i.e. at r = Rw, the stresses are sr = - δpw sθ = 2q þ ð2 - δÞpw

ð5:15Þ

sz = q þ ð1 - δÞpw Then for the bottom hole underbalance Δp = p0 - pw from (5.15) we have Δpw = p0 - ðsθ - 2qÞ=ð2 - δÞ

ð5:16Þ

Here p0 is the initial reservoir pressure. When the bottom hole pressure changes by the value Δpw the stresses on the borehole contour change by the value of Δsr = - δΔpw Δsz = ð1 - δÞΔpw Δsθ = ð2 - δÞΔpw

ð5:17Þ

5.2

Stress State in UGS Reservoirs Under Equal-Component Natural Stress State

5.2.2

43

Stress-Strain State of the Reservoir Under Cyclic Reservoir Pressure Changes and Equal-Component Natural Stress State

The second factor affecting the stress state in the vicinity of UGS wells is the cyclical change in reservoir pressure during gas injection/extraction (Karev et al. 2020). It is important to emphasise that the reservoirs of hydrocarbon deposits are overlain by ‘traps’, i.e. impermeable rocks. On the scale of geological times, all rocks can be considered ductile. Therefore, in the absence of significant tectonics over geological times, the deviatoric part of the stress tensor (causing shear stresses) of reservoir rocks should relax, and under these conditions, the initial stress state can hardly be expected to differ appreciably from the uniform all-round compression (Zheltov and Khristianovich 1955). If drawdowns and overbalances at the bottomhole for UGSF conditions are usually small, then the difference in reservoir pressure (and the fluctuations in the acting stresses in the soil matrix caused by them) at the end points of the injection/ extraction cycle are very significant. As a result, the initial stress state in the reservoir compressed initially uniform all-round becomes unequal—the effective vertical stresses will be differ and horizontal stresses. Since the pressure gradients created by operational underbalances and repressions are quite small and changes in reservoir pressure in the whole reservoir on a reservoir scale in the states of minimum and maximum saturation are quite significant, it is the latter that determine changes in the stress state in the reservoir during gas injection and extraction. The reservoir-scale stress state changes will be considered without taking into account local perturbations introduced by operating wells during gas injection/ extraction. In the working range of stresses and pressures, reservoir rocks deform elastically. Then, as indicated above, in the presence of a single fluid component (gas), the mechanical behaviour of the rock obeys the equations of poroelasticity. The basic system of equations can be written as (5.1), (5.2), (5.3), (5.4), (5.5). Let the UGS formation be an extended, horizontally located reservoir of a thickness h much smaller than the characteristic size of the reservoir in plan L. In the initial state (before the initial development of the field and its subsequent conversion to a UGS), the stresses and pressures within the reservoir can be considered homogeneous with high accuracy. In general case, the initial stress state is characterized by a stress tensor having six independent components σ 0ij (it is natural to choose as these components the ones of the tensor in Cartesian coordinate system with one of axes directed vertically) or three values of principal stress values and three angles characterizing directions of principal stresses relative to vertical and horizon sides. So, as a first approximation, we can assume an initial stress state in the form of

44

5

Geomechanical Modelling of the Stress-Strain State in UGS. . .

σ 0ij = qδij

ð5:18Þ

The initial effective stresses will then be s0ij = σ 0ij þ ð1 - δÞp0 δij

ð5:19Þ

It is convenient to consider the stress state in increments: Δσ ij, Δsij, Δεij, Δpr, Δui— changes in total and effective stresses, strains, reservoir pressure and displacements. Here, the change in reservoir pressure Δpr = pr - p0

ð5:20Þ

A change in the pore pressure in the reservoir results in its deformations, and the problem of determining the stresses and strains in the reservoir and surrounding rocks is mathematically fully equivalent to the problem about eigenstrains in the inclusion (Eshelby 1957; Mura 1987). In this case, according to the solution of the problem (Eshelby 1957; Mura 1987), the stresses and strains field in the inclusion may be assumed constant, and the displacement field homogeneous everywhere except for the points near the contour. As reservoir pressure changes, the nature of the change in boundary conditions is determined by reservoir geometry. On the lateral (vertical) surfaces, the boundary conditions correspond to the constancy of the normal (horizontal) displacements due to the large extent of the reservoir in its plane Δu1 = Δu2 = 0

ð5:21Þ

Due to the property of the constancy of the fields within the inclusion, the same conditions are maintained for all internal points in the reservoir. On horizontal surfaces, the stresses remain constant Δσ 3 = 0

ð5:22Þ

For an isotropic reservoir, substituting (2.20), (2.21) into Hooke’s law (5.4) using (5.2) results in the following system Δs3 = Δσ 3 þ ð1 - δÞΔpw 1 ν Δε1 = 0 = Δs1 - ðΔs3 þ Δs2 Þ E E Here ν is the Poisson’s ratio of the reservoir rock. Under the assumption Δs1 = Δs2, it turns out

ð5:23Þ

5.2

Stress State in UGS Reservoirs Under Equal-Component Natural Stress State

Δs3 = ð1 - δÞΔpw ν ν Δs1 = Δs2 = Δs = ð1 - δÞΔpw 1-ν 3 1-ν

45

ð5:24Þ

This solution can be obtained by directly applying the Eshelby formula for an inclusion under the action of homogeneous eigenvolume deformations (Eshelby 1957; Mura 1987). The relations (5.24) are derived from the assumption of an infinitely large ratio of the reservoir length to its thickness (inclusion size ratio in Eshelby’s terminology). However, the accuracy of the results obtained for finitesized reservoirs is quite high, since the relative error introduced by the failure to take into account the finiteness of this ratio is on the order of the square of the ratio of the reservoir thickness to its extent. Thus, it can be seen that changes in pore pressure in the UGS reservoir lead to changes in effective stresses acting in the formation in whole, to increase horizontal stress. This does not contradict the above justification of the assumption about uniformity of initial overall compression of the reservoir, because gas injection/ extraction cycles in UGSF are usually carried out every season, and during such time intervals, the stresses obviously do not have time to relax. The dependence of effective reservoir stresses on changes in reservoir pressure Δpw will be as follows (Δpw < 0 when drawdown, Δpw > 0 when repression) s3 = s03 þ ð1 - δÞΔpw ν ð1 - δÞΔpw s1 = s01 þ 1-ν ν s2 = s02 þ ð1 - δÞΔpw 1-ν

ð5:25Þ

Where si (i = 1, 2, 3) are effective principal stresses acting in the reservoir in whole. Here, a Cartesian coordinate system is chosen in which axis 3 points vertically (i.e., orthogonal to the layering plane) and axes 1 and 2 lie in the horizontal plane (the layering plane); s01 , s02 , s03 are initial effective principal stresses from rock pressure in the reservoir; s1 , s2 , s3 are current effective principal stresses from rock pressure in the formation; ν—Poisson’s ratio of reservoir rock. In the case of an equal-component initial stress state s01 = s02 = s03 = q þ ð1 - δÞp0 s3 = q þ ð1 - δÞpr = q þ ð1 - δÞ½p0 þ Δpw  ν s1 = s2 = q þ ð1 - δÞ p0 þ Δp 1-ν w

ð5:26Þ

Accordingly, the total stresses from rock pressure in the reservoir are (Zheltov and Khristianovich 1955)

46

5

Geomechanical Modelling of the Stress-Strain State in UGS. . .

q3 = s3 - ð1 - δÞpr q1 = q2 = s1 - ð1 - δÞpr

ð5:27Þ

Or with accounting (5.20) q3 = q q 1 = q2 = q -

1 - 2ν ð1 - δÞΔpw 1-ν

ð5:28Þ

It follows from relation (5.28) when injecting gas, when formation pressure becomes higher than the initial pressure, which corresponds to Δpw >0, total stresses in the reservoir depth in the horizontal plane q1 = q2 become greater in absolute value than total stress in vertical direction q3 by the value 11--2νν ð1 - δÞΔpw . When gas is withdrawn, however, when the reservoir pressure becomes lower than the initial pressure, corresponding to Δpr 0, and when withdrawing gas Δ pw < 0.

ð5:30Þ

5.2

Stress State in UGS Reservoirs Under Equal-Component Natural Stress State

47

Substituting in (5.29) the values q1 and q3 from (5.30), bearing in mind that the effective stresses are defined by the relationsi = σ i + (1 - δ)pw, gives sr = σ r þ ð1 - δÞpw = - δpw sz = σ z þ ð1 - δÞpw = q þ ð1 - δÞpw 1 - 2ν sθ = σ θ þ ð1 - δÞpw = 2q - 2 ð1 - δÞΔpr þ ð2 - δÞpw 1-ν

ð5:31Þ

Given Δpr = pr - p0, (5.31) can be rewritten as sr = σ r þ ð1 - δÞpw = - δpw sz = σ z þ ð1 - δÞpw = q þ ð1 - δÞpw sθ = σ θ þ ð1 - δÞpw = 2q - 2

ð5:32Þ

1 - 2ν ð1 - δÞðpr - p0 Þ þ ð2 - δÞpw 1-ν

Thus, it follows from (5.32) that in case of vertical well, a change of reservoir pressure during gas extraction and injection leads to a change of stresses acting in the rock on the well contour, stresses will be the same in all points of well contour. This is because stresses from rock pressure acting at depth in horizontal planes, although changing as gas is withdrawn and injected, remain equal to each other, so that the well is constantly in a state of uniform compression by horizontal stresses from rock pressure at the depth of the reservoir. It should be note that a similar stress state is realized near the walls of a perforation hole in a vertical cased well.

5.2.3.2

Horizontal Well

While in the case of a vertical well, both at gas injection and extraction, stresses are the same at all points on the well contour, the situation is different in the case of a horizontal well. In this case, the well is in the field of stresses from rock pressure at the depth of formation q1 and q3 which, according to (5.28), are not equal to each other. At the same time, vertical stress q3 remains equal to natural vertical rock pressure q, while stresses in the horizontal plane change – at pr < p0 they are less in absolute value of stress in the vertical direction q, and at pr > p0 they are greater. Accordingly, the stresses on the well contour will also be different. Both cases are discussed below. 1. If pr > p0. In this case jq3 j < jq1 j. At point M on the well contour, Fig. 5.2, the compressive stresses σ θ will be maximum modulo and at point N will be minimum. The solution can be obtained by the superposition of the solutions of two problems: problem 1 (Lamé problem, (Leav 1935)): All-round uniform compression

48

5

Geomechanical Modelling of the Stress-Strain State in UGS. . .

Fig. 5.2 Stress state near a horizontal well

Fig. 5.3 Stresses in the vicinity of the well for problem 1 (Lamé problem) when injecting gas

q 3 x3 M

pw (t)

N x1

q 3

q3 and well pressure pw (Fig. 5.3), problem 2 (Kirsh problem): uniaxial compression in horizontal direction with stress ðq1 - q3 Þ, pressure in the well is zero (Fig. 5.4). The total stresses on the well contour for this problem are. ð1Þ

ð1Þ q3 , σθ = 2~ q3 þ p w σð1Þ r = - pw , σ z = ~

ð5:33Þ

Point M on the well contour (Fig. 5.4). By analogy with the solution of the Kirsch problem (Timoshenko and Goodyear 1979) on uniform tension of a plate with a circular hole for the distribution of circular ð2Þ stress σ θ along the contour of the hole we have

5.2

Stress State in UGS Reservoirs Under Equal-Component Natural Stress State

49

x3 M

ϕ

N x1

q∼ 1

−

q∼ 3

Fig. 5.4 Stresses in the vicinity of the well for problem 2 (analogous to the Kirsch problem) during gas injection ð2Þ

σ θ = ðq1 - q3 Þð1 - 2 cos 2φÞ

ð5:34Þ

where φ is the angle counted from the direction of the compressive stress ðq1 - q3 Þ, i.e. axis x1. The stresses σ θ on the well contour will be maximum modulo at point M in Fig. 5.4, i.e. at φ = π /2, and stresses will be equal to. ð2Þ

ð2Þ q1 - ~ q3 Þ σð2Þ r = 0, σz = 0, σθ = 3ð~

ð5:35Þ

The superposition of stresses at point M gives. σ r = - pw , σ z = q3 , σ θ = 3q1 - q3 þ pw

ð5:36Þ

Accordingly, since si = σ i + (1 - δ)pw sr = - δpw sz = q3 þ ð1 - δÞpw sθ = 3 q1 - q3 þ ð2 - δÞpw Substituting expression (5.34) here gives

ð5:37Þ

50

Geomechanical Modelling of the Stress-Strain State in UGS. . .

5

sr = - δpw sz = q þ ð1 - δÞpw 1 - 2ν sθ = 2q - 3 ð1 - δÞΔpr þ ð2 - δÞpw 1-ν

ð5:38Þ

Or, with that in mind Δpr = pr - p0, sr = - δpw sz = q þ ð1 - δÞpw 1 - 2ν sθ = 2q - 3 ð1 - δÞðpr - p0 Þ þ ð2 - δÞpw 1-ν

ð5:39Þ

Point N on the well contour (Fig. 5.4). The stresses σ θ on the well contour will be minimum modulo at point N, i.e. at φ = 0 in Fig. 5.4, and stresses are equal to ð2Þ

σ ðr2Þ = 0, σ ðz2Þ = 0, σ θ = - ðq1 - q3 Þ

ð5:40Þ

The total stresses are equal to σ r = - pw , σ z = q3 , σ θ = 3q3 - q1 þ pw

ð5:41Þ

Accordingly sr = - δpw sz = q3 þ ð1 - δÞpw

ð5:42Þ

sθ = 3q3 - q1 þ ð2 - δÞpw Substituting expression (5.28) here gives sr = - δpw sz = q þ ð1 - δÞpw 1 - 2ν sθ = 2q þ ð1 - δÞΔpr þ ð2 - δÞpw 1-ν

ð5:43Þ

Or, with in mind that Δpr = pr - p0, sr = - δpw sz = q þ ð1 - δÞpw 1 - 2ν sθ = 2q þ ð1 - δÞðpr - p0 Þ þ ð2 - δÞpw 1-ν

ð5:44Þ

5.2

Stress State in UGS Reservoirs Under Equal-Component Natural Stress State

Fig. 5.5 Stresses in the vicinity of the well for problem 1 (Lamé problem) in gas extraction

51

q 1 x3 M

pw (t)

N x1

q 1

2. If pr < p0 In this case jq3 j > jq1 j. At point N on the well contour (Fig. 5.5) the stresses σ θ will be maximum modulo and at point M they will be minimum. The solution can be obtained by the superposition of the solutions of the two problems: Problem 1 (the Lamé problem): All-round uniform compression q and well pressurepw. The total stresses on the borehole contour for this problem are. ð1Þ

σ rð1Þ = - pw , σ ðz1Þ = q1 , σ θ = 2q1 þ pw

ð5:45Þ

Problem 2 (analogous to the Kirsch problem): Uniaxial compression in the vertical direction with stress ðq3 - q1 Þ, pressure in the well is zero. Point M. By analogy with the solution of the Kirsch problem (Timoshenko and Goodyear 1979) on uniform compression of a plate with a circular hole for the distribution of ð2Þ circular stress σ θ along the contour of the hole we have ð2Þ

σθ = ð~q3 - ~q1 Þð1 - 2cos2φÞ

ð5:46Þ

where φ is the angle measured from the direction of the compressive stress ðq3 - q1 Þ, i.e. the x3 axis. The stresses σ θ on the well contour will be minimum in modulo at point M, i.e. at φ =0 in Fig. 5.6, and are equal to

52

Geomechanical Modelling of the Stress-Strain State in UGS. . .

5

Fig. 5.6 Stresses in the vicinity of the well for problem 2 during gas extraction (analogous to the Kirsch problem)

q∼ 3 − q∼ 1 x3 M

ϕ

ð2Þ

σ ðr2Þ = 0, σ ðz2Þ = 0, σ θ = - ðq3 - q1 Þ

N x1

ð5:47Þ

Total stresses are equal σ r = - pw , σ z = q1 , σ θ = 3q1 - q3 þ pw

ð5:48Þ

sr = - δpw sz = ~q1 þ ð1 - δÞpw sθ = 3~q1 - q~3 þ ð2 - δÞpw

ð5:49Þ

Accordingly

Substituting here (5.28) gives sr = - δpw 1 - 2ν sz = q ð1 - δÞΔpr þ ð1 - δÞpw 1-ν 1 - 2ν sθ = 2q - 3 ð1 - δÞΔpr þ ð2 - δÞpw 1-ν

ð5:50Þ

sr = - δpw 1 - 2ν sz = q ð1 - δÞðpr - p0 Þ þ ð1 - δÞpw 1-ν 1 - 2ν sθ = 2q - 3 ð1 - δÞðpr - p0 Þ þ ð2 - δÞpw 1-ν

ð5:51Þ

Or

5.3

Stress State in the Near-Wellbore Zone of the Reservoir Under Unequal. . .

53

Point N. The stresses σ θ on the well contour will be maximum in modulo at point N in Fig. 5.6 and opposite to it, i.e. At φ = 3/2 π and φ = π /2, and stresses are equal to ð2Þ

σ ðr2Þ = 0, σ ðz2Þ = 0, σ θ = 3ðq3 - q1 Þ

ð5:52Þ

The total stresses at point M are σ r = - pw ðt Þ, σ z = q1 , σ θ = 3q3 - q1 þ pw

ð5:53Þ

sr = - δpw sz = q1 þ ð1 - δÞpw sθ = 3q3 - q1 þ ð2 - δÞpw

ð5:54Þ

Substituting expression (5.28) here gives sr = - δpw 1 - 2ν sz = q ð1 - δÞΔpr þ ð1 - δÞpw 1-ν 1 - 2ν sθ = 2q þ ð1 - δÞΔpr þ ð2 - δÞpw 1-ν

ð5:55Þ

sr = - δpw 1 - 2ν sz = q ð1 - δÞðpr - p0 Þ þ ð1 - δÞpw 1-ν 1 - 2ν sθ = 2q þ ð1 - δÞðpr - p0 Þ þ ð2 - δÞpw 1-ν

ð5:56Þ

And finally

5.3

Stress State in the Near-Wellbore Zone of the Reservoir Under Unequal Natural Stress State

Currently, a great deal of attention is being paid to the unequal initial stress state in the estimation and modelling of the stress state in reservoirs of fields under development. In general, the initial state should be characterised by six independent constants. The values of these constants should be determined from independent experiments, which in most cases causes great difficulties. Probably for this reason it is generally assumed, either explicitly or implicitly, that one of the principal stresses coincides with the vertical direction. If accept this assumption, the description of the initial stress state requires four constants instead of six: the value of the vertical stress

54

5

Geomechanical Modelling of the Stress-Strain State in UGS. . .

(assumed to be one of the three principal stresses), the values of the remaining two principal stresses (lying in the horizontal plane due to the initial assumption), and the angle determining the direction of one of these stresses in the horizontal plane. The mathematical modelling of rock deformation and fracture, taking into account its actual deformation and strength properties, is a difficult task, as the properties themselves need to be known, and these can vary widely through the reservoir. Physical modelling of real processes of rock deformation and fracture in the vicinity of horizontal boreholes in the presence of an uneven stress field away from the borehole can be carried out at the experimental facility of the Institute for Problems in Mechanics of the Russian Academy of Sciences—the Triaxial Independent Load Test System (TILTS). The task is to compose specimen load programs corresponding to real stress conditions occurring in the vicinity of boreholes during their construction and operation. In order to development them, it is necessary to know the change of stresses in the vicinity of the well during pressure reduction at its bottomhole. As noted above, the initial stress field is characterised by three principal stresses, a vertical stress and two horizontal stresses (maximum and minimum). The ratios between the principal stresses may be different; their magnitudes are determined by the weight of the overlying rocks, the geological structure of the massif and tectonic processes. Depending on the magnitudes of natural stresses, the stresses occurring near the wells will also be different. The values of these stresses also depend on the mutual orientation of the principal natural stresses and the direction of borehole drilling. Respectively, the corresponding programs of load the specimens during physical modeling of drilling and well operation processes on TILTS will also be different. In general case for their creation it is necessary to carry out rather complicated three-dimensional calculations on defining of stresses acting in the well vicinity. In the case where the direction of the borehole coincides with the directions of the principal natural stresses, the load programs can be compiled analytically.

5.3.1

Effect of Reservoir Pressure Changes on the Stress State in UGS Reservoirs Under Unequal Natural Stress State

So, let the reservoir be loaded in its original state (before drilling the well) uneven compressive stresses from rock pressure, which are characterized by three principal rock pressure stresses σ 01 , σ 02 , σ 03 . Here a Cartesian coordinate system is chosen in which axis 3 is directed vertically (i.e. orthogonal to the bedding plane) and axes 1 and 2 lie in the horizontal plane (the bedding plane). σ 03 -total vertical stress perpendicular to the bed, σ 01 , σ 02—horizontal stresses in the bedding plane. Hereafter, compressive stresses are assumed to be negative and pore pressure positive.

5.3

Stress State in the Near-Wellbore Zone of the Reservoir Under Unequal. . .

55

Then the initial effective stresses are s01 = σ 01 þ ð1 - δÞp0 s02 = σ 02 þ ð1 - δÞp0 s03

= σ 03

ð5:57Þ

þ ð1 - δÞp0

where p0 is the initial pore pressure. Changes in the pore pressure in the reservoir lead to deformations in it, and the problem of determining the stress and strain field in the reservoir bed and in the host rock is mathematically fully equivalent to the problem about eigenstrains in the inclusion (Tertsagi 1961; Lehnitsky 1977). In this case, according to the solution of the problem (Tertsagi 1961; Lehnitsky 1977), the stresses and strains field in the inclusion may be assumed constant, and the displacement field homogeneous everywhere except for the points near the contour. For convenience, notations for the initial state of stress in the reservoir are introduced q = σ 03 , q1 = σ 01 , q2 = σ 02

ð5:58Þ

q = σ 3 , q1 = σ 1 , q 2 = σ 2

ð5:59Þ

And for the current ones

For certainty, it is assumed that |q| > |q1| > |q2|. The initial total vertical stress from rock pressure σ 03 is equal to σ 03 = q = - γH, where H is the depth of the bed, γ is the average specific weight of the overlying rocks which is usually assumed to be γ = 2, 3  103 kg/m3. When the pore pressure changes, the pattern of the change in the boundary conditions is determined by the reservoir geometry. Similarly to how it was done in Sect. 5.2.2 for the equal-component natural stress state, considering that on the lateral (vertical) surfaces, the boundary conditions correspond to the constancy of the normal (horizontal) displacements due to the large extent of the reservoir in its plane Δu1 = Δu2 = 0

ð5:60Þ

and on the horizontal surfaces the stresses are kept constant, so that Δσ 3 = 0 S, the vertical stress from rock pressure in the reservoir is maintained when the reservoir pressure changes, i.e.

56

5

Geomechanical Modelling of the Stress-Strain State in UGS. . .

q = q = σ 03 = const

ð5:61Þ

and for the effective stresses acting on the reservoir ground skeleton it can be write s1 = q1 þ ð1 - δÞpr , s2 = q2 þ ð1 - δÞpr , s3 = q þ ð1 - δÞpr

ð5:62Þ

where pr is the current reservoir pressure. Consider how the stress state changes during gas injection/extraction. It is convenient to consider the stress state in increments. Let’s denote changes in total and effective stresses, deformations and displacements when reservoir pressure changes by the value Δpr = pr - p0 by Δσ i, Δsi, Δεi, Δpr, Δui. For an isotropic reservoir, substituting (5.62) into Hooke’s law with (5.60) and (5.59) results in the following system Δs3 = ð1 - δÞΔpr 1 Δε1 = 0 = Δs1 E 1 Δε2 = 0 = Δs2 E

ν ðΔs3 þ Δs2 Þ E ν ðΔs3 þ Δs1 Þ E

ð5:63Þ

Here ν is the Poisson’s ratio of the reservoir rock. Hence Δs1 = νðΔs3 þ Δs2 Þ Δs2 = νðΔs3 þ Δs1 Þ Δs3 = ð1 - δÞΔpr

ð5:64Þ

Solving system (5.64), we find Δs3 = ð1 - δÞΔpr ν ν Δs1 = Δs2 = Δs = ð1 - δÞ Δpr 1-ν 3 1-ν

ð5:65Þ

Thus, we see that a change in pore pressure in the UGS reservoir results in a change in effective stress, the relationship between the change in horizontal and vertical effective stresses is determined according to the well-known Dinnick formula. For changes in effective stresses in a reservoir with reservoir pressure change Δpr we have (Δpr < 0—at current reservoir pressure lower than the initial one, i.e. at pr < p0, Δpr > 0—at current reservoir pressure higher than the initial one, i.e. at p r > p0)

5.3

Stress State in the Near-Wellbore Zone of the Reservoir Under Unequal. . .

s3 = s03 þ ð1 - δÞΔpr ν ð1 - δÞΔpr s1 = s01 þ 1-ν ν ð1 - δÞΔpr s2 = s02 þ 1-ν

57

ð5:66Þ

Here s01 , s02 , s03 —initial effective stresses (acting in the soil skeleton) from rock pressure in the reservoir, given by (5.57); s1 , s2 , s3 —current effective stresses. Accordingly, the total stresses from rock pressure in the reservoir are (Lehnitsky 1977) q3 = s3 - ð1 - δÞpr q1 = s1 - ð1 - δÞpr

ð5:67Þ

q2 = s2 - ð1 - δÞpr From (5.66) and (5.65), considering (5.57), (5.59), (5.61), we find q3 = q 1 - 2ν ð1 - δÞΔpr q1 = q 1 1-ν 1 - 2ν q2 = q2 ð1 - δÞΔpr 1-ν

ð5:68Þ

It follows from relation (5.68) that when the reservoir pressure is higher than the initial pressure, which corresponds to pr > p0, the total stresses in the reservoir depth in the horizontal plane q1 and q2 increase modulo by the value 11--2ν ν ð1 - δÞΔpr . When the reservoir pressure becomes lower than the initial pressure, corresponding to Δpr < p0, the total stresses in the reservoir depth in the horizontal plane q1 and q2 decrease by 11--2ν ν ð1 - δÞΔpr . It should be emphasized that the change in stresses in the reservoir when the reservoir pressure changes in an equal-component natural stress state and an unequal-component natural stress state are equal to each other.

5.3.2

Stresses Occurring in the Near-Wellbore Zone of the Reservoir During Changes in Reservoir Pressure in an Unequal Natural Stress State

The change in stress state at the depth of the reservoir results in a change in the stresses acting in the vicinity of the wells, and these changes will be different for vertical and horizontal wells. In addition, the stress state occurring in the vicinity of horizontal wells will depend on the direction in which the horizontal well is being drilled relative to the direction of the minimum and maximum horizontal stresses. Section 5.3.1 shows that as reservoir pressure changes during gas injection/ extraction, the vertical stress in the UGS reservoir is kept equal to q, while the

58

5

Geomechanical Modelling of the Stress-Strain State in UGS. . .

Fig. 5.7 Horizontal section of a vertical well and natural stresses acting in the reservoir

Fig. 5.8 Lamé problem for a vertical well

horizontal stresses change, and can both increase and decrease. Current horizontal stresses corresponding to reservoir pressure pr, as before, we will denote q1 and q2 .

5.3.2.1

Vertical Well

Let’s find the stresses acting on the contour of the vertical well in the unequal natural stress state of the reservoir. Figure 5.7 shows a horizontal section of a vertical well and horizontal stresses acting in the reservoir away from the well. The problem under consideration can be represented as a superposition of the two problems: Problem 1. All-round uniform compression away from the borehole by stresses q2 . Inside the well the pressure is pw, Fig. 5.8. This problem is known as the Lamé problem.

5.3

Stress State in the Near-Wellbore Zone of the Reservoir Under Unequal. . .

59

Fig. 5.9 Kirsch problem for a vertical well

2. Uniaxial compression away from the hole in the vertical direction with tension q1 - q2 , Fig. 5.9. There is no pressure inside the hole. This problem is known as the Kirsch problem. For the first problem, it follows from the solution of the Lamé problem that after drilling the borehole, the radial, circumferential and axial stresses at all points on the contour of the vertical borehole will be the same and equal to σ r = - pw σ θ = 2q2 þ pw

ð5:69Þ

σz = q For the second problem, it follows from the solution of the Kirsch problem that the circumferential stresses σ θ vary along the contour of the well. They are tensile stresses at point M (and at the opposite point) and reach a maximum modulo at point N (and at the opposite point). At point M the stresses are equal: σr = 0 σ θ = - ð q1 - q 2 Þ

ð5:70Þ

σz = 0 The stresses acting on the borehole contour will be equal to the sum of the stresses got for each of the two above problems. A third principal stress σ z acting along the borehole axis equals to the vertical rock pressure, i.e. σ z = q = - γH. At point N the total stresses are equal:

60

Geomechanical Modelling of the Stress-Strain State in UGS. . .

5

σr = 0 σ θ = 3ðq1 - q2 Þ

ð5:71Þ

σz = 0 Then expressions for total stresses at point M from (5.69) and (5.70) can be written σ r = - pw σ θ = 3q2 - q1 þ pw

ð5:72Þ

σz = q The total stresses at point N from (5.69) and (5.71) are equal: σ r = - pw σ θ = 3 q1 - q 2 þ p w σz = q

ð5:73Þ

Accordingly, for a permeable formation, the effective stresses acting in the soil skeleton are equal: at point M Sr = - δpw Sθ = 3q2 - q1 þ ð2 - δÞpw

ð5:74Þ

Sz = q þ ð1 - δÞpw Substituting expressions (5.68) we find Sr = - δpw sθ = 3q2 - q1 - 2

1 - 2ν ð1 - δÞΔpr þ ð2 - δÞpw 1-ν

ð5:75Þ

Sz = q þ ð1 - δÞpw at point N

Sr = - δpw Sθ = 3 q1 - q2 þ ð2 - δÞpw Sz = q þ ð1 - δÞpw Given (5.68), we obtain

ð5:76Þ

5.3

Stress State in the Near-Wellbore Zone of the Reservoir Under Unequal. . .

Sr = - δpw sθ = 3q1 - q2 - 2

1 - 2ν ð1 - δÞΔpr þ ð2 - δÞpw 1-ν

61

ð5:77Þ

Sz = q þ ð1 - δÞpw

Introduce two lateral expansion coefficients α1 and α2 in the direction of maximum and minimum horizontal stress, so that q1 = α1 q, q2 = α2 q, α1 > α2

ð5:78Þ

Then for the effective stresses acting on the contour of the vertical well, Fig. 5.7, from (5.70) and (5.72) we have point M Sr = - δpw sθ = ð3 α2 - α1 Þq - 2 Sz = q þ ð1 - δÞpw

1 - 2ν ð1 - δÞΔpr þ ð2 - δÞpw 1-ν

ð5:79Þ

1 - 2ν ð1 - δÞΔpr þ ð2 - δÞpw 1-ν

ð5:80Þ

point N Sr = - δpw sθ = ð3 α1 - α2 Þq - 2 Sz = q þ ð1 - δÞpw

Assuming Δpr = pr - p0 and pw = pr + Δpw, where Δpw > 0 is the repression in the well during gas injection, and Δpw < 0 is the depression in the well during gas extraction, (5.79) and (5.80) can be rewritten as point M Sr = - δpw sθ = ð3 α2 - α1 Þq - 2 Sz = q þ ð1 - δÞpw

1 - 2ν ð1 - δÞðpr - p0 Þ þ ð2 - δÞpw 1-ν

ð5:81Þ

1 - 2ν ð1 - δÞðpr - p0 Þ þ ð2 - δÞpw 1-ν

ð5:82Þ

point N Sr = - δpw sθ = ð3 α1 - α2 Þq - 2 Sz = q þ ð1 - δÞpw

62

5

Geomechanical Modelling of the Stress-Strain State in UGS. . .

Fig. 5.10 Stresses acting in the vicinity of a horizontal well

5.3.2.2

Horizontal Well

Let’s find the stresses acting on the contour of the horizontal well in the unequal natural reservoir stress state, Fig. 5.10. In contrast to a vertical well, the distribution of stresses on the contour of a horizontal well will depend on the direction in which it is drilled relative to the directions of maximum and minimum horizontal principal stresses. Two extreme cases will be discussed below: • The axis of the horizontal borehole coincides with the direction of the maximum horizontal stress; • The axis of the horizontal borehole coincides with the direction of the minimum horizontal stress; Stress values at the well contour for other well orientations lie between these two cases.

5.3.2.2.1

The Well Is Drilled Along the Direction of Maximum Horizontal Stress

Figure 5.11 shows the vertical cross section of a horizontal well drilled along the maximum horizontal stress, and the rock pressure stresses acting in the reservoir away from the well. The third principal stress q2 acts along the axis of the well. The solution of this problem is reduced to the vertical well solution by replacing q1 with q and vice versa. Then from (5.72), (5.73), (5.74), (5.75), (5.76), (5.77), (5.78), (5.79), (5.80) at points M and N for full stress σ i and for the case of permeable formation for effective stress Si we have.

5.3

Stress State in the Near-Wellbore Zone of the Reservoir Under Unequal. . .

63

Fig. 5.11 Vertical section of a horizontal well drilled along the maximum horizontal stress

at point M σ r = - pw σ θ = 3 q2 - q þ pw

ð5:83Þ

σ z = q1 Sr = - δpw Sθ = 3 q2 - q þ ð2 - δÞpw Sz = q1 þ ð1 - δÞpw

ð5:84Þ

Substituting expressions (5.68) in (5.84), we find Sr = - δpw s θ = 3 q2 - q - 3 Sz = q1 -

1 - 2ν ð1 - δÞΔpr þ ð2 - δÞpw 1-ν

ð5:85Þ

1 - 2ν ð1 - δÞΔpr þ ð1 - δÞpw 1-ν

at point N σ r = - pw σ θ = 3q - q2 þ pw σ z = q1 Given (5.68), we obtain. Sr = - δpw

ð5:86Þ

64

5

Geomechanical Modelling of the Stress-Strain State in UGS. . .

Sr = - δpw Sθ = 3 q - q2 þ

1 - 2ν ð1 - δÞΔpr þ ð2 - δÞpw 1-ν

ð5:87Þ

1 - 2ν Sz = q1 ð1 - δÞΔpr þ ð1 - δÞpc 1-ν

Using (5.73), for the effective stresses acting at point M on the contour of the horizontal well, Fig. 5.10, from (5.85) we have Sr = - δpw sθ = ð3α2 - 1Þq - 3

1 - 2ν ð1 - δÞΔpr þ ð2 - δÞpw 1-ν

ð5:88Þ

1 - 2ν Sz = α1 q ð1 - δÞΔpr þ ð1 - δÞpw 1-ν

Using (5.78), for the effective stresses acting at point N on the contour of the horizontal well, Fig. 5.10, from (5.87) we have Sr = - δpw

1 - 2ν ð1 - δÞΔpr þ ð2 - δÞpw 1-ν 1 - 2ν Sz = α1 q ð1 - δÞΔpr þ ð1 - δÞpw 1-ν Sθ = ð3 - α2 Þq þ

ð5:89Þ

Here Δpr = pr - p0 and pw = pr + Δpw, where Δpw > 0 is the drawdown in the well during gas injection and Δpw < 0 is the repression in the well during gas extraction.

5.3.2.2.2

The Well Is Drilled Along the Direction of the Minimum Horizontal Stress

Figure 5.12 shows the vertical cross section of a horizontal well drilled along the minimum horizontal stress, and the rock pressure stresses acting in the reservoir Fig. 5.12 Vertical section of a horizontal well drilled along the minimum horizontal stress

5.3

Stress State in the Near-Wellbore Zone of the Reservoir Under Unequal. . .

65

away from the well. The third principal stress q2 acts along the axis of the well. The solution to this problem is reduced to the solution for a horizontal well drilled along the maximum horizontal stress by replacing q2 with q1 and vice versa. Then from (5.83), (5.84), (5.85), (5.86), (5.87) for total σ i and effective Si stresses at points M and N we have. at point M σ r = - pw σ θ = 3 q1 - q þ pw σ z = q2

ð5:90Þ

Sr = - δpw Sθ = 3 q1 - q þ ð2 - δÞpw

ð5:91Þ

Sz = q2 þ ð1 - δÞpw Substituting expressions (5.63) in (5.91), we find Sr = - δpw

1 - 2ν ð1 - δÞΔpr þ ð2 - δÞpw 1-ν 1 - 2ν Sz = q2 ð1 - δÞΔpr þ ð1 - δÞpw 1-ν sθ = 3q1 - q - 3

ð5:92Þ

at point N σ r = - pw σ θ = 3 q - q 1 þ pw σ z = q2 Sr = - δpc Sθ = 3 q - q1 þ ð2 - δÞpw

ð5:93Þ

ð5:94Þ

Sz = q2 þ ð1 - δÞpw Given (5.63), we obtain Sr = - δpw Sθ = 3 q - q1 þ Sz = q2 -

1 - 2ν ð1 - δÞΔpr þ ð2 - δÞpw 1-ν

ð5:95Þ

1 - 2ν ð1 - δÞΔpr þ ð1 - δÞpw 1-ν

Using (5.78) for the effective stresses acting at point M on the contour of the horizontal well, Fig. 5.12, from (5.92) we have

66

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Geomechanical Modelling of the Stress-Strain State in UGS. . .

Sr = - δpw sθ = ð3 α1 - 1Þq - 3

1 - 2ν ð1 - δÞΔpr þ ð2 - δÞpw 1-ν

ð5:96Þ

1 - 2ν Sz = α 2 q ð1 - δÞΔpr þ ð1 - δÞpw 1-ν

Using (5.87) for the effective stresses acting at point N on the contour of the horizontal well, Fig. 5.12, from (5.95) we have Sr = - δpw

1 - 2ν ð1 - δÞΔpr þ ð2 - δÞpw 1-ν 1 - 2ν Sz = α2 q ð1 - δÞΔpr þ ð1 - δÞpw 1-ν Sθ = ð3 - α1 Þq þ

5.4

ð5:97Þ

Conclusions

1. A geomechanical approach has been developed to justify rational technological regimes of UGS wells and optimise the well stock. 2. An important result which based on the developed geomechanical approach was an understanding of the mechanism by which stresses in UGS reservoirs change as gas is extracted or injected, and their magnitude is related to reservoir pressure levels. 3. The mechanism of how the stresses in UGS reservoirs change as gas is extracted or injected has been studied for both equal-component natural reservoir stresses and unequal-component natural stresses. 4. The stresses arising in the bottomhole zone of the reservoir during changes in pore pressure for vertical and horizontal wells have been calculated. The calculations are performed both for equal-component natural stress state in the formation and for unequal-component one.

References Biot MA (1935) Le problème de la consolidation des matières argileuses sous une charge. Ann Soc Sc de Brux 55:110–113 Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12:155–165 Eshelby JD (1957) Proc R Soc Lond A 241:376 Karev VI, Kovalenko YF (2006) Dependence of formation bottomhole zone permeability on underbalance and face design for different types of rocks (in Russian). Tekhnologii TEC 6: 59–63

References

67

Karev VI, Kovalenko YF, Zhuravlev AB, Ustinov KB (2015) A model of filtration into a borehole taking into account dependence of permeability on stresses (in Russian). Process Geospheres 4(4):35–44 Karev VI, Korolev DS, Kovalenko YF, Ustinov KB (2020. Special issue) Geomechanical and physical modelling of deformation processes in underground gas storage reservoirs under cyclic changes in reservoir pressure (in Russian). Gas Ind 4(808):46–52 Leav A (1935) Mathematical theory of elasticity (in Russian). ONTI NKGiP USSR, Moscow, Leningrad, p 676 Lehnitsky SG (1977) Theory of anisotropic body elasticity (in Russian). Nauka, Moscow, p 415 Mura T (1987) Micromechanics of defects in solids. Martinus Nijhoff, Dordrecht/Boston/Lancaster Tertsagi K (1961) Theory of soil mechanics (in Russian). Gosstroyizdat, Moscow Timoshenko SP, Goodyear J (1979) Theory of elasticity (in Russian). Nauka, Moscow, p 560 Zheltov YP, Khristianovich SA (1955) On hydraulic fracturing of an oil-bearing reservoir (in Russian). Izvestia of the USSR Academy of Sciences Dept of Technical Sciences 5:3–41

Chapter 6

Physical Modelling Programs of Rock Deformation and Fracture in the Near-Wellbore Zone of USG During Gas Injection and Extraction

It was noted above that during the operation of UGS, the stresses acting in the vicinity of wells are affected by two factors caused by gas injection and extraction. The first factor affecting the stress state in the vicinity of a well is the periodic underbalance (during gas extraction) and repression (during gas injection) created at the bottom of the well. The second factor is due to cyclical changes in reservoir pressure during gas injection/extraction. Both of these factors can be the cause of rock fracture in the UGS and sand production. At the injection or extraction stage, when the reservoir pressure is still close to the initial one, the stress state in the vicinity of the well depends mainly on the amount of underbalance or repression in the well. As reservoir pressure changes, its magnitude begins to have an increasing influence on the stress state in the borehole zone. Theoretically, it is extremely difficult to assess the degree of risk of rock fracture due to each of these factors, given the complexity of building a model for cyclically varying stresses, and their impact on rock strength properties. The solution is to experimentally simulate these processes under conditions of actually occurring stresses in the borehole zone. In order to determine the critical stresses that lead to rock failure, direct physical modelling of rock deformation and fracture processes in the vicinity of the USG wells under the action of real stresses occurring in the formation during cyclic pressure changes at the well bottom should be carried out on the TILTS. Modelling should also be carried out without taking into account changes in reservoir pressure in order to determine the contribution of each factor: cyclic changes in bottomhole and reservoir pressures during gas injection and pumping out. In order to conduct such experiments, load programs that modeling using the TILTS in specimens of rock the stresses actually occurring in the vicinity of the borehole must be created. The load programs are created on the basis of calculations. Relations for stresses on contour of horizontal and vertical wells are given in Chap. 2. Stresses calculations are performed for both equal-component natural © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Karev, Y. Kovalenko, Geomechanical Aspects of Operation of Underground Gas Storage, Springer Geology, https://doi.org/10.1007/978-3-031-34765-8_6

69

70

6 Physical Modelling Programs of Rock Deformation and Fracture in. . .

reservoir stresses and unequal-component natural stresses. The tests in each of the programs determine at what magnitude of underbalance the inelastic deformation of the rock occurs, with cracking or loosening.

6.1 6.1.1

Equilibrium Natural Stress State in the Reservoir Specimen Load Programs for Physical Modelling of Drawdown and Repressions on Bottom Hole Without Regard to Reservoir Pressure Changes

Experiments were carried out at the TILTS to physically simulate deformation in the borehole zone under the action of underbalance and repression during gas extraction and injection. The experiments simulated the cycling of 1 MPa underbalance and repression on bottom hole of well. After the stresses corresponding to the specified pressure drawdowns and repressions were created in the specimens, they were kept under the given stress state to measure the rock creep. In the course of experiments, strains of specimens in three directions and change of permeability of specimens in one of directions were measured. Appropriate load programs have been developed at the TILTS for rock specimens from the UGS №1, UGS №2, UGS №3 and UGS №4. The programs take into account the depths of the reservoirs of underground gas storage facilities and reservoir pressures in them. The programs simulate on specimens stress changes in the vicinity of wells at cyclic creation of repressions and drawdowns at their faces. When testing the specimens by the developed programs the deformation and destruction processes taking place in the reservoir during gas injection and extraction in the process of UGS operation are simulated. The programs provide for measurement of specimen permeability in the course of tests. Chapter 5 gives expressions (5.15) for stresses on the contour of a vertical uncased well at an equal-component initial stress state. In the same place, relations (5.17) are given for the change of stresses on the well contour as the pressure at the bottom hole changes. Based on relations (5.15), a program of specimen load has been compiled for physical simulation on TILTS of deformation and filtration processes taking place in bottomhole zone of underground gas storage well under the action of alternating pressures due to underbalance and repressions at well bottom during gas extraction and injection. Figure 6.1 shows such a program for one of the tested specimens. The stresses depicted on it s1, s2, s3 are stresses applied along axes 1, 2, 3 to the specimen. They correspond to the absolute values of stresses |sz|, |sθ|, |sr| acting in the vicinity of the well. In load step 1, the specimen is compressed uniformly on all sides to a stress equal to the difference between the absolute value of the rock pressure |q| and the reservoir pressure value p0. The stresses at the end of stage 1 correspond to the stresses acting in the soil skeleton before the borehole was drilled, provided that the vertical and

6.1

Equilibrium Natural Stress State in the Reservoir

71

Fig. 6.1 Specimen load program for simulating downhole pressure cycling during gas extraction and injection

lateral rock pressure are equal to the weight of the overlying rocks s1 = s2 = s3 = γH - p0. In the second stage of load, one stress component (s2) continues to increase, the second (s1) remains constant and the third (s3) decreases, with the load changing so that the average stress s = (s1 + s2 + s3)/3 remains constant throughout stage 2. The end point of the stage corresponds to the state when the well is drilled and the bottom hole pressure is equal to the reservoir pressure s1 = γH - p0, s2 = 2(γH - p0), s3 = δp0. The experiments assumed δ = 0.2. Stage 3 corresponds to the process of injecting gas into the well at repressure Δp+. The specimen was then held at constant load for about 10 min (stage 4), which simulates the period of UGS downtime. Steps 5, 6 then simulate the process of gas extraction at underbalance Δp-. In step 7, the specimen is held at constant load again and then the gas injection process is simulated again. Up to ten such load-unload cycles corresponding to gas injection-extraction were created in the course of experiments. The experiments recorded the strains in the three axes of the specimen and its permeability in one of the directions.

6.1.2

Physical Modelling Programs at the TILTS of Deformation and Filtration Processes in the Bottomhole Zone Under the Action of Alternating Loads Arising from Cyclic Changes in Reservoir Pressure

As indicated above, the main influence on the processes of rock deformation and fracture in the vicinity of wells (especially horizontal wells) is caused by stresses

72

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

arising from changes in the stress state at the depth of the reservoir during decreasing and increasing reservoir pressure in the UGS. Relationships for the resulting stresses at the horizontal and vertical well are given in Chap. 2. The programs of physical simulation of deformation and filtration processes in the bottomhole zone of UGS wells under the influence of alternating loads during gas extraction and injection, developed on the basis of these relations, are given below. Since the stresses arising in the vicinity of vertical and horizontal wells are different, the programs of rock specimen load during the tests will be different for them. In turn, the test programs for horizontal wells corresponding to stress changes at points M and N on the well contour (Fig. 5.2) will also be different because the stresses at these points during gas injection and extraction are different. Figure 6.2 shows a schematic view of a specimen load program simulating the change in stresses on an UGS well contour during a single gas injection/extraction cycle. The stresses depicted on it s1, s2, s3 are the stress values in MPa applied to the specimen during the experiment along load axes 1, 2, 3 of the TILTS, s1, s2, s3 > 0. They correspond to the stresses |sz|, |sθ|, |sr|, acting on the borehole contour.

Fig. 6.2 Schematic view of the specimen load program modelling the change in stresses on the UGS well contour during a single gas injection/extraction cycle

6.1

Equilibrium Natural Stress State in the Reservoir

s1 = jsz j, s2 = jsθ j, s3 = jsr j

6.1.2.1

73

ð6:1Þ

Vertical Uncased Well

1. Point A Point A corresponds to the initial condition in the reservoir away from the well. In the initial state, the soil skeleton is loaded with effective all-round compression stresses sAr = sAz = sAθ = q þ ð1 - δÞp0 , where p0 is the initial reservoir pressure. Accordingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have sA1 = sA2 = sA3 = jqj - ð1 - δÞp0 . 2. Point B Point B corresponds to the situation when the well is drilled and the pressure in the well is equal to the initial reservoir pressure, i.e. pw = p0. Then for the stresses at point B, assuming in (5.32) pw = p0 and pr = p0, we find sВr = - δp0 sВz = q þ ð1 - δÞp0 sВθ

ð6:2Þ

= 2q þ ð2 - δÞp0

Accordingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have sВ1 = jqj - ð1 - δÞp0 sВ2 = 2jqj - ð2 - δÞp0

ð6:3Þ

sВ3 = δp0

3. Point C—end of gas injection pr = pmax r . The BC section of the load program corresponds to the stage of gas injection at which the formation pressure is higher than the initial pressure, i.e. pr > p0. Point B corresponds to the start of gas injection by creating a repression at the bottom of the well Δpw, so pw = pr + Δpw. Stresses on the vertical well contour during gas injection at reservoir pressure pr are given by relations (5.32). In these relations one must put pw = pr + Δpw.

74

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

sr = - δðpr þ Δpw Þ sz = q þ ð1 - δÞðpr þ Δpw Þ 1 - 2ν ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr þ Δpw Þ sθ = 2 q - 2 1-ν

ð6:4Þ

Then at point C, which corresponds to the maximum reservoir pressure pr = pmax r one can write þ Δpw scr = - δ pmax r þ Δpw scz = q þ ð1 - δÞ pmax r 1 - 2ν scθ = 2q - 2 - p0 þ ð2 - δÞ pmax þ Δpw ð1 - δÞ pmax r r 1-ν

ð6:5Þ

And correspondingly for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have þ Δpw sc1 = jqj - ð1 - δÞ pmax r 1 - 2ν sc2 = 2 jqj þ 2 ð1 - δÞ pmax - p0 - ð2 - δÞ pmax þ Δpw r r 1-ν þ Δpw sc3 = δ pmax r

ð6:6Þ

4. Point D—at gas extraction pr = p0 The CD load program section corresponds to the gas extraction stage after gas injection is completed, at which the reservoir pressure still remains above the initial pressure, i.e. pr > p0. At this section, a depression is created at the bottom of the well Δpw. Stresses on the contour of the vertical well during gas extraction at reservoir pressure pr are given by relations (5.32). We have sr = - δðpr - Δpw Þ sz = q þ ð1 - δÞðpr - Δpw Þ 1 - 2ν sθ = 2q - 2 ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr - Δpw Þ 1-ν

ð6:7Þ

Then at point D, corresponding to a reservoir pressure equal to the initial pD r = p0 sD r = - δðp0 - Δpw Þ sD z = q þ ð1 - δÞðp0 - Δpw Þ

ð6:8Þ

sD θ = 2q þ ð2 - δÞðp0 - Δpw Þ And correspondingly for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have

6.1

Equilibrium Natural Stress State in the Reservoir

75

sD 1 = jqj - ð1 - δÞðp0 - Δpw Þ sD 2 = 2jqj - ð2 - δÞðp0 - Δpw Þ sD 3 = δðp0 - Δpw Þ

ð6:9Þ

5. Point E—end of gas extraction pr = pmin r The DE section of the load program corresponds to the gas extraction stage where the formation pressure is lower than the initial pressure, i.e. pr < p0. At this section, a depression Δpw is created at the well bottom. Stresses on the vertical well contour during gas extraction at reservoir pressure pr are given by relations (5.32). sr = - δðpr - Δpw Þ sz = q þ ð1 - δÞðpr - Δpw Þ 1 - 2ν sθ = 2 q - 2 ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr - Δpw Þ 1-ν

ð6:10Þ

Then at the point E corresponding to the minimum reservoir pressure pr = pmin r sEr = - δ pmin r - Δpw sEz = q þ ð1 - δÞ pmin r - Δpw 1 2ν min sEθ = 2q - 2 ð1 - δÞ pmin r - pw þ ð2 - δÞ pr - Δpw 1-ν

ð6:11Þ

And correspondingly for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have sE1 = jqj - ð1 - δÞ pmin r - Δpw 1 2ν min sE2 = 2 jqj þ 2 ð1 - δÞ pmin r - p0 - ð2 - δÞ pr - Δpw 1-ν sE3 = δ pmin r - Δpw

ð6:12Þ

6. Point F—at gas injection pr = p0 The EF section of the load program corresponds to gas injection stage after gas extraction end at which the reservoir pressure is lower than the initial pressure, i.e., pr < p0. At this section, a repression is created at the wellbore Δpw. Stresses on the vertical well contour during gas injection at reservoir pressure pw are given by relations (5.32). We have

76

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

sr = - δðpr ðtÞ þ Δpw Þ sz = q þ ð1 - δÞðpr þ Δpw Þ 1 - 2ν sθ = 2q - 2 ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr þ Δpw Þ 1-ν

ð6:13Þ

Then at the point F corresponding to the reservoir pressure equal to the initial pFr = p0 . sFr = - δðp0 þ Δpw Þ sFz = q þ ð1 - δÞðp0 þ Δpw Þ sFθ = 2q þ ð2 - δÞðp0 þ Δpw Þ

ð6:14Þ

And correspondingly for the stresses applied to the specimen during the experiment along the axes of the TILTS, we have sF1 = jqj - ð1 - δÞðp0 þ Δpw Þ sF2 = 2 jqj - ð2 - δÞðp0 - Δpw Þ sF3 = δðp0 þ Δpw Þ

6.1.2.2

ð6:15Þ

Horizontal Well

6.1.2.2.1

Lateral Point on Horizontal Well Contour

1. Point A Point A corresponds to the initial condition in the reservoir away from the well. The soil skeleton is loaded with effective all-round compression stresses sAr = sAz = sAθ = q þ ð1 - δÞp0 , where p0 is the initial reservoir pressure. Accordingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we havesA1 = sA2 = sA3 = jqj - ð1 - δÞp0 . 2. Point B Point B corresponds to the situation when the well is drilled and the pressure in the well is equal to the initial reservoir pressure, i.e. pw = p0. Then for the stresses at point B, assuming in (5.44) pw = p0 and pr = p0, we find sВr = - δp0

sВz = q þ ð1 - δÞp0 sВθ

ð6:16Þ

= 2q þ ð2 - δÞp0

Accordingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have

6.1

Equilibrium Natural Stress State in the Reservoir

77

sВ1 = jqj - ð1 - δÞp0

sВ2 = 2jqj - ð2 - δÞp0 sВ3 = δp0

ð6:17Þ

3. Point C—end of gas injection pr = pmax r The BC section of the load program corresponds to the stage of gas injection at which the reservoir pressure is higher than the initial pressure, i.e. pr > p0 and jq3 j < jq1 j. Point B corresponds to the start of gas injection by creating a repression at the bottom of the well Δpw. The stresses at point N on the contour of a horizontal well during gas injection at reservoir pressure pr > p0 are given by relations (5.44). We have sr = - δðpr þ Δpw Þ sz = q þ ð1 - δÞðpr þ Δpw Þ 1 - 2ν sθ = 2 q þ ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr þ Δpw Þ 1-ν

ð6:18Þ

Then at point C, which corresponds to the maximum reservoir pressure pr = pmax r scr = - δ pmax þ Δpw r c þ Δpw sz = q þ ð1 - δÞ pmax r 1 - 2ν scθ = 2q þ - p0 þ ð2 - δÞ pmax þ Δpw ð1 - δÞ pmax r r 1-ν

ð6:19Þ

And correspondingly for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have þ Δpw sc1 = jqj - ð1 - δÞ pmax r 1 - 2ν sc2 = 2 jqj - p0 - ð2 - δÞ pmax þ Δpw ð1 - δÞ pmax r r 1-ν þ Δpw sc3 = δ pmax r

ð6:20Þ

4. Point D—at gas extraction pr = p0 The CD section of the load program corresponds to the gas extraction stage after gas injection is complete, at which the reservoir pressure still remains above the initial pressure, i.e. pr > p0 and jq3 j < jq1 j. In this section, a depression Δpw is created at the bottom of the well. The stresses at point N on the contour of a horizontal well during gas extraction at reservoir pressure pr > p0 are given by relations (5.44). We have

78

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

sr = - δðpr - Δpw Þ sz = q þ ð1 - δÞðpr - Δpw Þ 1 - 2ν ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr - Δpw Þ sθ = 2q þ 1-ν

ð6:21Þ

Then at point D, corresponding to a reservoir pressure equal to the initial pr = p0 sD r = - δðp0 - Δpw Þ sD z = q þ ð1 - δÞðp0 - Δpw Þ sD θ = 2q þ ð2 - δÞðp0 - Δpw Þ

ð6:22Þ

And correspondingly for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have sD 1 = jqj - ð1 - δÞðp0 - Δpw Þ sD 2 = 2 jqj - ð2 - δÞ ðp0 - Δpw Þ sD 3 = δðp0 - Δpw Þ

ð6:23Þ

Point D corresponds to the moment when the reservoir pressure becomes equal to the initial pressure, i.e. pD r = p0 . 5. Point E—end of gas extraction pr = pmin r The DE section of the load program corresponds to the gas extraction stage where the reservoir pressure is lower than the initial pressure, i.e. pr < p0 and jq3 j > jq1 j. At this section, a depression is created at the well bottom at Δpw. Stresses on the contour of a horizontal well at point N during gas extraction at reservoir pressure pr < p0 are given by relations (5.56). We have sr = - δðpr - Δpw Þ 1 - 2ν sz = q ð1 - δÞðpr - p0 Þ þ ð1 - δÞðpr - Δpw Þ 1-ν 1 - 2ν ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr - Δpw Þ sθ = 2q þ 1-ν

ð6:24Þ

Then at the point E corresponding to the minimum reservoir pressure pr = pmin r sEr = - δ pmin r - Δpw 1 - 2ν min sEz = q ð1 - δÞ pmin r - p0 þ ð1 - δÞ pr - Δpw 1-ν 1 - 2ν min ð1 - δÞ pmin sEθ = 2q þ r - p0 þ ð2 - δÞ pr - Δpw 1-ν

ð6:25Þ

And, correspondingly, for the stresses applied to the specimen along the load axes of the TILTS, we have

6.1

Equilibrium Natural Stress State in the Reservoir

1 - 2ν min ð1 - δÞ pmin r - p0 - ð1 - δÞ pr - Δpw 1-ν 1 - 2ν min ð1 - δÞ pmin sE2 = 2 jqj r - p0 - ð2 - δÞ pr - Δpw 1-ν sE3 = δ pmin r - Δpw

79

sE1 = jqj þ

ð6:26Þ

6. Point F—at gas injection pr = p0 The EF section of the load program corresponds to the gas injection stage after the completion of gas extraction, where the reservoir pressure is lower than the initial pressure, i.e. pr < p0 and jq3 j > jq1 j. At this section, a repression is created at the wellbore Δpw. The stresses at point N on the contour of a horizontal well during gas injection at reservoir pressure pr < p0 are given by relations (5.56). We have sr = - δðpr þ Δpw Þ 1 - 2ν sz = q ð1 - δÞðpr - p0 Þ þ ð1 - δÞðpr þ Δpw Þ 1-ν 1 - 2ν sθ = 2 q þ ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr þ Δpw Þ 1-ν

ð6:27Þ

Then at the point F corresponding to the reservoir pressure equal to the initial pressure pr = p0 sFr = - δðp0 þ Δpw Þ sFz = q þ ð1 - δÞðp0 þ Δpw Þ sFθ = 2q þ ð2 - δÞðp0 þ Δpw Þ

ð6:28Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have sF1 = jqj - ð1 - δÞðp0 þ Δpw Þ sF2 = 2jqj - ð2 - δÞðp0 þ Δpw Þ sF3 = δðp0 þ Δpw Þ

ð6:29Þ

The point F corresponds to the moment when the reservoir pressure becomes equal to the initial pressure, i.e.pFr = p0 .

80

6

6.1.2.2.2

Physical Modelling Programs of Rock Deformation and Fracture in. . .

Upper Point on Horizontal Well Contour

1. Point A Point A corresponds to the initial state in the rock mass away from the well. The soil skeleton is in state all-round uniform compression sAr = sAz = sAθ = q þ ð1 - δÞp0 , where p0 is the initial reservoir pressure. Accordingly, for the stresses applied to the specimen in the course of the experiment along the load axes of the TILTS, we have sA1 = sA2 = sA3 = jqj - ð1 - δÞp0 . 2. Point B Point B corresponds to the situation when the well is drilled and the pressure in the well is equal to the initial reservoir pressure, i.e. pw = po. Then for the stresses at point B, assuming in (5.38) pw = p0 and pr = p0, we find sВr = - δp0

sВz = q þ ð1 - δÞp0 sВθ

ð6:30Þ

= 2q þ ð2 - δÞp0

Accordingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have sВ1 = jqj - ð1 - δÞp0

sВ2 = 2jqj - ð2 - δÞp0 sВ3 = δp0

ð6:31Þ

3. Point C—end of gas injection pr = pmax r The BC section of the load program corresponds to the stage of gas injection at which the formation pressure is higher than the initial pressure, i.e. pr > p0 and jq3 j < jq1 j. Point B corresponds to the start of gas injection by creating a repression at the bottom of the well Δpw. The stresses at point M on the contour of a horizontal well during gas injection at reservoir pressure pr > p0 are given by relations (5.39). We have sr = - δðpr þ Δpw Þ sz = q þ ð1 - δÞðpr þ Δpw Þ 1 - 2ν sθ = 2 q - 3 ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr þ Δpw Þ 1-ν

ð6:32Þ

At point C, which corresponds to the maximum reservoir pressure pr = pmax r , one can write

6.1

Equilibrium Natural Stress State in the Reservoir

81

scr = - δ pmax þ Δpw r þ Δpw scz = q þ ð1 - δÞ pmax r 1 2ν scθ = 2q - 3 - p0 þ ð2 - δÞ pmax þ Δpw ð1 - δÞ pmax r r 1-ν

ð6:33Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have þ Δpw sc1 = jqj - ð1 - δÞ pmax r 1 - 2ν sc2 = 2jqj þ 3 ð1 - δÞ pmax - p0 - ð2 - δÞ pmax þ Δpw r r 1-ν þ Δpw sc3 = δ pmax r

ð6:34Þ

4. Point D—at gas extraction pr = p0. The CD section of the load program corresponds to the gas extraction stage after gas injection is complete, at which the reservoir pressure still remains above the initial pressure, i.e. pr > p0 and jq3 j < jq1 j. At this section, a depression is created at the bottom of the well Δpw. Stresses at point M on the contour of a horizontal well during gas extraction at reservoir pressure pr > p0 are given by relations (5.39). sr = - δðpr - Δpw Þ sz = q þ ð1 - δÞðpr - Δpw Þ 1 - 2ν sθ = 2 q - 3 ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr - Δpw Þ 1-ν

ð6:35Þ

Then at point D, corresponding to a reservoir pressure equal to the initial pr = p 0 sD r = - δðpo - Δpw Þ sD θ

ð6:36Þ

= 2q þ ð2 - δÞðp0 - Δpw Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have sD 1 = jqj - ð1 - δÞðp0 - Δpw Þ sD 2 = 2jqj - ð2 - δÞðp0 - Δpw Þ

ð6:37Þ

sD 3 = δðp0 - Δpw Þ Point D corresponds to the moment when the reservoir pressure becomes equal to the initial pressure, i.e.pD r = p0 .

82

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

5. Point E—end of gas extraction pr = pmin r The DE section of the load program corresponds to the gas extraction stage where the reservoir pressure is lower than the initial pressure, i.e. pr < p0 and jq3 j > jq1 j. At this section, a depression is created at the well bottom at Δpw, so that at the DE sectionpw = pr - Δpw. Stresses on the contour of a horizontal well at point M during gas extraction at reservoir pressure pпл < p0 are given by relations (5.50). In these relations one must put pw = pr - Δpw. sr = - δðpr - Δpw Þ 1 - 2ν ð1 - δÞðpr - p0 Þ þ ð1 - δÞðpr - Δpw Þ sz = q 1-ν 1 - 2ν sθ = 2 q - 3 ð1 - δÞ ðpr - p0 Þ þ ð2 - δÞðpr - Δpw Þ 1-ν

ð6:38Þ

Then at the point E corresponding to the minimum reservoir pressure pr = pmin r sEr = - δ pmin r - Δpw 1 - 2ν min sEz = q ð1 - δÞ pmin r - p0 þ ð1 - δÞ pr - Δpw 1-ν 1 - 2ν min sEθ = 2 q - 3 ð1 - δÞ pmin r - p0 þ ð2 - δÞ pr - Δpw 1-ν

ð6:39Þ

And correspondingly for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have 1 - 2ν min ð1 - δÞ pmin r - p0 - ð1 - δÞ pr - Δpw 1-ν 1 - 2ν min sE2 = 2jqj þ 3 ð1 - δÞ pmin r - p0 - ð2 - δÞ pr - Δpw 1-ν sE3 = δ pmin r - Δpw sE1 = jqj þ

ð6:40Þ

6. Point F—at gas injection pr = p0 The EF section of the load program corresponds to the gas injection stage after the completion of gas extraction, at which the formation pressure is lower than the initial pressure, i.e. pr < p0 and jq3 j > jq1 j. At this section, a repression is created at the wellbore Δpw. The stresses at point M on the contour of a horizontal well during gas injection at reservoir pressure pr < p0 are given by relations (5.50). We have

6.2

Physical Modelling Programs for Unequal Natural Stress State

sr = - δðpr þ Δpw Þ 1 - 2ν sz = q ð1 - δÞðpr þ p0 Þ þ ð1 - δÞðpr þ Δpw Þ 1-ν 1 - 2ν sθ = 2q - 3 ð1 - δÞðpr þ p0 Þ þ ð2 - δÞðpr þ Δpw Þ 1-ν

83

ð6:41Þ

Then at the point F, the reservoir pressure equal to the initial pressure pr = p0 sFr = - δðp0 þ Δpw Þ sFz = q þ ð1 - δÞðp0 þ Δpw Þ sFθ

ð6:42Þ

= 2q þ ð2 - δÞðp0 þ Δpw Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the plant, we have sF1 = jqj - ð1 - δÞðp0 þ Δpw Þ sF2 = 2jqj - ð2 - δÞðp0 þ Δpw Þ sF3

ð6:43Þ

= δðp0 þ Δpw Þ

The point F corresponds to the moment when the reservoir pressure becomes equal to the initial pressure, i.e. pFr = p0 .

6.2

Physical Modelling Programs for Unequal Natural Stress State

As indicated above, the main influence on the processes of rock deformation and fracture in the vicinity of wells (especially horizontal wells) is caused by stresses arising due to changes in the natural stress state at the reservoir depth at decreasing and increasing reservoir pressure in the UGS reservoir. Relationships for arising stresses on the contour of horizontal and vertical wells at unequal natural stress are given in clause 5.3. The programs of physical simulation of deformation and filtration processes in the bottomhole zone of UGS wells under the action of alternating loads during gas extraction and injection under unequal natural stress state developed on the basis of these relations are given below. Since the principal stresses occurring in the vicinity of vertical and horizontal wells are different, the load programs of rock specimens during tests will be different for them. In turn, the test programs for vertical and horizontal wells corresponding to stress changes at points M and N on the well contour (Figs. 5.7 and 5.10) due to the unequal initial stress state will also differ from each other, since the stresses at these points during gas injection and extraction are different.

84

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

Figure 6.2 shows a schematic view of the specimen load program simulating the change in stresses on the UGS well contour during one gas injection/extraction cycle. The depicted stresses S1, S2, S3 are the values of stresses applied to the specimen during the experiment along load axes 1, 2, 3 of TILTS. They correspond to the stresses |sz|, |sθ|, |sr|, acting on the well contour. s1 = jsz j, s2 = jsθ j, s3 = jsr j

6.2.1

ð6:44Þ

Vertical Uncased Well

Due to the unequal initial stress state in the reservoir, the stresses along the contour of a vertical uncased well vary from point to point. The load programs for the different points will vary accordingly. The load programs for points M and N on the well contour are shown below.

6.2.1.1

Point M on the Borehole Contour (Fig. 5.7)

1. Point A Point A in Fig. 6.2 corresponds to the initial state in the reservoir away from the well. In the initial state, the soil skeleton is loaded with effective all-round compression stresses: in the vertical direction with the stress sAz = q þ ð1 - δÞp0 , in the direction of maximum horizontal stress with the stress sAθ = q1 þ ð1 - δÞp0 , in the direction of minimum horizontal stress with the stress sAr = q2 þ ð1 - δÞp0 , where p0 is the initial reservoir pressure. Accordingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have sA1 = jqj - ð1 - δÞp0 , sA2 = jq1 j - ð1 - δÞp0 , sA3 = jq2 j - ð1 - δÞp0 . Here, axis 1 of the TILTS is directed along the core axis. Or, assuming as above, q1 = α1q, q2 = α2q, α1 > α2 sA1 = jqj - ð1 - δÞp0 sA2 = α1 jqj - ð1 - δÞp0 sA2 = α2 jqj - ð1 - δÞp0

ð6:45Þ

2. Point B Point B corresponds to the situation when the well is drilled and the pressure in the well is equal to the initial reservoir pressure, i.e. pw = p0. Then for the stresses at point B, assuming in (5.81) pw = p0 and pr = p0, we find

6.2

Physical Modelling Programs for Unequal Natural Stress State

85

sВr = - δp0

sВθ = ð3α2 - α1 Þq þ ð2 - δÞp0 sВz = q þ ð1 - δÞp0

ð6:46Þ

Accordingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have sВ1 = jqj - ð1 - δÞp0

sВ2 = ð3α2 - α1 Þjqj - ð2 - δÞp0 sВ3 = δp0

ð6:47Þ

3. Point C—end of gas injection pr = pmax r The BC section of the load program corresponds to the stage of gas injection at which the reservoir pressure is higher than the initial pressure, i.e. pr > p0. Point B corresponds to the start of gas injection by creating a repression at the bottom of the well Δpw. Stresses on the vertical well contour during gas injection at reservoir pressure pr are given by relations (5.81). Then at point C corresponding to the maximum reservoir pressure pr = pmax r Sr = - δ pmax þ Δpw r 1 - 2ν sθ = ð3α2 - α1 Þq - 2 - p0 þ ð2 - δÞ pmax þ Δpw ð1 - δÞ pmax r r 1-ν þ Δpw Sz = q þ ð1 - δÞ pmax r

ð6:48Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SС1 = jqj - ð1 - δÞ pmax пл þ Δpс 1 - 2ν sС2 = ð3α2 - α1 Þjqj þ 2 - p0 - ð2 - δÞ pmax þ Δpw ð1 - δÞ pmax r r 1-ν SС3 = δ pmax þ Δpw r

ð6:49Þ

4. Point D—at gas extraction pr = p0 The CD section of the load program corresponds to the gas extraction stage after gas injection is completed, at which the reservoir pressure still remains above the initial pressure, i.e. pr > p0. At this section, a depression is created at the well bottom Δpw. Stresses on the vertical well contour at point M during gas extraction at reservoir pressure pr are given by relations (5.81). We have

86

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

Sr = - δðpr - Δpw Þ

1 - 2ν ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr - Δpw Þ 1-ν Sz = q þ ð1 - δÞðpr - Δpw Þ sθ = ð3α2 - α1 Þq - 2

ð6:50Þ

At the point D pr = p0. From (3.7), the stresses on the well contour in this case are sD r = - δp0 sD θ = ð3α2 - α1 Þq þ ð2 - δÞp0 sD z

ð6:51Þ

= q þ ð1 - δÞp0

Accordingly, for the stresses applied at point D on the specimen along the load axes of the TILTS, we have sD 1 = jqj - ð1 - δÞp0 sD 2 = ð3α2 - α1 Þjqj - ð2 - δÞp0 sD 3 = δp0

ð6:52Þ

5. 5. Point E—end of gas extraction pr = pmin r The DE section of the load program corresponds to the gas extraction stage, where the reservoir pressure is lower than the initial pressure, i.e. pr < p0. At this section, a depression is created at the bottom of the well Δpw. Stresses on the vertical well contour during gas injection at reservoir pressure pr are given by relations (5.81). Sr = - δðpr - Δpw Þ

1 - 2ν ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr - Δpw Þ 1-ν Sz = q þ ð1 - δÞðpr - Δpw Þ sθ = ð3α2 - α1 Þq - 2

ð6:53Þ

Then, at point E which corresponds to the lowest reservoir pressure pr = pmin r , one can write SEr = - δ pmin r - Δpw 1 - 2ν min ð1 - δÞ pmin r - p0 þ ð2 - δÞ pr - Δpw 1-ν SEz = q þ ð1 - δÞ pmin r - Δpw

SEθ = ð3α2 - α1 Þq - 2

ð6:54Þ

And, accordingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have

6.2

Physical Modelling Programs for Unequal Natural Stress State

SE1 = jqj - ð1 - δÞ pmin r - Δpw 1 - 2ν min sE2 = ð3α2 - α1 Þjqj þ 2 ð1 - δÞ pmin r - p0 - ð2 - δÞ pr - Δpw 1-ν SE3 = δ pmin r - Δpw

87

ð6:55Þ

6. Point F—at gas injection pr = p0 The EF section of the load program corresponds to the gas injection stage after the completion of gas extraction, at which the reservoir pressure is lower than the initial pressure, i.e. pr < p0. At this section, a repression is created at the wellboreΔpw. Stresses on the vertical well contour during gas injection at reservoir pressure pr are given by relations (5.81), in which it is necessary to put Sr = - δðpr þ Δpw Þ

1 - 2ν ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr þ Δpw Þ 1-ν Sz = q þ ð1 - δÞðpr þ Δpw Þ

sθ = ð3α2 - α1 Þq - 2

ð6:56Þ

Then at the point F corresponding to the initial reservoir pressurepr = p0. SFr = - δp0 SFθ = ð3α2 - α1 Þq þ ð2 - δÞp0 SFz

ð6:57Þ

= q þ ð1 - δÞp0

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SF1 = jqj - ð1 - δÞp0 sF2 = ð3α2 - α1 Þjqj - ð2 - δÞp0 SF3 = δp0

ð6:58Þ

Figure 6.3 shows the load program for a rock specimen from the UGS №3 reservoir, simulating stress changes at point M on the contour of a vertical well, Fig. 5.2. Stress values S1, S2, S3 along the ordinate axis are plotted in MPa. In compiling it, as well as in compiling the other load programs below, it has been assumed reservoir depth H = 750 m; initial reservoir gas pressure p0 = 7.6 MPa; = 10.5 MPa, reservoir reservoir gas pressure at the end of gas injection stage pmax r gas pressure at the end of gas extraction stage pmin = 6.5 MPa; δ = 0.2; lateral r pressure factor for maximum horizontal stress α1 = 0.8; ν = 0.2; lateral pressure factor for minimum horizontal stress α2 = 0.6.

88

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

Fig. 6.3 Load program for point M on the contour of the vertical well (Fig. 5.2) at UGS №3

6.2.1.2

Point N on Well Contour (Fig. 5.7)

1. Point A Point A on the load program for point N is the same as point A on the load program for point M. For the stresses applied at point A to the specimen along the load axes of the TILTS, we have SA1 = jqj - ð1 - δÞp0 SA2 = α1 jqj - ð1 - δÞp0

ð6:59Þ

SA3 = α2 jqj - ð1 - δÞp0 2. Point B Point B corresponds to the situation when the well is drilled and the pressure in the well is equal to the initial reservoir pressure, i.e. pw = p0. Then for the stresses at point B, assuming in (5.82) pw = p0 and pr = p0, we find

6.2

Physical Modelling Programs for Unequal Natural Stress State

89

SВr = - δp0

SВθ = ð3α1 - α2 Þq þ ð2 - δÞp0 SВz = q þ ð1 - δÞp0

ð6:60Þ

Accordingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SВ1 = jqj - ð1 - δÞp0

SВ2 = ð3α1 - α2 Þjqj - ð2 - δÞp0 SВ3 = δp0

ð6:61Þ

3. Point C—end of gas injection pr = pmax r The BC section of the load program corresponds to the stage of gas injection at which the reservoir pressure is higher than the initial pressure, i.e. pr > p0. Point B corresponds to the start of gas injection by creating a repression at the bottom of the well Δpw. Stresses on the vertical well contour during gas injection at reservoir pressure pr are given by relations (5.82). Then at point C corresponding to the maximum reservoir pressure pr = pmax r SСr = - δ pmax þ Δpw r 1 - 2ν SСθ = ð3α1 - α2 Þq - 2 - p0 þ ð2 - δÞ pmax þ Δpw ð1 - δÞ pmax r r 1-ν SСz = q þ ð1 - δÞ pmax þ Δpw r

ð6:62Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have þ Δpw SС1 = jqj - ð1 - δÞ pmax r 1 - 2ν SС2 = ð3α1 - α2 Þjqj þ 2 - p0 - ð2 - δÞ pmax þ Δpw ð1 - δÞ pmax r r 1-ν SС3 = δ pmax þ Δpw r

ð6:63Þ

4. Point D—at gas extraction pr = p0 The CD section of the load program corresponds to the gas extraction stage after gas injection is complete, at which the reservoir pressure still remains above the initial pressure, i.e. pr > p0. At this section, depression Δpw is created at the bottom of the wellbore. Stresses on the contour of the vertical well at point N in the process of gas extraction at reservoir pressure pr are given by relations (5.82). We have

90

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

Sr = - δðpr - Δpw Þ

1 - 2ν ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr - Δpw Þ 1-ν Sz = q þ ð1 - δÞðpr - Δpw Þ sθ = ð3α1 - α2 Þq - 2

ð6:64Þ

At point D pr = p0. From (6.64) for the stresses on the borehole contour we have sD r = - δp0 sD θ = ð3α1 - α2 Þq þ ð2 - δÞp0 sD z

ð6:65Þ

= q þ ð1 - δÞp0

Accordingly, for the stresses applied at point D on the specimen along the load axes of the TILTS, we have SD 1 = jqj - ð1 - δÞp0 SD 2 = ð3α1 - α2 Þjqj - ð2 - δÞp0 SD 3 = δp0

ð6:66Þ

5. Point E corresponds to the end of gas extraction pr = pmin r The DE section of the load program corresponds to the gas extraction stage, where the reservoir pressure is lower than the initial pressure, i.e. pr < p0. At this section, a depression is created at the bottom of the well Δpw. Stresses on the vertical well contour during gas injection at reservoir pressure pr are given by relations (5.82). Sr = - δðpr - Δpw Þ

1 - 2ν ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr - Δpw Þ 1-ν Sz = q þ ð1 - δÞðpr - Δpw Þ

Sθ = ð3α1 - α2 Þq - 2

ð6:67Þ

Then at the point E corresponding to the minimum reservoir pressure pr = pmin r SEr = - δ pmin r - Δpw 1 - 2ν min ð1 - δÞ pmin r - p0 þ ð2 - δÞ pr - Δpw 1-ν SEz = q þ ð1 - δÞ pmin r - Δpw

SEθ = ð3α1 - α2 Þq - 2

ð6:68Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have

6.2

Physical Modelling Programs for Unequal Natural Stress State

SE1 = jqj - ð1 - δÞ pmin r - Δpw 1 - 2ν min sE2 = ð3α1 - α2 Þjqj þ 2 ð1 - δÞ pmin r - p0 - ð2 - δÞ pr - Δpw 1-ν SE3 = δ pmin r - Δpw

91

ð6:69Þ

6. Point F corresponds to gas injection pr = p0 The section of the EF load program corresponds to the gas injection stage after the completion of gas extraction, at which the reservoir pressure is lower than the initial pressure, i.e. pr < p0. At this section, a repression is created at the wellbore Δpw. Stresses on the vertical well contour during gas injection at reservoir pressure pr are given by relations (5.82). Sr = - δðpr þ Δpw Þ

1 - 2ν ð1 - δÞðpr - p0 Þ þ ð2 - δÞðpr þ Δpw Þ 1-ν Sz = q þ ð1 - δÞðpr þ Δpw Þ

sθ = ð3α1 - α2 Þq - 2

ð6:70Þ

Then, at the point F corresponding to the initial reservoir pressure pr = p0, one can write SFr = - δp0 SFθ = ð3α1 - α2 Þq þ ð2 - δÞp0 SFz = q þ ð1 - δÞp0

ð6:71Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SF1 = jqj - ð1 - δÞp0 sF2 = ð3α1 - α2 Þjqj - ð2 - δÞp0 SF3 = δp0

ð6:72Þ

Figure 6.4 shows the load program for a rock specimen from the UGS №3 reservoir, simulating the change of stresses at point N on the contour of the vertical well, Fig. 5.2.

92

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

Fig. 6.4 Load program for point N on the contour of the vertical well (Fig. 5.2) at UGS №3

6.2.2

Horizontal Well

6.2.2.1

Load Program for a Horizontal Well Drilled Along the Direction of Maximum Horizontal Stress

6.2.2.1.1

Upper Point on the Contour of a Horizontal Well Drilled Along the Direction of Maximum Horizontal Stress

1. Point A. The stresses at point A corresponding to the initial effective stresses acting in the soil skeleton prior to drilling the well coincide with the stresses at point A for the vertical well (6.45). For the stresses applied at point A to the specimen along the load axes of the TILTS, we have

6.2

Physical Modelling Programs for Unequal Natural Stress State

SA1 = jqj - ð1 - δÞp0 SA2 = α1 jqj - ð1 - δÞp0 SA3

93

ð6:73Þ

= α2 jqj - ð1 - δÞp0

2. Point B Point B corresponds to the situation when the well is drilled and the pressure in the well is equal to the initial reservoir pressure, i.e. pw = p0. Then for the stresses at point B, assuming in (5.88) pw = p0 and pr = p0, we find SBr = - δp0 SBθ = ð3α2 - 1Þq þ ð2 - δÞp0 SBz = α1 q þ ð1 - δÞp0

ð6:74Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SB1 = α1 jqj - ð1 - δÞp0 SB2 = ð3α2 - 1Þjqj - ð2 - δÞp0

ð6:75Þ

SB3 = δp0 3. Point C corresponds to the end of gas injection pr = pmax r The BC section of the load program corresponds to the stage of gas injection at which the reservoir pressure is higher than the initial pressure, i.e. pr > p0. Point B corresponds to the start of gas injection by creating a repression at the bottom of the well. Stresses at the upper point on the horizontal well contour during gas injection at reservoir pressure pr are given by relations (5.88). Then, at point C corresponding to maximum reservoir pressure pr = pmax r , we obtain SCr = - δ pmax þ Δpw r 1 - 2ν SCθ = ð3α2 - 1Þq - 3 - p0 þ ð2 - δÞ pmax þ Δpw ð1 - δÞ pmax r r 1-ν 1 - 2ν ð1 - δÞ pmax - p0 þ ð1 - δÞ pmax þ Δpw SCz = α1 q r r 1-ν

ð6:76Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have

6 Physical Modelling Programs of Rock Deformation and Fracture in. . .

94

1 - 2ν ð1 - δÞ pmax - p0 - ð1 - δÞ pmax þ Δpw r r 1-ν 1 - 2ν ð1 - δÞ pmax - p0 - ð2 - δÞ pmax þ Δpw SC2 = ð3α2 - 1Þjqj þ 3 r r 1-ν C max S3 = δ pr þ Δpw

SC1 = α1 jqj þ

ð6:77Þ

4. Point D corresponds to stage of gas extraction pr = p0 The CD load program section corresponds to the gas extraction stage after gas injection is complete, at which the reservoir pressure still remains above the initial pressure, i.e. pr > p0. At this section, a drawdown Δpw is created at the bottom of the well. Stresses on the contour of the horizontal well at the top point during gas extraction at reservoir pressure pr are given by relations (5.88). At point D pr = p0 and for stresses on the well contour at the top point, we have SD r = - δp0 SD θ = ð3α2 - 1Þq þ ð2 - δÞp0 SD z

ð6:78Þ

= α1 q þ ð1 - δÞp0

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SD 1 = α1 jqj - ð1 - δÞp0 SD 2 = ð3α2 - 1Þjqj - ð2 - δÞp0 SD 3

ð6:79Þ

= δp0

5. Point E corresponds to the end of gas extraction pr = pmin r The DE section of the load program corresponds to the gas extraction stage, where the reservoir pressure is lower than the initial pressure, i.e. pr < p0. At this section, a drawdown Δpw is created at the bottom of the well. Stresses on the contour of a horizontal well at the top point during gas injection at reservoir pressure pr are given by relations (5.88). Then, at the point E corresponding to the minimum reservoir pressure, the stresses are SEr = - δ pmin r - Δpw 1 - 2ν min SEθ = ð3α2 - 1Þq - 3 ð1 - δÞ pmin r - p0 þ ð2 - δÞ pr - Δpw 1-ν 1 - 2ν min SEz = α1 q ð1 - δÞ pmin r - p0 þ ð1 - δÞ pr - Δpw 1-ν

ð6:80Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have

6.2

Physical Modelling Programs for Unequal Natural Stress State

1 - 2ν min ð1 - δÞ pmin r - p0 - ð1 - δÞ pr - Δpw 1-ν 1 - 2ν min ð1 - δÞ pmin SE2 = ð3α2 - 1Þjqj þ 3 r - p0 - ð2 - δÞ pr - Δpw 1-ν SE3 = δ pmin r - Δpw

95

SE1 = α1 jqj þ

ð6:81Þ

6. Point F corresponds to stage of gas injection pr = p0 The section EF of the load program corresponds to the gas injection stage after the completion of gas extraction at which the reservoir pressure is lower than the initial pressure, i.e. pr < p0. At this section, a repression Δpw is created at the wellbore. Stresses on the contour of a horizontal well at the upper point during gas injection at reservoir pressure pr are given by relations (5.88). Then at the point F corresponding to the initial reservoir pressure pw = p0. SFr = - δp0 SFθ = ð3α2 - 1Þq þ ð2 - δÞp0 SFZ = α1 q þ ð1 - δÞp0

ð6:82Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SF1 = α1 jqj - ð1 - δÞp0 sF2 = ð3α2 - 1Þjqj - ð2 - δÞp0 SF3 = δp0

ð6:83Þ

Figure 6.5 shows a load program for a rock specimen from the UGS №3 reservoir, simulating stress changes at the top point on the contour of a horizontal well drilled along the direction of maximum horizontal stress. It plots the stresses values along the axis of the TILTS.

6.2.2.1.2

Load Program for a Lateral Point on the Contour of a Horizontal Well Drilled Along the Direction of Maximum Horizontal Stress

1. Point A The stresses at point A of the load program for a horizontal well corresponding to the initial effective stresses acting in the soil skeleton prior to drilling the well coincide with the stresses at point A for a vertical well (6.45). For the stresses applied at point A on the specimen along the load axes of the TILTS, we have

96

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

Fig. 6.5 Load program for the upper point on the contour of the UGS №3 horizontal well drilled along the direction of maximum horizontal stress

SA1 = jqj - ð1 - δÞp0 SA2 = α1 jqj - ð1 - δÞp0 SA3

ð6:84Þ

= α2 jqj - ð1 - δÞp0

2. Point B Point B corresponds to the situation when the well is drilled and the pressure in the well is equal to the initial reservoir pressure, i.e. pr = p0. For the stresses at point B, assuming in (5.89) pr = p0 and pr = p0, we find SBr = - δp0 SBθ = = ð3 - α2 Þq þ ð2 - δÞp0 SBz

ð6:85Þ

= α1 q þ ð1 - δÞp0

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have

6.2

Physical Modelling Programs for Unequal Natural Stress State

97

SB1 = α1 jqj - ð1 - δÞp0 SB2 = = ð3 - α2 Þjqj - ð2 - δÞp0 SB3 = δp0

ð6:86Þ

3. Point C correspods to the end of gas injection pr = pmax r The BC section of the load program corresponds to the stage of gas injection at which the reservoir pressure is higher than the initial pressure, i.e. pr > p0. Point B corresponds to the start of gas injection by creating a repression Δpw at the bottom of the well. Stresses on the contour of a horizontal well at a lateral point during gas injection at reservoir pressure pr are given by relations (5.89). Then, at point C corresponding to maximum reservoir pressure pr = pmax r , we obtain SCr = - δ pmax þ Δpw r 1 - 2ν SCθ = ð3 - α2 Þq þ - p0 þ ð2 - δÞ pmax þ Δpw ð1 - δÞ pmax r r 1-ν 1 - 2ν - p0 þ ð1 - δÞ pmax þ Δpw SCz = α1 q ð1 - δÞ pmax r r 1-ν

ð6:87Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have 1 - 2ν ð1 - δÞ pmax - p0 - ð1 - δÞ pmax þ Δpw r r 1-ν 1 - 2ν ð1 - δÞ pmax - p0 - ð2 - δÞ pmax þ Δpw SC2 = ð3 - α2 Þjqj r r 1-ν C max S3 = δ pr þ Δpw SC1 = α1 jqj þ

ð6:88Þ

4. Point D corresponds to stage of gas extraction pr = p0 The CD section of the load program corresponds to the gas extraction stage after gas injection is complete, at which the reservoir pressure still remains above the initial pressure, i.e. pr > p0. At this section, a depression Δpw is created at the bottom of the well. Stresses on the contour of the horizontal well at the lateral point during gas extraction at reservoir pressure pr are given by relations (5.89). At point D pr = p0, and for stresses on the well contour, we have SD r = - δp0 SD θ = ð3 - α2 Þq þ ð2 - δÞp0 SD z = α1 q þ ð1 - δÞp0

ð6:89Þ

98

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SD 1 = α1 jqj - ð1 - δÞp0 SD 2 = ð3 - α2 Þjqj - ð2 - δÞp0

ð6:90Þ

SD 3 = δp0 5. Point E is the end of gas extraction pr = pmin r The DE section of the load program corresponds to the gas extraction stage, where the reservoir pressure is lower than the initial pressure, i.e. pr < p0. At this section, a drawdown Δpw is created at the bottom of the well. Stresses on the contour of a horizontal well at the lateral point during gas injection at reservoir pressure pr are given by relations (5.89). Then, at the point E corresponding to the minimum reservoir pressure, the stresses are SEr = - δ pmin r - Δpw 1 - 2ν min SEθ = ð3 - α2 Þq þ ð1 - δÞ pmin r - p0 þ ð2 - δÞ pr - Δpw 1-ν 1 - 2ν min SEz = α1 q ð1 - δÞ pmin r - p0 þ ð1 - δÞ pr - Δpw 1-ν

ð6:91Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have 1 - 2ν min ð1 - δÞ pmin r - p0 - ð1 - δÞpr - Δpw 1-ν 1 - 2ν min SE2 = ð3 - α2 Þjqj ð1 - δÞ pmin r - p0 - ð2 - δÞ pr - Δpw 1-ν SE3 = δ pmin r - Δpw SE1 = α1 jqj þ

ð6:92Þ

6. Point F corresponds to the stage of gas injection pr = p0 The section EF of the load program corresponds to the gas injection stage after the completion of gas extraction at which the reservoir pressure is lower than the initial pressure, i.e. pr < p0. At this section, a repression Δpw is created at the wellbore. Stresses on the contour of a horizontal well at the lateral point during gas injection at reservoir pressure pr are given by relations (5.89). Then at the point F corresponding to the initial reservoir pressurepr = p0.

6.2

Physical Modelling Programs for Unequal Natural Stress State

99

SFr = - δp0 SFθ = ð3 - α2 Þq þ ð2 - δÞp0 SFZ = α1 q þ ð1 - δÞp0

ð6:93Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SF1 = α1 jqj - ð1 - δÞp0 sF2 = ð3 - α2 Þjqj - ð2 - δÞp0

ð6:94Þ

SF3 = δp0 Figure 6.6 shows the load program for a rock specimen from the UGS №3 reservoir simulating the change in stress at the lateral point on the contour of a horizontal well drilled along the direction of maximum horizontal stress. It plots the stresses values along the axis of the TILTS.

Fig. 6.6 Load program for the lateral point on horizontal contour Uviazovskoye UGS well drilled along the direction maximum horizontal tension

100

6.2.2.2

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

Horizontal Well Drilled Along the Direction of the Minimum Horizontal Stress

6.2.2.2.1

Upper Point on the Contour of a Horizontal Well Drilled Along the Direction of the Minimum Horizontal Stress

1. Point A The stresses at point A corresponding to the initial stresses acting in soil skeleton before the borehole is drilled of the load program for the horizontal well coincide with the stresses at point A for the vertical borehole (6.45). For the stresses applied at point A to the specimen along the load axes of the TILTS, we have SA1 = jqj - ð1 - δÞp0 SA2 = α1 jqj - ð1 - δÞp0 SA3

ð6:95Þ

= α2 jqj - ð1 - δÞp0

2. Point B Point B corresponds to the situation when the well is drilled and the pressure in the well is equal to the initial reservoir pressure, i.e. pw = p0. Then, for the stresses at point B, assuming in (5.96) pw = p0 and pr = p0, we find SBr = - δp0 SBθ = ð3α1 - 1Þq þ ð2 - δÞp0 SBz

ð6:96Þ

= α2 q þ ð1 - δÞp0

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SB1 = α2 jqj - ð1 - δÞp0 SB2 = ð3α1 - 1Þjqj - ð2 - δÞp0 SB3 = δp0

ð6:97Þ

3. Point C is the end of gas injection pr = pmax r The BC section of the load program corresponds to the stage of gas injection at which the reservoir pressure is higher than the initial pressure, i.e. pr > p0. Point B corresponds to the start of gas injection by creating a repression Δpw at the bottom of the well. Stresses on the contour of the horizontal well at the upper point during gas injection at reservoir pressure pw are given by relations (5.96). Then, at point C corresponding to maximum reservoir pressure pr = pmax r , we obtain

6.2

Physical Modelling Programs for Unequal Natural Stress State

SCr = - δ pmax þ Δpw r 1 - 2ν SCz = α2 q - p0 þ ð1 - δÞ pmax þ Δpw ð1 - δÞ pmax r r 1-ν 1 - 2ν - p0 þ ð2 - δÞ pmax þ Δpw SCθ = ð3α1 - 1Þq - 3 ð1 - δÞ pmax r r 1-ν

101

ð6:98Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have 1 - 2ν ð1 - δÞ pmax - p0 - ð1 - δÞ pmax þ Δpw r r 1-ν 1 - 2ν - p0 - ð2 - δÞ pmax þ Δpw SC2 = ð3α1 - 1Þjqj þ 3 ð1 - δÞ pmax r r 1-ν C max S3 = δ pr þ Δpw SC1 = α2 jqj þ

ð6:99Þ

4. Point D corresponds to the stage of gas extraction pr = p0 The CD section of the load program corresponds to the gas extraction stage after gas injection is completed at which the reservoir pressure still remains above the initial pressure, i.e. pr > p0. At this section, a drawdown Δpw is created at the bottom of the well. Stresses on the contour of the horizontal well at the top point during gas extraction at reservoir pressure pr are given by relations (5.96). At point D pr = p0 and for stresses on the well contour, we have SD r = - δp0 SD θ = ð3α1 - 1Þq þ ð2 - δÞp0 SD z = α2 q þ ð1 - δÞp0

ð6:100Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SD 1 = α2 jqj - ð1 - δÞp0 SD 2 = ð3α1 - 1Þjqj - ð2 - δÞp0 SD 3 = δp0

ð6:101Þ

5. Point E is the end of gas extraction pr = pmin r The DE section of the load program corresponds to the gas extraction stage, where the reservoir pressure is lower than the initial pressure, i.e. pr < p0. At this section, a drawdown Δpw is created at the bottom of the well. Stresses on the contour of a horizontal well at the top point during the gas injection at reservoir pressure pr are given by relations (5.96). Then, at the point E corresponding to the minimum reservoir pressure, the stresses are

102

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

SEr = - δ pmin r - Δpw 1 - 2ν min SEθ = ð3α1 - 1Þq - 3 ð1 - δÞ pmin r - p0 þ ð2 - δÞ pr - Δpw 1-ν 1 - 2ν min ð1 - δÞ pmin SEz = α2 q r - p0 þ ð1 - δÞ pr - Δpw 1-ν

ð6:102Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have 1 - 2ν min ð1 - δÞ pmin r - p0 - ð1 - δÞ pr - Δpw 1-ν 1 - 2ν min SE2 = ð3α1 - 1Þjqj þ 3 ð1 - δÞ pmin пл - p0 - ð2 - δÞ pпл - Δpс 1-ν SE3 = δ pmin r - Δpw

SE1 = α2 jqj þ

ð6:103Þ

6. Point F corresponds to the stage of the gas injection pr = p0 The section EF of the load program corresponds to the gas injection stage after the completion of gas extraction at which the reservoir pressure is lower than the initial pressure, i.e. pr < p0. At this section, a repression Δpw is created at the wellbore. Stresses on the contour of the horizontal well at the upper point during gas injection at reservoir pressure pr are given by relations (5.96). Then, at the point F corresponding to the initial reservoir pressure pr = p0 SFr = - δp0 SFθ = ð3α1 - 1Þq þ ð2 - δÞp0 SFZ

ð6:104Þ

= α2 q þ ð1 - δÞp0

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SF1 = α2 jqj - ð1 - δÞp0 sF2 = ð3α1 - 1Þjqj - ð2 - δÞp0 SF3 = δp0

ð6:105Þ

Figure 6.7 shows the load program for a rock specimen from the UGS №3 reservoir simulating stress changes at the lateral point on the contour of a horizontal well drilled along the direction of the minimum horizontal stress. Figure 6.7 shows a load program for a rock specimen from the UGS №3 reservoir simulating stress changes at the lateral point on the contour of a horizontal well drilled along the direction of the minimum horizontal stress.

6.2

Physical Modelling Programs for Unequal Natural Stress State

103

Fig. 6.7 Load program for the lateral point on the contour of the horizontal well of the Uviazovskoye UGS drilled along the direction of the minimum horizontal stress

6.2.2.2.2

The Top Point on the Contour of a Horizontal Well Drilled Along the Direction of the Minimum Horizontal Stress

1. Point A. The stresses at point A corresponding to the initial stresses acting in the soil skeleton before drilling the horizontal well coincide with the stresses at point A for the vertical well (6.45). For the stresses applied at point A to the specimen along the load axes of the TILTS, we have SA1 = jqj - ð1 - δÞp0 SA2 = α1 jqj - ð1 - δÞp0

ð6:106Þ

SA3 = α2 jqj - ð1 - δÞp0 2. Point B Point B corresponds to the situation when the well is drilled and the pressure in the well is equal to the initial reservoir pressure, i.e. pw = p0.

104

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

Then for the stresses at point B, assuming in (5.97) pw = p0 and pr = p0, we find SBr = - δp0 SBθ = = ð3 - α1 Þq þ ð2 - δÞp0

ð6:107Þ

SBz = α2 q þ ð1 - δÞp0 And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SB1 = α2 jqj - ð1 - δÞp0 SB2 = = ð3 - α1 Þjqj - ð2 - δÞp0

ð6:108Þ

SB3 = δp0 3. Point C is the end of gas injection pr = pmax r The BC section of the load program corresponds to the stage of gas injection at which the reservoir pressure is higher than the initial pressure, i.e. pr > p0. Point B corresponds to the start of gas injection by creating a repression Δpw at the bottom of the well. Stresses on the contour of a horizontal well at a lateral point during gas injection at reservoir pressure pr are given by relations (5.97). Then, at point C corresponding to maximum reservoir pressure pr = pmax r , we obtain SCr = - δ pmax пл þ Δpс 1 - 2ν max SCθ = ð3 - α1 Þq þ ð1 - δÞ pmax пл - p0 þ ð2 - δÞ pпл þ Δpс 1-ν 1 - 2ν - p0 þ ð1 - δÞ pmax þ Δpw SCz = α2 q ð1 - δÞ pmax r r 1-ν

ð6:109Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have 1 - 2ν max þ Δpw ð1 - δÞ pmax w - p0 - ð 1 - δ Þ pr 1-ν 1 - 2ν - p0 - ð2 - δÞ pmax þ Δpw SC2 = ð3 - α1 Þjqj ð1 - δÞ pmax r r 1-ν SC3 = δ pmax þ Δpw r

SC1 = α2 jqj þ

ð6:110Þ

4. Point D corresponds to the stage of the gas extraction pr = p0 The CD load program section corresponds to the gas extraction stage after gas injection is completed at which the reservoir pressure still remains above the initial pressure, i.e. pr > p0. At this section, a drawdown Δpw is created at the bottom of the

6.2

Physical Modelling Programs for Unequal Natural Stress State

105

well. Stresses on the contour of the horizontal well at the lateral point during gas extraction at reservoir pressure pr are given by relations (5.97). At point D pr = p0 and for stresses on the well contour we have SD r = - δp0 SD θ = ð3 - α1 Þq þ ð2 - δÞp0

ð6:111Þ

SD z = α2 q þ ð1 - δÞp0 And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SD 1 = α2 jqj - ð1 - δÞp0 SD 2 = ð3 - α1 Þjqj - ð2 - δÞp0 SD 3 = δp0

ð6:112Þ

5. Point E is the end of the gas extraction pr = pmin r The DE section of the load program corresponds to the gas extraction stage, where the reservoir pressure is lower than the initial pressure, i.e. pr < p0. At this section, a drawdown Δpw is created at the bottom of the well. Stresses on the contour of a horizontal well at a lateral point during gas injection at reservoir pressure pr are given by relations (5.97). Then, at the point E corresponding to the minimum reservoir pressure, the stresses are SEr = - δ pmin r - Δpw 1 - 2ν min SEθ = ð3 - α1 Þq þ ð1 - δÞ pmin r - p0 þ ð2 - δÞ pr - Δpw 1-ν 1 - 2ν min SEz = α2 q ð1 - δÞ pmin r - p0 þ ð1 - δÞ pr - Δpw 1-ν

ð6:113Þ

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have 1 - 2ν min ð1 - δÞ pmin r - p0 - ð1 - δÞ pr - Δpw 1-ν 1 - 2ν min SE2 = ð3 - α1 Þjqj ð1 - δÞ pmin r - p0 - ð2 - δÞ pr - Δpw 1-ν SE3 = δ pmin r - Δpw

SE1 = α2 jqj þ

ð6:114Þ

106

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

6. Point F corresponds to the stage of the gas injection pr = p0 The EF section of the load program corresponds to the gas injection stage after the completion of gas extraction at which the reservoir pressure is lower than the initial pressure, i.e. pr < p0. At this section, a repression Δpw is created at the wellbore. Stresses on the contour of a horizontal well at a lateral point during gas injection at reservoir pressure pr are given by relations (5.97). Then at the point F corresponding to the initial reservoir pressure pr = p0 SFr = - δp0 SFθ = ð3 - α1 Þq þ ð2 - δÞp0 SFZ

ð6:115Þ

= α2 q þ ð1 - δÞp0

And, correspondingly, for the stresses applied to the specimen during the experiment along the load axes of the TILTS, we have SF1 = α2 jqj - ð1 - δÞp0 sF2 = ð3 - α1 Þjqj - ð2 - δÞp0 SF3 = δp0

ð6:116Þ

Figure 6.8 shows a load program for a rock specimen from the UGS №3 reservoir simulating the change in stresses at a lateral point on the contour of a horizontal well drilled along the direction of the minimum horizontal stress. It plots the stress values along the axis of the TILTS unit load S1, S2, S3 in MPa on the ordinate axis. Figure 6.8 shows the load program for a rock specimen from the UGS №3 reservoir, simulating stress changes at the top point on the contour of a horizontal well drilled along the direction of the minimum horizontal stress, Fig. 5.5. The graphs show that the highest stresses occur at the end of the extraction stage at the top point on the contour of a horizontal well drilled along the direction of maximum horizontal stress, and slightly lower stresses at the top point on the contour of a horizontal well drilled along the direction of minimum horizontal stress.

6.3

Physical Modelling Programs at the TILTS of the Deformation and Filtration Processes Taking Place in the Bottomhole Zone of UGS Wells Under Cyclical Changes in Reservoir Pressure for Specific UGS Fields

The developed load programs of rock specimens from UGS №1, UGS №2, UGS №3, UGS №4 and UGS №5 for physical modelling on TILTS of deformation and filtration processes taking place in bottomhole zone of UGSF wells under the action of alternating loads during gas extraction and injection are given below.

6.3

Physical Modelling Programs at the TILTS of the Deformation and. . .

107

Fig. 6.8 Load program for a top point on the contour of the UGS №3 horizontal well drilled along the direction of the minimum horizontal stress, Fig. 5.5

The programs are created under the assumption of the all-round uniform natural stress state in the reservoir and correspond to the stresses occurring at the lateral point of horizontal wells, since this point, as the relations in Chap. 5 suggest, is where the maximum stresses occur at the end of gas extraction from the reservoir. The same load programs also hold true for stresses occurring in the vicinity of perforation holes of vertical cased wells.

6.3.1

UGS №1

The depth of core sampling from the UGS №1 reservoir is approximately 1400 m corresponding to an initial uniform rock pressure of q = - γH = - 32.2 MPa at an average specific gravity of the overlying rocks γ = 2.3  103 kG/m3. The initial reservoir pressure is p0 = 14.0 MPa. The gas extraction phase is completed when the reservoir pressure drops to pmin r = 6:0 MPa and the injection phase is completed when the reservoir pressure increases to pmax = 14 MPa. r The Poisson’s ratio of reservoir rocks was taken as equal to ν = 0.2; δ = 0.2. Figure 6.9 shows the program of load rock specimens from wells of UGS №1 for

108

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

Fig. 6.9 Load program for specimens simulating change lateral point stresses on the contour of a horizontal well for the conditions of the UGS №1

lateral point of horizontal well constructed on the basis of relations of Sect. 5.2.3.2. On it the values of stresses along the axes of load of TILTS S1, S2, S3 in MPa are plotted on ordinate axis. Point B of the load program corresponds to the situation when the well is drilled and pressure in the well is equal to the initial reservoir pressure, i.e. pw = p0. Then for the stress at point B, assuming in (5.44) pw = pr = 14.0 МPа, we find. sВr = - 2:8 MPa,

sВz = - 21:0 MPa,

sВθ = - 39:2 MPa

Point C corresponds to the start of the gas extraction stage, and segment CD corresponds to the gas extraction stage of the reservoir. Point E of the load program corresponds to the moment when the reservoir pressure has decreased to a minimum value pmin r = 6:0 MPa, which corresponds to a maximum reservoir pressure drop of Δpr = - 8.0 MPa. At this point, stresses occurring on the wellbore contour at the end of extraction will be due only to a change in effective stresses in the depth of the massif, since after extraction the pressure in the well is equal to the reservoir pressure pmin r = 6:0 MPa.

6.3

Physical Modelling Programs at the TILTS of the Deformation and. . .

109

Stresses at the end of extraction at the lateral point on the contour of the horizontal well for conditions of UGS №1 we find from (5.56) Sr = - 1:2 MPa,

Sz = - 22:6 МPа,

Sθ = - 60:2 МPа

Consequently, as gas is extracted, the maximum stress Sθ on the contour of the horizontal well increases in absolute value from 39.2 MPa to 60.2 MPa, i.e. by 21.0 MPa.

6.3.2

UGS №2

The depth of core sampling from the UGS №2 reservoir is approximately 800 m which corresponds to the initial all-round uniform rock pressure q = - γH = 18.4 MPa with average specific gravity of the overlying rocks γ = 2.3  103 kG/m3. The initial reservoir pressure is p0 = 8.0 MPa. The gas extraction phase is completed when the reservoir pressure drops to pmin r = 5:5 MPa and the injection phase is completed when the reservoir pressure increases to pmax = 11 MPa. r Poisson’s ratio of reservoir rocks was taken as equal to ν = 0.2; δ = 0.2. Figure 6.10 shows the program of rock specimens load from Kasimovskiy UGS wells for lateral point of horizontal well constructed on the basis of relations of Sect. 5.2.3.2. On it on ordinate axis, the stress values along the load axes of TILTS S1, S2, S3 in MPa are plotted. Point B of the load program corresponds to the situation when the well is drilled and pressure in the well is equal to the initial reservoir pressure, i.e. pw = p0. Then, for stresses at point B assuming in (5.44) pс = pr = 8.0 MPa, we find. sВr = - 1:6 MPa,

sВz = - 12:0 MPa,

sВθ = - 22:4 MPa

The BC section corresponds to the stage of the gas injection. Point C corresponds to the end of the gas injection where the reservoir pressure is pmax = 11 MPa. r The CD section corresponds to the stage of gas extraction from the reservoir. At the point D, the gas pressure in the reservoir is equal to the initial pr = 8.0 МPа, and at the point E is equal to the minimum pressure at the end of gas extraction pmin r = 5:5 MPa. Stresses at the end of extraction at the lateral point on the contour of the horizontal well for conditions of UGS №2 we find from (5.56) sr = - 1:1 MPa,

sz = - 12:5 MPa,

sθ = - 25:4 MPa

Consequently, as gas is extracted, the maximum stress sθ on the contour of the horizontal well increases in absolute value from 22.4 MPa to 25.4 MPa, i.e. by 3 MPa.

110

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

Fig. 6.10 Specimen load program simulating lateral point stress variation on the contour of a horizontal well for UGS №2 conditions

6.3.3

UGS №3

The core sampling depth of the UGS №3 reservoir is approximately 760 m which corresponds to an initial all-round uniform rock pressure of q = - γH = 17.5 MPa with an average specific gravity of the overlying rocks γ = 2.3  103 kG/ m3. The initial reservoir pressure is p0 = 7.6 MPa. The gas extraction phase is completed when the reservoir pressure drops to pmin r = 6:5 MPa and the injection phase is completed when the reservoir pressure increases to pmax = 10:5 MPa. r Reservoir rock Poisson’s ratio was assumed to be ν = 0.2; δ = 0.2. Figure 6.11 shows the program of rock specimen load from UGS №3 wells for a lateral point of a horizontal well constructed on the basis of relations in Sect. 5.2.3.2. The values of stresses on the load axes of TILTS S1, S2, S3 in MPa are plotted on the axis of ordinates. Point B of the load program corresponds to the situation when the well is drilled and pressure in the well is equal to the initial reservoir pressure, i.e. pw = p0. Assuming for the stresses at the point B in (5.44) pw = pr = 7.6 МPа, we find.

6.3

Physical Modelling Programs at the TILTS of the Deformation and. . .

111

Fig. 6.11 Specimen load program simulating lateral point stress variation on the contour of a horizontal well for the conditions of the UGS №3

sВr = - 1:5 MPa,

sВz = - 11:4 MPa,

sВθ = - 21:3 MPa

The BC section corresponds to the stage of the gas injection. The point C corresponds to the end of the gas injection where the reservoir pressure is equal = 10:5 MPa pmax r The CD corresponds to the stage of the gas extraction from the reservoir. At the point D, the gas pressure in the reservoir is equal to the initial pressure pr = 7.6 МPа, and at the point E, it is equal to the minimum pressure at the end of gas extraction pmin r = 6:5 MPa. The stresses at the end of gas extraction at the lateral point on the contour of the horizontal well for conditions of UGS №3 we find from (5.56) Sr = - 1:3 МPа,

Sz = - 11:6 МPа,

Sθ = - 25:7 МPа

Consequently, as gas is extracted, the maximum stress Sθ on the contour of the horizontal well increases in absolute value from 21.3 MPa to 25.7 MPa, i.e. by 4.4 MPa.

112

6.3.4

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

UGS №4

The depth of core sampling from the UGS №4 reservoir is approximately 1195 m, corresponding initial all-round uniform rock pressure is equal q = - γH = 27.5 MPa. The initial reservoir pressure is p0 = 12.0 MPa. The gas extraction stage is completed when the reservoir pressure drops to pmin r = 9:6 MPa and the injection stage is completed when the reservoir pressure increases to pmax = 13:4 MPa. The r Poisson’s ratio of reservoir rocks was assumed to be ν = 0, 2; δ = 0, 2. Figure 6.12 shows the load program of rock specimen from UGS №4 wells for a lateral point of a horizontal well, constructed on the basis of relations in Sect. 5.2.3.2. The values of stresses acting along the axes of the TILTS S1, S2, S3 are plotted in MPa on the ordinate axis. Point B of the load program corresponds to the situation when the well is drilled and the pressure in the well is equal to the initial reservoir pressure, i.e. pw = p0. Then, assuming for the stresses at the point B in (5.44) pw = pr = 12.0 MPa, we find.

Fig. 6.12 Load program for specimens simulating change at the lateral point of stresses on the contour of a horizontal well for the conditions of the UGS №4

6.3

Physical Modelling Programs at the TILTS of the Deformation and. . .

sВr = - 2:4 MPa,

sВz = - 17:9 MPa,

113

sВθ = - 33:4 MPa

The BC section corresponds to the stage of gas injection. Point C corresponds to the = 13:4 MPa. end of gas injection where the reservoir pressure is pmax r The CD section corresponds to the stage of gas extraction from the reservoir. At the point D, the gas pressure in the reservoir is equal to the initial pr = 12.0 MPa, and at the point E, it is equal to the minimum pressure at the end of gas extraction pmin r = 9:6 MPa. The stresses at the end of gas extraction at the lateral point on the horizontal well contour for UGS №4 conditions, we find from (5.56) Sr = - 1:9 MPa,

Sz = - 18:4 MPa,

Sθ = - 40:9 MPa

Consequently, as gas is extracted, the maximum stress Sθ on the contour of the horizontal well increases in absolute value from 33.4 MPa to 40.9 MPa, i.e. by 7.5 MPa.

6.3.5

UGS №5

The depth of core sampling from the UGS №5 reservoir is approximately 1200 m, it corresponds to the initial all-round uniform rock pressure of q = - γH = 27.6 MPa. The initial reservoir pressure is p0 = 12.2 MPa. The Poisson’s ratio of the reservoir rocks is taken as ν = 0.2; δ = 0.2. Figure 6.13 shows the load program for rock specimens from UGS №5 for a lateral point on the contour of a horizontal well constructed on the basis of relations in Sect. 5.2.3.2. The values of stresses on the load axes of TILTS S1, S2, S3 are plotted in MPa on the ordinate axis. The point B of the load program corresponds to the situation when the well is drilled and pressure in the well is equal to the initial reservoir pressure, i.e. pw = p0. Then for stresses at the point B assuming in (5.44) pw = pr = 12.2 MPa, we find SBr = - 2:44 МPа,

SBz = - 17:84 МPа,

SBθ = - 33:24 МPа

For stresses in the lateral point on the contour of a horizontal well for conditions of UGS №5 at pmin r = 9 MPa from (5.56), we have Sr = - 1:7 МPа,

Sz = - 18:88 МPа,

Sθ = - 42:72 МPа

Consequently, as gas is extracted, the maximum lateral stress on the horizontal well contour sθ increases modulo from 33.24 MPa to 42.72 MPa, i.e. by 9.48 MPa.

114

6

Physical Modelling Programs of Rock Deformation and Fracture in. . .

Fig. 6.13 Specimen load program simulating lateral point stress changes on the contour of a horizontal well for the conditions of UGS №5

6.4

Conclusions

1. Based on the developed geomechanical approach, physical modelling programs have been developed at the TILTS for the deformation and filtration processes taking place in the bottomhole zones of the UGS reservoirs under the action of alternating pressures due to drawdowns and repressions during gas extraction and injection. 2. The programs have been constructed for both equal-component natural reservoir stresses and unequal-component natural stresses. 3. The programs correspond to changes in reservoir stresses as reservoir pressure changes during gas injection/extraction, and to changes in stresses in the vicinity of wells during cyclic creation of downhole repressions and depressions. 4. When testing the specimens using the developed programs, the deformation and fracture processes occurring in the reservoir during gas injection and extraction in the process of UGS operation are simulated. The experiments provide for measurement of specimen permeability in the course of tests. The experiments simulate the conditions occurring in the vicinity of both vertical and horizontal wells.

Chapter 7

Results of Physical Modelling of Deformation and Filtration Processes in the Bottomhole Zone of UGS Wells During Gas Extraction and Injection

7.1

Results of Physical Modelling at the TILTS of Downhole Repressions and Underbalances Without Regard to Changes in Reservoir Pressure

Physical modelling of deformation and filtration processes in the BHZ under the influence of alternating pressures due to depressions and repressions during gas extraction and injection was carried out at the TILTS. Such experiments were carried out on cubic rock specimens from the UGS №1, UGS №2, UGS №3 and UGS №4.

7.2

Test Specimens

A total of 17 experiments were carried out and 13 specimens from fours UGS were tested (four specimens were retested). Table 7.1 below shows the data of the tested specimens. All the specimens were first examined for anisotropy of elastic properties by sonication. Table 7.2 shows the elastic wave velocities in the specimens in each of the three axes. As an example, Figs. 7.1, 7.2, 7.3, and 7.4 show the results of a rock specimen from the UGS №4 tests. It was tested twice by simulating the creation of underbalances and overbalances Δp at the borehole face equal of 1 MPa and 2 MPa.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Karev, Y. Kovalenko, Geomechanical Aspects of Operation of Underground Gas Storage, Springer Geology, https://doi.org/10.1007/978-3-031-34765-8_7

115

116

7

Results of Physical Modelling of Deformation and Filtration Processes. . .

Table 7.1 Data on specimens for physical modeling

UGS UGS №4 UGS №1

UGS №2

UGS №3

7.3

Specimen numbers KR-1 KCH-3-1 KCH-3-2 KCH-5 KCH-6 KCH-7 KCH-8-1 KS-2 KS-3-1 KS-3-2 U-1-1 U-5-1 U-5-3

Well 95 317

312 447

55

Depths, m 1195.16 1401.50 1401.50 1406.70 1413.70 1429.49 1429.59 792.85 794.71 794.71 764.13 757.15 757.15

Test Results of KR-1 Specimen from UGS №4

Figure 7.1 shows the test program for the KR-1 specimen from the UGS №4 simulating one cycle of gas injection and extraction. The extraction drawdown and injection repression were equal Δp = 1 MPa. The KR-1 specimen was tested twice. First, it was placed in the load unit of the TILTS so that the load stress was applied along the axis 2 of the specimen, i.e. perpendicular to the core axis, and the permeability was measured along the core axis, in perpendicular to the bedding plane.

7.3.1

Test 1

Figure 7.2 shows the results of test 1. It shows the changes in the specimen strains along the three axes and the change in its permeability during the test. Six cycles of repression/depression were performed during the experiment. The specimen did not collapse. Therefore, it was carefully unloaded and the second experiment was performed on it. It differed from the first one in that the specimen was rotated in the load unit of the TILTS so that the stress load s2 was applied along the axis 1 of the specimen, i.e., perpendicular to the core axis, while the permeability was measured along the axis 3 of the core which corresponds to the radial direction into the wellbore.

7.3

Test Results of KR-1 Specimen from UGS №4

Table 7.2 Propagation velocity of longitudinal elastic waves in the specimens

NN 1

Specimen numbers KR-1

2

KCH-3-1

3

KCH -3-2

4

KCH -5

5

KCH -6

6

KCH -7

7

KCH -8-1

8

KS-2

9

KS-3-1

10

KS-3-2

11

U-1-1

12

U-5-1

13

U-5-3

a, m/s: speed of longitudinal waves

117 Axis number 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

a, m/s 2273 2150 2174 1960 1960 1792 2174 2174 1985 2564 2564 12,564 1785 1724 1638 2060 2173 2173 2857 2941 3030 1613 1613 1715 1492 1234 1265 1428 1370 1350 1156 1156 1250 1618 1538 1538 1608 1549 1549

118

7

Results of Physical Modelling of Deformation and Filtration Processes. . .

Fig. 7.1 Load program for specimen KR-1 Fig. 7.2 Test results of the specimen KR-1. Test 1

7.3

Test Results of KR-1 Specimen from UGS №4

119

Fig. 7.3 Test results of specimen KR-1. Test 2

Fig. 7.4 Picture of the specimen KR-1 after the test

7.3.2

Test 2

Figure 7.3 shows the results of test 2. During the experiment, 7 cycles of repression/depression were performed. The specimen did not fracture (Fig. 7.4). Figures 7.1 and 7.2 show that the initial permeability of the KR-1 specimen was high (3.7 D and 6.8 D respectively) and changed insignificantly during loading. Below is a summary table with data on all tests carried out. The columns indicate: (1) UGS from which the specimen was taken; (2) well number; (3) depth of sampling; (4) laboratory number of the specimen; (5) type of test at TILTS; (6) initial permeability of the specimen; (7) permeability of the specimen at the end of the test; (8) values of parameter δ and depression/repression at which the specimen was tested; test result (Table 7.3).

1401.50

№1

UGS №2

Specimen number KR-1

Sampling depth, м 1195.16

UGS №4

KS-3-1

KS-8-1

1429.59

792.85

KCH-7

1429.49

KS-2

KCH-6

1413.70

791.00

KCH-5

1406.70

KCH-3-2

0.64

2 is axis of cyclic loading permeability along axis 3 2 is axis of cyclic loading permeability along axis 3

2 is axis of cyclic loading permeability along axis 1

6.5

2 is axis of cyclic loading permeability along axis 1

6100

6000

69.6

4.5

4

6.1

5 Initial permeability, mD 3700

2 is axis of cyclic loading permeability along axis 3

Type of test 2 is axis of cyclic loading permeability 1—along axis 1 2—along axis 3 2 is axis of cyclic loading permeability along axis 3 2 is axis of cyclic loading permeability along axis 3 2 is axis of cyclic loading permeability along axis 3

4

4200

5200

0

3.9

21.7

3.9

1.0

4.2

Permeability at the end of the test, mD 3400

6

δ = 0 ΔP = 2 MPa has fractured before cyclic loading δ = 0.2 ΔP = 1 MPa has not fractured δ = 0.2 2 tests ΔP = 1 MPa ΔP = 2 MPa has not fractured δ = 0.2 ΔP = 2 MPa has fractured during 1 drawdown cycle δ=0 ΔP = 1 MPa has fractured during 1 drawdown cycle δ = 0.2 ΔP = 1 MPa has not fractured δ = 0.2 2 tests: ΔP = 1 MPa ΔP = 2 MPa has not fractured δ = 0.2, 2 tests: ΔP = 1 MPa ΔP = 3 MPa has not fractured

Notes δ = 0.2 ΔP = 1 MPa 2 tests: 1-top of horizontal well 2-side has not fractured

7

7

KCH-3-1

3

2

1

Table 7.3 Summary table of all tested specimens

120 Results of Physical Modelling of Deformation and Filtration Processes. . .

№3

U-1-1

U-5-1 Dark U-5-3 Dark

780.58

KS-3-2

764.13

794.71

2 is axis of cyclic loading permeability along axis 1 2 is axis of cyclic loading permeability along axis 1

2 is axis of cyclic loading permeability along axis 1 2 is axis of cyclic loading permeability along axis 3 2900 4900

6200

2400

9300

3500

2500

6400

δ= 0.2 ΔP = 1 MPa has fractured before cyclic loading δ = 0.2 ΔP = 1 MPa has fractured during 1 drawdown cycle δ = 0.2 ΔP = 1 MPa has not fractured δ = 0.2 ΔP = 1 MPa has not fractured

7.3 Test Results of KR-1 Specimen from UGS №4 121

122

7.4

7

Results of Physical Modelling of Deformation and Filtration Processes. . .

Results of Physical Modeling of Deformation and Filtration Processes in the Bottomhole Zone of UGS Wells Taking into Account Cyclic Changes in Reservoir Pressure

Physical modeling of deformation and filtration processes in the bottomhole zone under the action of alternating loads arising under cyclic changes in reservoir pressure was carried out at the TILTS using the developed load programs. The experiments were carried out on cubic rock specimens taken from the UGS №5 reservoir. The tests were carried out according to the load programs described in Chaps. 6 and 7. The test results of the specimen from UGS №2 and UGS №5 reservoirs are given below.

7.4.1

KS-5 Specimen Test Results from the UGS №2

The KS-5 specimen was taken from the depth of 798 m. The specimen was tested according to the load program corresponding to the lateral point N on the contour of the horizontal well and shown at Fig. 5.2. The test results are shown at Fig. 7.5. Figure 7.5a shows the test program for the specimen. The initial permeability of the specimen was equal to zero and did not appear during the test. The experiment simulated gas injection into the reservoir during which the reservoir pressure increased from 8.0 MPa to 11 MPa (point C in Fig. 6.2), and subsequent gas extraction. At the gas extraction stage, when the stress s2 applied to the specimen corresponding to the circumferential stress on the well contour sθ reached the value of 23.5 MPa which corresponds to a reservoir pressure of about 7.5 MPa, the specimen began to deform inelastically, Fig. 7.5b. When the stress s2 reached the value of 25.6 MPa corresponding to a reservoir pressure of pr = 6.7 MPa, the specimen has fractured, Fig. 7.5b.

Fig. 7.5 Test results of specimen KS-5: (a) specimen load program; (b) specimen strain curves

7.4

Results of Physical Modeling of Deformation and Filtration Processes. . .

7.4.2

Test Results of Rock Specimen AR-1.1 from UGS №5

7.4.2.1

Specimen AR-1-1

123

The specimen was tested according to the load program for lateral point N on the contour of the horizontal well. The test results of the specimen are shown at Fig. 7.6. Figure 7.6a shows the test program and the change in the permeability of the specimen during the test. The specimen had an initial permeability of 9.1 D which decreased to 7.5 D when the specimen was compressed to natural rock pressure. The experiment simulated gas injection into the reservoir during which the reservoir pressure increased from 13.1 MPa to 16 MPa (the point C in Fig. 7.6a), and subsequent gas extraction. During the gas extraction phase, until the stress applied to the specimen S1 corresponding to the circumferential stress Sθ on the well contour reached a value of 42.7 MPa, the specimen deformed almost elastically, Fig. 7.6b. When the stress S2 reached the value of 42.7 MPa (point D in Fig. 7.6a) corresponding to the reservoir pressure pr = 9.8 MPa, the creep of the specimen, i.e. its deformation under constant load began. Figure 7.6c shows creep curves of the specimen along two axes—the second one where the compressive stress is maximum, and the third one where it is minimum. With further increase in load, creep

Fig. 7.6 Test results of specimen AP1.1: (a) load program and permeability curve; (b) strain curves; (c) creep curves along two axes of specimen; (d) photo of specimen after test

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accelerated and at the load S2 = 43.7 MPa (the point E in Fig. 7.6a) corresponding to the reservoir pressure pr = 9.4 MPa, the specimen has fractured, Fig. 7.6c. The photo in Fig. 7.6d shows that the specimen crumbled almost to sand after the test.

7.4.3

Results of Physical Modeling of Deformation Processes in the Bottomhole Zone During Gas Injection and Extraction for the UGS №5

Table 7.4 shows the results of physical modeling on rock specimens from UGS №5 of the deformation processes in the bottomhole zone during gas injection and extraction. The last column of the table shows the value of reservoir pressure at which the creep of the specimens began in the experiments. Table 7.4 shows the UGS №5 reservoir is very heterogeneous in terms of deformation, strength and filtration properties. From this point of view they can be divided into three groups. The specimens of AR-1-1, AR-2-2, AR-5-5, AR-7-2, AR-8-2, AR-10-2, AR-10-3 and AR-11-3 were very weak and difficult to produce. Their permeability was very high (5–13 D). When physically simulating the gas injection and extraction processes on the specimens during the gas extraction stage at stresses corresponding to reservoir pressure of 9.4–14.2 MPa, the specimens began to deform “creep” intensively and then fractured. In contrast, rock specimens AR-15-1, AR-16-1, AR-13-2 and AR-14 were very strong and their permeability was equal zero indicating that the specimens tested were most likely from a non-permeability interlayer.

Table 7.4 Modeling results for deformation processes in the bottomhole zone during gas injection and extraction Specimen number AR-1-1 AR-2-2 AR-5 AR-15-1 AR-16-1 AR-7-2 AR-8-2 AR-10-2 AR-10-3 AR-11-3 AR-13-2 AR-14

Well 812 812 812 813 819 821 821 821 821 821 822 822

Depth, m 1249.43 1249.58 1252.22 1171.2 973.0 1199.85 1200.98 1201.34 1201.34 1201.70 1190.33 1214.0

Initial permeability, D 9.1 9.0 0.092 0 0 13.4 13.8 13.1 7.4 5.4 0 0

Creep initiation pressure, MPa 9.8 14.2 14.2 – – 9.8 – 9.4 9.8 9.8 – –

7.5

7.5

Conclusions

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Conclusions

1. Physical modeling of the deformation and filtration processes in the bottomhole zone under the influence of alternating pressures due to drawdowns and repressions during gas extraction and injection was carried out using the developed load programs at the TILTS. 2. The experiments simulated the cyclic creation of downhole underbalances and repressions values of 1 MPa and 2 MPa. The permeability of the specimen was measured in the direction corresponding to the radial direction into the borehole. Out of 17 experiments, in five cases, the specimens fractured during the tests while in the rest the specimens remained intact. The fracture occurred during the first cycle of simulated underbalance in the borehole. 3. Hence, we can conclude that cyclic creation of low drawdowns and repressions in wells mostly does not lead to rock fracture in the bottomhole zones of wells. Experiments also showed that in those cases where there was no specimen fracture, the change in rock permeability in the bottomhole zone was also not great. 4. The developed geomechanical approach was adapted to the conditions of UGS №5. The performed calculations showed that changes in reservoir pressure occurring during gas injection and extraction, lead to the fact that the stress state in the reservoir becomes significantly unequal-component. The unequalcomponent stress state in the reservoir leads to growth of tangential stresses in the vicinity of the wells when creating repressions and depressions, and this must be taken into account to select operation modes to reduce the risk of sand production. 5. Experimental studies and geomechanical modeling performed at the TILTS have shown that the change in reservoir pressure resulting from gas extraction and injection causes a maximum increase in tangential stresses on bottomhole of horizontal wells and a minimum on bottomhole of vertical wells. Therefore, to reduce the risk of sand production during well operations, it is advisable to consider replacing the horizontal well completion with a directional well completion which would significantly reduce responsible for rock fracture tangential stresses on bottomholes of wells. 6. Experiments on physical modeling of deformation and filtration processes under the influence of alternating loadings, performed at TILTS, showed when simulating conditions in BHZ corresponding to reservoir pressure of 9.4–9.8 MPa, creeping deformations appeared in the rock. When applied to the well, this means that at reservoir pressures below 10 MPa, sand may form in the BHZ and be carried into the well. Therefore, it is advisable not to reduce the reservoir pressure below 10 MPa to prevent sand production.

Chapter 8

Rationale for the Possibility of Using the Directed Unload Reservoir Method (DUR Method) to Increase Rock Permeability in the Bottomhole Zone of UGS Wells

The productivity and injectivity of a particular UGS well depend significantly on the permeability of the bottomhole zone. Its decline, even in a small vicinity of the well, significantly reduces well productivity (Beketov et al. 2008; Melnikova et al. 2015; Nifantov et al. 2014; Boronin et al. 2017). Reduced permeability is mainly due to well operation—intermittent injection of large volumes of gas containing solids and oil droplets. As a result, a fouling screen is created around the well, disrupting the hydrodynamic connection between the well and the reservoir. Chemical methods are widely used to clean the bottomhole zone, but their effect is short-lived and negligible. Recently, a mechanical method of removing the clogged reservoir zone and restoring its natural permeability by wellbore enlargement has gained popularity. But this method of well workover, firstly, is quite expensive and, secondly, is not applicable to horizontal wells. The mentioned problem of worsening of filtration properties of BFZ of UGS wells is in many respects similar to the problem of worsening of permeability of BFZ of injection wells of oil fields. Injection rate of injection wells worsens due to the contamination of the BHZ by mechanical impurities and hydrocarbon compounds contained in injected water. In Russia and abroad, drilling and workover operations are in most cases carried out in the mode of repression on reservoir. Under the influence of excess of bottomhole fluid pressure over reservoir pressure, the latter (drilling mud or other fluid) penetrates into BFZ and reduces its filtration-capacitative properties (FCP). In some cases (high repressions, low or very high FCP, inconsistency between properties of flushing fluid and properties of reservoir fluids), irreversible reduction of FCP in bottomhole zone occurs, as a result of which well completion is delayed and its design capacity is not achieved. Currently, more than 65% of Russia’s underground gas storage facilities are set up in depleted gas reservoirs that were at the time of their creation at reduced reservoir pressure. In these complex conditions, the opening of reservoirs was accompanied by intensive absorption of flushing fluid. Therefore, it is needed a long time to complete wells and they come on stream with reduced production rates. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Karev, Y. Kovalenko, Geomechanical Aspects of Operation of Underground Gas Storage, Springer Geology, https://doi.org/10.1007/978-3-031-34765-8_8

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Rationale for the Possibility of Using the Directed Unload. . .

To solve these problems, technologies based on the geomechanical approach to increase the permeability of reservoirs seem to be the most promising, given their efficiency, relatively low cost and environmental safety. Significant and irreversible increase in rock permeability can occur as a result of fracture and disintegration as mechanical stresses change. Many factors determine the value of stresses required for this—lithological composition, deformation and strength properties of rocks, depth of reservoir and rock structure, reservoir fluid pressure, bottomhole geometry and operating mode. Changes in the stress state in the bottomhole zone are carried out by technological operations on the well. A large number of experiments have been carried out recently to determine the effect of the stress-strain state of rocks on their filtration properties (Davies and Davies 2001; Fatt and Davis 1952; Hu et al. 2018; Nguyen et al. 2018; Holt 1990; Shi and Durucan 2009; Nasseri and Young 2016). For example, a series of triaxial loading and permeability measurement experiments on limestone specimens of the Coburg formation are presented in Nasseri and Young (2016). The dependence of permeability characteristics on stress-strain state of specimens under loading and unloading has been determined. It is shown that permeability decreases at small stresses, but a multiple increase compared with the initial one is observed at heavy loads. Moreover, after complete unloading the permeability of the specimens remains high. The authors attribute this to the occurrence of a network of microand macro-cracks in the rock. Change of permeability as a result of plastic deformation, as well as influence of non-uniform stress state and initial fracturing on rock creep, are considered in another article (Yang and Hu 2020). The research was carried out on cylindrical specimens of red sandstone with a single artificial crack using a conditional triaxial load system. During the experiment, the axial pressure was cyclic changed, creating a long-term non-uniform load in each state, while the permeability, elastic and plastic strains were measured in parallel and the strength and rheological characteristics were studied. The work concluded that the permeability of the rock studied was determined by the stresses level, the magnitude of deformation of the specimens and the duration of exposure under load. Multistage load/unload processes and creep deformation led first to a decrease in permeability and then to a sharp increase in filtration properties in the transition to the third stage of creep. For UGS reservoirs, studying the effect of stress state on rock permeability in the bottomhole zone can be a key towards their successful operation. Application of traditional well productivity enhancement technologies, such as hydraulic fracturing and acid treatment of wells at UGSs is not always effective. This is due to the fact that either depleted reservoirs of oil and gas fields or aquifers with very high permeability from tenths to several microns2 are chosen as UGS targets, and there is no need to increase the filtration surface area due to hydraulic fracturing. And acid treatment of the well is not always able to clean the filtration channels in the bottomhole zone from mechanical impurities and oil that got there during gas injection. One of the most promising ways to increase the permeability of reservoir rocks is to use the tremendous elastic energy stored in the rock mass due to the weight of

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Rationale for the Possibility of Using the Directed Unload. . .

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overlying rocks and reservoir pressure. The methods and parameters of such influence are determined on the basis of geomechanical approach. Use of technologies based on this approach will provide increase of efficiency of well operation and decrease of natural and technogenic risks during operations, including accidents and possible damage to the environment. One such technology is the directed unload reservoir method developed at IPMech RAS. The essence of the DUR method (Khristianovich et al. 2000; Kovalenko and Karev 2003; Klimov et al. 2015; Karev et al. 2020) is to create stresses in the vicinity of the well by unequal directional rock unload from rock pressure leading to rock cracking and formation of an artificial system of multiple micro- and macro-cracks in the reservoir. This system of cracks acts as an artificial grid of filtration channels, and its permeability is significantly (by an order of magnitude) higher than the natural permeability of the reservoir. Cracking, loosening of rock in the bottomhole zone can be caused by using the elastic energy stored in the rock mass (rock pressure) and the energy of reservoir fluid by creating drawdowns of a certain level at the bottomhole of well. The second important factor in the effectiveness of the DUR method is the need to maintain the required bottomhole pressure over time, as the fracture process develops gradually, spreading deep into the reservoir with time. This is due, firstly, to the rearrangement of the depression funnel in the vicinity of the well and, secondly, to the fact that at high stresses the rocks are no longer elastic and begin to deform over time (“creep”). Experience with the practical application of the DUR method has shown that it is effective to introduce stress concentrators into the formation in advance to reduce the bottomhole pressure at which stresses are built up in the formation to cause rock cracking. Perforation holes, vertical or horizontal slots can be such concentrators. The presence of stress concentrators not only initiates the fracture process in the vicinity of the well, but also makes it much more intense and extended through the formation. The DUR method has been successfully applied at a number of oil fields during well completion, workover of producing wells and workover of injection wells. Practice has shown that flow rate of uncased wells can be increase 2–4 times and cased wells 1.5–2 times. As for injection wells, geomechanics stimulation has allowed achieving 10–20 times increase in their injectivity, and the effect has been obtained in wells where other stimulation methods were used without success, including hydraulic fracturing. The DUR method has not yet been applied at UGS, however, the research results below suggest that it is feasible to use this method at wells of underground gas storage reservoirs.

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8.1

8

Rationale for the Possibility of Using the Directed Unload. . .

Test Methodology for Physical Modeling of Deformation Processes Using the TILTS on the Bottomhole of the Well During DUR Method Implementation

One of the key points of the developed method is to determine the stresses that must be created in the rock for it to crack and fracture. The analysis of mechanical behavior of UGS reservoir rocks which was carried out on the basis of geomechanical approach and given in Chap. 5 showed that change of pore pressure in UGS reservoir leads to change of stress state in the scale of reservoir. The initial stress state in the reservoir will be considered as equal-component compression by rock pressure q. The reason for this is the fact that UGS reservoirs are overlapped by “traps”, i.e. impermeable rocks, which are plastic in most cases. In addition, on a geological time scale, all rocks can be considered ductile. Therefore, in the absence of significant tectonic movements over considerable periods (tens and hundreds of millions of years), the deviatory part of the stress tensor (causing tangential stresses) of reservoir rocks should relax, and, under these conditions, the initial stress state can hardly be differ appreciably from the uniform all-round compression (Zheltov and Khristianovich 1955). It is known that shear stresses occurring in materials cause them to fail. A drawdown in a well is a decrease in downhole pressure compared to the current reservoir pressure. When an underbalanced in the well, for example, creates stress redistribution in the rock around the well, with stresses acting in radial direction to well decreasing by the value of the underbalance, and circumferential stresses increasing. As a result, with increasing underbalance, tangential stresses increase in the rock around the well which may eventually lead to cracking and even destruction. The calculations show how the stresses in the rock near the bottomhole change when the underbalance value changes for different variants of bottomhole design: analytical calculations in simple cases (open borehole) and numerical ones in more complicated cases (casing, perforation holes, slots, etc.) using threedimensional software. The found conditions of rock compression when varying the underbalance value for different face design variants are simulated on core specimens with the help of TILTS. This way, for each bottomhole design—open hole, cased hole, perforations, horizontal or vertical slots on the face—the stress distribution is calculated to change as the underbalance changes, i.e., a load program is created. Then, each of these programs is applied on the TILTS. The tests are used to determine at which part of the program, respectively at which underbalance value the rocks undergo inelastic deformation with cracking or loosening, accompanied by an increase in permeability. As a result, for each particular field, an optimal method of impact on reservoir rock is selected for increase permeability of bottomhole reservoir zone, including the type of bottomhole face design which is created by carrying out a number of technological operations (casing removal, perforation, cutting of slots) and the

8.1

Test Methodology for Physical Modeling of Deformation Processes Using. . .

131

Fig. 8.1 Stresses acting in the vicinity of the horizontal well in relation to the layered planes

magnitude of depression, required for cracking, loosening of rocks in the bottomhole reservoir zone. One of the main factors affecting the conditions and effectiveness of the DUR method is the anisotropy of the strength properties of the reservoir rocks in which the well was drilled. Anisotropy of strength properties of reservoir rocks is mainly related to the presence of layering planes in the formations, since layering planes are usually weakening planes. This explains the fact that rock cracking, which leads to an increase in permeability, largely depends not only on the magnitude of acting stresses, but also on their orientation relative to the rock strength anisotropy axes, i.e., in this case, relative to layering planes (Kovalenko et al. 2016). Figure 8.1 shows the circumferential and radial stress sθ, sr acting in the soil skeleton in the vicinity of a horizontal borehole at two points M and N. Along the axis of the borehole the stress is sz. As the fluid pressure in the well decreases, the circumferential stresses sθ at the points M and N will increase because they are proportional to the values of underbalance in the well. The difference when testing specimens at the points M and N is that at the point M, the maximum compressive stress sθ acts along the bedding, while the unload is perpendicular to the bedding. Conversely, at the point N, the maximum compressive stress sθ acts perpendicular to the bedding and the unload is parallel to the bedding. This explains the fact when modeling on the specimens conditions at the upper point M of a contour of a horizontal well, in the presence of layering and associated strength anisotropy, fracture usually occurs at significantly lower stress values sθ, and correspondingly lower underbalance, than when modeling the conditions at the lateral point N. This fact should be noted and it requires further study. Such studies may reveal the stress states that need to be created in the bottomhole zone to increase

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8 Rationale for the Possibility of Using the Directed Unload. . .

permeability and well productivity. This issue is particularly relevant when operating horizontal wells. With this in mind, the experiments simulate the location of the specimens on the contour of the horizontal borehole at point M (top) and the location of the specimens at point N (side), Fig. 8.1. This is achieved by appropriate positioning of the specimen in the loading assemble of TILTS. When modelling point N, the specimen is positioned so that axis 2 of the loading assemble, along which the load increases monotonically during the test, coincides with the axis of the core. When point M is simulated, the specimen is positioned in such a way that the axis 2 of the setup is perpendicular to the core axis. In the course of each experiment, the displacements of the specimen faces in time in each of the three directions are recorded, as well as the change in permeability of the specimen along one of its axes. The permeability of the specimen at the points N and M is also measured along different axes of the specimen: at the point N, perpendicular to the core axis, and at the point M, along it, that is, in both cases in the direction of the wellbore. This research allows us to draw an important conclusion regarding the choice of priority strain-strain and filtration characteristics of rocks of the productive section to be experimentally determined for creating and completing the geomechanical model of the field. The currently used conventional set of such data is based on the assumption of isotropic elastic and strength properties of rocks (Young’s modulus, Poisson’s ratio, Coulomb-More or Drucker-Pragueur strength constants, etc.). For their determination, installations based on the Carman principle are mainly used which do not allow the true stress states occurring in the formations in the vicinity of the boreholes to be created in rock cylindrical form specimens. At the same time, the deformation, strength and filtration properties of rocks depend significantly on the level and type of stresses created in them. Therefore, it can be argued that conclusions and forecast recommendations on stability of wellbore, maximum allowable underbalances and well rates which are made on the basis of geomechanical models that do not taking into account the anisotropy of rock deformation and strength properties as well as the dependence of their filtration properties on the acting stresses may be quite far from reality and do not solve the main problem—reducing risks and improving efficiency in well operation. The use of DUR method on UGS has a number of distinctive features compared to its application in conventional oil and gas fields. This is due to the cyclic changes in reservoir pressure during well operation. The question therefore arises as to whether it is more effective to use the DUR technique on stage of gas injection or extraction. It was shown above, in Chap. 5, that changes in reservoir pressure during injection and extraction of gas result in changes in stresses from rock pressure acting at depth in the reservoir. Therefore, the same underbalances created in the well will result in different stresses on the bottomhole during the gas injection and extraction stages.

8.2

Specimen Load Programs at the TILTS for Physical Modeling of. . .

133

Accordingly, load programs of rock specimens at TILTS during simulation of deformation processes in the bottomhole zone should be different as well. Following are the load programs for the conditions of UGS №5. Since horizontal wells are supposed to be used at operation of UGS №5, load programs are given just for them.

8.2

Specimen Load Programs at the TILTS for Physical Modeling of the Deformation Processes in the Vicinity of a Horizontal Borehole During the Implementation of the DUR Method

The stresses acting on the wall of a horizontal well, as shown in Sect. 5.3.1.2, vary from point to point of its contour. Calculations show that the maximum shear stresses occur at the well contour when drawdowns are created after gas injection. And the shear stresses τ = |sθ - sr|/2 acting in the ground skeleton, as well as the absolute values of circumferential stresses |sθ| reach a maximum at the top point M on the well contour (see Fig. 8.1) and a minimum at the lateral point N on the well contour at the end of gas injection, and conversely, reach a minimum at point M and a maximum at point N at the end of gas extraction. The load programs for rock specimens taken from the UGS №5 for points M and N at the end of gas injection (reservoir pressure Prmax pmах пл = 16 MPa) and at the end = 9 MPa) are given below. of gas extraction (reservoir pressure Pr min pmin пл As before, the stresses shown in the diagrams S1, S2, S3 are the stresses applied to the specimen during the experiment along load axes 1, 2, 3 of the TILTS. They correspond to the absolute values of the stresses |sz|, |sθ|, |sr|, acting in the soil skeleton on the borehole contour s1 = |sz|, s2 = |sθ|, s3 = |sr|. The ordinate axis on the graphs shows the stress values S1, S2, S3 in MPa. Point A on the load programs corresponds to the initial state in the rock massif away from the boreholes. As before, we assume that in the initial state in the rock, there is an equal-component compressive stress state under the rock pressure. Compressive stresses are assumed, as before, to be negative and pressure to be positive. In the initial state, the soil skeleton is loaded with effective all-round compression stresses sAr = sAz = sAθ = q þ ð1 - δÞp0 , where p0 is the initial reservoir pressure. Accordingly, for the stresses applied to the specimen along the axes of the TILTS, we have sA1 = sA2 = sA3 = jqj - ð1 - δÞp0 . The point B corresponds to the situation when the well is drilled and the pressure in the well is equal to the initial reservoir pressure, i.e. pw(t) = p0. Stresses at the point B are defined by relations (6.2), the values of stresses applied to the specimen along the axes of the TILTS are given by (6.3). The point C corresponds to the end of gas injection/extraction when pr ðt Þ = pmax = 16 MPa or pr ðt Þ = pmin r r = 9 MPa. The point D corresponds to the maximum underbalance value at the borehole bottom when pw = 0. The tests recorded the

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Fig. 8.2 Load program simulating change of stresses at the lateral point N on the contour of a horizontal well when creating a drawdown after gas injection (pmах = 16 MPa) r

change in the permeability of the specimens along one of the axes (Figs. 8.2, 8.3, 8.4, and 8.5). It can be seen from the above load programs that the greatest compressive stresses sθ and therefore shear stresses occur at the top point M on the contour of the horizontal well, if the underbalances in the well required to implement the DUR method are created after the gas is injected into the reservoir. It can be seen from the above load programs that the greatest compressive stresses sθ and therefore maximal shear stresses occur at the top point M on the contour of the horizontal well after the maximal gas injection and DUR method is expedient to apply the method at this stage. Therefore, in the experiments whose results are shown below, it was the load program shown in Fig. 8.3 that was implemented.

8.3

Results of TILTS Tests of Core Material from the UGS №5 to. . .

135

Fig. 8.3 Load program simulating change of stresses at the top point M on the contour of the horizontal well when creating a depression after gas injection (pmах = 16 MPa) r

8.3

Results of TILTS Tests of Core Material from the UGS №5 to Determine whether the Permeability of Reservoir Rocks Can Be Increased by Implementing the DUR Method

In order to determine the possibility of increasing the permeability of reservoir rocks by creating stresses of the required type and level in the bottomhole zone, four specimens from the UGS №5 were tested using the developed load programs at the TILTS. Figures 8.6, 8.7, 8.8, and 8.9 below show the test results.

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Fig. 8.4 Load program simulating change of stresses at the lateral point N on the contour of a horizontal well when creating a repression after gas extraction (pmin = 9 MPa) r

8.3.1

Specimen AR-7.1

The specimen was tested using the load program for the top point M on the contour of the horizontal borehole shown in Fig. 8.1. The test results of the specimen are shown in Fig. 8.6. The initial permeability of the specimen was 6.9 D and remained virtually unchanged until the specimen fractured. The experiment simulated a pressure drop Δpw at the well bottom after gas injection into the reservoir, i.e., at pmax = 16 MPa. r At pw = 11.5 MPa, intensive deformation of the specimen (creep) began, Fig. 8.6c. At pw = 8.2 MPa, the specimen fractured by macro-cracking.

8.3

Results of TILTS Tests of Core Material from the UGS №5 to. . .

137

Fig. 8.5 Load program for specimens simulating change of stresses at the top point M on the contour of the horizontal well when creating a repression after gas extraction (pmin = 9 MPa) r

8.3.2

Specimen AR-10.1

The specimen was tested using the load program for the top point M on the contour of the horizontal borehole shown in Fig. 8.1. The test results of the specimen are shown in Fig. 8.7. The initial permeability of the specimen was 9 D and remained virtually unchanged until the specimen fractured. The experiment simulated a pressure drop Δpw at the bottomhole after gas = 16 MPa. At pw = 8.2 MPa, intensive injection into the reservoir, i.e., at pmax r deformation of the specimen began (creep), Fig. 8.7. At pw = 4.3 MPa the specimen fractured by formation of macro-cracks.

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Fig. 8.6 Results of testing the specimen АР7.1: (a) the load program and the change in specimen permeability during the experiment; (b) strain curves; (c) creep curves along two axes of the specimen; (d) photo of specimen after test

Fig. 8.7 Test results of specimen AR-10.1: (a) load program and change of specimen permeability during the test; (b) strain; (c) creep curves along two axes of the specimen; (d) photo of specimen after the test

8.3

Results of TILTS Tests of Core Material from the UGS №5 to. . .

139

Fig. 8.8 Test results of specimen AP-11.1: (a) load program and specimen permeability change during the test; (b) strain; (c) photo of specimen after the test

8.3.3

Specimen AP-11.1

The specimen was tested using the load program for the top point M on the contour of the horizontal borehole shown in Fig. 8.1. The test results of the specimen are shown in Fig. 8.8. The initial permeability of the specimen was 15.7 D. When the specimen was compressed comprehensively, the permeability decreased to 14.5 D. The experiment simulated a pressure drop Δpw at the bottomhole after gas = 16 MPa. At pw = 7.1 MPa, intensive injection into the reservoir, i.e. at pmax r deformation of the specimen began and the specimen fractured without reaching the creep shelf by forming macro-cracks, Fig. 8.8b.

8.3.4

Specimen AP-12.3

The specimen was tested using the load program for the top point M on the contour of the horizontal borehole shown in Fig. 8.1. The test results of the specimen are shown in Fig. 8.9.

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Fig. 8.9 Test results of specimen AR-12.3: (a) load and change of specimen permeability during the test; (b) strain curves; (c) photo of the specimen after the test Table 8.1 Results of experiments to study the possibility of increasing rock permeability in the bottomhole zone of the Arbusovskoye UGS by creating the necessary stress conditions in it Specimen No. AR-7.1 AR-10.1 AR-11.1 AR-12.3

Initial permeability, D 6.9 9.0 15.7 0

Creep initiation pressure, MPa 11.5 8.2 7.1 –

Fracture pressure of the specimen, MPa 8.2 4.3 7.1 –

The initial permeability of the specimen was zero. The specimen deformed elastically and did not fractured throughout the test until the well was completely drained. The permeability of the specimen remained equal zero. Table 8.1 shows the results of experiments to study the possibility of increasing the permeability of rock in the bottomhole zone of gas storage reservoir by creating the necessary stresses in it. The last two columns of the table show the value of bottomhole pressure at which the experiments started creep of the specimens and their collapse, respectively.

References

8.4

141

Conclusions

1. Testing of core material from UGS №5 wells on TILTS allows us to make a conclusion about a real possibility to increase permeability of reservoir rocks by DUR method due to creation of required level of stresses in BHZ. While implementing this method the necessary stresses in the rock can be occurred by creating rather deep underbalances at the borehole bottom and choosing the necessary bottom hole design. Perforation, cutting of slots, cutting out casing in the productive interval of the wellbore affect stresses in the borehole zone and can initiate and facilitate the process of fracturing in the borehole zone. 2. The use of DUR method at a UGS has a number of distinctive features compared with its application at oil and gas fields. This is due to the cyclical change in reservoir pressure during the operation of the field. Changes in reservoir pressure during gas injection and extraction lead to changes in stresses from the rock pressure acting in the depth of the reservoir. Therefore, the same drawdowns created at the bottomhole will result in different stresses in the bottomhole zone during the gas injection and extraction phases. Consequently, the effectiveness of the DUR method will be different for gas injection or extraction. 3. Calculations based on the developed geomechanical approach have shown that the maximum shear stresses at the contour of vertical and horizontal wells when underbalanced at their bottomhole are generated at the maximum reservoir pressure in the UGS, i.e. at the end of the injection process. For horizontal wells, the maximum stresses occur at the top point M (see Fig. 8.1) on their contour. Consequently, the effectiveness of implementing the DUR method will be maximized if underbalance in wells are created after gas injection into the reservoir, Figs. 8.3 and 8.7. 4. The results of tests at TILTS of core material, which is specimen from UGS reservoir, showed that at implementation of DUR method at UGS №5 after gas = 16 MPa, at lowering pressure in the well to injection into reservoir, i.e. at pmах r about 7–8 MPa in rock should start growth of macro-cracks and, as a result, increase of permeability of BHZ.

References Beketov SB, Brazhnikov AA, Shebanov IM (2008) Methods of impact on reservoir in order to increase productivity of wells of underground gas storages (in Russian). Gornyi informationalanaliticheskii bulletin 7:129–134 Boronin SA, Tolmacheva KI, Osiptsov AA, Orlov DM, Koroteev DA, Sitnikov AN, Yakovlev AA, Belozerov BV, Belonogov EV, Galeev RR (2017) Modelling of injection well capacity with account for permeability damage in the near-wellbore zone for oil fields in Western Siberia (in Russian). Paper presented at the SPE Russian Petroleum Technology Conference, Moscow, Russia, October 2017 Davies JP, Davies DK (2001) Stress-dependent permeability: characterization and modeling. SPE J 6:224–235. https://doi.org/10.2118/71750-PA

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Fatt I, Davis DH (1952) Reduction in permeability with overburden pressure. J Pet Technol 4:16. https://doi.org/10.2118/952329-G Holt RM (1990) Permeability reduction induced by a nonhydrostatic stress field. SPE Form Eval 5(04):444–448. https://doi.org/10.2118/19595-pa Hu W, Wei Y, Bao J (2018) Development of the theory and technology for low permeability reservoirs in China. Pet Explor Dev 45(4):685–697. https://doi.org/10.1016/s1876-3804(18) 30072-7 Karev V, Kovalenko Y, Ustinov K (2020) Directional unload method is new approach to enhancing oil and gas well productivity. In: Advances in oil and gas exploration and production. Springer International Publishing, Switzerland, pp 155–166 Khristianovich SA, Kovalenko YF, Kulinich YV, Karev VI (2000) Increase of oil well productivity by means of geo-leaching method. Eurasia Oil Gas 2:90–94 Klimov DM, Karev VI, Kovalenko YF (2015) Experimental study of the influence of a triaxial stress state with unequal components on rock permeability. Mech Sol 50(6):633–640 Kovalenko YF, Karev VI (2003) Geoloosening method - new approach to the problem of increasing well productivity (in Russian). Technol Fuel Energy Complex 1:31–35 Kovalenko YF, Karev VI, Gavura AV, Shafikov RR (2016) On the need to account for anisotropy of strength and filtration properties of rocks in geomechanical modelling (in Russian). Pet Econ 11:114–117 Melnikova EV, Nifantov VI, Melnikov EA, Ivchenko OV, Ivchenko MV, Parfenov AM, Kaminskaya YV (2015) Results of well development at underground gas reservoirs and hydrocarbon fields (in Russian). Vesti gazovoy nauki Scientific and Technical Collection 3(23):57–54 Nasseri MHB, Young RP (2016) Laboratory triaxial and permeability tests on tournemire shale and cobourg limestone. Final Report, University of Toronto Rock Fracture Dynamics Facility Nguyen TS, Li Z, Su G, Nasseri MHB, Young RP (2018) Hydro-mechanical behaviour of an argillaceous limestone considered as a potential host formation for radioactive waste disposal. J Rock Mech Geotech Eng 10(6):1063–1081. https://doi.org/10.1016/j.jrmge.2018.03.010 Nifantov VI, Melnikova EV, Melnikov SA (2014) Increasing well productivity: experience, problems, prospects (in Russian). Gazprom VNIIGAZ, Moscow, p 242 Shi J-Q, Durucan S (2009). Exponential growth in San Juan Basin Fruitland coalbed permeability with reservoir drawdown-model match and new insights. In: SPE Rocky Mountain Petroleum Technology Conference https://doi.org/10.2118/123206-ms Yang S, Hu B (2020) Creep and permeability evolution behaviour of red sandstone containing a single fissure under a confining pressure of 30 MPa. Sci Rep 10:1900. https://doi.org/10.1038/ s41598-020-58595-2 Zheltov YP, Khristianovich SA (1955) On hydraulic fracturing of an oil-bearing reservoir//Izvestia of the USSR Academy of Sciences (in Russian). Branch Techn Sci 5:3–41

Chapter 9

Justification of Predictive Recommendations for Maximum and Minimum Allowable Drawdowns, Well Flow Rates and Prevention of Sand Production

One of the main complications in UGS operations is sand production. This leads to reduced gas extraction rates and damage to equipment at the surface and in the well. The main hypothesis of sand production, accepted by many researchers, is related to the stress-strain state of the rock in the bottomhole zone. Sand production in the borehole is caused by the failure of the bottomhole zone due to stresses in the rock during filtration of reservoir fluids (gas, water and their mixture) (Karimov 1981; Mirzajanzade et al. 2003; Bashkatov 1981; Vrachev et al. 1999; Basniev et al. 2005). The failure of the bottomhole zone occurs if the stresses exceed the rock strength limit. At present, sand control methods are mainly limited to the application of technologies aimed at preserving the bottomhole formation zone by reducing underbalance, by reducing well flow rates, by installing sand control filters, increasing the strength of rock in the bottomhole zone by securing it with various polymer binders, resins or cement, and others. However, despite many years of experience in operating wells with sand production, so far there is no sufficiently substantiated model of reservoir failure and no mathematical description of the processes taking place in the reservoir-well system. Developed geomechanical approach and physical modelling of deformation and filtration processes in bottomhole zone under alternating loads enable to give prognostic recommendations on maximum and minimum allowable repressions, well flow rates and prevention of sand production. They are aimed at prevention of stresses development in bottomhole zone, which lead to rock deformation (creep), its destruction and carrying of destroyed rock (sand) to the borehole. As shown in Chaps. 5 and 6 above, the main source of elevated stresses in the bottomhole zone of the UGS well is not the underbalance/overbalance created in the well, but the change in the initial stress state in the reservoir caused by changes in reservoir pressure. Calculations were performed both for equal-component initial stress state and for unequal-component one. It should be emphasized that the change in horizontal stresses in the reservoir occurs by the same amount when the reservoir pressure is changed. In addition, the change in stresses in the reservoir when the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Karev, Y. Kovalenko, Geomechanical Aspects of Operation of Underground Gas Storage, Springer Geology, https://doi.org/10.1007/978-3-031-34765-8_9

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9 Justification of Predictive Recommendations for Maximum and. . .

reservoir pressure changes in an equal-component and an unequal-component natural stress state are equal to each other. So, for the conditions of UGS №5, creation of underbalance 0.5 MPa at the borehole bottom leads to the increase of shear stresses responsible for rock deformation and destruction, on the borehole wall by 0.5 MPa. A decrease of reservoir pressure at the end of gas extraction stage to the value of 9 MPa leads to the increase of shear stresses by 5 MPa. It is not sufficient to know the stresses acting in the bottomhole zone of the borehole to answer the question of whether they will result in fracture of the rock. The value of stress at which creep begins depends on the lithology type of the rock, its strength, sandstone grain size distribution, bedding conditions, reservoir pressure and other factors. As the result, the approach traditionally applied to solving such problems, based on creating mechanical and mathematical models and making calculations based on them, faces great difficulties, overcoming which leads to the need to make certain simplifications and assumptions in the model. Therefore, practical conclusions based on the calculations on such models often being only of an estimative nature. To answer these questions, we used direct physical modelling of rock deformation and fracture processes in the bottomhole zone of the UGS well on a unique triaxial independent load test system (TILTS). The load program for the test specimens, which were 40 or 50 mm cubes, was determined on the basis of a developed geomechanical model. The calculations have shown that for a horizontal well, the maximum shear stresses occur on its contour at side point N, Fig. 5.2, at the minimum reservoir pressure at the end of gas extraction, which for UGS №5 is 9 MPa. The results of physical modeling of deformation and filtration processes occurring in the bottomhole zone under the action of alternating loads during gas injection and extraction both at constant reservoir pressure equal to the initial pressure and at changing during gas injection/extraction are given in Sect. 7.4.2. The experiments were carried out on core material taken from 5 UGS. The results of the experiments and their processing allow a number of practical conclusions to be drawn. Experiments on physical modelling of deformation processes in the vicinity of the borehole during cyclic underbalance/overbalance at constant reservoir pressure (Sect. 7.1) showed that the specimens deformed elastically and almost did not fracture when stresses corresponding to underbalance/overbalance up to 2 MPa. A different picture was observed during simulation of deformation processes in the bottomhole zone of the well during cyclic underbalanced/overbalanced wellbore with reservoir pressure changes (Sect. 7.4). In this case, most of the tested specimens began to deform (“creep”) intensively at the gas extraction stage at stresses corresponding to the reservoir pressure close to the minimum extraction pressure and then fractured. This means that when gas is extracted as long as the reservoir pressure is close to the initial reservoir pressure, or when gas is injected when the reservoir pressure is higher than initial one, the risks of intense sand production are relatively low.

9

Justification of Predictive Recommendations for Maximum and. . .

145

The risk of intense sand production increases during gas extraction when the reservoir pressure drops and reaches a minimum at the end of gas extraction. This conclusion is also confirmed by calculation using the Coulomb-More strength criterion. According to this criterion, fracture in rock occurs along planes with a normal n, at which the shear stress τn reaches the shear strength limit. And the normal stress acting in the fracture plane increases the shear resistance of the material by a value proportional to the normal stress. The condition for the rock reaching the limit state at the site with the normal n is (Tertsagi 1961) τn > с þ jσ n jtgρ

ð9:1Þ

where σ n and τn are the normal and shear stresses at the site, с is the adhesion coefficient and ρ is the angle of internal friction of the rock. In the principal stresses acting in the soil skeleton, which for the points on the contour of the horizontal borehole are the stresses |sθ| > |sz| > |sr|, the Coulomb-More criterion is written as sθ - sr s þ sr > θ sin ρ þ c cos ρ 2 2

ð9:2Þ

Since there is a correspondence between the stresses s1, s2, s3, applied along the load axes 1, 2, 3 of the TILTS during the experiment and the stresses sz, sθ, sr, acting on the contour of the horizontal well, s1 = |sz|, s2 = |sθ|, s3 = |sr|, then in terms of the stresses applied to the specimen along the three axes in the experiment, the strength test (9.2) can be rewritten as s þ s1 s2 - s1 > 2 sin ρ þ c cos ρ 2 2

ð9:3Þ

During gas injection and extraction for horizontal wells, as shown in Sect. 6.2.2, maximum stresses occur on the well contour at point M, Fig. 5.2, at the end of gas extraction from the reservoir, i.e. at pr ðtÞ = pmin r . Calculations of strength criterion for one of the wells of UGS №5 are given below using relation (9.3). According to the program of rock specimen load from UGS №5 for point N on the contour of the horizontal well at the end of gas extraction at pmin r = 9 MPa, the maximum stress is s2= 41.82 MPa and the minimum stress is 1.7 MPa. Assuming, as it was established above in Sect. 4.3 during triaxial tests of rock specimen AR9.2, Table 4.3, с =8 MPa, ρ= 380, from (9.3) we find. s þ s1 s2 - s1 = 20:06 MPa, 2 sin ρ þ c cos ρ = 18:82 MPa 2 2 This means that in this case there are areas on the contour of the horizontal borehole where the ultimate stress state is reached and where rock failure can begin and consequently sand production and transport into the borehole.

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Justification of Predictive Recommendations for Maximum and. . .

Let us now make similar calculations for the case where the minimum gas pressure in the reservoir at the end of gas extraction is pmin r = 10 MPa. In this case, from the relations in Sect. 6.3.2 for point N on the contour of the horizontal well at the end of gas extraction at pmin r = 10 MPa, the maximum stress is s2= 39.42 MPa and the minimum stress is 1.9 MPa. Then at the same values с = 8 MPa, ρ= 38°, from (9.3) we find. s2 - s1 s þ s1 = 18:76 MPa, 2 sin ρ þ c cos ρ = 19:12 MPa 2 2 Hence, at pmin r = 10 MPa, the ultimate stress state is not reached at the contour of the horizontal borehole and the risk of rock failure is minimal. Below 10 MPa, sand can form in the BHZ and be carried into the borehole. Therefore, it is advisable not to reduce the reservoir pressure below 10 MPa to avoid sand events associated with rock failure. Thus, to reduce the risk of sand production during field operation, it is important to determine the minimum pressure in the UGS wells, at which intensive destruction of rocks in their vicinity will not yet occur. This can only be done experimentally by simulating deformation processes in the vicinity of wells during cyclic creation of underbalances/repressions at its bottomhole, taking into account changes in reservoir pressure. Such experiments should be performed on true triaxial independent load facilities like TILTS, since only they allow to create real stress states in rock specimens, occurring in bottomhole reservoir zone during gas injection and extraction. From the above it follows that the studies carried out on the basis of the developed geomechanical approach allow to give prognostic recommendations for maximum overbalances and minimum allowable underbalances, well flow rates and prevention of sand production during operation of UGS. However, in order to clarify them it is necessary to conduct additional experiments according to developed load programs on rock specimens selected from reservoirs of the UGS under study. In conclusion, we should note that the geomechanical modeling performed allows us to draw one more practically important conclusion. Calculations made on its basis showed that changes in reservoir pressure, occurring during injection and extraction of gas from UGS reservoir, lead to the fact that the stress state in the reservoir becomes significantly unequal-component. The unequal stress state in the reservoir leads to growth of shear stresses in the vicinity of wells, with maximum shear stresses occurring in the vicinity of horizontal wells and minimum—in the vicinity of vertical wells. Therefore, in order to reduce the risk of sand production during well operation, it is advisable to consider replacing the horizontal well completion with a directional well completion, which would significantly reduce the shear stresses acting in the vicinity of the well, which are responsible for rock destruction.

References

9.1

147

Conclusions

1. Calculations made on the basis of geomechanical approach showed that changes in reservoir pressure, occurring during injection and extraction of gas from UGS reservoir, lead to the fact that the initial stress state in the reservoir becomes significantly unequal. This, in turn, leads to increased stresses in the vicinity of wells, especially horizontal wells, when creating repressions and depressions. Such calculation results must be taken into account to select well operation modes and reduce the risk of sand production. 2. Experiments on physical modeling of deformation and filtration processes under the influence of alternating loads performed at the TILTS facility confirmed that the source of rock failure in the borehole zone is redistribution of stresses from rock pressure in the depth of the reservoir when the reservoir pressure changes. In this case, the underbalance created at the borehole face determines only the outflow of already destroyed rock in the form of sand into the borehole. 3. In order to reduce the risk of sand production during well operation, it is advisable to use directional wells instead of horizontal wells which will significantly reduce the shear stresses acting in the vicinity of the well that are responsible for rock failure.

References Bashkatov AD (1981) Preventing sand production of wells (in Russian). Nedra, Moscow, p 176 Basniev KS, Budzulyak BV, Zinoviev VV (2005) Improvement of reliability and safety of underground gas storages operation (in Russian). Nedra-Business Centre, Moscow, p 391 Karimov MF (1981) Exploitation of underground gas storages (in Russian). Nedra, Moscow, p 248 Mirzajanzade AH, Kuznetsov OL, Basniev KS, Aliev ZS (2003) Fundamentals of gas production technology (in Russian). Nedra, Moscow, p 880 Tertsagi K (1961) Theory of soil mechanics (in Russian). Gosstroyizdat, Moscow Vrachev VV, Shafarenko VP, Shustrov VP (1999) Sand production during UGS operation (in Russian). Gas Industry 11:62