Flow mechanisms and numerical simulation of gas production from shale reservoirs

Transcription

1 Scholars' Mine Doctoral Dissertations Student Research & Creative Works 2015 Flow mechanisms and numerical simulation of gas production from shale reservoirs Chaohua Guo Follow this and additional works at: Part of the Petroleum Engineering Commons Department: Geosciences and Geological and Petroleum Engineering Recommended Citation Guo, Chaohua, "Flow mechanisms and numerical simulation of gas production from shale reservoirs" (2015). Doctoral Dissertations This Dissertation - Open Access is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact scholarsmine@mst.edu.

2 FLOW MECHANISMS AND NUMERICAL SIMULATION OF GAS PRODUCTION FROM SHALE RESERVOIRS by CHAOHUA GUO A DISSERTATION Presented to the Faculty of the Graduate School of the MISSOURI UNIVERSITY OF SCIENCE AND TECHNOLOGY In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY in PETROLEUM ENGINEERING 2015 Approved Mingzhen Wei, Advisor Ralph Flori Runar Nygaard Baojun Bai Xiaoming He

3 2015 Chaohua Guo All Rights Reserved

4 iii PUBLICATION DISSERTATION OPTION This dissertation has been prepared in the form of four papers for publication. Papers included are prepared as per the requirements of the journals in which they are submitted or will be submitted. Paper I Study on Gas Flow through Nano Pores of Shale Gas Reservoirs from page 61 to 95 has been published in Fuel. Paper II Pressure Transient and Rate Decline Analysis for Hydraulic Fractured Vertical Wells with Finite Conductivity in Shale Gas Reservoirs from page 96 to 120 has been published in Journal of Petroleum Exploration and Production Technology. Paper III Modeling of Gas Production from Shale Reservoirs considering Multiple Transport Mechanisms from page 121 to 159 has been published in Offshore Technology Conference held in Kuala Lumpur, Malaysia March And Paper IV Numerical Simulation of Gas Production from Shale Gas Reservoirs with Multi-stage Hydraulic Fracturing Horizontal Well from page 160 to 189 has been submitted to Journal of Petroleum Science and Engineering.

5 iv ABSTRACT Shale gas is one kind of the unconventional resources which is becoming an ever increasing component to secure the natural gas supply in U.S. Different from conventional hydrocarbon formations, shale gas reservoirs (SGRs) present numerous challenges to modeling and understanding due to complex pore structure, ultra-low permeability, and multiple transport mechanisms. In this study, the deviation against conventional gas flow have been detected in the lab experiments for gas flow through nano membranes. Based on the experimental results, a new apparent permeability expression is proposed with considering viscous flow, Knudsen diffusion, and slip flow. The gas flow mechanisms of gas flow in the SGRs have been studied using well test method with considering multiple flow mechanisms including desorption, diffusive flow, Darcy flow and stress-sensitivity. Type curves were plotted and different flow regimes were identified. Sensitivity analysis of adsorption and fracturing parameters on gas production performance have been analyzed. Then, numerical simulation study have been conducted for the SGRs with considering multiple mechanisms, including viscous flow, Knudsen diffusion, Klinkenberg effect, pore radius change, gas desorption, and gas viscosity change. Results show that adsorption and gas viscosity change will have a great impact on gas production. At last, the numerical simulation model for SGRs with multi-stage hydraulic fracturing horizontal well has been constructed. Sensitivity analysis for reservoir and fracturing parameters on gas production performance have been conducted. Results show that hydraulic fracture parameters are more sensitive compared with reservoir parameters. The study in this project can contribute to the understanding and simulation of SGRs.

6 v ACKNOWLEDGMENTS First of all, I would like to express my sincerest gratitude to my supervisors, Dr. Mingzhen Wei and Dr. Baojun Bai, for their outstanding guidance and generous support through my study at Missouri University of Science and Technology. They never hesitate to provide relentless support and motivation at all times. I am honored to have worked with them. And I am also eternally grateful to them. I would like to extend my thanks to my committee members Dr. Ralph Flori, Dr. Runar Nygaard, and Dr. Xiaoming He for providing me with technical assistance and advices during reviewing my work. I want to thank Dr. Xiaoming He for offering those useful suggestions and helping me go through the hard time. I am also extremely grateful to the Research Partnership to Secure Energy of America (RPSEA) for providing the financial support. I would like to thank all those faculties, staffs, friends, classmates, and colleagues who helped and encouraged me to complete this dissertation work. I want to thank Mrs. Paula Cochran and Mrs. Patricia Robertson for their help with my study affairs. I also want to thank Dr. Hao Zhang, Dr. Jianchun Xu, and Dr. Yongpeng Sun for helping me in discussing my research and sharing their suggestions and opinions about life. Finally, I wholeheartedly thank my parents and sister for their endless love and support. My family has always been the most powerful source of motivation for me to strive in the long and hard educational career. I hope my late father to feel happy about my hard work. He is a so great father that deserves my deep thanks and endless missing from the bottom of my heart through my whole life.

7 vi TABLE OF CONTENTS PUBLICATION DISSERTATION OPTION... iii ABSTRACT... iv ACKNOWLEDGMENTS... v LIST OF ILLUSTRATIONS... xi LIST OF TABLES... xv SECTION 1. INTRODCUTION STATEMENT AND SIGNIFICANCE OF THE PROBLEM RESEARCH OBJECTIVES AND SCOPE LITERATURE REVIEW SHALE GAS Importance of Shale Gas Reservoirs Pore Size Distribution of Shale Gas Reservoirs Gas Storage Forms in Shale Gas Reservoirs Permeability and Porosity of Shale Gas Reservoirs Accumulation and Production Characteristics of Shale Gas Reservoirs GAS FLOW MECHANISMS IN NANO PORES No-Slip Boundary Condition and Slip Boundary Condition Knudsen Number Gas Flow Mechanisms in the Nano Pore Simulation Methods for Gas Flow through Nano Pores SHALE PERMEABILITY MEASUREMENT METHODS Permeability and Intrinsic Permeability Permeability Measurement Methods Pulse-decay technique Canister desorption test Crushed samples dynamic pycnometry APPARENT PERMEABILITY MODEL FOR GAS FLOW THROUGH NANO PORES OF SHALE GAS RESERVOIRS Beskok and Karniadakis Model

8 vii Javadpour s Model Florence s Model Civan s Model Sakhaee-pour and Bryant s Model GAS FLOW MECHANISMS IN SHALE GAS RESERVOIRS Darcy Flow Flow in the Fracture Diffusion Gas Adsorption/Desorption Stress, Water, and Temperature Sensitivity SIMULATION MODEL FOR GAS PRODUCTION FROM SHALE GAS RESERVOIRS Equivalent Continuum Model Dual Porosity Model Multiple Interaction Continua Method Multiple Porosity Model Discrete Fracture Model CONCLUDING REMARKS REFERENCES PAPER PAPER I. STUDY ON GAS FLOW THROUGH NANO PORES OF SHALE GAS RESERVOIRS ABSTRACT INTRODUCTION EXPERIMENTS FOR GAS FLOW THROUGH NANO MEMBRANES SEM imaging of nano membranes Experimental Setup and Results SLIP EFFECT FACTOR DERIVATION AND APPARENT PERMEABILITY MODEL Mechanisms for gas flow in nanotubes Apparent permeability of gas flow in a nano capillary MODEL VALIDATION AND COMPARISON Model validation... 71

9 viii 4.2. Model comparison DISCUSSION CONCLUSION ACKNOWLEDGEMENTS Appendix A REFERENCES PAPER II. PRESSURE TRANSIENT AND RATE DECLINE NALYSIS FOR HYDRAULIC FRACTURED VERTICAL WELLS WITH FINITE CONDUCTIVITY IN SHALE GAS RESERVOIRS ABSTRACT NOMENCLATURE INTRODUCTION MATHEMATICAL MODEL Assumptions Mathematical model Solution TYPE CURVES AND ANALYSIS Type curves Sensitivity analysis CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES PAPER III. MODELING OF GAS PRODUCTION FROM SHALE RESERVOIRS CONSIDERING MULTIPLE FLOW MECHANISMS ABSTRACT INTRODUCTION PORE DISTRIBUTION AND GAS TRANSPORT PROCESS IN SGRs MODEL CONSTRUCTION Mass accumulation term Flow vector term Source and Sink term Gas viscosity change MODEL VERIFICATION EFFECT OF FLOW MECHANISMS ON SHALE GAS PRODUCTION Effect of Gas adsorption

10 ix Effect of non-darcy flow Effect of gas viscosity change Effect of pore radius change SENSITIVITY ANALYSIS Effect of initial pressure Effect of matrix Permeability and fracture permeability Effect of matrix porosity and fracture porosity CONCLUSION ACKNOWLEDGEMENTS NOMENCLATURE REFERENCES PAPER IV. NUMERICAL SIMULATION OF GAS PRODUCTION FROM SHALE GAS RESERVOIRS WITH MULTI-STAGE HYDRAULIC FRACTURING HORIZONTAL WELL ABSTRACT INTRODUCTION FLUID FLOW MECHANISMS IN TIGHT SHALE RESERVOIRS Viscous flow Knudsen diffusion Slip flow Gas adsorption and desorption APPARENT GAS PERMEABILITY IN THE MATRIX AND FRACTURE SYSTEM Apparent gas permeability in the shale matrix Apparent gas permeability in the shale fractures NUMERICIAL SIMULATION MODEL FOR SHALE GAS RESERVOIRS WITH MsFHW Model assumptions Mathematical model SENSITIVITY ANALYSIS Effect of matrix permeability Effect of matrix porosity Effect of fracture spacing Effect of fracture half-length

11 x 6. CONCLUSIONS ACKNOWLEDGEMENTS NOMENCLATURE REFERENCES SECTION 3. CONCLUSION VITA

12 xi LIST OF ILLUSTRATIONS Figure 1.1. Research scope of this project Figure 1.2. Research tasks of this project Figure 2.1. Energy production by Fuel in the U.S., (EIA 2013) Figure 2.2. Dry natural gas production in the U.S. (EIA, 2013) Figure 2.3. Comparison between conventional gas reservoir (a) and shale gas reservoir (b). (Modified from Javadpour et al., 2007) Figure 2.4. Variations in nanopore morphology in organic matter. (Loucks et al., 2009) Figure 2.5. Gas storage forms in shale gas reservoirs. Modified from Javadpour et al. (2007) Figure 2.6. Gas distribution in shale strata from macro-scale to micro-scale. In the fracture there exist free gas and in the matrix free gas and adsorption gas co-exist Figure 2.7. Survey of permeability and porosity for shale gas plays in North America In courtesy of Wang and Reed (2009) Figure 2.8. Formation Process of Shale Gas Reservoir Figure 2.9. Gas production process from macro to molecular scales. Flow to the wellbore is first initiated at the macroscale, followed by flow at increasingly finer scales. Modified from (Javadpour et al., 2007) Figure Comparison between no-slip and slip boundary condition Figure Flow regime classification based on Knudsen number (Heidemann et al., 1984; Karniadakis et al. 2006; Ziarani and Aguilera, 2012) Figure Langmuir Isotherm Curves for Five Typical Shale Gas Reservoirs PAPER I Figure 1. Gas distribution in shale strata from macro-scale to micro-scale. In the fracture, there exists free gas and in the matrix, free gas and adsorption gas co-exist Figure 2. SEM images of experimental membranes with pore size 20 nm, 55 nm, and 100 nm Figure 3. Schematic diagram and real picture of the experimental setup. In Fig. 3(a), from left to right the whole set up includes an aluminum cylinder, the membrane and an oring between the upstream and downstream parts Figure 4. Gas flow rate vs. the average pressure for nano membranes of different sizes. 88 Figure 5. Ratio of real gas flow rate to intrinsic flow rate vs. average pressure for different kinds of nano membranes

13 xii Figure 6. Ratio of slip length to pore radius vs. average pressure for nano membranes of different sizes Figure 7. Knudsen number vs. average pressure for nano membranes of different sizes. 90 Figure 8. Gas flow mechanisms in a nano pore. Red represents Knudsen diffusion, while purple represents viscous flow Figure 9. A schematic model for simulating gas flow in nanotubes Figure 10. Pressure drop versus flux for different values of σ Figure 11. Pressure drop versus flux for oxygen in a nanotube with a diameter of 235 nm and 220 nm Figure 12. Comparison of Kapp versus radius between new apparent permeability and that of Florence and Javadpour Figure 13. Pressure difference and gas mole flux relationship comparison among the proposed model, the analytical solution, Javadpour s solution and experimental data for Oxygen in 235 nm pores Figure 14. Comparison of gas permeability vs. pressure difference among the proposed model, the analytical solution, Javadpour s solution, and experimental data for oxygen in 235 nm pores Figure 15. (a) Effect of pore size on ratio of gas mole flux to equivalent liquid mole flux (Jg/Jl). (b) Effect of pore size on ratio of additional mole flux caused by Knudsen diffusion and slip flow to total mole flux. The upper figure shows how the ratio changes with the pore radius; the lower figure shows how the ratio changes with the Knudsen number Figure 16. Ratio of gas permeability to intrinsic liquid permeability vs. average pressure for nano membranes of different sizes PAPER II Figure 1. Schematic diagram of hydraulic fractured vertical well. The top and bottom boundary are closed Figure 2. Type curve for hydraulic fractured vertical wells with finite conductivity in shale gas reservoirs under different kinds of boundary conditions Figure 3. Effect of adsorption coefficient on PPR (pseudo pressure response) and RDR (Rate decline response) Figure 4. Effect of storage capacity ratio on PPR and RDR Figure 5. Effect of channeling factor on PPR and RDR Figure 6. Effect of fracture conductivity on PPR and RDR Figure 7. Effect of fracture skin factor on PPR and RDR Figure 8. Effect of stress sensitivity on PPR and RDR

14 xiii PAPER III Figure 1. Gas distribution in shale strata from macro-scale to micro-scale. In the fracture there exist free gas and in the matrix free gas and adsorption gas co-exist Figure 2. Gas production process from macro to molecular scales. Flow to the wellbore is first initiated at the macroscale, followed by flow at increasingly finer scales. Modified from (Javadpour et al., 2007) Figure 3. Idealization of the heterogeneous porous medium as DPM Figure 4. Transport scheme of shale gas production in DPM. Gas desorbed from the matrix surface and transferred to the fracture, then flow into the wellbore. Non-darcy flow, Knudsen diffusion, slip flow, and viscous flow have been considered Figure 5. Gas flow mechanisms in a nano pore. Red solid dots represent Knudsen diffusion, while blue ones represent viscous flow Figure 6. Pore radius change due to gas desorption. If single molecule gas desorption Langmuir isothermal is considered, when all the molecules have desorped from the surface, then the pore radius will increase as shown in the right part Figure 7. Gas viscosity variation with the Knudsen number (from 0.01 to 1), also means from slip flow to transition flow. Gas viscosity has changed a lot when the Knudsen number changes, which means it is necessary to consider the gas viscosity variation (Modified from Michalis et al., 2010). 152 Figure 8. Comparison between analytical solution and numerical simulation in this paper Figure 9. Real reservoir and simplified 2-D reservoir simulation model Figure 10. The effect of adsorption on gas production. (a) production rate vs. time; (b) cumulative production vs. time Figure 11. Matrix and fracture permeability change with time. (a) matrix permeability vs. time; (b) fracture permeability vs. time Figure 12. Effect of Non-Darcy flow on gas production. (a) production rate vs. time; (b) cumulative production vs. time Figure 13. Effect of gas viscosity change on gas production. (a) production rate vs. time; (b) cumulative production vs. time Figure 14. Effect of pore radius increase due to gas desorption on gas production. (a) production rate vs. time; (b) cumulative production vs. time Figure 15. Comparison of five different models: (1) basic model; (2) Considering adsorption; (3) Considering adsorption and non-darcy permeability change; (4) Considering adsorption and non-darcy permeability change, gas viscosity change; (5) Considering adsorption, non-darcy permeability change, gas viscosity change and pore radius change

15 xiv Figure 16. Effect of initial reservoir pressure on gas production. (a) production rate vs. time; (b) cumulative production vs. time Figure 17. Effect of matrix permeability on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time Figure 18. Effect of fracture permeability on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time Figure 19. Effect of matrix porosity on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time Figure 20. Effect of fracture porosity on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time PAPER IV Figure 1. Diagram of dual porosity model Figure 2. Synthetic model for production performance study Figure 3. Frequency distribution diagram of shale matrix permeability Figure 4. Effect of matrix permeability on the production performance of shale gas reservoirs. (a) cumulative gas production vs. time; (b) gas production rate vs. time Figure 5. Frequency distribution diagram of shale matrix porosity Figure 6. Effect of matrix porosity on the production performance of shale gas reservoirs. (a) cumulative gas production vs. time; (b) gas production rate vs. time Figure 7. Effect of fracture spacing on the production performance of shale gas reservoirs. (a) cumulative gas production vs. time; (b) gas production rate vs. time Figure 8. Effect of fracture half-length on the production performance of shale gas reservoirs. (a) cumulative gas production vs. time; (b) gas production rate vs. time

16 xv LIST OF TABLES Table 2.1. Knudsen s Permeability Correction factor for Tight Porous Media (Modified from Ziarani and Aguilera (2012)) PAPER I Table 1. Properties of AAO nano membranes used in the experiment Table 2. Experimental results for different kinds of nano membranes Table 3. Knudsen number and flow regimes classifications for porous media Table 4. Knudsen s permeability correction factor for tight porous media Table 5. Nanotube characteristics [20] Table 6. Model dimensions and fluid properties PAPER II Table 1. Definitions of the dimensionless variables PAPER III Table 1. Parameters used in the simulation model PAPER IV Table 1. Knudsen s Permeability Correction factor for Tight Porous Media (Guo et al., 2015) Table 2. Parameters used in the synthetic model for production performance study

17 SECTION 1. INTRODCUTION 1.1. STATEMENT AND SIGNIFICANCE OF THE PROBLEM Due to increasing energy shortage during recent years, gas producing from shale strata has played an increasingly important role in the volatile energy industry of North America and is gradually becoming a key component in the world s energy supply. Shale strata or shale gas reservoirs (SGR) are typical unconventional resources with numerous natural fractures and critically low permeability matrix which contains a large fraction of nano pores, which leads to an apparent permeability that is dependent on both the pressure, fluid type, and the pore structure. Study of gas flow mechanisms in nano pores and transport process in shale reservoirs is essential for the accurate numerical simulation of shale gas reservoirs. However, no comprehensive study has been done regarding gas flow mechanisms from micro to macro scales, also from experimental study to numerical simulation. This project will provide a solid foundation for later research on gas flow in shale strata and numerical simulation of unconventional resources with following main innovations: (1) A new mathematical model to characterize gas flow in nanotubes has been established and a new formulation to compute the apparent permeability for gas flow in nano pores has been derived; (2) The well test model for shale gas production with hydraulic fracturing considering finite conductivity has been rigorously constructed and analytically solved.

18 2 The model is different from any existing models. Sensitivity analysis has been carried out to study the effect of influencing factors for better fracturing design. (3) The mathematical model to characterize gas flow in shale strata from matrix to fracture, and to production well has been constructed. And the model has overcome the limitations in previous models: gas viscosity change, pore radius change, and multiplemechanism flow vector; (4) The numerical simulation model for gas production from shale gas reservoirs with multi-stage hydraulic fracturing horizontal well has been constructed. Sensitivity analysis have been conducted for both the reservoir parameters and the hydraulic fractures parameters based on real parameter s range. Therefore, this project will provide a comprehensive study of shale gas transport and production problems from micro scale to macro scale through a combined experimentsimulation methods. The results of this project are significant for accurate numerical simulation of shale gas production and also can provide a solid foundation for future work on shale gas RESEARCH OBJECTIVES AND SCOPE The primary objective of this project is to study gas transport mechanisms and behaviors in shale gas reservoirs from micro scale (nano pores and micro fractures) to macro scale (wellbore and reservoir) so that we can provide a comprehensive and rigorous study for accurate numerical simulation of gas production from shale strata, which is the main energy substitute and clean resource in the next decades. The research scope of this

19 3 project is illustrated in Figure 1.1. Specifically, the objectives of this project includes following three aspects: To study the gas flow mechanisms in nano pores using experimental and numerical methods, and obtain a physically accurate and ready to use model so that we can plug it into the numerical simulation model with easy access; To study the gas transport behaviors in shale strata. Gas flow mechanisms in shale matrix, natural fractures, and wellbore will be considered comprehensively into the simulation model which will be verified against analytical solution. To study the influencing factors in shale gas production process using numerical simulation and well test. The objective is to analyzing those influencing factors so that better fracturing design and production plan can be achieved. Figure 1.1. Research scope of this project.

20 4 This whole project is composed by four interrelated tasks which is shown in Figure 1.2. The four tasks can be classified into two categories: micro scale study and micro scale study. The first task is study of gas flow mechanisms in nano pores, which includes thress sub-tasks: Analysis of experimental results on gas flow through nano pores, Construction of fluid flow model for gas transport in nano pores, and Construction of fluid flow model for gas transport in nano pores. Through this task it is expected to have a comprehensive understanding of gas transport behavior in nano pores and derive the apparent permeability formulation which lay the foundation work for numerical simulation in Task 3 and 4. The second task is Study of gas transport mechanisms in reservoir scale using well test method. Through well test, we can have a general idea about the gas transport process in micro scale, which can provide the basis for the construction of numerical simulation model. The third task is construction, validation, and sensitivity analysis of numerical simulation model for shale gas production. In this task, we will based on Dual Porosity Model to characterize gas transport in reservoir. This task includes two sub-tasks: Construction of numerical simulation model for shale gas production and Validation and sensitivity analysis of the numerical simulation model of shale gas production. The last task is numerical simulation of gas production from shale gas reservoirs with multi-stage hydraulic fracturing horizontal well. In this task, we will construct the numerical simulation model and conduct sensitivity analysis for both reservoir and fracturing parameters based on the real parameter s range for shale gas reservoirs. The ultimate goal of this project is to has a clear understanding of gas transport mechanisms and behavior in shale reservoirs. There is a great difference compared with previous work and the work in this project will contribute some innovations to existing work in shale gas reservoir.

21 5 Figure 1.2. Research tasks of this project. Four journal articles in the following sections were written to address the four specific tasks listed above: (1) In the first paper, the apparent permeability model for gas flow through nano pore has been constructed. Experiments results for nitrogen flow through nano membranes (with pore throat size: 20 nm, 55 nm, 100 nm) have been analyzed. Obvious discrepancy between apparent permeability and intrinsic permeability has been observed. And relationship between this discrepancy and pore throat diameter (PTD) has been analyzed. Based on the advection-diffusion model, a new mathematical model to characterize gas

22 6 flow in nano pores has been constructed. We derived a new apparent permeability expression based on advection and Knudsen diffusion. A comprehensive coefficient for characterizing the flow process was proposed. Simulation results were verified against the experimental data for gas flow through nano membranes and published data. The model got verified using experimental data on nitrogen, and published data with different gases (oxygen, argon) and different PTDs (235 nm, 220 nm). Comparison among our results with experimental data, the Knudsen/Hagen-Poiseuille analytical solution, and existing data available in the literature have been conducted at the end. (2) In the second paper, the well test model for gas production from shale gas reservoirs with fractured vertical well with finite conductivity has been constructed with considering multiple flow mechanisms including desorption, diffusive flow, Darcy flow and stress-sensitivity. The pressure transient analysis and rate decline analysis models were established firstly. Then the source function, Laplace transform and the numerical discrete methods were employed to solve the mathematical model. At last, the type curves were plotted and different flow regimes were identified. At last, the sensitivity of adsorption coefficient, storage capacity ratio, inter-porosity flow coefficient, fracture conductivity, fracture skin factor and stress sensitivity were analyzed. (3) In the third paper, a comprehensive mathematical model which incorporates all known mechanisms for simulating gas flow in shale strata has been presented. Complex mechanisms including viscous flow, Knudsen diffusion, slip flow, and desorption are optionally integrated into different continua in the model. Effect of different mechanisms on the production performance of shale gas reservoirs have been analyzed by changing the corresponding term in mathematical model. At last, sensitivity analysis of reservoir

23 7 parameters have been conducted, such as initial pressure; matrix permeability; fracture permeability; matrix porosity; and fracture porosity. (4) In the fourth paper, the dual porosity model for gas production from shale gas reservoirs with multi-stage hydraulic fracturing horizontal well has been constructed. Gas flow mechanisms have been considered as the permeability tensor in the continuity equation for both matrix and fracture system. Sensitivity analysis on production performance of tight shale reservoirs with multi-stage hydraulic fracturing horizontal well have been conducted. Parameters influencing shale gas production were classified into two categories: reservoir properties including matrix permeability, matrix porosity and hydraulic fracture properties including hydraulic fracture spacing, fracture half-length. Typical ranges of matrix parameters have been reviewed. Sensitivity analysis have been conducted to analyze the effect of above factors on the production performance of shale gas reservoirs.

24 8 2. LITERATURE REVIEW 2.1. SHALE GAS Shale gas is the natural gas which is produced from those tight shale formations. Shale gas is becoming an increasingly important factor to secure energy over recent years for the considerable volume of natural gas stored and is gradually becoming a key component in the world s energy supply (Wang and Krupnick, 2013; Guo et al., 2014a). Shale strata or shale gas reservoirs are typical unconventional reservoirs with critically low transport capability in matrix and numerous natural fractures. Unconventional reservoirs are those geological formations which contain fossil energy but are uneconomical to produce (with recovery about 2%) currently when conventional recovery methods are applied (Arogundade and Sohrabi, 2012; Passey et al. 2010), such as shale oil/gas reservoirs, tight oil/gas reservoirs, and coalbed methane reservoirs. Shales are fine-grained (with average grain size below in) sedimentary rocks with ultra-low permeability (in the order of nanodarcy) (Kundert and Mullen, 2009). Typical shale gas reservoirs exhibits a net thickness of ft, porosity of 2-8%, total organic carbon of 1 to 14% and depth of 1,000-13,000ft (Cipolla et al., 2010). They are organic-rich formations and are both the source rock and the reservoir (Arthur et al., 2008). Natural gas have been stored in shale gas reservoirs with three forms (Guo et al., 2014b; Javadpour, 2009; Hill and Nelson, 2000): (a) in the micro fractures and nano pores, they are stored as free gas; (b) in the kerogen, they are stored as dissolved gas; (c) in the surface of the bedrock, they are stored as adsorption gas and about 20%~85% of gas in SGRs are stored in this kind of form.

25 Importance of Shale Gas Reservoirs. Natural gas, coal, and oil contribute 85% to the whole nation s energy structure, and natural gas contributes 22% of the total (Arthur et al., 2008; EIA, 2009). Natural gas is the cleanest fossil fuel compared with oil and coal. Most of the technically recoverable natural gas in North America is present in unconventional reservoirs such as tight sands, shale, and coalbed (Outlook, 2012). Natural gas plays an important role in satisfying U.S. energy demands and contribution of natural gas will continue to increase in the next twenty years, and is expected to increase to 38% of the total in 2040 (Figure 2.1). In 2012, shale gas has contributed 31% to the total U.S. natural gas production. Also, North America has the largest shale gas reserves. And the total shale gas volumes around the world are estimated to be about 16,000Tcf. (Fazelipour, 2010). Owing to new applications of horizontal well drilling, multi-stage hydraulic fracturing, and other technologies, shale gas has offset declines in production from conventional gas reservoirs, and contributes to major increases of U.S. natural gas production. Overall, natural gas production from unconventional reservoirs, especially shale gas reservoirs, is becoming an ever increasing portion of the U.S. main natural gas supplier, securing the bulk of the U.S. natural gas supply for the next twenty years which is shown in Figure 2.2 (Outlook, 2012). And the U.S. Energy Information Administration (EIA) states that natural gas from tight sand formations is currently the largest source of unconventional production, accounting for 30 percent of total U.S. production in 2030, but production from shale formations is the fastest growing source. (Sondergeld et al., 2010).

26 10 Figure 2.1. Energy production by Fuel in the U.S., (EIA 2013). Figure 2.2. Dry natural gas production in the U.S. (EIA, 2013).

27 Pore Size Distribution of Shale Gas Reservoirs. Understanding the pore size distribution is of great importance in evaluating the original gas in place and flow characteristics of the shale matrix. Javadpour found that most pore throat diameters are concentrated in the range of 4 ~ 200 nm through experimental analysis on 152 cores from nine reservoirs in North America (Javadpour et al., 2007; Javadpour, 2009; Curtis et al., 2010). Adesida et al. (2011) found that the pore sizes are in the range of micro to meso scale, with average less than 10 nm. Figure 2.3 is the pore distribution comparison between conventional and unconventional (shale) reservoirs (Javadpour et al., 2007). It can be found that in unconventional shale gas reservoirs, the number of nanopores is higher than those in conventional reservoirs. Due to this, the exposed surface area in shale gas reservoirs is larger than those in conventional reservoirs, which leads to more adsorption gas. Also, gas flow in those nano pores is different from Darcy flow. The diameter of pores in shale gas reservoirs are from a few nanometres to a few micrometres. And the morphology of the nano pores are also different. Figure 2.4 is the scanning electron images of the shale sample (Loucks et al., 2009). From the figure, we can find that for the nano pores, there are four different morphologies. (1) Very small pores. The pore size is from 18 to 46 nm. The nanopores are nearly spherical. Total porosity is about 5.2%. (2) Larger nanopores. The pore diameter is about 550 nm. (3) Tubelike pore throats connecting elliptical pores. The pore size is less than 20 nm. (4) Additional tubelike pore throats connecting elliptical pore. The pore-throat diameter is less.

28 12 Figure 2.3. Comparison between conventional gas reservoir (a) and shale gas reservoir (b). (Modified from Javadpour et al., 2007). Figure 2.4. Variations in nanopore morphology in organic matter. (Loucks et al., 2009). Wang and Reed (2009) have conducted pore structure study on Barnett shale in the Fort Worth Basin, North Texas, and have classified the shale formation into four different medium: organic matrix, nonorganic matrix, natural fractures, and hydraulic fractures. Researchers also classified the matrix pores according to the scale. Two types of matrix pores are nano-scale pores. (Davies et al., 1991; Reed et al., 2007; Bowker, 2007; Bustin

29 13 et al., 2008a) and micro-scale pores (Davies et al., 1991; Bustin et al., 2008a). Shale gas can not only be stored as adsorption gas on the surface of organic grains, but also can be stored as free gas inside the organic grains. They found that the pore network in the organic matrix are important gas flow paths. Sonderdeld et al. (2010) have found that the pores within the Kerogen of shale matrix are in the range of mesopores from 5 to 23 nm using scanning electron microscope imaging (SEM). Curtis et al. (2011) have detected pores about 2 nm using scanning transmission elctron microscope imaging (STEM). All these study have revealed the widely existence of nano pores in tight shale formations Gas Storage Forms in Shale Gas Reservoirs. Understanding gas storage forms is of great importance for the understanding of the gas transport process in the reservoir. Different from conventional reservoirs where gas is only stored in the pore space as free gas, unconventional shale gas reservoirs have a variety of storage forms (Swami et al., 2013). Aguilera (2010) suggested that there are three forms for gas to be stored in shale gas reservoirs. (a) Adsorption gas in the organic matter; (b) free gas in the organic pores, non-organic pores, and micro-fractures; (c) stored gas in the hydraulic fractures. Javadpour et al. (2007, 2009) suggested that there are also three forms (Figure 2.5) for gas to be stored in shale gas reservoirs: (a) in the micro fractures and nano pores, they are stored as free gas; (b) in the kerogen, they are stored as dissolved gas; (c) in the surface of the bedrock, they are stored as adsorption gas. Figure 2.6 illustrates the gas distribution and geometry of shale strata from micro to macro scale (Guo et al. 2015). It informs us that in the fracture, only free gas exists and in the matrix which is full of kerogen, free gas and adsorption gas co-exist.

30 14 Figure 2.5. Gas storage forms in shale gas reservoirs. Modified from Javadpour et al. (2007). Figure 2.6. Gas distribution in shale strata from macro-scale to micro-scale. In the fracture there exist free gas and in the matrix free gas and adsorption gas co-exist Permeability and Porosity of Shale Gas Reservoirs. Permeability and porosity are two critical parameters to determine the flow capacity and gas content in place. As hydrocarbon formation, shales have low permeability and low porosity, which are as

31 15 impermeable as concrete with permeability in the nanodarcy and porosity between 2-10% (Arogundade and Sohrabi, 2012). Also, through the experimental analysis of 152 cores of nine reservoirs in North America, Javadpour found that the average permeability of shale bedrock is 54 nd, and approximately 90% have permeability less than 150 nd (Javadpour et al., 2007; Javadpour, 2009). Wang et al. (2009) have a made a survey on the permeability-porosity relationship for shale gas plays in North America. The result of the survey is illustrated in the Figure 2.7. Figure 2.7. Survey of permeability and porosity for shale gas plays in North America In courtesy of Wang and Reed (2009). From Figure 2.7, it can be found that the porosity and permeability for North America shale gas reservoirs are extremely low. The matrix permeability is from 1 nanodarcy to 1 microdarcy, and the matrix porosity is 1%~5%. Porosity can determine the storage state of shale gas: free gas or adsorption gas. In those shale formations with high

32 16 porosity, gas will mainly be stored as free gas. And gas will be stored as adsorption gas in formations with low porosity. In the Ohio shale and Atrim shale with average porosity of 5-6% and maximum porosity of 15%, the free gas can be up to 50% of the total pore volume. Methods used to obtain the ultra-low shale gas permeability include pulse-decay technique, crushed samples dynamic pycnometry, canister desorption test (Alnoaimi and Kovscek, 2013; Luffel et al., 1993; Cui et al. 2004; Cui et al. 2009) Accumulation and Production Characteristics of Shale Gas Reservoirs. Understanding of the accumulation process is important for the understanding the gas distribution and the understanding of production process is key for the construction of numerical simulation model. The formation process of the shale gas reservoirs is totally different from conventional reservoirs. In conventional reservoirs, the hydrocarbons migrate from the source rock to the reservoir rock (Swami et al. 2013). When hydrocarbons formed, they will migrate and stops when it finds a structural trap which is called the reservoir rock. Then, the cap rock seals the hydrocarbons to prevent its further migration (Swami and Settari, 2012). Commonly, the cap rock are shale with extremely low permeability. However, due to the ultra-low permeability of shale gas reservoirs, there is no migratory path for the produced gas after it is formed in shale gas reservoirs. The shale formation are both the source rock and the reservoir rock in itself. The formation process of shale gas reservoir is illustrated in Figure 2.8, which can be divided into three periods: (1) The organic matter (Kerogen) gets deposited and the natural gas produced from natural fractured shale; (2) when the adsorption gas and dissolved gas become saturated, the extra gas will desorb and goes into the matrix pores; (3) with large amount of natural gas produced, the matrix pressure increase. Free gas flows into the fracture and accumulated.

33 17 The production process of shale gas reservoirs is a reverse process compared with the formation process. Javadpour has studied the production process of shale gas reservoirs, which is shown in Figure 2.9 (Javadpour et al., 2007). When producing the shale gas, firstly, the free gas in the fractures and large pores will be produced, see the step (4) in Figure 2.9. And then gas will be produced from the smaller pores, see the step (3) in Figure 2.9. With pressure depletion, the thermodynamic equilibrium between Kerogen and gas phase in the pores will change and gas will desorb from the surface of the Kerogen, see the step (2) in Figure 2.9. Due to the pressure difference, gas molecules will diffuse from the bulk of Kerogen to the surface, see the step (1) in Figure 2.9. Figure 2.8. Formation Process of Shale Gas Reservoir. Figure 2.9 Gas production process from macro to molecular scales. Flow to the wellbore is first initiated at the macroscale, followed by flow at increasingly finer scales. Modified from (Javadpour et al., 2007).

34 GAS FLOW MECHANISMS IN NANO PORES Due to the tiny pore size, the fluid continuum theory breaks down for shale pores with pore throat diameters in the range of nano scale (Wang & Reed, 2009). Some smaller pores, such as 5 to 50 nm are also found which is in the same scale of the kinetic diameter of methane molecules (Bowker, 2003; Heidemann et al., 1984). Except the conventional convective flow, many other flow mechanisms exist in the nano pores. In this section, the important gas flow mechanisms will be illustrated and presented No-Slip Boundary Condition and Slip Boundary Condition. Gas flow in the nano pores is different from conventional flow as the no-slip boundary condition will no longer be valid when the length scale of a physical system decreases (Javadpour, 2007; Roy et al., 2003). The difference between no-slip and slip boundary condition is shown in Figure In the no-slip boundary condition, the velocity profile in the pore is parabolic and the velocity on the wall is equal to zero. However, in the slip boundary such as in the nano pores, the gas molecules will slip on the wall and collide with other gas molecules, which means the velocity on the wall is not equal to zero. Figure Comparison between no-slip and slip boundary condition.

35 Knudsen Number. Knudsen number is a widely used dimensionless parameter which can be used to determine the degree of appropriateness of the continuum model (Bird, 1994; Hadjiconstantinou, 2006; Barber and Emerson, 2006). Also, it is a tool to determine the limitations of the Navier stokes equation according to the existence of slip or no-slip boundaries (Hadjiconstrantinou, 2006). The Knudsen number is defined as the ratio of gas mean-free-path λ to the pore throat diameter d. And the gas mean free path is defined as the distance required along the straight direction until that gas molecule collisions with other molecules or solid wall occurs (Civan, 2010). The formula to calculate gas mean free path is shown in Equation (1) which is provided by Heidemann et al. (1984). (1) (2) where is the Knudsen number, is the gas mean free path, is the pore diameter, is the boltzmann constant ( /), T is the temperature (K), is the pressure (Pa) and is the collision diameter of the gas molecular. It can be found that is related to pore throat diameter, gas pressure, and the temperature. It is positive related to temperature and negative related to pressure. Some scholars also proposed some other forms of equations to calculate the Knudsen number for simplification or application. Roy et al. (2003) have provided following formula to calculate the gas mean free path: (3)

36 20 Also, Loeb (2004) has proposed following formula to calculate gas mean free path considering ideal gas which is expressed as follows: (4) where is the gas bulk viscosity, is the gas density, is the gas molecular mass, is the universal gas constant, is the gas temperature, and is the pore pressure. A more fundamental definition of gas mean free path is presented as follows (Bird, 2002; Struchtrup, 2005): (5) where is the number of molecules, is the volume occupied by the molecules and is the diameter of a molecule. If the molecule cannot be described by a rigid sphere, should be replaced by the total scattering cross section (Huang, 1963). Knudsen number represents the ratio between viscous flow and diffusion. According to the Knudsen number, the gas diffusion can be classified into molecular diffusion, Knudsen diffusion, and surface diffusion (Kucuk and Sawyer, 1980; Smith et al., 1984). When the gas mean free path is less than the pore diameter, molecular diffusion dominates. The flow is dominated by the collision between gas molecules. When the pore diameters is greatly less than the gas mean free path, Knudsen diffusion dominates. The flow is dominated by the collision between gas molecules and the wall.

37 21 Also, Knudsen number is a measure of the gas rarefaction degree when gas flow through small pores. According to the Knudsen number, the gas flow regime can be classified into four kinds (Heidemann et al., 1984; Barber and Emerson, 2006; Schaaf and Chambré, 1961), which is illustrated in Figure Figure Flow regime classification based on Knudsen number (Heidemann et al., 1984; Karniadakis et al. 2006; Ziarani and Aguilera, 2012). According to Knudsen number, the fluid flow regimes can be classified into four types: (1). <0.01. Continuum regime. Under this Knudsen number range, the continuum assumption holds. The no-slip boundary condition is valid. And the flow is continuum flow or viscous flow. For simplification, gas transport can be characterized using conventional Darcy law. (2). 0.01< <0.1. Slip flow regime under this Knudsen number range. The no-slip boundary condition is not valid any longer and the rarefaction effects become more pronounced. The flow is called slip flow which cannot be characterized using continuum approach. The ratio of molecule-wall to molecule-molecule collisions increases as

38 22 Knudsen number increases. However, the latter is still dominant in the slip flow regime (Sakhaee-Pour and Bryant, 2012). Therefore, the Navier-Stokes equation is still valid for this domain. The flow can be described using Navier-Stokes equation combined with slip boundary condition. Also, the slip flow regime can be modeled using Dusty Gas Model (Mason and Malinauskas, 1983). (3). 0.1< <10. Much higher Knudsen number. The flow is called transition flow regime. The continuum assumption and Navier-stokes equation are both not applied. Molecular simulation method can be used to study this flow regime (Bird, 1994), in which the gas molecules are considered as a swarm of discrete particles. (4). >10. The intermolecular collisions become negligible and the flow regime is called free molecule flow (Michalis et al., 2010). In this Knudsen number range, the gas mean free path of the gas molecules is 10 times larger than the pore size. Boltzmann equation can be used to characterize such flow regime. However, it should be noted that this classification based on Knudsen number is empirical and the bounds of different regimes should not be exact values. Some literatures have set the limit of viscous flow region as (Sakhaee-Pour and Bryant, 2012; Civan, 2010) Gas Flow Mechanisms in the Nano Pore. Many investigators have studied micro-scale flow in shale nano pores (Javadpour et al., 2007; Passey et al., 2010; Florence et al., 2007; Freeman et al., 2011; Civan, 2010). Civan (2010) has separated the gas flow in a single capillary flow path according into three flow regimes according to the distance away from the pore wall. The flow path has been classified into three regions: inner flow region, intermediate region, and the near wall region. In the near wall region, gas transport can be described by a constant slip velocity and can be characterized as free molecular

39 23 regime. In the intermediate region, gas transport can be characterized using transition/knudsen flow regime. In the inner region which is away from the pore wall, the flow can be characterized as continuum flow. Akkutlu and Fathi (2011) developed a model to consider desorption and diffusion from Kerogen. Swami and Settari (2012) have proposed a model for gas flow through one nano pore in shale gas reservoirs considering the effects of non-darcy flow mechanism. According to the literature review, the most widely accepted flow mechanisms in gas shales are summarized as follows: (1). Viscous flow or convective flow: This is the region where Knudsen number has fairly low values such as in the conventional reservoirs where pore diameter is in micrometers. The gas mean free path is comparably negligible with the pore size. As Stops (1970) pointed out, when the gas mean free path is smaller than the pore throat diameter, the motion of gas molecules is determined by their collision with each other. Gas molecules collide with the wall less frequently. During this period, viscous flow exists, which is caused by the pressure gradient between single-component gas molecules. The mass flux of viscous flow can be calculated by the Darcy law, which can be expressed as Equation (6) provided by Kast and Hohenthanner (2000): (6) where is the mass flux caused by viscous flow (kg/(m 2 s)), is the intrinsic permeability of the nano capillary (m 2 ),p is the pressure of the nano capillary (Pa),and is the gas density (kg/ m 3 ).

40 24 (2). Knudsen Diffusion: When the diameter of the pore is very small, the mean free path lies relatively close to it. In this case, the collision between gas molecules and the wall becomes the dominant effect. The gas mass flux can be expressed by the Knudsen diffusion equation (Kast and Hohenthanner, 2000; Freeman et al., 2011; Gilron and Soffer, 2002) which is shown in Equation (7): (7) where is the mass flux caused by Knudsen diffusion (kg/(m 2 s)), is the gas mole concentration (mol/m 2 ), is the gas molar mass (kg/mol), is the Kundsen diffusion coefficient (m 2 /s), and can be expressed as Equation (8) provided by Florence et al. (2007):. (8) where is the porosity of a single nano capillary, which is equal to 1, is the gas constant of 8.314, is the temperature (), and is the constant. Also, Javadpour et al. (2007) proposed following expression for estimating the Knudsen diffusion coefficient. (9) (3). Gas slippage. Gas slippage is defined as an effect which leads the flow in the pores to deviate from viscous flow to non-laminar flow (Rushing et al., 2004). Gas slippage

41 25 occurs when the pore throat diameter is within the mean free path of gas molecules. Due to this, gas molecules will slip when they are in contact with pore surface and become fast. In low-permeability formations (less than md) or when the pressure is very low, gas slip flow cannot be omitted such as studying gas transport in tight reservoirs (Klinkenberg, 1941; Sakhaee-pour and Bryant, 2012). Under such kind of flow conditions, gas absolute permeability depends on gas pressure which can be expressed as follows: 1 (10) where is the apparent permeability, is the intrinsic permeability, is the average gas pressure, is the slip coefficient. Some empirical models have been developed to account for slip-flow, such as correlations developed from the Darcy matrix permeability (Ozkan et al., 2010); correlations developed based on flow mechanisms ((Brown et al., 1946; Javadpour et al., 2007; Javadpour, 2009); and semi-empirical analytical models by Moridis et al. (2010). Table 2.1 has listed the gas slippage factor proposed by different scholars. Table 2.1. Knudsen s Permeability Correction factor for Tight Porous Media (Modified from Ziarani and Aguilera (2012)). Model Klinkenberg (1941) Heid et al. (1950) Jones and Owens (1980) Correlation factor Sampath and Keighin (1982) Florence et al. (2007). Civan (2010)

42 Simulation Methods for Gas Flow through Nano Pores. Different modeling approaches have been adopted to simulate the flow of gas in nanotubes. Burnett (1935) introduced the Burnett equation type method in Bird (1994) and Bhattacharya and Lie (1991) both tried the molecular dynamics (MD) method. Tokumasu and Matsumoto (1999) and Karniadakis et al. (2002) used direct simulation Monte Carlo (DSMC) to study gas flow characteristics. Hornyak et al. (2008) used the Lattice- Boltzmann (LB) method to study gas flow. However, all of these modeling methods consume excessive space and time when systems are larger than a few microns, rendering them impracticable. The situation worsens when attempting to make accurate simulations with very small time steps and grid sizes, as convergence becomes a significant problem. Some researchers have attempted to derive an equation to characterize the law of gas flow. Klinkenberg (1941) introduced the Klinkenberg coefficient to consider the slip effect when gas flows in nano pores. And Beskok and Karniadakis (1999) derived a unified Hagen Poiseuille-type equation for volumetric gas flow through a single pipe. However, the applicability of these methods requires further investigation, and these modeling results have not yet been compared to real experimental data. The concept of apparent permeability for shale gas was first proposed by Javadpour (2009) to simplify simulation work. In 2009, he proposed the concept of apparent permeability, considering Knudsen diffusion and advection flow. Using this method, the flux vector term can be expressed simply in the form of a Darcy equation, which greatly reduces the computational complexity. Then, Civan (2010) and Ziarani and Aguilera (2012) derived the expression for apparent permeability in the form of a Knudsen number based on a unified Hagen-

43 27 Poiseuille equation (Beskok and Karniadakis, 1999). Shabro et al. (2011, 2012) applied the concept of apparent permeability further in pore scale modeling for shale gas SHALE PERMEABILITY MEASUREMENT METHODS Permeability and Intrinsic Permeability. Permeability is defined as the capacity of fluid flow in the reservoir rock. It is one of the most important properties of the rock and not the fluid (Tinni et al., 2012). Darcy law has been widely used to measure permeability which is shown as follows: (11) where is the fluid flow rate (cc/sec), is the fluid viscosity (cp), is the permeability (d), A is the surface area (cm 2 ), dp/dx is the pressure gradient along the fluid flow direction. In unconventional shale gas reservoirs, we often use the concept of intrinsic permeability. The intrinsic permeability measurement is important as it can be used to calculate many useful parameters such as gas mean free path, Knudsen number, average pore radius, and tortuosity (Alnoaimi, and Kovscek, 2013). The intrinsic permeability is only related with reservoir properties, which does not depend on the test fluids and flow conditions (Civan, 2009) Permeability Measurement Methods. Permeability is an important parameter controlling gas flow in the reservoir. Conventional steady state methods to obtain the gas permeability of shale will not be applicable due to ultra-low permeability. A

44 28 long time is needed for the flow rate to stabilize (1988; Cui et al., 2009). So, for unconventional shale permeability measurement, transient permeability measurement is commonly used, such as transient pressure pulse decay experiments (Alnoaimi, and Kovscek, 2013). Many scholars have studied the methods to measure gas permeability for low permeability samples (Brace et al., 1968; Swanson, 1981; Jones, 1997; Prince et al., 2009; Civan et al., 2012). Following are three widely used methods to obtain the shale gas permeability, such as pulse-decay Technique, crushed samples dynamic pycnometry, and canister desorption Test (Brace et al., 1968; Luffel et al., 1993; Cui et al., 2004; Egermann et al., 2005; Cui et al., 2009; Barral et al., 2010; Alnoaimi and Kovscek, 2013) Pulse-decay technique. Pulse-decay method can be used to determine tight/shale oil/gas permeability (Kranz et al., 1990; Fisher and Paterson, 1992; Cui et al., 2009). The intrinsic permeability can be measured from the apparent permeability and reciprocal pore pressure relationship (Alnoaimi and Kovscek, 2013). Alnoaimi and Kovscek have used pulse-decay method to determine the permeability of an Eagle Ford shale sample (Alnoaimi and Kovscek, 2013). In pulse-decay method, the apparatus consists of an upstream reservoir, a downstream reservoir, and a cell containing the rock sample with applying confining pressure. Pressure transducer is used to measure the pressure difference ( ) change between the upstream and downstream reservoir with time (t). Then the ~ data can be analyzed to obtain the sample permeability. The permeability obtained from pulse-decay method can represent the in situ permeability of the reservoir and is widely used in laboratories. However, this technic is dependent on the test gas. So,

45 29 using the commonly used He or N2 to measure the permeability of shale sample which is mainly composed of methane will be inappropriate (Cui et al., 2004) Canister desorption test. Desorption test is generally used to evaluate the gas content of shale gas reservoirs. Also, desorption test can be used to obtain the permeability of shale (Cui et al., 2004). When the drill sample is obtained from the shale well, it is instantly put into the canister with reservoir temperature to desorb. During this process, the cumulative desorption gas will be recorded with time. According to the desorption data, the shale permeability can be obtained using analytical methods. Canister desorption test is generally similar with the permeability measurement with crushed sample without exerting confining pressure. This method is comparably better than the crushed sample method as the test gas is the natural gas in the reservoir Crushed samples dynamic pycnometry. Crushed sample can be used to measure shale properties, such as density, porosity, and adsorption isotherms. This method is generally used to measure shale permeability as it can be used on small rock particles and faster than other permeability measurement methods (Tinni et al., 2012). Measuring shale permeability using crushed samples was designed by Luffel et al. (Luffel and Guidry, 1992; Luffel et al., 1993). Crushed sample can also be used to measure gas permeability of the shale sample (Luffel et al., 1993; Cui et al., 2004; Cui et al., 2009; Profice et al., 2012; Egermann et al., 2005). The methodologies of using crushed sample to measure shale gas permeability have been detailed by Cui et al. (2005). There are two reservoirs: the reference cell and the sample cell. Two pressure transduces are used to monitor the pressure in the two reservoirs. During the gas diffusing into the sample cell, the pressure data have been recorded until the equilibrium between the two reservoirs achieved. Then, the pressure vs.

46 30 time data can be used to calculate the porosity and permeability of the shale sample. The permeability measured using crushed sample can only represent the properties of primary micro-pores. For more accurate results, several experiments under different pressure conditions should be conducted using this method APPARENT PERMEABILITY MODEL FOR GAS FLOW THROUGH NANO PORES OF SHALE GAS RESERVOIRS Methods for reserve estimation, and numerical simulation of gas production from shale gas reservoirs require permeability information. Apparent permeability study for shale/tight gas reservoirs has drawn considerable attention as the conventional Darcy equation cannot characterize other flow regimes except the viscous flow regime (Civan, 2009), which courses the apparent permeability for gas flow in unconventional gas reservoirs deviates significantly from those in conventional gas reservoirs (Civan et al., 2011). And the apparent permeability method is a convenient way to be adopted in numerical simulation as it can be expressed in the form of Darcy equation. Many scholars have conducted study on the apparent permeability for gas flow in shale gas reservoirs. Klinkenberg (1941) first proposed that gas apparent permeability is a function of pressure for reservoirs with ultra-low permeability. Jones and Owen (1980) have derived an empirical static gas slippage factor and compared the results with Klinkenberg s slippage factor. Sampath and Keighin (1982) have proposed Klinkenberg coefficient correction for N2 in presence of water. Ertekin et al. (1986) incorporated Klinkenberg effect to consider the higher than expected gas production in shale gas reservoirs. Beskok and Karniadakis (1999) proposed a unified Hagen-poiseuille-type equation to characterize gas flow in

47 31 tight/shale reservoirs. The proposed model has considered multiple transport mechanisms, such as viscous flow, slip flow, transition flow, and free molecular flow. Based on the work of Beskok and Karniadakis (1999), Florence et al. (2007) proposed a theoretically derived apparent permeability model which is a function of Knudsen number. Freeman et al. (2011) modified the apparent permeability model of Florence to obtain a component-dependent apparent permeability. Recently, some researchers have proposed the methods to calculate the apparent permeability by coupling Darcy flow with Knudsen diffusion to describe gas flow in shale reservoirs (Javadpour, 2009; Akkutlu and Fathi, 2011; Shabro et al., 2011; and Darabi et al., 2012) Beskok and Karniadakis Model. The unified Hagen-poiseuille-type equation which is proposed by Beskok and Karniadakis (1999) can be expressed in Equation (12). This model can consider different flow regimes based on the Knudsen number. (12) 1 1 (13) tan (14) 1, 0, 0 (15) where is the pore diameter, is the length of flow path, is the volumetric gas flow rate through a single pipe, is the Knudsen number which based on gas mean free path and pore diameter, is the dimensionless rarefaction coefficient which can be calculated

48 32 using Equation (14) which is provided by Beskok and Karniadakis (1999) or using Equation (15) which is provided by Civan (2010). The parameter in the in the Equation (13) is the empirical slip coefficient, which is equal to 1 for fully developed slip flow condition.,,, and are all empirical constants which can be determined through experiments (Civan, 2009; Civan, 2010). Equation (12) can be used to calculate the flow rate for a single pore. For a bundle of tortuous flow paths such as the membranes, the volumetric gas flow rate can be calculated using Equation (16). (16) where is also the length of the flow path, is the number of the preferential hydraulic flow paths which can be calculated using the formula provided by Civan (2007). (17) where is the porosity, is the bulk surface area normal to the flow direction Javadpour s Model. Slip flow and Knudsen diffusion are two important flow mechanisms in SGRs. These two mechanisms can be combined in one equation to characterize gas flow for both conventional and unconventional SGRs (Javadpour, 2009; Shabro et al., 2009). Javadpour (2009) conducted a thorough study of gas flow in nano pores, deriving following formula for gas flow in nanotubes based on Knudsen diffusion and viscous force (Javadpour et al., 2007), as shown in Equation (18) and Equation (19):

49 (18) 1 (19) The apparent permeability provided by Javadpour is expressed in Equation (19):. (20) Florence s Model. Florence et al. (2007) have derived the apparent permeability expression by approximating the apparent permeability expression of Beskok and Karniadakis (1999) under slip flow condition (,) for <<1, which is shown in Equation (20). 1 1, 1 14 (21) Civan s Model. The apparent permeability model provided by Civan (2009) is shown in Equation (22), which is based on the work of Beskok and Karniadakis (1999). (22) (23)

50 34 where is the intrinsic permeability or the liquid permeability which is only related to the porous media and does not depend on the fluid types and flow conditions, is the tortuosity factor of the flow paths Sakhaee-pour and Bryant s Model. Sakhaee-pour and Bryant (2012) also proposed a model to characterize gas apparent permeability based on Knudsen number. They regarded the method proposed by Florence et al. (2007), which is one-order of Knudsen number is not accurate enough to characterize gas flow in the transition period. After fitting with the experimental data, they proposed a high-order equation of Knudsen number which is fit for the Knudsen number in the range from 0.1 to 0.8 (Shi et al., 2013). The apparent permeability expression is shown as follows: (24) 2.5. GAS FLOW MECHANISMS IN SHALE GAS RESERVOIRS Unlike conventional gas reservoirs, gas transports in shale gas reservoirs is a complex process. Gas diffusion, viscous flow due to pressure gradient, and desorption from Kerogen coexist. Also, the gas storage forms are complex in shale reservoirs. There are three forms for the gas to store in the shale: 1) free gas stored in the fractures and pores; 2) absorbed gas stored on the surface of the bedrock. Hill and Nelson estimated that 20%~85% of gas in shale might be stored as adsorbed gas (Hill and Nelson, 2000); 3) dissolved gas stored in the kerogen (Javadpour, 2009). Gas transport in unconventional shale strata is a multi-mechanism-coupling process which is different from conventional reservoirs. In micro fractures which are inborn or induced by hydraulic stimulation, viscous flow

51 35 dominates. And gas surface diffusion and gas desorption should be further considered in organic nano pores. Also, Klinkenberg effect should be considered when dealing with gas transport problem. Some scholars have studied gas transport behavior in SGRs. Bustin et al. (2008b) have studied effect of fabric on the gas production from shale strata. However, it is assumed that matrix does not have viscous flow and diffusion mechanisms. It is also assumed that gas transport in the fracture system obeys Darcy law which is not sufficiently accurate. Ozkan (2010) used a dual-mechanism approach to consider transient Darcy and diffusive flows in the matrix and stress-dependent permeability in the fractures for naturally fractured SGRs. However, they ignored effects of adsorption and desorption. Moridis et al. (2010) has considered Darcy s flow, non-darcy flow, stress-sensitive, gas slippage, non-isothermal effects, and Langmuir isotherm desorption. They found that production data from tight-sand reservoirs can be adequately represented without accounting for gas adsorption whereas there will be significant deviations if gas adsorption is omitted in SGRs. But they did not consider gas diffusion in the Kerogen. Wu and Fakcharoenphol (2011) has proposed an improved methodology to simulate shale gas production, but the gas adsorption-desorption were ignored in the model. It is necessary to develop a mathematical model which incorporates all known mechanisms to describe the gas transport behavior in tight shale formations. The simulation work was greatly simplified when the apparent permeability was first introduced by Javadpour. In 2009, the concept of apparent permeability considering Knudsen diffusion, slippage flow and advection flow was proposed (Javadpour, 2009). Through this method, the flux vector term can be simply expressed in the form of Darcy equation which will greatly reduce the computation complexity. Then the concept of apparent permeability was further applied in

52 36 pore scale modeling for shale gas (Shabro et al., 2011; Shabro et al., 2012). Civan and Ziarani derived the expression for apparent permeability in the form of Knudsen number (Civan, 2010; Ziarani and Aguilera, 2012) based on a unified Hagen-Poiseuille equation (Beskok and Karniadakis, 1999). Following are those mechanisms which have been considered in simulation of shale gas reservoirs according to the literature review Darcy Flow. The advection of gas phase in porous media is commonly characterized using Darcy equation. From Darcy equation, the gas velocity is directly proportional to the gas pressure gradient. Darcy equation is applicable when the gas velocity is not high or there is no boundary shear effect (Webb, 2006). Darcy flow has also been considered in simulation of shale gas reservoirs (Bustin et al., 2008; Moridis et al., 2010). For gas flow, more commonly used equation is the extended Darcy equation which considers the Klinkenberg effect (Klinkenberg, 1941). Klinkenberg pointed that for the same rock sample and test gas, the gas permeaiblity is different under different average pressure. Also, for the same rock sample and average pressure, the gas permeaiblity is different if the test gas are different. Equation (25) is the general Darcy equation which considers Klinkenberg effect (Florence et al., 2007; Javadpour, 2009): 1 (25) where is the gas flow velocity, m/s; is the Klinkenberg permeability, m 2 ; is the Klinkenberg coefficient, MPa; is the gas viscosity, Pa.s; is the average gas pressure, MPa.

53 Flow in the Fracture. At high flow velocities in the near wellbore hydraulic fractures, inertia effect cannot be neglected (Huang and Ayoub, 2008; Rushing et al., 2004). Due to inertia effect, the flow rate will keep constant when pressure increases. There are two reasons for this. One reason is gas deceleration within larger pores which slows down approaching gas molecules; another reason is though gas acceleration within tighter pores, gas velocity keeps constant due to viscous shearing effect (Arogundade and Sohrabi, 2012). So, the conventional linear Darcy s equation which assumes zero inertia effect will no longer be valid. Gas flow in hydraulic fracture follows Forchheimer flow model, which takes inertia effect into consideration (Huang and Ayoub, 2008; Moridis et al., 2010). Forchheimer equation can be expressed as follows: (26) where is the pressure gradient (Pa/m); is the gas viscosity (Pa.s); is the velocity (m/s); β is the Forchheimer coefficient; is the fluid density (kg/m 3 ); is the permeability (m 2 ). The first parameter on the right hand side of Equation (26) represents viscous flow and dominates at lower velocity and the second parameter on the right hand side of Equation (26) represents inertial forces and dominates at higher velocity where viscous forces become less dominant. Also, we can find that when β 0, Equation (26) changes to Darcy equation. Many scholars have proposed methods to compute the Forchheimer -parameter (Cooke, 1973; Kutasov, 1993; Frederick and Graves, 1994; Li and Engler, 2001). Kutasove (1993) computes the -parameter as a function of effective permeability to gas as well as

54 38 water saturation based on experiment results. Frederick and Graves (1994) have developed two empirical correlations for -parameter in two phase flow. Cooke (1973) has correlated -parameter coefficients to various proppant types Diffusion. The diffusion in porous media consists of continuum diffusion, and free molecule diffusion (Webb, 2006). Continuum diffusion can be characterized using Fick s law or Stefan-Maxwell equations (Bird, 2002). Free molecule diffusion, or Knudsen diffusion occurs when the gas mean free path is close to the pore diameter, such as in the nano pore, which has been detailed in the flow mechanisms of gas flow in nano pores. In shale formation, the diffusion refers to the gas flow from high concentration zone (shale matrix system) to the low concentration zone (shale micro-fracture system) due to concentration difference. When the equilibrium achieves, the diffusion process will end. The diffusion process in shale formation can be classified into pseudo steady diffusion and unsteady diffusion (Soeder, 1988; Shi and Durucan, 2008). The pseudo steady diffusion can be characterized using Fick first law, which states the mole or mass flux is proportional to a diffusion coefficient times the gradient of the mole or mass concentration (Webb, 2006).The unsteady diffusion is generally characterized using Fick s second law, which states the temporal evolution of gas concentration Gas Adsorption/Desorption. Adsorption gas is one main storage form in shale gas reservoirs (Cipolla et al., 2010). The adsorption gas is about 40-50% of the total gas in place (Arogundade and Sohrabi, 2012). Gas desorption is one important recovery mechanism in shale gas reservoirs. Gas transport through shale fracture network is the combination of desorption and diffusion (Biswas, 2011). Cipolla et al. (2010) showed that considering gas desorption will lead to improved gas recovery. Thomson et al. (2011) also

55 39 found that desorption gas accounts for about 17% of the expected ultimate recovery. Arogundade and Sohrabi (2012) concluded that 5-15% of the total gas production are desorption gas in shale gas reservoirs. The adsorption of gas on the surface of shale matrix is affected by temperature, pressure, gas type, and rock properties (Lane et al., 1989). To characterize the gas adsorption on the rock surface, there are three widely used adsorption equations: Henry equation; Freundlich equation; and Langmuir equation. Henry equation states that the adsorption volume is a linear function of pressure (King, 1993). With the increase of gas pressure, the adsorption volume increases. However, Henry equation assumes that the adsorption gas is ideal gas which is not valid for high pressure range (Dorris and Gray, 1980). Freundlich equation states there is exponential relationship between gas adsorption volume and pressure (Sheindorf et al., 1981). When the pressure is large enough, gas adsorption volume will increase much slowly (Cipolla et al., 2010; Cui et al., 2009). So, the pressure range for the Freundlich equation is only limited to low pressure range. The most widely used for characterizing shale gas adsorption is Langmuir isotherm equation (Langmuir, 1916; Langmuir, 1917; Freeman et al., 2010; Freeman et al., 2011; Freeman et al., 2013; Guo et al. 2014b). Langmuir assumes that there is monolayer gas molecules on the surface. The pressure depletion and gas desorption are synchronized and the system can achieve equilibrium instantly. As the shale gas are mainly composed of Methane. And the temperature of shale formation is higher than the critical temperature of Methane ( C). So, the methane will be adsorbed on the surface as monolayer. Based on the Langmuir equation (Equation (27)), the amount of gas adsorbed on the rock surface can be computed.

56 40 (27) where is the gas Langmuir volume, scf/ton; is the gas Langmuir pressure, psi; is the in-situ gas pressure in the pore system, psi. Langmuir volume means the maximum amount of gas that can be adsorbed on the rock surface under infinite pressure. Langmuir pressure is the pressure when the amount of gas adsorbed is 50% of the Langmuir volume. These two parameters play important roles in the gas desorption process. For different shale gas reservoirs, the differences of Langmuir volume and Langmuir pressure lead to distinct trend of gas content. Figure 2.12 shows the adsorbed gas content (scf/ton) versus pressure (psi) based on Langmuir parameters for five major shale plays in U.S. Adsorbed gas content (scf/ton) Marcellus shale New albany shale Haynesville shale Eagleford shale Barnett shale Pressure (psi) Figure Langmuir isotherm curves for five typical shale gas reservoirs.

57 Stress, Water, and Temperature Sensitivity. For gas flow in shale formations, some scholars also consider the effect of stress, water, and temperature (Peng and Robinson, 1976; McKee et al., 1988; Davies, J. P., and Davies, D. K., 1999; Reyes and Osisanya, 2000; Roy et al., 2003; Wang and Reed, 2009; Cho et al., 2013). Stress sensitivity refers to the change of shale permeability, porosity, total stress, effective stress, and rock properties (such as compressibility, Young s modules) with stress change (McKee et al., 1988; Davies, J. P., and Davies, D. K., 1999; Reyes and Osisanya, 2000). The effect of stress on matrix and fracture permeability has been studied in detail according to the literature review (Kwon et al., 2004; and Tao et al., 2010). Celis et al. (1994) have simulated the oil flow in stress sensitive naturally fractured reservoirs using permeability modulus. Gutierrez et al. (2000) have carried out experiments to investigate the stress dependent permeability of de-mineralized fractures in shale. Raghavan and Chin (2002) have proposed a geomechanical-fluid flow model to study the pressure transient response for stress sensitive reservoirs. Franquet et al. (2004) have investigated the effect of pressure dependent permeability on transient radial flow for tight gas reservoirs. Thompson et al. (2010) have concluded that pressure dependent permeability is responsible for performance behavior by analyzing the well performance data of Haynesville shale. Cho et al. (2013) have studied the pressure dependent natural fracture permeability in shale and its effect on shale gas well production. Pedrosa (1986) studied the pressure transient responses in stress sensitive reservoirs by defining a permeability modulus. Water sensitivity refers to the swelling of clay particles due to the existence of water. The effect can be characterized by correcting gas-water relative permeability curve or capillary pressure (Roy et al., 2003; Wang and Reed, 2009). Temperature permeability

58 42 refers to the formation and fluid properties change due to temperature. The effect can be characterized using Peng-Robinson equation (Peng and Robinson, 1976) SIMULATION MODEL FOR GAS PRODUCTION FROM SHALE GAS RESERVOIRS Reservoir simulation is the most useful technology for management of subsurface oil/gas reservoirs, which can help understand shale gas reservoirs and optimize the production decisions (Fazelipour et al., 2008; Fazelipour, 2011). Recent advances in simulation and modeling technologies have enable it possible to optimize and enhance the gas production from organic shale reservoirs (Lee and Sidle, 2010; Dahaghi et al., 2012). Many scholars have conducted modeling of gas production from unconventional shale gas reservoirs. The key problem in simulating fractured shale gas reservoirs is how to handle the fracture matrix interaction. Many different models have been proposed to simulate fluid flow in fractured reservoirs (Luffel et al., 1993; Mayerhoffer et al., 2006; Bustin et al., 2008b; Rubin, 2010; Ozkan et al., 2010; Fazelipour, 2011; Shabro et al., 2011; Alnoaimi and Kovscek, 2013;). These methods include: (1) equivalent continuum model (Wu, 2000); (2) dual porosity model, including dual porosity single permeability model, and dual porosity dual permeability model (Warren and Root, 1963; Kazemi, 1969); (3) Multiple Interaction Continua (MINC) method (Pruess, 1985; Wu and Pruess, 1988); (4) Multiple porosity model (Civan et al., 2011; Dehghanpour et al., 2011; Yan et al., 2013); (5) Discrete fracture model (Snow, 1965; Xiong et al., 2012). The details of each model are as follows: Equivalent Continuum Model. The Equivalent continuum model is first proposed by Snow (1968). Many different types of equivalent continuum models have been

59 43 proposed with different assumptions, such as for isothermal flow, coupled fluid and heat flow, single phase flow, and multi-phase flow (Wu, 2000). In the equivalent continuum model, the fractured porous media is considered as a continuum media. The properties of the fracture arrays, such as direction, location, permeability, porosity, etc., are averaged into the whole porous media. This method is focused on the study of the macro flow characteristics of the porous media without considering the specific flow condition in a single fracture. This method has not been widely used in shale gas reservoir simulation as it is not easy to obtain the averaged permeability tensor and it is too conceptual. Moridis et al. (2010) have constructed effective continuum shale gas reservoir simulation model with considering multi-component gas adsorption. In the effective continuum model, the shale gas reservoir is discretized and the fractures are characterized with grid cells as single planar planes or network of planar planes (Cipolla et al., 2009; Cheng, 2012) Dual Porosity Model. The concept of dual porosity was first proposed by Barenblatt and Zheltov (1960). Then, this concept was introduced into petroleum engineering by Warren and Root (1963). Dual porosity model is widely used to simulate the fluid flow in fractured porous media and many scholars have published a lot of literature based on dual porosity model (Kazemi, 1969; de Swaan O, 1976; Kucuk and Sawyer, 1990; Zimmerman and Bodvarsson, 1989; Zimmerman et al., 1993; Law et al., 2002; Shi and Durucan 2003; Ozkan et al. 2010). There are two interacting media: the matrix blocks with high porosity and low permeability, and the fracture network with low porosity and high conductivity. According to whether there is fluid flow in the matrix, the dual porosity model can be classified into dual porosity single permeability model and the dual porosity dual permeability model.

60 44 In the dual porosity single permeability model, the flow only occurs through the fracture system and matrix are treated as the spatially distributed sinks or sources to the fracture system (Wu et al., 2009). In the shale gas reservoirs, the matrix system serves as the gas storage place for the free and adsorption gas. And the fracture system provides high permeability (Zhang et al., 2009; Fan et al., 2010; Du et al., 2010; Du et al., 2011; Harikesavanallur et al., 2010). Watson et al. (1990) have performed production data analysis for Devonian shale gas reservoir using DPM and presented analytical reservoir production models for history matching and production forecasting. Bustin et al. (2008b) have constructed a 2D dual porosity model to simulated shale gas reservoirs which uses both experimental and field data as model input parameters. The fluid flow is governed by Darcy s law. The coupled governing equations are discretized implicitly in time and 2D space using finite difference method. Du (2010) simulated the hydraulic fractured shale gas reservoir as a dual porosity system. Microseismic responses, hydraulic fracturing treatments data and production history-matching analysis were applied. Ozkan et al. (2010) have developed a dual-mechanism dual-porosity model to simulate the linear flow for fractured horizontal well in shale gas reservoirs. Comparably same with dual porosity single permeability model, dual-porositydual-permeability model is also studied with considering the flow in the matrix additionally (Pereira et al., 2004; Yan et al., 2013). The matrix properties dominate the flow from matrix to fracture. Connell and Lu have proposed an algorithm for the dual-porosity model considering Darcy flow in the fractures and diffusion in the matrix (Lu and Connell, 2007; Connell and Lu, 2007). Moridis (2010) constructed a dual permeability model and compared it with dual porosity model and effective continuum model. Results showed that

61 45 dual permeability model offered the best production performance. But they also pointed out that the deviations become more obvious during the later time of production. Cao et al. (2010) have proposed a new simulation model for shale gas reservoir based on dual porosity dual permeability model with considering multiple flow mechanisms, such as adsorption/desorption, diffusion, viscous flow, and deformation Multiple Interaction Continua Method. As an extension for dual porosity model, Pruess (1992) have proposed a multiple interaction continua method (MINC) to more accurately characterize the interaction between matrix and fracture. The matrix system in every grid block for MINC method will be subdivided into a sequence of nested rings which will make it possible to more accurately calculate the interblock fluid flow. In comparison with dual porosity model, MINC method is able to describe the gradient of pressures, temperatures, or concentrations near matrix surface and inside the matrix which can provide a better numerical approximation for the transient fracture matrix interaction (Wu et al., 2009). Wu et al. (2009) have proposed a generalized mathematical model to simulate multiphase flow in tight fractured reservoirs using MINC method with considering following mechanisms: Klinkenberg effect, non-newtonian behavior, non- Darcy flow, and rock deformation Multiple Porosity Model. Some scholars also proposed multiple porosity model in which matrix is further separated into two or three parts with different properties. Civan et al. (2011) proposed a qual-porosity model to simulate shale gas reservoirs, including organic matter, inorganic matter, natural, and hydraulic fractures Dehghanpour et al. (2011) have proposed a triple porosity model to simulate shale gas reservoirs by incorporating micro fractures into the dual porosity models. They assumed that the matrix

62 46 blocks in dual porosity model is composed of sub-matrix with nano Darcy permeability, and micro fractures with milli to micro Darcy permeability. Result of sensitivity analysis showed that the rate of wellbore pressure drop has been significantly decreased using this model. Yan et al. (2013) presented a micro-scale model. In Yan s model, the shale matrix was classified into inorganic matrix and organic matrix. The organic part was further divided into two parts basing on pore size of kerogen: organic materials with vugs and organic materials with nanopores. Therefore, there are four different continua in this model: nano organic matrix, vug organic matrix, inorganic matrix, and micro fractures Discrete Fracture Model. In the discrete fracture model, the fractures are discretely modeled. The discrete fracture modeling requires unstructured gridding of the matrix fracture system using Delaunay triangulation. Each fracture is characterized as a geometrically well-defined entity. Baca et al. (1984) have proposed a 2D model for solute and heat flow in fractured reservoir using DFM. Juanes et al. (2002) investigated 2D and 3D single phase flow in fractured reservoir using finite element DFM. Karimi-Fard et al. (2003) have presented a simplified discrete fracture model using finite volume method combined with unstructured gridding, which can be used for 2D and 3D multiphase fluid flow simulation. Gong (2008) has proposed a workflow to enable the DFM to be applied in reservoir scale. In the modeling of shale gas reservoirs, gas flow from the matrix to the discrete fractures by diffusion and desorption (Ciplla et al., 2009; Rubin, 2010; Wang and Liu, 2011; Mongalvy et al., 2011; Gong et al., 2011). Mayerhoffer et al. (2006) have investigated the fluid flow in fractured shale gas reservoirs considering stimulated reservoir volume (SRV) using explicit fracture network (Mayerhofer et al., 2006). Gong et al. (2011)

63 47 have proposed a systematic methodology of building discrete fracture model based on microseismic interpretations for shale reservoirs with considering multiple mechanisms, such as gas adsorption, non-darcy effect, etc. The discrete fracture model is more rigorous, however, application of this method to reservoir scale is limited due to computational intensity and lack of detailed knowledge of fracture matrix properties and spatial distribution in reservoirs (Wu et al., 2009; Civan et al., 2011). And the dual porosity model can deal with the fracture matrix interaction more easily and less computationally demanding than discrete fracture method. So, dual porosity model is the main approach for modeling fluid and heat flow in fractured porous media (Wu et al., 1999; Wu et al., 2007; Wu et al., 2009) CONCLUDING REMARKS Shale gas is an important unconventional resources and also it is the main contributor to secure the natural gas supply in U.S. currently and in the future decades. Shale gas reservoirs (SGRs) are fine-grained sedimentary rocks with ultra-low permeability. Natural gas can be stored in shale gas reservoirs as three forms: as free gas stored in the micro fractures and nano pores; as dissolved gas stored in the kerogen; and as adsorption gas stored in the surface of the bedrock. Shales have low permeability from milidarcy to nanodarcy, and low porosity from 1% to 5%. The average grain size of the shale is less than in. Compared with conventional reservoirs, there are a large number of nano pores existed in the SGRs. There are multiple gas flow mechanisms during gas production process, such as viscous flow, Knudsen diffusion, slip flow, water and stress sensitivity, gas viscosity change, pore radius change, and gas adsorption/desorption. In the

64 48 nano scale, the continuum flow assumption breaks down and gas molecule s collision with the walls become dominant. Apparent permeability models are widely used in the simulation of gas flow through nano pores and many scholars have conducted related research in this area. For the simulation of SGRs, many models have been proposed to simulate shale gas production, such as equivalent continuum model, dual porosity model, multiple porosity model, and the discrete fracture model. The most widely used model is dual porosity model, which contains two interacting media: the matrix blocks with high porosity and low permeability, and the fracture network with low porosity and high conductivity. The study of fluid flow mechanisms and numerical simulation of shale gas reservoirs is still in the developing period. Many models were developed to more accurately characterize gas transport behavior in the nano pores and the SGRs.

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77 61 PAPER PAPER I. STUDY ON GAS FLOW THROUGH NANO PORES OF SHALE GAS RESERVOIRS Chaohua Guo; Mingzhen Wei Missouri University of Science and Technology (Published as an article in Fuel) ABSTRACT Unlike conventional gas reservoirs, gas flow in shale reservoirs is a complex, multiscale flow process having special flow mechanisms. Also, shale gas reservoirs contain a large fraction of nano pores, which leads to an apparent permeability that is dependent on both the pressure, fluid type, and the pore structure. Study of gas flow in nano pores is essential for the accurate numerical simulation of shale gas reservoir. However, no comprehensive study has been conducted pertaining to the flow of gas in nano pores. In this paper, experiments for nitrogen flow through nano membranes (with pore throat size: 20 nm, 55 nm, 100 nm) have been done and analyzed. Obvious discrepancy between apparent permeability and intrinsic permeability has been observed. And relationship between this discrepancy and pore throat diameter (PTD) has been analyzed. Then, based on the advection-diffusion model, we constructed a new mathematical model to characterize gas flow in nano pores. We derived a new apparent permeability expression based on advection and Knudsen diffusion. A comprehensive coefficient for characterizing the flow process was proposed. Simulation results were verified against the experimental data for gas flow through nano membranes and published data. By changing the comprehensive coefficient, we found the best candidate for the case of argon with a

78 62 membrane PTD of 235 nm. We verified the model using experimental data on nitrogen, and published data with different gases (oxygen, argon) and different PTDs (235 nm, 220 nm). The comparison shows that the new model matches the experimental data very closely. Additionally, we compared our results with experimental data, the Knudsen/Hagen-Poiseuille analytical solution, and existing data available in the literature. Results show that the model proposed in this study yielded a more reliable solution. Shale gas simulations, in which gas flowing in nano pores plays a critical role, can be made more accurate and reliable based on the results of this work. 1. INTRODUCTION With the growing shortage of domestic and foreign energy, producing gas from shale strata recently has become increasingly important in the volatile energy industry in North America and is gradually becoming a key component in the world s energy supply [1]. As more attention is given to these unconventional gas resources, understanding the rock and its permeability is crucial. The dynamics of gas in small pore throats in shale and other tight formations has become an important research topic during the current decade in the oil and gas industry [2]. Shale gas reservoirs are characterized by an organic-rich deposition with extremely low matrix permeability and clusters of mineral-filled natural fractures (Fig. 1). Through an experimental analysis of 152 cores of nine reservoirs in North America, Javadpour [3, 4] found that the average permeability of shale bedrock is 54 nd, and approximately 90% have permeabilities less than 150 nd. Most pore throat diameters (PTD) are concentrated in the range of 4 ~ 200 nm (10-9 m) [5].

79 63 Loucks et al. [6] found that gas shale strata is composed of micro and nano pores, with the majority being nano pores. These findings emphasize the importance of studying how gas flows in nano pores or nanotubes, which is critical for shale gas simulation and effective commercial production. Different modeling approaches have been adopted to simulate the flow of gas in nanotubes. Hornyak et al. [7] used the Lattice-Boltzmann (LB) method to study gas flow; Bird [8] and Bhattacharya et al. [9] both tried the molecular dynamics (MD) method; Tokumasu [10] and Karniadakis [11] used direct simulation Monte Carlo (DSMC) to study gas flow characteristics; and Burnett [12] introduced the Burnett equation type method in However, all of these modeling methods consume excessive space and time when systems are larger than a few microns, rendering them impracticable. The situation worsens when attempting to make accurate simulations with very small time steps and grid sizes, as convergence becomes a significant problem. Some researchers have attempted to derive an equation to characterize the law of gas flow. Beskok and Karniadakis [13] derived a unified Hagen Poiseuille-type equation for volumetric gas flow through a single pipe, and Klinkenberg [14] introduced the Klinkenberg coefficient to consider the slip effect when gas flows in nano pores. However, the applicability of these methods requires further investigation, and these modeling results have not yet been compared to real experimental data. The concept of apparent permeability was first proposed by Javadpour [4] to simplify simulation work. In 2009, he proposed the concept of apparent permeability, considering Knudsen diffusion and advection flow. Using this method, the flux vector term can be expressed simply in the form of a Darcy equation, which greatly reduces the

80 64 computational complexity. Then, Shabro et al. [15, 16] applied the concept of apparent permeability further in pore scale modeling for shale gas. Civan [17] and Ziarani and Aguilera [2] derived the expression for apparent permeability in the form of a Knudsen number based on a unified Hagen-Poiseuille equation [13]. In this paper, utilizing the ADM model [18] and considering the slip flow together with the Knudsen diffusion, we theoretically derived a new formula for the apparent permeability and the flux with a comprehensive coefficient to be determined from the experiment data. Different from what Javadpour [4] proposed, we assumed that the slip flow is part of Knudsen diffusion and proposed a new idea with the comprehensive coefficient in the Knudsen diffusion term instead of adding a theoretical dimensionless coefficient to correct for slip velocity in the Darcy term [4]. A comparison among the experimental data, the results of the proposed method, the K/HP analytical solution [19], and Javadpour s solution [4] shows that the proposed model produces simulation results that better match the experimental data provided by Cooper et al. [20]. 2. EXPERIMENTS FOR GAS FLOW THROUGH NANO MEMBRANES 2.1. SEM imaging of nano membranes Experiments for gas flow through nano membranes have been done in Missouri S&T. Nitrogen was used as the test gas. Three kinds of commercial AAO membranes (Fig. 2) have been used with pore size: 20 nm, 55 nm, 100 nm. The ceramic membranes are round slices with thickness of 100 μm. In order to have a better understanding of the nano membranes, images of the membranes have been analyzed using scanning electronic microscopy as shown in Fig. 2. Based on the images, the membrane pore size, pore density, and intrinsic permeability has been calculated and listed in the Table 1.

81 Experimental Setup and Results The schematic diagram and real picture of the experimental setup have been shown in Fig. 3. Ultra perm 600 was used to measure the pressure differences and flow rates. After applying confining pressure, the core holder with the membrane inside was injected by gas from upstream to downstream. Both upstream and downstream were measured by pressure meters and flow meters were set in downstream. By switching the upstream pressure to produce different pressure drops, the corresponding flow rate reading was recorded. In the experiments, we kept the outlet pressure as 1 atm. By changing the inlet pressure, we can obtain different flow rate. The gas flow rate was measured using gas flow meter. The gas permeability has been obtained using Eq. (1). The equivalent liquid permeability is calculated using Eq. (2). 100 (1) where, k is the gas permeability, md; A is the cross-sectional area, cm 2 ; p is the ambient pressure, 0.1 MPa; Q is the gas flow rate under ambient pressure, cm 3 /s; μ is gas viscosity, mpa.s; L is the length of the channel, cm. (2) The intrinsic permeability is calculated using the Eq. (2). The results of the experiments have been listed in the Table 2 as attached. Fig. 4 has shown the gas flow rate

82 66 vs. the average pressure for different kinds of nano membrane. As it can be found that 100 nm nano membrane has the largest flow rate, with 55 nm membrane second and 20 nm membrane last. This is easy to understand and no meaningful because 100 nm has the biggest pore size. So, we need to analyze the flow rate increase due to pore size effect. We can calculate the equivalent liquid flow rate based on Eq. (2) using Darcy equation. Then, we can obtain the increase of real flow rate compared with the intrinsic liquid flow rate. Fig. 5 shows the ratio of real flow rate (Q/Q0) to intrinsic liquid flow rate. Here, we can find that the 20 nm membrane actually has the largest flow rate increase compared with 55 nm membranes. And the 100 nm membrane nearly has no flow rate increase. Slip length is used to characterize the slip effect due to pore size decreasing. And the gas permeability can be expressed as the function of the slip length as shown in Eq. (3). 1 ) (3) Then, we can calculate the slip length for different kinds of membranes under different average pressure. As shown in following Fig. 6, we can find that the ratio between the slip length and hydraulic radius of the pore throat is increasing with decreasing pore size. Slip length is the largest in the 20 nm membrane and then the 55 nm membrane. For 100 nm, there is no slip length. Knudsen number is commonly used to characterize the gas flow period [2]. Table 3 has shown the gas flow regimes under different Knudsen number range. So, we can also calculate the Knudsen number for gas flow in the membranes under different pore size diameter and average pressure.

83 67 As shown in Fig. 7, for 100 nm membrane, under the experiment condition, the Knudsen number is less than 0.01, which is in the range of the continuum flow. However, for 20 nm and 55 nm membranes, the Knudsen number is in the range of 0.01 to 0.1, from Table 3, it is in the range of slip flow. So, that is the reason why 20 nm membranes has the largest permeability increase and slip length. And for 100 nm membranes, there is nearly no permeability increase and zero slip length. This has shown us that the dynamics of gas in small pore throats in shale and other tight formations (which is in the range of nano to micro) are totally different from those in conventional reservoirs. It is important to find a method to characterize gas flow in these pathways which can match the experiment data. 3. SLIP EFFECT FACTOR DERIVATION AND APPARENT PERMEABILITY MODEL 3.1. Mechanisms for gas flow in nanotubes We used the Advection-Diffusion Model to characterize gas flow behavior in nano pores, as shown in Fig. 8. However, we do not show the slip flow here because we will consider it in the proposed comprehensive coefficient. And also, rather than make correction on the Darcy term as Javadpour [4] has proposed, here we assumed that slip flow is combined in the Knudsen diffusion term. And correction has been made based on the Knudsen term. (1)Viscous flow: As Stops [21] pointed out, when the gas mean free path is smaller than the PTD, the motion of gas molecules is determined by their collision with each other. Gas molecules collide with the wall less frequently. During this period, viscous flow exists, which is caused by the pressure gradient between single-component gas molecules. The

84 68 mass flux of viscous flow can be calculated by the Darcy law, which can be expressed as Eq. (4) provided by Kast and Hohenthanner [22]: (4) where N is the mass flux caused by viscous flow (kg/(m 2 s)), k is the intrinsic permeability of the nano capillary (m 2 ),p is the pressure of the nano capillary (Pa),and is the gas density (kg/ m 3 ). (2)Knudsen Diffusion: When the diameter of the pore is very small, the mean free path lies relatively close to it. In this case, the collision between gas molecules and the wall becomes the dominant effect. The gas mass flux can be expressed by the Knudsen diffusion equation [22, 23]: (5) where and (6) where is the mass flux caused by Knudsen diffusion (kg/(m 2 s)), is the gas mole concentration (mol/m2), is the gas molar mass (kg/mol), is the Kundsen diffusion

85 69 coefficient (m 2 /s), and can be expressed as Eq. (7) provided by Florence et al. [25]:. (7) where is the porosity of a single nano capillary, which is equal to 1, is the gas constant of , is the temperature (), and c is the constant. In this article, following modified Kundsen diffusion coefficient will be used. (8) with a comprehensive coefficient, which we propose for the effect of the slip flow as part of the Knudsen diffusion. This coefficient will be determined and validated based on the experiment data [20] in the section of model validation and comparison Apparent permeability of gas flow in a nano capillary The Klinkenberg model was commonly used to correct the discrepancies between permeabilities measured with gases and liquids as flowing fluids. Klinkenberg effect or slip effect means when the mean free path of the gas molecules is within two orders of magnitude of the capillary radius, the velocity of the gas molecules at the surface of the porous medium will be non-zero [2]. Due to this non-zero velocity, the real flow rate will much higher than theoretically calculated. Klinkenberg proved that there exists an approximately linear relationship between the measured gas permeability (ka) and the reciprocal mean pressure (1/ ), which is given as Eq. (9):

86 70 1 (9) where is the "gas slippage factor" which is a constant that relates the mean free path of the molecules at the mean pressure ( ) and the effective pore radius (r). Table 4 has listed the gas slippage factor proposed by different scholars. In this paper, we will try to derive a new gas slippage factor to more accurately characterize the gas flow in nano pores through verification against the experimental data. As the gas mean free path [21] changes when gas flows in a nanotube, viscous flow and Knudsen diffusion co-exist. Here, we used the Advective-Diffusion Model (ADM) [18] to derive the mass flux firstly. Then compared with the Darcy law, we can obtain the slip effect factor and apparent permeability as follows (more detailed derivation process can be found in the Appendix A): Slip effect factor b: 4 (10) Apparent permeability : 1 (11) where is the intrinsic permeability of gas flow in cylinders can be calculated using Eq. (12) according to the Hagen-Poiseuille equation [19]:

87 71 (12) The mole flux can be expressed as: 1 (13) 4. MODEL VALIDATION AND COMPARISON A numerical model was constructed based on Eq. (13) for gas flow in the nanotube. For more accurately to verify the model, the simulation results were compared with the experimental data provided by Cooper [20]. The experimental data were collected from three kinds of nanotubes, which are listed in Table 5. The model dimensions and fluid properties are listed in Table 6. The simulation model is shown in Fig. 9. Changing the constant yielded different mole fluxes which as shown in Fig. 10(a). The best-fitting should lie in the range of 0.75 to Then, we were able to more accurately investigate the best-fitting, as shown in Fig. 10(b). We estimated the best-fitting to be Then, the new model was tested with the experimental data using different PTDs and gases Model validation The proposed model was verified using different gas and PTDs. The best-fitting was determined using argon with a PTD of 235 nm, as discussed above. During the verification process, varied gas types and PTDs were used to see whether the model still worked; first trying different gas types while maintaining the PTD at 235 nm; and then

88 72 trying different PTDs from 235 nm to 220 nm, while maintaining argon as the gas. As Fig. 11 shown, the model can fit the published experimental data under different diameters and gases. Fig. 11(a) demonstrates the match for oxygen in the diameter of 235 nm and Fig. 11(b) demonstrates the match for argon in the diameter of 220 nm. Later, we will discuss the results of comparison between these three models: the new model proposed in this paper, the model provided by Javadpour, and the theoretical analytical model. In the analytical model, solution is obtained when transport occurs because of a combination of Knudsen diffusion and forced viscous Hagen Poiseuille flow Model comparison We compared this new model with the theoretical analytical solution and some results available in the related literature. These models included those provided by Javadpour [4] and Florence et al. [25]. The analytical solution is derived when transport occurs because of a combination of Knudsen diffusion and forced viscous Hagen-Poiseuille flow. The theoretical analytical solution (K/HP Analytical) can be expressed using Eq. (14): (14) The first term is the Knudsen diffusion, and the second is the Hagen-Poiseuille flow [19]. Javadpour [4] conducted a thorough study of gas flow in nano pores, deriving a similar formula for gas flow in nanotubes based on Knudsen diffusion and viscous force [3], as shown in Eq. (15) and Eq. (16):

89 (15) 1 (16) The apparent permeability provided by Javadpour is expressed in Eq. (17):. (17) Florence et al. [25] approximated the apparent permeability derived by Karniadakis and Beskok [11] for Kn<<1 under a slip-flow condition, where the Knudsen number is a dimensionless number defined as the ratio of the molecular mean free path length to the PTD. It can be expressed as in Eq. (18): 1 4 (18) For a PTD of 235 nm, using argon in the simulation, the comparison between our new apparent permeability and that of Javadpour [4] and Florence et al. [25] is illustrated in Fig. 12. The comparison between our new model s numerical solution, experimental data, Javadpour s result and the analytical solution appears in Fig. 13. Fig. 12 indicates that the new model s apparent permeability most closely matched the permeability provided by Javadpour [4]. When compared with Florence s method [25], there was more divergence. However, more consideration was needed to determine which method more accurately characterized gas flow through nano pores because we did not

90 74 know which method was accurate. Therefore, it was necessary to compare these different methods against the experimental data. Fig. 13 illustrates the comparison of the different methods against the experimental data based on the flux for oxygen in 235 nm pores, which reveals that the proposed solution in this paper matched the experimental data more closely than the theoretical analytical solution or Javadpour s solution. As shown, the K/HP Analytical equation was not reliable when used to characterize the real gas flow because it did not consider the slip flow. Also, Javadpour s method yielded some error. Furthermore, Fig. 13 indicates that the flux did not increase proportionally with the pressure difference because the permeability actually changed with the average pressure which has been depicted in Fig. 14; this differs fundamentally from Darcy flow, in which permeability is a constant throughout the entire flow process. In addition, it is found that that in computing the flux, different permeability expressions generated very different results. This indicates that it is crucial to select the right model to compute the apparent permeability for gas flow through nano pores. Based on these comparison results, the new method is more reliable and can characterize the gas flow in nano pores more accurately. 5. DISCUSSION From Eq.11 1, it can be found that many factors affect the permeability of gas flowing through nano pores, such as the gas mole weight, temperature, pressure, pore size, and gas viscosity. The second part of Eq. 11 is the additional increased permeability due to the slip effect. We can directly find that when reservoir temperature is increased and pore pressure is decreased, the contribution of slip effect and Knudsen diffusion to permeability will increase. More slip flow will be expected in the

91 75 reservoirs at high temperatures and low pressures, which agrees with what Islam and Patzek had found [27]. The mean free path [2] of a molecule in a single-component gas can be computed by Eq. (19) and Knudsen number [3] can be computed by Eq. (20):, (19) (20) Here, is the molecular mean free path; is the Knudsen number; is the temperature (K); is the pressure (Pa); is the Boltzmann Constant ( J/K); is the hydraulic radius; and is collision diameter. The formulation in Eq. (19) reveals more clearly how temperature and pressure will affect the slip effect. When temperature is increased and pressure is decreased, the molecular mean free path will increase. Collision of gas moleculars with pore wall will increase, and thus more slip flow contributes to the total flux. However, in most gas reservoirs, temperature is assumed to be constant during the isothermal production process. Therefore, we choose to focus on the effect of the pore size on the gas behavior because heterogeneity is a common phenomenon in shale gas reservoirs which have a large range of pore size distribution. A typical shale gas reservoir is composed of organic nano pores, inorganic nano pores, micro pores, and micro natural fractures; therefore, showing the effect of the pore size on the gas behavior is interesting and critical. To better represent these effects, the ratio of the real mole flux to the equivalent liquid mole flux was plotted with regard to each factor.

92 76 Figure 15(a) shows the effect of the pore size on this ratio, indicating that as the pore size increased, the Jg/Jl decreased; this means that the deviation from viscous flow decreased as well. The lower part of Fig. 15(a) indicates that as the pore size increased, the Knudsen number decreased. The results obeyed the criteria to divide the gas flow stage [26]. When the Knudsen number is less than 0.001, the flow regime is continuum flow; when the Knudsen number is larger than 0.001, the flow will step into slip flow region. In micro pores, the viscous flow dominated and could be characterized by the Darcy flow. However, in nano pores, the gas slip effect and Knudsen diffusion dominated when the pore size decreased. Traditional Darcy law was invalid for characterizing the gas flow behavior in nanopores. Fig. 15(b) shows the fraction of additional mole flux caused by slip flow and Knudsen diffusion in the total mole flux, indicating that as the pore size decreased, the fraction of additional mole flux increased. When the pore size approached l nm, most of the mole flux actually was caused by the slip flow and Knudsen diffusion, and the flux caused by viscous flow was minor. This is because the exceedingly small pore throats (1 nm) restricts the free movement of gas moleculars. Therefore, slip flow dominates and contributes more to the flux as pointed by Islam and Patzek [27]. it is also found that when the PTD is 1000 nm (1e-6 m), Jg/Jl is equal to 1 and ratio of additional mole flux caused by Knudsen diffusion and slip flow to total mole flux is close to none. Bhatia and Nicolson [28] have analyzed the reason for this phenomenon. When the pore size is large enough, such as close to micro range, interparticle interactions will increase significantly. Then, it will be very easy and quick to achieve equilibrium for the reflected particles with the pore phase. Thus, the contribution of slip flow to the total flux will be reduced substantially.

93 77 Figure 16 has shown the relationship between ratio of real gas permeability and intrinsic permeability with average pressure for nano membranes of different sizes. It can be found that 20 nm has the biggest permeability increase and 100 nm membrane has nearly zero permeability increase. Although membrane with size of 100 nm has the largest intrinsic permeability due to its pore size, membrane with size of 20 nm has the largest permeability increase due to slip flow due to the tiny pore throat which restricts the free movement of gas moleculars. All the evidence proves that more slip flow contributes to the permeability increase. 6. CONCLUSION In this study, experiment for gas flow through diffent size of nano membranes has been done. And a new mathematical model to characterize gas flow in nanotubes has been established and a new formulation to compute the apparent permeability for gas flow in nano pores has been derived. By fitting values with the experimental data, the best-fitting value was found. Through validation with experiment data with nitrogen and published data with oxgen and argon, the new model was found to match the experimental data well. Additionally, the new model was compared with Javadpour s solution and the K/HP analytical solution, with the results showing that the new model matched the experimental data best. The effect of the pore size on the behavior of the gas was analyzed, and the results obeyed the criteria to divide the flow stage. This work will provide a solid foundation for later research on gas flow in shale strata and numerical simulation in nanotubes.

94 78 ACKNOWLEDGEMENTS Funding for this project is provided by RPSEA through the Ultra-Deepwater and Unconventional Natural Gas and Other Petroleum Resources program authorized by the U.S. Energy Policy Act of The authors also wish to acknowledge for the consultant support from Baker Hughes and Hess Company. Special thanks are also owed to the editors and anonymous reviewers of Fuel. Appendix A. Apparent permeability of gas flow in a nano capillary The Klinkenberg model was commonly used to correct the discrepancies between permeabilities measured with gases and liquids as flowing fluids. Klinkenberg effect or slip effect means when the mean free path of the gas molecules is within two orders of magnitude of the capillary radius, the velocity of the gas molecules at the surface of the porous medium will be non-zero [2]. Due to this non-zero velocity, the real flow rate will much higher than theoretically calculated. Klinkenberg proved that there exists an approximately linear relationship between the measured gas permeability (ka) and the reciprocal mean pressure (1/ ), which is given as Eq. (A.1): 1 (A.1) where is the "gas slippage factor" which is a constant that relates the mean free path of the molecules at the mean pressure ( ) and the effective pore radius (r). Table 4 has listed the gas slippage factor proposed by different scholars. In this paper, we will try to derive a

95 79 new gas slippage factor to more accurately characterize the gas flow in nano pores through verification against the experimental data. As the gas mean free path [2] changes when gas flows in a nanotube, viscous flow and Knudsen diffusion co-exist. Here, we used the Advective-Diffusion Model (ADM) [18] to derive the mass flux: (A.2) which we can transform to: 1 (A.3) With b as the slip effect factor, compared with the Darcy law: (A.4) We can obtain: 1 (A.5) Then, we can obtain:

96 80 4 (A.6) So, kapp can be expressed as Eq. (A.7): 1 1 (A.7) According to the Hagen-Poiseuille equation [19], the intrinsic permeability of gas flow in cylinders can be calculated using Eq. (A.8): (A.8) Additionally, kapp can be expressed as Eq. (A.9): (A.9) The mole flux can be expressed as: 1 (A.10) where is the apparent permeability (m 2 ), is the Klinkenberg coefficient or slip effect factor [29, 30], and is the comprehensive coefficient that needs to be fitted.

97 81 REFERENCES [1] Wang, Z. and A. Krupnick A Retrospective Review of Shale Gas Development in the United States. [2] Ziarani, A. S., and Aguilera, R Knudsen s permeability correction for tight porous media. Transport in porous media, 91(1), [3] Javadpour, F., D. Fisher, et al Nanoscale Gas Flow in Shale Gas Sediments. Journal of Canadian Petroleum Technology. 46(10). [4] Javadpour, F Nanopores and Apparent Permeability of Gas Flow in Mudrocks (Shales and Siltstone)." Journal of Canadian Petroleum Technology. 48(8): [5] Curtis, M. E., R. J. Ambrose, et al Structural Characterization of Gas Shales on the Micro- and Nano-Scales. Canadian Unconventional Resources and International Petroleum Conference. Calgary, Alberta, Canada, Society of Petroleum Engineers. [6] Loucks, R. G., R. M. Reed, et al Morphology, Genesis, and Distribution of Nanometer-Scale Pores in Siliceous Mudstones of the Mississippian Barnett Shale. Journal of Sedimentary Research. 79(12): [7] Hornyak, G. L., H. F. Tibbals, et al Introduction to Nanoscience and Nanotechnology. CRC Press, Boca Raton, FL. [8] Bird, G. A Molecular gas dynamics and the direct simulation of gas flows. Oxford University Press, Oxford, UK. [9] Bhattacharya, D. K. and G. C. Lie (1991). "Nonequilibrium gas flow in the transition regime: A molecular-dynamics study." Physical Review A 43(2): [10] Tokumasu, T. and Y. Matsumoto (1999). "Dynamic Molecular Collision (DMC) Model for Rarefied Gas Flow Simulations by the DSMC Method." Physics of Fluids 11(7): [11] Karniadakis, G. and A. Beskok Micro Flows: Fundamentals and Simulation. Springer-Verlag: Berlin. [12] Burnett, D. (1935). "Proc." London Math. Soc. 40: 382. [13] Beskok, A., & Karniadakis, G. E. (1999). Report: a model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Thermophysical Engineering, 3(1), [14] Klinkenberg, L. J The Permeability of Porous Media to Liquids and Gases, American Petroleum Institute. [15] Shabro, V., C. Torres-Verdin, et al Numerical Simulation of Shale-Gas Production: From Pore-Scale Modeling of Slip-Flow, Knudsen Diffusion, and Langmuir Desorption to Reservoir Modeling of Compressible Fluid. North American Unconventional Gas Conference and Exhibition. The Woodlands, Texas, USA, Society of Petroleum Engineers.

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100 84 Table 1. Properties of AAO nano membranes used in the experiment. Membrane type PTD (nm) Pore density ( nm -2 ) 20 nm membrane 55 nm membrane 100 nm membrane Intrinsic permeability (md) E E E E Table 2. Experimental results for different kinds of nano membranes. Inlet pressure 20 nm 55 nm 100 nm Flow Inlet Outlet Flow Inlet Outlet rate pressure pressure rate pressure pressure Outlet pressure Flow rate E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-17 NOTE: pressure unit is Pa. Flow rate unit is m 3 /s.

101 85 Table 3. Knudsen number and flow regimes classifications for porous media. Flow regime Continuum (Viscous) flow Slip flow Transition flow Free Molecular flow Knudsen number Kn< <Kn< <kn<10 Kn>10 NOTE: Modified from Ziarani and Aguilera [2] Model to be applied Darcy's equation for laminar flow and Forchheimer's equation for turbulent flow Darcy's equation with Klinkenberg or Knudsen's correction Darcy's law with Knudsen's correction can be applied. Alternative method is Burnett's equation with slip boundary conditions Knudsen's diffusion equation Alternative methods are DSMC and Lattice Boltzmann Model Klinkenberg [14] Heid et al. [29] Table 4. Knudsen s permeability correction factor for tight porous media. Jones and Owens [30] Correlation factor Sampath and Keighin [31] Florence et al. [25]. Civan [17] NOTE: Modified from Ziarani and Aguilera [2] Units: (psi), (md), (psi), r (nm), (psi), (nm) and (fraction)

102 86 Table 5. Nanotube characteristics [20]. Avg. PTD (nm) Avg. pore density ( m -2 ) Porosity Table 6. Model dimensions and fluid properties. Flow parameters Length L Diameter D Outlet Pressure Pout Pressure Drop P=Pin-Pout Absolute Viscosity μ Gas Molecular Weight Temperature Nanotube 60 μm 235 nm, 220 nm 4.8 kpa torr Pa.s Oxygen (0.032 kg/mol), Argon ( kg/mol) 300 K

103 87 fracture free gas macro scale matrix micro scale Kerogen dissoved gas adsorption gas Figure 1. Gas distribution in shale strata from macro-scale to micro-scale. In the fracture, there exists free gas and in the matrix, free gas and adsorption gas co-exist. Figure 2. SEM images of experimental membranes with pore size 20 nm, 55 nm, and 100 nm.

104 88 Figure 3. Schematic diagram and real picture of the experimental setup. In Fig. 3(a), from left to right the whole set up includes an aluminum cylinder, the membrane and an oring between the upstream and downstream parts. 1E-15 1E-16 Q (m 3 /s) 1E nm pore 55 nm pore 100nm pore 1E P avg (psi) Figure 4. Gas flow rate vs. the average pressure for nano membranes of different sizes.

105 Q 1 /Q nm pore 55 nm pore 100nm pore P avg (psi) Figure 5. Ratio of real gas flow rate to intrinsic flow rate vs. average pressure for different kinds of nano membranes l s /R nm pore 55 nm pore 100nm pore P avg (psi) Figure 6. Ratio of slip length to pore radius vs. average pressure for nano membranes of different sizes.

106 Slip flow 0.01<Kn< K n nm pore 55 nm pore 100nm pore 0.01 Continuum (Viscous) flow Kn< P avg (psi) Figure 7. Knudsen number vs. average pressure for nano membranes of different sizes. Figure 8. Gas flow mechanisms in a nano pore. Red represents Knudsen diffusion, while purple represents viscous flow.

107 91 Figure 9. A schematic model for simulating gas flow in nanotubes (a) 1400 (b) Delta P (torr) Experiment data Argon D=235nm =0.71 =0.75 =0.64 =0.89 Delta P (torr) Experiment data Argon D=235 nm Flux (mol/m 2 s) Flux (mol/m 2 s) Figure 10. Pressure drop versus flux for different values of σ.

108 Delta P (torr) Experiment data Oxygen D=235 nm Numerical result Delta P (torr) Experiment data Argon D=220nm Numerical result Flux (mol/m 2 s) Flux (mol/m 2 s) Figure 11. Pressure drop versus flux for oxygen in a nanotube with a diameter of 235 nm and 220 nm Oxygen T=300 K Pressure difference: 1000 torr 0.1 K app (m 2 ) E-3 1E-4 1E-5 Proposed method Javadpour's method Florence's method 1E-10 1E-9 1E-8 1E-7 1E-6 1E-5 r (m) Figure 12. Comparison of Kapp versus radius between new apparent permeability and that of Florence and Javadpour.

109 Delta P (torr) Experimental data Proposed method Javadpour's method K/HP Analytical method 200 Experimental gas: Oxygen T=300 K Pore diameter=235 nm Flux (mol/m 2 s) Figure 13. Pressure difference and gas mole flux relationship comparison among the proposed model, the analytical solution, Javadpour s solution and experimental data for Oxygen in 235 nm pores K g (m 2 ) Proposed method Javadpour's method Florence's method Delta P (torr) Figure 14. Comparison of gas permeability vs. pressure difference among the proposed model, the analytical solution, Javadpour s solution, and experimental data for oxygen in 235 nm pores.

110 J g /J l E-10 1E-9 1E-8 1E-7 1E-6 1E r (m) J g /J l E-3 K n Additional mole flux/ Total mole flux (%) E-10 1E-9 1E-8 1E-7 1E-6 1E r (m) E-3 K n Figure 15. (a) Effect of pore size on ratio of gas mole flux to equivalent liquid mole flux (Jg/Jl). (b) Effect of pore size on ratio of additional mole flux caused by Knudsen diffusion and slip flow to total mole flux. The upper figure shows how the ratio changes with the pore radius; the lower figure shows how the ratio changes with the Knudsen number.

111 K g /K l nm Experiment 55 nm Experiment 100 nm Experiment P avg (psi) Figure 16. Ratio of gas permeability to intrinsic liquid permeability vs. average pressure for nano membranes of different sizes.

112 96 PAPER II. PRESSURE TRANSIENT AND RATE DECLINE NALYSIS FOR HYDRAULIC FRACTURED VERTICAL WELLS WITH FINITE CONDUCTIVITY IN SHALE GAS RESERVOIRS Chaohua Guo a, Jianchun Xu b, Mingzhen Wei a, Ruizhong Jiang b a Missouri University of Science and Technology, Department of Petroleum Engineering, Rolla, MO, USA, b China University of Petroleum (East China), Department of Petroleum Engineering, Qingdao, CHINA, (Published as an article in Journal of Petroleum Exploration and Production Technology) ABSTRACT Producing gas from shale strata has become an increasingly important factor to secure energy over recent years for the considerable volume of natural gas stored. Unlike conventional gas reservoirs, gas transports in shale reservoir is a complex process. Gas diffusion, viscous flow due to pressure gradient, and desorption from Kerogen coexist. Hydraulic fracturing is commonly used to enhance the recovery from these ultra-tight gas reservoirs. It is important to clearly understand the effect of known mechanisms on shale gas reservoir performance. This article presents the pressure transient analysis (RTA) and rate decline analysis (RDA) on the hydraulic fractured vertical wells with finite conductivity in shale gas reservoirs considering multiple flow mechanisms including desorption, diffusive flow, Darcy flow and stress-sensitivity. The PTA and RDA models were established firstly. Then the source function, Laplace transform and the numerical discrete methods were employed to solve the mathematical model. At last, the type curves were plotted and different flow regimes were identified. The sensitivity of adsorption coefficient, storage capacity ratio, inter-porosity flow coefficient, fracture conductivity, fracture skin factor and stress

113 97 sensitivity were analyzed. This work is important to understand the transient pressure and rate decline behaviors of hydraulic fractured vertical wells with finite conductivity in shale gas reservoirs. NOMENCLATURE C =Wellbore storage coefficient (m 3 /Pa) cg =Gas compressibility (Pa 1 ) Cf D h =fracture conductivity =Diffusion coefficient (m 2 /s) =Reservoir thickness (m) kfh =Horizontal permeability of natural fracture (m 2 ) kfv =Vertical permeability of natural fracture (m 2 ) kf1 =Artificial fracture permeability(m 2 ) K0() L LfLi,; LfRi n p0 pi psc q q i =Modified Bessel function of second kind of order zero =Characteristic length (m) =Lengths of the left and right wings of the i th fracture (m) =Number of segments on the wing of each fracture =Reference pressure (Pa) =Initial reservoir pressure (Pa) =Standard state pressure (Pa) =Production rate from continuous point source (m 3 /s) =Flux per unit length of discrete segment(i,j), (m 2 /s)

114 98 qsc r R Sf t T Tsc Wf u =Production rate (m 3 /s) =Radial distance in natural fracture system (m) =External radius of matrix block (m) =Fracture skin factor, dimensionless =Time (s) =Reservoir temperature (K) =Standard state temperature (K) =Fracture width(m) =Laplace transform parameter, dimensionless V =Average volumetric gas concentration in the fracture (s-m 3 / m 3 ) VE =Equilibrium volumetric gas concentration in pseudo-steady diffusion (s-m 3 / m 3 ) Vi =Initial volumetric gas concentration in the fracture (s-m 3 / m 3 ) x,y,z xw,yw,zw Z Zi =Space coordinates in Cartesian coordinates (m) =Space coordinates of continuous point source (m) =Real-gas compressibility factor, dimensionless =Real-gas compressibility factor under the initial condition, dimensionless α λ ω τ ξ =Adsorption index, dimensionless =Inter-porosity flow coefficient, dimensionless =Storativity ratio, dimensionless =Sorption time constant (s) =Integration variable

115 99 μ μi φf ψ ψi _ D f i m sc w =Gas viscosity, (Pa s) =Initial gas viscosity, (Pa s) =Natural fracture porosity, fraction =Pseudo-pressure (Pa) =Initial pseudo-pressure (Pa) =Laplace domain =Dimensionless =Natural fracture =Initial condition =Matrix =Standard state =Wellbore INTRODUCTION With the growing shortage of domestic and foreign energy industry, producing gas from shale gas reservoirs are currently received great attention due to their potential to supply the entire world with sufficient energy for the decades to come (Wang and Krupnick, 2013). However, low gas recovery rate from unconventional shale resources remains the main technical difficulty. It is estimated that only % of GIP can be recovered from these unconventional reservoirs. Most of the gas production wells can be low or no production capacity without hydraulic fracturing or horizontal well drilling. A shale gas reservoir is characterized as an organic-rich deposition with extremely low matrix permeability and clusters of mineral-filled natural fractures. Through core experiment

116 100 analysis on 152 cores of nine reservoirs in North America, Javadpour found that the permeability of shale bedrock is mostly 54nd and about 90% are less than 150nd (Javadpour et al., 2007; Javadpour, 2009). Most of the pore diameters are in the range of 4 ~ 200 nm (10-9 m). There are three forms for the gas to store in the shale: 1) free gas stored in the fractures and pores; 2) adsorption gas stored on the surface of the bedrock, which Hill and Nelson estimated that 20%~85% of gas in shale is stored under this condition (Hill and Nelson, 2000); 3) dissolved gas stored in the kerogen (Javadpour, 2009). Hydraulic fracturing has been used to create high conductivity flow path to improve the productivity of low permeability shale reservoirs (Waters et al., 2009; Rahm, 2011; Arthur et al., 2009; Vermylen and Zoback, 2011). Well test is an efficient method to understand the gas flow mechanisms in unconventional shale and for better fracturing design (Prat 1990; Serra, 1981; Bumb and McKee, 1988; Brown et al., 2011; Bello and Wattenbarger, 2010; Bello, 2009). Bumb et al. have considered the effect of desorption for shale gas well test (Bumb and McKee, 1988). Brown et al. have studied PTA for fractured horizontal wells in unconventional shale reservoirs (Brown et al., 2011). Bello and Wattenbarger have studied RDA for multi-stage hydraulically fractured horizontal shale gas well (Bello and Wattenbarger, 2010; Bello, 2009). However, there are few studies on well test and rate decline analysis for fractured vertical wells in shale gas reservoirs. Most of the research focused on the pressure response of infinite conductivity fractured wells in infinite formation (Stanford, 1974; Rosa and Carvalho, 1989; Cinco-Ley and Samaniego, 1981). So, it is necessary to study the pressure response for hydraulic fractured vertical wells, which is meaningful for effective shale gas

117 101 production and deep understanding about gas flow mechanisms in unconventional shale reservoirs. In this paper, based on shale gas flow mechanisms, such as desorption, diffusion, and flow in the fractures, the mathematical model which considers the effect of outer boundary effects for hydraulic fractured vertical wells with finite conductivity has been constructed. This model also considers Fick's first law of diffusion, Langmuir isothermal adsorption (Yin, 1991), and stress-sensitivity of natural fracture. Through point-source function method (Wei et al., 1999), the single phase shale gas flow in matrix and fracture has been studied based on Fick s pseudo-steady state diffusion model (Cohen and Murray, 1981). Sensitivity analysis has been carried out for significant factors, such as desorption constant, fracture conductivity coefficient, storage capacity ratio, and inter-porosity flow coefficient. The effect of above factors on the PTA and RDA has been presented. MATHEMATICAL MODEL Assumptions (1)The shale gas formation consists matrix system and fracture system. And there exists heterogeneity in the fracture system: the horizontal permeability and vertical permeability of the fracture system are different k k ; (2)Free shale gas exists in the fracture system and can be described using Darcy law; (3)Free gas and adsorption gas coexist in the matrix system; fh fv

118 102 (4)The gas flow in the matrix system can be ignored. The gas in the matrix system first will desorb and then diffuse into the fracture system, which can be characterized using pseudo-steady state diffusion; (5)Single layer Langmuir isothermal curve can be used to describe the gas desorption from the matrix system; (6)Dynamic balance exists between adsorption gas and free gas; (7)The gas well is at constant production rate q sc (8)The effect of gravity and capillary pressure can be ignored. Mathematical model in standard condition; Based on above assumptions, combined with Darcy law, gas equation of state (EOS), and mass balance equation, the mathematical model can be established, which consists: desorption of shale gas, diffusion and flow characteristics. k x fh p f Z p f x k y fh p f Z p f y k z fv p f Z p f z f c gf p f Z p f t p T sc sc V t (1) In the above Eq. (1), the second term in the right hand reflected the effect of the gas desorption. The pseudo-pressure can be defined as follows: f p dp (2) Z p f p f 2 p0 Substitute Eq. (2) into (1), we can obtain: f f f f 2 psct V k fh k fh k fv f icgfi x x y y z z t Tsc t (3)

119 103 According to Fick's first law of diffusion, if the matrix is assumed as sphere, the gas diffusion rate of shale matrix per unit time and per unit volume can be expressed as follows: V t 2 6 D 2 R V E f V (4) Also, we defined dimensionless variables as shown in Table 1. The dimensionless forms for Eq. (3) and (4) can be given as follows: fd fd fd fd VD xd yd zd td td V t D D V ED fd V D (5) (6) Taking Laplace transformation to td, Eqs. (5) and (6) become: Then Eqs. (5) and (6) can be changed to: uv fd fd fd u fd 1 xd yd zd uv D= V ED fd V D D (7) (8) According to Langmuir isotherm adsorption equation: Vp Vp L f L i VED fd L VE Vi L p L p f p L p i (9) Taking Laplace transformation to td, Eq. (9) becomes: V ED fd V p L L p q T sc sc i i p 2 L p k ht f pl p i Z p fh sc i fd (10) With the definition of Adsorption index in Table 1, we can get: V ED fd fd (11)

120 104 α is pseudo-pressure related, which can be assumed as a constant in the discussion range. And it is equal to the value in the initial reservoir state (Ertekin and Sung 1989; Anbarci and Ertekin 1990). According to Eqs. (7-8) and (11), we can obtain: fd fd fd f ( u) x y z f u D D D u 1 u u fd (12) (13) Converting to spherical coordinates: 1 r 2 D 2 rd r D r fd D f u fd (14) Inner boundary conditions: fd lim rd 1 r rd 0 D (15) fd rd Three forms of outer boundary conditions have been considered in this paper: Infinite outer boundary (IOB): 0 (16) Constant pressure outer boundary (CPOB): 0 (17) fd Closed outer boundary (COB): r D fd 0 (18)

121 105 Apply the Laplace transform to both sides of the above equation from Eqs. (15) to (18). Then combine Eq. (14), we can obtain the line source solution under constant production rate condition: ~ psct q 1 K0 rd f u IOB Tsc kfhh u ~ 1 K0 red f u psct q f K0rD f u I0rD f u Tsc kfhh u I0 red f u ~ p 1 K1 red f u sct q K0rD f u I0rD f u Tsc kfhh u I1 red f u CPOB COB (19) Solution The schematic diagram of hydraulic fractured vertical well in shale gas reservoirs is shown in Fig. 1. There is one vertical well with finite conductivity after hydraulic fractured in a shale gas reservoir with top and bottom boundary closed. The well is produced at constant rate q sc. The reservoir pay zone is h. The hydraulic fracture totally fractured the whole reservoir, which means the fracture height is h ; The vertical cracks after hydraulic fracturing are symmetric against the wellbore. The fracture half length is x and fracture f width is W. Fracture permeability is f k ; Assume that the gas only flows into the wellbore f 1 through the hydraulic fractures. The fracture tips are closed with no gas flow. Taking the case with IOB as an example, the pressure response can be obtained by superimposing line source solutions for shale gas reservoirs with top and bottom boundary closed:

122 fd 0 2 q 1 fd u K xd yd f u d (20) 106 Here q fd q u fd Since the fracture is symmetric, following flow rate relationship can be obtained: 1 1 q fd dx D (21) 0 u Consider the skin effect by hydraulic fracturing, the fracture pseudo-pressure can be related with formation pseudo-pressure using following equation: D fd,,, x u x y u q x u S (22) D D D D fd D f Ignoring the compressibility of fluids in the fracture, following relationship can be obtained after combining boundary condition: x D wd f 1 D xd, yd 0, u xd u q fddxdd (23) uc 0 0 fd According to Eq. (20) and (22), above equation can be written in following form: fd f fd 0 0 fd D D C fd uc fd 1 1 wd q, 0 D 0 D 2 0 fd u K x f u K x f u d xd S q q dx d x (24) Equation (24) can be solved using discrete numerical methods. As shown in Fig. 1, the dimensionless fracture half-length can be divided into n fractions with step length. x Di is the center point of the i th discrete unit. x Di is the end point of the i th discrete unit. As the discrete unit is small enough, the flow rate can be assumed to be uniformly distributed. Eq. (24) can be discretilized into following form: x D

123 107 i1 Dj Dj n 1 xdi1 wd q fdi K0 x f u K0 x f u d S f q 2 x Di j x q fdi 0.5 x x x i x q x C fd i1 2 2 D D D Dj D fdj D 8 uc fd fdj (25) And the flow rate equation can be discretilized as follows: x n D i1 q fdi u 1 u (26) There are n 1 unknown variables in Eq. (25) and (26), that is wd and q fdi (with i 1, 2,..., n). By solving these equations, in Laplace space, we can obtain the bottomhole pseudo-pressure wd and the flow rate distribution in discrete units q fdi Consider the effect of wellbore storage, according to Duhamel's principle, the bottomhole pressure response is: wd wd (27) 2 1 u C D wd Using Stehfest inversion technique, we can obtain the bottomhole pseudo-pressure response in the real space. TYPE CURVES AND ANALYSIS Type curves Figure 2 is type curve for finite conductivity fractured vertical well in shale gas reservoirs with consideration of outer boundary effect in the log-log scale. The type curve can be divided into 8 stages: (1) the first stage is early afterflow period, the pseudo-pressure (PP) curve and pseudo-pressure-derivative (PPD) curve coincide with each other with the slope

124 108 equal to 1; (2) the second stage is afterflow transition period; (3) the third stage is the bilinear flow period with slope of PPD curve equal to ¼; (4) the fourth stage is formation linear flow period with slope of PPD equal to ½; (5) the fifth stage is pseudo radial flow of natural fractures with PPD equal to 0.5; (6) the sixth stage is the channeling period from matrix to fracture, there is a concave in the PPD curve; (7) the seventh stage is the formation pseudo radial flow period with PPD equal to 0.5; (8) the eighth period is boundary response period. There scenarios will appear for different kinds of boundary: PP curve and PPD curve will become upturned when the boundary is closed; otherwise, they will become downturned when the boundary is constant pressure. Sensitivity analysis Effect of adsorption coefficient Figure 3 shows the effect of adsorption coefficient on PPR (pseudo pressure response) and RDR (Rate decline response). Other parameters used are listed in the figure. From Fig. 3, it can be found that adsorption coefficient mainly affects the gas diffusion period from matrix to fracture. The larger the value, the concave in the PPD curve is wider and deeper. This suggested that the more gas desorbed, the more apparent the diffusion is. It can be also found that the larger the adsorption coefficient, the higher the gas production rate. This suggests that the more gas desorbed, the more apparent the diffusion is. Also, from the definition of the adsorption coefficient, we know that the large the Langmuir Volume, the larger the adsorption coefficient, which also denotes the effect of the Langmuir volume. Effect of storage capacity ratio Figure 4 shows the effect of storage capacity ratio on PPR and RDR. Other parameters used are listed in the figure. From Fig. 4, we can find that storage capacity ratio not only

125 109 affect the channeling period, but also has effect on transition period, linear flow period and formation linear flow period. The smaller the value, in the PPD curve, the wider and deeper the concave are. However, the position of PP and PPD curve in the transition period, linear flow period and formation linear flow period is more upper. Also, it can be found that the smaller the value, the smaller gas flow rate will be. Effect of channeling factor Figure 5 shows the effect of channeling factor on PPR and RDR. Other parameters used are listed in the figure. From Fig. 5, we can find that channeling factor mainly affects the diffusion time from matrix to fracture. The larger the channeling factor, the earlier the channeling start time, and the earlier the appearance of the concave in the PPD curve. The larger the inter-porosity factor, the earlier the matrix-fracture diffusion start time, and there will be a higher production rate. Effect of fracture conductivity Figure 6 shows the effect of fracture conductivity on PPR and RDR. Other parameters used are listed in the figure. From Fig. 6, we can find that the fracture conductivity mainly affects the transition period and early bilinear flow period. The larger the fracture conductivity, the characteristic of the early bilinear flow period is less clear. The PP and PPD curve are in lower position. When the value is larger enough, we cannot catch the early bilinear flow period. And the curve is close to the well testing curve for infinite conductivity fractured well. Also, it can be found that the higher the fracture conductivity, the higher the gas production rate is. Effect of fracture skin factor

126 110 Figure 7 shows the effect of fracture skin factor on PPR and RDR. Other parameters used are listed in the figure. From Fig. 7, we can find that the fracture skin factor mainly affects the flow from formation to fracture. The larger the value, the higher the PP curve is and more apparent the hump effect is. The larger the fracture skin factor, the more serious damage is during the hydraulic fracturing. And the lower the gas production rate is. Effect of stress sensitivity Figure 8 shows the effect of fracture stress sensitivity on PPR and RDR. Other parameters used are listed in the figure. From Fig. 8, we can find that the fracture stress sensitivity mainly affects the early period. The larger the value, the higher the PP curve is and the horizontal line of pressure curve is missing. And the larger the stress sensitivity, the higher the production rate is. CONCLUSIONS In this paper, the pressure transient model and rate decline model for fractured vertical wells in shale gas reservoir have been presented, and the parameters sensitivity analysis has been studied. The following conclusions can be summarized: (1) A mathematical model describing fluid flow of fractured vertical well in shale gas reservoir is established and the solutions, transient pressure and rate decline curves are analyzed. (2) When the well is produced at a constant rate, a bigger Langmuir volume (VL) will lead to a larger adsorption coefficient, which will cause a deeper depth of the trough during this period because big value of VL indicates more gas will be desorbed during the interporosity flow period.

127 111 (3) The number of fractures (MF) mainly affects the early flow regimes of the fractured vertical well, especially for the early linear and elliptical flow regimes. The more the MF is, the smaller the pressure drop and the pressure decreasing rate are. (4) When the well is produced at a constant bottom-hole pressure, the bigger the Langmuir volume (VL) and well length are, the greater the production rate will be in the later flow period. ACKNOWLEDGEMENTS Funding for this project is provided by RPSEA through the Ultra-Deepwater and Unconventional Natural Gas and Other Petroleum Resources program authorized by the U.S. Energy Policy Act of The authors also wish to acknowledge for the consultant support from Baker Hughes and Hess Company. REFERENCES Anbarci K, Ertekin T (1990) A comprehensive study of pressure transient analysis with sorption phenomena for single-phase gas flow in coal seams. SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. Arthur JD, Bohm B, Layne M (2009) Hydraulic fracturing considerations for natural gas wells of the Marcellus Shale. Bello RO (2009) Rate transient analysis in shale gas reservoirs with transient linear behavior (Doctoral dissertation, Texas A&M University. Bello RO, Wattenbarger RA (2010) Multi-stage hydraulically fractured horizontal shale gas well rate transient analysis. In North Africa Technical Conference and Exhibition. Society of Petroleum Engineers. Brown M, Ozkan E, Raghavan R, Kazemi H (2011) Practical solutions for pressuretransient responses of fractured horizontal wells in unconventional shale reservoirs. SPE Reservoir Evaluation & Engineering 14(06), Bumb AC, McKee CR (1988) Gas-well testing in the presence of desorption for coalbed methane and devonian shale. SPE formation evaluation 3(01), ]. Cinco-Ley H, Samaniego FV (1981) Transient pressure analysis for fractured wells. Journal of petroleum technology 33(9),

128 112 Cohen DS, Murray JD (1981) A generalized diffusion model for growth and dispersal in a population. Journal of Mathematical Biology 12(2), Ertekin T, Sung W (1989) Pressure transient analysis of coal seams in the presence of multi-mechanistic flow and sorption phenomena. SPE Gas Technology Symposium. Society of Petroleum Engineers. Javadpour F (2009) Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone). Journal of Canadian Petroleum Technology 48(8), Javadpour F, Fisher D, Unsworth M (2007) Nanoscale gas flow in shale gas sediments. Journal of Canadian Petroleum Technology 46(10), Prat G (1990) Well test analysis for fractured reservoir evaluation. Elsevier. Rahm D (2011) Regulating hydraulic fracturing in shale gas plays: The case of Texas. Energy Policy 39(5), Rosa AJ, Carvalho RD (1989) A Mathematical Model for Pressure Evaluation in an lnfinite-conductivity Horizontal Well. SPE Formation Evaluation 4(04), Serra K (1981) Well Test Analysis for Devonian Shale Wells. University of Tulsa, Department of Petroleum Engineering. Standford U (1974) Unsteady-state pressure distributions created by a well with a single infinite-conductivity vertical fracture. Transactions of the American Institute of Mining, Metallurgical and Petroleum Engineers 257. Vermylen J, Zoback, MD (2011) Hydraulic Fracturing Microseismic Magnitudes and Stress Evolution in the Barnett Shale Texas USA. In SPE Hydraulic Fracturing Technology Conference. Society of Petroleum Engineers. Wang Z, Krupnick A (2013) A retrospective review of shale gas development in the United States. Resources for the Future Discussion Paper. Waters GA, Dean BK, Downie RC, Kerrihard KJ, Austbo L, McPherson B (2009) Simultaneous hydraulic fracturing of adjacent horizontal wells in the Woodford Shale. In SPE Hydraulic Fracturing Technology Conference. Society of Petroleum Engineers. Wei, G, Kirby JT, Sinha A (1999) Generation of waves in Boussinesq models using a source function method. Coastal Engineering 36(4), Yin Y (1991) Adsorption isotherm on fractally porous materials. Langmuir 7(2),

129 113 Table 1. Definitions of the dimensionless variables. k fhhtsc fd i f p q T Dimensionless pseudo-pressure k fh t Dimensionless pseudo-time: t D ; 2 L Dimensionless coordinate: Dimensionless flow rate in the fracture system; sc x x D, L q fd sc c y y D, L 2Lq f x q Dimensionless gas concentration: V D V V i Dimensionless storage capacity ratio f ic gfi Channeling coefficients k fh ; 2 L Dimensionless fracture width W f W fd L Dimensionless fracture conductivity k f 1W f C fd k L Adsorption coefficient Dimensionless wellbore storage coefficient: C D fh sc L D V p, t D R 6 f 2 2 D pl pf pl pi L C 2 2 hl / i i gfi D 2k q z z L sc p q T k ht fh h k k fh fv Z p sc sc i i fh sc i z y x D1 x x D 2 Dn re x f h x x x y xd1 x D 2 D3 xdn x Dn 1 Figure 1. Schematic diagram of hydraulic fractured vertical well. The top and bottom boundary are closed.

130 114 Ψ wd,ψ, w D t D /C D infinite boundary closed boundary constant pressure boundary α=50,ω=0.5,λ=0.001,c f d=10,s=0.1,c D = ,R ed = t D /C D infinite boundary closed boundary constant pressure boundary 10 0 q D α=50,ω=0.5,λ=0.001,c fd =10,S=0.1,C D = ,R ed = t D /C D Figure 2. Type curve for hydraulic fractured vertical wells with finite conductivity in shale gas reservoirs under different kinds of boundary conditions.

131 α=10 α=20 α=30 ψ wd,ψ, wd t D /C D ω=0.5,λ=0.001,c fd =10,S=0.1,C D = t D /C D α=10 α=20 α=30 q D ω=0.5,λ=0.001,c fd =10,S=0.1,C D = t D /C D Figure 3. Effect of adsorption coefficient on PPR (pseudo pressure response) and RDR (Rate decline response).

132 ω=0.2 ω=0.4 ω=0.6 ψ wd,ψ, wd t D /C D α=50,λ=0.001,c fd =10,S=0.1,C D = t D /C D ω=0.2 ω=0.4 ω=0.6 q D α=50,λ=0.001,c fd =10,S=0.1,C D = t D /C D Figure 4. Effect of storage capacity ratio on PPR and RDR.

133 λ=0.001 λ=0.01 λ=0.1 P D,P, D t D /C D α=50,ω=0.5,c fd =10,S=0.1,C D = t D /C D λ=0.001 λ=0.01 λ= q D α=50,ω=0.5,c fd =10,S=0.1,C D = t D /C D Figure 5. Effect of channeling factor on PPR and RDR.

134 118 ψ wd,ψ, wd t D /C D C fd =10 C fd =30 C fd = α=50,ω=0.5,λ=0.001,s=0.1,c D = t D /C D C fd =10 C fd =30 C fd =80 q D α=50,ω=0.5,λ=0.001,s=0.1,c D = t D /C D Figure 6. Effect of fracture conductivity on PPR and RDR.

135 119 ψ wd,ψ, wd t D /C D S f =0.1 S f =0.3 S f = α=50,ω=0.5,λ=0.001,c fd =10,C D = t D /C D S f =0.1 S f =0.3 S f =0.5 q D α=50,ω=0.5,λ=0.001,c fd =10,C D = t D /C D Figure 7. Effect of fracture skin factor on PPR and RDR.

136 γ=0.0 γ=0.03 γ=0.06 ψ wd,ψ, wd t D /C D α=50,ω=0.5,λ=0.001,c fd =10,S=0.1,C D = t D /C D γ=0.0 γ=0.03 γ= q D α=50,ω=0.5,λ=0.001,c fd =10,S=0.1,C D = t D /C D Figure 8. Effect of stress sensitivity on PPR and RDR.

137 121 PAPER III. MODELING OF GAS PRODUCTION FROM SHALE RESERVOIRS CONSIDERING MULTIPLE FLOW MECHANISMS Chaohua Guo; Mingzhen Wei Missouri University of Science and Technology (Published as an article in in Offshore Technology Conference) ABSTRACT Gas transport in unconventional shale strata is a multi-mechanism-coupling process which is different from conventional reservoirs. In micro fractures which are inborn or induced by hydraulic stimulation, viscous flow dominates. And gas surface diffusion and gas desorption should be further considered in organic nano pores. Also, Klinkenberg effect should be considered when dealing with gas transport problem. In addition, following two factors can play significant roles under certain circumstances but have not been received enough attention in previous models. During pressure depletion, gas viscosity will change with Knudsen number; and pore radius will increase when the adsorption gas desorbs from the pore wall. In this paper, a comprehensive mathematical model which incorporates all known mechanisms for simulating gas flow in shale strata has been presented. The objective of this study is to provide a more accurate reservoir model for simulation based on flow mechanisms in the pore scale and formation geometry. Complex mechanisms including viscous flow, Knudsen diffusion, slip flow, and desorption are optionally integrated into different continua in the model. Sensitivity analysis has been conducted to evaluate the effect of different mechanisms on the gas production. Results show that adsorption and gas viscosity change will have a great impact on gas production. Ignoring

138 122 one of following scenarios, such as adsorption, gas permeability change, gas viscosity change and pore radius change, will underestimate gas production. INTRODUCTION Due to increasing energy shortage during recent years, gas producing from shale strata has played an increasingly important role in the volatile energy industry of North America and is gradually becoming a key component in the world s energy supply (Wang and Krupnick, 2013; Guo et al., 2014). Shale strata or shale gas reservoirs (SGR) are typical unconventional resources with critically low transport capability in matrix and numerous natural fractures. Core experiments have shown that 90% of shale bedrock permeability are less than 150 nd and diameters of the pore throat are 4~200 nm (Javadpour et al., 2007). Natural gas have been stored in SGRs with three forms (Guo et al., 2014): (a) in the micro fractures and nano pores, they are stored as free gas; (b) in the kerogen, they are stored as dissolved gas (Javadpour, 2009); (c) in the surface of the bedrock, they are stored as adsorption gas and about 20%~85% of gas in SGRs are stored in this kind of form (Hill and Nelson, 2000). Gas transport in extremely low-permeability shale formations is a complex process with many mechanisms co-existing, such as viscous flow, Knudsen diffusion, slip flow (Klinkenberg, 1941), and gas adsorption-desorption. Some scholars have studied gas transport behavior in SGRs. E. Ozkan used a dual-mechanism approach to consider transient Darcy and diffusive flows in the matrix and stress-dependent permeability in the fractures for naturally fractured SGRs (Ozkan et al., 2010). However, they ignored effects of adsorption and desorption. Moridis has considered Darcy s flow, non-darcy flow,

139 123 stress-sensitive, gas slippage, non-isothermal effects, and Langmuir isotherm desorption (Moridis et al., 2010). They found that production data from tight-sand reservoirs can be adequately represented without accounting for gas adsorption whereas there will be significant deviations if gas adsorption is omitted in SGRs. But they did not consider gas diffusion in the Korogen. Bustin studied effect of fabric on the gas production from shale strata. However, it is assumed that matrix does not have viscous flow and diffusion mechanisms. It is also assumed that gas transport in the fracture system obeys Darcy law which is not sufficiently accurate (Bustin et al., 2008). Yu-Shu Wu has proposed an improved methodology to simulate shale gas production, but the gas adsorption-desorption were ignored in the model (Wu and Fakcharoenphol, 2011). It is necessary to develop a mathematical model which incorporates all known mechanisms to describe the gas transport behavior in tight shale formations. The simulation work was greatly simplified when the apparent permeability was first introduced by Javadpour. In 2009, the concept of apparent permeability considering Knudsen diffusion, slippage flow and advection flow was proposed (Javadpour, 2009). Through this method, the flux vector term can be simply expressed in the form of Darcy equation which will greatly reduce the computation complexity. Then the concept of apparent permeability was further applied in pore scale modeling for shale gas (Shabro et al., 2011; Shabro et al., 2012). Civan and Ziarani derived the expression for apparent permeability in the form of Knudsen number (Civan, 2010; Ziarani and Aguilera, 2012) based on a unified Hagen-Poiseuille equation (Beskok and Karniadakis, 1999). In this paper, a general numerical model for SGRs was proposed which was constructed based on the dual porosity model (DPM). Mass balance equations for both matrix and

140 124 fracture systems were constructed using the dusty gas model. In the matrix, Knudsen diffusion, gas desorption, and viscous flow were considered. Gas desorption was characterized by the Langmuir isothermal equation. In the fracture, viscous flow and nondarcy slip permeability were considered. The increase of pore radius due to gas desorption was calculated from gas desorption volume from the pore wall. Gas viscosity was characterized as a function of Knudsen number. Weak form equations for the system has been derived based on constant pressure boundary and got solved using COMSOL. A comprehensive sensitivity analysis were conducted, and detailed investigation were done regarding their impact on the shale gas production performance. Results show that considering gas viscosity change will greatly increase gas production under given reservoir conditions and slow down the production decline curve. Considering pore radius increase due to gas desorption from the pore wall will obtain higher production, but the effect is not very significant under given reservoir condition. In SGRs, both the matrix and fracture permeability change with time during production. Ignoring one of these factors such as Knudsen diffusion, slippage effect, desorption, viscosity decrease, and porosity increase will lead to less cumulative production. Therefore, for more accurate shale gas production prediction, it is crucial to incorporate these factors into the existing models. PORE DISTRIBUTION AND GAS TRANSPORT PROCESS IN SGRs Understanding the pore distribution and geometry of shale strata is of great importance to understand the gas transport process in the media. The Fig. 1 shows the gas distribution and geometry of shale strata from micro to macro scale. It informs us that in the fracture, only free gas exists and in the matrix which is full of kerogen, free gas and adsorption gas

141 125 co-exist. In such a complicated system, gas will desorb from the pore wall and transport in the matrix system; then due to pressure difference between matrix and fracture system, gas transfer between matrix and fracture; then gas flow to the wellbore and is produced to the surface. This process is shown in Fig. 2. This study was inspired by the depicted gas transport process in shale strata in Fig. 2. MODEL CONSTRUCTION This theoretical study is based on the following basic assumptions: (1). Single component, one phase flow in SGRs; (2). Ignore the effect of gravity and heterogeneity on the gas flow; (3). Ideal gas behavior for the natural gas in shales, and gas deviation factor z=1; (4). Formation rocks are incompressible, and the porosity change due to rock deformation is ignored; (5). Isothermal flow process in the whole reservoir life; (6). Gas adsorption-desorption kinetics obey Langmuir curve, and can achieve equilibrium state quickly at any reservoir pressure (diffuse quickly). Consider a generalized mass balance equation (Pruess et al., 1999) in every grid block is as follows: ρ Q (1) where M is mass accumulation term; u is velocity; ρ is density; Q is source term; and t denotes time. For shale strata which are full of micro-fractures, two general methods can be used to deal

142 126 with the fracture-matrix interaction. One is the dual porosity continuum method (DPCM), including dual porosity and dual permeability (DPDM) (Warren and Root, 1963), or MINC (multiple interacting continua)(pruess et al., 1999). The other one is the explicit discrete fracture mode (Xiong et al., 2012). Though the later model is more rigorous, it is still limited in applications to field study due to computational intensity and lack of detailed knowledge of fracture distribution (Civan et al., 2011). In this study, the DPCM which is shown in the Fig. 3, is used to describe the complex fracture distribution. For the DPCM, there exist two mass balance equations corresponding for fracture and matrix systems respectively as indicated by Warren and Root (Warren and Root, 1963). With the subscript and representing fracture and matrix system respectively, the two set of equations are illustrated as follows: ρ u Q (2) ρ u Q (3) The first term on the left side is the mass accumulation term; the second term on the left side is the flow vector term; and the right side is the source/sink term. These terms will be addressed correspondingly in the following sub-sections. Mass accumulation term The general form of the mass accumulation term is: Mϕ S ρ (4)

143 127 where denotes the fluid phase, is porosity, is the faction of pore volume occupied by the phase, is the density of the phase. Specifically for gas reservoir, =1. However, in matrix system of shale strata, there are adsorption gas and free gas in the system which is shown in the Fig. 1. The free gas which occupies the pores of matrix can be represented by M ϕ S ρ, and the adsorded gas in the surface of the bulk matrix can be represented as 1, where is the adsorption gas volume per unit bulk volume. Gas desorption occurs when pressure drop exists in organic grids (kerogen). With free gas production, the pressure in the pores will decrease, which results in the pressure difference between the bulk matrix and the pores. Due to this pressure drop, gas will desorb from the surface of bulk matrix. The most commonly used empirical model describing sorption and desorption of gas in shale and providing a reasonable fit to most experimental data is the Langmuir single-layer isotherm model (Cui et al., 2009; Civan et al., 2011; Shabro et al., 2011; Freeman et al., 2012), which is expressed in Equation (5): V (5) where v is the Langmuir volume, denoting the amount of gas sorbed at infinite pressure. And p is the Langmuir pressure, corresponding to the pressure at which half of the Langmuir volume v is reached. v and p are normally measured under standard temperature and pressure (STP) conditions.

144 128 So, the adsorption gas volume per unit bulk volume can be expressed in Equation (6): q q (6) where q is the adsorption gas per unit area of shale surface, kg m ; V is the mole volume under standard condition (0, 1atm), m mol; q is the adsorption volume per unit mass of shale, m kg ; V is the Langmuir volume, m 3 /kg; p is the Langmuir pressure, ; ρ is the density of shale core, kg m. The mass accumulation considering free gas and adsorption gas can be represented as M ϕρ 1ϕq and corresponding partial differential form is M dv. Based on the equation of state (EOS): pv nrt RT, ρ pγ γφ (7) (8) In the fracture system, because there is only free gas, so the mass accumulation term can be described as follows: γφ (9)

145 129 Flow vector term This paper distinguishes gas flow in shale gas by flowing media: matrix and fracture systems. Due to the differences of pore sizes of those two media, gas flow mechanisms are different. Fig. 4 shows the general flow process from matrix to fracture, then to wellbore in shale strata. (1). Gas flow in the matrix nano pores The general Darcy law informs us the relation of velocity and pressure drop, as presented in the Equation (10): p ρg z (10) For low density fluids such as gases, it is assumed that effect of gravity can be ignored. Therefore, simplified empirical Darcy s law can be used for the flow vector term in pores in the range of tens to hundreds of microns:. ρ.ρ p (11) where K is the permeability tensor. In the matrix system where pores are in the range of nano-meter, conventional Darcy law cannot be used to describe the flow process. Bird et al concluded that gas transportation in nano pores is a multi-mechanism-coupling process including Knudsen diffusion, viscous convection, and slip flow (Bird et al., 2006). Fig. 5 shows the flow mechanisms of gas transport in the nano pores of shale strata (Guo et al., 2015). The following subsections provide more description on those flow mechanisms. Viscous flow. When the free path of the gas is smaller than the pore diameter (Knudsen

146 130 number is far less than 1), then the motion of the gas molecules is mainly affected by the collision between molecules. There exists viscous flow caused by the pressure gradient between the single component gas, which can be described by Darcy law (Kast and Hohenthanner, 2000): J p (12) where J is the mass flow (kg m s), ρ is gas density in bedrock (kg m ), k is the intrinsic permeability of bedrock (m ), μ is the gas viscosity (Pa s), p is pore pressure in the bedrock (Pa. Knudsen Diffusion. When the pore diameter is small enough so that the mean free path of the gas is close to the pore diameter (ie, Knudsen number > 1), the collision between the gas molecules and the wall surface dominates. The gas mass flow can be expressed by the Knudsen diffusion (Kast and Hohenthanner, 2000): J M D C (13) where J is the mass flow caused by Knudsen diffusion (kg/(m 2 s)), M is the gas molar mass (kg/mol), D is the diffusion coeffici-ent of the bedrock (m 2 /s), C is the gas mole concentration (mol/ m 3 ) Slip flow. In low-permeability formations (less than md) or the pressure is very low, gas slip flow cannot be omitted when studying gas transport in tight reservoirs (Klinkenberg, 1941; Sakhaee-pour and Bryant, 2012). Under such kind of flow conditions,

147 131 gas absolute permeability depends on gas pressure which can be expressed as follows: k k1 (14) where k is the apparent permeability, k is the intrinsic permeability, b is the slip coefficient. Some empirical models have been developed to account for slip-flow and Knudsen diffusion in the form of apparent permeability, including correlations developed from the Darcy matrix permeability (Ozkan et al., 2010), correlations developed based on flow mechanisms (Brown et al., 1946; Javadpour et al., 2007; Javadpour, 2009), and semiempirical analytical models by Moridis et al. in 2010 (Moridis et al., 2010). This research adopted Javadpour s method which considered slip flow, viscous flow and Knudsen diffusion (Javadpour, 2009). The apparent permeability is given as follows: k k 1 (15).. 1 (16) where is the tangential momentum accommodation coefficient which characterizes the slip effect. is a function of wall surface smoothness, gas type, temperature and pressure which varies in a range from 0 to 1. For simplification, in this paper, it is set as 0.8. So, the flow vector term in the matrix is as follows: ρ u.ρ p.γp p (17) For the matrix system, it should be noted that with the gas desorption from the pore wall, the pore radius is increasing. The volume of gas desorbed from the wall can be

148 132 characterized by the Langmuir isotherm curve. So the effective pore radius considering the gas desorption can be expressed as follows (Xiong et al., 2012): r r d (18) where r is the initial pore radius, d is the molecular diameter of methane, P is the Langmuir pressure, r is the initial pore radius which has deducted the molecular diameter, with the matrix pressure decreasing during the production process, gas desorbs from the pore wall. So, from Equation (18), we can find that the pore radius is increasing with production. If the matrix pressure can approach 0, which is the ideal scenario, the final pore diameter will be equal to initial radius r. The process is shown in Fig. 6. (2). Flow vector term in the fracture In this study, Knudsen diffusion and viscous flow are considered for gas flow in the frature system. The mass flux can be expressed as the summation of the two mechanisms. Therefore, (19) or, 1 (20) Using the form of conventional Darcy flow equation, the fracture apparent permeability can be expressed as:

149 133 k k 1 (21) where, D b (22). (23) where p is the fracture pressure, k is the initial fracture permeability, k is the apparent fracture permeability, b is the Klinkenberg coefficient for the fracture system, D is the Knudsen diffusion coefficient for the fracture system, is the fracture porosity. Combining all above, the flow vector term in the fracture is:. ρ u.ρ p.γp p (24) Source and Sink term For DPCM, it is of great importance to consider the interaction between the matrix and the fracture in simulation. There are different methods to handle this issue in reservoir simulation. The boundary condition approach explicitly calculates the amount of the matrix-fracture petroleum fluid transfer by imposing boundary condition at each time step. It is very useful in well testing (Kazemi et al., 1976; Ozkan et al., 2010). However, this method is impractical in full field simulation due to the high computational cost. Another method is the Warren-Root method (Warren and Root, 1963; Kazemi et al., 1976). The

150 134 Warren-Root method calculates the cross flux between the fracture and matrix systems by assuming pseudo-steady state flow between matrix and fracture when the no-flow boundary is confined. The gas transfer between the matrix and fracture systems is represented in the Equation (25): T (25) where σ 4 is the crossflow coefficient between the matrix and fracture systems, and L, L, L are the fracture spacing in the x,y, z directions. In the fracture system, there exists a source term which flows into fracture from the matrix and a sink term which flows out of fracture into the wellbore. Therefore, for the matrix system, the sink term which flows out of matrix to fracture can be described as follows: Q T (26) The sink term followed the model developed by Aronofsky and Jenkins (Aronofsky and Jenkins, 1954) that considers gas production from vertical well can be expressed using the Equation (27): q p p (27) In Equation (27), when the production well is in the corner, θ π 2 ; and when the production well in the center, θ 2π (Peaceman, 1983). In addition, p is the

151 135 bottomhole flowing pressure; p is the average pressure in the fracture system; r is the well radius; and r is the drainage radius which can be expressed as follows: 0.142Δx Δy r k k k k (28) where Δx, Δy, k, k are the length of grid and permeability in x and y direction. For the fracture sytem, Q Tq. Gas viscosity change Currently, some reservoir simulators have considered gas viscosity change, which is a function of pressure. However, here we considered gas viscosity change is function of Knudsen number (the ratio of the free molecular length to the pore diameter), which is dependent on gas transport period as discussed in part Gas viscosity will change with Knudsen number in the production process (Michalis et al., 2010). Figure 7 has shown the ratio of effective viscosity to initial viscosity changes with Knudsen number, which is a function of gas pressure. Equation (29) has shown the relationship between gas viscosity and the Knudsen number. More specifically, Civan has derived the expression of Knudsen number using a more general form as shown in Equation (29) which is easier to be considered in simulation (Civan et al., 2011). μ μ ; μ μ (29)

152 136 where μ0 is the initial viscosity at the initial reservoir pressure (pm=pf=pi). This suggested a Knudsen dependence of the rarefaction parameter and provided an analytical expression for this dependence. Equation (30) expresses the definition of Knudsen number which can be related to be the basic parameters that changes with pressure: K (30) The final mathematical model to characterize gas transport can be expressed as follows: Matrix: γφ γ p T (31) Fracture: γφ γ p Tq (32) with following boundary conditions considered in this paper: Initial condition: p p p (33) Boundary condition for matrix: F n =0 ( 0) (34) Boundary condition for fracture: F n 0 ( 0), p x,y,t p (35) In this paper, we will consider the model Equations (31)-(35) and four versions of its simplification: (1) considering the basic DPM which has considered none of adsorption, non-darcy flow, gas viscosity, and pore radius change. (2) considering the basic DPM with adsorption which corresponds to the first term of the left part of Equation (31) ; (3)

153 137 considering the basic DPM with adsorption and non-darcy permeability change. That is, bm 0 and bf0; (4) considering the basic DPM with adsorption, non-darcy permeability change, and gas viscosity change. Gas viscosity change which is represented in the Equation (29); and (5) considering adsorption, non-darcy permeability change, gas viscosity change, and pore radius change. Pore radius change is represented in the Equation (18), in which reff represents the changing radius as shown in bm (r= reff in Equation (18)). The finite element method was used to solve the PDE equations from Equations (31)-(35) which have been listed above. First, multiply a test function, on both sides of original Equation (31) where is the domain; is outside boundary; is the inner boundary: γφ v γ p v Tv (36) γφ v dxdy γ p v dxdy Tv dxdy (37) Using Green s Theorem (Divergence Theory and Integration by parts in multi-dimension), we obtain that: γ p v dxdy γφ γ p nvds γ p vdxdy (38) v dxdy γ p nvds γ p vdxdy Tv dxdy (39) If we just consider the solution on the domain, the weak form for the governing Equation (31) is:

154 138 γφ v dxdy γ p Similar process to governing Equation (31), we obtain: γφ v dxdy γ p vdxdy Tv dxdy (40) vdxdy T q v dxdy (41) MODEL VERIFICATION As there is no real field data which is same with this theoretical case, a common method to verify the righteousness is to compare the results of numerical simulation against the analytical results. Wu et al. have derived the analytical solution for 1D steady-state gas transport (Wu et al., 1998), which is shown in Equation (42). Thus, we can verify our model by comparing the results from simplification version of model in this paper with the analytical results. The reservoir properties and Klinkenberg properties used in this verification were obtained from the experimental study of the welded tuff at Yucca Mountain (Reda, 1987). 2 (42) where is the Klinkenberg factor, Pa; is the gas pressure at the outlet (x=l), Pa; is the air injection rate, kg/s; is the gas dynamic viscosity, Pa.s; is the length from the inlet (x=0) to the outlet, 10 m; x is the task location along the gas transport path, m; k is the intrisic permeability for welded tuff atyucca Mountain, m 2 ; is the compressibility factor, Pa -1 m -1. Fig. 8 has presented the match results between model in this paper and analytical solution. As

155 139 it can be found from Fig. 8, the results of the model in this paper agrees well with the analytical solution (Wu et al., 1998) provided by Wu et al, which has validated our mathematical model. EFFECT OF FLOW MECHANISMS ON SHALE GAS PRODUCTION The 2-D reservoir model is shown in Fig. 9. For simplification, we only study the ¼ area of the whole reservoir. The reservoir parameters are shown in Table 2. The parameters used in this paper are selected based on the literature review regarding shale gas simulation (Bustin et al., 2008). The reservoir simulated is located in the depth of 5463 ft with a pressure gradient of 0.53 / and temperature gradient of K / ft, which correspond to initial reservoir pressure and reservoir temperature of 20 and Other parameters are listed in Table 2. In this study, firstly we have investigated the impact of parameters such as initial reservoir pressure, matrix permeability, fracture permeability, matrix porosity, and fracture porosity on shale gas production. Then, the mechanisms discussed above are gradually incorporated into the simulation model gradually to investigate their impact on gas transport in shale strata. These mechanisms are adsorption, permeability change due to comprehensive slip effect, viscosity change, and radius change due to gas adsorption. Effect of Gas adsorption In this study, we have compared the scenarios of considering adsorption and without considering adsorption. When adsorption is considered, there will be an adsorption term in the left hand side of the Equation (31), while only free gas will be considered in the matrix system if adsorption is ignored. The following analysis are all based on the scenario of

156 140 considering adsorption. First, we need to know the effect to adsorption to the gas production. Fig. 10(a) and Fig. 10(b) illustrates the effect of adsorption on the production rate and cumulative production. It is obvious that adsorption has a great effect on the production rate and cumulative production. Ignoring the effect of adsorption gas will greatly underestimate the gas production in such situations. Effect of non-darcy flow Effect of non-darcy flow can be considered by adjusting the slip coefficient which appears in the Equation (31). If non-darcy flow is considered, slip coefficient bm=bf=0. Fig. 11 illustrates the matrix permeability and fracture permeability change with time. From the Fig. 11, it is clear that the gas permeabilities in the matrix and fracture are gradually increasing during pressure depletion. Matrix permeability has a greater increase compared with fracture permeability. As shown in Fig. 12(a) and Fig. 12(b), the impact of non-darcy and production rate and cumulative production are plotted on the log-log scales. It is clear that considering non-darcy flow will increase the production rate and cumulative production; however, there is no significant difference between these two scenarios. This is because though matrix permeability increases a lot as shown in Fig. 11(a), however, fracture permeability is critical in determining fluid flow rate from fracture system to the wellbore. Therefore, it is important to increase the fracture permeability rather than matrix permeability by including hydraulic fractures in using stimulations. Effect of gas viscosity change Effect of gas viscosity change is evaluated on the basis of previous study, which has considered adsorption and non-darcy flow. That is, if gas viscosity change is considered, then bm 0, bf0. Gas viscosity change is represented in the Equation (29). If gas viscosity

157 141 change is not considered, then bm 0, bf0, gas viscosity is a constant. Gas viscosity is a constant. As shown before, gas viscosity is decreasing with production and pressure depletion. Fig. 13 shows the comparison of production rate and cumulative production between these two scenarios, it can be seen that gas viscosity will have a great impact on production rate and cumulative production under the production condition of this paper. From Equation (29), we can find that gas viscosity is a non-linear function of the pressure; and during the gas production process, pressure decrease will lead to the viscosity decrease, which will result in the production increase. Therefore, ignoring the effect of gas viscosity change will decrease the estimated production. Effect of pore radius change Pore radius change is represented in the Equation (18), which will the change the radius shown in the part of bm (r= reff in Equation (17). When we analyze the effect of pore radius change, we still consider all the factors which have been considered before: adsorption, non-darcy flow, and gas viscosity change. From the Equation (18), we can find that the maximum pore radius in the production will be intrinsic radius plus 2/3 of diameter of CH4. Therefore, the pore radius actually does not change too much. However, reff is affected by matrix pressure, which is changing during reservoir depletion; therefore, it is still important to consider this effect. The effect of the pore radius change on the production rate and cumulative production are shown in Fig. 14. From Fig. 14(a) and Fig. 14(b), we can find that although there is a slightly increase in the production rate and cumulative production. The effect of radius change is not very significant. Fig. 15 shows the comparison of the above 5 models: (1) basic DPM which has not considered adsorption, non-darcy flow, gas viscosity and pore radius change; (2)

158 142 Considering adsorption; (3) Considering adsorption and non-darcy permeability change; (4) Considering adsorption and non-darcy permeability change, gas viscosity change; (5) Considering adsorption, non-darcy permeability change, gas viscosity change and pore radius change. It can be found that the mechanisms we have considered will increase the production. Ignoring any one of these factors will underestimate gas production. SENSITIVITY ANALYSIS For studying reservoir parameters effect on gas production, we used the model which has considered the effect of adsorption and non-darcy flow. The effect of the following reservoir parameters has been analyzed. (1) initial pressure; (2) matrix permeability; (3) fracture permeability; (4) matrix porosity; and (5) fracture porosity. For each scenario, we have compared the production rate and cumulative production for three stages. In this study, when discussing certain parameters effect, the other parameters are kept the same as listed in the Table 2. Effect of initial pressure Initial reservoir pressure is very important in the production, especially for gas reservoir which has no other driving force. Here, we have analyzed the effect of initial pressure on gas production. We have considered the scenarios of initial pressure as 0.2 MPa, 2.0 MPa and 20 MPa. Other parameters are kept the same as in the Table 2. Fig. 16 shows the comparison of production rate and cumulative production under different initial reservoir pressure. From the plots, we can find that initial pressure will have a great effect on the gas production; there is 3 to 5 folds increase in the cumulative production when the initial

159 143 pressure increase 10 folds. With the increase of reservoir initial pressure, gas production rate and gas cumulative production will increase. Effect of matrix Permeability and fracture permeability Permeability is the king in the reservoir production. For DPM, matrix permeability and fracture permeability are not the same and the effect of these two kinds of permeability are also different. In this study, effects of matrix permeability and fracture permeability on gas production are evaluated together to show which effect is more important. Fig. 17 shows the effect of matrix permeability with permeability varing from md to md to md. Fig. 18 shows the effect of fracture permeability with permeability changing from 0.1 md to1 md to 10 md. From these figures, we can find that the production rate and cumulative production will increase when the matrix permeability or fracture permeability is increasing. However, from the comparison between Fig. 17(a) and Fig. 17(b), and between Fig. 18(a) and Fig. 18(b), we can find that increase of the fracture permeability has a more apparent effect on the production rate and cumulative production. Effect of matrix porosity and fracture porosity For studying the effect of porosity, three levels of matrix porosity and fracture porosity are considered and evaluated. We have compared the difference when we change the matrix porosity from 5% to 10% to 20%. And by changing the fracture porosity from 0.01% to 0.1% to 1%. From Fig. 19 and Fig. 20, we can find that porosity increase will lead to production increase. However, matrix porosity has a more important effect in gas production as matrix is the main storage form of gas.

160 144 CONCLUSION This paper presents our theoretical model and mechanism study for shale gas production. Following conclusions have been obtained from this study: A new mathematical model which considers adsorption, non-darcy permeability change, gas viscosity change and pore radius increase due to gas desorption has been constructed; From the results analysis, considering one of the scenarios: adsorption, non-darcy permeability change, gas viscosity change and pore radius change will increase the production estimate. Among these mechanisms, adsorption and gas viscosity change have a great impact on gas production. Ignoring one of these effects will decrease gas production; From reservoir parameters sensitivity analysis, initial reservoir pressure has a great impact on gas production, fracture permeability has more important effect compared with matrix permeability. Porosity increase will lead to the increase of gas production. However, matrix porosity is more important than fracture porosity. ACKNOWLEDGEMENTS Funding for this project is provided by RPSEA through the Ultra-Deepwater and Unconventional Natural Gas and Other Petroleum Resources program authorized by the U.S. Energy Policy Act of The authors also wish to acknowledge for the consultant support from Baker Hughes and Hess Company.

161 145 NOMENCLATURE α =Tangential momentum accommodation coefficient b b b C =Slip coefficient =Non-darcy coefficient =Non-darcy coefficient for the fracture system =Gas mole concentration (mol m ) D =Diffusion coefficient of the bedrock (m s) d =Molecular diameter of methane (m) D =Knudsen diffusion coefficient for the fracture system (m s) J =Mmass flow caused by Knudsen diffusion (kg/(m s)) J =Mass flow caused by viscous flow (kg/(m s)) k =Initial permeability of bedrock (m ) k =Initial fracture permeability (m ) k =Apparent fracture permeability (m ) k =Apparent matrix permeability (m ) M M p p p p p Q =Gas molar mass kg mol =Mass accumulation (kg m.s) =Gas pressure in the fracture (Pa) =Gas pressure in the matrix (Pa) =Bottomhole pressure (Pa) =Langmuir pressure of CH4 (Pa) =Average pressure in the fracture (Pa) =Source term (kg m.s)

162 146 q =Adsorption volume per unit mass of shale (m kg) r r r =Effective pore radius (m) =Wellbore radius (m) =Drainage radius (m) S =Faction of pore volume occupied by phase β (%) V =Langmuir volume of CH4 (m kg) V =Gas mole volume under standard condition (0, 1atm) (m mol) ρ β =Density (kg m ) =Fluid phase ϕ =Matrix porosity (%) ϕ =Fracture porosity (%) ρ ρ μ σ =Density of shale core (kg m )) =Gas density in the bedrock (kg m )) =Gas viscosity (Pa s) =Crossflow coefficient REFERENCES Aronofsky, J. S., & Jenkins, R A Simplified Analysis of Unsteady Radial Gas Flow. Journal of Petroleum Technology, 6(07), Beskok A, Karniadakis G E Report: A Model for Flows in Channels, Pipes, and Ducts at Micro and Nano Scales. Microscale Thermophysical Engineering 3(1): Bird, R. B., Stewart, W. E., Lightfoot, E. N Transport Phenomena. Wiley. com. Brown, G. P., A. DiNardo, et al The Flow of Gases in Pipes at Low Pressures. Journal of Applied Physics.17(10):

163 147 Bustin A, Bustin R, Cui X Importance of Fabric on the Production of Gas Shales. Presented at Unconventional Reservoirs Conference in Colorado, USA, February. SPE MS. Civan F, Rai C S, Sondergeld C H Shale-gas Permeability and Diffusivity inferred by Improved Formulation of Relevant Retention and Transport Mechanisms. Transport in Porous Media. 86(3): Cui X, Bustin A M M, Bustin R M Measurements of Gas Permeability and Diffusivity of Tight Reservoir Rocks: Different Approaches and Their Applications. Geofluids. 9(3): Freeman C, Moridis G, Michael G, et al Measurement, Modeling, and Diagnostics of Flowing Gas Composition Changes in Shale Gas Well. Presented at Latin America and Caribbean Petroleum Engineering Conference in Mexico City, Mexico April. SPE MS. Guo, C., Xu, J., Wei, M., & Jiang, R Pressure transient and rate decline analysis for hydraulic fractured vertical wells with finite conductivity in shale gas reservoirs. Journal of Petroleum Exploration and Production Technology, 1-9. Guo, C., Xu, J., Wu, K., Wei, M., & Liu, S Study on gas flow through nano pores of shale gas reservoirs. Fuel, 143, Hill D G, Nelson C R Gas Productive Fractured Shales: An Overview and Update. Gas Tips. 6(3): Javadpour F Nanopores and Apparent Permeability of Gas Flow in Mudrocks (shales and siltstone). Journal of Canadian Petroleum Technology. 48(8): Javadpour F, Fisher D, Unsworth M Nanoscale Gas Flow in Shale Gas Sediments. Journal of Canadian Petroleum Technology. 46(10). Kast W, Hohenthanner C R Mass Transfer within the Gas-phase of Porous Media. International Journal of Heat and Mass Transfer. 43(5): Kazemi H, LS M, Porterfield K L, et al Numerical Simulation of Water-oil Flow in Naturally Fractured Reservoirs. SPE Journal. 16(6): Klinkenberg L J The Permeability of Porous Media to Liquids and Gases. Drilling and production practice. Michalis V K, Kalarakis A N, Skouras E D, et al Rarefaction Effects on Gas Viscosity in the Knudsen Transition Regime. Microfluidics and nanofluidics. 9(4-5): Moridis G, Blasingame T, Freeman C Analysis of Mechanisms of Flow in Fractured Tight-gas and Shale-gas Reservoirs. Presented at Latin American and Caribbean Petroleum Engineering Conference 1-3 December. SPE MS. Ozkan E, Raghavan R, Apaydin O Modeling of Fluid Transfer from Shale Matrix to Fracture Network. Presented at Annual Technical Conference and Exhibition, Lima, Peru, 1-3 December. SPE MS.

164 148 Peaceman D W Interpretation of Well-block Pressures in Numerical Reservoir Simulation with Nonsquare Grid Blocks and Anisotropic Permeability. SPE Journal. 23(3): Pruess K, Moridis G J, Oldenburg C TOUGH2 user's guide, version 2.0. Berkeley: Lawrence Berkeley National Laboratory. Reda D. C Slip-flow experiments in welded tuff: The Knudson diffusion problem. In: Chin-Fu Tsang (ed.). Coupled Processes Associated with Nuclear Waste Repositories, Sakhaee-Pour A, Bryant S Gas Permeability of Shale. SPE Reservoir Evaluation & Engineering. 15(4): Shabro V, Torres-Verdin C, Javadpour F Numerical Simulation of Shale-gas Production: From Pore-scale Modeling of Slip-flow, Knudsen Diffusion, and Langmuir Desorption to Reservoir Modeling of Compressible Fluid. Presented at North American Unconventional Gas Conference and Exhibition in Texas, USA June. SPE MS. Shabro V, Torres-Verdin C, Sepehrnoori K Forecasting Gas Production in Organic Shale with the Combined Numerical Simulation of Gas Diffusion in Kerogen, Langmuir Desorption from Kerogen Surfaces, and Advection in Nanopores. Presented at Annual Technical Conference and Exhibition in Texas, USA, 8-10 October. SPE MS. Wang Z, Krupnick A A Retrospective Review of Shale Gas Development in the United States. Warren J E, Root P J The Behavior of Naturally Fractured Reservoirs. SPE Journal. 3(3): Wu Y-S, Fakcharoenphol P A Unified Mathematical Model for Unconventional Reservoir Simulation. Presented at EUROPEC/EAGE Annual Conference and Exhibition in Vienna, Austria, May. SPE MS. Wu, Y. S., & Pruess, K Gas Flow in Porous Media with Klinkenberg Effects. Transport in Porous Media, 32(1), Xiong X, Devegowda D, Michel Villazon G, et al A Fully-coupled Free and Adsorptive Phase Transport Model for Shale Gas Reservoirs including Non-darcy Flow Effects. Presented at Annual Technical Conference and Exhibition in Texas, USA, 8-10 October. SPE MS. Ziarani A S, Aguilera R Knudsen s Permeability Correction for Tight Porous Media. Transport in porous media. 91(1):

165 149 Table 1. Parameters used in the simulation model. Parameter Value Unit Definition 5463 reservoir depth 0.54 reservoir pressure gradient reservoir temperature gradient k 1.00E-04 matrix initial permeability k 10 fracture initial permeability 0.05 Dimensionless matrix porosity Dimensionless fracture porosity R gas constant z 1 Dimensionless gas compressibility factor 10.4 initial reservoir pressure 3.45 bottom hole pressure mole weight of CH standard gas volume 2.07 langmuir pressure 2.83E-03 langmuir volume 2550 shale rock density 1.02E-05 initial gas viscoisty 0.1 wellbore radius 0.2 fracture spacing

166 150 Figure 1. Gas distribution in shale strata from macro-scale to micro-scale. In the fracture there exist free gas and in the matrix free gas and adsorption gas co-exist. Figure 2. Gas production process from macro to molecular scales. Flow to the wellbore is first initiated at the macroscale, followed by flow at increasingly finer scales. Modified from (Javadpour et al., 2007).

167 151 Figure 3. Idealization of the heterogeneous porous medium as DPM. Matrix Continuum(desorption, viscous flow, Knudsen flow, slip flow) Transfer function Fracture Continuum (viscous flow, Knudsen flow) Figure 4. Transport scheme of shale gas production in DPM. Gas desorbed from the matrix surface and transferred to the fracture, then flow into the wellbore. Non-darcy flow, Knudsen diffusion, slip flow, and viscous flow have been considered.

168 152 Figure 5. Gas flow mechanisms in a nano pore. Red solid dots represent Knudsen diffusion, while blue ones represent viscous flow. Figure 6. Pore radius change due to gas desorption. If single molecule gas desorption Langmuir isothermal is considered, when all the molecules have desorped from the surface, then the pore radius will increase as shown in the right part eff / Bahukudumal et al., 2003 NS-based model, Roodi and Darbandi, 2009 IP-based model, Roohi and Darbandi, 2009 Sun and Chan, 2004 Karniadakis et al., Kn Figure 7. Gas viscosity variation with the Knudsen number (from 0.01 to 1), also means from slip flow to transition flow. Gas viscosity has changed a lot when the Knudsen number changes, which means it is necessary to consider the gas viscosity variation (Modified from Michalis et al., 2010).

169 Gas Pressure (Pa) Results of the model in this paper Analytical Solution by Yu-Shu Wu (1988) Distance (m) Figure 8. Comparison between analytical solution and numerical simulation in this paper. p 1 n 0 p 1 n 0 p 1 n 0 p ( x, y, t) p f 2 p 1 n w 0 Figure 9. Real reservoir and simplified 2-D reservoir simulation model.

170 Production rate (x10 3 m 3 /d) time (days) Without adsorption With adsorption Cumulative production (x10 3 m 3 ) Production time (days) Without adsorption With adsorption Figure 10. The effect of adsorption on gas production. (a) production rate vs. time; (b) cumulative production vs. time. matrix permeability (md) fracture permeability (10-2 md) production time (days) production time (days) Figure 11. Matrix and fracture permeability change with time. (a) matrix permeability vs. time; (b) fracture permeability vs. time.

171 production rate (x10 3 m 3 /d) 10 1 Considering Non-Darcy Flow Without considering Non-Darcy Flow production time (days) Cummulative production (x10 3 m 3 ) Considering Non-Darcy Flow Without considering Non-Darcy Flow production time (days) Figure 12. Effect of Non-Darcy flow on gas production. (a) production rate vs. time; (b) cumulative production vs. time production rate (x10 3 m 3 /d) Without considering gas viscosity change Considering gas viscosity change production time (days) Cumulative production (x10 3 m 3 ) Without considering gas viscosity change Considering gas viscosity change production time (days) Figure 13. Effect of gas viscosity change on gas production. (a) production rate vs. time; (b) cumulative production vs. time.

172 156 production rate (x10 3 m 3 /d) Considering viscosity change Considering viscosity change and pore radius change Cumulative production (x10 3 m 3 ) Considering viscosity change Considering viscosity change and pore radius change production time (days) production time (days) Figure 14. Effect of pore radius increase due to gas desorption on gas production. (a) production rate vs. time; (b) cumulative production vs. time production rate (x10 3 m 3 /d) Cumulative production (x10 3 m 3 ) Production time (days) Basic model Adsorption Adsorption+Non-Darcy Adsorption+Non-Darcy+Viscosity change Adsorption+Non-Darcy+Viscosity change+radius change Figure 15. Comparison of five different models: (1) basic model; (2) Considering adsorption; (3) Considering adsorption and non-darcy permeability change; (4) Considering adsorption and non-darcy permeability change, gas viscosity change; (5) Considering adsorption, non-darcy permeability change, gas viscosity change and pore radius change.

173 production rate (x10 3 m 3 /d) pi=0.2 MPa pi=2.0 MPa pi=20 MPa cumulative production (x10 3 m 3 ) pi=0.2 MPa pi=2.0 MPa pi=20 MPa production time (days) production time (days) Figure 16. Effect of initial reservoir pressure on gas production. (a) production rate vs. time; (b) cumulative production vs. time production rate (m 3 /d) k mi =1E-4 md k mi =1E-5 md k mi =1E-6 md Cumulative production (x10 3 m 3 ) k mi =1E-4 md k mi =1E-5 md k mi =1E-6 md production time (days) production time (days) Figure 17. Effect of matrix permeability on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time.

174 production rate (x10 3 m 3 /d) k fi =0.1 md k fi =1.0 md k fi =10 md Cumulative production (x10 3 m 3 ) k fi =1.0 md k fi =10 md k fi =100 md production time (days) production time (days) Figure 18. Effect of fracture permeability on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time. 70 production rate (x10 3 m 3 /d) m =1.0% m =10% 20 m =20% 10 Cumulative production (x10 3 m 3 ) m =1% m =10% m =20% production time (days) production time (days) Figure 19. Effect of matrix porosity on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time.

175 production rate (x10 3 m 3 /d) f =0.01% f =0.1% f =1.0% Cumulative production (x10 3 m 3 ) f =0.01% f =0.1% f =1.0% production time (days) production time (days) Figure 20. Effect of fracture porosity on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time.

176 160 PAPER IV. NUMERICAL SIMULATION OF GAS PRODUCTION FROM SHALE GAS RESERVOIRS WITH MULTI-STAGE HYDRAULIC FRACTURING HORIZONTAL WELL Chaohua Guo; Mingzhen Wei Missouri University of Science and Technology (In submission to Journal of Petroleum Science and Engineering) ABSTRACT Unconventional tight shale reservoirs have been developed during recent years due to increasing shortage of conventional resources and a large number of multi-stage fractured horizontal wells (MsFHW) have been drilled to enhance reservoir production performance. Gas flow in tight shale reservoirs is a multi-mechanism process, including: desorption, diffusion, and non-darcy flow. The productivity of the shale gas reservoir with MsFHW is influenced by both reservoir condition and hydraulic fracture properties. In this paper, a dual porosity model was constructed to estimate the effect of parameters on shale gas production with MsFHW. The simulation model was verified with the available field data from the Barnett Shale. Following flow mechanisms have been considered in this model: viscous flow, slip flow, Knudsen diffusion, and gas desorption. Langmuir isotherm was used to simulate the gas desorption process. Sensitivity analysis on production performance of tight shale reservoirs with MsFHW have been conducted. Parameters influencing shale gas production were classified into two categories: reservoir properties including matrix permeability, matrix porosity and hydraulic fracture properties including hydraulic fracture spacing, fracture half-length. Typical ranges of matrix parameters have been reviewed. Sensitivity analysis have been conducted to analyze the effect of above factors on the production performance of shale gas reservoirs. Through

177 161 comparison, it can be found that hydraulic fracture parameters are more sensitive compared with reservoir parameters. And reservoirs parameters mainly affect the later production period. However, the hydraulic fracture parameters have significant effect on gas production from the early period. Result of this study can be used to improve the efficiency of history matching process. Also, it can contribute to the design and optimization of hydraulic fracture treatment design in unconventional tight reservoirs. 1. INTRODUCTION Exploitation of unconventional shale gas reservoirs has become an important component to secure natural gas supply of North America. Due to the improved stimulation techniques and multi-stage horizontal well drilling, economic development of ultra-low permeability shale gas reservoirs becomes viable (Cipolla et al., 2010). Shale gas is a kind of natural gas produced from tight shale reservoirs which are both source rock and storage media. In the shale gas reservoirs, gas is stored in three states: free gas in the nano pores and micro fractures; adsorbed gas on the surface of matrix; and dissolved gas in organic material (Yu et al. 2013; Guo et al. 2015). As shale formations have extremely low permeability which is about 54 to 150 nano-darcy (Cipolla et al. 2010; Guo et al. 2014a), economically development of shale had been regarded as impossible for a very long time until hydraulic fracturing technic has been used. Then, shale gas gradually becomes an important part of natural gas production and experienced a rapid growth since The combination of horizontal well and multistage hydraulic fracturing treatment can significantly improve production rate by enhancing the reservoir permeability and creating stimulated reservoir volume.

178 162 Significant progress has been achieved in modeling fluid flow in fractured rock since 1960s due to the increasing need to develop petroleum, geothermal, and other subsurface energy resources (Wu et al., 2009). Also, numerical modeling is a useful tool to understand reservoir performance and conduct production optimization. Compared with conventional reservoirs, shale gas reservoir contains large portion of nano pores and micro fractures. The key problem in simulating fractured shale gas reservoirs is how to handle the fracture matrix interaction. Many different models have been proposed to simulate fluid flow in fractured reservoirs (Alnoaimi and Kovscek, 2013; Rubin, 2010; Bustin et al., 2008; Fazelipour, 2011; Ozkan et al., 2010; Shabro et al., 2009). These methods include: (1) equivalent continuum model (Wu, 2000), (2) dual porosity model, including dual porosity single permeability model, and dual porosity dual permeability model (Warren and Root, 1963; Kazemi, 1969). (3) Multiple Interaction Continua (MINC) method (Pruess, 1985; Wu and Pruess, 1988). (4) Multiple porosity model (Civan et al., 2011; Dehghanpour et al., 2011; Yan et al., 2013); (5) Discrete fracture model (Snow, 1965; Xiong et al., 2012). Though the discrete fracture model is more rigorous than the first three models, it is still limited in applications to field study due to computational intensity and lack of detailed knowledge of fracture distribution (Civan et al., 2011). Equivalent continuum model. In the equivalent continuum model, the fractured porous media is considered as a continuum media. The properties of the fracture arrays, such as direction, location, permeability, porosity, etc., are averaged into the whole porous media. This method has not been widely used in shale gas reservoir simulation as it is not easy to obtain the averaged permeability tensor and it is too conceptual. Moridis et al. (2010) have constructed effective continuum shale gas reservoir simulation model with

179 163 considering muti-component gas adsorption. In the effective continuum model, the shale gas reservoir is discretized and the fractures are characterized with grid cells as single plannar planes or network of planar planes (Cipolla et al., 2009; Cheng, 2012). Dual Porosity Model. Dual porosity model is widely used to simulate the fluid flow in fractured porous media and many scholars have published a lot of literature based on dual porosity model (de Swaan O, 1976; Kazemi, 1969; Warren and Root, 1969; Shi and Durucan 2003; Ozkan et al. 2010). There are two interacting media: the matrix blocks with high porosity and low permeability, and the fracture network with low porosity and high conductivity. According to whether there is fluid flow in the matrix, the dual porosity model can be classified into dual porosity single permeability model and the dual porosity dual permeability model. In the dual porosity single permeability model, the flow only occur through the fracture system and matrix are treated as the spatially distributed sinks or sources to the fracture system (Wu et al., 2009). Watson et al. (1990) have performed production data analysis for Devonian shale gas reservoir using DPM and presented analytical reservoir production models for history matching and production forecasting. Bustin et al. (2008) have constructed a 2D dual porosity model to simulated shale gas reservoirs which uses both experimental and field data as model input parameters. Ozkan et al. (2010) have developed a dual-mechanism dual-porosity model to simulate the linear flow for fractured horizontal well in shale gas reservoirs. In the dual porosity dual permeability model, the flow in the matrix is considered additionally (Pereira et al., 2004; Yan et al., 2013). Connell and Lu have proposed an algorithm for the dual-porosity model considering Darcy flow in the fractures and diffusion in the matrix (Lu and Connell, 2007; Connell and Lu, 2007). Moridis (2010) constructed a dual permeability model and

180 164 compared it with dual porosity model and effective continuum model. Results showed that dual permeability model offered the best production performance. Cao et al. (2010) have proposed a new simulation model for shale gas reservoir based on dual porosity dual permeability model with considering multiple flow mechanisms, such as adsorption/desorption, diffusion, viscous flow, and deformation. Multiple Interaction Continua method. As an extension for dual porosity model, Pruess (1985) have proposed a multiple interaction continua method (MINC) to more accurately characterize the interaction between matrix and fracture. The matrix system in every grid block for MINC method will be subdivided into a sequence of nested rings which will make it possible to more accurately calculate the interblock fluid flow. In comparison with dual porosity model, MINC method is able to describe the gradient of pressures, temperatures, or concentrations near matrix surface and inside the matrix which can provide a better numerical approximation for the transient fracture matrix interaction (Wu et al., 2009). Wu et al. (2009) have proposed a generalized mathematical model to simulate multiphase flow in tight fractured reservoirs using MINC method with considering following mechanisms: Klinkenberg effect, non-newtonian behavior, non- Darcy flow, and rock deformation. Multiple Porosity Model. Some scholars also proposed multiple porosity model in which matrix is further separated into two or three parts with different properties. Civan et al. (2011) proposed a qual-porosity model to simulate shale gas reservoirs, including organic matter, inorganic matter, natural, and hydraulic fractures Dehghanpour et al. (2011) have proposed a triple porosity model to simulate shale gas reservoirs by incorporating micro fractures into the dual porosity models. They assumed that the matrix blocks in dual

181 165 porosity model is composed of sub-matrix with nano Darcy permeability, and micro fractures with milli to micro Darcy permeability. Yan et al. (2013) presented a micro-scale model. In Yan s model, the shale matrix was classified into inorganic matrix and organic matrix. The organic part was further divided into two parts basing on pore size of kerogen: organic materials with vugs and organic materials with nanopores. Therefore, there are four different continua in this model: nano organic matrix, vug organic matrix, inorganic matrix, and micro fractures. Discrete Fracture Model. In the discrete fracture model, the fractures are discretely modeled. Each fracture is characterized as a geometrically well-defined entity. In the modeling of shale gas reservoirs using discrete fracture model, gas flow from the matrix to the discrete fractures by diffusion and desorption (Rubin, 2010; Mayerhofer et al., 2006; Gong et al., 2011). Mayerhoffer et al. (2006) have investigated the fluid flow in fractured shale gas reservoirs considering stimulated reservoir volume (SRV) using explicit fracture network (Mayerhofer et al., 2006). Gong et al. (2011) have proposed a systematic methodology of building discrete fracture model based on microseismic interpretations for shale reservoirs with considering multiple mechanisms, such as gas adsorption, non-darcy effect, etc. The discrete fracture model is more rigorous, however, application of this method to reservoir scale is limited due to computational intensity and lack of detailed knowledge of fracture matrix properties and spatial distribution in reservoirs (Civan et al., 2011; Wu et al., 2009). And the dual porosity model can deal with the fracture matrix interaction more easily and less computationally demanding than discrete fracture method. So, dual

182 166 porosity model is the main approach for modeling fluid and heat flow in fractured porous media (Wu et al., 2009). In this paper, a numerical simulation model for shale gas reservoirs with multistage hydraulic fracturing horizontal well was constructed based on the dual porosity model. Mass balance equations for both matrix and fracture systems were constructed using the dusty gas model. In the matrix, Knudsen diffusion, gas desorption, and viscous flow were considered. Gas desorption was characterized by the Langmuir isothermal equation. In the fracture, viscous flow and non-darcy slip permeability were considered. The model got solved using commercial finite element software COMSOL. A comprehensive sensitivity analysis were conducted, and detailed investigation were done regarding their impact on the shale gas production performance. The rest of this paper are organized as follows: Section 2 has presented the gas flow mechanisms in tight shale reservoirs. In section 3, the apparent permeability for gas flow in shale matrix and fracture system have been derived. In section 4, the mathematical model for gas production from shale reservoirs with multi-stage hydraulic fracturing horizontal well has been constructed. Section 5 provided the model verification. And lastly, the sensitivity analysis have been conducted for reservoir and hydraulic fracture parameters based on a synthetic model. 2. FLUID FLOW MECHANISMS IN TIGHT SHALE RESERVOIRS Due to the complexity of flow in shale gas reservoirs, three flow mechanisms should be considered in the dual-permeability model: viscous flow, Knudsen diffusion, slip flow, and gas adsorption Viscous flow This is the region where Knudsen number (The ratio of gas mean free path to pore

183 167 throat diameter) has fairly low values such as in the conventional reservoirs with pore diameter in micrometers. In this Knudsen number range, the free path of the gas is smaller than the pore diameter (Knudsen number is far less than 1), so the motion of the gas molecules is mainly affected by the collision between molecules. The molecule s collision with the pore walls are negligible. The gas flow is driven by the pressure gradient between the single component gas, which can be characterized using the Darcy law: J p (1) where J is the mass flow (kg m s), ρ is gas density in bedrock (kg m ), k is the intrinsic permeability of bedrock (m ), μ is the gas viscosity (Pa s), p is pore pressure in the bedrock (Pa Knudsen diffusion This is the region where the Knudsen number is larger than 1 where the pore diameter is small enough so that the mean free path of the gas is close to the pore diameter. Under this Knudsen range, the gas flow is dominated by the collision between the gas molecules and the wall surface. The gas mass flow can be characterized using the Knudsen diffusion equation (Kast and Hohenthanner, 2000) which is shown in Equation (2): J M D C (2)

184 168 where J is the mass flow caused by Knudsen diffusion (kg/(m 2 s)), M is the gas molar mass (kg/mol), D is the diffusion coefficient of the bedrock (m 2 /s), C is the gas mole concentration (mol/ m 3 ) Slip flow In low-permeability formations (such as shale gas reservoirs) or when the pressure is very low, gas slip flow (or Klinkenberg effect) cannot be ignored when studying gas transport in tight reservoirs (Klinkenberg, 1941; Florence et al., 2007; Sakhaee-pour and Bryant, 2012). Under such kind of flow conditions, gas permeability depends on gas pressure which can be expressed as follows: k k1 (3) where k is the apparent permeability, k is the intrinsic permeability, b is the slip coefficient. Many scholars have proposed different expressions for the slip coefficient which are listed in the Table 1. (Guo et al., 2015) 2.4. Gas adsorption and desorption Gas desorption occurs when pressure drop exists in organic grids (Kerogen). With free gas production, the pressure in the pores will decrease, which results in the pressure difference between the bulk matrix and the pores. Due to this pressure drop, gas will desorb from the surface of bulk matrix. The most commonly used empirical model describing sorption and desorption of gas in shale and providing a reasonable fit to most experimental data is the Langmuir single-layer isotherm model (Freeman et al., 2013). Based on the

185 169 Langmuir equation (Equation (4)), the amount of gas adsorbed on the rock surface can be computed. (4) where is the gas Langmuir volume denoting the amount of gas adsorbed at infinite pressure, scf/ton; is the gas Langmuir pressure corresponding to the pressure at which half of the Langmuir volume v is reached, psi; is the in-situ gas pressure in the pore system, psi. Langmuir volume means the maximum amount of gas that can be adsorbed on the rock surface under infinite pressure. Langmuir pressure is the pressure when the amount of gas adsorbed is half of the Langmuir volume. These two parameters play important roles in the gas desorption process. For different shale gas reservoirs, the differences of Langmuir volume and Langmuir pressure lead to distinct trend of gas content. The adsorption gas volume per unit bulk volume can be expressed in Equation (5): q q (5) where q is the adsorption gas per unit area of shale surface, kg m ; V is the mole volume under standard condition (0, 1atm), m mol; q is the adsorption volume per unit mass of shale, m kg ; V is the Langmuir volume, m 3 /kg; p is the Langmuir pressure, ; ρ is the density of shale core, kg m.

186 APPARENT GAS PERMEABILITY IN THE MATRIX AND FRACTURE SYSTEM 3.1. Apparent gas permeability in the shale matrix Some empirical models have been developed to account for slip-flow and Knudsen diffusion in the form of apparent permeability, including correlations developed from the Darcy matrix permeability (Ozkan et al., 2010); correlations developed based on flow mechanisms (Javadpour et al., 2007; Javadpour, 2009); correlations as a function of Knudsen number based on Beskok and Karniadakis s work (Sakhaee-Pour and Bryant, 2012; Beskok and Karniadakis); and semi-empirical analytical models by Moridis et al. in 2010 (Moridis et al., 2010). This research adopted Javadpour s method which considered slip flow, viscous flow and Knudsen diffusion (Javadpour, 2009). The apparent permeability is given as follows: k k 1 (6).. 1 (7) where is the tangential momentum accommodation coefficient which characterizes the slip effect. is a function of wall surface smoothness, gas type, temperature and pressure which varies in a range from 0 to 1. For simplification, in this paper, it is set as Apparent gas permeability in the shale fractures In this study, Knudsen diffusion and viscous flow are considered for gas flow in the frature system. The mass flux can be expressed as the summation of the two mechanisms. Therefore,

187 171 (8) or 1 (9) Using the form of conventional Darcy flow equation, the fracture apparent permeability can be expressed as: where k k 1 (10) D b (11). (12) where p is the fracture pressure, k is the initial fracture permeability, k is the apparent fracture permeability, b is the Klinkenberg coefficient for the fracture system, D is the Knudsen diffusion coefficient for the fracture system, is the fracture porosity. 4. NUMERICIAL SIMULATION MODEL FOR SHALE GAS RESERVOIRS WITH MsFHW 4.1. Model assumptions Assumptions are as follows: (1) the flow is isothermal;(2) there are only one component, one phase flow in the shale reservoir;(3) the gas diffuse quickly and can

188 172 achieve phase equilibrium instantaneously;(4) Formation rocks are incompressible and the porosity change due to rock deformation is ignored; (5) The gas in the shale reservoir can be assumed ideal gas with gas deviation factor equal to 1; (6) Gas adsorption-desorption kinetics obey Langmuir isotherm equation Mathematical model (1) Continuity equation In the dual porosity model, there exist two mass balance equations corresponding for fracture and matrix systems respectively as indicated by Warren and Root (Warren and Root, 1963). The diagram of dual porosity model is shown in the Figure 1. The continuity equation for the matrix and fracture system are as follows: In the matrix: In the fracture: ρ v Q (13) ρ v Q (14) The first term on the left side M is the mass accumulation term; the second term on the left side v is the flow vector term; and the right side Q is the source/sink term. In the matrix, there are adsorption gas and free gas co-existed. So, in the matrix, the mass accumulation term: Mϕ S ρ M ϕ S ρ 1 (15)

189 173 Based on the Equation of State (EOS) and Equation (5), Equation (15) can be transformed as follows: γφ (16) where γ. In the fracture system, only free gas existed, so the mass accumulation term can be described as follows: γφ (17) (2) Motion equation In the matrix and fractures, the permeability can be characterized using the modified Darcy s model. The motion equations are as followed: (18) (19) In the matrix, 1, where is shown in Equation (7). In the fracture, k mi 1 b f, where b f is shown in Equation (11). 3) Gas production from Multi-stage hydraulic fracturing horizontal well

190 174 In the horizontal well with multi-stage transverse hydraulic fractures, the fractures are the main flow channels for the fluid flow from the shale micro-fractures into the horizontal wellbore. So, the gas production of horizontal wellbore is the sum of the gas production from each hydraulic fractures. For gas production from one single fracture, we assumed each fracture as a horizontal well along the y direction. Peaceman s model was used in this study (Peaceman, 1983). The following equation assume the horizontal well is along the x direction: ) (20) expressed as: where is the bottom-hole pressure, and re is the effective radius and can be when, other (21) 4) Numerical simulation model By put the motion equations into continuity equation, the complete numerical simulation model was derived which is composed of flow equation, boundary conditions and initial condition. using the results of Guo et al. (2014b): Equation in the matrix system:

191 175 γφ γ p T (22) Equation in the fracture system: γφ γ p TQ (23) where T is the gas transfer between the matrix and fracture systems (Warren and Root, 1963; Kazemi et al., 1976) is represented in the Equation (24): T (24) where σ 4 is the crossflow coefficient between the matrix and fracture systems, and L, L, L are the fracture spacing in the x,y, z directions. In the fracture system, there exists a source term which flows into fracture from the matrix and a sink term which flows out of fracture into the wellbore. Q is the sum of gas production from the fractures of them multi-stage hydraulic fractured horizontal well. Initial condition: p p p Boundary condition for matrix: F n =0 ( 0) Boundary condition for fracture: F n 0 ( 0), p x,y,t p 5. SENSITIVITY ANALYSIS In order to conduct sensitivity analysis, we constructed a shale gas reservoir model with a volume of 2500 ft x 2000 ft x 300 ft, which is called synthetic model. The parameters

192 176 used in the synthetic model have been listed in the Table 2. In the middle of the reservoir, there is a horizontal well with six stages of hydraulic fractures which is shown in Figure 2. Sensitivity analysis in this paper was constructed based on the synthetic model and two categories of parameters have been studied on the production performance of shale gas reservoir: the reservoir parameters and the hydraulic fracture parameters. For the reservoir parameters, we have studied the effect of matrix permeability and matrix porosity on the production performance of the shale gas reservoir. For the hydraulic fracture parameters, two parameters have also been studied: fracture spacing and fracture half-length Effect of matrix permeability Shale gas reservoirs are typical unconventional reservoirs with ultra-low permeability from 10 to 1000 nanodarcies (10-6 md) (Cipolla et al., 2010). According to the literature review, it can be found that matrix permeability of shale gas reservoirs in the U.S. are in the range of 10 nd (1 nd =10-9 Darcy= 10-6 md) to 1000 nd. The permeability distribution is shown in Figure 3. Based on the reviewed range of matrix permeability, sensitivity analysis of matrix permeability on production performance of shale gas reservoirs has been conducted. In this study, we have considered five cases with matrix permeability close to the reviewed matrix permeability range. The matrix permeability of the five cases are: 10 nd, 200 nd, 500 nd, 800 nd, and 1000 nd. The effect of matrix permeability on cumulative gas production and gas production rate are shown in Figure 4. From this figure, it can be found that the effect of matrix permeability on shale gas production performance is not significant. Also, it can be found that in the early period, the matrix permeability barely has influence on gas production. In the late period of gas production, the difference began to appear. However, the difference

193 177 is not significant. When the matrix permeability changes from 10 nd to 1000 nd, the cumulative gas production in 20 years only increases from 2540 MMSCF to 2750 MMSCF, which is about 7.65% increase Effect of matrix porosity Sensitivity analysis of matrix porosity on production performance of shale gas reservoirs has also been conducted based on the reviewed range of matrix porosity, which is shown in Figure 5. It can be found that matrix porosity are in the range of 2% to 8%. So, in this study, we have considered five cases with matrix porosity close to this range. The matrix porosity of the five cases are: 2%, 4%, 6%, 8%, and 10%. The effect of matrix porosity on cumulative gas production and gas production rate are shown in Figure 6. From Figure 8, it can be found that the effect of matrix porosity on the production performance of shale gas reservoirs is also not very significant. When the matrix porosity changes from 2% to 10%, there is no significant change in the early and late production period. The cumulative gas production changes from MMSCF to MMSCF, which is 12% increase Effect of fracture spacing When there are multiple transverse fractures in shale gas reservoirs, it is important to consider the effect of fracture spacing. Cipolla et al. (2009) found that in order to achieve the commercial production rates and optimum depletion of shale gas reservoir, the fracture spacing should be minimized. However, when the fracture spacing is too small, the interaction between fractures will become more apparent, which is not good for production. Also, more fractures will require more investment to create fractures. So, it is important to study the effect of fracture spacing on shale gas production performance. Sensitivity

194 178 analysis of fracture spacing on production performance of shale gas reservoirs has been conducted in this paper. Due to limitations of the reservoir size, we only studied four cases with fracture spacing equal to 100 ft, 200 ft, 300 ft, and 400ft. The effect of fracture spacing on cumulative gas production and gas production rate are shown in Figure 7. From the figure, it can be found that the effect of fracture spacing on gas production performance is very significant. There is difference not only in the early period, but also in the late period. And when the fracture spacing changes from 100 ft to 400 ft, the cumulative gas production increases from MMSCF to MMSCF, which is about 52% increase. Also, from the results, we can find that the smaller the fracture spacing, the more the gas production is. This findings coincide with the results of Cipolla et al. (2009) Effect of fracture half-length Fracture half-length is the horizontal distance from horizontal wellbore to the end of hydraulic fracture. Fracture half-length is important in deciding production from multistage hydraulic fracturing horizontal well as this is the main high conductivity channels for fluid flow from the reservoir into the horizontal wellbore. Sensitivity analysis of fracture half-length on production performance of shale gas reservoirs has been conducted in this study. We have considered five possible cases with considering the limitation of the reservoir size. The fracture half-length of the five cases are: 100 ft, 200 ft, 300 ft, 400 ft, and 500 ft. The effect of fracture half-length on cumulative gas production and gas production rate are shown in Figure 8. From the figure, it can be found that the larger the fracture half-length, the higher the cumulative gas production and the gas production rate. This is because the hydraulic fractures are the main flow channels for fluid flow. When the fracture half-length changes

195 179 from 100 ft to 500 ft, the cumulative gas production changes from MMSCF to MMSCF for the 20 years gas production, which is about 42% increase. From the comparison of above four parameters, it can be found that hydraulic fracture parameters are the dominate factors in shale gas production with multi-stage hydraulic fracturing horizontal well. Matrix porosity is more dominant compared with matrix permeability which denotes the matrix is mainly the storage media instead of the fluid flow media. And fracture spacing is more dominate compared with fracture half-length. In order to optimize gas production from shale gas reservoirs, it is better to decrease the fracture spacing. However, it should also be noted that when the fracture spacing is decreased, more investment will be needed to create the fractures. So, the optimal plan will be the balance between technology and the economic investment. 6. CONCLUSIONS In this work we make an attempt to study the influence of various parameters on the production performance of ultra-low permeability shale gas reservoirs with multi-stage hydraulic fracturing horizontal well. A dual porosity model was constructed to simulation gas production from shale reservoirs with multi-stage hydraulic fracturing horizontal well with considering multiple flow mechanisms, including: viscous flow, slip flow, Knudsen diffusion, and gas desorption. A synthetic model has been constructed to analyze the effect of reservoir parameters and hydraulic fracture parameters on gas production performance based on the reviewed ranges of corresponding parameters.

196 180 According to the sensitivity analysis, it can be found that hydraulic fracture parameters are the dominate factors which affect the gas production. Fracture spacing is more dominant compared with the fracture half-length. And matrix porosity is more dominant compared with the matrix permeability. ACKNOWLEDGEMENTS This work is funded by RPSEA through the Ultra-Deepwater and Unconventional Natural Gas and Other Petroleum Resources program authorized by the U.S. Energy Policy Act of The authors also wish to acknowledge for the consultant support from Baker Hughes and Hess Company. NOMENCLATURE p = pressure gradient, Pa/m p pm K = fracture pressure, Pa =matrix pressure, Pa = permeability, m 2 kmi = intrinsic matrix permeability, m 2 kfi = initial fracture permeability, m 2 J J = mass flow rate, kg m s = mass flow caused by Knudsen diffusion, kg/(m 2 s) C = gas mole concentration, mol/ m 3 ν σ = velocity, m/s = crossflow coefficient between the matrix and fracture systems

197 181 μ rw re = Langmuir volume, m 3 /kg = Langmuir pressure, Pa = gas viscosity, Pa s = wellbore radius, m = effective radius, m q = mass production rate, m 3 /s qv = volume production rate, kg/s = fracture porosity, % = matrix porosity, % Γ Γ t = bottom hole pressure, Pa = average pressure in the fracture system, Pa = Outer boundary domain = Inner boundary domain = time, s x, y, z = cartesian coordinate L, L, L = fracture spacing in the x, y, z directions ρ = density of shale core, kg/m 3 q = adsorption gas per unit area of shale surface, kg m q = adsorption gas volume per unit mass of shale, m kg V = gas mole volume under standard condition, m mol = tangential momentum accommodation coefficient

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201 185 Table 1. Knudsen s Permeability Correction factor for Tight Porous Media (Guo et al., 2015). Model Correlation factor Klinkenberg Heid et al. Jones and Owens Sampath and Keighin Florence et al.. Civan Table 2. Parameters used in the synthetic model for production performance study. Parameters Barnett Shale case Unit Model dimensions 2500x2000x300 (LxWxH) (762x609.6x91.44) ft (m) Initial reservoir pressure 2400 (1.65x10 7 ) psi (Pa) Bottom-hole pressure 500 (3.45x10 6 ) psi (Pa) Production period 20 (6.31x10 8 ) year (s) Reservoir temperature 200 (93.3) o F ( o C) Gas viscosity 0.02 ( ) cp (Pa*s) Bulk density 158 ( ) lb/ft 3 (kg/m 3 ) Reservoir top depth 6800 (2072.6) ft (m) Langmuir pressure 650 (4.48 x10 6 ) psi (Pa) Langmuir volume 100 (0.0028) SCF/ton (m 3 /kg) Fracture conductivity 9 (1.22x10-14 ) md-ft (m 2 -m) Matrix permeability (6.125x10-19 ) md (m 2 ) Matrix porosity 0.06 fraction Horizontal wellbore length 1000 (304.8) ft (m)

202 186 Figure 1. Diagram of dual porosity model. Figure 2. Synthetic model for production performance study.