Fundamentals and Practical Aspects of Gas Injection 9783030771997, 9783030772000

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Petroleum Engineering

Reza Azin Amin Izadpanahi Editors

Fundamentals and Practical Aspects of Gas Injection

Petroleum Engineering Editor-in-Chief Gbenga Oluyemi, Robert Gordon University, Aberdeen, Aberdeenshire, UK Series Editors Amirmasoud Kalantari-Dahaghi, Department of Petroleum Engineering, West Virginia University, Morgantown, WV, USA Alireza Shahkarami, Department of Engineering, Saint Francis University, Loretto, PA, USA Martin Fernø, Department of Physics and Technology, University of Bergen, Bergen, Norway

The Springer series in Petroleum Engineering promotes and expedites the dissemination of new research results and tutorial views in the field of exploration and production. The series contains monographs, lecture notes, and edited volumes. The subject focus is on upstream petroleum engineering, and coverage extends to all theoretical and applied aspects of the field. Material on traditional drilling and more modern methods such as fracking is of interest, as are topics including but not limited to: • • • • • • • • •

Exploration Formation evaluation (well logging) Drilling Economics Reservoir simulation Reservoir engineering Well engineering Artificial lift systems Facilities engineering

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Reza Azin · Amin Izadpanahi Editors

Fundamentals and Practical Aspects of Gas Injection

Editors Reza Azin Petroleum Engineering Persian Gulf University Bushehr, Iran

Amin Izadpanahi Oil and Gas Research Center Persian Gulf University Bushehr, Iran

ISSN 2366-2646 ISSN 2366-2654 (electronic) Petroleum Engineering ISBN 978-3-030-77199-7 ISBN 978-3-030-77200-0 (eBook) https://doi.org/10.1007/978-3-030-77200-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 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

Preface

Gas injection has been used as a means of pressure maintenance in the secondary oil recovery and in the tertiary oil recovery as Enhanced Oil Recovery (EOR) techniques. The concept and operation of gas injection started nearly two decades after industrial oil production in the nineteenth century when the operators faced with the pressure decline in the reservoir and found the produced gas as an abundant source to reinject and overcome the problem. Many field cases used various injecting gases, ranging from non-hydrocarbon to hydrocarbon, lean to enriched, or immiscible to miscible. The operating conditions vary from one case to another, depending on the geological conditions of the reservoir, its rock and fluid properties, and the availability of gas sources to inject. The objective of gas injection may also be different from one project to another. This book reviews some of the applications of gas injection and combines the new features as well as some key challenges associated with gas injection. The book is organized into ten chapters. Chapter 1 provides an overview of the hydrocarbon resources worldwide as well as a historical review of the gas injection operation. In Chap. 2, some of the key issues related to Pressure–Volume– Temperature (PVT) are reviewed. The phase diagram of injection gas, reservoir oil, and their mixtures are presented and discussed. Also, important PVT experiments, including CCE, DL, CVD, flash separation, and swelling test are discussed and supported by worked examples. Chapter 3 deals with basic concepts of oil and gas flow in porous media. In this chapter, single- and two-phase flow concepts are reviewed. Different flow equations, including viscous, pre-Darcy, Darcy, and Brinkman flow are introduced and the boundary between these flow regimes is presented. The well-known diffusivity equation which is a combination of continuity equation and Darcy velocity is also introduced and discussed. The relative permeability as an important concept in two-phase flow through porous media is described and the forces affecting oil flow in porous media are introduced as well as different dimensionless numbers which reflect the relative importance of forces during oil flow. Chapter 4 focuses on Underground Gas Storage (UGS) as a special gas injection applied for peak shaving and balancing the supply and demand of natural gas. In Chap. 5, gas injection into the gas cap known as pressure maintenance is introduced. The concept of pressure maintenance, historical approach, mechanistic v

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modeling, and reservoir material balance are discussed, followed by pressure maintenance models and mechanisms in fractured reservoirs. Chapter 6 deals with gas recycling and addresses conceptual, economical, historical, and operational aspects of the gas recycling operation. Chapter 7 focuses on the design of surface facilities for gas injection. In this chapter, conceptual design, design considerations and basis, and the general layout of a gas injection facility are described. Then, elements of a surface facility are introduced and described in detail. Chapter 8 presents methods for predicting the water content of gases, mostly based on equations of state and rigorous thermodynamic models. Different methods of predicting the water content of acid gas systems are evaluated and the water content diagrams compatible with the experimental data for pure CO2 , H2 S, CH4 , and their mixture are presented. Also, the hydrocarbon and non-hydrocarbon gases solubility in water, calculation of gas solubility in water using Henry’s constant law and the effect of salinity on gas solubility are presented in this chapter. The structure of Chap. 8 is based on engineering procedures and worked examples and may be used as an engineering guide to predict the water content of gases and gas solubility in brine. Chapter 9 focuses on the most important challenges including the corrosion in different types of gas injection, gravity override, mobility control, cap rock integrity, trapping, and Health, Safety and Environment (HSE) of gas injection. These concepts must take into account before, during, and after a gas injection project to increase the efficiency of the project. Chapter 10 deals with capillary trapping in porous media. The concepts, types, models, and correlations for phase trapping are presented and discussed in this chapter. References are added at the end of each chapter. Also, some chapters contain useful information as appendix. In preparing the contents of this book, more than one thousand references were reviewed comprehensively. Also, many chapters include worked examples to make a clear content and field application Bushehr, Iran March 2021

Reza Azin Amin Izadpanahi

Acknowledgements

A number of scientists, professors, and industrial experts contributed to the chapters of this book. These include Prof. Mahmood Moshfeghian, Prof. Shahriar Osfouri, Dr. Ahmad Jamili, Dr. Ali Ranjbar, Dr. Alireza Shahkarami, Dr. Mohamad MohamadiBaghmolaei, Ahmad Banafi, Amin Izadpanahi, Fatemeh Kazemi, Parviz Zahedizadeh, and Pooya Aghaee Shabankareh. Also, the authors visited several oil and gas injection and production facilities and had technical meetings with industrial experts to get better insight into details of gas injection operations. I’d like to acknowledge Reza Heidari, Mohammad Abdali, and Dr. Hamid Khedri and appreciate their technical support and valuable suggestions. Reza Azin Professor of Petroleum Engineering [email protected]

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Contents

1

Introduction to Gas Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reza Azin, Amin Izadpanahi, and Alireza Shahkarami

1

2

PVT of Gas Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reza Azin, Amin Izadpanahi, and Shahriar Osfouri

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3

Basics of Oil and Gas Flow in Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . Reza Azin, Amin Izadpanahi, and Parviz Zahedizadeh

73

4

Gas Injection for Underground Gas Storage (UGS) . . . . . . . . . . . . . . . 143 Reza Azin and Amin Izadpanahi

5

Gas Injection for Pressure Maintenance in Fractured Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Ahmad Jamili, Amin Izadpanahi, Pooya Aghaee Shabankareh, and Reza Azin

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Gas Recycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Reza Azin, Amin Izadpanahi, and Mohamad Mohamadi-Baghmolaei

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Design of Subsurface and Surface Facilities for Gas Injection . . . . . . 319 Reza Azin and Ahmad Banafi

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Water-Hydrocarbons System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Amin Izadpanahi and Reza Azin

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Challenges of Gas Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Reza Azin, Amin Izadpanahi, and Ali Ranjbar

10 Capillary Phase Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Fatemeh Kazemi, Reza Azin, and Shahriar Osfouri

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Chapter 1

Introduction to Gas Injection Reza Azin, Amin Izadpanahi, and Alireza Shahkarami

Abstract This chapter describes the different methods and the application of gas injection in the oil and gas industry and environmental purposes. For this purpose, first the worldwide hydrocarbon distribution is studied which categorized as conventional and unconventional resources. Also, the graphical distribution is reported using the latest statistics of conventional and unconventional hydrocarbon resources. Second, the steps of recovery of oil reservoirs are studied which consist of primary, secondary and tertiary oil recovery. Gas injection is performed for improvement of oil recovery in the secondary and tertiary oil recovery. Environmental protection and reduction of greenhouse gases is another goal of gas injection. Third, various relevant aspects including, historical evolution, screening criteria, sources of gas and economic of gas injection are investigated.

1.1 Introduction Rapid economic growth and population increment have led to a severe increase in energy demand in all around the world. According to the BP statistical review of world energy in 2016, fossil fuels among different natural energy resources, have the largest contribution (nearly 80%) in supplying the world’s energy needs. Also, it has been estimated that fossil fuels will remain the predominant sources of energy by 2035 [1]. Nonetheless, these most widely used energy sources have numerous R. Azin (B) Department of Petroleum Engineering, Faculty of Petroleum, Gas and Petrochemical Engineering, Persian Gulf University, Bushehr, Iran e-mail: [email protected] A. Izadpanahi Oil and Gas Research Center, Persian Gulf University, Bushehr, Iran e-mail: [email protected] A. Shahkarami Saint Francis University, Loretto, PA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 R. Azin and A. Izadpanahi (eds.), Fundamentals and Practical Aspects of Gas Injection, Petroleum Engineering, https://doi.org/10.1007/978-3-030-77200-0_1

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negative aspects such as the economic dependency of some regions and countries on fossil fuels, their reserves restriction and most importantly, environmental effects [2]. With regard to the global energy consumption in the last three decades, oil and gas are the most important hydrocarbons and fossil resources [1]. These resources are mainly classified into two categories, named conventional and unconventional. Global distribution of these two groups happens at a ratio of about 1:4 [3]. Various definitions have been appointed for conventional and unconventional reservoirs by numerous experts. Nevertheless, the most prevalent description of conventional resources is supreme quality and high permeability reservoir which may be withdrawn economical volumes of hydrocarbons without needing substantial excitation and only by drilling a vertical hole and perforating the productive interval [4]. As reported by the international energy agency (IEA), these hydrocarbon reservoirs are largely concentrated in the Middle East and Russia. Global proven reserves of conventional oil and gas are prognosticated about 1.3 trillion barrels and 220 trillion cubic meters, respectively [5]. Any source of oil and gas which has a low quality reservoir due to the low permeability or high oil viscosity and needs production technologies remarkably different from those used to produce from conventional reservoirs is commonly illustrated as unconventional [6]. Canada, Venezuela and Asia Pacific are the main regions with these types of hydrocarbon resources. Oil shale, oil sands (heavy oils, bitumen and tar deposits), light tight oil, tight gas, shale gas, coal-bed methane (CBM) and methane hydrates are the categories of unconventional resources as defined by IEA [5]. Estimated unconventional oil proven reserves are around 400 billion barrels in the world. Due to the high heterogeneity of rock formations, assessment of proven reserves is very arduous in unconventional gas resources, and its exact value is uncertain [5]. As mentioned, another important drawback of a sharp increase in fossil fuel consumptions is CO2 and other greenhouse gases (GHGs) emissions. The increase of GHGs in the atmosphere, has led to environmental disruption through global warming and climate changes [7]. Therefore, environmental protection with regard to the fossil fuels domination on worldwide energy supply, has become a global challenge in petroleum industry. Gas injection has been suggested and employed as an efficacious approach almost since 1930 to extend hydrocarbon production in different stages of exploitation, called secondary and tertiary recovery operations [8–11]. Also, in the case of CO2 or acid gas injection, this procedure has been considered as useful tools to reduce the concentration of atmospheric GHGs and its destructive effects [12]. Oil reservoirs due to the very low primary recovery factor, around 20–40% for conventional and up to 15% for unconventional, in comparison with gas reservoirs were aimed at most gas injection projects in all around the world [13]. This chapter provides an overview of worldwide hydrocarbon resource distribution, followed by a review on the state-of-the art of gas injection technologies into oil reservoirs as secondary and tertiary recovery processes. Various relevant aspects, including historical evolution, the goal of gas injection, sources of gas for injection, challenges and problems of gas

1 Introduction to Gas Injection

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injection will be presented, and some key research findings and the gaps in current understanding will be highlighted.

1.2 Worldwide Distribution of Hydrocarbon Reservoirs Hydrocarbon reservoirs generally categorized as conventional and unconventional reservoirs. This grouping is defined based on the required technology for extracting the hydrocarbon and geological reservoir conditions. Unconventional oil and gas extraction needs new and costly technologies comparing to conventional ones. Conventional oil is identified as the oil produced by the reservoir’s energy and pressure maintenance methods (primary and secondary recovery). In contrast, unconventional oil cannot produce without the enhanced oil recovery methods. Extra heavy oil, natural bitumen, tar sands, and shale oil are categorized as unconventional oil [14]. Conventional natural gas is defined as the gas produced by reservoir pressure, but the production of unconventional gas requires complicated technologies. Tight gas, coal-bed methane, and gas hydrates are known as unconventional gas resources [14]. Figure 1.1 shows the resource triangle of conventional and unconventional resources [6]. In this section, the worldwide distribution of each resource is explained.

Fig. 1.1 Resource triangle which comprises the volume of conventional and unconventional resources [6]

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1.2.1 Conventional There are some annual reports that contain the amounts of energy resources which published by the British Petroleum (BP) company [15, 16]. In this chapter, the amounts of crude oil and natural gas are reported from these annual reports of 2010 and 2018. As it is obvious, these amounts are changed, and the share percent of each region increased or decreased. Crude Oil. The Middle East, Latin America and North America have the most conventional oil resources. Table 1.1 shows the proven oil reserves in different regions of the world in 2010 and 2018. Natural Gas. Europe and Eurasia and the Middle East have the most amount of natural gas reserves which reported in Table 1.2. In this table, the amount of conventional gas reserves in 2010 and 2018 has been reported. Table 1.1 Distribution of crude oil reserves all over the world [15, 16] Area

Proven oil reserves Percentage (%) Proven oil reserves Percentage (%) 2010 (109 bbl) 2018 (109 bbl)

North America

73.7

5.5

226.1

13.3

Latin America

198.9

14.9

330.1

19.5

Europe and Eurasia

136.9

10.3

158.3

9.3

Middle East

754.2

56.6

807.3

47.6

42.2

3.2

48

2.8

127.7

9.6

126.5

7.5

1333.6

100

1696.3

100

Asia Africa Total (109 bbl)

Table 1.2 Distribution of natural gas reserves all over the world [15, 16] Area

Proven natural gas Percentage (%) Proven natural gas Percentage (%) reserves 2010 reserves 2018 (1012 m3 ) (1012 m3 )

North America

9.16

4.9

10.8

5.6

Latin America

8.06

4.3

8.2

4.2

Europe and Eurasia

63.09

33.7

62.2

32.1

Middle East

76.18

40.6

79.1

40.9

Asia

16.24

8.7

19.3

10

14.76

7.9

13.8

187.49

100

193.4

Africa Total (1012 m3 )

7.1 100

1 Introduction to Gas Injection

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1.2.2 Unconventional The unconventional gas resources include coal-bed methane, tight gas, aquifer gas, and gas hydrates. Oil sands, heavy oil, natural bitumen and shale oil are categorized as the unconventional oil resources [14]. Oil Sands. Oil sands (also known as natural bitumen and tar sands) are mixtures of water, sands, bitumen and clay. The density of oil sands is greater than 1 gr/cm3 (less than 10° API) and viscosity is greater than 10,000 cp. The methods of producing oil sands include surface mining, underground mining or thermal methods depending on the reservoir depth. Cyclic steam stimulation (CSS) and steam-assisted gravity drainage (SAGD) are applied to reduce oil viscosity and make the fluid mobile. In CSS, the viscosity is reduced by injecting the steam and held the steam in the reservoir for a specified time. In SAGD, two horizontal wells are drilled from a vertical well. The steam is injected from the upper well and the oil is produced from the lower well [14]. Heavy Oil. The definition and classification of heavy oil differ from one country to other. In countries having large resources of conventional crude oil, classification of heavy oil refers to oils with 20–25 API. However, in some regions, heavy oil refers to oils below 20 API and viscosity lower than that (200 cp). The conventional, as well as thermal methods, may be applied to extract heavy oil from reservoirs. Shale Oil. A calcareous mudstone which contains kerogen is known as shale oil. The methods of producing shale oil are similar to oil sands. About 35–245 Liters of oil can be extracted from 1-ton shale oil rock [14]. Table 1.3 shows proven reserves of unconventional oil in the world [17, 18]. Also, Fig. 1.2 shows the distribution of unconventional oil in the world. Shale Gas. The gas trapped in the fine-grained sedimentary rock (shale) is known as shale gas. Normally, hydraulic fracturing is required for the production of this kind of gas. Table 1.3 Distribution of unconventional oil around the world [17, 18] Region

Oil sands (109 bbl)

North America

530.9

Latin America 0.1

Heavy oil (109 bbl) 35.3

Shale oil (109 bbl)

Total (109 bbl)

80

646.2

265.7

59.7

325.5

Europe and Eurasia

33.9

18.3

90.1

142.3

Middle East

0

78.2

42.9

121.1

Asia

42.8

29.6

73.9

146.3

Africa

43

7.2

0.1

Total (109 bbl)

650.7

434.3

346.7

50.3 1431.7

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Fig. 1.2 Distribution of unconventional oil in different regions of the world (references are the same as Table 1.3)

Tight Gas. The gas reservoirs with the permeability less than 0.1 millidarcy are known as tight gas reservoirs. Low permeability sandstone or limestone is the main rock in the tight gas reservoirs [19]. The main method of producing tight gas is hydraulic fracturing technique that entails injecting high pressure mixtures of water, proppants, and chemical to crack the reservoir rock and consequently enhance reservoir permeability [14]. Coal-bed Methane. The coal seams contain rich methane and often high proportions of nitrogen and carbon dioxide. These kinds of gas resources are known as coal-bed methane, where coal is the source rock of methane. The method of producing coal-bed methane is drilling a well in the coal seam and creating artificial fractures and filling them with a sand-water mixture. Table 1.4 shows the amount of unconventional gas resources all around the world. Also, Fig. 1.3 shows the distribution of unconventional gas in the world. Table 1.4 Unconventional gas resources all around the world [19] Area

Tight gas (1012 m3 )

Coalbed methane (1012 m3 )

North America

22.3

6.3

Shale gas (1012 m3 ) 42.7

Total 71.3

Latin America

3.7

0.2

35.6

39.5

Europe and Eurasia

4.9

1.7

51.5

58.1

Middle East

2.3

0

15.5

17.8 79.1

Asia

17

18.8

43.3

Africa

2.3

0.9

29.3

32.5

Total (1012 m3 )

52.5

27.9

217.9

298.3

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Fig. 1.3 Distribution of unconventional gas in different regions of the world (references are the same as Table 1.4)

Figures 1.4 and 1.5 compare the conventional and unconventional oil and gas resources worldwide, respectively.

Fig. 1.4 Comparison of distribution of conventional and unconventional oil in different regions of the world (references are the same as Tables 1.1 and 1.3)

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Fig. 1.5 Comparison of the distribution of conventional and unconventional gas in different regions of the world (references are the same as Tables 1.2 and 1.4)

1.3 Historical Evolution of Gas Injection Gas injection is the oldest method of injection fluid into the hydrocarbon reservoirs, with various applications in oil and gas industry and it has been done for different purposes such as pressure maintenance, enhanced oil recovery (EOR), underground gas storage (UGS), carbon capture and sequestration (CCS), gas recycling, and etc. Pressure maintenance, EOR and gas recycling have been done for improving the ultimate recovery from oil and gas reservoirs. These methods can result in recovery improvement by affecting different parameters such as reservoir pressure, oil viscosity, miscibility conditions, capillary forces, sweep efficiency, and etc. Underground gas storage is a method to store the hydrocarbon gases in natural reservoirs to meet the energy demands during high consumption seasons. Carbon capture and sequestration is a necessary process for decreasing the effects of greenhouse gases and preventing global warming. In this section, a brief history of each method is expressed. The concept of gas injection was first suggested as early as 1864 [20]. In 1903, Dunn expressed the results of gas injection into a well. Then, studies continued on gas injection and the first re-pressuring project was carried on in Ohio by air injection into depleted and partially depleted wells in 1913. The results of this project demonstrated that the production from these wells greatly increased after gas injection. After this project, pressure maintenance was widely used in different fields worldwide as an improving oil recovery method [21]. Underground gas storage started with an experiment as early as 1915 in Ontario, Canada. Gas was injected into several reconditioned old wells. Gas production started from that field in the following winter to meet peak requirements. This experiment

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repeated the next year (1916) in the near Buffalo, New York. After the success of these two projects, underground gas storage became a common method for storing hydrocarbon gases in low consumption seasons and producing these gases in high consumption seasons [22]. Gas recycling projects began in 1941. In one case, gas recycling was done to improve the recovery from wet-gas reserves. In another study, gas recycling was proposed as a production method based on 8 years of field experiments [23, 24]. The growth of EOR processes has been continued since the end of world war II. In that time, operators who had faced with declining reservoirs production after primary and secondary recovery found out that large quantities of oil stayed in their reservoirs. Decreasing production from reservoirs, increasing oil consumption, and decreasing success of new reservoir discoveries motivated the research and field activities involving EOR processes to respond in specific the oil boycott of 1973 and the subsequent energy crisis. Economic efficiency is one of the most important parameters for planning an EOR project, in other words, the costs of producing oil able to compete economically with the cost of EOR processes [25]. In the 1950s, miscible flooding was shown as one of the most favorable methods for enhancing ultimate oil recovery from reservoirs [26–32]. The significant viscosity reduction effect of carbon dioxide injection was reported as early as 1963. It has been shown that the viscosity reduction of heavy oil using carbon dioxide could be up to 98% [33]. In the 1970s, studies were conducted in the field of CO2 viscosity reduction and CO2 huff ‘n’ puff process [34, 35]. The first cyclic stimulation with CO2 was done in 1981 in Canada [34]. Application of carbon capture to mitigate climate change was first discussed in 1982 by focusing on control CO2 production from power plants [36, 37]. There are other alternatives accessible in the case of CO2 storage. The first suggested option for carbon storage was as early as 1977. In this proposal, the injection of carbon dioxide into the ocean was expressed. In this method, CO2 is transferred into deep water where it would stay for many years [38]. The other options for storing carbon are geological formations, oil and gas reservoir for EOR purposes, saline formations and coal seams [39–42]. The world’s first CO2 storage plant began to operate in 1996. In this plant, CO2 injected into a deep saline formation under the North Sea [43]. Figure 1.6 shows the historical timeline of gas injection methods.

1.4 The Need for Gas Injection Figure 1.7 shows the conceptual trend of production rate and cumulative production with time. As shown in this figure, throughout the lifecycle of an oil field, various stages of production are encountered. Primary crude oil production frequently proceeds with natural energies of a reservoir to let the hydrocarbons flow to the surface. In this stage, production rate experiences a buildup phase followed by plateau rate. The primary production mechanisms include rock and fluid expansion, solution gas drive, gravity segregation, gas cap drive, natural water influx, or a combination

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Fig. 1.6 Historical evolution of gas injection

of two or more energies. Contribution of each of these driving forces in pushing the oil towards and out of producer well is different from one formation to another and depends on a number of factors such as geological structure, petrophysical characteristics, physical and thermodynamic properties of reservoir fluids, and etc. By declining the reservoir pressure, the natural drive energies of reservoir decrease and external forces are required to keep the reservoir production. Studies indicate that solution gas drive exists in the most oil reservoirs and can recover less than 25% of original oil in place (OOIP) for conventional reservoirs [44, 45]. Also, natural water drives exist almost in one-third of world reservoirs. In the most optimistic case, a strongly water driven conventional reservoir with high permeability, up to 50% of OOIP can be extracted [45, 46]. For gas-cap expansion and gravity drainage, the recovery is only around 5–30% [46]. Usually, the primary recovery of unconventional reservoirs is much lower than conventional ones. Studies on shale formations and extremely low-permeable oil reservoirs show that the primary recovery is below 10% [47–49]. Recovery factors of heavy oil reservoirs (35

6000

Hydrocarbon

>23

4000

CO2

>22

2500

References

[72, 73]

Huff n puff

>0.775

0.026

18.9–42

43–1800

2000–11,500

[51, 52, 80–86]

Gas recycling

NM

NM

21–1080

5855–13,205

[24, 87–98]

NM = Not mention

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1.6 Sources of Gas for Gas Injection The basic requirement for a gas injection operation is the availability of a certain source of gas. Generally, either hydrocarbon or non-hydrocarbon gases may be used for injection. Some sources like CO2 as the main greenhouse gas with increasing emission trends have been found an attractive source for injection. Also, N2 is another abundant source of gas and has been considered as a suitable gas for pressure maintenance projects in both oil and gas condensate reservoirs. Nitrogen is a safe, noncorrosive and environment friendly gas which can be separated from air. However, it raises the saturation pressure of reservoir fluid; moreover, it can’t dissolve in heavy hydrocarbon components and has a high minimum miscibility pressure. Therefore, N2 injection as an enhanced oil recovery process can be applied to high pressure, deep and light oil reservoirs [99–103]. The mixture of N2 and CO2 which is known as flue gas can be used as an alternative to pure CO2 and pure N2 for gas injection. Injection of this mixture has some advantages such as eliminating the pure gas separation costs, postponing the gas breakthrough compared to pure N2 injection, low compressibility and occupying more space in the reservoir compared to other gases and enhance the production because of CO2 displacement [104–106]. The mass transfer mechanism is also included in the flue gas injection which is in favor of valuable oil components recovery [107]. It was shown in the literature that the flooding efficiency is increasing by increase the CO2 fraction in the flue gas due to CO2 ability to crude oil swelling [108]. Another gas injection source is the hydrocarbon stream supplied either by associate gas and used for gas recycling or from independent gas reservoirs. The hydrocarbon gas may be used before or after treatment in gas plant. In other words, the injection gas may contain hydrocarbons, mostly methane, and non-hydrocarbon components like CO2 , H2 S, N2 , noble gases, and etc. It was shown that hydrocarbon gas injection works better than N2 and CO2 injection because of the adaptability and miscibility between the injected gas and reservoir fluids. The disadvantage of this method is that a large amount of natural gas will be back into the reservoir and this amount is unusable until the gas recycling project is stopped [106]. Therefore, hydrocarbon gas injection is attractive in areas with abundant sources of natural gas. Also, hydrocarbon gas injection has been used in gas condensate reservoirs to reduce the amount of condensate near wellbores, maintain the reservoir pressure above the saturation pressure and decrease the oversupply operation in the low market seasons [109]. The mixture of CO2 and H2 S, which is called Acid gas, could be considered as an injection fluid. This mixture can be released with the upstream production and the decrement gases from natural gas and petroleum processing. Acid gas has many irreparable effects on human health because of that in order to dispose, acid gas injection must be taken into account [110–112].

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1.7 Economics of Gas Injection For developing a gas injection process an economic evaluation must take into account for measuring the feasibility of the method. In this section, a review of the economic evaluation of different modes of gas injection was done.

1.7.1 Pressure Maintenance For planning a pressure maintenance project an economic analysis should be done. The expenditures of a pressure maintenance project include, well operation, plant and lines costs. The well operation costs are different for a flowing well with a pumping well. The costs of operating a flowing well will be lower than that of a pumping well and the average cost of a flowing well will increase more slowly than that of a pumping well. Also, replacements and workovers must be considered for well operation costs. In a pressure maintenance project, the operation of some plants and lines are necessary such as, compressor plant, injection lines, de-sulphurizer, and dehydrator. The costs of labor, supervision, repairs and materials must be added to the costs of building and operating the mentioned plants and lines [113].

1.7.2 CCS Some factors influence the costs of CCS projects such as fuel prices and the costs for monitoring and other regulatory requirements. The costs of technology developments, the regulatory environment, the base for storage potential and etc. must be added to the above expenditures. The main costs of a CCS project are divided into three components including, capture and compression, transport and storage. Capture and compression for large resources of CO2 like plants is the largest component of the whole CCS costs. The most economical and common way of CO2 transportation is through pipelines. Construction costs (material, labor, and etc.), operation and maintenance costs (maintenance, monitoring, energy costs and etc.) and other costs (insurance, design, fees and etc.) are the three main cost components for transporting pipelines. Geological storage, ocean storage and storage via mineral carbonation are three main methods for storing CO2 . In this section, the costs of storing in geological formation have been considered. The cost estimates of storage in geological formation can be made with certainty because the equipment and technologies which use in geological formation storage widely used in oil and gas industries. However, some specific factors like onshore vs offshore, reservoir depth and geological characteristics like thickness, permeability, and porosity create a considerable variability and range of costs [66].

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1.7.3 EOR The costs of an EOR gas injection process is divided into investment and operation costs. Investment costs include initial gas acquiring costs, compressors, injection plants, gas injection facilities, production facilities and wells work-over, transmission pipelines, new production and injection wells and surface facilities. The most expensive part of an EOR gas injection is building gas recovery plants. Well treatments (corrosion inhibitors, scale inhibitors and paraffin inhibitors), well workovers and servicing, power, labor, and facility maintenance are the most important operating costs. A large percentage of field operating costs usually spent on well workovers [114].

1.7.4 Underground Gas Storage The costs of underground gas storage contain gas purchasing costs, well costs, and dehydration and compression facilities. In the comparison of other methods of storage, aquifers and depleted fields supply more storage capacity per invested dollar. Storage in dry gas fields costs lower than in aquifers because of the lower investment in wells and surface facilities [115].

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Chapter 2

PVT of Gas Injection Reza Azin, Amin Izadpanahi, and Shahriar Osfouri

Abstract This chapter introduces the PVT challenges of gas injection. First, the phase diagram of various gases, oil samples and mixtures of oil and gas are investigated. In the next part, the important PVT experiments are discussed in details. These experiments consist of CCE (constant composition expansion), DL (differential liberation), CVD (constant volume depletion), flash separation and swelling test. Some calculation must be considered for designing the gas injection process as it is discussed in Sect. 2.4. In Sects. 2.5, 2.6 and 2.7, three cases are studied about change of phase behavior due to gas injection, MMP calculation of gas injection and asphaltene precipitation due to gas injection, respectively. The optimum design of gas injection and PVT challenges associated with gas injection are presented in Sects. 2.8 and 2.9, respectively.

Nomenclature Acronyms GIIP MMP

Gas Initially in place Minimum Miscibility Pressure

R. Azin (B) Department of Petroleum Engineering, Faculty of Petroleum, Gas and Petrochemical Engineering, Persian Gulf University, Bushehr, Iran e-mail: [email protected] A. Izadpanahi Oil and Gas Research Center, Persian Gulf University, Bushehr, Iran e-mail: [email protected] S. Osfouri Department of Chemical Engineering, Faculty of Petroleum, Gas and Petrochemical Engineering, Persian Gulf University, Bushehr, Iran e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 R. Azin and A. Izadpanahi (eds.), Fundamentals and Practical Aspects of Gas Injection, Petroleum Engineering, https://doi.org/10.1007/978-3-030-77200-0_2

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Variables fs K L m Mw p R S SEO Ss t T V vs x z

Fugacity Dispersion coefficient Reservoir length Mass of initial gas in-place, lb Molecular Weight, g/gmol Pressure, psi Ideal Gas Constant, ft3 .psi/lbmol.R Saturation Enriched oil slug size Solvent slug size Time Temperature, °F Volume, m3 or ft3 Molar volume of solid phase Mole fraction Compressibility factor

Subscripts bt d EO/O EO/S i inj l O/S origin rel sat S/CG

Breakthrough Dew-point Enriched-oil and oil Enriched-oil and solvent Initial Injected Liquid Oil-Solvent Original Relative Saturation Solvent and chase gas

2.1 Introduction Understanding the thermodynamic interaction between the injection gas and the original reservoir fluid is a fundamental step in gas injection. Distribution of components in phases is dependent on equilibrium conditions reflected by equality of chemical potential for each component in each phase. However, the main challenge in any

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gas injection operation is that thermodynamic equilibrium is rarely met due to high rate of injection and limited contact time between injected gas and reservoir fluid. Despite this fact, many gas injection processes are designed by assuming a thermodynamic equilibrium is happening. Subsequently, the process is overdesigned to take into account the uncertainties arisen by deviation from equilibrium conditions. Co-injection of contaminants such as CH4 , H2 S, N2 , O2 and SO2 with CO2 stream as acid gas and flue gas provides an extra benefit to operators and let them handle greenhouse gas emission from reservoirs or plants in addition to miscible EOR and CO2 sequestration projects. However, each contaminant may increase or decrease in MMP. The combined effect of components depends on the composition of injection gas and needs be examined and modeled. This chapter introduces the PVT challenges of gas injection. First, the phase diagram of various gases, oil samples and mixtures of oil and gas are investigated. In the next part, the important PVT experiments are discussed in details. These experiments consist of CCE, DL, CVD, flash separation and swelling test. Some calculation must be considered for designing the gas injection process as it is discussed in Sect. 2.4. In Sect. 2.5, 2.6 and 2.7, three cases are studied about change of phase behavior due to gas injection, MMP calculation of gas injection and asphaltene precipitation due to gas injection, respectively. The optimum design of gas injection and PVT challenges associated with gas injection are presented in Sects. 2.8 and 2.9, respectively.

2.2 Phase Diagram When a gas stream is injected into an oil reservoir, it mixes with the oil in the gas-oil front. Also, the gas composition approaches an equilibrium state assuming infinite mixing given enough time. This, however, does not mean that the bulk gas stream is in equilibrium with the bulk oil in reservoir. In this case, thermodynamics is a useful tool for phase behavior study and modeling injecting gas and reservoir oil system at the mixing zone. Different modes of injection in terms of miscibility, i.e. miscible, immiscible, near-miscible, first-contact miscible (FCM), and multiple-contact miscible (MCM) gas injection, are referred to phase behavior study and modeling of gas injection for EOR purposes. The state of miscibility depends on the composition of injection gas and reservoir oil, as well as operating temperature and pressure. In terms of the phase envelope, the PT diagrams of injection gas and reservoir oil are getting closer to each other as the degree of miscibility increases. The term “bubble point” stands for a pressure at which the first bubble of vapor gas evolves from liquid hydrocarbon at reservoir temperature. This point is important in reservoir engineering, as it is the intermediate between single and two-phase regions. Practically, the actual bubble point of a reservoir hydrocarbon mixture may occur at a pressure above bubble point measured experimentally, as the gas evolves from oil forms as micro volumes or smaller size; the volumes of

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gas need to grow enough to shape a visible “bubble”. Similar to bubble point, the term “dew point” applies mostly to light hydrocarbon mixtures, gas and gas condensate reservoirs. This term refers to a pressure at which the first drop of liquid hydrocarbon appears at reservoir temperature. Also, the formation of liquid hydrocarbon initiates at a pressure above the experimentally measured dew point, long before the liquid droplet or foggy atmosphere is visible by the operator from sight glass of a PVT cell. Both bubble point and dew point pressures are determined from thermodynamic phase equilibrium calculations. The phase diagram is often used to determine the type of reservoir fluid. Based on the oil composition, its phase diagram, and reservoir temperature and pressure conditions, the hydrocarbon reservoir may be distinguished as black oil, near-critical oil, volatile oil, gas condensate, wet gas or dry gas.

2.2.1 Hydrocarbon Gas Typical phase envelopes for different hydrocarbon injection gases are shown in Fig. 2.1. Details of injection gas compositions for samples of this figure are given in Table 2.1. Composition of light hydrocarbon (CH4 ) is dominant in all samples. A typical phase diagram is comprised of two parts, bubble point line (curve) and dew point line (curve). 1400 Sample 1

1200

Sample 2 Sample 3

Pressure (Psi)

1000

Sample 4 Sample 5

800 600 400 200 0

-320

-270

-220

-170

-120

-70

-20

Temperature (F)

Fig. 2.1 Typical PT diagram for different injection gases with composition given in Table 2.1

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Table 2.1 Composition of typical hydrocarbon-dominant injection gases Component

Sample 1

Sample 2

Sample 3

Sample 4

Sample 5

C1

90

85

80

85

80

CO2

5

10

15

5

5

H2 S

5

5

5

10

15

2.2.2 Non-hydrocarbon Gas Table 2.2 shows the composition of non-hydrocarbon gas which the relevant phase diagram is shown in Fig. 2.2. Compared to hydrocarbons, the phase diagram of CO2 rich injection gas differs significantly from hydrocarbon-based gas. The PT diagram of CO2 -rich gas shifts towards right compared to CH4 -rich samples. Table 2.2 Composition of typical CO2 -dominant injection gases Component

Sample 6

Sample 7

Sample 8

C1

20

10

10

Sample 9 5

Sample 10 5

CO2

80

80

90

90

95

H2 S

0

10

0

5

0

1400 Sample 6

1200

Sample 7 Sample 8

Pressure (psi)

1000

Sample 9 Sample 10

800 600 400 200 0 -200

-150

-100

-50

0

50

100

Temperature (F)

Fig. 2.2 Typical PT diagram for different injection gases with composition given in Table 2.2

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2.2.3 Oil Samples Table 2.3 presents the composition of typical reservoir oil which are from the Iranian oil reservoirs. Petroleum is a complex mixture of hydrocarbon compounds, some non-hydrocarbon compounds like CO2 , H2 S, N2 , H2 O, noble gases, and some hybrid compounds called oganometallic groups. Understanding an exact composition of a reservoir oil helps to predict its properties. However, the exact analysis of crude oil is practically impossible, as it contains hundreds to thousands of compounds. The major hydrocarbon families of petroleum include paraffin (alkane), naphthene (cycloalkane), aromatic, resin, and asphaltene. Its chemical composition varies from one reservoir to another, depending on the source and origin of kerogen, the basic hydrocarbon complex which further converts into oil, migration, depth and geothermal conditions. Based on the concentration of acid gas components (CO2 , H2 S), the oil may be considered sweet (having little or no acid gas or sulfur-containing component) or sour (having large amounts of acid gas or sulfur-containing components). Figure 2.3 presents the PT diagram for different oil with composition given in Table 2.3. Table 2.3 Composition of typical reservoir oil Sample 1 Component

Sample 2 Mole%

Component

Sample 3 Mole%

Component

Mole%

H2 S

0.44

H2 S

0.63

CO2

1.22

N2

1.66

N2

1.85

CH4

38.63

CO2

2.19

CO2

1.87

C2 H6

6.97

CH4

48.43

CH4

49.07

C3 H8

4.77

C2 H6

5.38

C2 H6

5.82

IC4

1.8

C3 H8

3.91

C3 H8

4.1

NC4

3.98

IC4

1.03

IC4

1.21

IC5

1.5

NC4

1.87

NC4

2.68

NC5

1.62

IC5

0.85

IC5

0.74

C6+

39.51

NC5

0.98

NC5

0.65

FC6

3.06

FC6

4.07

FC7

3.62

FC8

2.05

FC9

2.78

FC10

1.85

C7+

30.2

FC11

1.53

C12+

15.48

2 PVT of Gas Injection

29

4500 Sample 1 Sample 2

4000

Sample 3

Pressure (psi)

3500 3000 2500 2000 1500 1000 500 0

0

100

200

300

400

500

600

700

800

900

Temperature (F) Fig. 2.3 Typical PT diagram for different oil samples with the composition given in Table 2.3

2.2.4 Mixtures of Oil and Gas When gas is injected into an oil reservoir, the combined PT diagram between injection gas and reservoir oil changes dynamically, as the volume ratio of oil and gas in contact is varying. Complete mixing of injection gas and reservoir fluid rarely occurs in field operation; thus, one may expect a set of phase envelopes rather than a unique one. In other words, the composition at gas/oil interface approaches equilibrium and a spectrum of compositions can be detected from bulk gas to gas/oil interface and into bulk oil. As a result, a spectrum of dew point pressure from bulk gas to gas/oil interface and a spectrum of bubble point pressure from gas/oil interface to the bulk of the oil is formed corresponding to compositional changes upon gas injection. The in-situ enrichment of injected gas upon multiple contacts with reservoir oil enables the operator to use relatively cheap dry gas for injection and let the gas-oil system approach miscibility at reservoir conditions. Depending on the equilibrium between the injection gas and reservoir fluid, three different zones are formed. Figure 2.4 presents the zones that are formed due to gas injection. An important point on the phase diagram of reservoir oil is that it is for original composition only. Pressure decline upon oil production causes removal of light hydrocarbon from reservoir oil, leading to a dynamic shift in the phase diagram. Based on the physical phenomena occurring in a gas injection, three fluid phases may form as well as a solid phase. Fluid phases include bulk gas (either as a gas phase or supercritical fluid), bulk oil, and gas-oil mixing zone. Also, formation and precipitation of asphaltene as solid phase is reported in some gas injection projects

30

R. Azin et al.

Bulk Gas

Injection Gas/Bulk Reservoir Fluid Interaction

Bulk Reservoir Fluid

Fig. 2.4 Formation of zones due to gas injection

[1–8]. The solid phase may be precipitated asphaltene, wax, or combination of asphaltene, wax and salt, which separate from reservoir oil upon contact and mixing with injection gas and change in thermodynamic conditions of original fluid.

2.3 Pressure-Volume-Temperature (PVT) Experiments Based on the type and purpose of gas injection, different PVT tests may be required for a complete analysis of fluid phase behavior. In gas injection for pressure maintenance, the interaction between injection gas and reservoir oil involves gas diffusion and dissolution (solubility test) into oil at the gas-oil contact, the former is known as transport phenomenon rather than thermodynamic property. Like pressure maintenance, gas dissolution and diffusion is the main measured PVT and transport properties in the case of CO2 sequestration. However, usually a more comprehensive PVT study is ordered before going through gas injection for EOR and UGS purposes to fully understand fluid phase behavior and characterize reservoir oil and its interaction with injecting gas. These tests include, but not limited to, CVD, BPP (or DPP), Flash, Swelling, DL, CCE, miscibility tests, IFT reduction. For under-saturated reservoirs, oil formation volume factor and solution gas ratio are also measured. A detailed description and procedure of important PVT tests are presented here.

2.3.1 Constant Composition Expansion Constant composition expansion is accomplished on the crude oil or gas condensate samples to obtain the P-V relations of these systems. No hydrocarbon is removed from the cell during this test, and the composition of the mixture in the cell remains constant. This test is performed to reach the following experimental data: a.

Saturation pressure (bubble-point or dew-point pressure).

2 PVT of Gas Injection

b. c. d.

31

Isotherm compressibility coefficient of the single-phase fluid above the saturation pressure. Gas phase compressibility factors. Total hydrocarbon volume as a function of pressure.

The experimental procedure of this test is shown in Fig. 2.5 and describes as follow: Step 1. A hydrocarbon fluid sample (oil or gas) is placed in a visual PVT cell at reservoir temperature and pressure bigger than the initial reservoir pressure. Step 2. The pressure is decreased at constant temperature by discharging mercury from the cell. Step 3. The hydrocarbon volume is recorded. Step 4. Steps 2 and 3 are repeated until the pressure of the cell is reached to the intended pressure. The volume in saturation pressure is chosen as the reference volume. The relative volume in this test expressed as the ratio of the volume of hydrocarbon system to the volume at saturation pressure: Vr el =

Fig. 2.5 Schematic of CCE experiments steps

V VSat

(2.1)

32

R. Azin et al.

2.3.2 Differential Liberation In differential liberation, the liberated gas from an oil sample during the pressure reduction is removed from the cell before establishing the equilibrium with the liquid phase. Composition of total hydrocarbon system is varying in this process. This test is a representative of the separation process in the reservoir. Also, it can simulate the flowing behavior of hydrocarbon systems above the critical gas saturation. The experimental data attained from this test include: a. b. c. d.

Solution gas-oil ratio as a function of pressure. Oil shrinkage as a function of pressure. Remaining oil density as a function of pressure. Evolved gas properties such as, composition, compressibility factor and the specific gravity.

The procedure of this test is summarized in the following steps and is shown in Fig. 2.6: Step 1. The reservoir fluid sample is placed in a PVT cell at the reservoir temperature. Step 2. The cell is pressurized to saturation pressure by injection of mercury and the volume of liquid is recorded. Step 3. The cell is depressurized by removing the mercury from the cell. Step 4. The liberated gas is discharged from the cell and the pressure is maintained constant by reinjecting mercury. Step 5. The volume of removed gas and remaining oil are measured. Step 6. Step 4 and 5 are repeated in different pressures until the atmospheric pressure. Step 7. The final volume of remaining oil is referred as the residual oil.

Fig. 2.6 Schematic of DL experiments steps

2 PVT of Gas Injection

33

2.3.3 Constant Volume Depletion Constant volume depletion generally is conducted on gas condensate fluid samples and it is necessary for a prediction of pressure depletion performance. Determining reserves, evaluating field separation methods, planning for increasing liquid recovery and designing future operations need reliable and accurate pressure depletion performance. These predictions can be done with the help of experimental data collected from the CVD test. These important laboratory measurements which are provided by the CVD test include: a. b. c. d. e.

Dew-point pressure Variation of composition in gas phase Compressibility factor in reservoir conditions Original in-place hydrocarbons recovery at any pressure Accumulation of retrograde condensate

CVD test consists some steps which are shown in Fig. 2.7 and summarized as follow: Step 1. A specific amount of representative reservoir fluid is charged to the visual PVT cell. The overall composition (zi ) of this fluid must be known. PVT cell is maintained at reservoir conditions (reservoir pressure and temperature). Reference volume is defined as the initial volume (Vi ) of the saturated fluid. Step 2. The initial gas compressibility factor and Gas Initially In place are calculated by:

Fig. 2.7 Schematic of CVD experiments steps

34

R. Azin et al.

zd =

pd Vi n i RT

G I I P = 379.4n i

(2.2) (2.3)

where pd = dew-point pressure, psia. Vi = initial gas volume, ft3 . ni = initial number of moles of the gas = m/MWa . m = mass of initial gas in-place, lb. R = gas constant, 10.731. zd = compressibility factor at dew-point pressure. GIIP = Gas Initially In place. Step 3. The cell pressure is decreased to a specified pressure (P). The pressure variations are achieved by add or remove the mercury from the cell which illustrated in Fig. 2.7b. Decreasing the pressure causes the formation of a liquid phase. When the equilibrium is established between the fluids in the cell, the total volume (Vt ) and retrograde liquid volume (Vl ) are visually measured. Retrograde liquid saturation is defined as the ratio of the volume of retrograde liquid to the initial volume and mathematically shows as: Sl =

Vl × 100 Vi

(2.4)

Step 4. The gas is removed while an amount of mercury reinjected to the cell to maintain the cell pressure (at the constant pressure P) and to reach the initial volume. This process is shown in Fig. 2.7c. The volume of the removed gas is measured at the cell conditions. Step 5. The properties and composition of the removed gas are determined. Also, the volume of removed gas is measured in standard conditions. These steps are repeated until the minimum test pressure is attained after which the composition and quantity of the liquid and gas are measured.

2.3.4 Flash Separation Flash separation is also known as the separator test which is conducted to evaluate the volumetric behavior as the reservoir fluid passes through separators and then into the stock tank. The results of this test are used to design the optimum surface facilities and conditions. The combination of this test and differential liberation test data can obtain the PVT parameters (Bo , Rs and Bt ) which are necessary for engineering

2 PVT of Gas Injection

35

calculations. The details of this test are not reported as this test is not involving in the gas injection process.

2.3.5 Swelling Test The swelling test is related to the reservoirs under gas injection or dry gas cycling scheme. This test determines the dissolution of the injected gas in the reservoir sample. It can be concluded from this test, each addition of injection gas causes the relative volume and saturation pressure to increase. The following experimental data are obtained from this test: a. b.

The relation between saturation pressure and the amount of injected gas. The variation in fluid volume due to contact with the injected gas.

Swelling test experimental procedure is summarized in the following steps and schematically is shown in Fig. 2.8: Step 1. A visual PVT cell is filled with a reservoir fluid sample with a known overall composition (zi ). The cell is maintained at reservoir temperature and fluid saturation pressure. The volume of the fluid in this condition is recorded as (Vsat )origin . Step 2. A predesignate volume of proposed injection gas with known composition is added to the cell. The cell pressure is increased until only one phase is present. This pressure is defined as the new saturation pressure (ps ). Also, the new fluid volume (Vsat ) is recorded in this condition. The relative volume in this test is

Fig. 2.8 Schematic of DL experiments steps

36

R. Azin et al.

defined as the ratio of the volume of fluid in new saturation pressure (Vsat ) to the original fluid volume (Vsat )origin and mathematically shows: Vr el =

Vsat (Vsat )orig

(2.5)

Step 3. Step 2 is repeated until the mole percent of injected gas attains a preset value.

2.4 PVT Calculation in Gas Injection Processes 2.4.1 Equation of State Results of PVT tests are used to tune equations of state (EOS), which are powerful tools for mathematical modeling of phase behavior and thermodynamic studies for reservoir fluid and their mixtures with injection gases. The common types of EOS in reservoir engineering are cubic for compressibility factor or volume, which originate from the well-known Van der Waals cubic EOS developed in 1873. Development of this EOS was a turning point in modeling gas phase behavior and included the role of attractive and repulsive forces in the ideal gas equation. Many modifications were imposed on the parameters of Van der Waals-type EOS during the past century to suggest accurate mathematical model of thermodynamic data and phase behavior studies for a wide range of fluid systems. A general form of cubic EOS may be written as follow. f (P, V, T ) = 0

(2.6)

Parameters of this general form are then customized for any specific cubic EOS [9]. Table 2.4 shows a series of EOS known as cubic EOS. When tuned, the EOS is used to model and simulate phase envelope and PVT experiments. As mentioned, measured PVT data are used to optimize the EOS parameters in a systematic procedure known as EOS tuning. For a proper tuning Table 2.4 Series of EOS

Reference

Equation of state

Redlich_Kwong (1949)

p=

RT V −b

−

Soave_Redlich_Kwong (1972)

p=

RT V −b

−

a V (V +b)T 0.5 a(T ) V (V +b)

Peng_Robinson (1976)

p=

RT V −b

−

a(T ) V (V +b)+b(V −b)

Stryjek_Vera_Peng_Robinson (1986)

p=

RT V −b

−

a(θ,T ) V (V +b)+b(V −b)

Patel_Teja (1982)

p=

RT V −b

−

a(θ,T ) V (V +b)+c(V −b)

2 PVT of Gas Injection

37

approach, it is assumed that PVT experiments are performed at controlled conditions and fluid samples are consistent and their mixture represents original reservoir fluid. However, such conditions may not exist in some cases. Any failure in proper quality check (QC) of selected fluids or deviation from equilibrium conditions may result in erroneous PVT modeling [10, 11]. The first steps in PVT studies is taking fluid samples and recombining them to obtain original reservoir fluid. These steps are shown schematically in Fig. 2.9. Osfouri and Azin reviewed the challenges associated with sampling and recombination of hydrocarbon fluids [10]. According to their study, main sources of error during sampling include separator instability, lack of vapor-liquid equilibrium, volume ratio of separator outlets, and the presence of contaminants. Sometimes, analysis of separator outlet streams indicates the deviation from equilibrium for some wells which can have direct impact on saturation pressure and oil (condensate) to gas ratio prediction of well stream. Also, improper fluid handling, oil and gas leakage, and the presence of corrosive compounds can severely affect the quality of the recombined fluid. Overall, Thermodynamic equilibrium is a key factor for a successful sampling. Osfouri et al. proposed an algorithm for QC of PVT data [12]. Schematic of this algorithm is shown in Fig. 2.10. The first step of PVT QC is checking the validity of the samples of oil and gas for recombination. The second step is the verification thermodynamic equilibrium conditions of the samples. The Rs should be corrected in the case that oil carries over in the gas stream in the third step. In the next step, the material balance should be checked for the oil and gas phases. Finally, for gas condensate samples the validity of CVD tests is checked. In the case of negative composition for one or more components, the compositions that reported in CVD test should not be used tuning the EOS and regression [12].

Fig. 2.9 Schematic of fluid sampling and recombination [12]

38

R. Azin et al.

Fig. 2.10 Schematic of the algorithm for QC of PVT data [12]

Also, there are components in the fluid composition whose composition are not clear; these “pseudo components” are lumped form of other pure components and their physical and critical properties need to define before going through EOS tuning. This procedure is known as “heavy end” or “plus fraction” fluid characterization Sometimes, the users decide to lump selected components to reduce the total number of components and avoid large runtime in reservoir simulation. The EOS tuning is a nonlinear, multivariable optimization problem which seeks for optimized EOS parameters to fit best with measured PVT data and predict accurate thermodynamic properties. The EOS tuning has been a topic of interest for chemical and petroleum engineers since 80s and nearly all field development studies considered comprehensive studies. Osfouri et al. proposed an integrated strategy for fluid characterization and EOS tuning [11]. Schematic of this integrated approach is shown in Fig. 2.11 According to this approach, either dynamic (D) or static (S) strategy is described by a “path” which specifies the class of fluid or SCN of plus fraction, splitting and lumping (if any), type of empirical correlation, and grouping step (in dynamic strategy). For class B, Splitting C12+ to C12 and C13+ was done by either exponential (EX) or gamma (GA) distribution function. For example, the path named as S-B-GA-L refers to a case in static strategy in which grouping class B with gamma distribution function and Lee-Kesler [13] correlation is studied. The abbreviation “NO” in classes A and

2 PVT of Gas Injection

39

Fig. 2.11 Schematic of an integrated approach for fluid characterization and EOS tuning [11]

C indicate that no splitting was made on them and both are run with composition up to C12+ . For dynamic strategy, for example, D7-EX-L-1 refers to a case in which C7+ is split by exponential distribution function and Lee-Kesler [13] correlation is used to characterize SCN and lumped groups. The last number (1 in this example) refers to the grouping step of Whitson’s procedure [6]. Therefore, proper EOS tuning is highly dependent on the quality of measured PVT data and fluid characterization. However, there are debates on the overall process of EOS tuning and fluid characterizations in gas injection. First, the PVT experiments are conducted at controlled and near-equilibrium conditions, while reservoir conditions are rarely approached equilibrium. Second, a loop between invalid PVT data and improper EOS tuning will result in improper fluid phase determination with inherent errors.

2.4.2 Gas-Liquid Miscibility Component transfer between phases is occurred through mass transfer due to the lack of equilibrium. According to Danesh [9], miscibility and thermodynamic equilibrium at the gas-liquid interface is just an assumption and may not be observed in real-world field application. The equilibrium between phases can be violated by accelerated interphase mass transfer. This transport phenomenon continues until the composition of gas and liquid come into equilibrium and chemical potential of components become equal at the gas-liquid front. If the properties of two phases

40

R. Azin et al.

are similar in terms of composition and intermolecular forces and interfacial tension approaches zero, the two phases become miscible. Miscibility is a thermodynamic state which is affected by composition of injection gas and reservoir oil, temperature, and pressure. At a given temperature, miscibility increases with pressure for gas-hydrocarbon systems.

2.4.3 Different States of Miscibility 2.4.3.1

First-Contact Miscibility (FCM)

Miscibility is a dynamic process and may occur at the first contact between phases, known as first-contact miscibility (FCM). In operation of this process, a relatively small primary slug is injected which is miscible with reservoir fluid. Then a larger and less expensive slug is followed the primary slug. The size of each slug depends on the economics of operation. Also, the primary and secondary slug should reach the miscibility conditions for an efficient FCM process [14]. Also, it is necessary to inject enough solvent to ensure that FCM is maintained to the end of the process. A simple dispersion model can determine the amount of required solvent to meet this criterion. The following equation can be used to determine the minimum slug size in the FCM process [15]: (Ss )min =

  4.652  K O/S × tbt + K S/C G × tbt 2L

(2.7)

where (Ss )min = minimum solvent slug size. L = reservoir length. KO/S = dispersion coefficient between oil and solvent. KS/CG = dispersion coefficient between solvent and chase gas. and tbt = breaking through time. On the other hand, methane flooding at reservoir conditions is an immiscible flood because methane and crude oil are partially soluble. But, crude oil can dissolve hydrocarbons with higher molecular weight such as propane and LPG at most reservoir conditions. In this case, LPG and propane become liquid when pressure and temperature are increased. If the methane is brought to contact with propane at specific conditions (above the critical point) propane changes from liquid to gas and become soluble in methane. In this case, propane and methane will mix in all proportions and formed a single phase gas [16].

2 PVT of Gas Injection

2.4.3.2

41

Multiple-Contact Miscibility (MCM)

For most systems, the composition of injection gas and reservoir oil is not miscible at the first contact and gradually approach miscible conditions through successive contacts, which is known as multiple-contact miscibility (MCM). Therefore, the MCM refers to gas-oil systems which are immiscible initially, but approach miscibility upon successive contacts which may be achieved by vaporization of light and intermediate components of reservoir oil into the gas phase, or condensation of intermediate components of injection gas into reservoir oil. In the case of lean gas injection, mass transfer of components between phases may lead to enrichment of the gas phase with intermediate heavy components vaporized from liquid hydrocarbon. In this case, the miscibility process is known as vaporizing drive miscibility or vaporizing gas drive (VGD). Alternatively, when rich gas is injected into oil reservoir, the concentration gradient may force transfer of intermediate components from the gas phase into oil, and intermediate components condense into the liquid phase. This mechanism is known as condensing drive miscibility or condensing gas drive (CGD). When both condensing and vaporizing mechanisms are active in a gas-oil contact, the process is known as condensing/vaporizing gas drive (CVGD). The effects of dispersion and dispersive mixing as well as the amount of required solvent for enriching the oil must be considered for calculation the slug size for a condensing solvent MCM displacement. Also, slug size must enriched oil by providing enough intermediate molecular weight components. Slug size calculation can be summarized in the following steps [15]: Step 1. Determine the necessary amount of enriched oil to dominate the dilution effect using the following equation: (S E O )min =

  4.652  K E O/O × tbt + K E O/S × tbt 2L

(2.8)

where (SEO )min = minimum enriched oil slug size. L = reservoir length. KEO/O = dispersion coefficient between enriched-oil and oil. KEO/S = dispersion coefficient between enriched-oil and solvent. and tbt = breaking through time. Step 2. Calculate the amount of required solvent to create the volume of enriched oil which calculated in step 1 using the equation of state in a phase behavior software modelling and verified in the laboratory. Step 3. Determine the volume of the required solvent to dominate the dilution process using the following equation (Ss )min =

  4.652  K S/E O × tbt + K S/C G × tbt 2L

(2.9)

42

R. Azin et al.

where (Ss )min = minimum solvent slug size. L = reservoir length. KS/EO = dispersion coefficient between solvent and enriched-oil. KS/CG = dispersion coefficient between solvent and chase gas. and tbt = breaking through time. Step 4. Add the volumes determined in the step 2 and 3.

2.4.4 Minimum Miscibility Pressure (MMP) The lowest pressure at which miscible conditions prevail is known as minimum miscibility pressure (MMP). For pressures above MMP, the system is certainly miscible. When the injection pressure high enough that the injected gas can become miscible with reservoir oil at the first contact. This pressure is defined as the first contact miscibility pressure (Pmax ) [17]. Temperature, oil composition, gas composition and initial gas-oil ratio are the four important parameters that affect Pmax and MMP. These pressures are increased almost linearly with increasing the temperature that concludes reservoir temperature has a strong effect on the crude oil-CO2 miscibility development. The effect of oil composition (dead and live oil) is weak in the injection of pure CO2 . The effect of gas composition was studied by adding CH4 to the pure CO2 which causes to increase the Pmax and MMP. Also, the injection of produced gas can reach miscibility with reservoir oil at the actual reservoir conditions. The initial GOR has a strong effect on the Pmax and MMP which depend on the amount of CH4 component of the live oil-impure CO2 system [18].

2.4.5 MMP Measurements Experimental techniques used for measuring miscibility conditions between phases include slim tube test and rising bubble apparatus (RBA) [9]. A drawback of the slim tube experiment is that physical dispersion is likely to occur and cause misleading results [19]. Visual PVT Cell is also used for detecting MMP and miscibility conditions at successive pressures. Johnson and Pollin described the process of MMP determination using PVT cell [20]. They pointed out that MMP is associated with critical pressure of gas/oil mixtures when pure hydrocarbon is used as oil. They injected CO2 into pure alkanes and figured out that measured MMP was close to critical pressure of the mixture. However, such a conclusion may not be valid in determining MMP for a complex reservoir fluid. Apart from the experiment, MMP may be determined through compositional simulation, mixing cell models, analytical methods and empirical correlations.

2 PVT of Gas Injection

43

2.4.6 MMP Correlations Many correlations have been proposed to predict MMP of different gases in crude oil, some of which will be reviewed here. The correlations are developed for nonhydrocarbon and hydrocarbon gases. In general, developed correlations are a function of reservoir temperature, the composition of reservoir oil and injection gas, properties of oil heavy end (plus fraction), and the ratio of light to intermediate components in reservoir oil. Some correlations consider proportions and properties of intermediate components (C2 –C6 ) [21] and C5+ properties [21, 22] The mole% of volatile components (C1 , N2 ) and volume ratio of volatile to intermediate components are also considered in some correlations [22]. A critical review, evaluation and comparison of MMP correlations is addressed by Shokir [23, 24], Emera and Sarma [25], and Yuan et al. [19].

2.4.6.1

Non-hydrocarbon Gas Correlations

Alston et al. Correlation Alston et al. developed MMP correlations for CO2 -live oil (LO) and CO2 -stock tank oil (STO) [21]. CO2 -LO: pC O2 −L O = 8.78 × 10−4 (TR )1.06 (MwC5+ )1.78 (xvol /xint )0.136

(2.10)

Which, TR : Reservoir Temperature, °F. MC5+ : Molecular weight of C5+ xint : mole fraction of intermediate (C2 –C4 ), CO2 and H2 S in oil. xvol : mole fraction of volatile (C1 , N2 ). CO2 -STO: pC O2 −ST O = 8.78 × 10−4 (TR )1.06 (MwC5+ )1.78

(2.11)

In this model, equation, N2 , CO2 are volatile components and C2 –C4 are intermediate components. The MMP is determined in psia. The authors extended their LO correlation to predict MMP of impure streams of CO2 and other non-condensable gases using a correction factor: (M M PC O2 )imp−Lo = (M M PC O2 ) × Fimp

(2.12)

F imp is the correction factor for impure CO2 :  Fimp =



87.8 Tcm



1.935×87.8 Tcm



(2.13)

44

R. Azin et al.

Table 2.5 Range of applicability for Alston et al. [21]

Parameter

Ranges

Temperature, °F [K]

90–243 [305–390]

Volatile/intermediate ratio

0.14–13.61

C5+ molecular weight

169.2–302.5

Weight average critical temperature, °F [K]

70.7–115.7 [294.7–319.7]

Experimental MMP, psia [MPa]

948–4930 [6.5–34]

In this equation, T cm is the pseudo critical temperature of gas mixture (°F) calculated by summing the component critical temperature, Tci (°R), multiplied by its weight factor, wi : Tcm =

n 

wi Tci − 459.7

(2.14)

i=1

Range of applicability for Alston et al. correlation [21] is given in Table 2.5.

Johnson and Pollin Correlation Johnson and Pollin proposed an accurate and reliable empirical correlation for predicting the minimum miscibility pressure. This correlation can be used for a wide range of live oil and stock tank oil and pure or diluted CO2 as injection gas [20]. PM M P − Pci = αi (T − Tci ) + I (β M − Mi )2

(2.15)

where PMMP and Pci are the miscibility pressure and the injection gas critical pressure. T, Tc , M and Mi are the reservoir temperature, injection gas critical temperature, the average molecular weight of the oil and molecular weight of injection gas, respectively. The other parameters of this equation are defined as follow: I = C11 + C21 M + C31 M 2 + C41 M 3 + (C12 + C22 M)A P I + C13 A P I 2

(2.16)

Constants of equations are listed in Table 2.6. αi depends on the type of injecting gas:   psia For pure CO2 : αi = 18.9 K

(2.17)

2 PVT of Gas Injection

45

Table 2.6 Constants of Eqs. 2.15 and 2.16

C11 = − 11.7300

C21 = 0.06313

C31 = −1.954

C41 = 2.052 × 10−7

C12 = 0.136

C13 = −7.222 × 10−5

C22 = −1.138 × 10−5

β = 0.285

  1000y2 For CO2 + N2 : αi = 10.51 1.8 + (T − Tci )   100y2 For CO2 + CH4 : αi = 10.5 1.8 + (T − Tci )

(2.18) (2.19)

In this correlation, Pci , Tci , and Mi are critical pressure, critical temperature and molecular weight of injection gas. M and API refer to molecular weight and API gravity of reservoir oil.

Shokir Correlation Shokir used a graphical user interface program called GRACE to develop his model. The proposed MMP correlations depend on reservoir temperature, vol% of light and intermediate hydrocarbons, the molecular weight of C5+ , and mole % of C1 , C2 –C4 , N2 , and H2 S in the injected gas [23, 24]. Details of Shokir correlation is as follows: M M P = −0.068616 × Z 3 + 0.31733 × Z 2 + 4.9804 × Z + 13.432 Z=

8 

Zn

(2.20)

(2.21)

i=1

Z n = A3n xn3 + A2n xn2 + A1n x2 + A0

(2.22)

This correlation applies for pure and impure CO2 streams. The impurities include CH4 , N2 , H2 S, and C2 –C6 . Constants of Shokir correlation are given in Table 2.7.

Kamari et al. Correlation Kamari et al. correlation is developed based on Gene Expression Programming (GEP) approach for CO2 -oil MMP. This correlation is a function of Tc , TR, vol/int (the ratio of volatile to intermediate components)and MwC5+ in live oil system [26]. The details of this correlation given as follow:

46

R. Azin et al.

Table 2.7 Constants of Shokir correlation [23] n

x

A3

A2

A1

A0

1

TR

2.3660E−06

−5.5996E−04

7.5340E−02

−2.9182E+00

2

%vol

−1.3721E−05

1.3644E−03

−7.9169E−03

−3.1227E−01

3

%interm

3.5551E−05

−2.7853E−03

4.2165E−02

−4.9485E−02

4

MwC5+

−3.1604E−06

1.9860E−03

−3.9750E−01

2.5430E+01

5

%C1

1.0753E−04

−2.4733E−03

7.0948E−02

−2.9651E−01

6

%C2 −%C4

6.9446E−06

−7.9188E−05

−4.4917E−02

7.8383E−02

7

%N2

0

3.7206E−03

1.9875E−01

−2.5014E−02

8

%H2 S

3.9068E−06

−2.7719E−04

−8.9009E−03

1.2344E−01

M M P = 1.138A1 A2 − +

A4 1.1384 A1 A2 A3 2.6354

A1 A2 A3 (1.138A1 A2 −

A4 1.138A1 A2 A3

+ 0.0613)

+ 0.0613

(2.23)

where A1 , A2 , A3 , A4 and A5 are defined as follow:

A3 =

Tc2



A1 = A5 + A25 − 0.3016

(2.24)

A2 = A25 − 0.3016

(2.25)

A4 2.8381  (2MwC5+ +   )2 V ol. V ol. I nt. I nt.

− 2.6558 2MwC5+ 2.8381   +  Tc T c V ol. I nt. ⎛ ⎞2 2MwC5+ 2.8381  ⎠ +⎝ +  Tc T c V ol. I nt.

(2.26)

A4 =

MwC5+ TR A4 + − TC TC TC A3 ⎛ ⎞2 2MwC5+ A4 2.8381 ⎝  ⎠ −  +   Tc TC3 V ol. I nt. T c V ol. I nt.

A5 =

(2.27)

2 PVT of Gas Injection

+

47

2.6558 TC

(2.28)

Kaydani et al. This correlation is developed using Multi-gene genetic programming (MGGP) [27]. Details of this correlation is as follow: Kaydani et al. are reported that the correlation (2.29) is the best MGGP model with the minimum errors for pure CO2 injection. This correlation is a function of reservoir temperature, molecular weight of C5+ , and mole fraction of intermediate and volatile components. M M P = 0.2683 × TR + 0.2683 × MwC5+ + 0.2683 × TR (TR − X int )2 + 0.3339 × TR (TR + X vol − X int ) + 0.1161

(2.29)

Kaydani et al. also proposed a new correlation for Fimp as follows √ 4 Fimp =

X Cont × X C O2 + 0.04 × X C O2 + 0.003 TR

(2.30)

where Xcont , XCO2 , and TR are the concentration contaminant components (N2 , C1 , H2 S, and C2 –C4 ), CO2 concentration and reservoir temperature. The value of concentration must be entered based on 1 mol and normalized temperature.

Emera and Sarma Correlation Emera and Sarma developed a correlation for estimating the MMP when CO2 is mixed with other gases. This correlation is developed using genetic algorithm (GA) [25], as follows:   pr,impureC O2 1.8TC W + 32 = 3.406 + 5.786 × pr, pur eC O2 1.8TC,C O2 + 32     1.8TC W + 32 2 1.8TC W + 32 3 − 23.0 × + 20.48 × 1.8TC,C O2 + 32 1.8TC,C O2 + 32 4  1.8TC W + 32 − 5.7 × (2.31) 1.8TC,C O2 + 32 where pr,impureC O2 =

M M PimpureC O2 pC W

(2.32)

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R. Azin et al.

Table 2.8 Constants of Emera and Sarma correlation [25] Components

SO2

H2 S

CO2

C2

C1

N2

Other components

MFi

0.3

0.59

1

1.1

1.6

1.9

1

Table 2.9 Constants of Yuan correlation for Pure CO2 [19]

MC7+ (g/mol)

139–319

a5

1.1667E−01

PC2−6 (%)

2–40.3

a6

8.1661E+03

a1

−1.4634E+03

a7

−1.2258E−01

a2

0.6612E+01

a8

1.2283E−03

a3

−4.4979E+01

a9

−4.0152E−06

a4

0.2139E+01

a10

−9.2577E−04

pC W =

n 

wi pCi

(2.33)

i=1

pr, pur eC O2 = TC W =

M M Ppur eC O2 pC,C O2

n 

M Fi wi Tci

(2.34)

(2.35)

i=1

Values of MFi for various components are reported in Table 2.8. In the case of MMP < oil bubble point pressure (pb ), pb is chosen as the MMP.

Yuan et al. Correlation Yuan et al. reviewed and evaluated common CO2 -MMP correlations, and proposed their correlation for pure and impure CO2 streams [19]. For pure CO2 : M M Ppur e = a1 + a2 MwC7+ + a3 PC2−6 PC2−6 + (a4 + a5 MC7+ + a6 )T MwC7+ 2 + (a7 + a8 MwC7+ + a9 MwC + a10 PC2−6 )T 2 7+

(2.36)

where T, MwC7+ and PC2−6 are the reservoir temperature, C7+ molecular weight and total molar percentage of C2 –C6 , respectively. The range of MC7+ , PC2−6 and the best fit coefficient for Eq. (2.36) are reported in Table 2.9. Impure CO2 :

2 PVT of Gas Injection

49

Table 2.10 Constants of Yuan correlation for impure CO2 [19]

a1

−6.5996E−02

a6

−2.7344E−02

a2

−1.5246E−04

a7

−2.6953E−06

a3

1.3807E−03

a8

1.7279E−08

a4

6.2484E−04

a9

−3.1436E−11

a5

−6.7725E−07

a10

−1.9566E−04

M M Pimp = 1 + m(PC O2 − 100) M M Ppur e

(2.37)

where m = a1 + a2 MwC7+ + a3 PC2−6 PC2−6 + (a4 + a5 MwC7+ + a6 )T MwC7+ 2 + (a7 + a8 MwC7+ + a9 MwC + a10 PC2−6 )T 2 7+

(2.38)

The coefficients of Eq. (2.38) are reported in Table 2.10. Equation (2.38) is strictly reliable only for methane contents in the gas up to 40%.

Glaso Correlation Glaso’s correlation is suggested for the MMP calculation in miscible injection with N2 . This correlation is a function of reservoir temperature, molecular weight of C7+ and the mole percent of methane and intermediate components in reservoir fluid [28]. Equation (2.39) is the proposed MMP correlation for the oil with the API lower than 40. M M P = 80.14 + 35.35H + 0.76H 2 H=

A P I < 40

0.88 T 0.11 MwC 7+

[(C2−6 )0.64 (C1 )0.33 ]

(2.39)

(2.40)

The proposed MMP correlation for the oil with the API higher than 40 is as follows: M M P = −648.5 + 2619.5H − 1347.6H 2 A P I > 40 H=

0.48 MwC 7+

[T 0.25 (C2−6 )0.12 (C1 )0.42 ]

(2.41)

(2.42)

50

2.4.6.2

R. Azin et al.

Hydrocarbon Gas Correlations

Glaso Correlation Glaso presented a generalized MMP correlation which derived from graphical correlations. This correlation can predict the MMP for MCM displacement by hydrocarbon, N2 or CO2 gas and it is a function of reservoir temperature, molecular weight of C7+ , mole percent of CH4 and the molecular weight of intermediates in the injection gas [29]. This correlation is shown as: M M P = 810 − 3.404MwC7+ −1.058 3.730 + (1.7 × 10−9 MwC exp(786.8MwC ))T 7+ 7+

FR > 18%

(2.43)

M M P = 2947.9 − 3.404MwC7+ −1.058 3.730 + (1.7 × 10−9 MwC exp(786.8MwC ))T 7+ 7+

− 121.2FR FR < 18%

(2.44)

FR is the mole percent of C2 –C6 in reservoir fluid. T is reservoir temperature in °F, and MMP is expressed in psig.

Firoozabadi and Khalid Correlation Firoozabadi and Khalid correlation can predict the MMP for VGD process. The effective parameters in this correlation are the ratio of intermediates (excluding C6 ), reservoir temperature and molecular weight of C7+ [30]. Correlation () is the proposed correlation by Firoozabadi and Khalid. 

CC2 − CC5 MwC7+ T 2.5   CC2 − CC5 2 + 1430 × 103 MwC7+ T 2.5



M M P = 9433 − 188 × 103

(2.45)

Kuo Correlation Kuo presented a correlation based on MMP prediction of four reservoir fluids and several enriched-gas. This correlation can be used for prediction of methane concentration at a given operating pressure or MMP for a fixed slug composition [31]. The proposed correlation is as follow:

2 PVT of Gas Injection

51

Table 2.11 Constants of Kuo correlation

A

B

C

0.19899861

0.00055769

0.58347828

D

E

F

0.62406453

0.57821035

0.00058948

log(C1 ) = (A + B × T ) × log(T ) + C × log(P) + D × log(MwC5+ ) + (E + F × MwC2+ ) × log(MwC2+ )

(2.46)

where, C1 : maximum allowable methane concentration in the injection gas, mol% T: Temperature, °F. P: pressure, psia. MWC5+ : molecular weight of C5+ fraction in the reservoir fluid. MWC2+ : molecular weight of intermediate (C2 –C4 ) fractions in the displacing gas. Table 2.11 represent the constant for Kuo’s correlation.

Pedrood Correlation Pedrood simulated the rich gas injection and displacement process which is investigated in one dimension and compositional model [32]. PM = 49.15 − 0.6863θ + 2.482 × 10−4 θ 2 − 0.2054 2

(2.47)

where  and θ present the properties of reservoir oil and injected gas respectively and mathematically show as: =

[MwC5+

106 yC2−4 + (1.8T − 460)]

θ = 100(yC4 + 0.8yC3 + 0.5yC2 +C O2 )

2.4.6.3

(2.48) (2.49)

The Impact of Optimization on the Accuracy of Correlations

As seen, the MMP correlations contain parameters which can be optimized for specific systems. For instance, Zirrahi et al. used a large set of published data to optimize parameters of Alston et al. [21] correlation to fit the measured MMP data

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R. Azin et al.

for acid gas and flue gas streams [33]. They showed that a classic correlation with optimized parameters works better than existing correlations.

2.4.7 Ternary Diagram It is a common practice to study the miscibility concept using the ternary diagram. Basically, ternary diagram is constructed for systems containing three components, each laid on a corner of an equilateral triangle. In the case of a multi-component hydrocarbon system, the whole mixture is divided into three subgroups or “pseudo” components, usually named as light (L), intermediate (I), and heavy (H), as shown in Fig. 2.12. The composition of reservoir oil and injection gas can be shown as a “point” either on a side or inside the triangle. In Fig. 2.12 the example fluid includes 40% component A, 40% component B, and 20% component C. The overall composition can be easily divided into three parts based on defined L, I, and H components. These components may be either pure or a combination of two or more pure components. As an example, light components like CH4 , N2 and their mixture are normally regarded as light; CO2 , H2 S, C2 –C6 as intermediate, and C7+ as the heavy group. A ternary diagram may have two or more two-phase regions which represent the degree of miscibility between these pseudo components. It should be mentioned that any ternary diagram is constructed for a given temperature and pressure. Changing either temperature or pressure causes the two-phase region to expand or shrink and make the injection gas and reservoir oil approach or deviate from miscibility, as well as a change in the slope of tie lines that determine equilibrium compositions. Fig. 2.12 A typical ternary diagram

2 PVT of Gas Injection

53

Ternary diagram may be developed either by experiment or using EOS and PVT modeling for a given fluid system. The question which arises in the PVT study of gas injection is to what extent ternary diagram can represent the real behavior of reservoir fluid and injection gas. As stated by Danesh [9], ternary diagrams and the procedure of determining miscibility conditions between injection gas and reservoir oil using ternary diagram should not be used for a real field operation and gas injection design. In other words, the composition of reservoir fluid is so complex that reducing it into simple subgroups of light, intermediate and heavy pseudo components may overlook its real behavior when contacted injection gas. In field practice, ternary diagrams are used in the preliminary design of gas composition enrichment for miscible injection, which will then be overdesigned to guarantee the miscible conditions throughout the gas flow in well and reservoir.

2.5 The Impact of Gas Injection on Phase Behavior of Hydrocarbons A case study of gas injection was done in this section. Gas cap was assumed as the injection place. The changes in phase behavior of gas cap were studied first. Then, the effect of new composition of gas cap and reservoir oil is studied. 1, 10, 30 years of gas injection were considered as the injection time. CO2 , CH4 , N2 were selected as the gas injection candidates. Fluid data of an Iranian oil reservoir are used in this case study. Original gas cap composition was selected from the DL test of the studied reservoir fluid. The original oil composition and reservoir pressure were assumed constant during the gas injection process. Table 2.12 represents the gas cap and oil composition, the initial volume of gas cap and gas injection rate into the gas cap in all cases. The procedure of calculation and modelling the phase behavior is as follows: Step 1. Plot the P-T diagram for original oil and gas cap composition. Step 2. Calculate the composition change in gas cap due to the gas injection using the volume balance. yi−new =

(VOrig × yi−Orig ) + (VI n j × yi−in j ) VOrig + VI n j

(2.50)

where yi = molar fraction of ith Component. V = Volume (m3 ). Step 3. Plot the P-T diagram for new gas cap composition. Step 4. Specify the primary (oil) and secondary (gas) composition for evaluating the phase behavior of the oil and gas mixture. Step 5. Calculate the mole fraction of oil and gas in gas cap.

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R. Azin et al.

Table 2.12 Composition of gas in the gas cap and reservoir oil, initial volume of gas cap and injection rate

Components

Gas cap composition

Oil composition

H2 S

0.513

0.63

N2

3.2946

1.85

CO2

2.21

1.87

CH4

81.69

49.07

C2 H6

5.7649

5.82

C3 H8

2.68

4.1

IC4

0.62

1.21

NC4

1.2075

2.68

IC5

0.42

0.74

NC5

0.49

0.65

FC6

0.46

4.07

FC7

0.34

3.62

FC8

0.29

2.05

C09

7.05E−03

2.78

C10

4.56E−03

1.85

C11

2.96E−03

1.53

C12+

5.43E−03

15.48

Initial gas cap volume

(m3 )

3,936,041,676

Injection rate (m3 /day) 1,000,000

Step 6. Repeat steps 2–5 for the specified injection times. The results of phase behavior modelling are presented in the following sections.

2.5.1 N2 Injection Figure 2.13 shows the effects of Nitrogen injection phase behavior of gas in gas cap and reservoir oil. As indicated in this figure, the saturation pressure of gas in gas cap at reservoir temperature increases due to increase the amount of Nitrogen. Also, Nitrogen injection decreases the criconder-therm and increases the criconder-bar of the gas in gas cap. It can be understood that Nitrogen has significant effect on the fluid saturation pressure.

2 PVT of Gas Injection

55

7000

Original Oil Critical Point- Original Original Gas Cap Mixture Composition- 1 year Critical Point-Mixture 1 year Gas Cap Composition- 1 year Mixture Composition- 10 years Critical Point-Mixture 10 years Gas Cap Composition- 10 years Mixture Composition- 30 years Critical Point-Mixture 30 years Gas Cap Composition- 30 years

6000

Pressure (psi)

5000 4000 3000 2000 1000 0

-100

100

300

500

700

900

1100

1300

Temperature (F)

Fig. 2.13 Phase behavior of gas cap and reservoir oil for different times of Nitrogen injection

2.5.2 CO2 Injection Figure 2.14 describe the changes of fluid phase behavior due to CO2 injection. In opposite of Nitrogen injection, CO2 injection decreases the saturation pressure, criconder-bar and criconder-therm of the gas in gas cap. It cannot be seen significant changes in the reservoir fluid behavior by increasing the time of CO2 injection. 7000

Original Oil Critical Point- Original Original Gas Cap Mixture Composition- 1 year Critical Point-Mixture 1 year Gas Cap Composition- 1 year Mixture Composition- 10 years Critical Point-Mixture 10 years Gas Cap Composition- 10 years Mixture Composition- 30 years Critical Point-Mixture 30 years Gas Cap Composition- 30 years

6000

Pressure (psi)

5000 4000 3000 2000 1000 0

-100

100

300

500

700 Temperature (F)

900

1100

Fig. 2.14 Phase behavior of gas cap and reservoir oil for different times of CO2 injection

1300

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R. Azin et al.

2.5.3 CH4 Injection The phase behavior of gas in gas cap and oil in methane injection are shown in Fig. 2.15. This figure indicates that the methane has not a significant effect in gas cap phase behavior in comparison of N2 and CO2 injection. But, the saturation pressure of reservoir fluid increases more than CO2 injection and less than N2 injection.

2.6 Case Study of the Minimum Miscibility Pressure for Different Injected Gases A case study was conducted to determine the minimum miscibility pressure (MMP) of the FCM and MCM processes for different injected gases. The oil sample was introduced in Table 2.12. First, the minimum miscibility pressure was calculated for the pure gas injection. Then, the effects of combining various injected gases were investigated. In this case study, cell to cell method is applied for calculation of MMP. This method determines the MMP by forming a ternary diagram. Figure 2.16 presents ternary diagrams for the mentioned oil sample and a mixture of CO2 and CH4 as solvent. N2 and CH4 are categorized as pseudo component 3, C12+ as the pseudo component 1 and other components as pseudo component 2. This diagram shows the phase boundaries of combined fluid under various pressures at a constant temperature. As it is obvious, the two-phase region, defined by the area in-between the vapor and liquid lines, begins to close and decrease in size as pressure increases. The miscibility condition is closed to the FCM by decreasing the size of the two 7000

Original Oil Critical Point- Original Original Gas Cap Mixture Composition- 1 year Critical Point-Mixture 1 year Gas Cap Composition- 1 year Mixture Composition- 10 years Critical Point-Mixture 10 years Gas Cap Composition- 10 years Mixture Composition- 30 years Critical Point-Mixture 30 years Gas Cap Composition- 30 years

6000

Pressure (psi)

5000 4000 3000 2000 1000 0 -100

100

300

500

700

900

1100

1300

Temperature (F)

Fig. 2.15 Phase behavior of gas cap and reservoir oil for different times of Methane injection

2 PVT of Gas Injection

57

Fig. 2.16 A sample of ternary diagram for oil sample and a mixture of CO2 and CH4 as solvent in different pressures

phase region. Also, the non-miscible condition is changed to the MCM as pressure increases. Figure 2.17 shows the MMP for pure injected gas in FCM and MCM mechanisms. This figure demonstrates that the Nitrogen injection has the highest MMP in both FCM and MCM. Also, the use of Methane as the injected gas need very high pressure to reach the miscibility condition with the studied fluid. CO2 and Ethane 30000

Minimum Miscibility Pressure (psi)

FCM

25000

MCM

20000

15000

10000

5000

0

CH 4

Fig. 2.17 MMP for pure injected gas

CO 2

Injected Gas

N2

C 2 H6

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R. Azin et al.

Minimum Miscibility Pressure (psi)

25000

20000

FCM MCM

15000

10000

5000

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CH4- Fraction

Fig. 2.18 MMP for CH4 -CO2 solvent

needs low pressure to meet the miscibility condition which can be reached in the reservoir pressure and temperature. Also, it was found that Methane and Nitrogen reach miscibility condition by the Vaporizing Gas Drive (VGD). In contrast, the Condensing Gas Drive (CGD) is the main mechanism in CO2 and Ethane injection.

2.6.1 CH4 /CO2 Solvent Figure 2.18 demonstrates the changes in MMP by increasing the fraction of methane in injection of CH4 /CO2 solvent. As shown, the MMP would increase by increasing the methane fraction in injected in fractions lower than 0.7 in both MCM and FCM processes. The MMP of the MCM reaches a constant value and MMP of FCM increase rapidly in the fraction upper than 0.7. Also, the CGD changes to VGD in the 0.7 fraction of methane in injected gas.

2.6.2 CO2 /N2 Solvent Figure 2.19 shows the MMP for FCM and MCM processes for CO2 and Nitrogen solvent injection as a function of CO2 fraction in the solvent. The MMP of MCM process in this case has a constant value until it reaches 0.5 fraction of CO2 in the solvent. Also, in this value, the mechanism changes from VGD to CGD which cause the MMP of MCM decreases from this value. The slope of decreasing the MMP of FCM is lower in the higher fractions of CO2 .

2 PVT of Gas Injection

59

Minimum Miscibility Pressure (psi)

30000 25000

FCM MCM

20000 15000 10000 5000 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CO2- Mole Fraction

Fig. 2.19 MMP for CO2 -N2 solvent

2.6.3 CH4 /N2 Solvent Figure 2.20 shows the MMP for Nitrogen and Methane solvent. In this case, the VGD is the mechanism for reaching the miscibility condition. The MMP for MCM process has a constant value due to the values of MMPs for pure Nitrogen and Methane which almost similar as it is shown in Fig. 2.17.

Minimum Miscibility Pressure (psi)

30000 25000

FCM MCM

20000 15000 10000 5000 0 0

0.1

0.2

0.3

0.4

0.5

N2- Mole Fraction

Fig. 2.20 MMP for N2 -CH4 solvent

0.6

0.7

0.8

0.9

1

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R. Azin et al.

Minimum Miscibility Pressure (psi)

25000

20000

FCM MCM

15000

10000

5000

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CH4- Fraction Fig. 2.21 MMP for C2 H6 -CH4 solvent

2.6.4 CH4 /C2 H6 Solvent Figure 2.21 show the MMP for Ethane and Methane solvent as a function of Methane fraction. As it is shown, the MMP increases by increasing the fraction of Methane in the solvent in both FCM and MCM mechanisms. Also, it was found that the fraction 0.2 is the threshold for changing the CGD to VGD. This change causes that the MMP of FCM increased rapidly but, the MMP for MCM process increased slowly in a higher amount of Methane in the solvent.

2.6.5 C2 H6 /CO2 Solvent Figure 2.22 describes the MMP of the Ethane and CO2 solvent as a function of CO2 fraction. Obviously, the MMP of FCM has constant value due to the close MMP of pure CO2 and Ethane. But, the MMP of MCM increases slightly by increasing the CO2 fraction in the solvent. Also, the mechanism of reaching miscibility in this case is CGD.

2.6.6 C2 H6 /N2 Solvent Figure 2.23 shows the MMP for Ethane and Nitrogen solvent in FCM and MCM process. The MMP of FCM process increases by increasing the Nitrogen fraction.

2 PVT of Gas Injection

Minimum Miscibility Pressure (psi)

5000

61

FCM

4500

MCM

4000 3500 3000 2500 2000 1500 1000 500 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

CO2- Fraction

Fig. 2.22 MMP for C2 H6 -CO2 solvent

Minimum Miscibility Pressure (psi)

30000 25000

FCM MCM

20000 15000 10000 5000 0 0

0.1

0.2

0.3

0.4

0.5

N2- Fraction

Fig. 2.23 MMP for C2 H6 -N2 solvent

Also, the MMP of MCM process increases until the fraction of Nitrogen reaches 0.5. But, the MMP becomes constant in the higher fractions of Nitrogen. It is found out that in the 0.2 fraction of Nitrogen CGD turns into the VGD. This fact shows that the Nitrogen has higher power to control the miscibility process.

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2.7 The Effect of Gas Injection on Asphaltene Precipitation In this section, a case study was conducted on asphaltene precipitation due to injection of different gases into an Iranian oil reservoir [34]. For this purpose, the composition of reservoir fluid is split and regrouped. Then, the new heavy component (C36+ ) is divided into precipitating and non-precipitating components. After that, the equation of state is tuned using the experimental data of the studied fluid. Tables 2.13 and 2.14 show the original composition and modified composition of reservoir fluid. Table 2.15 shows the SARA analysis of the studied fluid. Multi-phase flash can model the asphaltene precipitation and the solid model can predict the fugacities of components in solid phase. The solid phase can contain one or more components. The precipitated phase is assumed as an ideal mixture of solid components. Precipitating component fugacity is calculated by the following relationship: Table 2.13 Reservoir oil composition and heavy fraction properties

Components

Reservoir oil (Mol %)

H2 S

0.27

N2

0.71

CO2

1.55

C1

40.01

C2

8.12

C3

5.35

iC4

1.02

nC4

2.45

iC5

1.04

nC5

1.32

C6

4.94

C7

4.67

C8

2.92

C9

3.79

C10

2.95

C11 C12

2.75 +

16.15

Total

100.00

Molecular weight of residual oil +

285

Molecular weight of C12 fraction

548

Molecular weight of reservoir oil

128

Sp.Gr. of C12 + fraction @ 60 °F

0.9286

2 PVT of Gas Injection Table 2.14 Reservoir oil composition after splitting and regrouping

63 Components

Reservoir oil (Mol %)

N2

0.2699678

CO2

0.70991534

H2 S

1.5498152

CH4

40.005229

C2 H6

8.1190317

C3 H8

5.349362

IC4

1.0198784

NC4

2.4497078

IC5

1.039876

NC5

1.3198426

C6 –C11

Table 2.15 Results of SARA analysis

22.017374

C12 –C20

4.4833402

C21 –C29

3.2387371

C30 –C35

1.6426098

C36A+

6.3776687

C36B+

0.4076444

Saturates (wt. %)

Aromatics (wt%)

Resins (wt%)

Asphaltenes (wt%)

66.70

29.80

0.70

2.80

ln f s = ln f s∗ +

vs ( p − p ∗ ) RT

(2.51)

where fs * and fs are the fugacities of pure solid asphaltenes at pressure p* and p, respectively. vs is the molar volume of pure solid asphaltenes. R and T are the universal gas constant and temperature, respectively. The mole fraction of C36A+ and C36B+ is computed as follow: z C36B+ =

wC36B+ × MwOil MwC36B+

z C36A+ = z C36+ − z C36B+

(2.52) (2.53)

After these calculations, the asphaltene precipitation is predicted by defining a secondary composition. Five cases were selected as composition of injection gas including, N2 , CO2 , CH4 , and two mixtures of gases (one CH4 dominant and one CO2 dominant). In the following sections, the results of the asphaltene precipitation are described.

64

R. Azin et al. 0.5

Solvent Mole Fraction = 0.00 Solvent Mole Fraction = 0.05 Solvent Mole Fraction = 0.1 Solvent Mole Fraction = 0.15 Solvent Mole Fraction = 0.2

Asphaltene Precipitation (wt%)

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

2000

4000

6000

8000

10000

12000

14000

Pressure (psi)

Fig. 2.24 Curve of asphaltene precipitation due to Nitrogen injection

2.7.1 Pure N2 Figure 2.24 represents the impact of pure N2 injection on asphaltene precipitation which is investigated on 0, 0.05, 0.1, 0.15 and 0.2 mol fraction of N2 . It is shown that the maximum amount of asphaltene precipitation is increased slightly by increasing the mole fraction of injected N2 . Also, the pressure range of asphaltene precipitation is increased due to the presence of N2 . This figure indicates that by increasing the mole percent of N2 , the maximum amount of asphaltene precipitation can be reached in a higher pressure in N2 injection scheme. As the maximum amount of asphaltene precipitation occurs in the saturation pressure, it can be concluded that N2 can significantly increase the saturation pressure.

2.7.2 Pure CO2 The injection of pure CO2 increases the maximum asphaltene precipitation greatly, as shown in Fig. 2.25. The onset pressure of asphaltene precipitation is also increased due to increasing the mole percent of CO2 . In this scheme, the pressure range of asphaltene precipitation increases due to increasing the onset pressure. It can be concluded that the CO2 effect on the saturation pressure is lower than N2 injection.

2 PVT of Gas Injection

65

Asphaltene Precipitation (wt%)

0.8

Solvent Mole Fraction = 0.00 Solvent Mole Fraction = 0.05 Solvent Mole Fraction = 0.1 Solvent Mole Fraction = 0.15 Solvent Mole Fraction = 0.2

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

2000

4000

6000

8000

10000

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Pressure (psi) Fig. 2.25 Curve of asphaltene precipitation due to CO2 injection

2.7.3 Pure CH4 Figure 2.26 represents the effect of pure CH4 injection on the asphaltene precipitation. The impact of pure CH4 on the maximum amount of asphaltene precipitation is similar to N2 . This amount increases slightly in this scheme. Also, the onset pressure of asphaltene precipitation increases slightly in pure CH4 injection in comparison to

Asphaltene Precipitation (wt%)

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Solvent Mole Fraction = 0.00 Solvent Mole Fraction = 0.05 Solvent Mole Fraction = 0.1 Solvent Mole Fraction = 0.15 Solvent Mole Fraction = 0.2

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Pressure (psi) Fig. 2.26 Curve of asphaltene precipitation due to Methane injection

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Solvent Mole Fraction = 0.00 Solvent Mole Fraction = 0.05 Solvent Mole Fraction = 0.1 Solvent Mole Fraction = 0.15 Solvent Mole Fraction = 0.2

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0

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Pressure (psi) Fig. 2.27 Curve of asphaltene precipitation due to Methane dominant gas injection

N2 and CO2 . The impact of CH4 on the saturation pressure is higher than CO2 and lower than N2 .

2.7.4 CH4 Dominant In this scheme, the mole percent of CH4 , CO2 and H2 S are 0.9, 0.05 and 0.05, respectively. Figure 2.27 shows the effect of this scheme of gas injection on asphaltene precipitation. This effect is almost similar to CH4 injection due to high mole fraction of methane. But, the maximum of asphaltene precipitation and onset pressure are lower in this scheme in comparison to the pure methane injection.

2.7.5 CO2 Dominant The mole fraction of CO2 , CH4 and H2 S are 0.8, 0.1 and 0.1, respectively. As it is shown in Fig. 2.28, the maximum amount of asphaltene precipitation and the onset pressure are lower than pure CO2 injection scheme which is the impact of impurity in reducing the effect of pure CO2 on asphaltene precipitation.

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Asphaltene Precipitation (wt%)

0.8

Solvent Mole Fraction = 0.00 Solvent Mole Fraction = 0.05 Solvent Mole Fraction = 0.1 Solvent Mole Fraction = 0.15 Solvent Mole Fraction = 0.2

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

2000

4000

6000

8000

10000

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Pressure (psi) Fig. 2.28 Curve of asphaltene precipitation due to CO2 dominant gas injection

2.7.6 Comparison of Different Gases Figure 2.29 comprises the effect of 0.1 mol fraction of different injection gases on asphaltene precipitation in reservoir temperature (174 °F). As it obvious, CO2 injection increases the maximum amount of precipitated asphaltene in comparison of other injection gases. Also, the onset pressure of asphaltene precipitation increases in the case of CO2 injection. In CO2 dominant gas injection, the maximum asphaltene precipitation decreased due to the presence of CH4 and H2 S. The lowest onset

Asphaltene Precipitation (wt%)

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CH 4

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CH 4 Dominant

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Pressure (psi)

Fig. 2.29 Curve of asphaltene precipitation for 0.1 mol fraction of different injected gas

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pressure belongs to CH4 dominant scheme. Also, it can be concluded, that the best choice for gas injection is CH4 dominant for preventing a high amount of asphaltene precipitation and a significant increase in saturation pressure.

2.8 Optimum Design of Gas Injection Prediction of miscibility conditions includes MMP determination and optimum gas composition for miscible injection into a given oil reservoir. The latter is known as minimum miscibility enrichment (MME) which refers to gas composition. It is worth noting that MMP is sometimes known as minimum dynamic miscibility pressure (MDMP) [20]. This term is used to reflect the dynamic behavior of gas oil contact and gradual enrichment which eventually leads to miscibility conditions. It is well known that miscible gas injection works best and gives highest ultimate oil recovery factor (RF). Gases like CO2 and H2 S lower the MMP upon contact with reservoir oil, while light components like N2 and CH4 increase the mixture MMP [33]. Similar trend was also observed for oxygen, and it was found that this component increases MMP when present in CO2 stream, but not as much as when CO2 stream is contaminated with N2 [35]. Therefore, although co-injection of contaminants such as CH4 , H2 S, N2 , O2 and SO2 with CO2 stream as acid gas and flue gas provides an extra benefit to operators and let them handle greenhouse gas emission from reservoirs or plants in addition to miscible EOR and CO2 sequestration projects, each individual contaminant may increase or decrease MMP. The combined effect of components depends on composition of injection gas and needs be examined and modeled.

2.9 PVT Challenges Associated with Gas Injection From what we have presented here, PVT studies play a crucial role in the design and optimization of the gas injection process. However, there are challenges associated with PVT, part of which is reflected in fluid characterization, QC of the reservoir and injection fluids, as discussed above. In addition, there are challenges with constructing and using ternary diagram when there is a risk of asphaltene precipitation and deposition during gas injection, as it alters the whole composition and properties of reservoir oil when contacts the injected gas. The dynamic behavior of phase envelope may be simulated through PVT experiments. Results of these tests can be used to determine the method of secondary/tertiary recovery. For example, if differential liberation (DL) test reveals a shift of phase diagram to higher temperature, the fluid composition becomes heavier as a result of light hydrocarbon removal as vapor phase. Such a change in composition acts as a drive toward higher liquid condensation, and the reservoir is a good candidate for either pressure maintenance by lean gas injection to keep intermediate hydrocarbons or for gas cycling to vaporize and recover liquid hydrocarbons [36]. Orr and Taber

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describe more suggestions for secondary/tertiary recovery techniques based on phase behavior of oil reservoirs. Usually, PVT experiments are performed on bulk of fluid, whereas the reservoir fluid exists in porous media. As long as the pores are in the order of micrometers or larger, the measured PVT properties at bulk conditions are close to the porous medium. For smaller pores in the order of micrometers to nanometers, the pore size affects PVT properties. Such small sizes eventuate considerable curvature effect which results in significant capillary forces [37]. The effects of these forces are in opposite to gravity such that instead of a sharp change in phase, there will be a transition zone with two or three phases in equilibrium with each other [38]. The capillary effect is remarkable in formations with pore radius of micro and nanoorders and high interfacial tension. One of the important outcomes of this fluid confinement is variation in phase behavior. A detailed review and analysis of the effect of capillary pressure on phase behavior studies of gas condensate systems is given by Kiani et al. [39]. They found that presence of porous media has significant effects on phase behavior and VLE calculations. These effects could be due to shift in critical properties and also difference between phase pressures related to capillary forces. Their results indicate that interfacial curvature will suppress bubble point pressure; therefore, molar fraction of light components will decrease, while heavy components will increase in their composition. Because of low order of magnitude for C12+ composition, increase in composition is not significant. Also, gas phase pressure will decrease. It is investigated that porous media has a remarkable effect on GOC location, as pore radius decreases, bubble point pressure decreases too, resulted to change phases in lower depths. With an increase in pore sizes, decrease in compositional gradients occurs. Due to suppression in bubble point pressure and GOC appearance in upper depths, the thickness of the transition zone decreases and consequently, compositional gradients calculated in this zone decreases. Asphaltene deposition is another challenge associated with PVT of gas injection, which in turn affects reservoir petro-physical properties and alters wetting conditions. This subject is studied in experimental, numerical and field level. For example, Mousavi-Dehghani et al. studied asphaltene deposition in porous media during a miscible gas injection [40]. They observed a decrease in porosity of porous medium upon gas injection and interpreted this by asphaltene deposition as a result of changes in oil composition upon gas injection. Change in reservoir permeability due to asphaltene precipitation in core samples put under gas injection is reported by Minssieux et al. [41]. They suggested a mechanism for core damage by asphaltene deposition which describes rate of asphaltene separation and deposition on rock surface. The wells in Ghawar oil field, Saudi Arabia’s main oil field, have also faced with asphaltene problem upon gas injection [42]. In Iran, asphaltene deposition is reported in a number of gas injection projects, including Kupal [3], and Aghajari [43]. A detailed case study on the impact of gas injection on asphaltene precipitation was presented in Sect. 2.7. Another issue involves scale up of experimental data for miscibility measurements. For example, there are debates on the applicability of MMP measurements obtained from slim tube. This apparatus is made of an unconsolidated sand pack to

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mimic porous media in a capillary tube. However, it is too simple to represent a real reservoir and quite far from a real porous medium. Therefore, the oil recovery factor obtained from the slim tube has little or no correlation with that obtained in reservoir. In addition, the slim tube experiment is conducted at “local” rather than “overall” thermodynamic equilibrium between reservoir oil and injection gas. Therefore, slim tube can be considered as an “experimental simulation” of miscibility test.

References 1. Bennion DB, Thomas EB, Bennion DW, Bietz RE. Formation screening to minimize permeability impairment associated with acid gas or sour gas injection/disposal. In: Annual technical meeting. Petroleum Society of Canada; 1996. 2. Hustad OS, Jia N, Pedersen KS, Memon A, Leekumjorn S. High-pressure data and modeling results for phase behavior and asphaltene onsets of Gulf of Mexico oil mixed with nitrogen. SPE Reserv Eval Eng. 2014;17:384–95. 3. Kalantari-Dahaghi A, Gholami V, Moghadasi J, Abdi R. Formation damage through asphaltene precipitation resulting from CO2 gas injection in Iranian carbonate reservoirs. SPE Prod Oper. 2008;23:210–4. 4. Yonebayashi H, Al Mutairi AM, Al Habshi AM, Urasaki D. Dynamic asphaltene behavior for gas-injection risk analysis. SPE Reserv Eval Eng. 2011;14:493–504. 5. Negahban S, Bahamaish JNM, Joshi N, Nighswander J, Jamaluddin AKM. An experimental study at an Abu Dhabi reservoir of asphaltene precipitation caused by gas injection. SPE Prod Facil. 2005;20:115–25. 6. Srivastava RK, Huang SS, Dong M. Asphaltene deposition during CO2 flooding. SPE Prod Facil. 1999;14:235–45. 7. Yonebayashi H, Masuzawa T, Dabbouk C, Urasaki D. Ready for gas injection: asphaltene risk evaluation by mathematical modeling of asphaltene precipitation envelope (APE) with integration of all laboratory deliverables. In: SPE reservoir characterisation and simulation conference. European Association of Geoscientists & Engineers; 2009. p. cp-170. 8. Yonebayashi H, Takabayashi K, Iizuka R, Tosic S. Managing experimental-data shortfalls for fair screening at concept selection: case study to estimate how acid-gas injection affects asphaltene-precipitation behavior. Oil Gas Facil. 2016;5. 9. Danesh A. PVT and phase behaviour of petroleum reservoir fluids. Elsevier; 1998. 10. Osfouri S, Azin R. An overview of challenges and errors in sampling and recombination of gas condensate fluids. J Oil Gas Petrochemical Technol. 2016;3:1–14. 11. Osfouri S, Azin R, Rezaei Z, Moshfeghian M. Integrated characterization and a tuning strategy for the PVT analysis of representative fluids in a gas condensate reservoir. Iran J Oil Gas Sci Technol. 2018;7:40–59. 12. Gerami S, Kiani Zakheradi M, Azin R, Osfouri S. A unified approach for quality control of drilled stem test (DST) and PVT data. Gas Process. 2014;2:40–50. 13. Kesler MG, Lee BI. Improved prediction of enthalpy of fractions: hydrocarbon Processing; 1976. 14. Green DW, Willhite GP. Enhanced oil recovery, vol. 6. Henry L. Doherty Memorial Fund of AIME, Society of Petroleum Engineers Richardson, TX; 1998. 15. Chen SM, Olynyk J, Asgarpour S. Effect of multiple-contact miscibility on solvent slug size determination. J Can Pet Technol. 1986;25. 16. Clark NJ, Shearin HM, Schultz WP, Garms K, Moore JL. Miscible drive—its theory and application. J Pet Technol. 1958;10:11–20. 17. Holm LW. Miscibility and miscible displacement. J Pet Technol. 1986;38:817–8.

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18. Gu Y, Hou P, Luo W. Effects of four important factors on the measured minimum miscibility pressure and first-contact miscibility pressure. J Chem Eng Data. 2013;58:1361–70. 19. Yuan H, Johns RT, Egwuenu AM, Dindoruk B. Improved MMP correlations for CO2 floods using analytical gas flooding theory. In: SPE/DOE symposium on improved oil recovery. Society of Petroleum Engineers; 2004. 20. Johnson JP, Pollin JS. Measurement and correlation of CO2 miscibility pressures. In: SPE/DOE enhanced oil recovery symposium. Society of Petroleum Engineers; 1981. 21. Alston RB, Kokolis GP, James CF. CO2 minimum miscibility pressure: a correlation for impure CO2 streams and live oil systems. Soc Pet Eng J. 1985;25:268–74. 22. Cronquist C. Carbon dioxide dynamic miscibility with light reservoir oils. In: Proceedings of the Fourth Annual US DOE Symposium, Tulsa, vol. 1; 1978. p. 28–30. 23. Shokir EME-M. Precise model for estimating CO2 -oil minimum miscibility pressure. Pet Chem. 2007;47:368–76. 24. Shokir EME-M. CO2 -oil minimum miscibility pressure model for impure and pure CO2 streams. J Pet Sci Eng. 2007;58:173–85. 25. Emera MK, Sarma HK. A reliable correlation to predict the change in minimum miscibility pressure when CO2 is diluted with other gases. SPE Reserv Eval Eng. 2006;9:366–73. 26. Kamari A, Arabloo M, Shokrollahi A, Gharagheizi F, Mohammadi AH. Rapid method to estimate the minimum miscibility pressure (MMP) in live reservoir oil systems during CO2 flooding. Fuel. 2015;153:310–9. 27. Kaydani H, Najafzadeh M, Hajizadeh A. A new correlation for calculating carbon dioxide minimum miscibility pressure based on multi-gene genetic programming. J Nat Gas Sci Eng. 2014;21:625–30. 28. Glaso O. Miscible displacement: recovery tests with nitrogen. SPE Reserv Eng. 1990;5:61–8. 29. Glaso O. Generalized minimum miscibility pressure correlation (includes associated papers 15845 and 16287). Soc Pet Eng J. 1985;25:927–34. 30. Firoozabadi A, Khalid A. Analysis and correlation of nitrogen and lean-gas miscibility pressure (includes associated paper 16463). SPE Reserv Eng. 1986;1:575–82. 31. Kuo SS. Prediction of miscibility for the enriched-gas drive process. In: SPE annual technical conference and exhibition. Society of Petroleum Engineers; 1985. 32. Pedrood P. Prediction of minimum miscibility pressure in rich gas injection. MSc thesis, Tehran Univ Tehran; 1995. 33. Zirrahi M, Azin R, Malakooti R. Prediction of minimum miscibility pressure for carbon capture and sequestration and acid gas disposal applications; n.d. 34. Izadpanahi A, Azin R, Osfouri S, Malakooti R. Asphaltene precipitation in a light oil reservoir with high producing GOR: case study. Adv Nanomater Technol Energy Sect. 2019;3:270–9. 35. Yang F, Zhao G-B, Adidharma H, Towler B, Radosz M. Effect of oxygen on minimum miscibility pressure in carbon dioxide flooding. Ind Eng Chem Res. 2007;46:1396–401. 36. Orr Jr FM, Taber JJ. Phase diagrams (1987 PEH Chapter 23). Pet Eng Handb. 1987. 37. Firoozabadi A. Thermodynamics of hydrocarbon reservoirs. McGraw-Hill; 1999. 38. Wheaton RJ. Treatment of variations of composition with depth in gas-condensate reservoirs (includes associated papers 23549 and 24109). SPE Reserv Eng. 1991;6:239–44. 39. Kiani M. Modeling development of compositional grading in hydrocarbon reservoirs. Persian Gulf University; 2019. 40. Mousavi DSA, Vafaei SM, Mirzaei B, Fasih M. Experimental investigation on asphaltene deposition in porous media during miscible gas injection; 2007. 41. Minssieux L, Nabzar L, Chauveteau G, Longeron D, Bensalem R. Permeability damage due to asphaltene deposition: experimental and modeling aspects. Rev l’Institut Français Du Pétrole. 1998;53:313–27. 42. Sunil K, Abdullah A-G, Dimitrios K. Asphaltene precipitation in high gas-oil ratio wells. In: Middle east oil show; 2003. 43. Hashemi R, Kshirsagar LK, Nandi S, Jadhav PB, Golab EG. Experimental and CMG study of asphaltene precipitation under natural depletion and gas injection conditions. Pet Coal. 2019;61.

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44. Izadpanahi A, Azin R, Osfouri S, Malakooti R. MODELING OF ASPHALTENE PRECIPITATION IN A LIGHT OIL RESERVOIR WITH HIGH PRODUCING GOR: CASE STUDY. In2nd International Biennial Oil, Gas and Petrochemical Conference 2018 Nov. 45. Izadpanahi A, Azin R, Osfouri S, Malakooti R. Optimization of Two Simultaneous Water and Gas Injection Scenarios in a High GOR Iranian Oil Field. In82nd EAGE Annual Conference & Exhibition 2020 Dec 8 (Vol. 2020, No. 1, pp. 1-5). European Association of Geoscientists & Engineers.

Chapter 3

Basics of Oil and Gas Flow in Reservoirs Reza Azin, Amin Izadpanahi, and Parviz Zahedizadeh

Abstract This chapter deals with basic concepts of oil and gas flow in porous media. In the first section (oil flow), the basics of oil flow in porous media and different flow equations, including viscous, Darcy, and Brinkman flow are introduced. Then, based on a new work the boundary between these kinds of flow are reported. Then, the wellknown diffusivity equation, combination of continuity equation and Darcy velocity, is introduced. The relative permeability as an important concept in two-phase flow through porous media is described as well. After that, the forces affecting oil flow in porous media are introduced as well as different dimensionless numbers which reflect the relative importance of forces during oil flow. The gas flow through porous reservoir can occur as single or multiphase. A second phase releases as a result of pressure decline in reservoir and starts flowing when its saturation exceeds a minimum value, known as critical saturation. At these conditions, the relative permeability concept applies to take into account the multiphase flow. Also, the classic Darcy’s law equation for flow in porous media may fail in certain flowing conditions, and the non-Darcy flow equations need to be studied and applied for proper modelling of gas flow in porous medium. In addition, the role of wettability alteration on production enhancement of gas condensate reservoirs is a novel approach in gas reservoir performance. Also, the active forces during CO2 storage and the challenges of gas flow in unconventional gas reservoirs are two important subjects. These subjects will be covered in the second section (gas flow) of this chapter.

R. Azin (B) Faculty of Petroleum, Gas and Petrochemical Engineering, Department of Petroleum Engineering, Persian Gulf University, Bushehr, Iran e-mail: [email protected] A. Izadpanahi · P. Zahedizadeh Oil and Gas Research Center, Persian Gulf University, Bushehr, Iran e-mail: [email protected] P. Zahedizadeh e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 R. Azin and A. Izadpanahi (eds.), Fundamentals and Practical Aspects of Gas Injection, Petroleum Engineering, https://doi.org/10.1007/978-3-030-77200-0_3

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Nomenclature Acronyms ANN BHP IPR LSSVM MGGP TPR

Artificial neural network Bottom-hole pressure, psi Inflow performance curve Least square support vector machine Multi-gene genetic programming Tubing performance curve

Variables C Ct E Fo f G Grw h I0 I1 k K0 K1 M p p¯ p0 pp Pi Pe PD Pi Pwf PDwf q qD R Ra

Isothermal compressibility, pa− 1 Total compressibility, 1/psi Non-Darcy effect Forchheimer number Fractional flow Cumulative gas (HCPV) Diffusional Grashof number Formation thickness, m Modified Bessel function, first kind, zero-order Modified Bessel function, first kind, the first order Permeability, md Modified Bessel function, the second kind, zero-order Modified  function, the second kind, the first order  √ Bessel M = zf(z) Pressure, psi Average reservoir pressure, pa Initial Pressure, psi Pseudo-pressure Initial reservoir pressure, pa Peclet number Dimensionless pressure, (PD = 2πKf h(Pqμi −P(r,t)) ) Initial reservoir pressure, pa Wellbore pressure during the drawdown period, pa i −Pwf ) Dimensionless wellbore pressure, (PDwf = 2πKf h(P ) qμ 3 −1 Production rate, m s Dimensionless production, (qD = kf h(PqμB ) i −Pwf) Cylinder Radius, m Rayleigh number

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Re r rc re rw rD rDe S Sc t tD U U* U v v vg vp z

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Reynolds number Radius, ft Pore radius, ft Reservoir outer boundary radius, m Wellbore radius, m Dimensionless radius, (rD = r/rw ) Dimensionless outer boundary radius, (reD = re /rw ) A characteristic coefficient of the fractured rock proportional to the specific surface of the block Schmidt number Time, s Kf t Dimensionless time, (tD = [(∅C) +(∅C) ]μr2 ) m

f

w

Velocity, ms− 1 Rate of mass flow per unit volume, represents the transfer of fluid between blocks and fractures, kg/s m3 , (U∗ = ρSKm (Pμm −Pf ) ) Flux, S− 1 Velocity, ft/s Average velocity (m/s) General solution Particular solution Compressibility factor/Laplace space variable

Greek Letters μ β γ σ ρ v Ø λ ω θ α

Viscosity, cp Coefficient of quadratic velocity term Coefficient of cubic velocity term Interfacial tension, dyne/ft Density, lb/ft3 Velocity (ft/d) Porosity αK r2 Dimensionless matrix/fracture permeability ratio, (λ = K12 w ) C2 ) Fracture storage, (ω = ∅1 C∅12+∅ 2 C2 Rock wettability Angle of deviation/interporosity flow shape factor, m− 2

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Fracture Dry gas Wet gas Dry gas injected Matrix Recovery of wet gas Relative Wellbore Critical water

3.1 Introduction This chapter deals with basic concepts of oil and gas flow in porous media. In the first section (oil flow), the basics of oil flow in porous media and different flow equations, including viscous, Darcy, and Brinkman flow are introduced. Then, based on a new work the boundary between these kinds of flow are reported. Then, the wellknown diffusivity equation, combination of continuity equation and Darcy velocity, is introduced. The relative permeability as an important concept in two-phase flow through porous media is described as well. After that, the forces affecting oil flow in porous media are introduced as well as different dimensionless numbers which reflect the relative importance of forces during oil flow. The gas flow through porous reservoir can occur as single or multiphase. A second phase releases as a result of pressure decline in reservoir and starts flowing when its saturation exceeds a minimum value, known as critical saturation. At these conditions, the relative permeability concept applies to take into account the multiphase flow. Also, the classic Darcy’s law equation for flow in porous media may fail in certain flowing conditions, and the non-Darcy flow equations need to be studied and applied for proper modelling of gas flow in porous medium. In addition, the role of wettability alteration on production enhancement of gas condensate reservoirs is a novel approach in gas reservoir performance. Also, the active forces during CO2 storage and the challenges of gas flow in unconventional gas reservoirs are two important subjects. These subjects will be covered in the second section (gas flow) of this chapter.

3.2 Oil Flow 3.2.1 Basics of Oil Flow in Porous Media The original form of Navier–Stokes equation, (shown in Eq. (3.1)) describes the balance of pressure forces, viscous forces, and external forces. A specific solution

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of Navier–Stokes equation is presented in Eq. (3.2). The assumptions behind the Navier–Stokes equation include incompressible fluid with constant fluid properties and Newtonian fluid. According to the equation of motion, the rate of increase of momentum equals the sum of rate of momentum addition by convection, molecular transport, and external forces on fluid, all expressed per unit volume [1]. These are the main forces acting in a flowing continuum. Flow of fluids through porous media may be treated by simplifying the porous media as a bundle of capillary tubes. The Poiseuille equation, also known as Hagen–Poiseuille equation, shown in Eq. (3.3) and is used to formulate the flow of fluids in simplified porous medium. Also, fluid flow through porous media was studied experimentally by Darcy on an unconsolidated pack of sandstone [2]. ∂ 2v 1 dp ∂ 2v + = ∂x2 ∂ y2 μ dz v=

(3.1)

 1 dp  2 R − r2 4μ dz

(3.2)

1 R 2 dp 8 μ dz

(3.3)

v=

He constructed a vertical physical model, packed it with unconsolidated sand, and saturated the medium with water. Then, he performed experiments at atmospheric conditions to derive his famous equation, now widely used in the oil industry (Eq. (3.4)). v=

K dP μ dz

(3.4)

He introduced the term “permeability” to the context of fluid flow through porous media. The assumptions made by Darcy referred to the physical model in which he conducted his experiments. These assumptions include incompressible fluid, steady state conditions, complete saturation, single phase flow, constant physical properties (mainly viscosity and density), laminar flow, homogeneous formation, and constant temperature. This equation is now widely applied in all types of fluid flow through porous media, including groundwater flow, oil flow, and gas flow. By equating Poiseulle and Darcy equations, an expression for permeability is obtained as a function of capillary radius. The Poiseulle equation can be written for both pressure gradient and gravity flow [3]. The Darcy velocity is sometimes referred to infiltration velocity [4]. The Darcy equation was later extended to conditions which were not originally considered by Darcy. For example, cases like unsteady state and pseudosteady state flow of slightly compressible and compressible fluids, as well as turbulent flow and multiphase flow in heterogeneous and layered formations quite deviate from original assumptions made by Darcy. However, the researchers and petroleum engineers preferred to keep the original form of Darcy’s equation

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and modify it whenever necessary and extend its application as the foundation for oil flow in reservoir. The Darcy’s law equation can be expressed in both pressure gradient and potential gradient forms to account for gravity effects and vertical flow of fluids [5]. Though derived from experiments, the Darcy’s law may be derived theoretically from the Navier–Stokes equation using a formal averaging procedure [6]. According to Saidi [3, 7], Darcy equation can be derived by ignoring the inertia terms in the Navier–Stokes equation and taking the average velocity of the flow. Also, Neuman showed that the Hagen-Poiseuille equation, as well as the expression describing Couette flow between parallel plates, can be derived from the equations presented in his work and may thus be viewed as special cases of Darcy’s law [6]. The Darcy flow is sometimes referred to slip flow. This means that flow on the solid walls, here on grain surfaces of porous medium, is not zero, and the fluid molecules slip on the solid walls. This is the main difference between Darcy flow and viscous flow governed by Navier–Stokes equation. The viscous flow stops when the radius of a capillary tube (or capillary pore) is close to the thickness of adsorbed layer. Under these conditions, slip flow prevails and the viscous flow is no longer valid. When the pore diameter is small enough to fit within the boundary layer of viscous flow, the fluid will be unable to flow under viscous regime, and slip flow appears. Darcy flow occurs when slip flow fully develops. Saidi states that flow through a circular tube with a diameter of about one micron is probably non-viscous and probably follows Darcy Law [3, 7]. In recent years, there have been debates and questions on the applicability and validity of Darcy’s law in low-permeable formations, where fluid velocity is typically low. Many researches addressed several limitations of Darcy’s law at certain conditions where this equation is no longer valid [8–11]. These conditions are high velocities, known as non-Darcy flow in the context of petroleum engineering, low gas pressure, known as slip flow or Klinkenberg effect [12], and Non-Newtonian flow as observed in certain heavy oil, which is addressed by Gavin [9]. He analyzed typical velocities in oil and gas reservoirs and realized significant differences from an order of magnitude or more, between these velocities and those used in original Darcy’s experiments [2]. In all oil and gas cases, the flow velocity was much lower than the ranges of velocities reported by Darcy. Going farther from producing well reduces the oil and gas velocity so that there is little difference between linear and non-linear solutions of diffusivity equation [13]. Longmuir mentioned that the pre-Darcy flow may occur both in consolidated, tight systems, and in unconsolidated sand packs, and came to a conclusion that there is lack of accessible data which confirms Darcy’s law governs flow of fluid in porous media at reservoir conditions [9]. When fluid velocity in porous media is lower than a minimum value, it cannot be treated by Darcy’s law, and a linear relationship between pressure gradient and velocity does not exist. This is called pre-Darcy flow [9, 14–17]. A simple interpretation of this concept is that there is no need to expect the linear relationship between velocities versus pressure gradient to pass through the origin. In other words, care should be paid when thinking of Darcy equation as general flow mechanism in porous media, as this equation may apply at low velocity single-phase flow of some fluids in some cases, but not to other fluids in other porous media. Both pre-Darcy and post-Darcy

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flowing conditions may be attributed as non-Darcy Flow, although the post-Darcy, i.e. high flow velocity, has been studied by many researchers. Studies of Prada and Civan [18] revealed that pressure gradient in porous media should be higher than a threshold for the Darcy’s law to apply. The fluid force needs to be sufficient to overcome threshold pressure gradient. They suggest that Darcy’s law needs correction to account for this effect. Evidences of pre-Darcy flow were reported by others, mostly in the fields of water and soil science and engineering [19–23]. This concept is emerging in petroleum reservoir engineering, and numerical software has been developed to study low-velocity flow behavior in porous media. For example, Yu et al. [24] applied a low-velocity, nonDarcy flow numerical simulator to study a low-permeability oil formation. Farmani et al. investigated the pre-Darcy flow for gas and liquid phases in unconsolidated sand pack. N-heptane, n-hexane, water and condensate were used as liquid phase and N2 , CH4 and CO2 were used as gas phase. They observed pre-Darcy, Darcy and postDarcy flow regimes for these substances. Also, they investigated these flow regimes in a porous media with different particle size. They reported that the pre-Darcy flow depends on different parameters such as low velocity, porous medium properties, fluid properties and interaction between medium and fluid. Also, the mobility of the system was found as an important parameter on the onset of pre-Darcy flow [25]. In another work, Farmani et al. investigated the pre-Darcy flow and Klinkenberg effect in the calcite, dolomite and chalk-salt core samples. These cores have low permeability (0.011–18.53 md) and low porosity (12.33–28.21%). The onset of preDarcy flow was investigated in these cores. N2 was used as the flowing gas in core flood experiments. They reported the onset of pre-Darcy flow in each core. Also, the pre-Darcy, Darcy and post-Darcy flow regimes were determined using the Reynolds number analysis and reduced pressure drop analysis [26]. Another form of flow through porous media is defined by Brinkman and is known as Brinkman equation, shown in Eq. (3.5) [27]. Brinkman flow describes a flow pattern combined of Darcy and viscous flow [3, 7]. The Brinkman equation considers extra terms in the form of second derivative of velocity to account for additional pressure drop compared to laminar Darcy pressure drop. This equation applies in unconsolidated porous medium with large permeability, where the viscous shearing stresses acting on a volume element of fluid cannot be neglected [27]. This type of flow frequently occurs when experiments are designed and conducted in unconsolidated sand pack, glass beads, catalyst pack, and etc. where the flow is neither fully Darcy nor fully viscous. The Brinkman equation approaches Darcy law for low rock permeability, and approaches viscous flow when permeability approaches infinity. 1 dp ∂ 2v v ∂ 2v + 2− = 2 ∂x ∂y k μ dz

(3.5)

However, there is no clear idea about the range of flow rates where the flow regime changes from one state to another state in porous media. According to de Lemos, a distinction between laminar, nonlinear, and fully turbulent flow in porous media is not as evident as it is observed in unobstructed flow, and adequate models which

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cover a wide range of medium (K, ϕ) and flow properties (ReH ) still have to be developed [28]. Dybbs and Edwards classified the flow regimes into four regions, starting with laminar or Darcy flow, which is characterized by Re = 1 to turbulent flow characterized by Re > 300 [29]. The modified Reynolds number based on Ergun equation takes into account the porosity and is used to calculate the friction factor for flow through porous media [29]. As discussed earlier, Darcy’s law arises when the flow on the microscopic scale is Stokes flow, which is governed by the Navier–Stokes equations without the nonlinear advective terms. Under these conditions, the microscopic Reynolds is negligible. Once the microscopic Reynolds number becomes sufficiently large, then steady recirculating regions form downstream of particles, and even before this happens, the flow past, say, a regular array of cylinders or spheres, become non-symmetric due to the increasing influence of the advective terms in the Navier–Stokes equations. This is still well within the laminar regime and the Reynolds is still single digit or perhaps a little larger. Therefore, it seems that while turbulent microscopic flow is definitely non-Darcy, non-Darcy flow is not necessarily turbulent and there remains quite a large range of values of Re for which the microscopic flow remains laminar. In other words, the non-Darcy flow does not necessarily correspond to change in flow regime from laminar to turbulent flow, and deviation from linear Darcy flow is not initiated by turbulence. As will be discussed later, the onset of non-Darcy flow occurs at Reynolds numbers much lower than the onset of turbulent flow. Also, Hubbert addressed some of the Darcy’s law deficiencies [30]. He performed a dimensional analysis on Darcy equation and explored that this equation is a purely kinematic expression in its primitive form which involves explicitly only the dimensions of length and time, while the flow of a viscous fluid through a three-dimensional network of channels in porous solids is a dynamic phenomenon that involves forces and energies. According to Hubbert, it is not clear how the fluid properties, density and dynamic viscosity are involved in the flow, and how the flow rate is affected by the geometrical properties of the porous solid [30]. There are many forms of Darcy equation, each referring to a special gradient form. The two common forms of Darcy equation are expressed as pressure gradient and fluid potential gradients [5]. Hubbert introduced the potential form of Darcy’s law in terms of energies and forces, although the potential is indeterminate or nonexistent for some cases. The potential form of Darcy’s law and the concept of equipotential surfaces are used to analyze such phenomena as oil migration, hydrocarbon entrapment, identification of oil–water interface, and etc. [30]. In addition to the pre-Darcy flow, there are other conditions where flow through porous media, in this case oil reservoirs, may deviate from Darcy’s law. This phenomenon may occur when flow velocity increases, so that the linear relationship between flow rate and pressure gradient will be no longer valid. Rather, flow equation is parabolic at high velocities and occurs mostly near wellbore. In the context of fluid mechanics, this phenomenon may be expressed as transition from laminar to turbulent flow, as expressed similarly in the context of reservoir engineering [7, 11]. The non-Darcy flow observed in oil and gas reservoirs implies a high pressure gradient near wellbore, which causes the diffusivity equation to be non-linear at this

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region [13]. The transition from Darcy, i.e. linear flow, to non-Darcy, i.e. turbulent flow depends on reservoir rock and fluid properties and oil production rate. It should be noted that the transition or change in flowing regime from Darcy to non-Darcy depends mainly on well production rate and reservoir permeability. For oil reservoirs, Azin et al. showed that non-Darcy flow is expected at flow rates above 500 bbl/day [13]. In general, there are two criteria which determine flow regime and transition from Darcy to non-Darcy flow in porous media. These criteria are the Reynolds number and the Forchheimer number. The Forchheimer number represents the ratio of pressure drop consumed by liquid–solid interactions to that by viscous resistance. This number has a direct relation to non-Darcy effect [31]. The critical Forchheimer number is a criterion for the onset of non-Darcy flow and may be selected according to the features of the problem, and should be scientifically more reasonable in compared to a fixed critical value. Zimmerman et al. performed flow experiments and simulations to evaluate transition between Darcy-non-Darcy flow regimes [32]. They defined a critical Reynolds number, at which the non-Darcy pressure drop contributes 10% of the total pressure drop. For practical purposes, this transition seems to occur at about Re = 10 [32]. Hassanizadeh and Gray conducted a survey of the literature to reveal that the critical Reynolds number which determines the transition between Darcy and non-Darcy flow ranges between 1 and 15 [33]. Zeng and Grigg reviewed the two types of non-Darcy criteria and proposed a revised Forchheimer number [31]. They suggested a reference value of 0.11 for the critical Forchheimer number, which corresponds to a 10% non-Darcy effect [32]. In another study, Hassanizadeh and Gray showed that the onset of nonlinearities occurs at Reynolds numbers around 10, where macroscopic viscous and inertial forces become negligible in comparison with microscopic viscous forces [33]. They analyzed different theories, empirical correlations, and mechanisms that support the non-Darcy flow. They suggested that microscopic viscous forces, i.e. drag forces, grow at high flow velocities, so that the solid–fluid drag force becomes much larger than inertial and viscous stress forces at the onset of nonlinear flow.

3.2.2 Boundary Determination Between Darcy, Brinkman and Viscous Flow Zahedizadeh et al. solved the main flow equations (Darcy, Brinkman and Viscous) in cylindrical systems. The solution of Viscous and Brinkman equations are presented in Appendix 3.1 and 3.2, respectively. In their study, the boundary of each flow is reported. They expressed that flow regime is affected by variation of radius and permeability. Other characteristics like pressure drop and viscosity do not have any impact on the flow regime [34]. Figure 3.1 shows the viscous-Brinkman and Darcy-Brinkman flow regime boundaries as a function of permeability and radius for different deviation of resulted

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1E+11 1E+10 1E+09 100000000 10000000 1000000 100000 10000 1000 100 10 1 0.1 0.01 0.001 0.0001 0.00001

S=0.1 S=0.5 S=1 S=2

K (md)

Viscous-Brinkman Darcy-Brinkman

Darcy Flow 0.1

1

10

100

1000

Radius (μm) Fig. 3.1 Darcy-Brinkman and viscous-Brinkman flow regime boundaries in cylindrical coordinate for different deviation of velocity profiles (S) [34]

velocity profiles (S). Based on this figure, by increasing R at a constant permeability, the type of flow regime will change from viscous to Brinkman and then Brinkman to Darcy. The viscous flow is dominated in the small radius due to the effect of wall on the velocity profile. The wall effect is considerable only on the boundary layer in large flow path dimensions. In these kinds of flow path, most parts of the velocity profile are not affected by the wall. So, the Darcy flow will be dominated in these conditions. Based on Fig. 3.1, by increasing the S parameter in the viscous-Brinkman flow boundary, the flow type changes to Brinkman flow. Therefore, S = 0.1 indicates the exact boundary of viscous and Brinkman flow [34]. In Fig. 3.2, the permeabilities related to the different S parameter values for a constant radius (R = 10 μm) in the Brinkman-viscous flow boundary are obtained by using Fig. 3.1. Figure 3.2 shows that the Brinkman velocity profile approaches the viscous flow velocity profile by decreasing the S parameter. At S = 0.1, the velocity distribution diagram which calculated from the Brinkman and viscous equations have a high agreement. Therefore, in S ≤ 0.1, the viscous equation can be employed instead of the Brinkman flow equation [34]. Based on Fig. 3.1, by an increase in the S parameter in the Darcy-Brinkman flow boundary, the flow regime type tends to Brinkman. In this condition, it can be stated that S = 0.5 shows the exact boundary of Darcy and Brinkman flow regimes. In Fig. 3.2, the permeabilities related to the different S values for a constant radius (R = 10 μm), in the Brinkman-Darcy flow regime boundary are gained using Fig. 3.1, and the velocity profile is achieved, based on the Brinkman equation. Also, the velocity profile for the Darcy flow regime is shown for permeability values related to different S values [34].

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Fig. 3.2 Viscous velocity profile and Brinkman velocity profile for different deviation of velocity profiles (S) in cylindrical coordinates [34]

Based on Fig. 3.3, it can be seen that the effect of the boundary layer reduces by decreasing the S parameter. In this case, the Brinkman velocity profile tends to Darcy velocity. At S = 0.5, the fluid flow velocity profile achieved from the Brinkman equation corresponds to the Darcy equation in more than 99% of the dimensionless radius. Therefore, the wall effect is at its minimum value in S ≤ 0.5 values, and the flow regime can be assumed as Darcy flow. In this condition, in the regions, lower of the line S = 0.5 in the Brinkman-Darcy flow boundary in Fig. 3.1, the Darcy equation can be used to calculate the fluid flow [34]. 4

Fig. 3.3 Darcy and Brinkman velocity profile for different S values in cylindrical coordinates [34]

S=2

3.5

Darcy flow Brinkman flow

Velocity (cm/hr)

3 2.5

0.1

2

R=10 μm

1.5

S=1

1 0.5

S=0.5

0 0.9

0.95

r/R

1

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3.2.3 Flowing Fluids and Diffusivity Equation in Porous Media The flowing fluid in reservoir may be incompressible, slightly compressible, or compressible in nature. The oil flowing through porous media is regarded as slightly compressible, and its density changes slightly with pressure. This is especially important for flow of oil in reservoir, as the reservoir is typically subject to pressure change, normally pressure depletion. Therefore, a slightly compressible fluid tends to expand upon pressure depletion and occupy more space at lower pressure. Expansion of fluid will assist in oil recovery, as the reservoir volume is assumed constant, so the expanded fluid will contribute to reservoir drive and pressurize the formation. Reservoir engineers apply a simple exponential equation to define pressure dependency of reservoir fluid [35]. This equation is applicable at constant temperature, typically for thin formations where temperature gradient is negligible. In addition, the oil flow may occur as single phase or associated with one (water/gas), two (water and gas), or even more (water, supercritical fluid, gas, solids like precipitated but not deposited asphaltene and scale) phases. The two-phase and three-phase flow through porous media has been more addressed in the literature. It should be mentioned that the oil flow may be originally single phase, but switch to two-phase as a result of pressure depletion and release of solution gas. In this case, single phase flow continues until the gas saturation rises beyond critical gas saturation and gas starts flowing with oil. The flowing gas may have been injected for pressure maintenance or EOR purposes. In addition, water can flow with oil as separate phase. Sources of water are injected water for displacement (EOR) and/or pressure maintenance, and natural water influx from aquifer. There are cases when three phases are present and flow all together, e.g. in the case of combined water and gas injection in water alternating gas (WAG) process, or in the case of natural water influx into a saturated oil reservoir. Generally, analysis of fluid flow becomes more complex when the number of flowing phases increase. For example, density difference between flowing phases may cause gravity segregation, lead to gas or water coning, improper oil sweep, and trapping problems. Geometrically, flow may occur in linear, radial, spherical or semispherical forms. Radial flow is a common flow geometry, as it resembles oil flow towards a producing well. In the case of fluid flow through naturally or induced fractures, linear flow is dominant. Also, when a well is partially penetrated into producing formation or partially perforated, spherical flow becomes dominant. Many classic reservoir engineering textbooks describe different flow geometries, of which the works of Craft et al. [35], Dake [5], and Ahmed [8]. The fluid velocity in reservoir depends on the type of fluid, its viscosity and production rate at the well. For radial configuration, the fluid velocity is low at distances far from well, and increases as fluid approaches wellbore. Generally, flowing velocity is higher in gas reservoirs compared to oil reservoirs. Longmuir calculated superficial velocity for a number of oil and gas wells [9]. Typical fluid velocities at a radius of 10 ft from wellbore were about 275 ft/day, 7 ft/day, and 0.1 ft/day for gas well, high-rate oil well, and low-rate oil well, respectively. The fluid velocities decreased

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drastically at distances far from wellbore. For the same reservoirs, velocities at a radius of 700 ft from wellbore were calculated about 3 ft/day, 0.11ft/day, and 0.002 ft/day for gas well, high-rate oil well, and low-rate oil well, respectively. Production of oil from reservoir depends basically on flow of oil through porous reservoir. The wells drilled in the formation need be fed by fluid to produce hydrocarbon. Flow of fluid through porous medium is the fluid pass towards well. Understanding fundamentals of fluid flow and flow mechanisms are essential for proper reservoir management. Flow of fluids through porous media is associated with changes in fluid pressure as a function of time and position, in the case radial flow as a function of time and radius with respect to production well. This pressure behavior in oil reservoir is described in the form of diffusivity equation derived from material balance or continuity equation in the reservoir. The linear form of diffusivity equation is shown as Eq. (3.6): φμct ∂p ∂2 p 1 ∂ p = + ∂r 2 r ∂r 0.0002637k ∂t

(3.6)

Equation (3.6) is written in radial form, as is the usual flow pattern in oil flow towards a producing well. The linear diffusivity equation is the simplified form of general diffusivity equation in which the non-linear term is removed for simplicity. This equation is composed of continuity equation combined with Darcy’s law as the auxiliary equation. The diffusivity equation shown in Eq. (3.6) is written for slightly compressible fluid. Equation (3.6) is linear, which implies that pressure distribution during oil flow can be described by a linear PDE. This equation can be solved for different boundary conditions that define transient and pseudo steady state solutions. This is why the diffusivity equation is regarded as the basis for almost all reservoir engineering calculations, including pressure distribution in transient and pseudosteady state conditions, well test analysis, and etc. Analytical solution of the linear diffusivity equation is given in Appendix 3.3. The nonlinear PDE form of diffusivity equation arises when pressure gradient, ∂∂rp , in reservoir becomes significant. In terms of velocity, Darcy velocity is determined directly by pressure gradient. A higher ∂∂rp implies larger Darcy velocity. Meanwhile, a larger flow velocity which is frequently the case in near-wellbore region, implies nonlinear PDE as shown in Eq. (3.7):   1 ∂ρ ∂ p 2 ∂p φμct ∂2 p 1 ∂ p + + = 2 ∂r r ∂r ρ ∂ p ∂r 0.0002637k ∂t

(3.7)

It is worth noting that a nonlinear diffusivity equation does not mean that there is a change in flow regime from laminar to turbulent; rather, it just implies a higher pressure gradient and flowing velocity which is still within the laminar flow regime. The analytical solution of Eq. (3.7) is presented in Appendix 3.4.

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3.2.4 Relative Permeability Although the Darcy equation was developed for single phase flow, it was later extended to formulate and model multiphase flow through porous media by introducing the effective and relative permeability concepts. Multiphase flow in porous media is significant in Soil Science, Hydrology, Chemical Engineering and Petroleum Engineering applications. For reservoir engineers, multiphase flow is essential in reservoir simulation, determining reservoir performance, well productivity, well injectivity and reservoir ultimate recovery. Phenomena such as diffusion, dispersion, viscous fingering are more dominant in porous medium especially in the presence of heterogeneities. These factors are the main reasons for complexity of multiphase flow in porous media. When there is more than one phase flowing through porous media, each phase flows by its effective permeability, and saturation of one phase affects rock permeability to other phases. Effective permeability is a relative measure of the conductance of the porous medium for one fluid while it is saturated with more than one fluid [8]. Thus, for example, the porous medium has an effective permeability to oil and an effective permeability to water in the two-phase oil–water flow. The ratio of effective to absolute permeability is called relative permeability. This parameter is a function of fluid saturation, rock and fluid properties, including pore structure, pore size distribution, porosity, absolute permeability, saturation history, wettability of surfaces, interfacial tension, viscosity of fluid, temperature, flow rate, and overburden pressure [36–38]. Relative permeability data are shown either in tabulated form or as relative permeability curves which are plotted against fluid (usually the wetting phase) saturation. For oil/water systems, relative permeability curves are tabulated and plotted against water saturation, for gas/oil and gas/water systems, these data are demonstrated against liquid saturation. The relative permeability data are generated either in laboratories using the analysis of multiphase fluid flow in the core or predicted through mathematical correlations and models. Overall, laboratory methods are categorized into two major groups, steady state and unsteady state methods [39]. In steady state method, the immiscible fluids flow simultaneously in core plugs until saturation and pressure equilibrium is attained. The unsteady state method involves injecting a fluid at constant rate or constant pressure to displace insitu fluid. Laboratory measurements of relative permeability curves are usually very sensitive, time consuming and costly [40]. Hence, researchers prefer to obtain these data from other methods that are quick and accurate. Dake discussed the challenges that arise during laboratory measurements of relative permeability, mainly due to differences between mobility ratio of flowing phases and reservoir heterogeneity [41]. In addition to laboratory measurements, empirical and analytical mathematical correlations have been proposed yet for predicting or estimating the relative permeability of two-phase systems, that each of them has its own characteristics. Empirical correlations and analytical mathematical models are widely used for predicting relative permeability data. Generally, most of these models have been developed as a

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function of wetting phase saturation by using a stepwise linear and nonlinear regression analyses or analytical/numerical approach such as capillary models, statistical models and network models which are based on the assumption that a porous media consists of a bundle of capillary tubes. Mohammadi-Baghmolaei et al. reviewed a number of common approaches for relative permeability prediction models and correlations. They stated that the relations for estimation of relative permeability of two phase systems are divided into two categories. The first category includes such correlations which predict relative permeability as function of capillary pressure, absolute permeability, porosity, fluid saturation, interfacial tension and etc. [42]. The following correlations are in the first category which summarized in Table 3.1. Purcell in 1949 introduced the first analytical mathematical models to estimate the relative permeability of water–oil and gas-oil systems using capillary pressure data [43]. In 1951, Fatt and Dykstra presented correlations to calculate the wetting phase relative permeability using developed equations by Purcell as a base model [44]. They considered a lithology factor for including a fluid path that is a function of saturation. In 1953, Burdine developed similar equations with Purcell’s models through the introduction of a tortuosity factor as a function of wetting phase saturation [45]. In 1954, Corey developed an empirical correlation to estimate relative permeability of gas-oil systems based on relative permeability measurements on a large number of cores from several formations [46]. Corey also realized that the capillary pressure curve could be approximated by a linear equation and replaced in Purcell’s model. In 1958, Wiley and Gardner presented equations to estimate the relative permeability Table 3.1 The first category of relative permeability correlations Year

Relative permeability correlations

System

Rock type

Wettability

References

1949

Purcell

OW/GL

–

–

[43]

1951

Fatt and Dykstra

OW

Sandstone

–

[44]

1953

Burdine

OW/GL

Sandstone

–

[45]

1954

Corey

GL

Sandstone

–

[46]

1958

Wyllie and Gardner

OW/GL

Sandstone

–

[47]

1958

Wahl

GL

Sandstone

–

[48]

1958

Torcaso and Wyllie

GL

Sandstone

–

[49]

1966

Brooks-Corey

OW/GL

Sandstone

–

[50]

1982

Honarpour et al.

OW/GL

Sandstone/Carbonate

Any wettability

[51]

2000

Ibrahim and Koederitz

OW/GL

Sandstone/Carbonate

Any wettability

[52]

2001

Mulyadi, Amin, and Kennaird

GL

Sandstone

–

[53]

2013

Mosavat et al.

OW

Sandstone

–

[36]

2015

Xu et al.

OW

Sandstone

Water-wet

[40]

O Oil; W Water; G Gas; L Liquid

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oil–water and gas-oil systems in drainage process using capillary pressure data [47]. Also, Wahl et al. presented equation to calculate the ratio of relative permeabilities in gas-oil systems based on sandstone field measurements in drainage process [48]. Torcaso and Wyllie developed a simple formula to calculate the relative permeability of oil phase in gas-oil systems [49]. Prison presented correlations for estimation of the relative permeabilities of wetting and non-wetting phases in drainage and imbibition processes and in clean, water-wet rocks [8]. In 1966, Brooks and Corey improved Corey’s capillary pressure model and proposed correlations for capillary pressure and relative permeability of wetting and non-wetting phases [50]. In 1982, Honarpour et al. developed empirical correlations for water–oil and gas-oil systems in which the impacts of wettability and rock type had been considered [51]. In 2000, Ibrahim and Koederitz presented correlations for estimating relative permeability in 4 different systems through implementation of a linear regression model by using 416 sets of measured relative permeability data [52]. In 2001, Mulyadi, Amin, and Kennaird proposed MAK correlation to estimate relative permeability in the gas– water system. This correlation is based on Corey’s model for (gas-oil system) [53]. In 2005, Lomeland et al. proposed smooth and flexible three-parameter analytical correlations for relative permeability of water–oil, gas-oil and gas–water systems. Their model has flexibility for entire saturation range and also shows good agreement with experimental data in high or low fluid saturation [54]. The impact of pressure, fluid viscosity and flow rate were studied by Mosavat et al. [36]. Their proposed correlation estimates relative permeability of water–oil systems based on Corey’s model. Xu et al. proposed an empirical model to consider the impact of displacement pressure gradient for relative permeability of water phase in water–oil system [40]. The second category refers to the parametric models. The required parameters are obtained by implementing measured experimental data. Indeed, these models cannot be used in the absence of laboratory measured data. Also, some models have adjustable parameters that need be tuned based on measured data. As an example, correlations developed by Corey’s [46], Brooks and Corey [50], Lomeland et al. (LET) [54], Sigmund and McCaffery [55], Chierici [56, 57], Van Genuchten and Maulem [58], and Li et al. [59] have one or more adjustable parameters that need be adjusted before they can be applied to a reservoir model. The information of these correlations are summarized in Table 3.2. The decision tree shown in Fig. 3.4 can act as a guideline to select a proper model to predict relative permeability for a given system. Based on this figure, the LET model applies in most cases of oil and gas phases, both in carbonate and in sandstone systems. In addition to empirical correlations, intelligent systems such as artificial neural network (ANN), adaptive neuro-fuzzy inference system (ANFIS) and least square support vector machine (LSSVM) have been applied to predict of petrophysical properties, including relative permeability. Also, Ahmadi et al. used ANN intelligent model to predict the oil/water relative permeabilities with different learning algorithms [60]. The gas/oil relative permeabilities were also modelled by Ahmadi using LSSVM model [61]. Despite this growing rate of predicting petrophysical properties using intelligent systems, it should be mentioned that dealing with these

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Table 3.2 The second category of relative permeability correlations Year

Relative permeability correlations

System

Rock type

References

1954

Corey generalized correlation

OW/GL

–

[46]

1966

Brooks and Corey generalized correlation

OW/GL

–

[50]

1979

Sigmund and McCaffery

OW/GL

Carbonate

[55]

1984

Chierici

OW/GL

Sandstone

[57]

1999

Van Genuchten-Maulem

OW/GL

–

[58]

2005

LET

OW/GL

Composite rock

[54]

2006

Li et al.

OW

–

[59]

O Oil; W Water; G Gas; L Liquid

Fig. 3.4 Decision tree for proper model selection of relative permeability

systems cause some severity. The intelligent systems are highly structural-dependent and require an available source of data for development of models. Moreover, these systems do not provide an applicable clear mathematical function that can be employed in modelling and simulation as well. It is worth mentioning that among intelligent models, those which provide mathematical relationship between input data and output are more attractive for engineers. This will eliminate the sense of “passive” role in “black-box” calculations of intelligent techniques and let the user play a more “active” role and handle equations and correlations in engineering problems. Mohamadi-Baghmolaei et al. applied the multi-gene genetic programming (MGGP) approach to develop a new consolidated robust equation which predicts relative permeability of gas/oil systems [42]. Their proposed correlation requires rock and fluid properties such as gas saturation, gas molecular weight, oil API gravity, rock porosity, and absolute permeability. These properties are routinely measured on rock

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and fluid samples. In the following, the relative permeability correlations obtained by genetic algorithm are proposed by Farmani et al. and Mohamadi-Baghmolaei et al. Mohamadi-Baghmolaei et al. gas phase relative permeability correlation [42]: ∗ = 0.2308ag − 1.091 krg

ag = 4.947( Sg∗

2

+ bg + cg + dg + eg

(3.8) (3.9)

  0.000993 × f g × A P I + (Ln(A P I ))2 (3.10) M Wg       kabs ∗ Ln(Ln(A P I )) cg = Ln A P I − 5.741Sg Ln M Wg + 0.003574 ϕ (3.11)   4 0.005159 (M W g 29.5531 + 0.034718 kabs + M Wg ϕ dg = (3.12) (A P I )4 ⎛ ⎞ 41 109.4206 ⎠   eg = ⎝ (3.13) M Wg kabs ϕ bg = −

   f g = A P I E x p −6.996025Sg∗    4   + 9.701419 Sg∗ Ln M Wg Ln(A P I ) Ln(Ln(A P I )) − A P I

(3.14)

Mohamadi-Baghmolaei et al. oil phase relative permeability correlation [42]: kr∗o = 0.09595ao + 0.03296 

ao =

So∗ (3.057

− bo )  So∗

co ∗ ( So∗ +ε)((So +ε) )



(3.333 − do )

(3.16)

7.841M W

− M Wg + M Wg −2 A Pg I   bo = + 2.135A P I + 2.372 5.504M Wg − 0.7117 kabs ϕ (M W g

(3.15)

(3.17)

7.815 − 2 A P I + eo   (3.18) + 1.751A P I + 2.006 4.682M Wg − 0.5836 kabs ϕ   15.47 + E x p 6.143So∗ + f o     do =   kabs − A P I 0.7117 kabs − 5.504M Wg − 2.135A P I − 5.562 ϕ ϕ co =

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Farmani et al. gas phase relative permeability correlation [62]: ∗ = 2.284ag + 3.331 krg

(3.19)

  ag = (20.53 Sg∗0.25 + bg cg dg

(3.20)

   μg bg = [ Sg∗ + 4.105 3.159Sg∗ − 4.159Nca 17.6 − 20.53 − 217.1Sg∗ μo

∗ Sg   ∗ + 3I F T − 4Sg + 4.457 4.159 I F T (k/ϕ − 4.866)(k/ϕ Nca )   (3.21) + 20.53 k/ϕ)1/8 − 481.8 cg =

[55.18Sg∗

− 325.1Nca +

√

I FT k/ϕ

(k/ϕ Nca )

  Sg∗ 0.25 − Sg∗ + 4.875

+ 8.403))4.247 + 21.96(k/ϕ)1/8 − 331.3]    ⎤   ∗ I F T − μg + 4.402 + 4.159

S   g μ μ μ g g o N  ⎣ ca

dg = I F T − I F T + + + 10.53⎦ μo  μo Mwg 

(3.22)

⎡

(3.23)

Farmani et al. oil phase relative permeability correlation [62]: kr∗o = 0.1588ag + 0.07801

(3.24)

  Nca Nca 1.238Nca − ao = ln ( + 0.1472) do bo co μg        0.221 I F T −5.774 μo −4.889 + exp(Sg∗ ) Nca Sg∗ − 0.221 + exp exp Sg∗    +10.59Mwg Nca exp − exp Sg∗ (3.25)

   ∗      Mw  Sg Nca g + + Sg∗ + 0.4007 Sg∗ + 0.007532 ∗ kϕ Sg + 0.1472 k/ϕ    ∗     4.956  Sg N ca − 0.334 (17.98Nca ) + co = Sg∗ + 0.1732 + k/ϕ Sg∗ + 0.1472 k/ϕ 

bo = 



(3.26) (3.27)

Their results show that these correlation can predict the gas and oil relative permeability with high accuracy. More information on the procedure of proposing these correlations are available on [42, 62].

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3.2.5 Flow Through Fractures and Fissures Natural fractures play crucial role in facilitating flow in tight reservoirs. Depending on the size and extension of fractures, analysis of fluid flow through these conduits may be treated as viscous flow (Navier–Stokes approach), Darcy flow, or Brinkman flow. When the size of fracture approaches typical pore size, fluid flow can be treated as it was Darcy in porous media. On the other side, the flow of fluid will be treated as viscous flow when the fracture opening is macroscopic with minimized capillary effect. The Brinkman flow occurs in fracture opening between porous and macroscopic scales. This type of flow is frequently observed when the researchers conduct flow experiments on unconsolidated sand pack, and some may wrongly interpret the results simply by Darcy law. The viscous flow gradually converts to Brinkman and then to Darcy flow as the fracture opening decreases, and for fractures opening at the order of one micrometer, the flow is almost completely Darcy type [3, 7]. A common mistake to reservoir engineers is that they assume Darcy flow in fractures, which arises when they equate the flow equation derived for a single fracture to Darcy flow in order to obtain fracture permeability. This approach implies that the flow through fractures is always of Darcy type, regardless of fluid type and fracture dimensions. Moreover, the apparent permeability obtained by this approach is valid for a single fracture and is not applicable for a set of fractures or a fracture network. Golf-Racht proposed a different equation for average permeability of a fracture set, which is typically different from that of a single fracture [63]. Barenblatt et al. introduced the concept of dual porosity. Based on this concept, the fracture is the main flow path of the fluid, however, the matrix stores the hydrocarbon [64]. Warren and Root modified the Barenblatt model by using the fracture compressibility. In this model, fluid storage capacity (ω) and inter-porosity flow (λ) [65]. Zahedizadeh et al. provide a solution for Warren and Root model based on the reservoir with different boundary conditions [66]. A schematic of fractured reservoir and its boundaries is shown in Fig. 3.5. The Warren-Root model obtains pressure distribution in fractured reservoirs Eqs. 3.28 and 3.29 introduce the fluid flow mathematical model through matrix and fractures [63]:  m = − Km grad(Pm ) U μ

(3.28)

 f = − Kf grad(Pf ) U μ

(3.29)

Fractures have a great impact on the porous medium permeability. Great knowledge of fracture network and flow equations are required for the reservoir simulation. There are two main methods to model the fluid flow through a fractured reservoir known as equivalent continuum modelling and discrete fracture modelling (DFM) [67]. Warren and Root introduced the dual porosity concept as one of the basic equivalent continuum modelling methods to simulate the fractured reservoirs. In the

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Fig. 3.5 Schematic of the fractured reservoir model

Warren-Root model, the flow in both fracture and matrix is assumed as Darcy flow [65]. However, the fluid flow in the fractures is more complex due to the high velocity. In the later works, a non-Darcian flow was considered as a modification in the calculations of the dual porosity/dual permeability. Choi et al. employed the Forchheimer equation as a non-Darcy term in the fractures. This term shows that the results of these two models differ significantly. However, the results of these two models show good agreement in some specific range of parameters [68]. DFM is known as one of the most suitable methods for explaining the explicitly of fracture and matrix systems [69, 70]. Nevertheless, the main disadvantage of this method is that meshing a complex three-dimensional fracture network is very difficult. Also, this method has high computational cost due to a huge number of grids and convergence problems. Therefore, DFM application in the field scale is not realistic at present. But it seems that the DFM is becoming more applicable in the future according to its improvements [71, 72]. Equation (3.30) is resulted as a reservoir dimensionless pressure equation in Laplace space: PDf (z, rD ) = C1 I0 (M rD ) + C2 K0 (M rD )

(3.30)

where, C 1 and C 2 are defined as the Bessel equation constants. These constants are are first and second type calculated by applying boundary conditions. I 0 and K 0 √ modified Bessel functions in zero order. M is defined as zf(z). This parameter is used to simplify the calculations. F(z) is shown mathematically by Eq. (3.31): f(z) =

ω(1 − ω)z + λ (1 − ω)z + λ

(3.31)

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Table 3.3 demonstrates the solution of Eq. (3.30) by different inner and outer boundary conditions. The solution of Eqs. 3.32 through 3.37 will be presented in Appendix 3.5. Figure 3.6 indicates the result of production behavior in various outer and inner boundary conditions. The first region in Fig. 3.6 shows the production at initial times, and production of fluid is mainly through fractures. The second region in Fig. 3.6a shows the transition zone. In this region, both fractures and matrix contribute to fluid production. Also, the dimensionless pressure (PD ) stays almost constant in this region. In the transition zone and in the case of constant pressure production (Fig. 3.6b), both fractures and matrixes play the same role in fluid production where the dimensionless rate (qD ) stays almost constant. In the third region, the matrix has a higher contribution in reservoir production rate because of fracture capacity for fluid storage ends up. The fourth region in Fig. 3.6 refers to reservoir boundary effect, and its behavior depends on outer boundary condition. Figure 3.7a, b show the effect of dimensionless reservoir radius (rDe ) on wellbore pressure profile and production rate, respectively. This figure demonstrates that the onset of region IV depends on the selected reservoir radius. The reservoir boundary effect gets away from the transition zone, as the reservoir radius increases. Figure 3.8 shows the plots of PD -tD and qD -tD at constant rDe and ω for closed outer boundary condition. According to these figures, by reducing the matrix-fracture inter-porosity flow (λ), the transition zone accosts the reservoir boundary effect. High λ values imply high matrix ability to move fluid toward the fractures and show higher matrix to fracture permeability ratio. Therefore, a lower pressure difference is needed to remove fluid from the matrix to fracture in high λ values. Figure 3.8 also indicates Table 3.3 Solution of PDE for fractured reservoir with different inner and outer boundary conditions Inner BC

Outer BC

Solution

Equations

Constant production rate

Closed

PDf (z · rD ) =

(3.32)

Constant production rate

Constant pressure

PDf (z · rD ) =

(3.33)

Constant pressure production

Closed

qD (z · rD ) =

(3.34)

Constant pressure production

Constant pressure

qD (z · rD ) =

(3.35)

Constant production rate

Infinite

Laplace space form:

K1 (M rDe ) I0 (M rD )+I1 (M rDe )K0 (M rD ) z M [K1 (M) I1 (M rDe )− K1 (M rDe ) I1 (M)] K0 (M rD )I0 (M rDe )−K0 (M rDe )I0 (M rD ) z M[K0 (M rDe )I1 (M )+K1 (M )I0 (M rDe )]

M[ K1 (M) I1 (M rDe )− K1 (M rDe ) I1 (M)] z[K1 (M rDe ) I0 (M rD )+I1 (M rDe )K0 (M rD )] M[K0 (M rDe )I1 (M )+K1 (M )I0 (M rDe )] z [K0 (M rD )I0 (M rDe )−K0 (M rDe )I0 (M rD )]

PD (z · rD ) =

(3.36)

K0 (M rD ) z M K1 (M)

Warren-Root analytical solution: PD (tD · 1) = 21 Ln (2.25tD ), M < 0.01

(3.37)

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Fig. 3.6 Different regions in a—constant production rate, b—constant pressure production

Fig. 3.7 Different reservoir dimensionless radius in a—constant production rate, b—constant pressure production

Fig. 3.8 Different λ values in constant rD and ω for closed outer boundary condition a—constant production rate, b—constant pressure production

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Fig. 3.9 Different ω values in a—constant production rate, b—constant pressure production condition

that the transition zone interferes with reservoir boundary effect in λ smaller than 10–9 . Therefore, it may be hard to specify the transition zone in the PD -tD or qD -tD curves in a finite natural fractured reservoir with low λ values. Figure 3.9 shows the PD -tD and qD -tD curves for different ω values at constant production rate and constant pressure production conditions, respectively. The transition zone extends by decreasing fracture storage capacity (ω), as shown in Fig. 3.9. Based on this figure, the reservoir boundary effect and transition zone do not interfere with variation of ω. As mentioned, only λ variation can influence the transition zone and the reservoir boundary interference. It is impossible to obtain the transition zone and fractures, when interference occurs between the reservoir boundary effect and transition zone in a finite reservoir. Prediction of the interference area between reservoir boundary and transition zone is discussed by Zahedizadeh et al. They introduced λmin for this purpose where, at λ values less than λmin , it is impossible to identify fractured finite reservoirs. In other words, λmin for a certain rDe is the smallest λ value in which the fractured reservoirs are identifiable. Figure 3.10 indicates the λmin as a function of rDe in a logarithmic scale which shows the interface points of the reservoir boundary effect and transition zone. The relationship between λmin and rDe can be written as Eqs. (3.38) and (3.39) for constant production rate and constant pressure production conditions, respectively: λmin =

30.8319 2.0599 r De

(3.38)

λmin =

12.4394 2.0001 r De

(3.39)

In the regions above the curve, i.e., λ > λmin , fractures can be detected. On the other side, in the regions below the curve, i.e., λ < λmin , the reservoir fractures cannot be detected as usual.

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Fig. 3.10 λmin values in different rDe in a—constant production rate, b—constant pressure production condition

3.2.6 Forces Acting on Fluid Flow in Oil Reservoirs Flow of fluids is governed by forces acting on fluid in reservoir. As expressed before, the main forces acting on a flowing continuum in classical fluid mechanics include convection, molecular transport, and external forces [1]. In the case of fluid flow in reservoir, gravity, capillary, and buoyant forces are active as well. There are three main forces acting in an oil reservoir, i.e. viscous forces, capillary forces, and gravity forces. Each force may originate from certain conditions. For instance, capillary forces arise when two or more phase are present in the capillary pore space, and may support or resist flow of each phase, depending on the wetting tendency of rock and type of flowing fluid. The capillary forces and capillary pressure is frequently defined by Young–Laplace equation: pc =

2σ cos θ rc

(3.40)

According to this equation, Pc is a function of interfacial tension, σ, rock wettability, θ, and pore radius, rc , which is a measure of pore size, pore size distribution, and permeability of porous media. Capillary forces are believed to be a main mechanism in oil migration and formation of oil reservoirs. In tight reservoirs with low pore radii and low permeability, capillary effects and capillary forces become more dominant. In addition, capillary forces provide capillary continuity in oil flow through fractured reservoirs [3], which is also known as block-to-block interaction and matrix reinfiltration. As a result of capillary continuity, the oil flowing out of a matrix block re-enters another matrix block rather than flowing through a fracture network towards producing well. This is mainly due to wetting tendency of matrix block and preferred wettability towards oil. Capillary forces are observed in the pore scale, while such phenomena as Buoyancy forces originate from geothermal and pressure gradients in reservoir, which cause changes in fluid density and result in convective flow in reservoir, provided

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there is sufficient fracture system available in reservoir. According to Saidi [3], thermal convection is not likely to occur in an oil field lacking fracture system, even though it has quite large matrix permeability as high as 1 Darcy. Gravity flow is an important production mechanism especially in fractured reservoirs with low matrix permeability. Oil production rate under gravity flow depends on density difference between oil and gas, oil viscosity, elevation between the two contacts, rock permeability, and capillary holdup. The driving force for gravity drainage is density difference between phases, which provides a pressure difference and causes instability within the oil column. Although it is theoretically possible that such an instability occurs in any porous medium, gravity flow is practically observed in fractured formations only, and presence of fractures is crucial in assisting oil recovery under gravity drainage [3]. In a conventional reservoir the capillary pressure curve controls fluid distribution through the reservoir; therefore, the transition zone will come between the water– oil contact and the oil zone which, in the case of important capillary forces (tight formation), may be very thick. In a fractured reservoir this situation is completely different. The discontinuity of the matrix caused by the fracture network cutting the continuum of the matrix bulk into small individual matrix blocks, explains why the water table is only related to the fracture network. In addition, since the fractures are large channels with negligible capillary forces, the transition zone disappears in a fractured reservoir, and water–oil contact becomes a horizontal plane. On the other hand, capillary and gravitational forces (through the capillary pressure curve and gravitational curve) control the static and dynamic equilibrium of each matrix block. The basic element which relates individual block behavior to reservoir behavior is the water–oil contact in fractures and is called water table level. These water–oil contacts in fractures, together with the oil–water contacts inside the matrix, the last corresponding to displacement front level, are essential reference planes for the evaluation of the driving mechanism of capillary and gravity forces. An analogical situation will take place in the case of a gas-cap for both gas-oil contacts in fractures and matrix blocks, where the first is called gas-cap table and the second gas displacement front. If there are more than one force dominating fluid flow, their relative magnitude would be important to reservoir engineer. For example, when more than one phase flows through porous medium, viscous and capillary forces are active. In this case, the ratio of viscous to capillary force makes a dimensionless number called capillary number, which determines the relative magnitudes of these forces in reservoir. Bond number is another common dimensionless number, defined as the ratio of gravity to capillary forces. These are frequently used to interpret EOR experiments and water flooding operation, although there are questions on adequacy and theoretical validity of them [3]. For instance, increasing capillary number, i.e. viscous forces against capillary forces, has been well known as a strategy for increasing ultimate oil recovery and recovery factor during an EOR process. Determining a threshold value for each dimensionless number for a certain process depends on rock, i.e. porosity, permeability, wetting tendency and fluid properties, as well as experimental conditions. In one experiment, the critical capillary number required to remobilize the

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trapped phase was determined to be 2E−8 in gas–water systems. A similar order was observed to remobilize trapped gas in gas-oil systems [73]. Løvoll et al. conducted experiments on unconsolidated glass beads to investigate the competition of gravity, capillary and viscous forces during drainage [4]. The synthetic porous medium had porosity of 63% and permeability of 1915 D, quite high compared to that for normal oil reservoirs. Although they described some scaling equations and proposed stability criteria based on generalized Bond number, they did not provide a clear distinction between gravity, capillary and viscous forces in their experiments. Convection arises as two modes, Benard convection and Taylor convection [3]. Benard convection takes place when a liquid filled in a container is under negative density gradient due to either thermal or diffusion process. According to Saidi [3], Benard convection starts from a stable to unstable conditions, while Taylor convection starts from an unstable and goes to stable conditions. Convective flux may occur as a result of concentration gradient, pressure gradient, temperature gradient, or a combination of two or more gradients. Existence of one gradient can make another gradient(s) to emerge. For example, combination of pressure and temperature gradient in an oil reservoir may result in compositional changes throughout the whole depth of reservoir, which is normally known as compositional grading or compositional gradient [74]. Kiani et al. studied the two dimensional compositional gradient in mixtures of oil and gas. They also improved a thermodynamic model for prediction of the methane and plus fraction distribution and compositional gradient in reservoirs. They used the data of a giant gas condensate reservoir to validate their model. Their results showed that the developed model could accurately predict the experimental data. Also, they reported that natural convection could change the vertical and horizontal compositional gradient and also it affects the pattern of species distribution [75]. Another example of convection arises as a result of density differences caused by concentration gradient during CO2 disposal into saline aquifer [76, 77]. Whatsoever the origin of gradient, convection is described by Rayleigh number, Ra, which describes the ratio of convective to viscous forces. Saidi demonstrated convective flow and resulting compositional gradient in a number of Middle East oil fields and calculated Ra in these fields [3]. The minimum Ra, known as critical Ra, required for onset of convection depends on the flow geometry and boundary conditions of problem under study. The convective flow may couple with other transport phenomena, i.e. mass transfer and/or heat transfer. This coupling is known as thermal diffusion, cross effect, Soret effect or Dufour effect [1]. Thermal diffusion ratio, thermal diffusion factor, and Soret coefficient are used to determine molecular flux under the coupled influence of heat and mass transfer [1]. The pressure gradient can have a more complex impact on oil reservoirs, as the amount of solution gas dissolved in oil is directly dependent on pressure and temperature. It is well established that convective flux acts as a driving force for mass transfer in oil at different points of reservoir, which in turn imposes compositional gradient characterized by variations in fluid bubble point pressure, phase diagram, composition, density, and other physical properties. Ra is sometimes expressed as the product of Grashof and Schmidt numbers [1]:

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Ra = Grw · Sc

(3.41)

Grw is the diffusional Grashof number, arises due to buoyant forces caused by concentration inhomogeneities. Sc is defined as the ratio of viscous to diffusive forces, also known as the ratio of momentum diffusivity to mass diffusivity [1]. Molecular diffusion is normally neglected in oil production and its magnitude is small compared to other forces. In many practical field operation, impact of this parameter is considered as minimal due to relatively high production and/or injection rates in field scale. However, it can compete with other mechanisms in certain cases. Significance of diffusion in oil recovery from oil reservoirs has been addressed in the literature, e.g. [3, 7, 78, 79]. Molecular diffusion is also important when applying solvent-based heavy oil recovery [80–82]. Peclet Number, Pe, is another important dimensionless number which compares the relative magnitude of convective to diffusive forces in porous medium. Pe arises when molecular diffusion as a transport phenomenon contributes to oil production, e.g. miscible displacement [83, 84] and solvent-based heavy oil recovery [80, 82, 85]. This criterion is applied in cases with competing roles of diffusion and dispersion, i.e. convection. A high Pe implies dominant role of convection and decreased role of molecular diffusion, such as miscible displacement. In mathematical form, the mass transfer Pe is sometimes expressed as the product of Reynolds and Schmidt numbers [1]: Pe = Re · Sc

(3.42)

Pe is also used to estimate dispersion coefficients [1]. When there is a high capillary pressure in an oil reservoir, a difference between gravity and capillary forces act as a driving force for oil flow and reservoir depletion. A second phase, gas or water, fills the pore space during oil production and diminishes this driving force. At a balance between capillary and gravity forces, there is no extra force available for further oil production, although there is oil remaining in reservoir which cannot be produced due to capillary resistance. This region is known as capillary hold-up zone. For an oil reservoir, the ratio between capillary height, hc , and block height, L, is an important factor that affects ultimate oil recovery. Normally, low permeability reservoirs have higher Pc, thus remaining higher oil saturation in the capillary hold-up zone. On the other side, reservoirs with higher permeability have lower Pc and produce more oil, thus showing lower remaining oil saturation.

3.3 Gas Flow The gas reservoirs are classified as conventional and unconventional. The conventional gas reservoirs contain either dry gas, wet gas, or gas condensate, lean or rich depending on the composition of intermediate components in gas. Unconventional gas reservoirs include gas shales, coal bed methane, natural gas hydrates, and tight

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gas reservoirs. If the reservoir gas contains CO2 and/or H2 S, the gas will be sour, otherwise it is regarded as sweet gas when there is little or no acid gas component in its composition. Proper evaluation of a gas reservoir performance requires such information as gas in place, average reservoir pressure, and recoverable gas. These are crucial information and can be extracted through reservoir simulation, and/or production data analysis techniques. However, due to their unusual phase behavior, reservoir studies of gas condensate fluids are associated with challenges. A detailed methodology and analysis of challenges in gas condensate reservoirs are described in [86–88].

3.3.1 The Diffusivity Equation for Gas Reservoirs Flow of gas through porous media is described by a modified diffusivity equation, in which the pressure is replaced by pseudo-pressure, pp defined by Eq. (3.43) for radial system: ∂pp ∂2 pp φμct 1 ∂pp = + ∂r 2 r ∂r 0.0002637k ∂t

(3.43)

Equation (3.43) is derived by assuming laminar flow, constant temperature, and constant reservoir parameters. pp describes the relationships between PVT and transport properties, compressibility factor and viscosity, with pressure in the reservoir operating range, as is defined as Eq. (3.44): p p ( p) =

1 2 ( p − p02 ) μz

(3.44)

Details of converting general form of diffusivity equation from oil to gas systems are given by Wattenbarger [89]. Equation (3.43) is nonlinear, as the product of viscosity and compressibility is a function of pressure as well. However, the solution to this nonlinear PDE is obtained similar to linear PDE developed for flow of slightly compressible fluid like oil in porous reservoir. At certain conditions, m(P) may be simplified to pressure squared (Eq. (3.45)) or pressure form (Eq. (3.46)): φμct ∂ p2 1 ∂ p2 ∂ 2 p2 = + ∂r 2 r ∂r 0.0002637k ∂t

(3.45)

∂2 p 1 ∂ p φμct ∂p = + ∂r 2 r ∂r 0.0002637k ∂t

(3.46)

Equation (3.45) applies at low pressure, i.e. P < 2000 psia, where the product (μZ) is constant. Also, Eq. (3.46) is valid at high pressure, typically P > 3000 psia, where P/(μZ) is essentially constant. The exact pressure condition for Eqs. (3.45) and

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(3.46) to apply depends on gas specific gravity. The general solution of Eq. (3.46) is similar to oil reservoir, where P is replaced by m(P). The original form of gas diffusivity equation (Eq. (3.43)) applies to dry gas flow through porous media. In the case of multiphase gas flow, such as gas condensate two-phase flow at pressures between upper and lower dew-point pressure, relative permeability to each phase and capillary pressure affect the flow pattern. Details of multiphase flow equations are given by Ertekin et al. [90].

3.3.2 High Velocity Effects in Gas Flow When a gas well starts production from a gas reservoir, gas flowing velocity in reservoir increases as the stream approaches production well in a radial flow and reaches a maximum at the wellbore. The high velocity at the wellbore implies a transition from laminar to turbulent flow. At these conditions, the Darcy flow equation is no longer valid. The high-velocity phenomena are characteristic of gas flow through porous media and fractures, as well as gas flow during injection/withdrawal of gas in underground gas storage (UGS). The high-velocity region is characteristic of gas flow through porous media and gas flow through fractures and formation, sometimes modelled by adding a quadratic or cubic velocity term which is frequently added to Darcy’s law, and is commonly called the Forchheimer term [91–94]. This additional term accounts for higher pressure drop compared to Darcy flow and is associated with a coefficient called turbulence factor or slip factor. The cubic velocity term is used to correct inertia term to Darcy equation for low Reynolds numbers [95]. For higher Reynolds numbers, laminar flow switches into a transition regime and then into the Forchheimer flow regime. This regime is described by the Forchheimer equation, in which Darcy equation is corrected by adding a quadratic velocity term [95]. The difference between cubic and quadratic velocity correction is that the coefficient of cubic velocity is a combination of fluid properties only, while the quadratic velocity correction has a coefficient composed of fluid density multiplied by rock parameter. This parameter is known as termed the inertial resistance, β-factor, high-velocity flow coefficient, or non-Darcy flow coefficient. The Forchheimer equation is defined as: −

μν dp = + βρν 2 dX k

(3.47)

The cubic form of non-Darcy equation is written as follows [96]: −

dp μν = + βρν 2 + γρ 2 ν 3 dX k

(3.48)

Figure 3.11 show the proposed formulas for calculation of β in the literature [97].

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Fig. 3.11 Proposed formulas for calculating β [97]

Wu et al. provided analytical solutions to the gas flow equation including Klinkenberg effect [98]. They also proposed values for the turbulence factor and tested the validity of the conventional assumption used for linearizing the gas flow equation. Zeng and Grigg reviewed models and correlations for non-Darcy flow and discussed theoretical foundations of Forchheimer number and Reynolds number [31]. They defined non-Darcy effect (E) as the ratio of pressure gradient consumed in overcoming liquid–solid interactions to the total pressure gradient, and Forchheimer number, Fo, as follow: Fo = E=

kβρν μ βρν 2 − ddpX

(3.49)

(3.50)

Combining above equations and considering the Forchheimer equation, the relationship between Fo and E is obtained as follows: E=

Fo (1 + Fo )

(3.51)

Zeng and Grigg suggested that the Forchheimer number is directly connected to the error of ignoring the non-Darcy behavior of flow in porous media [31].

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3.3.3 Flow Behavior of Gas Condensate Reservoirs Any hydrocarbon reservoir experiences a pressure decline upon depletion and hydrocarbon production. Due to its composition, the gas condensate fluid tends to become two-phase when reservoir pressure drops below upper dew-point pressure. The intermediate components of gas condensate liquefy upon pressure drop and form a film of liquid hydrocarbon, i.e. condensate, which propagates farther into reservoir. The condensate bank reduces gas relative permeability and well productivity. In its worst condition, accumulation of condensate bank around wellbore may block flow of gas. This phenomenon is referred to condensate blockage. According to Fevang and Whitson [99], Three regions are expected to form upon gas production from a gas condensate reservoir. The first region, just connected to wellbore contains liquid and gas phases flowing simultaneously. The middle region has immobile liquid phase saturation below a critical value necessary to flow. The third region contains single phase gas and no liquid hydrocarbon saturation. The flow is single phase in regions 2 and 3, while two-phase flow prevails in region 1. Therefore, flow equations for each region should be assigned according to the flowing phases. Extension of condensate bank in region 2 can vary from tens of feet in lean gas condensate to hundreds of feet in rich gas condensate reservoirs [100]. When a fluid flows through porous medium, its momentum changes continuously as a result of changes in pore size. According to Newton’s first law, the fluid inertia opposes momentum change and causes pressure drop against fluid flow. This resistive force is called inertia force, which increases with flow velocity, followed by lowering gas and liquid flow. As a result of inertia force in two-phase flow, the relative permeability to both gas and liquid decreases. This is known as inertia effect and may be expressed as Forchheimer coefficient, β. The two-phase Forchheimer coefficient is used for gas-condensate systems as a function of gas saturation and relative permeability to gas [101–103]. Similar to inertia force caused by momentum changes in flow path, viscous forces which arise as a result of friction between fluids and pore walls act against flow and cause pressure drop. Combination of inertia and viscous forces produces a higher pressure drop which may result in formation of condensates in porous media. In twophase flow, capillary forces become active as well as viscous and inertia forces. If the inertia forces overcome capillary forces, gas flows through bank of condensate liquid and breaks the condensate bridge in pores. Thus, the path opens for gas and liquid flow. This is known as positive coupling [101]. The positive coupling phenomenon is associated with improvement in relative permeability of gas and condensate and counteracts the negative role of inertia. Therefore, positive coupling introduces a new concept against negative inertia in two-phase flow of gas and condensate. That is, the two-phase relative permeability to gas and liquid is not fixed; rather, it may increase with flow velocity as observed in long cores filled with condensate [104]. It was further shown that the positive coupling phenomenon is a function of interfacial properties. At large interfacial tension, the capillary force is high and phases flow in distinct pathways in porous media. In this case, the relative permeability is mainly a function of fluid saturation [101]. However, at low interfacial tension, i.e. near

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miscible conditions, relative permeability depends on both liquid saturation and interfacial tension [105]. Development of an accurate correlation is useful in predicting relative permeability as a function of gas velocity, fluid saturation and interfacial tension. Such a correlation will help in avoiding costly relative permeability curves. Different models have been developed for relative permeability as a function of interfacial tension. Coats proposed correlations for relative permeability of gas and condensate on the basis of changes in interfacial tension [106]. He used an interpolating correlation between base curves and miscible curve which approaches a straight line. Available correlations that consider the positive coupling are obtained by two methods: The Corey functions with the coefficients are interpolated between miscible and immiscible ranges, and interpolation between miscible and immiscible relative permeability curves obtained from Coats method. Many studies support the idea that relative permeability is a function of capillary number which depends on viscosity, velocity and interfacial tension [105, 107, 108]. Correlations of gas/oil relative permeability as a function of interfacial tension are also proposed by Harbert [108], Jamiolahmady et al. [109, 110], and Fulcher et al. [111]. Also, experiments are reported which aim at the effect of interfacial tension on relative permeability curves in gas condensate systems. Asar and Handy studied the effect of interfacial tension on gas-oil relative permeability in gas condensate systems [112]. They showed that relative permeability curves approach a 45° line as interfacial tension reduces to zero. Also, by increasing interfacial tension, relative permeability of oil (liquid phase) drops faster than relative permeability of gas. The residual oil and gas saturation increases by increase in interfacial tension on the other hand, at low interfacial tension near the critical value, liquid can flow at low liquid saturation. Munkerud and Torsaeter studied the role of interfacial tension and spreading on relative permeability of gas condensate systems [113]. They used different fluid systems including methane, ethane, propane, pentane, hexane, and decane with low interfacial tension and high potential for liquid drop out. Their flow experiments showed a strong relationship between relative permeability and gas-condensate interfacial tension. Similar experiments were reported by Henderson et al. [114, 115], Mott et al. [103, 116], and Chen et al. [117], with a conclusion that relative permeability of wetting phase is more sensitive to interfacial tension, while the non-wetting phase changes slightly with decrease in interfacial tension. On the other hand, some studies showed that interfacial tension has more influence on relative permeability of the non-wetting phase [108, 111, 115, 118, 119]. Meanwhile, some studies suggested that relative permeability is not affected by interfacial tension [105, 107, 112, 120, 121]. Such a contradictory finding may be due to type of fluids and prevailing experimental conditions. In other words, it appears that the range of experimental conditions, i.e. temperature, pressure, and composition affect the magnitude and range of interfacial tension which may cause reverse impact on relative permeability from a certain point. Henderson et al. studied the role of negative inertia and positive coupling at high flow velocity on relative permeability of gas condensate reservoirs [102]. Their results show that inertia forces prevail in cores totally saturated with gas, while positive

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coupling becomes active with condensate saturation and subsequent improvement in relative permeability was observed in all ranges of interfacial tension. According to their experiments, positive coupling was detected at velocities up to 700 m/s and interfacial tension up to 0.78 mN/m [102]. Jamiolahmady et al. reported the negative inertia and positive coupling effects on gas-condensate relative permeability in lowpermeability core samples [122]. They conducted difference tests in velocity range of 22–183 m/s and three interfacial tension values of 0.036, 0.15, and 0.85 mN/m. Results showed that for interfacial tension of 0.85 mN/m and condensate-gas-flow ratio (CGFR) less than 0.05, inertia effects prevail and gas relative permeability decreases as velocity increases. Positive coupling becomes more important at higher CGFR, resulting in relative permeability improvement. For interfacial tension of 0.15 mN/m, positive coupling was observed in velocity range of 22.8–45.7 m/s, and inertia effect was dominant for all CGFR. For the lowest interfacial tension case (0.036 mN/m), improvement in gas relative permeability was limited to CGFR = 0.4 and velocity range of 22.8–45.7 m/s [122]. Based on current studies, the role of positive coupling has been approved as a mechanism of improvement in gas flow against negative inertia force which causes additional pressure drop at high flow velocity. Approaching miscible conditions by reducing interfacial tension between gas and condensate can highlight the role of positive coupling. There is still much elaboration remaining to clarify the impact of physical and transport properties like condensate API, gas and condensate viscosity, interfacial tension, composition of gas and condensate phases, as well as operating conditions such as pressure, temperature, gas and condensate flow rate, etc. Fruitful research on these topics may enhance our future understanding of the exact, detailed role of positive coupling and negative inertia in gas condensate flow through porous media.

3.3.4 Gas Flow in Multilayer Reservoirs The multilayer hydrocarbon formations are frequently found in nature. These formations are characterized by such features as heterogeneity, anisotropy, and sometimes are associated with abnormal pressure profile. The gas flow in these formations is affected by horizontal and vertical porosity and permeability variations. Generally, a dominant vertical overburden pressure implies a lower vertical permeability compared to the horizontal permeability. In this case, the upward gas flow and migration between layers is less likely to occur, especially in the presence of impermeable layer and each layer may be considered independent of the others. The impermeable layers act as barriers and non-communicating layers can be studied independently (Fig. 3.12). However, natural phenomena like tectonic stresses may generate micro and macro fractures which can lead to excellent pathways which transfer the gas vertically between layers. Besides, diffusion and dispersion of gas as a result of concentration difference into upper layers acts as another means of vertical flow. Several observations are reported

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Fig. 3.12 Schematic of a multilayer formation with barriers in between

in the hydrocarbon reservoirs that confirm vertical gas diffusion and gradual change in the pressure and composition of reservoirs. The most common diffusion is the diffusion which is dependent on concentration gradients. There are other kinds of diffusion which are dependent on pressure gradient (pressure diffusion), composition gradient (ordinary diffusion), external forces (forced diffusion) and temperature gradient (thermal diffusion). In some situations, diffusion may compete with the gravity forces. The gravity segregates the lighter and heavier components but, the diffusion makes the mixture uniform [123]. Patzek et al. studied the vertical diffusion in ideal gas mixtures and gravity stable in gas reservoirs. Their results show good approximation before the concentration of diffusing gases at bottom and top boundaries of the reservoir reach considerable concentration [124]. This type of flow between layers requires a special attention in averaging reservoir properties. There are three common average methods for permeability including, weighted, harmonic and geometric average permeability. Weighted average method is used to calculate the average permeability for some parallel layers which separate from each other with impermeable layers. Also, each layer has its own flow rate. But, the layers have only one flow rate in the harmonic average method. In this method, layers face different pressure drop due to different permeability. Geometric average permeability is introduced by Warren and Price. They reported that the permeability of a heterogeneous reservoir could be shown by a geometric average method. The mathematical formulation of each method and more information is available in reservoir engineering handbook by Ahmed [8]. Kuchuk and Saeedi studied the inflow performance of horizontal wells in multi-layer reservoirs with crossflow. They showed that if the variations of horizontal and vertical permeability are large, the system should not be treated as a single layer with average properties. They stated that the productivity index for an equivalent single layer system are much lower than productivity index for actual (multi-layer) system [125]. However, in some cases, the

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average properties can be calculated with the Eqs. 3.52 through 3.55. These equations could be used when the surface rate is constant and pressure behavior is assumed to be equal to a single layer with average properties [126]. n ki h i k = i=1 n i=1 h i n ∅i h i ∅ = i=1 n i=1 h i n i=1 C t i ki h i Ct =  n i=1 ki h i n i=1 kr o i h i kr o =  n i=1 h i

(3.52) (3.53) (3.54) (3.55)

Nikjoo and Hashemi reported that three types of flow regimes could be identified in a single phase layered reservoir with crossflow: (1) the behavior of layers without crossflow, (2) a transition regime due to the fluid transfer between the layers, and (3) the pressure of two layers become equal. They reported that the average properties of equivalent single layer reservoirs can be calculated with the following equations [127]: k=

1 h

h kdz

(3.56)

∅Ct dz

(3.57)

0

1 ∅Ct = h

h 0

Fetkovich et al. stated that in the cases that crossflow exists between layers the system could be treated as an equivalent one layer reservoir with average properties. But, in the lack of crossflow, the multi-layer system can be combined to one equivalent layer if: (1) the diffusivity properties of the layers are the same or (2) the ratio of flow rate to initial gas in place is the same for each layer [128]. Park and Horne suggested two new methods for initial estimation of the parameter’s value of a multi-layer reservoir with crossflow. Their first method can be applied in a reservoir with two isotropic layers. But, their second method is applicable for N layer reservoirs. More information and mathematical description of each method are available in Park and Horne [129]. It is important to carefully consider these phenomena in the study of gas flow in communicating multilayer reservoirs. The question that arises is whether a multilayer formation can be averaged into a single reservoir with an averaged properties or not. Dake described the methods of calculating the average of dry gas saturation and

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pseudo-relative permeabilities for the flow of wet and dry gas in non-communicating multilayer reservoirs for gas recycling applications. Also, he reported the mathematical expression of the vertical sweep efficiency calculation. The cumulative recovery of wet gas (G p D ) is a function of average dry gas saturation (Sgde ), fractional flow of dry gas ( f gde ), cumulative dry gas injected (G id ) and critical water saturation (Swc ). Equation 3.58 shows the recovery equation reported by Dake [41]: G pD

  Sgde + 1 − f gde G id = 1 − Swc

(3.58)

where the fractional flow for dry gas is shown as follows: ρ sin α 1 − 2.743 × 10−3 ϑμrgw gw kk

f gde =

1+

μgd krgw μgw krgd

(3.59)

It is a common practice in reservoir engineering to combine two or more layers and consider the whole formation as a uniform layer with averaged properties. Azin et al. studied the gas flow in a multilayer gas condensate reservoir. They comprised the IPR curves for different number of layer. The actual reservoir has 28 production layers. They simplified model into 1 layer and 4 layer reservoir. This simplification can significantly decrease the run time of the model. But, the accuracy of results is an important matter which has to be checked. Figure 3.13 shows the inflow performance curve (IPR) for the actual reservoir and a single layer reservoir by averaging the reservoir properties. As observed in this figure, the results of single layer reservoir are inaccurate. The gas flow rate for a specific bottom-hole pressure (BHP) in single layer reservoir is higher than the actual reservoir with 28 layers [130]. They observed that three layers have low permeability which can act as hinder for pressure exchange between other layers. Also, the pressure distribution in the layers between two impermeable layers are same because of their high vertical permeability. They assumed that the layers between each two impermeable layer as one layer with average reservoir properties. Figure 3.14 shows the IPR curve of actual reservoir and the simplified four layer reservoir. As observed in this figure, the IPR curves for both models are showed good agreement. They concluded that with this simplification the accuracy of the model increases and also the run time of the model decreases [130]. Figure 3.15 shows the IPR and TPR curves for their studied reservoir. The intersection of these two curves at a given well flowing pressure and average reservoir pressure shows the optimum operating condition. As observed from this figure, the operating conditions of single layer model are significantly different from the 4 layers reservoir [130].

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7000

Single Layer (Pr=5000 psi) Single Layer (Pr=4000 psi) Single Layer (Pr=3000 psi) Single Layer (Pr=2000 psi) 28-Layer (Pr=5000 psi) 28-Layer (Pr=4000 psi) 28-Layer (Pr=3000 psi) 28-Layer (Pr=2000 psi)

Bottom-hole Pressure (psi)

6000 5000 4000 3000 2000 1000 0 0

200

400

600

800

Gas Flow Rate (MMSCFD) Fig. 3.13 IPR curves for single layer and actual model [130] 7000

4-Layer (Pr=5000 psi) 4-Layer (Pr=4000 psi) 4-Layer (Pr=3000 psi) 4-Layer (Pr=2000 psi) 28-Layer (Pr=5000 psi) 28-Layer (Pr=4000 psi) 28-Layer (Pr=3000 psi) 28-Layer (Pr=2000 psi)

Bottom-hole Pressure (psi)

6000 5000 4000 3000 2000 1000 0 0

200

400

Gas Flow Rate (MMSCFD) Fig. 3.14 IPR curves for 4 layers and actual model [130]

600

800

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6000

Single Layer (Pr=5000 psi) Single Layer (Pr=4000 psi) Single Layer (Pr=3000 psi) Single Layer (Pr=2000 psi) 4-Layer (Pr=5000 psi) 4-Layer (Pr=4000 psi) 4-Layer (Pr=3000 psi) 4-Layer (Pr=2000 psi) TPR (Pwh=3500) TPR (Pwh=2500) TPR (Pwh=1500) TPR (Pwh=600)

Bottom-hole Pressure (psi)

5000

4000

3000

2000

1000

0 0

100

200

300

400

500

600

700

800

900

Gas Flow Rate (MMSCFD)

Fig. 3.15 IPR and TPR curves for single and 4 layer model [130]

3.3.5 Wettability Alteration Towards Gas Wetting of Reservoirs As discussed above, formation and growth of condensate liquids near the wellbore cause operation problems as well as economic disadvantages of losing valuable condensate in reservoir. The most common methods used for eliminating condensate blockage phenomenon include injection of gases like nitrogen, methane, propane, carbon dioxide and gas recycling [131–136], hydraulic fracturing [137], drilling horizontal wells [138] and injection of methanol [139] as a solvent injection. These methods are regularly applied to remove condensate accumulation in near wellbore region. On the other hand, these methods have become less favorable due to high cost and lack of stability in long term. Therefore, finding an economic method with high permanency to eliminate of this common problem is of special importance among reservoir engineers. Changing the wettability of near wellbore region from strong liquid-wet to intermediate-wet or even strong gas-wet is a key strategy in preventing condensate dropout and improving production from gas condensate reservoirs. Therefore, a recent technique which aims at controlling liquid drop out and keeping condensate production from gas condensate reservoirs is wettability alteration of near-wellbore formation towards gas wetting. Liquid repellency refers to the ability of surface to repel liquid (such as water and oil) in the presence of other phase

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[140]. Knowing the fact that trapping condensate in pore spaces near the production well causes low mobility due to the intense liquid-wetting of rock [141], several researchers tried to investigate the effects of wettability alteration using simulation, modelling of the process or experimental approaches. Li and Firoozabadi [142] were the first who changed the wettability of gas condensate reservoirs to gas-wet by implementing chemicals. They used fluorinated surfactants to alter wettability of near wellbore region. Their results showed that permanent gas-wetting state could be established in Berea sandstone and Kansas chalk via chemical treatment. Zhang et al. [143] examined the mobility of gas and liquid phases before and after wettability change. They showed that fluorinated polymers as wettability improvers can present a good level of water and oil repulsion from the surface of rock, convert its wettability to intermediate gas-wet durably, increase gas and liquid phase relative permeabilities and thereby reduce the amount of condensate blockage and increase the productivity of wells in the reservoir condition. Also, Mousavi et al. [141] used fluorinated nano-silica to alter wettability of limestone in the region near wellbore of gas condensate reservoirs from strong liquid-wet to intermediate gas-wet. Mostly, the main purpose of studies involving wettability alteration of gas condensate reservoirs, has been finding the best chemicals to alter wettability of rock surface durably from strong liquid-wet to gas-wet in lab scale [144, 145] and few studies have examined the effect of wettability alteration on production factors in field scale. One of these studies can be cited Zoghbi et al. [146]. They investigated the optimal wettability condition to reach the maximum amount of gas condensate wells productivity through simulation process of production. Also, Sheydaeemehr et al. [147] evaluated three different relative permeability curves that represented three states of wettability in their model. They showed that changing wettability of reservoir rock from liquid-wet to gas-wet and intermediate gas-wet leads to reduce condensate saturation in near wellbore and a significant increase in condensate recovery factor. An example of wettability alteration in field application is the research done by Butler et al. [148], where a research team from Trueblood resources Inc., the University of Texas at Austin and 3M Company used chemicals for wettability alteration purpose on a gas condensate well located in Oklahoma. Reported results showed that near wellbore chemical stimulation could increase gas production rate by a factor of three. In addition, their preliminary field studies indicate that wettability alteration using chemicals can be proposed as a new effective approach while it is economically feasible as just near wellbore is needed to be treated. The most widely used nanoparticles such as TiO2 [149], Al2 O3 [150] and SiO2 [151] alter the rock wettability from oil-wet to water-wet in water/oil system, from condensate-wet to water-wet in condensate/water system [152], and from hydrophilic (water-wet) to hydrophobic in water/air system. However, none of these nanoparticles can alter the contact angle in the oil/air or liquid/air systems. The main reason can be attributed to the surface forces, free energy and the interfacial tension between the phases and surfaces [153, 154]. Many researchers have attempted to achieve wettability alteration towards gas wetting. Basu et al. prepared the superhydrophobic and oleophobic surface coating by Polydimethyl siloxe. The contact angle of water and lubricant oil reached to 158– 160° and 79° on glass side, respectively. Zheng et al. studied wettability alteration of

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glass side using silica nanoparticle modified by polytetrafluoroethylene and the result was increased in contact angle of water to 156°. Also, the heat resistance was studied on water contact angle [154]. Nimittrakoolchai et al. experimentally investigated the effect of silica nanoparticle coated by trichloro (1H,1H,2H,2H-perfluorooctyl) silane group on wettability alteration surface. Their experiments indicated water, EG and seed oil contact angle were 173.2°, 146.7° and 147.6° [155]. Gao et al. modified silica nanoparticle by hexadecyltrimethoxysilane group and coated on cotton and polyester fabrics surface. Their results showed that surfaces were superhydrophobic, so that the contact angle of cotton and polyester fabrics reached from zero to 155° and 143°, respectively [156]. Lakshmi et al. used silica nanoparticle coated by fluoroalkylsilane group to study the wettability alteration. The contact angle of water, EG, oil and n-decane have reached to 157.4°, 153.7°, 126.3° and 63.4° [157]. Li et al. synthesized fluorinated-polyacrylate/silica core shell by emulsion polymerization method and its effect was investigated on wettability alteration of surface [158]. They indicated that the core–shell resulted in superhydrophobic property of glass side. Hsieh et al. synthesized fluorinated titanium dioxide nanoparticle by chemical vapor deposition (CVD) method [159]. They coated fluorinated titanium dioxide nanoparticle on surface and evaluated the effect of nanoparticle on wettability alteration towards hydrophilic to superhydrophobic. Also, superhydrophobic stability of surface was investigated. Results showed that the surface was water repellency in the presence of nanoparticle, so that the contact angle of water was about 166.1° and the superhydrophobic stability was about 60 min. So that the surface was superhydrophobic and oleophobic. Pazokifard et al. synthesized TiO2 nanoparticle coated by 1H,1H,2H,2H-Perfluorooctyltriethoxysilane group [160]. They investigated the effect of pH (2, 6 and 11) on synthesized nanoparticle, and concluded that fluorosilane was adsorbed on surface of TiO2 at neutral (pH = 6) and basic (pH = 11) conditions. Also, Yildirim et al. synthesized SiO2 nanoparticle coated by 1H,1H,2H,2H perfluorooctyltriethoxysilane group and studied its impact on wettability alteration effect of glass side [161]. The water contact angle was 161.6°. Brassard et al. reported SiO2 nanoparticle modified by ethanolilc fluoroalkylsilane group and coated on surface. The contact angle of water was found to be 122165° that showed super hydrophobicity of surface [162]. Mousavi et al. used fluorinated nanosilica to change wettability from strongly condensate-wet to gas-wet in water/condensate/gas systems, in vicinity of condensate gas wells in limestone rock. Their results indicated change in contact angle for water and gas condensate from zero to 124–147 and 50–70, respectively. In similar studies, Ramezani et al. [163], Schaeffer et al. [164] Wang et al. [140], Wang et al. [165] and Valipour Motlagh et al. [166] modified SiO2 nanoparticle by isooctyltrimethoxysilane, tridecafluoro-1,1,2,2-tetrahydrooctyltrichlorosilane, tridecafluorooctyltriethoxysilane, polytetrafluoroethylene and 1H,1H,2H,2H-perfluorodecyltriethoxysilane, respectively [144]. Their results showed alteration of contact angle in water, EG and fuel oil and liquid repellency of surface. Jin et al. prepared SiO2 nanoparticle and included fluoro surfactant (FG 40) and a polymer (FP-2) and investigated changes in contact angle of brine and n-decane.

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Their results indicated that contact angle of brine increased from 23° to 138°, 126° and 150° for surfaces coated by FG40, FP-2 and modified nanosilica FG40 and the contact angle of n-decane altered from zero to 91°, 56°, 127° coated by FG40, FP2 and modified nanosilica FG40, respectively. Recently, Saboori et al. synthesized Fluorine-doped silica and Fluorine-doped silica-coating by fluorosilane nanofluid and tested them on surface of carbonate and sandstone rock samples to study their potential for converting surface properties towards ultrahydrophobic and ultraoleophobic behavior [167]. They studied effect of nanofluid on wettability alteration and stability of ultrahydrophobic and ultraoleophobic properties by measuring contact angles of water, oil, condensate, n-decane and ethylene glycol in air. Their results showed that adding 0.05 wt% of nanoparticle changes contact angle from liquid-wet to strongly gas-wet in all systems. The original contact angle of water, oil, condensate, n-decane and Ethylene glycol were 37.95°, 0°, 0°, 0°, 0° for carbonate rock, which altered to 145.59°, 142.73°, 138.24°, 139.06° and 146.52° after treatment. For sandstone, the original contact angles were 40.40°, 0°, 0°, 0°, 0° for water, oil, condensate, n-decane and ethylene glycol, which altered to 160.01°, 151.40°, 131.85°, 140.27° and 151.7° after treatment [167]. They further tested novel nanoparticles on core samples and compared results of spontaneous imbibition before and after treatment. Imbibition of water, crude oil, and condensate into dry core changed from 0.77, 0.76 and 0.82 PV to lower than 0.10, lower than 0.10 and lower than 0.15 PV for untreated and treated carbonate rocks [168]. In addition, the potential of synthesized nanofluid on enhancement of gas production was studied by Sakhaei et al. [169]. They conducted contact angle, spontaneous imbibition and core flooding experiments to investigate the effect of synthesized nanofluid adsorption on wettability of rock surface and liquid mobility. Results of contact angle experiments revealed that wettability of rock could alter from strongly oil-wetting to the intermediate gas-wetting even at elevated temperature. Imbibition rates of oil and brine were diminished noticeably after treatment, indicating a promising modification of relative permeability towards gas wetting and decreased condensate drop-out around wellbore. They reported 60 and 30% enhancement in pressure drop of condensate and brine floods after wettability alteration with modified nanofluid which confirms successful field potential of this chemical [169]. The time to initiate wettability alteration, radius of treatment, and ultimate state of wettability alteration, i.e. moderate or strong gas wetting are issues to be studied for a candidate reservoir. Sakhaei et al. conducted a simulation study to figure out the role of wettability alteration in a gas reservoir [170]. They studied effects of treatment radius and time to optimize wettability alteration and reach the maximum condensate production. Results indicate that near-wellbore wettability alteration leads to lower critical condensate saturation which has a brilliant impact on improving production parameters and reservoir recovery factors. Also, they found that highest recovery factor in optimal condition is achieved when the wettability state of reservoir rock is altered from strong liquid-wet to intermediate-wet, at the small radius around the production well in early times. The strong gas-wetting, however, showed a reverse effect and reduced the ultimate recovery of both gas and condensate. Based on this study, wettability alteration was shown to influence inflow performance relationship

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(IPR) curves. Curve moves downward by decreasing the average reservoir pressure, meaning that for specific bottom-hole pressure, the well produces a lower rate at lower average reservoir pressure. Also, by altering wettability of near wellbore zone, the IPR curve moves upward in a given average reservoir pressure. That is to say, by improving the wettability, the gas phase relative permeability is increased. So, gas production rate is higher than the base case where no treatment is made. The main reason for this favorable phenomenon is the reduction of adverse effect of condensate blockage and pressure drop.

3.3.6 Gas Flow in Unconventional Reservoirs Unconventional reservoirs and their distribution in the world was presented in Chap. 1. As stated in Chap. 1, the unconventional gas reservoirs have larger reserves in comparison with conventional ones. But, production from these kinds of reservoirs needs specific technologies. Gas flow in unconventional reservoirs is multiscale flow and complex process, unlike the gas flow in conventional reservoirs. The gas flow in these reservoirs is subject to coupled processes, more non-linear, non-linear adsorption/desorption, non-Darcy flow at both high and low flow rate, rock deformation within micro-fractures and nano-pores and strong fluid/rock interaction [171]. Shale gas reservoir is one of the unconventional gas reservoirs which consists of large fraction of nano-pores. An apparent permeability is caused by these nano-pores. This apparent permeability depends on fluid type, pore pressure and pore structure. Guo et al. studied the nitrogen flow through nano membranes. Also, they constructed a new mathematical model for characterizing the gas flow in nano pores. This model is developed based on the advection–diffusion model. Also, they derived a new apparent permeability based on the Knudsen diffusion and advection. Their results show that the model can predict the experimental data with high accuracy [172]. Sun et al. reported that in the presence of organic pores smaller than 2 nm, surface diffusion dominates the transport capacity. In contrast, the larger pore radius gives the stronger transport capacity for inorganic pores. Humidity, stress dependence and gas desorption affect the effective radius of nano scale pores which have significant effects on transport capacity [173]. Slip flow, continuum flow, surface diffusion for adsorbed gas and transition flow of bulk gas are the main mechanisms that coexist in nano pores of shale gas reservoirs. Wu et al. developed a model by coupling an adsorbed gas surface diffusion model and a bulk gas transport model. Their results showed that their model could describe bulk gas transport more accurately in comparison with other models. Also, they reported that surface diffusion has an important role in nano pores with diameter less than 2 nm (as the results reported by Sun et al. [173]). They expressed that in shale gas reservoirs the stress dependence has high impact on fluid flow and it is related to the effective stress, shale matrix mechanical properties and gas transport mechanisms [174].

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Another kind of unconventional gas reservoir is coal-bed methane (CBM) which contain large reserves of methane. Horizontal wells are widely used in development of these reservoirs. The production from these reservoirs may face a period of single phase water production followed by two phase water and gas production transient flow. Many researchers were work on modelling the water and gas flow in CBM reservoirs [175, 176]. Clarkson and Qanbari proposed a methodology for prediction of under-saturated and low permeability CBM wells based on the concept of dynamic drainage area (DDA). Their model results show a reasonable agreement with field data [176]. Variation of coal cleat width due to coal matrix shrinkage and effective stress which causes the variation of Klinkenberg coefficient is another challenge in recovery process of CBM reservoirs. In most of researches, this important matter is ignored by researchers. Wang et al. proposed two improved models, one of these models is under constant effective stress and another is under reservoir conditions. The results of first model show that the proportion of permeability variation is greater than the original model due to Klinkenberg effect. Also, the Klinkenberg coefficient change substantially however, the effect of effective stress is vanished. The second model attached the coal porosity and apparent permeability together. The results of second model show good agreement with field data, in particular when the gas pressure is low [177].

Appendix 3.1: The Viscous Flow Equation in Cylindrical Coordinates The viscous equation is obtained in cylindrical coordinates according to Eq. (3.60) where B = μ1 ddzP . 1 dv d 2v =B + dr 2 r dr

(3.60)

This equation is an inhomogeneous ordinary differential equation because of its non-zero right-hand side. To solve this equation, first, the homogeneous part (B = 0) is solved and the general solution (vg ) is obtained, then the inhomogeneous part (B = 0) is calculated, and the particular solution (vp ) is achieved, and finally, the total answer is resulted as in Eq. (3.61): v(r ) = vg + v p

(3.61)

To solve the homogeneous part, the right-hand side of Eq. (3.60) is considered equal to zero and the general solution (vg ) is calculated. Therefore, Eq. (3.62) is obtained as follows:

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d 2 vg 1 dvg + =0 dr 2 r dr

(3.62)

Using the variation of variables in Eq. (3.63), Eq. (3.62) rewritten as Eq. (3.64): u=

d 2 vg dvg du . = dr dr dr 2 1 du + u=0 dr r

(3.63) (3.64)

By arranging and integrating from Eq. (3.64), Eq. (3.65) is obtained as follows: u=

C1 r

(3.65)

where C 1 is a constant. By combining Eqs. (3.63) and (3.65) and reintegration, Eq. (3.66) is achieved: vg = C1 ln(r ) + C2

(3.66)

where C 2 is constant. Based on the point that the left-hand side of Eq. (3.60) is a second-order differential equation, the particular solution of equation (vp ) is a relationship in the order of two. Therefore, the particular solution of the equation and its derivatives are obtained as Eq. (3.67): v p = ar 2 + br + c ·

d 2v p dv p = 2ar + b · = 2a dr dr 2

(3.67)

where a, b, and c are constant. Using Eq. (3.67), Eq. (3.60) can be rewritten as Eq. (3.68): 1 b (2a) + (2ar + b) − B = 2a + 2a + − B = 0 r r

(3.68)

Assuming b = 0 in Eq. (3.68), the relationship of constants B and a results as Eq. (3.69): a=

B 4

(3.69)

Therefore, assuming c = 0 in Eq. (3.67), the particular solution of the differential equation is obtained as Eq. (3.70): vp =

B 2 r 4

(3.70)

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The overall solution of the differential equation is expressed as Eq. (3.71) using Eqs. (3.61), (3.66) and (3.70): v=

B 2 r + C1 ln(r ) + C2 4

(3.71)

Boundary conditions are employed to calculate the constants C 1 and C 2 . Applying the boundary conditions of Eqs. (3.72) and (3.73) in Eq. (3.71), these constants are calculated according to Eq. (3.74): r = ±R, v = 0

(3.72)

dv =0 dr

(3.73)

r = 0,

C1 = 0.C2 = −

B 2 R 4

(3.74)

Therefore, by substitution of Eq. (3.74) and B value in Eq. (3.71), the viscous flow velocity profile equation in a cylinder can be obtained as Eq. (3.2). v=

 1 dp  2 R − r2 4μ dz

To calculate the average value of a function in cylindrical coordinates, Eq. (3.75) is used. R vr dr v = 0 R (3.75) 0 r dr Therefore, by substituting Eq. (3.2) in Eq. (3.75), the value of the average velocity can be calculated according to Eq. (3.76). 1 dp v= 4μ dz

 R   4   2 2 1 dp R2 − 0 r R − r dr = R R2 4μ dz 2 0 r dr

R4 4

 (3.76)

By simplifying Eq. (3.76), the viscous flow average velocity equation in a cylinder is obtained as Eq. (3.3). v = 0.125

R 2 dp μ dz

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Appendix 3.2: The Brinkman Equation in Cylindrical Coordinates Brinkman equation in cylindrical coordinates is known as Eq. (3.77): r2

d 2v dv − Ar 2 v = r 2 B +r 2 dr dr

(3.77)

In Eq. (3.77), parameter B is equal to the constant value of μ1 ddzP , and parameter A is equal to K −1 . This equation is an ordinary inhomogeneous differential equation. To solve this equation, both the general (vg ) and the particular (vp ) solutions which respectively indicate the homogeneous (r 2 B = 0) and inhomogeneous part (r 2 B = 0) are calculated. According to Eq. (3.61), the sum of the particular and general solutions is equal to the total solution of the equation. To calculate vg , first Eq. (3.77) is homogenized as Eq. (3.78): r2

d 2v dv − Ar 2 v = 0 +r dr 2 dr

(3.78)

The answer of Eq. (3.78) with its similarity to the modified Basel differential equation (reference) is obtained as Eq. (3.79):  √   √  v g = C 1 I0 r A + C 2 K 0 r A

(3.79)

where I 1 and K 1 are the Modified Bessel functions of the first and second kind in the zero-order. According to the point that the left-hand side of Eq. (3.77) is a second-order differential equation, the particular solution of equation (vp ) is in the same order. In this case, the particular solution and its derivatives can be assumed as Eq. (3.80): v p = ar 2 + br + c ·

d 2v p dv p = 2ar + b · = 2a dr dr 2

(3.80)

where a, b, and c are constant. Therefore, Eq. (3.77) can be rewritten as Eq. (3.81) using Eq. (3.80).   r 2 (2a) + r (2ar + b) − Ar 2 ar 2 + br + c = Br 2

(3.81)

According to Eq. (3.79), it can be seen that for the validity of this equation, the values a and b must be equal to zero (a = b = 0) and the value of c must be equal to –B/A (c = −B/A). Therefore, by placing these constants in Eq. (3.80), the particular solution can be calculated as Eq. (3.82).

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vg = −

B A

(3.82)

Finally, providing particular and general solutions and also considering Eq. (3.61), the velocity function is calculated as Eq. (3.83).  √   √  B v(r ) = C1 I0 r A + C2 K 0 r A − A

(3.83)

Boundary conditions are employed to calculate the constants C 1 and C 2 in Eq. (3.81). By applying the boundary conditions of Eq. (3.73) in Eq. (3.83), Eq. (3.84) can be written as follows:  √   √  √ √ (3.84) C1 AI1 0 ∗ A − C2 AK 1 0 ∗ A = 0 According to the Bessel function properties, the values I 1 (0) and K 1 (0) are equal to zero and infinity, respectively. Therefore, for the validity of Eq. (3.82), the value of C 2 must be equal to zero (C 2 = 0). Also, by applying the boundary conditions of Eq. (3.72) and the value of C 2 in Eq. (3.83), Eq. (3.85) is achieved. By solving Eq. (3.85), the C 1 constant results in Eq. (3.86).  √  B C 1 I0 R A − = 0 A C1 =

(3.85)

B  √  AI0 R A

(3.86)

Finally, by placing the constants C 1 and C 2 as well as the values A and B in Eq. (3.83), the velocity function is calculated as Eq. (3.87). ⎛ v(r ) =

 I0

√r K

⎞

K dP⎝ 1 −  ⎠ μ dz I0 √RK

(3.87)

To calculate the average value of a function in a cylindrical system Eq. (3.75) is used. Therefore, by placing Eq. (3.140) in Eq. (3.75), Eq. (3.88) is obtained as follows: ⎛  √   R ⎞   ⎞ ⎛ R R √r √r r K I r I 1 0 r dr 0 K dP⎝ 0 K dP⎜ K K 0 ⎟ ⎠   1 − v= − = ⎝ R  2 R   ⎠ R R μ dz μ dz r R √ 0 r dr I0 √ K 0 r dr I0 K 2 0

(3.88)

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where I 1 is the modified Bessel function of the first kind in the order of one. By solving and simplification of Eq. (3.88), the average velocity function for the Brinkman flow in a cylindrical system is calculated as Eq. (3.89).  ⎞ √ √R K I 2 1 dP K ⎝ K   ⎠ v= 1− R dz μ R I0 √ K ⎛

(3.89)

Appendix 3.3: Linear Diffusivity Equation Solution This solution is provided by John Lee for radial flow solution in an infinite-acting homogenous reservoir [178]. The basic partial differential equation is given in dimensionless format ∂ 2 pD 1 ∂ pD ∂ pD + = r D ∂r D ∂t D ∂r D2

(3.90)

where rD = p D = p DC

r rw

kh ( pi − pr ) q Bμ

t D = t DC

k t QμCt rw2

(3.91) (3.92) (3.93)

where t DC and p DC are given by The “initial” condition is given as p D (r D · t D = 0) = 0 (uniform pressure distribution)

(3.94)

The constant rate inner boundary condition is  ! ∂ pD rD = −1 (constant flow rate at the well) ∂r D r D =1

(3.95)

The “infinite-acting” outer boundary condition is given by p D (r D → ∞ · t D ) = 0

(3.96)

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Rewriting Eq. (3.90):  ! ∂ pD ∂ pD 1 ∂ rD = r D ∂r D ∂r D ∂t D

(3.97)

The Boltzmann transform variable, ε D , is defined as t D = ar Db t Dc

(3.98)

where in this problem a = 1/4, b = 2, c = −1. Which yields εD = Expanding

r D2 4t D

(3.99)

∂ 2 pD ∂r D2

 ! 1 ∂ pD ∂ ∂ pD ∂ pD + = ∂r D ∂r D r D ∂r D ∂t D

(3.100)

Applying the chain rule ∂t D ∂ p D ∂ pD = ∂x ∂ x ∂t D

(3.101)

Which combined with Eq. (3.100) gives  ! ∂ε D ∂ ∂ε D ∂ p D 1 ∂ε D ∂ p D ∂ε D ∂ p D + = ∂r D ∂ε D ∂r D ∂ε D r D ∂r D ∂ε D ∂t D ∂ε D

(3.102)

Expanding    !  ! ∂ε D ∂ p D 1 ∂ε D ∂ε D ∂ 2 p D ∂ε D ∂ p D ∂ε D ∂ + + − =0 ∂r D ∂ε D ∂r D ∂ε D ∂r D ∂ε2D r D ∂r D ∂t D ∂t D

(3.103)

Isolating terms 

∂ε D ∂r D

!2

  !  ∂ε D 1 ∂ε D ∂ 2 pD ∂ε D ∂ ∂ε D ∂ p D + + − =0 ∂r D ∂ε D ∂r D r D ∂r D ∂t D ∂t D ∂ε2D

Dividing through by (∂ε D /∂r D )2 gives

(3.104)

3 Basics of Oil and Gas Flow in Reservoirs

   ! ∂ε D ∂ ∂ε D 1 ∂ε D ∂ 2 pD 1 ∂ε D ∂ p D + + − =0 r D ∂r D ∂t D ∂t D ∂ε2D (∂ε D /∂r D )2 ∂r D ∂ε D ∂r D

123

(3.105)

Reducing the ∂()/∂ε D term we have    ! ∂ ∂ε D 1 ∂ε D 1 ∂ε D ∂ p D ∂ 2 pD + + − =0 r D ∂r D ∂t D ∂t D ∂ε2D (∂ε D /∂r D )2 ∂r D ∂r D Which can be further reduced to  2 ! ∂ 2 pD ∂ εD 1 1 ∂ε D ∂ε D ∂ p D + + − =0 r D ∂r D ∂t D ∂t D ∂ε2D (∂ε D /∂r D )2 ∂r D2

(3.106)

(3.107)

Completing the factorization of the (∂ε D /∂r D )2 gives !  ∂ 2εD 1 ∂ε D ∂ p D 1 1 1 ∂ 2 pD + + − =0 r D ∂ε D /∂r D ∂ε2D (∂ε D /∂r D )2 ∂r D2 (∂ε D /∂r D )2 ∂t D ∂t D (3.108) Using Eq. (3.99) the following derivation is taken

∂ 2εD ∂r D2

 2    rD r D2 ∂ 1 −1 r D2 ∂ε D ∂ −1 = = = = εD (3.109) ∂t D ∂t D 4t D 4 ∂t D t D t D 4t D tD  2  rD 2 ∂ε D ∂ 2 r D2 2 = = rD = = tD (3.110) ∂r D ∂r D 4t D 4t D r D 4t D rD   2 !   ∂ rD ∂ 2 2 ∂ 2 r2 2 = = rD = = 2 D = 2 t D (3.111) ∂r D ∂r D 4t D ∂r D 4t D 4t D r D 4t D rD

Substituting Eqs. (3.109)–(3.111) into Eq. (3.108) gives  !  −1 ∂ pD 1 2 1 1 1 ∂ 2 pD − + ε + ε =0 D D 2 2 2 2 r D (2ε D /r D ) (∂ε D /∂r D ) tD ∂ε D ∂ε D (2ε D /r D ) r D (3.112) Reducing  ! 1 1 ∂ pD ∂ 2 pD + + + 1 =0 2ε D 2ε D ∂ε D ∂ε2D

(3.113)

!  1 ∂ pD ∂ 2 pD + 1 + =0 ε D ∂ε D ∂ε2D

(3.114)

Finally

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where Eq. (3.114) is the “Boltzmann” transformed differential equation. Initial and boundary condition in terms of Boltzmann transform are p D (r D · t D = 0) = 0

(3.115)

where for t D → 0; ε D → ∞, which gives p D (ε D → ∞) = 0

(3.116)

The outer boundary condition, Eq. (3.94), p D (r D → ∞.t D ) = 0

(3.117)

Or as r D → ∞; ε D → ∞ which yields p D (ε D → ∞) = 0

(3.117)

where Eqs. (3.116) and (3.118) are the same which illustrates that the Boltzmann transform “collapses” 2 conditions into 1. Combining this observation with the inner boundary condition, we have 2 “boundary” conditions. Coupling this observation with the fact that Eq. (3.114) is only a function of the Boltzmann variable, ε D , we can solve Eq. (3.100) uniquely. Note that the “collapsing” of the initial and outer boundary conditions must occur or the Boltzmann transform is technically invalid. Recalling the constant rate inner boundary condition, Eq. (3.95)  ! !  ∂ pD ∂ pD rD + = −1 or r D + = −1 (line source condition) ∂r D r D =1 ∂r D r D →0 (3.119) Or  !  ! !    2 ∂ pD ∂ε D ∂ p D ∂ pD rD = rD εD = 2 εD + = −1 ∂r D ∂ε D r D →0 rD ∂ε D r D →0,ε D →0 ∂ε D ε D →0 (3.120) Which can be rearranged to yield  εD

∂ pD ∂ε D

! ε D →0

=

−1 2

(3.121)

Making the following variable of substitution v=

dp D dε D

(3.122)

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Substituting Eq. (3.122) into Eq. (3.124) and noting that use of ordinary derivatives !  1 dv v=0 + 1+ dε D εD !  1 1 1 dε D = −dε D − dv = − 1 + dε D v εD εD

(3.123) (3.124)

Integrating ln(v) = −ε D − ln(ε D ) + β; β = constant of integration

(3.125)

Exponentiation v = exp[−ε D − ln(ε D ) + β]

(3.126)

 "  " v = exp −ε D exp − ln(ε D ) exp[β]

(3.127)

Or

Which reduces to v=

α exp[−ε D ] εD

(3.128)

where α = exp[β], i.e., the constant of integration. Recalling Eq. (3.120) and combining gives α dp D = exp[−ε D ] dε D εD

(3.129)

Multiplying through by ε D gives εD

dp D = αexp[−ε D ] dε D

(3.130)

Substitution of Eq. (3.128) into Eq. (3.119) gives "  −1 α lim exp[−ε D ] = ε D →0 2

(3.131)

α = −1/2

(3.132)

Or

Substitution of Eq. (3.132) into Eq. (3.129) gives

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dp D −1 = exp[−ε D ] dε D 2ε D

(3.133)

Separating and integrating Eq. (3.133) gives p D p D =0

−1 dp D = 2

ε D ε D =0

1 −ε D e dε D εD

(3.134)

where we note that p D = 0 at ε D = ∞ is the initial outer boundary condition. Completing the integration and reversing the limits we have ∞

1 pD = 2

1 −y e dy y

(3.135)

r2 ε D = 4tD D

We note that the integral in Eq. (3.135) is the exponential integral, E1 (x), which is given by ∞ E1 (x) =

1 −y e dy y

(3.136)

X

Combining Eqs. (3.135) and (3.136) gives our final result p D (r D , t D ) =

 2  r 1 E1 D 2 4t D

(3.137)

Appendix 3.4: Solution of Non-linear Diffusivity Equation The analytical solution of non-linear diffusivity equation needs changing variables in order to perform linear equation to be solved analytically.  2 ∂p ∂2 p 1 ∂ p ∅μc ∂ p + c + = 2 ∂r r ∂r ∂r kt ∂t p = p − pi · T =

1 kt and p = ln p ∗ ∅μc c

The linearized form is presented as [179]:

(3.138) (3.139)

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∂ p∗ ∂ 2 p∗ 1 ∂ p∗ = + ∂r 2 r ∂r ∂T

(3.140)

The linear equation can be solved using Laplace transform and concludes:  p =

⎛ ⎞       −1 ⎝ 0.5772λ T  ⎠ + ln 1 + λ ln 2 + 0.2319 c rw 1 + λ lnT + 0.2319 2 r w

(3.141) λ=

qμc 4π kh

(3.142)

where, h is formation thickness.

Appendix 3.5: The Solution of the Warren-Root Equation in the Different Boundary Condition Warren-Root Equations Reservoir pressure distribution is achieved according to the mathematical model. Different reservoir boundary conditions can be applied to this end. The WarrenRoot model is used to calculate pressure distribution in fractured reservoirs. Equations (3.28) and (3.29) are used for fluid velocity through matrix and fractures media, respectively:  m = − Km grad(Pm ) U μ  f = − K2 grad(Pf ) U μ Subscripts m and f denote to matrix and fracture media, respectively. K is absolute permeability in (m2 ), U is velocity in (ms−1 ), P is pressure in (Pa), and μ is dynamic viscosity in (Pa s). To calculate matrix and fracture pressure distribution, the mass conservation equation is formulated as follows:   ∂(∅m ρ) + div ρUm + U∗ = 0 ∂t

(3.143)

  ∂(∅f ρ) + div ρUf + U∗ = 0 ∂t

(3.144)

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U∗ =

ρ SKm (Pm − Pf ) μ

(3.145)

The rate of mass flow per unit volume is defined by U * in (kg s−1 m−3 ), which indicates fluid transfer between matrix and fracture in quasi-steady-state condition. U is fluid velocity in (ms−1 ), and ϕ is porosity, which represents fluid storage capacity in each media. S is a characteristic coefficient of fractured rock proportional to the specific surface of a block, and ρ is fluid density in (kg m−3 ). For slightly compressible fluids, density is calculated by Eq. (3.146). ρ = ρ0 (1 + CP)

(3.146)

where C is fluid compressibility in (Pa−1 ), and indicates dependency of fluid volume on the pressure. Equations (3.147) and (3.148) are derived by combining Eqs. (3.143) to (3.146). SKm ∂Pm + (Pm − Pf ) = 0 ∂t μ    SKm ∂Pf Kf 1 ∂ ∂Pf r + ∅f Cf − (Pm − Pf ) = 0 ∂t μ r ∂r ∂r μ ∅m Cm

(3.147) (3.148)

Dimensionless variables in Eqs. (3.149)–(3.153) are used to reduce the number of variables. PD =

2πKf h(Pi − P(r, t)) qμ

(3.149)

r rw

(3.150)

Kf t (Cm ∅m + Cf ∅f )μr2w

(3.151)

αKm r2w Kf

(3.152)

∅f Cf ∅m Cm + ∅f Cf

(3.153)

rD = tD =

λ= ω=

Dimensionless pressure is defined by PD , r D is dimensionless radius, and t D is dimensionless time. Equations (3.152) and (3.153) represent fracture reservoir parameters, which indicate interporosity (λ) and fracture storage capacity (ω). At the initial time, there is a pressure equilibrium state in the reservoir. Overall reservoir pressure is equal to reservoir initial pressure in this condition. Therefore, at the initial times, Eq. (3.154) is used as follows:

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P = Pi t = 0

(3.154)

Dimensionless pressure at initial time is resulted by a combination of Eqs. (3.149) and (3.154). PDm = 0 tD = 0

(3.155)

Equations (3.147) and (3.148) are rewritten in dimensionless forms by combining Eqs. (3.149)–(3.153) as follows: ∂PDm − λ(PDf − PDm ) = 0 ∂tD   ∂PDm ∂PDf 1 ∂ rD + (1 − ω) + =0 rD ∂rD ∂rD ∂tD

(1 − ω) −ω

∂PDf ∂tD

(3.156) (3.157)

Equation (3.157) is rewritten as Eq. (3.158): 1 ∂PDf ∂PDm ∂PDf ∂P2Df + = (1 − ω) +ω ∂ 2 rD rD ∂rD ∂tD ∂tD

(3.158)

There are several methods for solving partial differential equations. Laplace transform is one of these methods. Laplace transform is applied to solve differential Eqs. (3.156) and (3.158). Consequently Eq. (3.159) is resulted by Eq. (3.156) Laplace transform: (1 − ω)zPDm (z, rD ) − PDm (t = 0) = λ(PDf (z, rD ) − PDm (z, rD ))

(3.159)

The Laplace transform variable is known as z. By substituting Eq. (3.159) in the initial condition (Eq. (3.155)), matrix dimensionless pressure is resulted as the following equation: PDm (z) =

λPDf (z · rD ) (1 − ω)z + λ

(3.160)

To obtain the fracture pressure equation, the Laplace transform of Eq. (3.158) is rewritten as Eq. (3.161). ∂ 2 PDf (z · rD ) 1 ∂PDf (z.rD ) + = (1 − ω)zPDm (z.rD ) + ωzPDf (z.rD ) rD ∂rD ∂r2D

(3.161)

Equation (3.160) is replaced in Eq. (3.161) and Eq. (3.162) is resulted as follows: d2 PDf (z.rD ) 1 dPDf (z.rD ) + = [zf(z)] PDf (z, rD ) rD drD dr2D

(3.162)

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In Eq. (3.162), f(z) is a function of ω and λ and is described as Eq. (3.31): f(z) =

ω(1 − ω)z + λ (1 − ω)z + λ

The solution of the partial differential Eq. (3.162) is necessary to calculate fracture dimensionless pressure. Therefore, r D 2 is multiplied by Eq. (3.162) and Eq. (3.163) is obtained as a standard form of Bessel equation. r2D

 d2 PDf (z.rD ) dPDf (z.rD )  2 + rD − rD zf(z) PD2 (z.rD ) = 0 2 drD drD

(3.163)

Therefore, the solution of Eq. (3.163) is written as Eq. (3.30) by I 0 and K 0 parameters, which are the first and second type of modified Bessel functions in zero-order, respectively. PDf (z.rD ) = C1 I0 (MrD ) + C2 K0 (MrD ) Bessel equation constants are introduced by C 1 and C 2 , and boundary √ conditions are applied to obtain these values. To simplify calculations, M = zf(z) is considered. Constants of Eq. (3.30) are calculated by different boundary conditions substitution, and pressure distribution equations are obtained as a result.

Constant Production Rate in Closed Outer Boundary Well production rate is kept constant in constant production rate conditions. A choke valve is usually used to stabilize the production rate in operating conditions. This case is more common in hydrocarbon reservoirs production. The wellbore boundary condition is written as Eq. (3.164) in this case: 

∂P ∂r

 = constant r = rw

(3.164)

rw

Some hydrocarbon reservoirs are confined with faults or impermeable layers. No fluid flows through the outer boundary in these closed reservoirs. Also, for a reservoir with several production wells, the closed boundary can be assumed among the wells. The closed boundary conditions are expressed in Eq. (3.165): 

∂P ∂r

 = 0 r = re

(3.165)

re

Darcy equation is applied to make a relationship between flow and pressure for an inner boundary condition (the well condition) as Eq. (3.166). In this case, C 1 and

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131

C 2 are calculated by replacing the boundary conditions (Eqs. (3.164) and (3.165)) in Eq. (3.30). q = −2πrw h

Kf ∂Pf r = rw μ ∂r

(3.166)

Equation (3.165) is written for closed boundary condition and dimensionless form of Eqs. (3.165) and (3.166) is resulted as Eqs. (3.168) and (3.167), respectively. ∂PDf = −1 rD = 1 ∂rD

(3.167)

∂PDf = 0 rD = rDe ∂r D

(3.168)

By differentiating Eq. (3.30) and integrating with Eq. (3.167), results can be represented as the following equations: C1 M I1 (M rD ) − C2 M K1 (M rD ) = C1 MI1 (M ) − C2 M K1 (M ) =

∂PDf ∂rD

−1 z

(3.169) (3.170)

The first and second types of modified Bessel functions in the first order are known as I 1 and K 1 . Equation (3.171) is resulted by differentiating Eq. (3.30) and integrating with Eq. (3.168). C1 M I1 (M rDe ) − C2 M K1 (M rDe ) = 0

(3.171)

A system of two linear Eqs. (3.170) and (3.171) is formed to calculate C 1 and C 2 : C1 =

−K1 (M rDe ) z[(K1 (M rDe )) × M I1 (M ) − I1 (M rDe ) × M K1 (M )]

(3.172)

C2 =

−I1 (MrDe ) z[K1 ( M rDe ) × M I1 (M ) − I1 (M rDe ) × M K1 (M )]

(3.173)

Equation (3.32) is resulted by substitution C 1 and C 2 in Eq. (3.30) as follows: PDf (z.rD ) =

K1 (M rDe )I0 (M rD ) + I1 (M rDe )K0 (M rD ) zM[K1 (M)I1 (M rDe ) − K1 (M rDe )I1 (M)]

Therefore, the dimensionless pressure equation in a closed fractured reservoir and constant production rate is given by Eq. (3.32).

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Constant Production Rate in Constant Pressure Outer Boundary For infinite reservoirs with high-pressure supplier, there is no pressure drop at the reservoir outer boundary, and the outer boundary pressure equals to reservoir initial pressure, as shown by Eq. (3.174). The dimensionless form of this equation is presented as Eq. (3.175). Pe = Pi r = re

(3.174)

PDm = PDf = 0 rD = rDe

(3.175)

Equation (3.175) is replaced in Eq. (3.30) and Eq. (3.176) is resulted as follows: C1 I0 (M rDe ) + C2 K0 (M rDe ) = 0

(3.176)

A system of two linear Eqs. (3.170) and (3.176) is formed to calculate C 1 and C 2 in this case: K0 (M rDe ) z M[K0 (M rDe )I1 (M ) + I0 (M rDe )K1 (M )]

(3.177)

I0 (M rDe ) z M[K0 (M rDe )I1 (M ) + I0 (M rDe )K1 (M)]

(3.178)

C1 = − C2 =

Equation (3.33) is resulted by replacing C 1 and C 2 in Eq. (3.30). PDf (z.rD ) =

K0 (M rD )I0 (M rDe ) − K0 (M rDe )I0 (M rD ) zM[K0 (M rDe )I1 (M) + K1 (M)I0 (M rDe )]

Equation (3.33) is applicable for dimensionless flow rate calculation at constant pressure boundary in case of constant pressure production condition.

Constant Pressure Production in Closed Outer Boundary The wellbore pressure is kept constant, and flow rate changes i.e., it decreases with time in constant pressure production condition. In operating conditions, well pressure is maintained constant by using some gauges that alter the production rate. This production condition is mostly used in reservoirs with a high risk of gas or water coning. The wellbore boundary condition, in this case, is written as Eq. (3.179): P = constant r = rw

(3.179)

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Production rate equation is obtained using the wellbore pressure equation suggested by Van Everdingen and Hurst [180]: qD (z.rD ) =

1 z2 PDwf (z, rD )

(3.180)

The dimensionless production equation (Eq. (3.34)) is obtained by combining Eqs. (3.32) and (3.180). qD (z.rD ) =

M[K1 (M)I1 (M rDe ) − K1 (M rDe )I1 (M)] z[K1 (M rDe )I0 (M rD ) + I1 (M rDe )K0 (M rD )]

The dimensionless flow rate at a closed boundary fractured reservoir with constant pressure production is calculated by Eq. (3.34).

Constant Pressure Production in Constant Pressure Outer Boundary In this case, the inner and outer boundary conditions are described by Eqs. (3.179) and (3.174). The Van Everdingen and Hurst relations are applied as the same as the previous section. Equation (3.35) is concluded by combining Eqs. (3.33) and (3.180): qD (z.rD ) =

M[K0 (M rDe )I1 (M) + K1 (M)I0 (M rDe )] z[K0 (M rD )I0 (M rDe ) − K0 (M rDe )I0 (M rD )]

Equation (3.35) is applicable for dimensionless flow rate calculations at constant pressure reservoir boundary in the case of constant pressure production.

Warren-Root Analytical Solution The original solution provided by Warren and Root considers infinite outer boundary conditions and also the constant production rate for an inner boundary condition. The inner boundary condition is presented as Eq. (3.164) and the outer boundary condition is defined as Eq. (3.181): P = Pi r = ∞

(3.181)

This assumption is applied to constant production rate conditions [65]. This simplification is applied to Eq. (3.32). The infinite values of the first and second-order Bessel function tend to infinite and zero, respectively. Thus, Eq. (3.36) is a simplified

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form of Eq. (3.32) which is resulted by using the Warren-Root assumptions: PD (z.rD ) =

K0 (M rD ) zM K1 (M)

The Laplace inverse calculation of Eq. (3.36) is not possible by conventional analytical methods. An approximate method for calculating K 0 and K 1 is possible by considering primary terms of Bessel functions: K0 (M) = −γ − Ln K1 (M) =

r

D

2

M2



1 M2

(3.182) (3.183)

In Eq. (3.182), 7 is Euler number and equals 0.5772. This approximation is applicable only for M < 0.01 values. Also, r D = 1 is assumed in the above equations. Equation (3.184) is obtained by substituting Eqs. (3.182) and (3.183) in Eq. (3.36) and dimensionless pressure is calculated by numerical inverse Laplace method, as follows:     −λtD 1 −λtD − Ei (3.184) PD (tD .1) = [0.80908 + Ln (tD ) + Ei 2 ω(1 − ω) (1 − ω) Exponential integral, which is called Ei function, is defined as Eq. (3.185): ∞ Ei(−X) = −

e−u du u

(3.185)

X

The Ei function values limit to zero at a very long time. Therefore, Eq. (3.184) is converted to Eq. (3.37): PD (tD .1) =

1 1 [0.80908 + Ln(tD )] = Ln(2.25tD ) = 1.15 log(2.25tD ) 2 2

Therefore, Warren and Root solved the inverse Laplace of dimensionless pressure equation in a fractured reservoir with simple assumptions in specific conditions (M < 0.01). The solution provided by Warren and Root is a simplified form of the general solution shown in Eq. (3.32) for M < 0.01 at wellbore (r D = 1) [63].

References 1. Bird RB, Stewart WE, Lightfoot EN. Transport phenomena, 2nd ed. (2002) n.d. 2. Brown GO. Henry Darcy and the making of a law. Water Resour Res. 2002;38:11.

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170. Sakhaei Z, Mohamadi-Baghmolaei M, Azin R, Osfouri S. Study of production enhancement through wettability alteration in a super-giant gas-condensate reservoir. J Mol Liq. 2017;233:64–74. 171. Wu Y-S, Li J, Ding D, Wang C, Di Y. A generalized framework model for the simulation of gas production in unconventional gas reservoirs. SPE J. 2014;19:845–57. 172. Guo C, Xu J, Wu K, Wei M, Liu S. Study on gas flow through nano pores of shale gas reservoirs. Fuel. 2015;143:107–17. 173. Sun Z, Li X, Shi J, Zhang T, Sun F. Apparent permeability model for real gas transport through shale gas reservoirs considering water distribution characteristic. Int J Heat Mass Transf. 2017;115:1008–19. 174. Wu K, Chen Z, Li X, Guo C, Wei M. A model for multiple transport mechanisms through nanopores of shale gas reservoirs with real gas effect–adsorption-mechanic coupling. Int J Heat Mass Transf. 2016;93:408–26. 175. Sun Z, Shi J, Wang K, Miao Y, Zhang T, Feng D, et al. The gas-water two phase flow behavior in low-permeability CBM reservoirs with multiple mechanisms coupling. J Nat Gas Sci Eng. 2018;52:82–93. 176. Clarkson CR, Qanbari F. A semi-analytical method for forecasting wells completed in low permeability, undersaturated CBM reservoirs. J Nat Gas Sci Eng. 2016;30:19–27. 177. Wang G, Ren T, Wang K, Zhou A. Improved apparent permeability models of gas flow in coal with Klinkenberg effect. Fuel. 2014;128:53–61. 178. Lee J. Well testing. Society of Petroleum Engineers; 1982. 179. Odeh AS, Babu DK. Comparison of solutions of the nonlinear and linearized diffusion equations. SPE Reserv Eng. 1988;3:1–202. 180. Van Everdingen AF, Hurst W. The application of the Laplace transformation to flow problems in reservoirs. J Pet Technol. 1949;1:305–24. https://doi.org/10.2118/949305-G.

Chapter 4

Gas Injection for Underground Gas Storage (UGS) Reza Azin and Amin Izadpanahi

Abstract In this chapter, some of the key aspects of UGS as a sustainable energy supply infrastructure were reviewed. This type of gas injection is associated with the reuse of depleted oil and gas formations as a natural underground storage volumes with proved reservoir characteristics and cap rock integrity, as well as available surface facilities and wells. The operation has its distinct features in terms of planning, gas injection/withdrawal, and etc. Some of the common criteria for screening the UGS were also reviewed and discussed in this chapter.

Abbreviations Acronyms CHR CPE GE I/W OGIP

Condensate holding ratio Condensate production enhancement Gas equivalent Injection/withdrawal Original gas in place

R. Azin (B) Faculty of Petroleum, Gas and Petrochemical Engineering, Department of Petroleum Engineering, Persian Gulf University, Bushehr, Iran e-mail: [email protected] A. Izadpanahi Oil and Gas Research Center, Persian Gulf University, Bushehr, Iran e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 R. Azin and A. Izadpanahi (eds.), Fundamentals and Practical Aspects of Gas Injection, Petroleum Engineering, https://doi.org/10.1007/978-3-030-77200-0_4

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Variables B bpss c c1 c2 ce G h K k Mw N P P PD PDi Pp Ppwf Pr Pw q Qp Rep re rw S T tca tDA tp V W x y Z Z

Formation volume factor, reservoir conditions/standard conditions Pressure loss due to the steady-state inflow of gas, psia/ft3 /day Compressibility, 1/psi Matrix pore volume compressibility, 1/psi Fracture pore volume compressibility Effective compressibility, 1/psi Gas volume, scf Net pay thickness, ft K-Value Permeability, md Molecular weight, g/gmol Condensate volume, STB Average drainage area pressure, psia Pressure, psia Dimensionless pressure Dimensionless pressure integral derivative Pseudopressure, psia2 /cp Dimensionless bottom-hole pseudopressure Pressure in the reservoir at the drainage radius, psia Bottomhole pressure, psia Flow rate, ft3 /day Production rate of a given phase, SCF/D Drainage radius, ft Exterior radius, ft Wellbore radius, ft Saturation Temperature, F Material balance pseudotime for gas, day Dimensionless time Production time, day Volume, acre-ft Water Volume, bbl Liquid phase mole fraction Gas phase mole fraction Average gas compressibility factor Gas compressibility factor

Greek Letters φ

Porosity

4 Gas Injection for Underground Gas Storage (UGS)

φ1 φ2 γ μ

145

Matrix void space to bulk volume Fracture void space to bulk volume Specific gravity Viscosity

Subscripts b e f g i inj ng p sc so sw t w

Bulk Influx Formation Gas Initial Injected Non-gas Produced Standard condition Sour gas injection Sweet gas injection Total Water

4.1 Introduction Natural gas has now become one of the world’s main energy source in only less than two decades. According to international energy prospect scenario, the global natural gas consumption during years 2001–2025 will experience an average growth rate of 2.9–3.2% per year which is comparable to an annual growth rate of 1.8% for oil and 1.5% for coal [1]. Countries must plan and consider further investments to guarantee a constant rate of natural gas supply for the coming years. Worldwide resources of natural gas and the need for clean and low GHG emission fuel has made a shift towards consumption of natural gas. Underground gas storage (UGS) projects are becoming an essential part of gas and market chain in countries to resolve seasonal demand and mitigate pick shaving problems efficiently. UGS plays a crucial role in energy management, balancing demand and supply chain as well as regulating the gas price. This chapter will review the key features of UGS technology as a special application of gas injection.

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4.2 The Concept of UGS Underground gas storage (UGS) is defined as the storage of large quantities of natural gas in a storage formation in order to support the natural gas demand in domestic, commercial, industrial, or space heating as the most crucial application especially in winter. The strong trend towards increasing the number of UGS facilities is observed in gas importing and hub countries an economical, safe and environment-friendly technology operating tool. Generally, UGS includes two steps of injection and withdrawal (I/W) in each cycle. In the injection phase, natural gas is injected into subsurface formation when market demand drops below the supply available from the pipeline. Natural gas is then withdrawn out of storage reservoir to provide a steady supply from the pipeline when demand exceeds supply. Some UGS reservoirs are designed and operated to provide seasonal gas to designated and predictably constant markets. They are called base load types of reservoirs. Other UGS projects are designed to respond only at extreme demand for gas, called peak shaving reservoirs. There are three basic requirements attributed to the performance of UGS reservoirs, including verification of inventory, assurance of deliverability, and containment against migration [2]. The inventory presents existing gas in the storage reservoir. It is made up of base gas (or cushion gas) and top gas (or working gas). Deliverability refers to the ability of UGS field to deliver gas to its dedicated market and depends on the equalized pressure prevailing underground. Since the pressure is a function of the amount of gas in the storage container, it follows that deliverability is a function of inventory. If the container does not hold the gas, it becomes subject to the attrition of its inventory through the migration of gas. Based on the source of gas and final destination of stored gas, the UGS may be operated in the production zone, hub, or consumption zone. The UGS in production zone involves long-term gas storage into depleted gas reservoirs located in the gas production zone. This type of gas injection is of special importance for rapid development and depletion of gas reservoirs shared between two or more countries. Unless there is a valid agreement between holding countries, shared reservoirs are subject to depletion by each country independently. At these conditions, there is a possibility for unbalanced gas production and gas migration as a consequence. This unbalanced production may be a result of a delay in the development of production facilities, pipeline, surface and processing facilities, gas boosting stations, unbalanced gas consumption, and etc. At these conditions, one solution for balancing gas withdrawal from a shared reservoir will be UGS in a production zone. Another option for UGS is storing gas in geological formations located in export hubs. For example, countries like Ukraine act as a hub for gas export between Russia and the rest of European countries. These UGS hubs store gas volumes in larger amounts required by the country itself, and the UGS is designed mainly for gas export. The third scenario for UGS is storing gas in underground formations to compensate peak demands in the consumption and populated areas. This scenario necessitates UGS in

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formations near the consumption areas and is known as the UGS at the consumption zone. The UGS is a mature technology both in surface facilities and underground. Yet, there are uncertain features in cap-rock integrity, reservoir inventory, well design, and etc. when UGS is planned for a new reservoir.

4.3 History of UGS in the World The first experimental project of UGS was conducted in Welland County, Ontario, Canada in 1915 [3]. One year later in 1916, the first operational UGS project on the bases of depleted field was created in Zoar field near Buffalo, New York, USA. The first UGS project in the aquifer was operated in 1946 in Kentucky, USA. In 1961, the first UGS project in salt cavern was started in Michigan, USA, and the first UGS operation in an abandoned mine and lined rock cavern were 1963 in Colorado, USA and 2002 in Skallen, Sweden [4]. The idea of storing natural gas in underground reservoirs during low-consumption for use in high-demand seasons and meet the peak rates has found worldwide application since 1950s.

4.4 Candidate Reservoirs The most common type of gas storage projects are operated in underground reservoirs, classified in three principal types, i.e. depleted gas reservoirs, aquifer reservoirs, un-minable coal beds and salt cavern reservoirs. Most early storage pools and a high percentage of all pools today were developed in depleted reservoirs [5]. Besides these types of UGS, Tureyen et al. describe other means of gas storage, i.e. storage in pipelines, storage in high pressure steel reservoirs, storage as liquefied natural gas, and storage in manmade caverns [5]. Malakooti and Azin describe the main features, advantages and disadvantages of three main UGS approaches. Storage in salt caverns is quite different than in a depleted oil and gas reservoir and aquifer. The salt cavern is initially washed by water and a large cave is formed underground to ensure high deliverability of gas during peak season. Whereas the gas flow occurs through porous media in the case UGS in depleted reservoir and saline aquifer. The key properties of a UGS plan can be summarized as follow [3]: 1. 2. 3.

Supply of natural gas flow for gas consumers is guaranteed especially during the cold season for peak shaving Natural gas provided for gas consumers when technical problems occur in gas processing facilities Balance between natural gas production and consumption is established during all time

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Significant decrease in required investments to build new capacities for gas transportation system of the country Establish a strategic resource for emergency and critical situations like war, earthquake, and etc. Increased withdrawal from shared gas fields and reinjection into domestic reservoirs Providing necessary facilities in business markets such as buying and storing cheap natural gas for domestic consumption, obtaining higher income by importing stored gas at high gas price increases in market.

A real case study of a UGS project involves subsurface and surface design and engineering studies. The surface facility for UGS is similar to other gas injection applications and will be discussed in Chap. 7. The subsurface studies include all aspects of geosciences, i.e. geology, geophysics, petrophysics, reservoir engineering, and production engineering. From the reservoir engineering aspect, the following parameters are important: • • • • • •

Determine reservoir storage capacity and reservoir performance Optimize well design, well number and well location in the UGS reservoir Compare the performance of horizontal versus vertical well in the UGS reservoir Analyze well productivity and deliverability Design injection/withdrawal rate scenarios Determine oil/condensate recovery in UGS operated in depleted oil or gas reservoir • Determine the role of impurities like acid gas (CO2 /H2 S) in the injecting stream on reservoir and produced gas compositions. • Perform and evaluate the reservoir material balance.

4.5 Worldwide UGS Projects Nowadays, underground gas storage projects have become an essential part of the gas chain due to supply-demand unbalances of natural gas globally, more gas consumption in the world, and more usage of underground reservoirs to store natural gas. Statistical studies show that 432 underground reservoirs have been designed for UGS plans in 11 countries by 1974, 300 of which were located in the United States. Based on reports by CEDIGAZ Company, the total number of 554 UGS reservoirs were operating worldwide in 1993. These storage reservoirs were able to deliver 243 MMMm3 working gas which amounts to 11% of world gas usage in 1993. In 2009, the number of underground storage facilities increased to 630 with 352.48 MMMm3 working gas. Statistical studies illustrate that UGS plans last averagely eight to ten years for each reservoir. UGS projects represent the fact that a production capacity of 100 MMMm3 is installing in the world. This capacity will not be enough because 2% growth in gas consumption increases the value of gas

4 Gas Injection for Underground Gas Storage (UGS) Table. 4.1 Number of active projects based on the region

149

Region

Number of active projects

USA

392

Canada

62

Europe

143

Common Wealth of Independent States (CIS)

48

Asia-Oceania

23

Middle East

3

Argentina

1

consumption up to 3100 MMMm3 in 2010. CEDIGAZ predicted that a capacity equal to 200 MMMm3 is required with the investment of $b40. At the end of 2016, 672 UGS facilities were in operation worldwide, representing a working gas capacity of 424 bcm equal to 12% of 2016 world gas consumption. Table 4.1 shows the UGS projects based on the region. Global UGS capacity is expected to increase from 413 bcm in 2015 to between 547 and 640 bcm in 2035, which requires an investment of $b123–270 [6]. Recognition of appropriate underground structures to store gas has started in Iran since 1989. Due to high consumption pattern, average daily gas consumption in four months of winter is about 2.5 times that in the rest of year in Iran. This rate may reach 4 times during the extreme cold days of winter. Consequently, supplementary sources of gas are needed in peak demand season which can be provided by UGS. NIGC established an independent management called UGS company with the mission to develop UGS projects in 2005. Currently, there is a UGS facility operating in Sarajeh formation, Iran. Sarajeh is an anticline reservoir located 40 km far from south east of Qom, about 160 km south of Tehran. The structure sizes are 25 × 5 km. This field has a storage capacity of 3.3 MMMm3 gas. Subsurface studies determined that 7.3 MMm3 gas should be injected into Sarajeh field during the summer season. After that 9.8 MMm3 gas should be produced from the reservoir during four cold months of the year. It was established if the reservoir performance satisfied the results obtained by researches, the injection and production rates will increase up to 22 and 33 MMm3 per day, respectively [3].

4.6 Screening Criteria for UGS Generally, UGS reservoirs consist of good to excellent quality formations which are often located spatially close to the ultimate demand source (i.e. major population centers). Most of these reservoirs represent natural gas pools which have been depleted near to or below abandonment pressure during normal production operations, then are used on a seasonal basis for gas storage. For a reservoir to be a candidate for gas storage, the following criteria must be satisfied [7]:

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Sufficient reservoir volume allows for storage of the required amount of gas without exceeding containment pressure constraints and without requiring uneconomic compression to high pressure levels. Satisfactory containment of the gas by upper and lower sealing cap rock. Sufficient inherent permeability to allow injection and production at required delivery rates during peak demand periods. Limited sensitivity to reductions in permeability, well injectivity/productivity due to the presence of in-situ water (mobile or immobile), presence of liquid hydrocarbons (mobile or immobile), plugging of the near injector region by compressor lubricants or other introduced fluids, and reservoir stress fluctuations during successive pressure cycles. Absence of hydrogen sulfide gas (in-situ or bacterially generated). Ability to drill and complete additional wells in the formation as required with causing severe formation damage (due to the highly depleted pressure condition which may often exist in these reservoirs).

Gas storage reservoirs are generally high permeability clastics or carbonates (1000–10,000 mD in-situ permeability is common) existing at intermediate depths and temperatures. In general, these reservoirs are depleted formations which originally contained dry (non-retrograde), sweet (no H2 S) natural gas. Typically, these zones do not contain mobile water or active or partially active aquifers, oil legs or residual liquid hydrocarbon saturation, although this is not always the case [7]. Another issue which must be considered in reservoir screening is yearly cyclic gas injection-shut-in-withdrawal pattern. According to Fig. 4.1, a complete cycle of UGS involves injection, intermediate shut-in, withdrawal, and final shut-in. The intermediate shut-in period lets the gas to evenly distribute in reservoir and relax the whole reservoir before withdrawal. In terms of reservoir inventory, the UGS is theoretically expected to move on a linear pattern on P/z versus Gp plot for a volumetric reservoir. However, there are frequently disturbances observed in UGS pressure profile indicating deviation from linear pattern. According to Fig. 4.1, a shut-in period between successive injection and withdrawal may relax deviations in

Fig. 4.1 A complete cycle of UGS involves injection, intermediate shut-in, withdrawal, and final shut-in

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reservoir pressure. Normally, a 0.5–2 month shut-in time is suggested by simulation studies [8] and field operations. In a UGS operation in Iran, a stable foam production was observed which severely affected the surface facility, especially separators, instruments, and flow meters, so that the operator decided anti-foam injection at early cycles of I/W. In next cycles, the foaming phenomenon was greatly resolved by extending the intermediate shut-in period and letting the reservoir to relax before withdrawal. Figure 4.2 shows the pressure behavior of a UGS cycle and its relationship with the reservoir inventory. The ideal general material balance equation suggests a linear pressure trend versus gas in place shown by dash line X–Y. However, the real field operation is expected to follow a cycle like ABCD in an aquifer-supported reservoir and A B C D volumetric reservoir. The UGS cycle starts from point A at the start of injection season and the reservoir is charged with injected gas up to point B, where a shut-in time causes a slight reduction in the measured pressure. During the shut-in time, pressure distribution becomes uniform throughout a reservoir with a permeability pattern typical of a sandstone formation. Therefore, the reservoir pressure at the end of injection is expected to be at point C for an aquifer-supported reservoir and point C for a volumetric reservoir. For a tight formation with low 1600 1500

B Reservoir with Aquifer

Reservoir Pressure (psia)

1400

C'

Volumetric Reservoir

B' C

Y

1300 1200 1100 1000 900 A'

800

A

D 700

D' X

600

12

14

16

18

20

22

24

26

Total Gas in Place (Bcf) Fig. 4.2 Comparison of cycles in volumetric reservoir with a reservoir with aquifer

28

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permeability, the difference between pressure at points B and C is small. Also, The difference between points C and C implies the potential of aquifer to support the reservoir pressure. The withdrawal stage starts from point C and ends at point D (D for a volumetric reservoir). Again, the reservoir pressure becomes uniform during a shut-in time and point D approaches point A at the end of withdrawal. In an ideal reservoir with uniform rock properties, the cycle may approach the X–Y line. This figure also suggests that part of the pressure decline at the end of withdrawal stage is restored by the aquifer and the pressure at point D is higher than point D for a volumetric reservoir. The degree of communication between reservoir and aquifer determines the difference between pressures at the end of injection and withdrawal for two reservoirs. Table 4.2 summarizes the screening criteria for UGS obtained from field operations. The formation thickness, depth, porosity, permeability, reservoir pressure and temperature are among the important factors that should be taken into account for UGS planning and selecting the candidate formations. In the studied cases, most formations were depleted oil reservoirs. Two formations were gas condensate, one gas reservoir, and a rock salt.

4.7 UGS Key Aspects 4.7.1 Threshold Pressure The minimum pressure required for displacing the cap-rock pore water is defined as the threshold pressure. First, the gas enters the cap rock but it doesn’t have the ability to flow through the cap rock. After continuous flow to the cap rock, the gas saturation exceeds the critical value. In this stage, the gas flows through the cap rock. Threshold pressure can measure the sealing ability of cap rock. This characteristic is directly measured in lab experiments which has different procedures and approaches. Also, some theoretical relations has been developed based on the rock properties such as porosity, permeability and formation resistivity for predicting the threshold pressure [19].

4.7.2 Well Testing The well testing for UGS project needs some specific requirements in comparison to the conventional well testing. Repeated evaluation must be done to predict the formation damage due to the high risk of clogging of gravel pack or screen and sand production. Also, well productivity and injectivity is tested during the well testing for characterizing the effect of turbulence flow on well performance during the storage cycle. The turbulence flow in this stage is caused by the significant

Weathered Zone

Bockfleis Formation

NM = Not mentioned Ph = Horizontal permeability Pv = Vertical permeability

Tiehchenshan

Triassic Hauptdolomite

Oil field

Gas condensate

65.6

Sarajeh

Gas field

Sockeye Field

164

NM*

Gas condensate

Jintan Salt Mine

229.6

New York City

Oil field

Oil field

2200

Niger Delta

Gas field

Rock salt

300

IZ-2 Reservoir

Oil field

164

70

656

NM

NM

NM

Pecorade

Oil field

Formation thickness (ft)

Field

Reservoir fluid

9022

5600

3260

NM

NM

NM

5800

6300

6100

1860

NM

8202

Formation depth (ft)

Table. 4.2 Screening criteria of some UGS projects and studies

0.18

0.235

NM

0.085

0.062

0.037

0.065

Low

0.155

0.354

0.25

0.06

Porosity

2494 3938

Ph * = 50–300, Pv * = 25–150

NM

4235

5699

NM

3214.7

816

2260

3843

Reservoir pressure (psi)

20–300

NM

400

90

50

20

Low

NM

NM

NM

0.05

Permeability (md)

223

NM

NM

NM

228

NM

142

119

140

210.2

Reservoir temperature (F)

[18]

[17]

[16]

[15]

[14]

[13]

[12]

[11]

[10]

[9]

Refs

4 Gas Injection for Underground Gas Storage (UGS) 153

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variation of reservoir pressure. Identification of the lateral and vertical variations of reservoir petrophysical properties are another important subject in well testing of a UGS project. These variations can cause uneven pressure distribution. Dynamic response of the system can be affected by minor heterogeneities due to high rates constrained by storage operations [19].

4.7.3 Rock Mechanics Extension and magnitude of fault stability and ground movements are evaluated with geo-mechanical analyses. Geo-mechanical behavior of the system is characterized under existing reservoir conditions by in-situ measurements and lab experiments on cores. Three most important properties of rock mechanics are in-situ stresses, intact rocks and faults. More details of this section can be found in Verga et al. [19].

4.8 Reservoir Fluid Phase Behavior in UGS PVT study in UGS reservoirs depends on the type of reservoir under study. For depleted oil and gas reservoirs, PVT study involves characterization of original fluid remaining in place at the end of depletion, selection and tuning of a proper EOS using detailed PVT experiments to match the experimental data. This approach is reported in the works of Khamehchi and Rashidi [14], Xiao et al. [4], Azin et al. [3, 20–22]. Details of PVT study techniques were reviewed in Chap. 2. In the case of UGS in depleted gas condensate reservoir, condensate revaporization may occur which results in fluctuations of CGR in successive I/W cycles [21]. For UGS in saline aquifers, solubility of gas into aquifer and water content of injected gas upon contact with reservoir water must be determined either through experiments or predicted by models. Zirrahi et al. proposed exact models for prediction of water content [23] and solubility of gas in saline aquifer [24]. Azin et al. reported that substantial changes in compressibility factor of reservoir fluid occurs in successive I/W cycles [25]. This is due to sharp compositional changes in reservoir fluid as a result of contact and mixing with injection gas.

4.8.1 Injection Gas Versus Reservoir Fluid During UGS, the injected gas contacts reservoir fluid and mixes with it. Depending on the composition of injection and reservoir fluids, extent of contact, time of contact, and level of mixing, the phase behavior of reservoir fluid is subject to change. The cyclic injection of dry gas can vaporize the light and intermediate components of

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reservoir oil. Also, it may re-vaporize the condensate bank in gas-condensate reservoirs. However, the radius and depth of these effects are limited in large reservoirs and it takes time for the reservoir boundary to “sense” the effect of gas storage. The effects of injected gas on the reservoir fluid phase behavior was discussed in Chap. 2.

4.8.2 Sour Gas Versus Sweet Gas In addition to sweet gas injection, sour gas, a blend of natural gas, hydrogen sulfide (H2 S) and carbon dioxide (CO2 ) streams, can be considered as another option for the purpose of UGS in gas production hub and in formations with sufficient gas treating facilities. Doing so can increase the gas withdrawal from shared reservoirs and increase the strategic reserves of country without the need to gas treatment. Underground sweet gas storage causes double costs in the industry, as sour gas stream is treated before injection and after withdrawal to remove contamination with fluid remaining in reservoir. Storage of sour gas is a more economic option for UGS projects because the sweetening and treating process is performed once. In this manner, the rate of gas withdrawal would be higher from shared reservoirs since operators do not need to construct treatment facilities before reinjection of the produced gas. When UGS is implemented in depleted gas condensate reservoirs, it can enhance condensate recovery through gas injection. Injection of dry gas increases the dew point pressure and helps revaporization of heavy hydrocarbon fractions into gas phase [26]. Pure CO2 and separator gases containing H2 S and CO2 are common injection streams to minimize condensate drop-out in near-wellbore zones [27]. A decrease in the compressibility factor is another impact of acid gases present in the injection gas. Based on compositional simulation, CO2 is found to be an efficient dry gas stream that can be injected into the retrograde reservoir to improve condensate recovery [28]. Changes in the wetting behavior of reservoir rock and interfacial tension between the storage gas and reservoir fluid are other important issues during UGS. Melean et al. [29] investigated the influence of gas/condensate interfacial tension and wetting behavior of condensate on the porous substrate. They found that CO2 is more advantageous than CH4 and N2 in reducing capillary forces and promoting the spreading of the condensate phase on water. Sour natural gas mixtures cause problems like corrosion in wellbore and surface facilities, and increased tendency for hydrate formation at elevated pressures [30]. These issues imply that all aspects of sour gas mixtures including their advantages and disadvantages must be considered before commencing UGS using sour gas. Malakooti and Azin compared the sweet gas injection with three sour gas streams. The compositions of each stream are reported in Table 4.3. The sweet gas stream contains no CO2 and H2 S. Sour gas stream 3 has the highest mole fraction of H2 S [20].

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Table. 4.3 Compositions of the injected gas streams [20] Composition

Sweet gas

Sour gas stream 1

Sour gas stream 2

Sour gas stream 3

C1 –N2

0.975

0.955116

0.906784

0.851784

C2 –CO2

0.0246

0.034498

0.07742

0.12742

C3 –nC4

0.0004

0.006978

0.008613

0.008613

iC4 –nC5

0

0.000045

0.000047

0.000047

FC6

0

0.000001

0.000001

0.000001

C7 –C11

0

0

0

0

C13+

0

0.000012

0.000013

0.000013

H2 S

0

0.00335

0.007122

0.012122

0.97

Sweet Gas Stream Sour Gas Stream 1 Sour Gas Stream 2 Sour Gas Stream 3

Z Factor

0.95 0.93 0.91 0.89 0.87 0.85 0

1000

2000

3000

4000

Pressure (psia)

Fig. 4.3 Gas compressibility factor versus pressure [20]

Figure 4.3 shows the gas compressibility factor of different gas streams. Obviously, the presence of CO2 and H2 S components in the injection gas stream decreases the compressibility z-factor of the injection stream [20]. Figure 4.4 indicates the difference between the compressibility factor of reservoir fluid and injected gas. The difference between the compressibility z-factor of injected gas and reservoir fluids is smaller in the case of sour gas streams. Also, decreasing trend of each stream is because of decreasing the difference between the compositions of reservoir fluid and injected gas [20]. Figure 4.5 shows the pressure rise at the end of I/W cycle. This figure indicates that by increasing the mole fraction of CO2 and H2 S the reservoir pressure at the end of I/W cycle will be decreased. This fact is another reason of the smaller difference between the reservoir and injected gas compressibility factor [20]. Figure 4.6 shows the ratio of difference between heating value (HV) of produced gas and injected gas to HV of the injected gas. This figure illustrates the effect of H2 S on the HV of produced gas. This ratio is increased by increasing the mole percent of H2 S. A stream with 0% of H2 S mole fraction has the lowest difference

4 Gas Injection for Underground Gas Storage (UGS) 0.03

Sweet Gas Stream Sour Gas Stream 1 Sour Gas Stream 2 Sour Gas Stream 3

0.025

Zinj-Zres

157

0.02 0.015 0.01 0.005 0 0

2

4

6

8

10

I/W Cycle

Fig. 4.4 Difference between the compressibility factor of reservoir fluid and injected gas during I/w cycles [20] 2400 Sweet Gas Stream Sour Gas Stream 1 Sour Gas Stream 2 Sour Gas Stream 3

2390

Pressure (psia)

2380 2370 2360 2350 2340 2330 2320 1

3

5

7

9

I/W Cycle

Fig. 4.5 Pressure at the end of I/W cycle [20]

between the injected and produced gas in comparison to other streams. Also, heating value of produced gas increases during each I/W cycle due to the increasing of heavy component mole fraction in produced gas. Heavy component mole fraction in gas phase increases due to the pressure decline of the reservoir. As the total heavy component of the reservoir decreases, the production of heavy components is also decreased. As a results, the heating value of production gas stream decreases [20]. Figure 4.7 demonstrates the heating value for injection and production streams with different percentage of H2 S. The sweet gas stream has the highest heating value. By increasing the H2 S percent, the heating value is decreased in both injection and production streams [20].

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100*(HVprod-HVinj)/HVinj (%)

25 Sweet Gas Sour Gas 5% H2S Sour Gas 10% H2S

20 15 10 5 0

0

2

4

6

8

10

I/W Cycle Fig. 4.6 Ratio of difference between HV of produced and the injected gas streams to HV of injected gas stream [20]

Heating Value (Btu/Cu ft)

1150 Prod (Sweet Gas) Prod (Sour Gas 5% H2S) Prod (Sour Gas 10% H2S) Inj (Sweet Gas) Inj (Sour Gas 5% H2S) Inj (Sour Gas 10% H2S)

1100 1050 1000 950 900 850

1

2

3

4

5

6

7

8

9

10

11

I/W Cycle Fig. 4.7 HV of production and injection streams versus successive I/W cycles [20]

4.8.3 Condensate Re-vaporization During UGS Liquid condensate formed by retrograde condensation during primary depletion in gas condensate reservoirs is trapped in porous media due to its low saturation. Injection of pipeline gas into this reservoir increases pressure and cause trapped condensate to evaporate and mix with the injected and residual gases. The withdrawn gas will be a mixture of injected gas, evaporated condensate and residue gas. As a result, the withdrawn gas will contain heavy hydrocarbons and must be processed to prevent condensation in the gathering and transmission distribution systems. It should be

4 Gas Injection for Underground Gas Storage (UGS)

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noted that significant volume of liquid can be produced from the withdrawn gas even though the liquid yield during storage withdrawal cycle is significantly lower than the liquid yield during the primary production. This is mainly due to high gas withdrawal rates during storage operation. Proper design of surface and processing facilities to handle the produced liquid is essential for reliable operation of the storage field. Condensate yield depends on the composition of withdrawn gas from storage. The degree of mixing among residue gas, evaporated condensate, and the injected gas determines the withdrawn gas composition and depends on several factors including well configuration, the residence time of injected gas, reservoir structure, and I/W schedule. Performance of UGS in depleted gas-condensate reservoirs is strongly composition-dependent. Therefore, it is necessary to predict reservoir fluid compositional changes under varying pressures and depletion processes. To define the fluid properties at the reservoir and surface conditions, usually a laboratory PVT analysis, such as constant volume depletion (CVD) test, is performed on the original reservoir fluid. Detailed PVT studies are costly, tedious and time consuming. As a result, an equation of state (EOS) is often used to match the reported experimental data. Once matched, the EOS can be used to predict the fluid properties over a wide range of pressure and temperature conditions. The most successful EOS for natural gas property calculation is the one proposed by Peng and Robinson. The compositional analysis of the reservoir fluid is the primary input data to the EOS. Many studies have emphasized the C7+ characterization as the key element in attaining agreement between EOS and laboratory results. Studies show that the composition of injection gas affects condensate recovery during I/W cycles. Azin et al. defined condensate production enhancement (CPE) by Eq. (4.1), to determine the role of sour compounds on the condensate recovery in UGS [20]: CPE =

Npso − Npsw Npsw

(4.1)

where Npso : Cumulative condensate production with sour gas injection Npsw : Cumulative condensate production with sweet gas injection. Condensate holding ratio (CHR) of injected gas is defined as the ratio of condensate in injected sour gas in equilibrium with reservoir fluid to that in pure CH4 (as a base component for storage gas) in equilibrium with reservoir fluid (4.2) [20]: in j.

CHR =

yC5+ C H4 yC5+

(4.2)

The CHR can be interpreted as a function of the condensate equilibrium ratio. The K-value of condensate (C5+ ) in the mixture of injected gas is defined as Eqs. (4.3) and (4.4) for storage of sour and pure methane, respectively:

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R. Azin and A. Izadpanahi sour K C5+ =

sour yC5+ sour xC5+

pur e

yC5+

(4.3)

pur e

K C5+ =

pur e

xC5+

(4.4)

Also, the mole fraction of condensate in the gas phase for each state is obtained by rearrangement of Eqs. (4.5) and (4.6). sour sour sour = K C5+ xC5+ yC5+ pur e

pur e

pur e

yC5+ = K C5+ xC5+

(4.5) (4.6)

By substituting Eqs. (4.5) and (4.6) in (4.2), the condensate holding ratio is obtained, as follows: CHR =

sour sour K C5+ xC5+ pur e

pur e

K C5+ xC5+

(4.7)

C5+ is the major constituent of the reservoir liquid phase and accounts for more than 85 mol%, regardless of the type of storage injection gas. Therefore, changes in the mole fraction of C5+ in sour gas injection and pure methane injection can be assumed to be negligible. By this assumption Eq. (4.7) changes as: CHR =

sour K C5+ pur e

K C5+

(4.8)

Malakooti and Azin conducted simulation on sour and sweet gas injection into a depleted lean gas condensate reservoir. They investigated the condensate revaporization during the gas storage with different compositions. Figure 4.8 shows the K-value of the condensate increases in the presence of H2 S and CO2 . Also, it can be concluded that the effect of H2 S is more significant than CO2 [20]. Figure 4.9 indicates the condensate holding ratio (CHR) for different gas compositions. The trend of this figure is similar to the previous figure. This figure shows that pure CO2 can significantly increases the condensate production. In contrast, pure methane has the highest condensate holding ratio in comparison to other gas streams [20]. Figure 4.10 shows the cumulative field condensate production during 10 I/W cycles. As it is obvious from this figure, the sweet gas stream has the lowest condensate production. Also, the condensate production increases by increasing the CO2 and H2 S in the injected gas [20]. Figure 4.11 shows the condensate production enhancement (CPE) versus 10 I/W cycles. The trend of this figure is similar to previous figure. In this figure, the streams

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161

0.04 0.035

K Value

0.03 0.025 0.02 0.015

Sweet Natural Gas Pure CO2 Pure CH4 0.8CH4+0.2CO2 0.8CH4+0.2H2S

0.01 0.005 0

600

1100

1600

2100

2600

3100

Pressure (psia) Fig. 4.8 K-value of heavy components for different types of storage gas [20] 4

CH4

Condensate Holding Ratio

3.5

0.8CH4+0.2CO2 0.8CH4+0.2H2S

3

CO2

2.5 2 1.5 1 0.5 0 400

900

1400

1900

2400

2900

3400

Pressure (psia)

Fig. 4.9 CHR versus pressure [20]

with 8.3, 2.8 and 0.85% mole percent of CO2 + H2 S have 7, 5 and 3% of condensate production enhancement, respectively [20]. According to Figs. 4.10 and 4.11 the heavy hydrocarbons have high tendency to vaporize during the withdrawal stage by increasing the percentage CO2 and H2 S in the injection gas. The molecules of these gases make the London force with heavy hydrocarbons. Also, the sulfur atoms and oxygen of H2 S and CO2 are electronegative and both molecules have a polar structure. Besides, the uneven distribution of molecular electron density of heavy hydrocarbon causes the multi-pole state which helps to H2 S and CO2 to attract the heavy hydrocarbon. Also, the uneven distribution is stronger in the presence of H2 S and CO2 . These are the reasons of higher condensate

R. Azin and A. Izadpanahi Cumulative Field Condensate Production (STB)

162 2.40E+06

Sweet Gas Stream Sour Gas Stream 1 Sour Gas Stream 2 Sour Gas Stream 3

2.20E+06 2.00E+06 1.80E+06 1.60E+06 1.40E+06 1.20E+06 1

3

5

7

9

I/W Cycle

Fig. 4.10 Cumulative condensate production for the different type of injected gas [20]

Condensate Production Enhancement (%)

8 H2s+CO2=8.3%

7

H2s+CO2=2.8% H2s+CO2=0.85%

6 5 4 3 2 1 0

1

3

5

7

9

I/W Cycle

Fig. 4.11 CPE caused by different mole fraction of CO2 and H2 S [20]

production and higher CPE by increasing the mole percent of H2 S and CO2 in the injected gas [20].

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4.9 Reservoir Material Balance in UGS 4.9.1 Reserve Estimate in UGS There are three broad categories in the estimation of reserves; that is, volumetric, material balance and production decline analysis. Selection of appropriate reserve estimation technique depends on the available information. Generally, the range of uncertainty for a reserve estimate decreases and confidence level increases when more information is available and the reserve estimate is validated by more than one technique. Volumetric methods involve calculation of reservoir hydrocarbons in place contained in a rock volume. The key unknown in volumetric reserve estimate may be rock volume, porosity, or fluid saturation. Material balance methods involve the analysis of pressure behavior as reservoir fluids are withdrawn. Reserve estimate by material balance requires sufficient production and pressure data, as well as rock and fluid properties, characteristics of aquifers, and accurate average reservoir pressure and is more reliable than volumetric method. Production decline methods of reserves estimation involve the analysis of production behavior as reservoir fluids are withdrawn. Confident application of decline analysis methods requires sufficient period of stable operating conditions after wells established drainage areas. Factors affecting production decline behavior include reservoir rock and fluid properties, transient versus stabilized flow, changes in operating conditions and depletion mechanism. Reserves may be assigned based on decline analysis when sufficient production data is available. The decline relationship used in projecting production should be supported by all available data. For a gas reservoir with known bulk volume Vb (acre-feet), average porosity φ, average connate water Sw , gas formation volume factor Bg , the standard volume of gas in place is calculated by Eq. (4.9): G=

43560Vb φ(1 − Sw ) Bg

(4.9)

The general material balance equation for a gas reservoir is written as follows by applying the law of conservation of mass to the reservoir and associated production:  G(Bg − Bgi ) + G Bgi

 Cw Swi + C f P + We = G p Bg + Bw W p 1 − Swi

(4.10)

For most gas reservoirs, the gas compressibility term is much greater than the formation and water compressibility, and the second term on the left-hand side of Eq. (4.10) becomes negligible: G(Bg − Bgi ) + We = G p Bg + Bw W p

(4.11)

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When reservoir pressures are abnormally high, this term is not negligible and should not be ignored. When there is neither water encroachment into nor water production from a reservoir of interest, the reservoir is said to be volumetric. For a volumetric gas reservoir, Eq. (4.11) reduces to: G(Bg − Bgi ) = G p Bg

(4.12)

Using expressions for Bg and Bgi substituting them into Eq. (4.12) results in Eq. (4.13) at isothermal process, i.e. reservoir temperature: P Z

=−

Pi Pi Gp + Zi G Zi

(4.13)

Because Pi , Z i and G are constant for a given reservoir, Eq. (4.13) suggests that a plot of PZ versus G p yields a straight line with slope of − ZPi Gi and intercept of Pi Zi

. If PZ is set equal to zero, which would represent the production of all the gas from a reservoir, then the corresponding G p equals OGIP, G. As long as the UGS reservoir is volumetric with no indication of inventory loss, the material balance straight line approach describes the behavior of UGS reservoir for I/W cycles. In reality, pressure-inventory relationship deviates from straight line which is a result of fluctuation in reservoir pressure at the end of I/W. The I/W end effect caused by rapid injection and withdrawal is reflected as deviations in pressure-inventory plots. Flanigan discussed different pressure-inventory scenarios which occur for volumetric and water drive UGS reservoirs [31]. He described cases with loss of inventory due to gas leakage through gas cap or as a result of improper casing design, migration to nearby formations, dissolution into saline aquifer, and etc. In water-drive reservoirs, the relation between G p and PZ is not linear, and water influx causes less pressure drop than under volumetric control. In the case of condensate production, the gas equivalent of the condensate in SCF/STB must be added to G p and two-phase compressibility factor must be placed instead of single phase compressibility [32]. G E = 133000

γ Mwo

For the volumetric naturally fractured gas reservoir, the familiar be represented as [33]:   Gp Pi 1− ∗ = Z i∗ G Z P ∗

Z =

Z 1 − ce (Pi − P)

(4.14) P Z

equation can

(4.15) (4.16)

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165

Gerami et al. [33] introduced another equation for total compressibility that has a term accounting for the effects of fracture porosity and fracture compressibility: ct = cg + cng

(4.17)

cng is related to the compressibility of matrix pore volume, fracture pore volume, and connate water (non-gas components) compressibility: cng = ce [1 − cg (Pi − P)]

(4.18)

where ce is effective compressibility which is defined as follows: ce =

φ1 (cw Swi + c1 ) + c2 φ2 φ1 (1 − Swi ) + φ2

(4.19)

For the stress-sensitive, naturally fractured reservoirs, fracture porosity φ2 and fracture compressibility c2 can be calculated at the new reservoir pressure Pi+1 [34]: φi+1 = φi



ki+1 ki

c2i+1 =

 13

=

log Pki+1 − log Ph log Pki − log Ph

−1 pki+1 ln(Pki+1 /Ph )

(4.20) (4.21)

where Pk is the stress on the fractures minus the average reservoir pressure, i.e., the net stress on the fractures and Ph is an apparent healing pressure which is in the order of 40,000 psia [34]. In conventional p/z material balance, the extrapolated straight-line trend of measured shut-in pressures (for gas reservoirs) is used to predict OGIP. Since most gas wells do not produce at constant flow rate, a constant pressure difference between average reservoir conditions and flowing conditions cannot be assumed. For a variable rate system, the difference between reservoir p/z and flowing p/z is not constant, but a function of flow rate. Normalized Rate/Normalized Cumulative approach is an alternative to the flowing p/z plot which is applicable to both oil and gas reservoirs, and works for constant or variable rate systems. Flexibility and simplicity are the main advantages of this analysis. The procedure is similar to the pseudo-steady-state approach, but involves plotting the inverse of pseudo-steady-state equation, so that a declining trend is produced: 2qtca Pi 1 1 q =− + Pp b pss G i (ct μz)i Pp b pss

(4.22)

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R. Azin and A. Izadpanahi

tca

μi cti = q(t)

t 0

q(t) μ(P)ct (P)

dt

(4.23)

The normalized pressure integral (NPI) is used to develop a vigorous analytical method for data drawdown without any noise, as is typical of standard well test derivative. The solution involves a pressure integral curve as the base curve for noisy drawdown analysis. The dimensionless pressure integral is defined as follows: PDi =

1

t D A PD (t)dt

(4.24)

khPp 1.417 × 106 T q

(4.25)

0.00634ktca π φμi cti re2

(4.26)

tD A 0

PD =

tD A =

Conceptually, pressure integral is a cumulative average flowing pressure drop. The smoothness of pressure integral makes it as an ideal base curve for the standard pressure derivative, if the raw data contains any degree of noise or scatter.

4.9.2 Drainage Radius in UGS The drainage radius in gas storage wells is described as the distance from the flow boundary in the reservoir to the center of production well. Ligen et al. represented a mathematical relation for calculating gas drainage radius of UGS. Their model was developed for gas production based on the following assumptions: • • • • • •

The reservoir is assumed as a cylindrical shape A production well is located at the center of the cylinder The vertical thickness of the reservoir is sufficient The reservoir is homogeneous and isotropic and also has good connectivity Gas flow considered as single-phase fluid flow Gas produces at a constant rate.

They presented the following equation for calculating the drainage radius in gas storage wells. More information about the solution techniques and deriving equations are available in Ligen et al. [35]. 

ηt 14.682 2p Re p Pp max − Ppw f p Bg Tsc ηt p 3 Re p = ln − − 0.84e −2 Psc (Pmax − Pmin )T Re2p rw 4

(4.27)

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Table. 4.4 The accuracy of Eq. 4.27 reported by Ligen et al. [35] Well number Permeability, mD Cal drainage radius, m Actual radius, m Relative error, % 1

−11.9

4

151

169

2

9

220

197

10.5

3

13

245

266

−8.6

4

16

289

312

−8

5

25

356

328

7.9

6

30

365

402

−10.1

7

36

422

437

−3.6

8

42

427

447

−4.7

9

49

488

463

5.1

10

55

484

513

−6.0

11

64

553

583

−5.4

12

70

542

577

−6.5

13

81

618

634

−2.6

14

86

598

626

−4.7

15

92

617

669

−8.4

16

96

629

674

−7.2

17

100

683

672

2.6

where η=

K μφct

Table 4.4 indicates the Ligen et al. study of drainage radius for 17 UGS wells. Their results showed that the radius values calculated by the mathematical model is accurately matched to the field data. Also, they expressed that the drainage radius depends on multiple variables. By increasing the rock permeability and production time, the drainage radius increases [35].

4.10 Well Pattern in UGS Proper design of wells is an important step in the development of UGS. Well number, well location, well completion, and well type, i.e. horizontal or vertical are key factors in an optimum well pattern study. Application of horizontal drilling in UGS is emerging in a few fields. In 2000, about 100 horizontal wells were operating out of some 10,000 storage wells in 600 UGS facilities in the world [36]. Simultaneously, at least more than 100 new horizontal wells were planned, pointing out the growth in the use of this technique to increase maximum deliverability. Studies show that

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a horizontal well intercepting vertical fractures has a better productivity index than a vertical well intercepting a vertical fracture. Horizontal wells have productivity index 1.5–6 times vertical well in the same reservoir, depending on reservoir quality and horizontal drain length [36]. Horizontal wells can also minimize water conning during operation, if the drain trajectory is always above the gas–water interface and pressure drop inside the drains is smaller than in a vertical well, so less water coning during withdrawal [36]. Tureyen et al. [5] investigated the effects of well properties on the performance of UGS reservoir in Northern Marmara gas field in Turkey. Their modeling studies showed that the wellhead pressure has significant effect on the performance of UGS reservoir. For a fixed number of wells, working gas capacity increases as the wellhead pressure is decreased. They found that the number of horizontal wells required to obtain the desired working gas capacity strongly depends on the wellbore damage, horizontal lengths and partial completion characteristics. Efforts to minimize the wellbore damage help to improve the production/injection performance of the storage reservoirs. Recently, application of horizontal well is commenced in the development of UGS in LiaoHe oil field, China [37]. The authors address technical challenges for converting UGS in complex fault-block reservoir, including local environmental restrictions and imperfect gas storage well cementing technology in LiaoHe oil region. Under these circumstances, horizontal wells and branching wells drilling and completion technologies are suggested to complete gas storage drilling. For instance, a working gas capacity of 1 billion sm3 is estimated to be obtained by using 10 horizontal wells each with a horizontal length of 100 m if there is no wellbore damage. Keeping the well number the same, 1 billion sm3 is obtained by increasing the horizontal length nearly five times if the mechanical skin factor is 20. The advantages of using horizontal wells include intercepting natural fractures, higher recoveries, reduction of water conning and reduction of sand production problems. On the other side, horizontal wells are costly compared to vertical wells. In a UGS plan, new wells may be drilled near the crests and in high gas saturation and permeability zones, especially in the high permeability fractured zones. Having high permeability leads to increase connection transmissibility factor; this in turn results in well productivity or injectivity improvement. Productivity index is defined as the ratio of production rate of a given phase to pressure drawdown defined as follows: J=

Qp Pr − Pw

(4.31)

where Q p indicates the production rate of a given phase and Pr presents the pressure in the reservoir at the drainage radius. Equation (4.31) can be applied for injection wells if injection rates are treated as negative.

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4.11 Monitoring of UGS Projects 4.11.1 Surface Monitoring Surface monitoring techniques are divided into conventional monitoring and synthetic aperture radar (SAR). Conventional techniques include geotechnical monitoring (clinometers, distometers, extensometers and etc.), GPS (global positioning system) or conventional topographic survey. These conventional techniques can be applied to limited set of benchmarks and also it needs the access to the area. SAR technology has high potential for evaluating ground surface displacement over wide areas. This technology collects a vast number of data from remote sensing satellites which prepare a deformation map. This technology doesn’t need any physical contact and ground access for preparing the deformation map of the monitoring area [19].

4.11.2 Well Monitoring Static bottom-hole pressure is one of the characteristics that measures at the end of each injection cycle and at the end of each withdrawal cycle. This characteristic evaluates the reservoir connectivity which specifies the performance of the UGS field and gas inventory. Saturation variation is recorded to verify the place of gas water contact using RST logs. Also, the well head pressure and gas rates are monitored using telemetry for well performance evolution. Mini-frac test or leak-off test is performed for ensuring the mechanical integrity of the seal rock [19].

4.11.3 Wellbore Integrity Monitoring A most important requirements for UGS wells is the efficient zonal isolation which catered by the cement between casing and wellbore. The zonal isolation may loss due to high injection pressure and production rates. Also, pressure variations may induce micro-annulus or micro cracks in the cement over time. Cement failure can caused the loss of efficient zonal isolation, gas migration behind the casing and corrosion problems. Cement distribution can be monitored by wireline logging with ultra-sonic and sonic tools. Also, these tools can identify the discrete defects such as micro-annulus and channeling [19]. Azin et al. presented a new formulation for the cement used in UGS wells. The advantages of this formulation include improvement of cement ductility and elasticity, preventing cement cracking and decreasing the crack propagation. A polymer and an elastomer will be added to the common cement composition. Their results showed that by adding the latex to the cement formulation, the hydration, density, fluid loss and bonding of the cement are decreased. Also, they expressed

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that polypropylene fiber can significantly improve the stress–strain properties and compressive strength of the cement. On the other hand, the combined flexibility of microfibers and cohesive effect of latex can increase the resistance of the cement against excessive pressures [38].

4.11.4 Micro-Seismic Monitoring Pressure variation and gas movements can be tracked with micro-seismicity method. In this method, micro-seismic events are detected by passive seismic monitoring. This detection is done by the creation or reactivation of small fractures or by the modification of pore pressures. This method is provided a real-time picture of the passage of gas at specific points where the appearance of micro-seismicity and the increase in pore pressure are linked together. Also, the weakness zones in the reservoir or cap rocks can be characterized by this method [19].

4.12 Numerical Simulation and Dynamic Modelling of UGS The objective of reservoir simulation in UGS is to predict reservoir performance, pressure distribution, storage capacity, productivity index, and etc. In general, numerical simulation is used to evaluate the potential of depleted reservoirs for switch to gas storage reservoirs. Parameters affecting UGS simulation include reservoir geometry, rock properties, PVT of injecting and reservoir fluid, well pattern and well placement. Optimum base pressure and optimum time to start UGS, gas volume remaining in reservoir at the start of UGS, I/W scenarios, optimum number, location and type of wells, and etc. are determined by numerical reservoir simulation. Details of governing equation, discretizing the partial differential equations, and solution techniques in the context of reservoir simulation are discussed by Ertekin et al. [39]. Some issues must be considered in the dynamic modelling of UGS operations, including gas composition variation, reservoir temperature variation due to injection of cold gas, pressure variation due to the repeated I/W cycles at high gas rates, hysteresis of capillary pressures, rock compressibility and gas water relative permeabilities. The mentioned issues are not considered in the common reservoir simulation. Another difference between UGS and conventional reservoir simulation is the time steps. In the UGS simulation, the frequent switches between injection and withdrawal should be correctly accounted for. But, in the conventional reservoir simulation, the time steps are typically set (yearly, monthly and etc.) to optimize the simulation run time [19]. Xiao et al. [4] carried out a simulation study using both 3D full-field black oil reservoir model and full-field multi-component reservoir model to evaluate and consider the feasibility of creating UGS plan in a buried hill fractured oil depleted carbonate reservoir with bottom water, sufficient closure, a good quality and tight cap rock.

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Khamehchi and Rashidi [14] studied simulation of UGS in Sarajeh gas field, Iran. Based on available information and those obtained by the model, they concluded this field could be a candidate for UGS plan. Later, this reservoir was converted in to the first operating UGS reservoir in Iran. Compositional model was used to create the model in their studies. Azin et al. [21] simulated underground gas storage in a partially depleted gas reservoir using a compositional simulator. The purpose of their research was optimization of performance of UGS with different reservoir depletion scenarios, gas injection and aquifer strength. They concluded that if the gas reservoir is depleted down to its ultimate recovery, it may contain less base gas than required by gas storage operations. Under these conditions, the withdrawal rate from reservoir may not meet the target rate in the high-consumption seasons. At these conditions, operator may set the withdrawal at a lower target rate and let the reservoir take enough pressure before raising the target pressure. Alternatively, it can prolong the injection period in the first cycle in order to inject sufficient reserve into reservoir before starting withdrawal. In separate studies, effect of sour gas injection as a strategic UGS, feasibility of UGS in a fractured reservoir, and optimization of well pattern and well location was presented by Azin et al. [20], Jodeyri Entezari et al. [22], and Malakooti and Azin [8]. Soroush and Alizadeh [13] presented a real case study of an Iranian gas condensate reservoir for UGS. The feasibility of doing a storage project was supported by the simulator results, and the reservoir is capable of storing the gas and withdrawing it in wintertime without any problem. Based on the simulation results, they concluded that it is possible, and even suitable, to maintain gas storage with higher field pressures. McVey and Spivey [40] used computer reservoir simulation to maximize the deliverability of the gas storage reservoir and minimize the cost to meet a particular I/W schedule. Presence of naturally occurring fractures incorporates additional complexity to the behavior of UGS reservoir. Gerov and Lyomov [41] analyzed the performance of UGS in a multi-layered, fractured carbonate and sandstone formation. Simulation confirmed that water encroachment through fractures and water cut result in the unstable UGS performance and gas dis-balance. Bilgesu and Ali [42] presented a numerical model to predict the impacts of important formation properties such as matrix permeability, fracture spacing, and fracture orientation on the efficiency of well design for UGS in a fractured reservoir. Jodeyri Entezari et al. [22] investigated UGS in a partially depleted, naturally fractured gas reservoir through compositional simulation. Effects of fracture parameters, i.e. fracture shape factor, fracture permeability and porosity were studied. They concluded that water production from reservoir is affected by the distribution of fracture density. Also, fracture shape factor affects water flow through porous medium, and active aquifer can reduce condensate drop out around the well bore. For a complete UGS case study, the reader is referred to Malakooti and Azin which the results of this study were explained in Sect. 4.8 [3].

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23. Zirrahi M, Azin R, Hassanzadeh H, Moshfeghian M. Prediction of water content of sour and acid gases. Fluid Phase Equilib. 2010;299:171–9. 24. Zirrahi M, Azin R, Hassanzadeh H, Moshfeghian M. Mutual solubility of CH4, CO2, H2S, and their mixtures in brine under subsurface disposal conditions. Fluid Phase Equilib. 2012;324:80– 93. 25. Azin R, Nasiri A, Entezari AJ, Montazeri GH. Investigation of underground gas storage in a partially depleted gas reservoir. In: CIPC/SPE gas technology symposium 2008 Joint Conference. 2008. p. 18. https://doi.org/10.2118/113588-MS. 26. Adel H, Tiab D, Zhu T. Effect of gas recycling on the enhancement of condensate recovery, case study: Hassi R’Mel South field, Algeria. In: International oil conference exhibition in Mexico. Society of Petroleum Engineers; 2006. 27. Zaitsev IY, Dmitrievsky SA, Norvik H, Yufin PA, Bolotnik DN, Sarkisov GG, et al. Compositional modeling and PVT analysis of pressure maintenance effect in gas condensate field: comparative study. In: European petroleum conference. Society of Petroleum Engineers; 1996. https://doi.org/10.2118/36923-MS. 28. Seto CJ, Jessen K, Orr Jr FM. Compositional streamline simulation of field scale condensate vaporization by gas injection. In: SPE reservoir simulation symposium. Society of Petroleum Engineers; 2003. 29. Melean Y, Bureau N, Broseta D. Interfacial effects in gas-condensate recovery and gas injection processes. In: SPE annual technical conference exhibition. Society of Petroleum Engineers; 2001. 30. Wichert E, Aziz K. Calculate Zs for sour gases. Hydrocarb Process. 1972;51:119. 31. Flanigan O. Underground gas storage facilities: design and implementation. Elsevier; 1995. 32. Terry RE, Rogers JB, Craft BC. Applied petroleum reservoir engineering. Pearson Education; 2015. 33. Gerami S, Pooladi-Darvish M, Mattar L. Decline curve analysis for naturally fractured gas reservoirs: a study on the applicability of “pseudo-time” and “material balance pseudotime”. In: International petroleum technology conference. International Petroleum Technology Conference; 2007. 34. Jones FO Jr. A laboratory study of the effects of confining pressure on fracture flow and storage capacity in carbonate rocks. J Pet Technol. 1975;27:21–7. 35. Ligen T, Guosheng D, Chunhui S, Jieming W. Mathematical description of gas drainage radius for underground gas storage. Chem Technol Fuels Oils. 2018;54:500–8. 36. Bagci AS, Ozturk B. Performance analysis of horizontal wells for underground gas storage in depleted gas fields. In: Eastern regional meeting. Society of Petroleum Engineers; 2007. 37. Zhu Z, Yang Y. Horizontal drilling and cementing technology of the gas storage in LiaoHe oil region. In: IADC/SPE Asia Pacific drilling technology conference exhibition. Society of Petroleum Engineers; 2012. 38. Shahvali A, Azin R, Zamani A. Cement design for underground gas storage well completion. J Nat Gas Sci Eng. 2014;18:149–54. 39. Ertekin T, Abou-Kassem JH, King GR. Basic applied reservoir simulation. Reservoir: Society of Petroleum Engineers; 2001. 40. McVay DA, Spivey JP. Optimizing gas storage reservoir performance. In: SPE annual technical conference exhibition. Society of Petroleum Engineers; 1994. 41. Gerov LG, Lyomov SK. Gas storage performance in a fractured formation. In: SPE gas technology symposium. Society of Petroleum Engineers; 2002. 42. Bilgesu HI, Ali W. Effect of reservoir properties on the performance and design of gas storage wells. In: SPE eastern regional meeting. Society of Petroleum Engineers; 2004.

Chapter 5

Gas Injection for Pressure Maintenance in Fractured Reservoirs Ahmad Jamili, Amin Izadpanahi, Pooya Aghaee Shabankareh, and Reza Azin

Abstract Gas injection into the gas cap which is known as pressure maintenance or crestal gas injection is done to increase the reservoir pressure. Different types of gas may be injected in this method including producing gas, N2 , CO2 etc. The injected gas is chosen base on the field development studies. Each of these gases has some advantage and disadvantages. Gas injection in naturally fractured reservoirs is a challenge which needs more investigation on this subject. This chapter summarizes the basic concepts of pressure maintenance and active mechanisms during pressure maintenance in naturally fractured reservoirs. Also, this chapter provides the essential concepts in simulation of pressure maintenance in fractured reservoirs.

Abbreviations Acronyms AD-OO BHP EC EDFM EOR

Automatic differentiation-object oriented Bottom-hole pressure European commission Embedded discrete fracture model Enhanced oil recovery

A. Jamili Saint Francis University, Loretto, PA, USA e-mail: [email protected] A. Izadpanahi · P. Aghaee Shabankareh Oil and Gas Research Center, Persian Gulf University, Bushehr, Iran e-mail: [email protected] R. Azin (B) Department of Petroleum Engineering, Faculty of Petroleum, Gas and Petrochemical Engineering, Persian Gulf University, Bushehr, Iran e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 R. Azin and A. Izadpanahi (eds.), Fundamentals and Practical Aspects of Gas Injection, Petroleum Engineering, https://doi.org/10.1007/978-3-030-77200-0_5

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GMRES GOR HFM ILU IOR IPR MINC MRST MsRSB SAIGUP

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Generalized minimal residual method Gas oil Ratio Hierarchical fracture module Incomplete lower/upper Improved oil recovery Inflow-performance relation Multiple Interacting Continua Matlab reservoir simulation toolbox Multiscale restriction smoothed basis Sensitivity analysis of the impact of geological uncertainties on production forecasting

Variables A B C c cf D Dc,o , Dc.g Di j Dg De,c De,i d e f o,c , f g,c f m,i F G H k kc Kc ki,j kro , krg ,krw krgcw krocw krwro L

Area, m2 Formation volume factor, vol/vol Concentration, mole/m3 Component Formation Compressibility, 1/psi Depth, m Diffusion coefficient of component c in oil and gas, cm2 /s Binary diffusion coefficient of components i and j, cm2 /s Gas diffusion coefficient, cm2 /s Effective diffusion coefficient for component c at matrix-fracture boundary, cm2 /s Effective diffusion coefficient for component i, cm2 /s Correlation coefficients Correlation coefficients Fugacity of component c in oil and gas, psi Fugacity of component i in phase m, psi Formation resistivity factor Gas in place, ft3 Fracture thickness in z-direction, m Permeability, md Diffusion mass transfer coefficient of component c at matrix-fracture boundary, mole/(m2 s) Equilibrium ratio of component c Binary interaction coefficient Relative permeability of oil, gas, and water Gas relative permeability at connate water Oil relative permeability at connate water Water relative permeability at residual oil saturation Moles of oil per unit mole feed

5 Gas Injection for Pressure Maintenance in Fractured Reservoirs

l m MWi n1 nc N Nc,p nog ,ng ,nw ,now Pcog , Pcow Pc0 p Pc pc,i Pi po ,pg ,pw pr e f p q D, f m,c qC, f m,c q R Rs Rs Sgg So ,Sg ,Sw Sgr Sorg Sorw Swc Swir Si t T Tc,i To ,Tg ,Tw M M , Tg,c To,c Tr,i V Vp v vo vg v x , v y , vz

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Length of fracture, m Cementation factor Molecular weight of component i, g/gmole Exponent Number of components Oil in place, bbl Diffusion molar flux of component c at phase p, mole/(m2 s) Exponents on relative permeability curves Capillary pressure (oil–gas and oil–water), psi Reference capillary pressure at reference interfacial tension, psi Pressure, psi Capillary pressure, psi Critical pressure of component i, psi Parachor of component i Pressure of oil, gas, and water, psi Reference pressure, psi Pressure gradient, psi/ft Diffusion rate of component c at the matrix-fracture boundary, mole/s Convection mass transfer rate of component c at the matrix-fracture boundary, mole/s Flow rate, ft3 /day Universal gas constant, cm3 MPa/(K. mole) Solution gas oil-ratio, scf/stb Produced gas oil-ratio, scf/stb Geometric mean of matrix and fracture gas saturation Saturation of oil, gas, and water Residual gas saturation Residual oil saturation to gas Residual oil saturation to water Critical water saturation Irreducible water saturation Volume shift parameter in PR EOS Time, day Temperature, K Critical temperature of component i, K Transmisibilities of oil, gas, and water, mole.md/(m2 .cp) Molecular transmisibilities of component c in oil and gas, mole/s Reduced temperature of component i Moles of vapor per unit mole feed Pore volume Average gas stream velocity in the fracture, m/s Oil bulk velocity, m/s Gas bulk velocity, m/s Fluid bulk velocities in x, y, and z directions, m/s

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Vr Vp Vc,i W x,y,z xc xj xi,m ,xj,m yc yj yc,m f yc, f (yi )m , (yi ) f Zc Zj Zo ,Zg ,Zm zr e f We

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Bulk volume, m3 Pore volume, m3 Critical volume of component i, cm3 / mole Fracture width in y-direction, m Cartesian coordinates Mole fraction of component c in oil phase Mole fraction of component j in oil phase Mole fraction of component i and j in phase m Mole fraction of component c in gas phase Mole fraction of component j in gas phase Mole fraction of component c in the gas phase at matrix-fracture boundary Mole fraction of component c at the entrance of the fracture Mole fraction of component i in gas phase in matrix and fracture Overall composition of component c Overall composition of component j Compressibility factor of oil, gas, and phase m Reference elevation, m Water influx

Greek αs i j σi j t x, y, z γo , γ g , γw γo , γ g , γw μo , μg , μw φ φ0 φo,c , φg,c ρo , ρg , ρw ρr ρm ρC,s ρmr λ σ σ0 τ

Factor for considering skin-effect at matrix-fracture boundary Collision diameter of the Lennard–Jones potential Collision integral of the Lennard–Jones potential Time step, day Grid cells dimensions, m Specific gravity of oil, gas, and water, psi/ft Average specific weight of oil, gas, and water, psi/ft Viscosity of oil, gas, and water, cp Porosity Porosity at a reference pressure Fugacity coefficient of component c in oil and gas Molar densities of oil, gas, and water, mole/cm3 Reduced density Mixture molar density, mole/cm3 Critical density of component c, mole/cm3 Reduced density of the mixture Mobility Interfacial tension, dyne/cm Initial interfacial tension corresponding to the read-in capillary pressure, dyne/cm Tortuosity of the porous medium

5 Gas Injection for Pressure Maintenance in Fractured Reservoirs

ωi , ωc

Acentric factor of component i and c

Subscripts c c c f g i i i j k m m m o p p p r ref t x,y,z w

Component Capillary Critical Fracture Gas Component Grid block number in x-direction Initial Grid block number in y-direction Grid block number in z-direction Mixture Phase Matrix block Oil Phase Produced Pore Reduced Reference Total X,y,z directions Water

Superscript l L M v

Iteration level Time step Molecular diffusion Vapor

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5.1 Introduction Gas injection into the gas cap which is known as pressure maintenance or crestal gas injection is done to increase the reservoir pressure. Different types of gas may be injected in this method including producing gas, N2 , CO2 etc. The injected gas is chosen base on the field development studies. Each of these gases has some advantage and disadvantages. Gas injection in naturally fractured reservoirs is a challenge which needs more investigation on this subject. This chapter summarizes the basic concepts of pressure maintenance and active mechanisms during pressure maintenance in naturally fractured reservoirs. Also, this chapter provides the essential concepts in simulation of pressure maintenance in fractured reservoirs. It is common to inject gas in naturally fractured reservoirs to maintain the reservoir pressure and increases oil recovery primarily by gravity drainage and to a lesser extent by mass transfer between the flowing gas in the fracture and the porous matrix. In most cases, mass transfer is considered to contribute a small amount to the oil displacement. Mass transfer could be an important recovery mechanism in the case of a low permeable and/or small matrix block size. The mechanism is aided by the high area is available for mass transfer in naturally fractured reservoirs. Although gravity drainage has been studied extensively, there has been limited research on mass-transfer mechanisms between the gas flowing in the fracture and fluids in the porous matrix. Some mathematical models were developed which describes the mass transfer in between a gas flowing in the fracture and resident fluid in a matrix block. The injected gas diffuses into the porous matrix through gas and liquid phases causing the vaporization of oil in the porous matrix which is transported by convection and diffusion to the gas flowing in the fracture. Mass transfer between the fracture and the matrix is assumed to occur by diffusion mass transfer and fluid flow between the matrix and the fracture. The model accounts for diffusion and convection mechanisms in both gas and oil phases in the porous matrix driven by capillary pressure gradients which are generated due to changes in phase behavior as the gas dissolves in the oil phase.

5.2 The Concept of Pressure Maintenance Hydrocarbons can be produced using natural reservoir energy. This stage of production is called natural depletion or primary production. In this stage, production accomplished under various schemes such as gravity drainage, solution gas drive, rock and fluid expansion, gas cap expansion or aquifer drive. The dominant primary production mechanism depends on the reservoir properties and the production regime. The reservoir pressure and energy decrease due to the fluid production. At this stage, it is necessary to augment the natural energy with an external source. IOR and EOR methods are used as the external sources. IOR methods accomplished by water

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and/or gas injection. The main purpose of IOR is to maintain reservoir’s pressure. Hence, IOR methods are usually called pressure maintenance [1]. In the case of strong reservoir aquifer, it is necessary to maintain gas cap pressure equal to aquifer pressure to retard the water encroaching the reservoir and controlling water–oil ratio. Gravity difference between oil and injected gas plays an important role through the oil production. Gravity force determines fluids distribution and through a balance with capillary force leads to oil production [2]. Oil production can be high from a reservoir in which pressure is maintained by gas injection and still have an active gravity segregation. The velocity of gas flow must be controlled so that it does not dissolve in the oil but push the oil downward. This velocity is called critical velocity of counterflow which determines oil extraction. Counterflow velocity highly depends on the rate of pressure drop. It is better to maintain reservoir’s pressure above solution pressure where no gas would evolve and there would be no counterflow of gas and oil. In gas-oil gravity process, gas is injected in the gas cap through vertical wells and oil is produced by the horizontal wells at the bottom of oil zone. Gas fills the voidage which was created by oil production and overcomes pressure drop in both oil and gas zone. Injected gas also recompresses dispersed gas into oil zone which leads to gravity drainage process [3]. As the rate of injection or production decreases, recovery by this scheme increases. Capillary force act opposes of gravity force, so as the pores are depleted, gravity drainage rate will be slower [4, 5]. If there is a considerable difference between mobility ratio of gas and oil, the counter-flow rate is largely governed by the smaller mobility ratio [6].

5.2.1 Gas Injection in Gas-Cap Pressure maintenance is a process which either all the reservoir’s gas or part of it or an internal gas will be injected into the gas-cap at any stage of production to supply production energy and overcome the pressure drop [7]. Pressure maintenance partakes unused gas to maintain pressure and keep natural depletion mechanisms process [8]. The best recovery is obtained in reservoirs with high vertical permeability, a thick column of oil or in fractured reservoirs [9]. Natural gases in oil reservoirs may be present in free or solution condition. In the case of existing free gas, the reservoir may have primary gas cap. But, in the case of no free gas, the solution gas is evolved from the oil by decreasing the pressure below the bubble point pressure and secondary gas cap is formed [10, 11]. Solution gas drive starts under bubble point pressure, where gas evolves from oil. This evolved gas causes oil to flow in the direction of pressure declines [5]. The evolved gas is also helped the gravity drainage mechanism for producing more oil [12]. Craig and Geffen [13] compared oil recovery from sandstone reservoirs from five producing programs as (a) solution drive, (b) full pressure maintenance at original

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Cumulative Surplus Gas at Atomospheric Conditions (pore volumes)

180 Solution Drive PPM (pi=2000 psig) PPM (pi=2370 psig) FPM (pi=2000 psig) FPM (pi=2370 psig)

160 140 120 100 80 60 40 20 0 -20 0

5

10

15

20

25

30

Oil Recovery (%)

Fig. 5.1 Oil recovery under partial pressure maintenance (PPM) and full pressure maintenance (FPM) [13]

bubble point (2370 psi), (c) partially pressure maintenance at 2370 psi, (d) starting full pressure maintenance after depletion to 2000 psi, (e) starting partially pressure maintenance after depletion to 2000 psi. Figure 5.1 shows the recovery under different schemes which indicates that best recovery will be obtained by full pressure maintenance at original bubble point with 10,000 cu ft/bbl GOR. The gas injection for pressure maintenance is not recommended in some reservoirs especially in reservoirs with high producing gas-oil ratio. Izadpanahi et al. studied different scenarios for improving oil recovery in an Iranian fractured oil reservoir. Their results showed that by increasing the rate of gas injection in the studied reservoir the final recovery decreased [14].

5.2.2 Gas Recycling Produced gas is reinjected into the reservoir in the gas recycling projects [15, 16]. In gas condensate reservoirs, when the reservoir pressure decreases to below the dew point pressure, light components (such as methane) leave the solution and a heavy liquid phase called condensate form [17]. The maximum pressure drop occurs near production wells, so the maximum amount of condensate is expected to form in this region [18]. The main purpose of gas recycling is to prevent the loss of retrograde liquid phase formed in the reservoir. Increasing pressure above dew point can postpone the formation of condensate [19–21]. Also, the produced gas in the oil reservoirs is reinjected with the purposes of pressure maintenance. More information about gas recycling will be provided in Chap. 6.

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5.2.3 Water Injection Water is injected into the reservoir to maintain pressure and to sweep the oil. Water injection is known as a method of secondary oil recovery. This method has been widely used around the world, San Jorge Basin [22], Natih field [23], Zarzaitine field, Algeria [18], Mamou field [17], the Fort Chadbourne field [24], Adena sand field [25], Greater Burgan field [26], Carbonate oil field, Abu Dhabi [27], wainwright and Wildmere fields, Canada [28, 29], the Elk Hills oil field, California [30], Southern Louisiana gas reservoir [31], Barmer basin of Rajasthan, India [32], Wilmington field, Los Angles [33], the Battrum Northeast pool, Saskatchewan [34], Inglewood field, California [35], Siri oil field, Iran [36] and many others [37–44].

5.3 History of Pressure Maintenance Irwin L. Dunn was the first one to demonstrate the benefits of gas injection into a shallow producing well in 1903. He believed that pressure maintenance retains reservoir energy. Gas and/or air was injected in Pennsylvania and West Virginia before 1910 and, by 1911 pressure maintenance was started in Ohio. Significant increasing in oil production was observed in Ohio which makes pressure maintenance a proven method for increasing recovery [8]. In the early days of pressure maintenance, injection was performed at low pressure of below 100 psi using vacuum pumps. Later it becomes obvious that using vacuum pumps does not increase ultimate recovery appropriately. Lease operation limitation was the main problem in the early days of repressuring. The first operation with a large portion of a field as a unit was in Nowata County, Oklahoma. Finally the first case of gas injection with unitization in the usual sense was in Kettleman Hills reservoir in California [4]. The first pressure maintenance by gas injection was started in the 1920s in the Macksburg field, Washington improved techniques and theories [15]. In 1980’s in Venango County, Pennsylvania, an operator, James Dinsmoor, purposely allowed communication between a partially depleted oil sand and an underlying gas sand. A few years, he installed a gas pump for injecting gas into depleted wells [4].

5.4 Sources of Injected Gas Produced gas. Produced gas is easy to get and needs no additional facilities for capturing. Reinjecting reservoir gas conserves reservoir energy. The economic feasibility of natural gas injection is doubted in some area due to the price and usage of natural gas in that area. Today, gas plays an important role in the industry. As mentioned earlier, the produced gas is injected into the reservoir to maintain reservoir pressure and increase final recovery of hydrocarbons. Produced gas injection in

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the Shoats Creek field recovered 50 percent of oil in place [45]. The Pickton field Bacon Lime reservoir [46] was under pressure maintenance by gas and water injection. Only 6.6 MSTB oil in place of total 34 MMSTB can be produced by primary production. After gas injection, GOR increased which caused shut-in of 100 wells of total 166 wells. By controlling high GOR, final recovery from this reservoir reached to 25 MMSTB. In another case, all of the produced gas from Ekofisk reservoir was reinjected which increased recovery by 18% of oil in place [47]. Reinjecting produced gas needs a true estimate of gas in place. Kaye [48] concluded that calculating gas in place depending on data about acreage, reservoir pressure and temperature, sand thickness, porosity and connate water. Brickner [49] described the Bryans Mill field’s gas as sour gas. In the case of cycling, sweetening sour gas is needed. Tarner et al. [50] reported that injection of sour gas of Smackover reservoir was stopped due to corrosion and deposit effects on facilities. Jacoby and Berry [51] studied the effects of dry gas injection on a volatile oil reservoir. After injection, a phase equilibrium between volatile oil and dry gas will be established and lead to oil vaporization. Initially lighter components such as propane, butanes, and pentanes will be vaporized and after that heavier components such as the hexanes, heptanes and octanes will be transferred to the gas phase. Flue Gas. Flue gases contain hydrocarbons (such as methane and propane), hydrogen, carbon monoxide, carbon dioxide, nitrogen, steam and oxygen [52, 53]. The first successfully system for flue gas injection was installed and put on production in 1959 in an oil field in Louisiana state. From then injection of flue gas became a usual method. Access to flue gases needs surface plants which carry out the responsibility of capturing it. Building surface plants need investments and investigation but final NPV of the project can be higher. However, it can have corrosion effects which corrosion will be investigated in Chap. 9. This method is a cheap potential gas storage program. Also, injecting flue gases reduce air pollution [54]. Barstow [55] reported that surface plant was built in Texas in 1945 in order to inject flue gas into the reservoir. This project failed due to corrosion. The compressors and relevant equipment were badly corroded. By the time that equipment broke down, nearly 20 years of injection had been carried out. Nitrogen. The most available source of nitrogen is in the air. Air is made up of a mixture of gases. The major component of air is nitrogen gas, followed by oxygen. Nitrogen injection has performed widely around the world [9]. In the case of miscibility, nitrogen should be removed from the natural gas. Nitrogen injection may have negative effects in some reservoirs. This method increases the dew-point pressure of gas condensate which causes the early condensate formation [9]. Economic of capturing nitrogen, nitrogen emission, reservoir heterogeneity, the effect of nitrogen on phase behavior and stock tank liquid composition are the most important factors in planning a nitrogen injection project [9]. The Cantarell Complex, Mexico was under pressure maintenance by nitrogen injection since 1996. Pure nitrogen injection was started at the rate of 1200 MMSCFD which makes this project the largest nitrogen injection in the world at the time [56].

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Stewart et al. [57] reported that in the Elk Basin field lies in park county, Wyo. Injected gas composed of 90% of nitrogen and 10% of carbon dioxide which started in 1949 resulted in 60% of the recovery. This reservoir did not have a gas cap but after pressure drop under 2128 psi, it was formed. Nitrogen was injected in the crest of the Ekofisk field in Norwegian sector at 200 MMscf/D. A 3D, three-phase flow with a capillary pressure function was used to simulate the performance of this fractured and multiple layered reservoirs [58]. Carbon Dioxide. Carbon dioxide’s emission is a world challenge. CO2 emission is resulting from the burning of fossil flues and coals. CO2 emissions will be increased by 35.6 billion metric tons in 2020 [59]. One way for reducing CO2 emissions is to capture and storage (CCS) it underground. Pressure maintenance can be a way for (CCS) [60]. Law [61] studied the Athabasca bitumen reservoir in Northern Alberta, Canada. Pressure maintenance steam-assisted gravity drainage by CO2 and N2 and flue gas was modelled. CO2 injection leads to bitumen viscosity reduction due to CO2 solubility. The best recovery is enhanced by CO2 injection. This mechanism decreases oil viscosity and interfacial tension and has swelling effect. Zheng and Yang [62] evaluated a three-dimensional (3D) model in a heavy oil reservoirs which have been discovered in western Canada (Alberta and Saskatchewan). Three horizontal and five vertical wells were used to inject CO2 to maintain reservoir pressure after waterflooding. This process increased oil recovery by 8.9–12.4%. Pressure maintenance using CO2 is reported as a suitable approach to increase oil recovery of the heavy oil reservoirs by Zheng et al. They studied this method experimentally and numerically. They also reported that the horizontal wells have a better performance in comparison to the conventional 5-spot well configuration [63]. Pourhadi and Hashemi studied injection of different gases for enhanced oil recovery. Their results showed that water-alternating- CO2 injection could significantly increase the oil recovery due to improve of reservoirs pressure maintenance and extend the production plateau [64].

5.5 Active Mechanisms During Pressure Maintenance There are three basic mechanisms to transport miscible and immiscible fluids in porous media: convection (or bulk flow), molecular diffusion, and mechanical dispersion. Mechanical dispersion is neglected in the model. A brief description of convection and molecular diffusion mechanisms follow.

5.5.1 Convection (Bulk Flow) Convection is the transport of the component as it is carried along within bulk fluid movement. The driving force for convection (bulk flow) is the potential gradient. Darcy’s law is an empirical relationship between fluid flow rate in porous media and the potential gradient:

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qp =

K kr p A ∇∅ μp

∇∅ = ∇ p + γ p ∇ D

(5.1) (5.2)

where q p is flow rate of phase p (oil, gas, or water). K is absolute permeability. A is cross sectional area. kr p is relative permeability of phase p (oil, gas, or water). μ p is viscosity of phase p (oil, gas, or water). γ p is specific weight of phase p (oil, gas, or water). ∇∅ is potential gradient. ∇ D is depth gradient.

5.5.2 Molecular Diffusion Molecular diffusion is the mechanism of a component transport by random molecular motion. Molecular diffusion is the tendency to mix due to chemical potential gradient. Bird et al. [65] showed that concentration gradient instead of chemical potential gradient could be used as the driving force for ideal or near ideal mixtures. Diffusion is a slow process caused by random molecular motion. But diffusion can cause convection or bulk movement [66]. Convective flow can have many causes, such as pressure gradients and temperature differences. However, even in isothermal and isobaric systems, diffusion can produce convection. CO2 -n-decane system is a very good example of diffusion causes convection in an isothermal and isobaric system as experimental work of Chukwuma [67] showed. Therefore, combined mass transfer flux of diffusion and convection must be used in modelling diffusion: n i = ρi vi = ρi (vi − v0 ) + ρi v0 = ji + ρi v0

(5.3)

In Eq. 5.3 n i is the combined mass transfer flux (total diffusion flux), ji is the diffusion mass transfer, and ρi v0 is convective mass transfer. v0 is some convective reference velocity and should be chosen so that v0 is zero as frequently as possible. v0 could be mass average velocity (good for constant density liquids), molar average velocity (good for ideal gases where the molar concentration is constant), or volume average velocity (best overall for constant-density liquids and ideal gases) [66]. Usually, convective part of mass transfer is neglected, which means that there is no bulk movement (or the bulk is stagnant) because of diffusion. Basically, for some cases (ideal mixtures where there is not huge difference in component properties) neglecting convective part of mass transfer is not a bad assumption, but for non-ideal mixtures, neglecting convective part of mass transfer may not be a good assumption.

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For example, Chukwuma [67] recognized that in the CO2 diffusion process in ndecane, there is a density change that causes free convection in the system. The free convection enhances CO2 diffusion and makes diffusion occurs in 45 min while for other gases it takes several hours. Density of CO2 and n-decane mixture increases until 70% mol CO2 and then decreases. Diffusion process is modelled based on the driving force: concentration gradient or chemical potential gradient. The modelling process will be discussed next.

5.5.2.1

Concentration Gradient Driving Force

Diffusion for ideal mixtures is driven by concentration gradient (Fick’s law): J A = −cD AB ∇x A

(5.4)

where D AB is binary diffusion coefficient. C is the mixture concentration. ∇x A is the concentration gradient of component A. 5.5.2.2

Chemical Potential Gradient Driving Force

Diffusion for non-ideal mixtures is driven by chemical potential gradient and is derived by irreversible thermodynamics as Bird et al. [68]: ⎡ ⎢ J A = − cD AB ⎢ ⎣

x ∇ ln a  A  A Concentration Diffusion

⎤

+

1 [(ϕ A − ϕ B )∇ p − ρω A ω B (g A − gb )]   c RT Pressure−Forced Diffusion

+ k T ∇ ln T ⎦    Thermal Diffusion

where a A is activity of component A. ϕ A = c A V A and ϕ B = c B V B are volume fractions of species of A and B. c A and c B are concentrations of A and B. ω A and ω B are mass fractions of A and B. ρ is the mixture density. g A and g B are body force per unit mass acting on species A and B. k T is thermal diffusion ratio.

(5.5)

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By dropping pressure, thermal and forced diffusion: ⎡

⎤

J A = −cD AB ⎣

⎦

x ∇ ln a  A  A

(5.6)

Concentration Diffusion

Activity is a function of composition and a A = x A γ A , so: 

  ∂ ln a A ∂ ln x A ∂ ln a A ∇x A = −cD AB ∂ ln x A T,P ∂ x A ∂ ln x A T,P

  ∂ ln γ A ∇x A = − cD AB 1 + (5.7) ∇x A ∂ ln x A T,P J A = − cD AB x A

If the mixture is “ideal”, then the activity coefficient is equal to unity and the equation becomes similar to Fick’s law. If the mixture is “nonideal”, then activity corrected diffusion coefficient is [68]:  D aAB

= D AB

∂ ln a A ∂ ln x A





= D AB 1 + T,P

∂ ln γ A ∂ ln x A





(5.8) T,P

To replace activity with fugacity, we know: 

aA =

fA f A0

(5.9)

where 

f A is the fugacity of component A in the mixture. f A0 is the value of the fugacity at standard state. Also, we know that 

dμ A = RT d ln( f A ) = RT d ln(a A )

(5.10)

Substituting Eq. (5.10) into Eq. (5.6): ⎡ J A = −cD AB ⎣

⎤ x ∇ ln a  A  A

⎦

Concentration Diffusion 

= −cD AB x A ∇ ln f A =

−cD AB x A ∇μ A RT

(5.11)

5 Gas Injection for Pressure Maintenance in Fractured Reservoirs





∂ ln f A J A = −cD AB x A ∂ ln x A   ∂ ln f A = −cD AB ∂ ln x A



 T.P

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∂ ln x A ∇x A ∂xA



∇x A

(5.12)

T.P

By comparing Eqs. (5.4) and (5.12), the activity corrected diffusion coefficient is:  D aAB

= D AB



∂ln f A ∂lnx A

 (5.13) T,P

Equation (5.13) shows corrected diffusion coefficient for non-ideality. The diffusive flux of component c in phase p (oil, gas, or water), J cp , is given by Firoozabadi (2000) as:   Jc. p = −ϕ S p ρ p Dc.Mp ∇xc. p + Dc.T p ∇T + Dc.P p ∇ p

(5.14)

where Dc.Mp .Dc.T p .Dc.P p are concentration, temperature and pressure diffusion coefficients of component c in phase p (oil, gas, or water). ∇xc. p is concentration gradient of component c in phase p (oil, gas, or water). ∇T is temperature gradient. ∇ p is pressure gradient. ϕ is porosity. S p is saturation of phase p (oil, gas, or water). ρ p is density of phase p (oil, gas, or water). Considering the dispersion term (or correcting the concentration diffusion coefficients for the effect of porous media) in previous equation, it becomes:   Jc.∗ p = −ϕ S p ρ p K c. p ∇xc. p + Dc.T p ∇T + Dc.P p ∇ p

(5.15)

where K c. p is the combined diffusion and dispersion coefficient of component c in phase p (oil, gas, or water). 5.5.2.3

Molecular Diffusion Coefficients in Hydrocarbons

Molecular diffusion coefficients are calculated by the method of Da Silva and Belery [69]. This method is based on the published work of Sigmund [70]. From kinetics theory, the diffusion coefficients for binary systems are related to pressure, temperature, and composition through the Hirschfelder et al. equation [65], which gives the low-pressure limit of the density-diffusivity product (Sigmund [70]):

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ρm0 Di0j



2.2648 × 10−5 = σi2j Ωi j

1 1 + Mwi Mwj

0.5 T 0.5

(5.16)

i j and σi j are collision diameter and collision integral of the Lennard–Jones potential in Eq. (5.16). They are related to the component critical properties (TC,i , PC,i , Vc,i , and ZC,i ) of component i through the following set of equations: −6/5

1/3

σi = 0.1866Vci Z ci

i = 1 : nc

  σi j = 0.5 σi + σ j i. j = 1 : n c 18/5

εi = 0.1866Tci Z ci

i = 1 : nc

0.5  εi j = εi ε j Ti j =

(5.17) (5.18) (5.19) (5.20)

T εi j

(5.21)

  1.06036 + 0.193 exp −0.47635Ti j 0.1561 Ti j   + 1.03587 exp −1.52996Ti j   + 1.76474 exp −3.89411Ti j

Ωi j =

(5.22)

The density-diffusivity product as given by Eq. (5.16) does not remain valid for the high pressures encountered in hydrocarbon reservoirs. A polynomial correction expressed as a function of the reduced molar density has to be used as in the following equation: Di j =

Di j =

ρm0 Di0j

(0.99589 + 0.096016ρmr ρm  2 3 ρmr < 3 −0.22035ρmr + 0.032874ρmr ρm0 Di0j ρm

(0.18839 exp(3 − ρmr )) ρmr > 3

where ρm is the mixture molar density. ρmr = ρρmcm is reduced density of the mixture. and is critical density of the mixture ρmc = zi is the mixture composition.

nc 1nc 1

2/3

z i VC.i

5/3

z i VC.i

.

(5.23)

(5.24)

5 Gas Injection for Pressure Maintenance in Fractured Reservoirs

191

Finally, effective diffusion coefficients for each component of the mixture are estimated on the basis of the Wilke formula (Sigmund [70]): 1 − zi De. j = n c j =1 j = i

5.5.2.4

zi Di j

.

(5.25)

Molecular Diffusion Coefficients of CO2 in Water

Average diffusion coefficients for CO2 in water have been measured by Unver and Himmelblau [71] and Thomas and Adams [72] at atmospheric pressure and for a range of temperatures. Grogan and Pinczewski [73] mentioned that the reported measurements are well correlated by the Stokes–Einstein equation as follows: DCO2 −Water = 5.72 × 10−12

T μw

(5.26)

The constant in the equation is dimensional and the equation therefore requires 2 that T be in K, μ in centipoises, and DCO2 −Water in ms . Grogan and Pinczewski [73] 2 reported a value of 3.6 × 10−9 ms for DCO2 −Water at 13.1 MPa and 54 °C which is   2 close to DCO2 −Oil 5 × 10−9 ms . Campbell and Orr [74] conducted an experiment where CO2 diffused in water and mobilized the trapped oil in a dead-end pore. Figure 5.2 shows the experiment. Oil (red Soltrol) was trapped by water at the beginning of the experiment. CO2 flows in the main channel and has no direct contact with the oil. The CO2 dissolves in the water and diffuses toward the oil. When CO2 reaches the oil, it dissolves into the oil causing oil swelling. The swelled oil volume increases and pushes the water toward the main channel. At the final stage, oil starts to flow in the main channel.

Fig. 5.2 Mobilization of trapped oil by CO2 in water [74]

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5.5.2.5

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The Effect of Tortuosity on Molecular Diffusion

Tortuosity is a characteristic of a porous medium and defined as the ratio of the true length of the flow path of a fluid particle and the straight-line distance between the starting and finishing point of that particle’s motion. Tortuosity depends on porosity of the porous medium. If the porosity is high, tortuosity is low and vice versa. Because of the tortuosity in a porous medium, effective diffusion coefficient is lower than their values without a porous medium. This effect is shown by the following relation: Deffective =

D τ

(5.27)

where D is a diffusion coefficient for a component. τ is tortuosity of the porous medium. Deffective is the effective diffusion coefficient corrected for porous medium which should be used in calculations. Tortuosity is related to porosity through the formation resistivity factor (F) with the following form: τ = Fφ

(5.28)

F = φ −m

(5.29)

where

where m is cementation factor which depends on the nature of porous media and usually varies from 1.5 to 2.5. Amyx et al. [75] and Langness et al. [76] presented a good review of the relationship between tortuosity and porosity. They also gave the following relation based on experimental results: τ = (Fφ)2/1.67

(5.30)

Substituting Eq. (5.28) into Eq. (5.29) gives one relation between tortuosity and porosity as: τ = φ 1.2−1.2m

(5.31)

Equation (5.31) is used to estimate the effective diffusion coefficient for oil and gas components in the porous medium. The tortuosity is often treated as an adjustable parameter. The tortuosity is used to modify the molecular diffusion coefficients, adapting it for use in porous media.

5 Gas Injection for Pressure Maintenance in Fractured Reservoirs

193

5.6 Worldwide Pressure Maintenance Projects Table 5.1 shows some pressure maintenance project all around the world. In this table, the reservoir and fluid properties, injection date and injection rate are reported for each project. It is understandable that pressure maintenance can be applied in reservoirs with different characteristics. Gas injection in some Iranian reservoirs was proposed in the 1950s after observing the decline in production. The aim of the gas injection is for increasing reservoir pressure and consequently increasing the recovery factor to achieve maximum extraction of oil reserves. In 1976, after conducting studies, gas injection operations in Haftkel field (Asmari reservoir) began as the first gas injection project in this country. The second gas injection project was launched in July 1977 in the Asmari reservoir of Gachsaran oil field. In these two projects, two other oil reservoirs were selected as sources of gas supply due to the lack of suitable infrastructure for collecting associated gases and the lack of development of independent gas reservoirs. This led to a waste of oil in these reservoirs and a reduction in their reserves. Other gas injection projects in Iran started after these two projects which reported in Table 5.2.

5.7 Reservoir Material Balance for Pressure Maintenance Material balance which is a well-known calculation for chemical engineers was introduced by a reservoir material balance equation by Schilthuis. Schilthuis presented equations to relate pressure drop and reservoir’s energy to oil and gas production [87]. The overall balance below bubble point depends on pore space volume, volumetric equilibrium fluid properties at reservoir temperature, and uniform pressure with phase equilibrium temperature and fluid composition. Build-up pressure test gives a fair average data for the formation.

5.7.1 Muskat’s Material Balance Consider a porous media that saturated by oil, gas and immobile water which is shown in Fig. 5.3. The flow equations in the mentioned system are as follow: Oil phase:  ∇ Gas phase:

 ∂ So λo ∇( po + ρo gz) = ∅ Bo ∂t Bo

(5.32)

1969

–

Coalinga Nose

Northwes Avard

–

–

200,000

–

1941

–

–

1976

1942

1943

1944

1950

1948

1944

Canal

Gachsaran

Bibi-Hakimeh

Haftkel

Coles Levee main vestern

Coles Levee vestern 35

Coyote

Coyama

Cymeric

Newhall-Potrero

32,300

3300

119,700

21,300

30,400

170,200

400,000

–

–

11,600

11,900

1976

Elk Hills

1959

1939

Bahrain

–

Buena Vista 555

1933

Hilbig

–

36,900

1949

Elk Basin Tensleep

Injection rate (Mcf/D)

Buena Vista 27B 1949

Injection starting time

Field

192

112

141

303

94

309

110

1600

1600

110

193

43

61

386

1800

110

–

250

Formation thickness (ft)

Table 5.1 Worldwide Pressure maintenance projects

7000

2400

4500

5300

9100

8800

2087

5000

7400

8200

5300

3900

4660

6750

5800

2175

2000

4538

Formation depth (ft)

0.18

0.336

0.265

0.214

0.205

0.195

Low

0.075

0.09

0.15

0.23

0.26

0.0758

19.9

0.236

0.25

0.241

0.107

Porosity

75

1300

600

83

108

115

Low

–

–

200

1063

250

127

421

88

40

570

91

Permeability (md)

3129

1210

1661

2614

4014

3974

1100

–

–

3564

2288

1577

2594

3504

2461

1236

1228

1327

Reservoir pressure (psi)

170

135

146

185

235

235

123

–

–

210

184

145

150

212

210

140

120

127

Reservoir temperature (F)

0.72

6.2

1.6

1.05

0.45

0.45

-

–

–

0.37

1.2

1.02

0.95

0.65

0.42

2.25

1.86

1.4

Oil viscosity (cp)

35

18.9

32

33

34

34

37

30

–

37

29

28.7

36

31.9

36

38

29

Oil API gravity

(continued)

[83]

[83]

[83]

[83]

[83]

[83]

[85]

[84]

[84]

[83]

[83]

[83]

[82]

[81]

[80]

[79]

[78]

[77]

References

194 A. Jamili et al.

Injection starting time

1946

1948

1958

Field

Rio Bravo

Greely

West Guara

Table 5.1 (continued)

9560

16,200

31,000

Injection rate (Mcf/D)

100

131

134

Formation thickness (ft)

5120

11,500

11,400

Formation depth (ft)

0.29

0.2

0.224

Porosity

5000

712

500

Permeability (md)

2280

5029

5014

Reservoir pressure (psi)

–

252

250

Reservoir temperature (F)

0.58

0.23

0.2

Oil viscosity (cp)

42

37

39.9

Oil API gravity

[86]

[83]

[83]

References

5 Gas Injection for Pressure Maintenance in Fractured Reservoirs 195

196

A. Jamili et al.

Table 5.2 Pressure maintenance projects in Iran Injected volume in one year (millionm3 /day)

Field

Starting date

A

1985

0.02

B

1990

15.12

C

1992

5.61

D

1995

3.78

E

1999

2.12

F

2001

2.01

G

2003

5.1

H

2010

16.41

I

2013

0.11

Fig. 5.3 Schematic of porous media system

∇

    ∂ R s λo Rs So ∇ pg + ρg gz = ∅ ρ g Sg + ρg λg + Bo ∂t Bo

(5.33)

Capillary pressure: pc = p g − po

(5.34)

So + So + Swc = 1

(5.35)

Saturation:

Oil and gas mobility:

5 Gas Injection for Pressure Maintenance in Fractured Reservoirs

197

λo =

ko μo

(5.36)

λg =

kg μg

(5.37)

By integrating Eq. (5.32) in the absence of gravitational and capillary pressure over the entire reservoir volume Vr and use divergence theorem.  Ar

λo ∂ p d Ar = ∅ Bo ∂n

 Vr

∂ ∂t



So d Vr Bo

(5.38)

∂ denotes differentiation along the normal to the reservoir area Ar of the where ∂n volume Vr and directed away from the interior of Ar . By Assumption of average values over the integral, Eq. (5.37) can be written as below.

∂ λo A r ∂ p = ∅Vr Bo ∂n ∂t



So Bo

(5.39)

The left side of Eq. (5.39) is known as the Darcy’s law for oil phase, so Eq. (5.39) may write as below. ∂ qo = −∅Vr ∂t



So Bo

(5.40)

Equation (5.40) for gas phase is written as follows:  ∂ Rs So ρ g Sg + qg = −∅Vr ∂t Bo

(5.41)

The surface gas oil ratio is written as: R=

qg Bo Bo + Rs = λ + Rs qo B g Bg

(5.42)

At last, by substitution qo and qg from Eqs. (5.40) and (5.41) into Eq. (5.42) and changing the time variable of differentiation into pressure variable, Eq. (5.43) is achieved.   Sg d B g Bg So d Rs λSo d Bo d So = fo − + + (5.43) dp Bg dp Bo dp Bo dp where f o = (1 + λ)−1

(5.44)

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Equation (5.43) is recognized as Muskat’s material balance equation or the zerodimensional approximation of the depletion problem. This equation is also known as the semi-steady state depletion equation. Since it represents an integral average of the true flow equation where saturation and pressure gradients are considered uniform [88].

5.7.2 Integral Material Balance Equations Muskat’s material balance equation is a linear relation which can be written as the following relationship. {E x pansion} − {W ithdrawals} + {I n f lux} = 0

(5.45)

Consider a saturated oil reservoir with an active aquifer where gas-oil contact remains constant as the injected gas in the gas cap slowly and freely disseminate into the oil zone, the only region limited to fluid withdrawals. The initial gas saturation in the oil saturation and the oil saturation in the gas cap is zero and residual water through the hydrocarbon zone is constant. All the fluids are incompressible. The material balance of this system based on Eq. (5.45) may be stated as:  Bg − Bgi {E x pansion} = N (Bt − Bti ) + N m Bti Bgi  N (1 + m)Bti Swi Btw − Btwi + 1 − Swi Btwi N Bti + (1 + m)c f p 1 − Swi   {W ithdrawals} = N p Bo + N p Bg R p − Rs + W p Bw {I n f lux} = G i Bg + We

(5.46) (5.47) (5.48)

Formation compressibility and two-phase volume factor are defined as follows: 1 V p V pi p   Bt = Bo + Rsi − Rs Bg cf = −

(5.49) (5.50)

By substituting Eqs. (5.46), (5.47) and (5.48) into Eq. (5.45), and the following material balance equation is obtained:

5 Gas Injection for Pressure Maintenance in Fractured Reservoirs

199

    N p Bt + R p − Rsi Bg − We + W p Bw    B −B  N= B −B (1+m)B S (1+m)Bti c f p tw twi + Bt − Bti + m Bti gBg gi + 1−Swti wi Bt 1−Sw i

i

wi

(5.51)

i

Rp is the net cumulative gas-oil ratio and Wp is the net cumulative produced water which respectively includes the injected gas and water [89].

5.7.3 Differential Material Balance Equation The same unified approach used for integral equation is followed. Differential MBE is not applicable for under saturated reservoirs and the expansion of connate water and pore volume will be neglected. The treatment of water influx and gas injection will be incorporated after the basic MBE is derived. System E x pansion     m 1 − Swi d Bg So Bg d Rs Sg d B g So d Bo + + − = Vp − Bo dp Bo dp Bg dp Bg dp

(5.52)

Oil Pr oduction     m 1 − Swi d Bg So Bg d Rs Sg d B g So d Bo + + − × fo = Vp − Bo dp Bo dp Bg dp Bg dp

(5.53)

where in the absence of gas reinjection, gravitational and capillary effects: f o = (1 + λ)−1

(5.54)

If a fraction r of the produced gas is reinjected, the above mobility ratio term must be replaced as follows: λ = (1 − r )λ − r Rs

Bg Bo

(5.55)

The rate of oil desaturation is calculated as follows:     m 1 − Swi d Bg So Bg d Rs Sg d B g d So So d Bo = Vp − + + − Vp dp Bo dp Bo dp Bg dp Bg dp × fo + V p

So d Bo Bo dp

(5.56)

where the left side of Eq. (5.56) is the rate of oil desaturation. The right side of this equation is the summation of oil production and oil shrinkage. The final form of

200

A. Jamili et al.

the differential material balance is expressed as follows:     1 d Bg Bg d Rs 1 d Bg λ d Bo d So = So + + − (1 + m) 1 − Swi dp Bo dp Bg dp Bo dp Bg dp  dWe We d B g − + (5.57) (1 + λ)−1 Bg dp dp e where dW is water encroachment term directly into the system expansion and We is dp the net water influx per unit original hydrocarbon system pore volume. The injected gas in must be of the dispersed type [90].

5.8 Pressure Maintenance in Fractured Reservoirs 5.8.1 Block-To-Block Process The block-to-block interaction is governed by fracture permeability, fracture capillary pressure, fracture relative permeability, re-infiltration phenomena, and capillary continuity. These concepts will be discussed in detail next.

5.8.1.1

Fracture Permeability

There are two fracture permeability types: Intrinsic Fracture Permeability, Kff . The intrinsic fracture permeability is associated to the conductivity measured during the flow through a single fracture or through a fracture network, independent of the surrounding rock (matrix). It is, in fact, the conductivity of a single channel (fracture) or a group of channels (fracture network). In this case the flow cross section is represented only fractures void areas (extending the surrounding matrix area) [91]. Conventional Fracture Permeability, Kf . The intrinsic fracture permeability disregards the rock bulk volume associated to the single fracture; on the contrary, in the conventional fracture permeability the fracture and the associated rock bulk form a hydrodynamic unit [91].

5.8.1.2

Fracture Capillary Pressure

Three models are used for fracture capillary pressure as follows: 1: 2:

Pcf = 0, Pcf = constant, and

5 Gas Injection for Pressure Maintenance in Fractured Reservoirs

3:

201

Pcf = Pcf (S w )

The first model has been widely used in numerical simulation, but it seems that this model does not have physical sound. One may consider zero fracture capillary pressure when only one phase flowing in the fracture or the pressures of two phases is equal (like capillary end effect) which can be for very large fracture that may not existed [92, 93]. The second model is based on the relationship between two flat parallel plates. The equation for capillary pressure is: Pc f =

2σ cos θ r

(5.58)

where r is the fracture aperture. The third model assumes that the fracture capillary pressure curve has a shape similar to that of porous medium but with major differences in curvature. It is reasonable to expect the fracture capillary pressure to have a very low irreducible wetting-phase saturation compared with the matrix. The consequence of saturation-dependent fracture capillary pressure is that full capillary continuity between stacks of matrix blocks may be expected. Dindoruk et al. [94] proposed the following expression for the fracture capillary pressure: Pc f =

Pc0f

   So − Sor f − σ f ln 1−S or f

n f   

Pc f ≥ Pc0f

(5.59)

where S orf is the residual liquid saturation in the fracture space, Pcf 0 is the fracture threshold capillary pressure and σf is the logarithmic slope of the fracture capillary pressure. Pcf 0 is obtained from the adjustment of the initial production data, while it is estimated by matching calculated and measured data of the entire test period. Dindourk et al. [94] matched Pcf 0 and σf for every experiment even for similar experimental setup. They justified using Pcf 0 and σf different and for similar situation to the scratch of the fracture surface. Fracture capillary pressure has a significant effect on drainage performance across a stack of matrix blocks. The flow across a fracture is very sensitive to fracture capillary pressure due to capillary pressure driving force for liquid film flow [94, 95]. Firoozabadi et al. [95] reported 30 to 40 Psi (207 to 276 kPa) fracture capillary pressure by assuming fracture surface covered with cones.

5.8.1.3

Fracture Relative Permeability

The relative permeability of the fractured reservoirs is one of the important problems in studying these kind of reservoirs. Some of the published experimental results indicate using two relative permeability curves during the simulation of the fractured reservoirs with wide fractures (Zero capillary pressure). One for matrix and the other one (two straight lines) for fracture system. In this case, the capillary pressure in

202

A. Jamili et al.

the fracture is zero so, the relative permeability becomes two diagonal straight lines for the fracture or fracture relative permeability is a linear function of saturation. For the matrix, the usual relative permeability curves are being used depend on the wettability of the matrix. On the other hand, some other publications are based on using the same relative permeability curve for reservoirs with narrow fractures, which explains the behavior of the fluid in matrix and fracture. In this case, the capillary pressure of the fracture is not zero, but the permeability of the fracture is huge. The line with a little curvature or no curvature will be used for the non-wetting fluid’ relative permeability curve and a curved line will be used for the wetting phase because, the wetting phase is continuous and fills the small pores and the other fluid flows through the larger pores. Edgar et al. [96] believed that generalized Darcy model might not be appropriate for flow in filled fractures, because the porous medium does not allow the possibility of ‘blobs’ of one phase transported by another continuous phase. It states that one phase can only move upon establishing a continuous flow path. Substituting the equation for permeability of fractures proposed by Witherspoon et al. [97]: k f = 84 × 106 × w 2f

(5.60)

wf is the fracture width. Into Darcy’s law results in the following equation for water relative permeability in a fracture: kr w f = −12μw

vw w 2f ddpx

(5.61)

wf is the fracture width. μw is water viscosity. vw is water velocity. dp is pressure gradient. dx Equation (5.60) shows that different relative permeability curves result from different pressure gradient in the fracture. When capillary pressure in the matrix is large relative to viscous force in the fracture, oil is expelled into the fracture at a rate independent of the viscous forces. The velocity of the water, the oil blob growth, and the roughness of the fracture wall determine the pressure gradient at which the blobs are pushed downstream; however most numerical reservoir simulators are written under the assumption of a Darcy flow model for the fractures. Therefore, specific relative permeability curves must be obtained for a given injection flow rate [96]. Because matrix-fracture interaction is controlled by injection rate, it is apparent that the relative permeability relationships for wetting and non-wetting phases are rate dependent. Wetting and non-wetting fluids flow simultaneously in both fractures and the matrix. The combination of these two flow processes results in a combined relative permeability behavior that has not been determined [96].

5 Gas Injection for Pressure Maintenance in Fractured Reservoirs

203

McDonald et al. [98] mentioned that pressure change effects fracture relative permeability which does not consider in the simulators. Jones et al. [99] showed that it does whenever flow is laminar in the fracture and relative permeability depends on fracture aperture, and on roughness of the fracture surface.

5.8.1.4

Capillary Continuity and Re-infiltration Phenomena

In the dual-porosity concept, fracture acts as a conduit for fluid flow and matrix acts as a source or sink. It means that produced oil from the matrix will flow through the fracture. In 1979, Saidi et al. [100] observed that the drained oil from the upper matrix is sucked or re-infiltrated (or re-imbibed) into the lower matrix instead of flowing through the fracture network. They questioned the validity of the dualporosity concept, which assumed that matrix is a discontinuous medium and drained oil is produced through the fracture only. This phenomenon is called block-to-block interaction. Prediction of naturally fractured reservoir behavior without considering block-to-block interaction will be optimistic. From that date, a lot of work has been done and re-infiltration and capillary continuity terms are used frequently in the naturally fractured reservoirs literature. First, let’s define capillary continuity and re-infiltration terms.

Definition of Re-infiltration and Capillary Continuity Re-infiltration can be defined as a fraction of drained oil from the upper matrix block that re-infiltrates (re-imbibes) into the lower matrix block, but the re-infiltrated phase could be a continuous or discontinuous phase. If there is not phase continuity between matrices, then every block produces separately. Therefore, the effective height is equal to individual matrix block height and the overall recovery is the same as individual matrix block recovery. On the other hand, capillary continuity is an especial case of re-infiltration where the re-infiltrated phase has to be a continuous phase. Figure 5.4 shows the difference between capillary continuous and discontinuous cases. As one may see from Fig. 5.4, phase continuity increases the effective block height during gravity drainage with respect to the discontinuous case, where each block drains independently. Figure 5.5 shows how re-infiltration and capillary continuity occurs. In Fig. 5.5a, re-infiltration occurs through contact areas and liquid bridges, where capillary continuity exists, because in this case re-infiltrated phase (oil) is a continuous phase. In Fig. 5.5b, reinfiltration occurs by liquid film, but capillary continuity (phase continuity) may not exist necessary especially toward the end of gravity drainage process, where the oil drainage rate is small, and it is hard for the re-infiltrated phase (oil) to be continuous. So it is possible that there is complete re-infiltration between blocks without having capillary continuity as Saidi [85] mentioned for history matching of Haft Kel field (Iran). Since the drained oil from the upper block prefers to re-infiltrate into the lower

204

A. Jamili et al.

Fig. 5.4 Continuous and discontinuous matrices [101]

Fig. 5.5 Schematic of re-infiltration and capillary continuity [91]

block instead of producing through the fracture, as observed by Saidi et al. [100], re-infiltration and capillary continuity delay the oil production. In re-infiltration, the re-infiltrated phase (usually oil) does not have to be a continuous phase; therefore, re-infiltration does not necessarily change the ultimate recovery from a stack of matrix blocks. But in capillary continuity, since the reinfiltrated phase (usually oil) is continuous, so the ultimate recovery will be increased significantly from a stack of matrix blocks as one may see from Fig. 5.4. Figure 5.6 shows the definition and difference between capillary continuity and re-infiltration. As one may see, capillary continuity is a special case of re-infiltration. The problem

5 Gas Injection for Pressure Maintenance in Fractured Reservoirs

205

Fig. 5.6 Definition of Re-infiltration and capillary continuity

Fig. 5.7 Constriction coefficient method [92]

is that these two phenomena overlaps especially during the early life time of a naturally fractured reservoir, which makes it hard to distinguish between them and predict the reservoir behavior correctly. Some methods have been proposed to analyze and distinguish between re-infiltration and capillary continuity which will be discussed next. All methods are similar in terms of assigning saturation to the fracture. Constriction Coefficients Method. Saidi et al. [102] used constriction coefficients to simulate capillary continuity and re-infiltration processes. Constriction coefficient between two blocks is defined as: C=

qc qo

(5.62)

where qo is the total flow from the upper block and qc is the flow through the contact area or liquid bridges between two blocks (Fig. 5.7). Basically, this method provides capillary continuity and re-infiltration between blocks by assigning fracture saturation. Saidi [103] often assumed capillary discontinuity between a stack of matrix blocks. He argued that if the fracture aperture (critical fracture aperture) is about

206

A. Jamili et al.

0.002 in. (0.05 mm) or more, capillary continuity between a stack of blocks cannot be realized. It seems that he found the above number by balancing gravity and capillary forces of a liquid droplet between two flat parallel plates just before detachment. His model has nothing to do with direct contact areas between two matrixes. The results of experimental work of Sajadian et al. [104] showed that there is a critical fracture aperture for stabilizing liquid bridge. They derived the following equation for critical fracture aperture (tcf ) by equating the weight of the droplet and capillary force just before detachment as:  tc f =

8σ 3ρg



1/2 =

8 × 30 × 10−3 mN ×

N 3 × (0.724 − 0.001186) cmg 3 × 9.0806 kg ×

= 0.336 cm

1/2

1m 100 cm 1 kg 1000 g

(5.63)

They assumed capillary continuity for fracture aperture less than Eq. (5.63) and discontinuity for fracture aperture more than Eq. (5.63). Capillary continuity cannot be described adequately by using Eq. (5.63) for Firoozabadi et al. [105] experiments. Pcf –krf method. This method treats a fracture as a porous media like the matrix and assign Pcf and krf to the fracture. Kazemi et al. [106] used fracture capillary pressure as a matching parameter in reservoir history matching. Saidi et al. [107] interpreted using fracture capillary pressure as a matching parameter as capillary continuity between blocks and he wrote: “application of fracture capillary pressure for better reservoir history match by Kazemi may mean an effective taller block height”. Basically, fracture saturation will be assigned by using this method; therefore, capillary continuity will be achieved between two blocks that surround the horizontal fracture. Firoozabadi et al. [95, 105], Horie et al. [101], and Dindourk et al. [94] used this method to investigate the capillary continuity phenomenon. Zero fracture capillary pressure means there is no capillary continuity between blocks and each block drains independently [92, 93]. Labastie [108] investigated capillary continuity by changing the contact area between blocks. He concluded that fracture permeability is the controlling parameter of capillary continuity between blocks. Festoy et al. [109] modified single porosity simulator for fractured reservoirs to consider the effect of contact area between blocks. They showed that even with a small contact area between blocks, the upper blocks will drain during the life of the reservoir. They assumed that all of the drained oil from the upper block will be sucked by the lower block. The contact area between the blocks affects the drainage time schedule. It means with small contact area the drainage will be longer that if there is larger contact area. However, the ultimate recovery for small or larger contact areas does not change. Pseudo-capillary pressure curve. Assigning pseudo-capillary pressure curve to describe capillary continuity in a stack of matrix blocks is another method. This method has been used by Thomas et al. [110] and Fung [111] and is based on vertical equilibrium (VE) concept [112]. Fung [111] included re-infiltration in a computational grid cell of a dual-porosity reservoir simulation that contains a stack of matrix

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blocks. He first refined the grid into the level of individual matrix blocks, and then adopted a dual-porosity approach to calculate the drainage performance of the entire stack (computational grid cell). In this model, the re-infiltration effect is considered by allowing the communication between each fracture and the matrix block below. The amount of re-infiltration is characterized by a fractional re-infiltration parameter. Once the drainage rate versus the average stack saturation is obtained, the information is then used to construct a pseudo-capillary pressure curve for the full-scale reservoir simulation. Fung [111] showed a reasonable agreement between his model and fine grid simulation results. As one may notice, fracture saturation will be assigned between blocks by constructing a pseudo-capillary pressure curve for the entire stack of matrix block and this fracture saturation will achieve the continuity between blocks. Connection-dependent relative permeability method. Por et al. [113] used connection-dependent relative permeability between matrix and fracture nodes. When fracture oil saturation is below threshold saturation, oil will flow to the lower matrix block instead of flowing through the fracture toward the producing wells. Once threshold fracture oil saturation is exceeded, then relative permeability will increase to one linearly. Similar to Fung [111], Por et al. [113] used the dual-permeability approach to account for capillary continuity. Again, by using this method, assigned fracture saturation will establish the continuity between every two blocks. Discussion. Capillary continuity between blocks will be achieved by assigning saturation in the fracture and maintaining phase continuity between blocks, which is the common point of the discussed methods. This point will significantly help understanding and analyzing the re-infiltration and capillary continuity literature. There are several factors affecting re-infiltration and capillary continuity. Coats [114] investigated the effect of horizontal pressure gradient in a horizontal fracture on re-infiltration by using fine grid simulation. Coats [114] calculations showed no re-infiltration for a fracture with 11° with horizontal plane. Horie et al. [101] studied the effect of contact points between matrix blocks and showed that there will be continuity between matrix blocks in the case of direct contact or fine sand grain between matrixes. Festoy et al. [109] modified single porosity simulator for fractured reservoirs to consider the effect of contact area between blocks. They showed that even a small contact area between blocks helps phase continuity (or capillary continuity), and the upper blocks will drain during the life of the reservoir. Fung and Collins [111] suggested that when a reservoir shows a high degree of capillary continuity, the dual-permeability model is more appropriate. In fact, the effects of partial matrix continuity may be approximately accounted for if an effective matrix block height that is taller than the actual matrix block height is used. However, using taller matrix block alters the amount of recoverable oil resulting from gravity. It seems that pressure drop between matrices is neglected in this model. Vidal [115] did experiments and showed that the falling film is an important mechanism in oil recovery from a stack of matrix blocks by gravity drainage. Vidal [115] concluded that:

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– A good knowledge of the history of cycles of saturation and desaturation in the stack of matrix blocks. – The accurate description of the block shape and boundary conditions, which affect the ratio of influence of viscous and capillary forces. – Reliable and accurate curves of relative permeabilities and capillary pressures for the successive cycles. Saidi et al. [100] observed that drained oil from the upper matrix preferred to travel through the lower block instead of passing along the fractures and by this way, re-infiltration only delays the oil production by gravity drainage as shown in Fig. 6 of their paper. Re-infiltration did not change the ultimate recovery from a stack of matrix blocks. In other words, they did not consider the effect of capillary continuity which increases the ultimate recovery from a stack of matrix blocks. They just modified the single block concept. In single block concept, the drained oil from a matrix is produced, so each matrix drains independently. They defined the degree of block-block interaction (α) as the fraction of drained oil from the upper block which re-infiltrate into the lower block and can be varied between 0 and 1. Figure 2 of their paper is a special case of continuity Eq. (5.46) that was derived by Cardwell and Parsons [116] for constant instant of time. ∂ Vy ∂ So = −ϕ ∂Z ∂t

(5.64)

From Fig. 6 of Saidi et al. [100] paper, one may expect that if a matrix is supplied by a certain infiltration rate, the saturation profile does not change until the re-infiltration rate changes. They investigated the effect of re-pressurizing on oil recovery and concluded that re-pressurizing could increase oil recovery; however, it will take a longer time to respond to re-pressurizing in the case of strong degree of block to block interaction. At the end, they concluded that the actual degree of block to block interaction could not be estimated from primary performance (because at early stage the performance is dominated by relative permeability curve and not by block-block interaction). Good geological and geometrical properties of both the matrix blocks and the fracture are required. Firoozabadi et al. [105] studied the re-infiltration process in fractured reservoirs. They concluded that at early time of production, falling film is the dominant oil production mechanism, after a while re-infiltration will be dominant mechanism. However there still be falling film flow which cannot be neglected. However, some of the oil in the falling film may be re-infiltrating (or re-imbibing) into the matrix. This phenomenon has not received attention in the literature. Firoozabadi et al. [117] tried to develop a theory for re-infiltration phenomenon and add it to the current simulators. They assumed incompressible fluids, infinite gas mobility and zero fracture capillary pressure. Firoozabadi et al. [117] performed analytical investigations of the re-infiltration process in vertical and tilted matrix blocks. They concluded that in a two-phase gas–liquid system, the fracture network is basically a conduit for the gas and the liquid path is through the matrix blocks.

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Tan et al. [118] approximated the drainage rate for all the matrix blocks in a stack by using only two drainage curves: (1) the drainage curve for the first block where reinfiltration is absent, and (2) the drainage curve for the rest of the blocks (represented by the drainage curve of the second block) where re-infiltration can take place. They assumed that oil drained from the upper matrix blocks will completely re-infiltrate into the lower blocks, provided the stack is surrounded by gas. After finding the drainage curve for first and second block by fine grid simulation, the drainage curve of the stack is obtained by material balance. Also, they assumed infinite gas mobility, zero fracture capillary pressure and kr o f changes linearly with So f . Tan et al. [118] scaled drainage curve by using initial drainage rate (maximum rate) and the average gas saturation of the block approaches Sge at an infinite time to obtain a similar form of solution for the block saturation history. Actually, they followed Coats [114] proposal for scaling. They postulated that, under the condition that there is no fracture capillary pressure, if the matrix blocks are tall or the threshold pressure is small, a single relationship between qi , S g + and t + can be obtained to approximate the drainage rate when matrix capillary pressure (due to reservoir pressure) and block heights vary. They defined qi , S g + and t + as follows: Sg+ =

Sg Sge

 0 ∗ ρgh − Pcm  t =t  0 ρgh − Pcm   0 ∗ hi   ki ρgh − Pcm i qi∗ Sg+ qi = ∗  0 k ρgh − Pcm i h ∗

(5.65)



+

(5.66)

(5.67)

Superscript * refers to the data of the reference matrix block. They concluded that there are three distinct drainage characteristics curve for each case. Those are: (1)

(2) (3)

the first curve belongs to the drainage of the top block where the matrix block is not subject to re-infiltration, but it is in capillary contact with the second block, the second series of curves represent the drainage performance of blocks 2 to N–1 where both re-infiltration and capillary continuity process take place, and the third curve belongs to the drainage of the bottom matrix block where re-infiltration exists, but fracture capillary pressure is zero at the bottom face.

To account for capillary continuity between blocks they used Pcf 0 = 0.0088 psi and σ f = 0.0023 psi. They showed a good agreement between their results and fine grid simulation results. It seems that by increasing the number of matrix blocks in the stack their model will have significant deviation from fine grid simulation. Horie et al. [119] showed that there would be continuity between matrix blocks in the case of direct contact or fine sand grain between matrices. Firoozabadi et al. [105] investigation showed that at early production time there would be continuity

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through the matrix blocks and after a while building liquid bridge and detachment will continue between blocks. However, the rate of building liquid bridge and detachment will decrease by increasing production time. They concluded that fracture gas/oil capillary pressure and fracture-liquid relative permeability (the combined effect is fracture-liquid transmissibility) is the controlling factors of production mechanism. Saidi et al. [107] argued that the presence and degree of capillary continuity in a fractured reservoir could be estimated by analyzing by the variation of oil production rate versus variation of oil column thickness or the variation of oil–water contact. When oil column thickness reaches a constant thickness for some years, oil production either drops sharply or reaches a small rate. It then means either near fully or negligible capillary continuity. The experimental work of Firoozabadi et al. [105] verified their statement. Based on Fig. 5.4, for continuous case, the saturation is continuous function of block height or it differentiable. For noncontinuous case, the function of saturation versus block height is not differentiable at fractures, because there is a jump in saturation. The degree of continuity depends on how much the saturation at the bottom face of the upper block is close to the saturation at the top face of the lower block. It is questionable that if a new differentiate function version could be defined which will be able to determine the degree of continuity.

5.8.2 Gravity Drainage It is believed that gravity drainage is the most important mechanism for oil recovery in naturally fractured reservoirs [91, 103], especially in gas invaded zone, and water invaded zone. Gravity drainage is caused by fluid density difference in the fracture and adjacent matrix. Gravity drainage mechanism can be categorized as: free-fall or forced gravity drainage versus equilibrium or non-equilibrium gravity drainage. In free-fall gravity drainage, gravity difference is the driving mechanism, while in forced gravity drainage, gravity difference and viscous pressure difference are the driving forces. In equilibrium gravity drainage, there is no mass transfer (diffusion) between oil and gas, but there is mass transfer (diffusion) between oil and gas in nonequilibrium gravity drainage. Mass transfer occurs by convective transport, molecular transport (molecular diffusion, thermal diffusion, and pressure diffusion), and mechanical dispersion mechanisms. The combination of molecular diffusion and mechanical dispersion mechanisms is called hydrodynamics dispersion mechanism, or convective mixing mechanism. Mass transfer between oil and gas can affect interfacial tension, capillary pressure and relative permeability curves and reduce the holdup zone height. Because of these reasons, gas injection can be very efficient process in naturally fractured reservoirs (oil swelling). Depending on what type of gas is injecting interfacial tension gradient can be favorable or not. The differences of capillary pressure lead to flow either from or towards the fracture which is called “Capillary pumping”. In case of nitrogen

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injection, capillary pumping helps the production from the matrix by decreasing the interfacial tension in the matrix, while CO2 injection tends to decrease the production by increasing the interfacial tension in the matrix [120]. Based on simulation, Saidi concludes that early gas injection can increase oil recovery. Qasem et al. [121] and Aniefiok et al. [122] simulated gas injection and concluded that early gas injection is an efficient method, but low matrix permeability oil recovery deteriorates. No experimental data is available to confirm the increment of oil recovery by early gas injection. Miscibility of gas into oil can be achieved by increasing reservoir pressure to minimum miscibility pressure and in this sense miscible gas injection in naturally fractured reservoirs is not economic, because a huge amount of gas has to be injected to increase reservoir pressure to minimum miscibility pressure. Therefore, enhancing miscibility of gas into oil by economic methods, such as, diffusion and convection mechanisms is a very important method. Effect of injected gas composition can be investigated by using single block model [123] to simulate the drainage behavior of a single matrix block surrounded by fractures by injection of different gases, such as, N2 , CO2 , dry gas (mainly C1 ), and wet gas into the fracture at the top and producing oil from the matrix bottom. Also, matrix block height is very important in gravity drainage behavior. Parachor method can be used to assign interfacial tension to grid nodes and investigate interfacial tension gradient between the matrix and surrounded fractures caused by injecting different gases. Dykstra [124] investigated oil recovery by gravity drainage. He mentioned there is a certain height in the matrix that above that height gas saturation is below critical saturation and only oil drains and when gas-oil front passes that height, gas becomes mobile and there will be two phase flow. Therefore, one may conclude that at early time production, since only oil drains, fracture capillary pressure could be zero and when gas-oil front passes that certain height there will be two phase flow and hence fracture capillary pressure will not be zero. Then oil bridges forming and breaking takes place [105] and at this stage fracture capillary pressure will oscillate between zero (breaking bridge moment) and non-zero (forming bridge moment). At the end only, gas fills the fracture and fracture capillary pressure will become zero again. Of course, fracture aperture is a key parameter that should be considered. Schechter et al. [125] reviewed the gravity drainage modelling literature and concludes that there are four main models for equilibrium gravity drainage process modelling. They are: 1. 2. 3. 4.

Cardwell-Parsons-Dykstra (CPD) model (this model will be reviewed in details) [116, 124] Nenniger-Storrow (NS) model [126] Pavone-Bruzzi-Verre (PBV) model [127] and Luan model [128] Also, there are empirical models proposed by Aronofsky et al. [129] as: η = 1 − e−βt

(5.68)

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Cardwell and Parsons [116] were the first who describe gravity drainage process with theoretical sound basis. They divided the matrix into two parts. The upper part is unsaturated region where So is less than 100% and the lower region is 100% oil saturated. Oil velocity for upper region is:  dp ∂ So k ρg − Vu = μ d So ∂z

(5.69)

and for lower region (saturated): Vu =

 H kρg 1− μ h0

(5.70)

where H is the height of the top of the 100% saturated region after equilibrium has been reached and h is the height of the top of the 100% saturated region before equilibrium has been reached. They ignored capillary pressure and gas pressure to develop a simple form of continuity equation to solve. They calculated front velocity as:     H dz ρg k1 1 − L−z − kd = (5.71) dt μϕ (1 − So ) where k1 is the permeability of the medium to the fluid at 100% saturation, kd is the permeability at the saturation just above the front, L is the draining column length and z is the front location. Above the front, the following relationship was derived between z and So : z=

ρg dk t μϕ d So

(5.72)

To find the initial condition, they differentiated Eq. (5.71) and put it equal to zero to find the maximum front velocity which will be initial velocity. They assumed that at zero oil saturation, oil permeability is zero. Dykstra [124] modified this assumption by defining new saturation variable and assuming zero oil permeability at residual oil saturation as: So =

So − Sor 1 − Sor

(5.73)

He argued that this definition changed B in the permeability equation: k = SoB

(5.74)

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He found B equal to 3.2, while Cardwell and Parsons [116] found 3.5. Then he followed their approach and derived front velocity equation as: ⎛$ B %⎞   μϕ  z  B−1  H kr (Si ) 1 − L−z − t ⎟ dz ρgk ⎜ ⎜ ⎟ =  B   ⎝ ⎠  dt μϕ μϕ  z B−1 Si − t

(5.75)

Also, he calculated oil recovery as follows: R =1−

(L−z)Soi Boi

+

)z

So 0 Bo dz

L Soi Boi

=

1 Boi z − L L Soi Bo



z 0

So dz Bo

(5.76)

The relation between So and z above the front is:  So =

zμϕ  Bkρgt

1 B−1

(5.77)

Dykstra mentioned that the above equations are valid when S g < S gc . If S g > S gc , then gas becomes mobile and the possibility exists that two gas-oil fronts will occur as described by Martin [130] -one moving up and one moving down. Based on the author’s knowledge, there is not research work on this idea. Most of research has been done with dead oil, so it seems that this idea worth looking. Hasanzadeh et al. studied the gravity drainage in a fractured physical model. They investigated two types of gravity drainage including free fall gravity drainage (FFGD) and forced gravity drainage (FGD). They also used nitrogen, carbon dioxide and air as non-wetting phase and condensate and water as the wetting phase in their experiments. A detailed sensitivity analysis was performed by them that investigated the effect of fracture blockage, injection rate, dip angle, and type of wetting phase and injection fluid on the recovery factor (RF). A numerical simulation was conducted to simulate the behavior of their experiments. Their results showed that FGD using gas injection has a better performance than the FFGD. Also, they showed that in the cases with the variation of the vertical orientation, the RF is decreased. They stated that in the immiscible gas assisted gravity drainage obtained higher condensate recovery in comparison to water recovery due to lower interfacial tension and lower viscosity of condensate [131].

5.8.3 Capillary Hold-up Gas injection in fractured reservoirs increases oil recovery by gravity drainage, diffusion, and oil swelling. Gravity drainage is a mechanism that would help to recover oil due to density differences between oil in the matrix and gas in the fracture. However,

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Fig. 5.8 Capillary holdup in fractured reservoirs

the performance of gravity drainage is limited by threshold height, which depends on matrix capillary pressure, and matrix block size. Figure 5.8 shows the concept of capillary holdup. A typical capillary pressure versus saturation is shown in left side of Fig. 5.8. Threshold capillary height (hc ) can be calculated from the equilibrium between gravity and capillarity as: Pc = Pg − Po  hc =  ρo − ρg gh

(5.78)

where Pc is the capillary pressure. Pg is the gas pressure. Po is the oil pressure. ρo is the oil density. ρg is the gas density. In order to assure displacement of oil by gas in fracture, block height must be higher than threshold height (hc ). In the case of low permeable and small matrix block size, gravity drainage may not be an efficient mechanism. In such cases, diffusion for small matrix block size such as Ekofisk field (North Sea) or low permeable Asmari limestone reservoirs (Iran) can be an important recovery mechanism during high pressure gas injection. Diffusion reduces high interfacial tension which results in threshold height reduction. However, in case of lean gas injection, IFT will increase during the process which is the opposite of the first case. As oil in the matrix dissolves

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the injected gas from the fracture, the oil swells and produces into the fracture system. In fact, in several predominantly oil wet Iranian reservoirs such as “Ahwaz” reservoir where the height of the matrix blocks is normally between 3 and 5 m, the residual oil saturation in the water-invaded zone after 40 years remains very high and gas-oil gravity drainage seems to be an efficient mechanism for releasing the oil from such reservoirs. The matrix block height in Iranian reservoirs varies between 1 and 20 ft. Therefore, in some Iranian reservoirs such as “Gachsaran” in which the block height is less than 2 m, diffusion could be the dominant recovery mechanism.

5.8.4 Diffusion Few laboratory publications have been devoted to describing the diffusion mechanism in naturally fractured reservoirs. No laboratory work was published on the diffusion effects in naturally fractured reservoirs before 1990. During that period (before 1990), simulation studies were conducted to investigate the diffusion mechanism on oil recovery in naturally fractured reservoirs. The laboratory and simulation studies of diffusion as a recovery mechanism in naturally fractured reservoirs are discussed in what follows. Van Golf-Racht [91] and Saidi [103] were among the firsts that provided good explanation of molecular diffusion role in naturally fractured reservoirs.

5.8.4.1

Laboratory Studies

Morel et al. [132] conducted laboratory studies of the effect of diffusion in 1dimension on oil recovery in naturally fractured reservoirs. Figure 5.9 shows the layout of the experiments. The experiments were performed with cores of Paris Basin Chalk (0.105*0.105*1.1811 ft3 ). The permeability and porosity of the samples were Fig. 5.9 Diffusion experiment layout [120]

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Table 5.3 Description of 1-dimension diffusion experiments by Morel et al. [132] Experiment no.

Composition Initial gas Pressure of the mixture saturation in (MPa) (mole %) core (%)

Total experiment time (days)

Gas injection flow rate in the fracture (cm3 /h)

Injected gas

M3

C1 (44.1%), C5 (55.9%)

0

10.1

24

4 then 8

Methane

M4

C1 (52.4%), C5 (47.6%)

25

10.1

16

4 then 8

Methane

M8

C1 (45.8%), C5 (54.2%)

7.2

9.8

15

4

Methane

M5

C1 (52.4%), C5 (47.6%)

29

10.1

16

4 then 8

Nitrogen

M6

C1 (44.1%), C5 (55.9%)

0

10.1

73

4

Nitrogen

M7

C1 (50.7%), C5 (49.3%)

0

11.7

13

8, 12 then Nitrogen 16

2 md and 40%, respectively. Cores were saturated with a binary mixture of C1 –C5 . Methane or nitrogen was injected in the fracture. They investigated the effects of the diffusing gas (N2 or C1 ), gas flow rate in the fracture, and initial gas saturation in the core. The experiments were performed at 38.5 °C. Table 5.3 shows the details of the experiments. They concluded the following: 1. 2. 3.

4.

5.

Initial gas saturation has little effect on oil recovery. Pentane recovery is linear with time, which indicates that the recovery process in not governed by a pure diffusion mechanism. Pentane recovery by methane injection is 1.6 times faster than recovery by nitrogen injection at corresponding times. The pentane concentration in the gas phase in the core was 1.6 times higher for methane injection than for nitrogen injection. In the nitrogen injection case, saturation profiles along the core revealed a strong capillary end effect resulting in accumulation of oil in the matrix near the fracture (Fig. 5.10). When nitrogen is the diffusing gas, the flow rate has a small effect on pentane recovery; whereas flow rate greatly affects methane production.

Chukwuma [67] studied diffusion of CO2 into n-decane at 100 °F and 206 psia. Figure 5.11 shows the experimental setup. Glass rods of different diameters (2 mm, 3 mm, 4 mm, etc.) and Pyrex glass beads of 4 mm diameter were used as packing in the study. CO2 diffuses into n-decane from the top. He recognized that the density of a CO2 and n-decane mixture had an unusual behavior increasing up to 70% mol CO2 and then decreasing at higher concentrations of CO2 . The density change causes free convection in the vertical direction with the denser fluid flowing down. Free

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80 Case a (C1-C5)

70

Gas Saturation (%)

Case b (N2-C5) 60 50 40 30 20 10

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Distance from Fracture (m)

Fig. 5.10 Gas saturation profile after 39 days along the core for experiment no. M6 of Table 5.2 [120]

Fig. 5.11 Schematic of diffusion experiment [121, 122]

convection enhances CO2 mass transfer. It took only 45 min to saturate n-decane by density induced vertical flow while for other gases, such as methane, it takes several hours. He concluded that the free convection causes the effective diffusivity to be much higher than a typical molecular diffusion. For example, the asymptotic value of the effective diffusivity for carbon dioxide in n-decane is about 0.2 cm2 /sec at 206 psia and 1000 F whereas the molecular diffusivity for ethane in n-decane at the same temperature and pressure is about 5.0 × 10–5 cm2 /sec. Renner [133] used an experimental setup similar to Chukwuma’s [67] experimental setup (Fig. 5.11) to study CO2 and ethane diffusion into n-decane at 100 °F temperature and pressures up to 846 psia. As CO2 diffuses into the oil (n-decane),

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the pressure tends to drop in the CO2 space. As this occurs, the pressure raised by compressing the CO2 . From the movement of the piston rod and the linear position transducer on the gas metering vessel, the volume of CO2 injected to maintain constant pressure over the rock face as a function of time may be readily determined. Because CO2 mass transfer into oil (n-decane) results in swelling of the oil, the gas/oil interface will move as a function of time. Horizontal and vertical Berea core (2-in diameter and 6-in long) setups were used in the experiments to investigate the effect of gravity-induced convection on the observed mass transfer. He observed that the effective diffusivity of CO2 in n-decane in vertical cores is more than in horizontal cores which appears to be because of combined diffusion and gravity-induced convection processes. On the other hand, diffusivity of ethane in n-decane is not affected by the orientation of the core. Thiebot and Sakthikumar [134] studied gravity drainage and mass transfer in cylindrical cores surrounded by fractures (Fig. 5.12). They used limestone and chalk cylindrical core with a length of 40 cm and permeabilities of 60 md and 2 md, respectively. First, the core was saturated with live oil, representative North Sea light oil with a bubble point pressure of 180 bar at a reservoir temperature of 132 °C. Second, equilibrium gas was injected at the top of the core in the fracture and oil was produced from the bottom. Equilibrium gas is a gas in thermodynamic equilibrium with the live oil used in the experiment. Therefore, there is no mass transfer between the equilibrium gas and the live oil. Gravity is the recovery mechanism in this stage. The step was continued until oil production ceased (gravity drainage equilibrium). Third, methane or nitrogen was injected instead of the equilibrium gas. Mass transfer between nitrogen and methane as non-equilibrium gases and live oil in the core occurs in this stage. They concluded that injection of non-equilibrium gas leads to significant additional oil recovery even after gravity drainage equilibrium. Le Romancer et al. [135, 136] performed similar experiments as Morel et al. in 1-D conditions (Fig. 5.9) on chalk cores saturated with a methane-pentane mixture in the presence of different water saturations and with three diffusing gases: nitrogen, Fig. 5.12 Experiment setup of Thiebot and Sakthikumar [123], Darvish et al. [124], and Karimaie [125]

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methane, and carbon dioxide. Table 5.4 shows the details of the experiments. Similar to Morel et al., it was observed that there is an accumulation of oil in the matrix near the fracture surface when nitrogen is injected. The gas saturation profiles were similar to Fig. 5.10. Figure 5.13 shows the influence of the diffusing gas type on the pentane recovery. Based on Fig. 5.13, Le Romancer et al. [135] claimed that only nitrogen injection allows the obtaining of a constant and high pentane production rate. Therefore, it was concluded that in a diffusion process it is essential to keep the highest oil saturation near the fracture. From this point of view, nitrogen is an interesting candidate. Their carbon dioxide diffusion experiment is simulated in this dissertation. Riazi et al. [137] conducted a laboratory experiment to study the diffusion mechanism at reservoir conditions (Fig. 5.14). In their experiment, diffusion of N2 into a mixture of oil and gas (in matrix) at 270 bar and 403 K was studied. The oil components were N2 , CO2 , C1 , C2 , C3 , iC4 , nC4 , iC5 , nC5 , C6 , C7+ . Cylindrical core samples (8.3 cm height and 5.1 cm diameter) from the Ekofisk field in the North Sea were used in a vessel with limited free-volume which was purged with nitrogen immediately following depressurization from the initial bubble point at 382.8 bar to 275.9 bar. Porosity and permeability of a core sample were 0.31 and 0.29 md, respectively. A core sample was supported by the vessel so that all the surfaces were open to the free-volume. The diffusion process was monitored by analysis of the gas composition in the free volume with time. Their simulation of the experiment will be discussed in the next section. The results showed the importance of diffusion in recovery of oil components. Le Gallo et al. [120] used the same setup (Fig. 5.9) as Morel et al. [132] to study diffusion in 1-dimension in Paris Basin Chalk. A description of the experiments is given in Table 5.5. Le Gallo et al. [120] concluded that capillary phenomenon inside the matrix contributes to liquid flow towards the fracture and may be enhanced if interfacial tension is increased by injecting of a gas such as nitrogen. Table 5.4 Description of 1-dimension diffusion experiment by Le Romancer et al. [135, 136] Test

Injected gas

Water saturation (%)

Initial gas saturation (%)

Composition Gas flow of the mixture rate in the (mole %) fracture (cm3 /hr)

Pressure (MPa)

Total test time (days)

Mure

C1

0

0

10.1

23

M6

N2

0

C1 = (44%) C5 = (56%)

M10

N2

30

60

M11

C1

30

39

M12

C1

13

52

M13

N2

13

M25

CO2

11

4

73

49 C1 = (28%) C5 = (72%)

6.3

95

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Pentane Recovery (initial mass %)

100

CO2/C1-C5

90

C1/C1-C5

80

N2/C1-C5

70 60 50 40 30 20 10 0

0

500

1000

1500

2000

2500

Diffusion Time (hour)

Fig. 5.13 Effect of diffusing gas type on pentane recovery [126]

Fig. 5.14 Schematic of high-pressure experimental cell [137]

Darvish et al. [138] conducted an experiment to study the effect of CO2 injection into cylindrical cores (60 cm long and 4.6 cm diameter) from North Sea (Maastrichtian chalk) surrounded by fractures (Fig. 5.12) at reservoir conditions. Permeability and porosity of the core were 4 md and 44%. The oil components were N2 , CO2 , C1 , C2 , C3 , C4 , C5 , C6 , C7+ . The volume between core and core holder (fracture) was filled by Wood’s metal. After saturating the core with the oil mixture, a fracture volume surrounding the core was created by heating the solid core and melting the wood’s metal and draining the melted wood’s metal from the space between the core

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Table 5.5 Description of Le Gallo et al. 1-dimension diffusion experiment [120] Test

Injected gas

Water saturation (%)

Initial gas saturation (%)

Composition Gas flow of the mixture rate in the (mol%) fracture (cm3 /h)

Pressure (MPa)

Total test time (days)

M5

N2

0

29.5

C1 = (52.4%) 4–8 C5 = (47.6%)

10.2

16

M29

C1

0

0

C5 = (100%)

4

10.2

65

M30

C1

0

0

C1 = (37%) C5 = (49%) C15 = (14%)

4

10

95

and the core holder. The oil circulation was continued until fracture and core both were completely saturated with oil. Once the sealing material from the fracture was removed, the oil in the fracture was displaced by injecting CO2 at high flow rate. Then CO2 was injected at the top of the core and oil was produced from the bottom. The experiment was performed at 300 bars at 130 °C. The Eclipse compositional simulator was used to simulate the experiment. Mass transfer between gas in the fracture and oil in the matrix is not considered in Eclipse. Gas–gas and oil-oil diffusion are allowed in Eclipse only. Therefore, Darvish et al. [138] had to initialize the fracture with oil and gas phases of rich CO2 to initiate diffusion between oil in the matrix and oil in the surrounding fractures. The fracture was initialized with a mixture of 95 mol% CO2 and 5 mol% of the heaviest component. The fluid inside the fracture has a two-phase condition in which liquid phase has a very high concentration of CO2 . The presence of two-phase condition in the fracture with a high concentration of CO2 in its liquid phase would start the liquid–liquid diffusion from the fracture to the matrix. A zero gas and oil diffusion coefficient were assigned for the heaviest component. The simulation results showed that the key mechanism to recover oil from a tight matrix block is diffusion and gravity drainage has no significant effect. They recommended that the existing compositional simulators should be updated to consider gas (in the fracture)-oil (in the matrix) mass transfer on oil recovery. Karimaie [139] investigated gas injection (secondary recovery) and gas injection after water injection (tertiary recovery) in oil-wet carbonate cores. The objective was to investigate an EOR process for oil-wet carbonate fractured rocks. The core samples were 20 cm long and 3.8 cm diameter. He used C7 -C1 as oil. Porosity and permeability values were in the ranges of 8–25% and 1.5 to 130md, respectively. His experimental setup and procedure were the same as for the Darvish et al. [138] experiments (Fig. 5.12). Secondary gas injection experiments were done at 220 bars and 85 °C. In secondary gas injection experiments, equilibrium gas was initially injected to displace oil by gravity. Equilibrium gas was in equilibrium with the oil in the core and therefore, there was no mass transfer between the equilibrium gas and the oil. Once oil production ceased, the second period of pure CO2 or N2 injection followed. In tertiary gas injection, first oil was displaced by water injection at 220

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bars and 85 °C. Then equilibrium gas injection started at 210 bars and 85 °C followed by the second period of equilibrium gas, N2 or CO2 injection at 220 bars and 85 °C. He claimed that diffusion plays an important role in both secondary and tertiary oil recovery. He showed experimentally that tertiary oil recovery increased by increasing injection pressure from 210 to 220 bar at 85 °C. However, the efficiency of the process strongly depends on the type of gas. Injecting CO2 resulted in higher recovery than equilibrium gas or nitrogen injection in tertiary recovery. He claimed this is due to the fact that, in CO2 injection, several mechanisms such as gravity drainage, diffusion, swelling, and IFT reduction are contributing to oil recovery. Also, in secondary recovery, when nitrogen is injected, ultimate recovery is lower than CO2 injection. No detailed simulation was done.

5.8.4.2

Simulation Studies

Coats [114] included the effect of diffusion in dual-porosity models. Diffusion coefficients for liquid–liquid diffusion are about 100 times smaller than those for gas–gas diffusion. Liquid–gas diffusion coefficients are larger than liquid–liquid diffusion coefficients but still less than gas–gas diffusion coefficients. Therefore, Coats (1989) neglected gas-oil and oil-oil diffusion between fracture and matrix in his formulation and only gas–gas diffusion was considered as:     Diffusion between matrix and fracture = ϕSgg Dg ρg yi m − ρg yi f

(5.79)

where Sgg is the geometric mean of matrix and fracture gas saturation. Dg is gas diffusion coefficient. ρg .ρo are molar densities of gas and oil. (yi )m .(yi ) f are mole fraction of component i in gas phase in matrix and fracture. ϕ is matrix porosity. Da Silva and Belery [69] simulated the effect of diffusion on oil recovery from highly fractured reservoirs with low matrix permeability in the North Sea and in Africa. The oil components were C1 , C2 –C6 , and C7+ . The injected gas was nitrogen. The simulation studies were done at 266 °F and 4415 psia. The maximum matrix block height was 4 ft in their simulations. The diffusion equation for a matrix block was solved analytically for a step change in concentration at the matrix boundary. The analytical solution provided the concentration of each component as a function of time. Their analytical simulation results showed the significant effect of diffusion on the oil recovery, especially for small matrix block size of the order of several feet or less. They suggested considering the effect of diffusion on oil recovery in simulation of naturally fractured reservoirs. Thomas et al. [140] conducted a simulation study of nitrogen injection into the highly fractured Ekofisk field in the North Sea. The model temperature and pressure

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were 268 °F and 4000 psig. Bubble point pressure was 5545 psia. They defined the diffusion time as the time required to increase concentration of N2 in the core to 99% by diffusion. They showed that diffusion time for 1 and 10 ft blocks are 10 days and 5 years, respectively. The interfacial tension was increased by nitrogen diffusion. Hua and Whitson [141] simulated experiment no. M5 shown in Table 5.3. Their model combined an analytical solution for mass transfer in the fracture with a numerical model in the core. An analytical solution in the fracture was used to define a mass transfer coefficient between matrix and flowing gas in the fracture. Convection (driven by pressure gradient) between matrix and fracture is not considered in the model. They showed that diffusion is an important mechanism for transporting N2 and C1 in the porous media. C5 is transported to the fracture face mainly by oil convection inside the core. They also recognized the importance of correction of capillary pressure for the variation of interfacial tension due to gas diffusion in oil recovery calculations. They used a ternary diagram to explain why pentane recovery is not only by pure diffusion. They explained that as core fluid contacts nitrogen, the amount of pentane in the oil phase should increase, which means that pentane will diffuse from the fracture into the core. This is impossible since the injected gas does not contain pentane from the fracture. The only way to keep phases in equilibrium is to have pentane supplied from the lower part of the core. This is the reason behind oil convection from the matrix towards the fracture. Fayers et al. [142] simulated experiment no. M5 (nitrogen diffusion experiment) of the Morel et al. diffusion experiments (Table 5.3) to test their compositional simulator. The computational mesh had 20 grid blocks along the core and 3 grid blocks along the fracture, which allowed its inlet, mass transfer region, and outlet to be represented. The mass transfer coefficient between matrix and fracture was evaluated using a laminar flow theory similar to that described by Hua and Whitson. They showed the importance of correcting capillary pressure with interfacial tension in the calculations. Also, they recognized that the shapes of the calculated saturation profiles are strongly dependent on the selection of a capillary pressure curve and on the accuracy of determining variations of interfacial tension. Riazi et al. [137] solved the diffusion equation (Eq. 5.61) analytically to simulate their experiments (Fig. 5.14). They treated the fracture as a boundary condition for the matrix. Two boundary conditions were studied for the fracture-matrix interface. They were a stagnant condition and high flow in the fracture. Their simulation results showed good agreement with experimental data (composition of methane versus time) for both cases. They recognized that diffusion is a very important mechanism in oil recovery. Saidi [85] simulated performance of the Haft Kel field at Iran. Matrix block size varies from 8 to 14 ft in the Haft Kel field. Permeability changes between 0.05 to 0.8 md. He showed the importance of diffusion during history matching of the Haft Kel reservoir. Lenormand et al. [143] developed a mass transfer coefficient between matrix and fracture similar to the Hua and Whitson [141] model. The model was used successfully to simulate the following experiments:

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M5 nitrogen diffusion experiment in Table 5.3 [132]. M12 methane diffusion experiment in Table 5.4 [135, 136]. M29 and M30 methane diffusion experiments in Table 5.5 [120].

Hoteit and Firoozabadi [144] simulated gas injection using finite element methods. The domain of the model is a 2-D vertical cross-section (xz) with 500 m length and 100 m height with different fracture spacing of 100 m*10 m, 10 m*10 m, and 10 m*5 m. Matrix permeability was set 1md or 0.1md in the simulation studies. Matrix porosity was 20%. Fracture relative permeability was linear. Capillary pressures in matrix and fracture were assumed zero. Table 5.6 presents the details of their simulation study. One injection well and one production well were defined in the model. The injection well was located on top right corner and the production well was located at the lower opposite corner. They considered the effect of non-ideality to calculate the diffusion coefficients in a multi-component mixture. They concluded that for a low permeability matrix (1 and 0.1 md) the effect of diffusion is much more than what current models predict. They treated the fracture as a boundary between adjacent matrices. In their simulation, the pressure, saturation, and mole fraction in the fracture were calculated by interpolation between adjacent matrices. Their simulation results showed 25% increase in oil recovery by including diffusion with Table 5.6 Simulation examples of Hoteit and Firoozabadi [144] Example no P(bar) at top T (K) Oil of the model composition

Injected gas composition

Gas injection Production rate (PV/day)

1

38

366

C3

C1

1.30E−04

BHP = 38 bar

2

320

366

CO2 & C3 , C1 , C2 , C3, C4, C5, C5, C6, C7–9, C10+

C1

6.80E−05

BHP = 320 bar

3

175

366

CO2 , N2 –C1 , CO2 H2 S–C2 –C3 , C4 –C6 , C7 –C9 , C10 –C14 , C15 –C18 , C19+

6.20E−05

BHP = 175 bar

4

437

410

CO2 , N2 –C1 , C2 , C3 , C4 –C5 , C6 –C7 , C8 –C11 , C12 –C19 , C20 –C29 , C30+

BHP = Bottom hole pressure

CO2 –N2 –C1 , 5.76E−05 C2 , C3 , C4 –C5 , C6 –C7

Constant rate = 7.2e–5 PV/day

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their method relative to the case without diffusion. The effect of diffusion was more pronounced for smaller fracture spacing. Alavian et al. [145] simulated a secondary CO2 injection experiment of Karimaie. In the Karimaie [139] experiments, equilibrium gas was in equilibrium with the oil in the core and therefore, there was no mass transfer between the equilibrium gas and the oil. The SENSOR compositional model (single porosity) was used to simulate the experiment. SENSOR does not have a diffusion mechanism in the single porosity model. A cylindrical model (single porosity) with 10 grids in radial direction and 51 grids in vertical direction was used to simulate the experiment. The simulation results showed the following results: 1.

2. 3.

Darcy displacement is the dominant recovery mechanism in the Karimaie [139] secondary experiment during the equilibrium gas injection period because of a low conductivity in the surrounding fracture. The fracture space in the Karimaie [139] experiments was created by melting wood’s metal initially filled the space. Simulation results indicated that the fracture had low conductivity (20–30 md) which means melting wood’s metal was not a successful process. It was concluded that near-miscible displacement was the dominant production mechanism during secondary CO2 injection. Gravity-capillary forces had a minor effect in Karimaie [139] experiment.

Moortgat et al. [146] simulated the Darvish et al. [138] CO2 experiment by finite element methods. Their simulation method is the same as Hoteit and Firoozabadi [144] method. A Cartesian model with 19*1*40 grids in x, y, and z direction was used to simulate the experiment. It was found that diffusion was an important recovery mechanism. However, the impact of diffusion on oil recovery was not as significant as the Darvish et al. [138] simulation results showed.

5.8.4.3

Multi Component Diffusion

Multi component diffusion occurs when the flux of one component is influenced by the concentration gradient of a second component [147]. In some cases, the first component’s flux can be accelerated or decelerated by as much as an order of magnitude. In other words, the diffusion can cause a temporary unmixing, exactly opposite to the effect commonly expected. In some cases, the first component can diffuse against its concentration gradient, from a region of low concentration into a region of high concentration. Cussler [147] mentioned the lack of experimental data as the major deficiency to verify multi component mathematical models. Cussler [147] believed that all available evidence suggests that all mathematical models with same number of diffusion coefficients give similar results.

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5.8.4.4

A. Jamili et al.

Diffusion Coefficient

Basically, diffusion coefficients are functions of concentration, temperature, and pressure. Their functionality of concentration, temperature, and pressure is derived from non-equilibrium thermodynamics and known as Maxwell–stefan equation. Bird et al. [68] showed that there are three driving forces for a diffusion process: concentration gradient (molecular diffusion), temperature gradient (temperature diffusion), and pressure gradient (pressure diffusion). Firoozabadi et al. [148] studied the role of diffusion in oil composition variation in fractured reservoirs and they used Bird classification of diffusion driving mechanisms. They claimed that by using their model the oil composition variation can be explained better, but they didn’t show how. Molecular diffusion coefficients are 2 to 3 orders of magnitude lager than temperature diffusion coefficients and 7 to 9 orders of magnitude larger than pressure diffusion coefficients. Firoozabadi et al. [144] used a tensor to represent diffusion coefficients. They calculated the diffusion flux by multiplying diffusion coefficient by concentration gradient and because of the effect of multicomponent diffusion some of the off diagonal diffusion coefficients are negative. Diffusion coefficients in their work are functions of concentration, temperature, and pressure and unlike their previous work [148]; they did not follow Bird’s calcification by considering separate concentration diffusion coefficient, temperature diffusion coefficient, and pressure diffusion coefficient. Again, the major deficiency of this mathematical model like other mathematical models is lack of experimental verification. On the other hand, diffusion flux of every component is the algebraic summation of that component diffusion flux in the other components, which for some of them are negative, and will results to a number and because of this, in the literature, the investigators do not care about individual component of diffusion coefficient tensor. They care about the total diffusion flux of that component. Diffusion coefficients are usually obtained from Sigmund correlations [149], which are made for reservoir conditions. It has shown that Sigmund correlations [149], are able to predict the diffusion coefficients with reasonable accuracy. Da silva et al. [69] extended and used Sigmund correlations to study the effect of diffusion on the reservoirs at North Sea and Africa. Their simulation results showed the huge effect of diffusion on the oil recovery and because of that they suggested studying effect of diffusion carefully and including it in the matrix-fracture fluid transfer. Darvish et al. [138] experimentally studied CO2 injection into fractured cores at reservoir conditions. They were able to match their experimental results using extended Sigmund correlations [149], proposed by Da Silva et al. [69]. Sigmund correlations have been used in most studies.

5.8.4.5

Field Examples

Table 5.7 shows field examples where diffusion is an important mechanism in oil recovery [85, 139].

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Table 5.7 Field examples where diffusion is important Field name

Location

Matrix block height (ft)

K (md)

Lithology

Ekofisk

North sea