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Doctoral Dissertations Student Theses and Dissertations

2015

Flow mechanisms and numerical simulation of gas production Flow mechanisms and numerical simulation of gas production

from shale reservoirs from shale reservoirs

Chaohua Guo

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Department: Geosciences and Geological and Petroleum Engineering Department: Geosciences and Geological and Petroleum Engineering

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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

2015

Chaohua Guo

All Rights Reserved

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. 25-28 March 2014. 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.

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.

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.

vi

TABLE OF CONTENTS

PUBLICATION DISSERTATION OPTION.................................................................... iii 

ABSTRACT ....................................................................................................................... iv 

ACKNOWLEDGMENTS .................................................................................................. v 

LIST OF ILLUSTRATIONS ............................................................................................. xi 

LIST OF TABLES ............................................................................................................ xv

SECTION

1. INTRODCUTION ................................................................................................. 1 

1.1. STATEMENT AND SIGNIFICANCE OF THE PROBLEM ...................... 1 

1.2. RESEARCH OBJECTIVES AND SCOPE ................................................... 2 

2. LITERATURE REVIEW ...................................................................................... 8 

2.1. SHALE GAS ................................................................................................. 8 

2.1.1. Importance of Shale Gas Reservoirs.. ................................................. 9 

2.1.2. Pore Size Distribution of Shale Gas Reservoirs ................................ 11 

2.1.3. Gas Storage Forms in Shale Gas Reservoirs. .................................... 13

2.1.4. Permeability and Porosity of Shale Gas Reservoirs.. ........................ 14

2.1.5. Accumulation and Production Characteristics of Shale Gas Reservoirs .......................................................................................... 16 

2.2. GAS FLOW MECHANISMS IN NANO PORES ...................................... 18 

2.2.1. No-Slip Boundary Condition and Slip Boundary Condition ............ 18 

2.2.2. Knudsen Number. .............................................................................. 19 

2.2.3. Gas Flow Mechanisms in the Nano Pore .......................................... 22 

2.2.4. Simulation Methods for Gas Flow through Nano Pores ................... 26 

2.3. SHALE PERMEABILITY MEASUREMENT METHODS ...................... 27 

2.3.1. Permeability and Intrinsic Permeability ............................................ 27 

2.3.2. Permeability Measurement Methods. ................................................ 27 

2.3.2.1. Pulse-decay technique. ..........................................................28 

2.3.2.2. Canister desorption test ..........................................................29 

2.3.2.3. Crushed samples dynamic pycnometry .................................29 

2.4. APPARENT PERMEABILITY MODEL FOR GAS FLOW THROUGH NANO PORES OF SHALE GAS RESERVOIRS ................. 30 

2.4.1. Beskok and Karniadakis’ Model. ...................................................... 31 

vii

2.4.2. Javadpour’s Model ............................................................................ 32 

2.4.3. Florence’s Model. .............................................................................. 33 

2.4.4. Civan’s Model ................................................................................... 33 

2.4.5. Sakhaee-pour and Bryant’s Model .................................................... 34 

2.5. GAS FLOW MECHANISMS IN SHALE GAS RESERVOIRS ..................... 34 

2.5.1. Darcy Flow ........................................................................................ 36 

2.5.2. Flow in the Fracture .......................................................................... 37 

2.5.3. Diffusion ............................................................................................ 38 

2.5.4. Gas Adsorption/Desorption ............................................................... 38 

2.5.5. Stress, Water, and Temperature Sensitivity ...................................... 41 

2.6. SIMULATION MODEL FOR GAS PRODUCTION FROM SHALE GAS RESERVOIRS.................................................................................................. 42 

2.6.1. Equivalent Continuum Model ........................................................... 42 

2.6.2. Dual Porosity Model ......................................................................... 43 

2.6.3. Multiple Interaction Continua Method. ............................................. 45

2.6.4. Multiple Porosity Model ................................................................... 45 

2.6.5. Discrete Fracture Model .................................................................... 46 

2.7. CONCLUDING REMARKS ............................................................................ 47 

REFERENCES ........................................................................................................ 49

PAPER

PAPER I. STUDY ON GAS FLOW THROUGH NANO PORES OF SHALE GAS RESERVOIRS .................................................................................................................. 61 

ABSTRACT............................................................................................................. 61 

1. INTRODUCTION ............................................................................................... 62 

2. EXPERIMENTS FOR GAS FLOW THROUGH NANO MEMBRANES ........ 64 

2.1. SEM imaging of nano membranes .............................................................. 64 

2.2. Experimental Setup and Results .................................................................. 65 

3. SLIP EFFECT FACTOR DERIVATION AND APPARENT PERMEABILITY MODEL ............................................................................. 67 

3.1. Mechanisms for gas flow in nanotubes ....................................................... 67 

3.1.1. Apparent permeability of gas flow in a nano capillary ..................... 69 

4. MODEL VALIDATION AND COMPARISON ................................................ 71 

4.1. Model validation .......................................................................................... 71 

viii

4.2. Model comparison ....................................................................................... 72 

5. DISCUSSION ...................................................................................................... 74 

6. CONCLUSION .................................................................................................... 77 

ACKNOWLEDGEMENTS ..................................................................................... 78

Appendix A .............................................................................................................. 78 

REFERENCES ........................................................................................................ 81 

PAPER II. PRESSURE TRANSIENT AND RATE DECLINE NALYSIS FOR HYDRAULIC FRACTURED VERTICAL WELLS WITH FINITE CONDUCTIVITY IN SHALE GAS RESERVOIRS .................................................. 96 

ABSTRACT............................................................................................................. 96 

NOMENCLATURE ................................................................................................ 97 

INTRODUCTION ................................................................................................... 99 

MATHEMATICAL MODEL ................................................................................ 101 

Assumptions ..................................................................................................... 101 

Mathematical model ......................................................................................... 102 

Solution ..................................................................................................... 105 

TYPE CURVES AND ANALYSIS ...................................................................... 107 

Type curves ..................................................................................................... 107 

Sensitivity analysis ........................................................................................... 108 

CONCLUSIONS ................................................................................................... 110 

ACKNOWLEDGEMENTS ................................................................................... 111 

REFERENCES ...................................................................................................... 111 

PAPER III. MODELING OF GAS PRODUCTION FROM SHALE RESERVOIRS CONSIDERING MULTIPLE FLOW MECHANISMS ................... 121 

ABSTRACT........................................................................................................... 121 

INTRODUCTION ................................................................................................. 122 

PORE DISTRIBUTION AND GAS TRANSPORT PROCESS IN SGRs ........... 124 

MODEL CONSTRUCTION ................................................................................. 125 

Mass accumulation term ................................................................................... 126 

Flow vector term ............................................................................................... 129 

Source and Sink term ........................................................................................ 133 

Gas viscosity change ........................................................................................ 135 

MODEL VERIFICATION .................................................................................... 138 

EFFECT OF FLOW MECHANISMS ON SHALE GAS PRODUCTION .......... 139 

Effect of Gas adsorption ................................................................................... 139 

ix

Effect of non-Darcy flow .................................................................................. 140 

Effect of gas viscosity change .......................................................................... 140 

Effect of pore radius change ............................................................................. 141 

SENSITIVITY ANALYSIS .................................................................................. 142 

Effect of initial pressure ................................................................................... 142 

Effect of matrix Permeability and fracture permeability .................................. 143 

Effect of matrix porosity and fracture porosity ................................................ 143 

CONCLUSION ...................................................................................................... 144 

ACKNOWLEDGEMENTS ................................................................................... 144 

NOMENCLATURE .............................................................................................. 145 

REFERENCES ...................................................................................................... 146 

PAPER IV. NUMERICAL SIMULATION OF GAS PRODUCTION FROM SHALE GAS RESERVOIRS WITH MULTI-STAGE HYDRAULIC FRACTURING HORIZONTAL WELL .................................................................... 160 

ABSTRACT........................................................................................................... 160 

1. INTRODUCTION ............................................................................................. 161 

2. FLUID FLOW MECHANISMS IN TIGHT SHALE RESERVOIRS .............. 166 

2.1. Viscous flow .............................................................................................. 166 

2.2. Knudsen diffusion ..................................................................................... 167 

2.3. Slip flow .................................................................................................... 168 

2.4. Gas adsorption and desorption .................................................................. 168 

3. APPARENT GAS PERMEABILITY IN THE MATRIX AND FRACTURE SYSTEM .................................................................................. 170 

3.1. Apparent gas permeability in the shale matrix .......................................... 170 

3.2. Apparent gas permeability in the shale fractures ...................................... 170 

4. NUMERICIAL SIMULATION MODEL FOR SHALE GAS RESERVOIRS WITH MsFHW ..................................................................... 171 

4.1. Model assumptions .................................................................................... 171 

4.2. Mathematical model .................................................................................. 172 

5. SENSITIVITY ANALYSIS .............................................................................. 175 

5.1. Effect of matrix permeability .................................................................... 176 

5.2. Effect of matrix porosity ........................................................................... 177 

5.3. Effect of fracture spacing .......................................................................... 177 

5.4. Effect of fracture half-length ..................................................................... 178 

x

6. CONCLUSIONS ............................................................................................... 179 

ACKNOWLEDGEMENTS ................................................................................... 180 

NOMENCLATURE .............................................................................................. 180 

REFERENCES ...................................................................................................... 182

SECTION

3. CONCLUSION .................................................................................................... 190 

VITA .............................................................................................................................. 194 

xi

LIST OF ILLUSTRATIONS

Figure 1.1. Research scope of this project. ......................................................................... 3 

Figure 1.2. Research tasks of this project. .......................................................................... 5 

Figure 2.1. Energy production by Fuel in the U.S., 1980-2040 (EIA 2013). ................... 10 

Figure 2.2. Dry natural gas production in the U.S. (EIA, 2013). ...................................... 10 

Figure 2.3. Comparison between conventional gas reservoir (a) and shale gas reservoir (b). (Modified from Javadpour et al., 2007). .................................. 12 

Figure 2.4. Variations in nanopore morphology in organic matter. (Loucks et al., 2009). ..................................................................................... 12 

Figure 2.5. Gas storage forms in shale gas reservoirs. Modified from Javadpour et al. (2007). ............................................................................................................ 14 

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. ................................................................................................... 14 

Figure 2.7. Survey of permeability and porosity for shale gas plays in North America In courtesy of Wang and Reed (2009). ........................................... 15 

Figure 2.8. Formation Process of Shale Gas Reservoir. ................................................... 17 

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). ............... 17 

Figure 2.10. Comparison between no-slip and slip boundary condition. ......................... 18 

Figure 2.11. Flow regime classification based on Knudsen number (Heidemann et al., 1984; Karniadakis et al. 2006; Ziarani and Aguilera, 2012). ............... 21 

Figure 2.12. Langmuir Isotherm Curves for Five Typical Shale Gas Reservoirs. ............ 40

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. ................................................................................. 87

Figure 2. SEM images of experimental membranes with pore size 20 nm, 55 nm, and 100 nm. ................................................................................................... 87

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. ........ 88

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. .............................................................. 89

xii

Figure 6. Ratio of slip length to pore radius vs. average pressure for nano membranes of different sizes. ........................................................................ 89

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. ............................................................ 90

Figure 9. A schematic model for simulating gas flow in nanotubes. ................................ 91

Figure 10. Pressure drop versus flux for different values of σ. ........................................ 91

Figure 11. Pressure drop versus flux for oxygen in a nanotube with a diameter of 235 nm and 220 nm. ...................................................................................... 92

Figure 12. Comparison of Kapp versus radius between new apparent permeability and that of Florence and Javadpour. .............................................................. 92

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. ............................................ 93

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. ............................................ 93

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. ......................................... 94

Figure 16. Ratio of gas permeability to intrinsic liquid permeability vs. average pressure for nano membranes of different sizes. ........................................... 95

PAPER II

Figure 1. Schematic diagram of hydraulic fractured vertical well. The top and bottom boundary are closed. ........................................................................ 113

Figure 2. Type curve for hydraulic fractured vertical wells with finite conductivity in shale gas reservoirs under different kinds of boundary conditions. ........ 114

Figure 3. Effect of adsorption coefficient on PPR (pseudo pressure response) and RDR (Rate decline response). ...................................................................... 115

Figure 4. Effect of storage capacity ratio on PPR and RDR. .......................................... 116

Figure 5. Effect of channeling factor on PPR and RDR. ................................................ 117

Figure 6. Effect of fracture conductivity on PPR and RDR. .......................................... 118

Figure 7. Effect of fracture skin factor on PPR and RDR. ............................................. 119

Figure 8. Effect of stress sensitivity on PPR and RDR.................................................. 120

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. ................................................................................................. 150

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). ............. 150

Figure 3. Idealization of the heterogeneous porous medium as DPM. ........................... 151

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. .......................................................................................... 151

Figure 5. Gas flow mechanisms in a nano pore. Red solid dots represent Knudsen diffusion, while blue ones represent viscous flow. ...................................... 152

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. ................................................................................ 152

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. ........................................................................................................... 153

Figure 9. Real reservoir and simplified 2-D reservoir simulation model. ...................... 153

Figure 10. The effect of adsorption on gas production. (a) production rate vs. time; (b) cumulative production vs. time. ............................................................. 154

Figure 11. Matrix and fracture permeability change with time. (a) matrix permeability vs. time; (b) fracture permeability vs. time. ........................... 154

Figure 12. Effect of Non-Darcy flow on gas production. (a) production rate vs. time; (b) cumulative production vs. time. .............................................. 155

Figure 13. Effect of gas viscosity change on gas production. (a) production rate vs. time; (b) cumulative production vs. time. .............................................. 155

Figure 14. Effect of pore radius increase due to gas desorption on gas production. (a) production rate vs. time; (b) cumulative production vs. time. ............... 156

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. ................................ 156

xiv

Figure 16. Effect of initial reservoir pressure on gas production. (a) production rate vs. time; (b) cumulative production vs. time. .................................................... 157

Figure 17. Effect of matrix permeability on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time. ............... 157

Figure 18. Effect of fracture permeability on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time. ............... 158

Figure 19. Effect of matrix porosity on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time. ............... 158

Figure 20. Effect of fracture porosity on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time. ............... 159

PAPER IV

Figure 1. Diagram of dual porosity model. ..................................................................... 186

Figure 2. Synthetic model for production performance study. ....................................... 186

Figure 3. Frequency distribution diagram of shale matrix permeability. ....................... 187

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. ................................................................................................. 187

Figure 5. Frequency distribution diagram of shale matrix porosity. ............................... 188

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. ................................................................................................. 188

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. ................................................................................................. 189

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. ................................................................................................. 189

xv

LIST OF TABLES

Table 2.1. Knudsen’s Permeability Correction factor for Tight Porous Media (Modified from Ziarani and Aguilera (2012)). .............................................. 25

PAPER I

Table 1. Properties of AAO nano membranes used in the experiment. ............................ 84

Table 2. Experimental results for different kinds of nano membranes. ............................ 84

Table 3. Knudsen number and flow regimes classifications for porous media. ............... 85

Table 4. Knudsen’s permeability correction factor for tight porous media. ..................... 85

Table 5. Nanotube characteristics [20]. ............................................................................ 86

Table 6. Model dimensions and fluid properties. ............................................................. 86

PAPER II

Table 1. Definitions of the dimensionless variables. ...................................................... 113

PAPER III

Table 1. Parameters used in the simulation model. ........................................................ 149

PAPER IV

Table 1. Knudsen’s Permeability Correction factor for Tight Porous Media (Guo et al., 2015). ................................................................................................. 185

Table 2. Parameters used in the synthetic model for production performance study. .... 185

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.

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 multiple-

mechanism 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 experiment-

simulation 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.

1.2. 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

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.

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.

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

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

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.

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 0.0025 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 50-600 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.

9

2.1.1. 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).

10

Figure 2.1. Energy production by Fuel in the U.S., 1980-2040 (EIA 2013).

Figure 2.2. Dry natural gas production in the U.S. (EIA, 2013).

11

2.1.2. 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.

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

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.

2.1.3. 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.

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.

2.1.4. 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

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

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).

2.1.5. 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.

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).

18

2.2. 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.

2.2.1. 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 2.10. 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 2.10. Comparison between no-slip and slip boundary condition.

19

2.2.2. 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 (1.3805 10 / ), 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)

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.

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 2.11.

Figure 2.11. 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

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 0.001 (Sakhaee-Pour and Bryant, 2012; Civan, 2010).

2.2.3. 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

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/(m2·s))， is the intrinsic

permeability of the nano capillary (m2)，p is the pressure of the nano capillary (Pa)，and

is the gas density (kg/ m3).

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/(m2·s))， is the gas mole

concentration (mol/m2)， is the gas molar mass (kg/mol), is the Kundsen diffusion

coefficient (m2/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

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 0.001 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 Correlation factor

Klinkenberg (1941) 4 ⁄

Heid et al. (1950) 11.419 .

Jones and Owens (1980) 12.639 .

Sampath and Keighin (1982) 13.851 ⁄ .

Florence et al. (2007) ⁄ .

Civan (2010) 0.0094 ⁄ .

26

2.2.4. 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 1935. 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-

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.

2.3. SHALE PERMEABILITY MEASUREMENT METHODS

2.3.1. 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 (cm2), 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).

2.3.2. 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

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).

2.3.2.1. 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,

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).

2.3.2.2. 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.

2.3.2.3. 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.

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.

2.4. 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

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).

2.4.1. 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

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.

2.4.2. 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):

33

.

(18)

1.

1 (19)

The apparent permeability provided by Javadpour is expressed in Equation (19):

.

(20)

2.4.3. 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

≪1 4 (21)

2.4.4. 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)

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.

2.4.5. 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:

0.8453 5.4576 0.1633 (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

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

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.

2.5.1. 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, m2; is the

Klinkenberg coefficient, MPa; is the gas viscosity, Pa.s; is the average gas pressure,

MPa.

37

2.5.2. 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/m3); is the permeability

(m2). 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

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.

2.5.3. 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.

2.5.4. 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

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 (-

47.22 0C). 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.

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.

0 1000 2000 3000 4000 50000

50

100

150

200

250

Ads

orb

ed

ga

s co

nte

nt (

scf/

ton)

Pressure (psi)

Marcellus shale Eagleford shale New albany shale Barnett shale Haynesville shale

Figure 2.12. Langmuir isotherm curves for five typical shale gas reservoirs.

41

2.5.5. 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

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).

2.6. 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:

2.6.1. Equivalent Continuum Model. The Equivalent continuum model is first

proposed by Snow (1968). Many different types of equivalent continuum models have been

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).

2.6.2. 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.

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-porosity-

dual-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

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.

2.6.3. 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.

2.6.4. 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

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.

2.6.5. 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)

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).

2.7. 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 0.0025 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

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.

49

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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

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].

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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

1935. 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

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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.

65

2.2. 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, cm2; p is the

ambient pressure, 0.1 MPa; Q is the gas flow rate under ambient pressure, cm3/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

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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.

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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

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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/(m2·s))， k is the intrinsic

permeability of the nano capillary (m2)，p

is the pressure of the nano capillary (Pa)，and is the gas density (kg/ m3).

(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/(m2·s))， is the gas mole

concentration (mol/m2)， is the gas molar mass (kg/mol), is the Kundsen diffusion

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coefficient (m2/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 8.314 . ⁄ ⋅   , 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.

3.1.1. 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):

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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]:

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(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 0.89. Then, we were able to more accurately investigate

the best-fitting , as shown in Fig. 10(b). We estimated the best-fitting to be 0.82. Then,

the new model was tested with the experimental data using different PTDs and gases.

4.1. 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

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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.

4.2. 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):

73

.

(15)

1.

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

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

FromEq. 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

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 (1.3808×10-23J/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.

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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.

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.

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 2005. 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

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:

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):

   1

1 ⁄

1

(A.9)

The mole flux can be expressed as:

1

 

(A.10)

where   is the apparent permeability (m2)， is the Klinkenberg coefficient or slip

effect factor [29, 30], and is the comprehensive coefficient that needs to be fitted.

81

REFERENCES

[1] Wang, Z. and A. Krupnick. 2013. A Retrospective Review of Shale Gas Development in the United States.

[2] Ziarani, A. S., and Aguilera, R. 2012. Knudsen’s permeability correction for tight porous media. Transport in porous media, 91(1), 239-260.

[3] Javadpour, F., D. Fisher, et al. 2007. Nanoscale Gas Flow in Shale Gas Sediments. Journal of Canadian Petroleum Technology. 46(10).

[4] Javadpour, F. 2009. Nanopores and Apparent Permeability of Gas Flow in Mudrocks (Shales and Siltstone)." Journal of Canadian Petroleum Technology. 48(8): 16-21.

[5] Curtis, M. E., R. J. Ambrose, et al. 2010. 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. 2009. Morphology, Genesis, and Distribution of Nanometer-Scale Pores in Siliceous Mudstones of the Mississippian Barnett Shale. Journal of Sedimentary Research. 79(12): 848-861.

[7] Hornyak, G. L., H. F. Tibbals, et al. 2008. Introduction to Nanoscience and Nanotechnology. CRC Press, Boca Raton, FL.

[8] Bird, G. A. 1994. 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): 761-767.

[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): 1907-1920.

[11] Karniadakis, G. and A. Beskok. 2002. 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), 43-77.

[14] Klinkenberg, L. J. 1941. The Permeability of Porous Media to Liquids and Gases, American Petroleum Institute.

[15] Shabro, V., C. Torres-Verdin, et al. 2011. 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.

82

[16] Shabro, V., C. Torres-Verdin, et al. 2012. 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. SPE Annual Technical Conference and Exhibition. San Antonio, Texas, USA, Society of Petroleum Engineers.

[17] Civan, F. 2010. A Review of Approaches for Describing Gas Transfer through Extremely Tight Porous Media. AIP Conference Proceedings.

[18] Jinno, K., A. Kawamura, et al. 1993. Real-time Rainfall Prediction at Small Space-time Scales using a Two-dimensional Stochastic Advection-diffusion Model. Water Resour. Res. 29(5): 1489-1504.

[19] Rutherford, S. W., & Do, D. D. (1997). Review of time lag permeation technique as a method for characterisation of porous media and membranes. Adsorption, 3(4), 283-312.

[20] Cooper, Sarah M., et al. "Gas transport characteristics through a carbon nanotubule." Nano Letters 4.2 (2004): 377-381.

[21] Stops, D. W. 1970. The Mean Free Path of Gas Molecules in the Transition Regime. Journal of Physics D: Applied Physics. 3(5): 685.

[22] Kast, W. and C. R. Hohenthanner. 2000. Mass Transfer within the Gas-phase of Porous Media. International Journal of Heat and Mass Transfer. 43(5): 807-823.

[23] Gilron, J. and A. Soffer. 2002. Knudsen Diffusion in Microporous Carbon Membranes with Molecular Sieving Character. Journal of Membrane Science. 209(2): 339-352.

[24] Freeman, C. M., Moridis, G. J., & Blasingame, T. A. (2011). A numerical study of microscale flow behavior in tight gas and shale gas reservoir systems. Transport in porous media, 90(1), 253-268.

[25]Florence, F. A., J. Rushing, et al. 2007. Improved Permeability Prediction Relations for Low Permeability Sands. Rocky Mountain Oil & Gas Technology Symposium. Denver, Colorado, U.S.A., Society of Petroleum Engineers.

[26] Andrade, J., Civan, F., & Devegowda, D. (2011). Design and Examination of Requirements for a Rigorous Shale-Gas Reservoir Simulator Compared to Current Shale-Gas Simulators. Paper SPE 144401 presented at the SPE North American Unconventional Gas Conference and Exhibition, The Woodlands, Texas, 14–16 June.

[27] Islam, A., & Patzek, T. (2014). Slip in natural gas flow through nanoporous shale reservoirs. Journal of Unconventional Oil and Gas Resources, 7, 49-54.

[28] Bhatia, s., Nicolson, D., 2003. Molecular transport in nanopores. J. Chem. Phys. 119, 1719-1730.

[29] Kaluarachchi, J. (1995). "Analytical Solution to Two-Dimensional Axisymmetric Gas Flow with Klinkenberg Effect." Journal of Environmental Engineering 121(5): 417-420.

[30] Wu, Y.-S., K. Pruess, et al. (1998). "Gas Flow in Porous Media With Klinkenberg Effects." Transport in Porous Media 32(1): 117-137.

83

[31] Heid, J.G., et al. 1950. Study of the Permeability of Rocks to Homogeneous Fluids," in API Drilling and Production Practice. 230-246

[32] Jones, F.O. and Owens, W.W. 1979. A Laboratory Study of Low Permeability Gas Sands. Paper SPE 7551 presented at the 1979 SPE Symposium on Low-Permeability Gas Reservoirs, May 20- 22, 1979, Denver, Colorado.

[33] Sampath, K., Keighin, C.W. 1982. Factors affecting Gas Slippage in Tight Sandstones of Cretaceous Age in the Uinta Basin. J. Pet. Technol. 34(11), 2715–2720.

84

Table 1. Properties of AAO nano membranes used in the experiment.

Membrane type PTD (nm) Pore density ( nm-2) Intrinsic permeability

(mD) 20 nm

membrane 20 9.42E-05 3.60E-04

55 nm membrane

55 5.49E-05 0.01

100 nm membrane

100 2.64E-05 0.078

Table 2. Experimental results for different kinds of nano membranes.

20 nm 55 nm 100 nm Inlet

pressure Outlet

pressure Flow rate

Inlet pressure

Outlet pressure

Flow rate

Inlet pressure

Outlet pressure

Flow rate

218149.9 101008.1 2.95E-17 121623.4 101008.1 1.06E-16 374936.5 101008.1 4.55E-16

246280.5 101008.1 3.47E-17 173816.6 101008.1 1.85E-16 414098.7 101008.1 5.45E-16

315296.9 101008.1 4.41E-17 204429.3 101008.1 2.14E-16 438092.4 101008.1 5.95E-16

366869.6 101008.1 4.95E-17 221528.3 101008.1 2.29E-16 466016.2 101008.1 6.46E-16

382313.9 101008.1 5.48E-17 233318.3 101008.1 2.38E-16 543513.1 101008.1 7.76E-16

420028.2 101008.1 5.84E-17 240419.9 101008.1 2.41E-16 628042.8 101008.1 9.02E-16

447262.4 101008.1 6.09E-17 247521.5 101008.1 2.47E-16 714089.3 101008.1 1.02E-15

506626.2 101008.1 6.57E-17 241592 101008.1 2.42E-16 770902.0 101008.1 1.09E-15

533308.9 101008.1 6.78E-17 218356.7 101008.1 2.16E-16 843365.8 101008.1 1.15E-15

583709.5 101008.1 7.16E-17 936720.7 101008.1 1.23E-15

503868.3 101008.1 6.56E-17 623905.9 101008.1 8.95E-16

439264.5 101008.1 6.03E-17 523104.7 101008.1 7.41E-16

375626.0 101008.1 5.45E-17 413753.9 101008.1 5.48E-16

287993.7 101008.1 4.38E-17

166301.4 101008.1 1.9E-17 NOTE: pressure unit is Pa. Flow rate unit is m3/s.

85

Table 3. Knudsen number and flow regimes classifications for porous media.

Flow regime Knudsen number

Model to be applied

Continuum (Viscous) flow Kn<0.01 Darcy's equation for laminar flow and Forchheimer's equation for turbulent

flow

Slip flow 0.01<Kn<0.1 Darcy's equation with Klinkenberg or

Knudsen's correction

Transition flow 0.1<kn<10

Darcy's law with Knudsen's correction can be applied. Alternative method is Burnett's equation with slip boundary

conditions

Free Molecular flow Kn>10 Knudsen's diffusion equation Alternative

methods are DSMC and Lattice Boltzmann

NOTE: Modified from Ziarani and Aguilera [2]

Table 4. Knudsen’s permeability correction factor for tight porous media.

Model Correlation factor

Klinkenberg [14] 4 ⁄

Heid et al. [29] 11.419 .

Jones and Owens [30] 12.639 .

Sampath and Keighin [31] 13.851 ⁄ .

Florence et al. [25] ⁄ .

Civan [17] 0.0094 ⁄ . NOTE: Modified from Ziarani and Aguilera [2]

Units: (psi), (md), (psi), r (nm), (psi), (nm) and (fraction)

86

Table 5. Nanotube characteristics [20].

Avg. PTD (nm) Avg. pore density (×1012 m-2) Porosity

212 8 0.28

235 10.2 0.22

220 7.2 0.27

169 9.4 0.21

Table 6. Model dimensions and fluid properties.

Flow parameters Nanotube

Length L 60 μm

Diameter D 235 nm, 220 nm

Outlet Pressure Pout 4.8 kPa

Pressure Drop ∆P=Pin-Pout

100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 1200.0 torr

Absolute Viscosity μ 2.22 10-5 Pa.s

Gas Molecular Weight Oxygen (0.032 kg/mol), Argon (0.039948 kg/mol)

Temperature 300 K

87

fracture

matrix

free gas

adsorption gas

macro‐scale micro‐scale

Kerogen

dissovedgas

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.

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.

20 30 40 50 60 70 80 90

1E-18

1E-17

1E-16

1E-15

20 nm pore 55 nm pore100nm pore

Q (

m3 /s

)

Pavg

(psi)

Figure 4. Gas flow rate vs. the average pressure for nano membranes of different sizes.

89

20 30 40 50 60 70 80 900.1

1

510

25

50

75

9095

99

99.9

20 nm pore 55 nm pore100nm pore

Q1/

Q0

Pavg

(psi)

Figure 5. Ratio of real gas flow rate to intrinsic flow rate vs. average pressure for different kinds of nano membranes.

20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

20 nm pore 55 nm pore100nm pore

l s/R

Pavg

(psi)

Figure 6. Ratio of slip length to pore radius vs. average pressure for nano membranes of different sizes.

90

20 30 40 50 60 70 80 900.00

0.01

0.02

0.03

0.04

0.05

Continuum (Viscous) flow Kn<0.01

20 nm pore 55 nm pore100nm pore

Kn

Pavg

(psi)

Slip flow 0.01<Kn<0.1

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.

91

Figure 9. A schematic model for simulating gas flow in nanotubes.

0 10 20 30 40 50 600

200

400

600

800

1000

1200

1400

Del

ta P

(to

rr)

Flux (mol/m2s)

Experiment data Argon D=235nm

=0.71 =0.75 =0.64 =0.89

0 10 20 30 40 50 600

200

400

600

800

1000

1200

1400

De

lta P

(to

rr)

Flux (mol/m2s)

Experiment data Argon D=235 nm

0.78 0.82 0.85

(a) (b)

Figure 10. Pressure drop versus flux for different values of σ.

92

0 10 20 30 40 50 600

200

400

600

800

1000

1200

De

lta P

(to

rr)

Flux (mol/m2s)

Experiment data Oxygen D=235 nm

Numerical result

0 5 10 15 20 25 30 35 40 45 50 550

200

400

600

800

1000

1200

Del

ta P

(to

rr)

Flux (mol/m2s)

Experiment data Argon D=220nm

Numerical result

Figure 11. Pressure drop versus flux for oxygen in a nanotube with a diameter of 235 nm and 220 nm.

1E-10 1E-9 1E-8 1E-7 1E-6 1E-5

1E-5

1E-4

1E-3

0.01

0.1

1

10

Kap

p (m

2)

r (m)

Proposed method Javadpour's method Florence's method

Oxygen T=300 KPressure difference: 1000 torr

Figure 12. Comparison of Kapp versus radius between new apparent permeability and that of Florence and Javadpour.

93

0 10 20 30 40 50 600

200

400

600

800

1000

1200

Experimental gas: Oxygen T=300 KPore diameter=235 nm

Experimental data Proposed method Javadpour's method K/HP Analytical method

Del

ta P

(to

rr)

Flux (mol/m2s)

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.

0 200 400 600 800 1000 12000.00

0.05

0.10

0.15

0.20

0.25

Kg (m

2)

Delta P (torr)

Proposed method Javadpour's method Florence's method

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.

94

10 1 0.1 0.01 1E-30.1

1

510

25

1E-10 1E-9 1E-8 1E-7 1E-6 1E-50.1

1

510

25

J g/J l

Kn

J g/J l

r (m)

10 1 0.1 0.01 1E-3

0

20

40

60

80

1001E-10 1E-9 1E-8 1E-7 1E-6 1E-5

0

20

40

60

80

100

Kn

Add

ition

al m

ole

flux/

Tot

al m

ole

flux

(%)

r (m)

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.

95

20 30 40 50 60 70 80 900

20

40

60

80

100

120

140

20 nm Experiment 55 nm Experiment 100 nm Experiment

Kg/

Kl

Pavg

(psi)

Figure 16. Ratio of gas permeability to intrinsic liquid permeability vs. average pressure for nano membranes of different sizes.

96

PAPER II. PRESSURE TRANSIENT AND RATE DECLINE NALYSIS FOR HYDRAULIC FRACTURED VERTICAL WELLS WITH FINITE

CONDUCTIVITY IN SHALE GAS RESERVOIRS

Chaohua Guoa, Jianchun Xub, Mingzhen Weia , Ruizhong Jiangb a Missouri University of Science and Technology, Department of Petroleum Engineering,

Rolla, MO, USA, 65401 b China University of Petroleum (East China), Department of Petroleum Engineering,

Qingdao, CHINA, 266555 (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

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 (m3/Pa)

cg =Gas compressibility (Pa−1)

Cf =fracture conductivity

D =Diffusion coefficient (m2/s)

h =Reservoir thickness (m)

kfh =Horizontal permeability of natural fracture (m2)

kfv =Vertical permeability of natural fracture (m2)

kf1 =Artificial fracture permeability(m2)

K0() =Modified Bessel function of second kind of order zero

L =Characteristic length (m)

LfLi,; LfRi =Lengths of the left and right wings of the ith fracture (m)

n =Number of segments on the wing of each fracture

p0 =Reference pressure (Pa)

pi =Initial reservoir pressure (Pa)

psc =Standard state pressure (Pa)

q =Production rate from continuous point source (m3/s)

iq =Flux per unit length of discrete segment(i,j), (m2/s)

98

qsc =Production rate (m3/s)

r =Radial distance in natural fracture system (m)

R =External radius of matrix block (m)

Sf =Fracture skin factor, dimensionless

t =Time (s)

T =Reservoir temperature (K)

Tsc =Standard state temperature (K)

Wf =Fracture width(m)

u =Laplace transform parameter, dimensionless

V =Average volumetric gas concentration in the fracture (s-m3/ m3)

VE =Equilibrium volumetric gas concentration in pseudo-steady diffusion

(s-m3/ m3)

Vi =Initial volumetric gas concentration in the fracture (s-m3/ m3)

x,y,z =Space coordinates in Cartesian coordinates (m)

xw,yw,zw =Space coordinates of continuous point source (m)

Z =Real-gas compressibility factor, dimensionless

Zi =Real-gas compressibility factor under the initial condition,

dimensionless

α =Adsorption index, dimensionless

λ =Inter-porosity flow coefficient, dimensionless

ω =Storativity ratio, dimensionless

τ =Sorption time constant (s)

ξ =Integration variable

99

μ =Gas viscosity, (Pa·s)

μi =Initial gas viscosity, (Pa·s)

φf =Natural fracture porosity, fraction

ψ =Pseudo-pressure (Pa)

ψi =Initial pseudo-pressure (Pa)

_ =Laplace domain

D =Dimensionless

f =Natural fracture

i =Initial condition

m =Matrix

sc =Standard state

w =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 10-30 % 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

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

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 differentfvfh kk ;

(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;

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 scq in standard condition;

(8)The effect of gravity and capillary pressure can be ignored.

Mathematical model

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.

t

V

T

p

t

p

Z

pc

z

p

Z

pk

zy

p

Z

pk

yx

p

Z

pk

x sc

scffgff

fffv

fffh

fffh

(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:

0

2fp

f f p

pp dp

Z

(2)

Substitute Eq. (2) into (1), we can obtain:

2f f f f scfh fh fv f i gfi

sc

p T Vk k k c

x x y y z z t T t

(3)

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：

2

2

6E f

V DV V

t R

(4)

Also, we defined dimensionless variables as shown in Table 1.

The dimensionless forms for Eq. (3) and (4) can be given as follows:

2 2 2

2 2 21fD fD fD fD D

D D D D D

V

x y z t t

(5)

DfDED

D

DVV

t

V

(6)

Taking Laplace transformation to tD, Eqs. (5) and (6) become:

Then Eqs. (5) and (6) can be changed to:

2 2 2

2 2 21fD fD fD

DfDD D D

u uVx y z

(7)

ED DfDD

V VuV =

(8)

According to Langmuir isotherm adsorption equation:

L f L iED fD E i

L f L i

V p V pV L V V L

p p p p

(9)

Taking Laplace transformation to tD, Eq. (9) becomes:

2sc sc i iL L

ED fDfDfh sc iL f L i

p q T ZV pV

k hT pp p p p

(10)

With the definition of Adsorption index in Table 1, we can get:

fDfDEDV (11)

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:

2 2 2

2 2 2( )fD fD fD

fDD D D

f ux y z

(12)

u

uuuf

1 (13)

Converting to spherical coordinates:

fDD

fDD

DD

ufr

rrr

2

2

1 (14)

Inner boundary conditions:

0lim 1D

fDD

rD

rr

(15)

Three forms of outer boundary conditions have been considered in this paper:

Infinite outer boundary (IOB):

0D

fDr

(16)

Constant pressure outer boundary (CPOB):

0f D (17)

Closed outer boundary (COB):

0fD

Dr

(18)

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:

~

0

~0

0 0

0

~1

0 0

1

1IOB

1CPOB

1COB

scD

sc fh

eDsc

f D Dsc fh eD

eDsc

D Dsc fh eD

p T qK r f u

T k h u

K r f up T qK r f u I r f u

T k h u I r f u

K r f up T qK r f u I r f u

T k h u I r f u

(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

ratescq . 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 fx and fracture

width is fW . Fracture permeability is

1fk ; Assume that the gas only flows into the wellbore

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:

106

1 2 201

1

2fD D DfDq u K x y f u d

(20)

Here u

qq fD

fD

Since the fracture is symmetric, following flow rate relationship can be obtained:

udxq DfD

11

0 (21)

Consider the skin effect by hydraulic fracturing, the fracture pseudo-pressure fD

can be related with formation pseudo-pressure D using following equation:

, , ,fD DD D D D ffDx u x y u q x u S (22)

Ignoring the compressibility of fluids in the fracture, following relationship can be

obtained after combining boundary condition:

Dx

DfDDfD

DDDfwD ddxquxuC

uyx0 01 ,0,

(23)

According to Eq. (20) and (22), above equation can be written in following form:

1

0 00

0 0

1,

2D

wD D DfD

x

f D DfD fDfD fD

q u K x f u K x f u d

S q q dx d xC uC

(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 Dx

. Dix is the center point of the i th discrete unit. Dix 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:

107

1

0 01

22

1

1

2

0.58

Di

Di

n x

Dj DjwD ffDi fDjxi

jD

DjD D D DfDi fDjifD fD

q K x f u K x f u d S q

xq x x x i x q x

C uC

(25)

And the flow rate equation can be discretilized as follows:

u

uqxn

ifDiD

1

1

(26)

There are 1n unknown variables in Eq. (25) and (26), that is wD and fDiq （with

1, 2, ...,i n ）. By solving these equations, in Laplace space, we can obtain the bottomhole

pseudo-pressure wD and the flow rate distribution in discrete units fDiq 。

Consider the effect of wellbore storage, according to Duhamel's principle, the

bottomhole pressure response is:

wDD

wDwD

Cu

21

(27)

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

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

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

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.

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 2005. The authors also wish to acknowledge for the consultant

support from Baker Hughes and Hess Company.

REFERENCES

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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 pressure-transient responses of fractured horizontal wells in unconventional shale reservoirs. SPE Reservoir Evaluation & Engineering 14(06), 663-676.

Bumb AC, McKee CR (1988) Gas-well testing in the presence of desorption for coalbed methane and devonian shale. SPE formation evaluation 3(01), 179-185.].

Cinco-Ley H, Samaniego FV (1981) Transient pressure analysis for fractured wells. Journal of petroleum technology 33(9), 1749-1766.

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Cohen DS, Murray JD (1981) A generalized diffusion model for growth and dispersal in a population. Journal of Mathematical Biology 12(2), 237-249.

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), 16-21.

Javadpour F, Fisher D, Unsworth M (2007) Nanoscale gas flow in shale gas sediments. Journal of Canadian Petroleum Technology 46(10), 55-61.

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), 2974-2981.

Rosa AJ, Carvalho RD (1989) A Mathematical Model for Pressure Evaluation in an lnfinite-Conductivity Horizontal Well. SPE Formation Evaluation 4(04), 559-566.

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), 271-299.

Yin Y (1991) Adsorption isotherm on fractally porous materials. Langmuir 7(2), 216-217.

113

Table 1. Definitions of the dimensionless variables.

Dimensionless pseudo-pressure fiscsc

scfhfD Tqp

hTk

Dimensionless pseudo-time: 2L

tkt fh

D ;

sc

fhgfiif q

hkc

2

Dimensionless coordinate: L

xx D ，

L

yy D ，

fv

fhD k

k

L

zz

Dimensionless flow rate in the fracture system;

sc

DDffD q

txLqq

,2

Dimensionless gas concentration: iD VVV

Dimensionless storage capacity ratio

gfiif c

Channeling coefficients 2L

k fh

;

D

R2

2

6

Dimensionless fracture width L

WW f

fD

Dimensionless fracture conductivity Lk

WkC

fh

fffD

1

Adsorption coefficient sc sc i iL L

fh sc iL f L i

p q T ZV p

k hT pp p p p

Dimensionless wellbore storage coefficient: i

D Lh

CC

/2 2

x

z

y

fxer h x

y

1Dx 2Dx 3Dx Dnx1Dnx

1Dx 2Dx Dnx

Figure 1. Schematic diagram of hydraulic fractured vertical well. The top and bottom boundary are closed.

114

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

10910

1010

1110

1210

-3

10-2

10-1

100

101

102

tD/CD

q D

infinite boundary

closed boundary

constant pressure boundary

α=50,ω=0.5,λ=0.001,Cf D

=10,S=0.1,CD

=0.00001,ReD

=100

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

10910

1010

1110

1210

-3

10-2

10-1

100

101

102

tD/CD

Ψw

D,Ψ

, wD

·tD

/CD

infinite boundary

closed boundary

constant pressure boundary

α=50,ω=0.5,λ=0.001,Cfd=10,S=0.1,C

D=0.00001,R

eD=100

①

② ③④

⑤⑥

⑧⑦

Figure 2. Type curve for hydraulic fractured vertical wells with finite conductivity in shale gas reservoirs under different kinds of boundary conditions.

115

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

10910

1010

1110

1210

-3

10-2

10-1

100

101

102

tD/CD

q D

α=10

α=20

α=30

ω=0.5,λ=0.001,Cf D

=10,S=0.1,CD

=0.00001

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

10910

1010

1110

1210

-3

10-2

10-1

100

101

102

tD/CD

ψw

D,ψ

, wD

·tD

/CD

α=10

α=20

α=30

ω=0.5,λ=0.001,Cf D

=10,S=0.1,CD

=0.00001

Figure 3. Effect of adsorption coefficient on PPR (pseudo pressure response) and RDR (Rate decline response).

116

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

10910

1010

1110

1210

-3

10-2

10-1

100

101

102

tD/CD

q D

ω=0.2

ω=0.4

ω=0.6

α=50,λ=0.001,Cf D

=10,S=0.1,CD

=0.00001

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

109

1010

1011

1012

10-3

10-2

10-1

100

101

102

tD/CD

ψw

D,ψ

, wD

·tD

/CD

ω=0.2

ω=0.4

ω=0.6

α=50,λ=0.001,Cf D

=10,S=0.1,CD

=0.00001

Figure 4. Effect of storage capacity ratio on PPR and RDR.

117

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

10910

1010

1110

1210

-3

10-2

10-1

100

101

102

tD/CD

q D

λ=0.001

λ=0.01

λ=0.1

α=50,ω=0.5,Cf D

=10,S=0.1,CD

=0.00001

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

10910

1010

1110

1210

-3

10-2

10-1

100

101

102

tD/CD

PD

,P, D

·tD

/CD

λ=0.001

λ=0.01

λ=0.1

α=50,ω=0.5,Cf D

=10,S=0.1,CD

=0.00001

Figure 5. Effect of channeling factor on PPR and RDR.

118

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

10910

1010

1110

1210

-3

10-2

10-1

100

101

102

tD/CD

ψw

D,ψ

, wD

·tD

/CD

CfD

=10

CfD

=30

CfD

=80

α=50,ω=0.5,λ=0.001,S=0.1,CD

=0.00001

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

10910

1010

1110

1210

-3

10-2

10-1

100

101

102

tD/CD

q D

CfD

=10

CfD

=30

CfD

=80

α=50,ω=0.5,λ=0.001,S=0.1,CD

=0.00001

Figure 6. Effect of fracture conductivity on PPR and RDR.

119

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

10910

1010

1110

1210

-3

10-2

10-1

100

101

102

tD/CD

q D

Sf=0.1

Sf=0.3

Sf=0.5

α=50,ω=0.5,λ=0.001,Cf D

=10,CD

=0.00001

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

109

1010

1011

1012

10-3

10-2

10-1

100

101

102

tD/CD

ψw

D,ψ

, wD

·tD

/CD

Sf=0.1

Sf=0.3

Sf=0.5

α=50,ω=0.5,λ=0.001,Cf D

=10,CD

=0.00001

Figure 7. Effect of fracture skin factor on PPR and RDR.

120

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

10910

1010

1110

1210

-3

10-2

10-1

100

101

102

tD/CD

q D

γ=0.0

γ=0.03

γ=0.05

α=50,ω=0.5,λ=0.001,Cf D

=10,S=0.1,CD

=0.00001

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

10910

1010

1110

1210

-3

10-2

10-1

100

101

102

tD/CD

ψw

D,ψ

, wD

·tD

/CD

γ=0.0

γ=0.03

γ=0.06

α=50,ω=0.5,λ=0.001,Cf D

=10,S=0.1,CD

=0.00001

Figure 8. Effect of stress sensitivity on PPR and RDR.

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

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,

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

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 non-

darcy 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

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

tdenotes time.

For shale strata which are full of micro-fractures, two general methods can be used to deal

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:

M ϕ S ρ (4)

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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.

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, m3/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 1 ϕ q and corresponding partial differential form is M dV

.

Based on the equation of state (EOS):pV nRT RT, ρ pγ

∗ ∗ (7)

γφ (8)

In the fracture system, because there is only free gas, so the mass accumulation term can

be described as follows:

γφ (9)

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

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/(m2·s)), M is the gas molar

mass (kg/mol), D is the diffusion coeffici-ent of the bedrock (m2/s), C is the gas mole

concentration (mol/ m3)

Slip flow. In low-permeability formations (less than 0.001 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,

131

gas absolute permeability depends on gas pressure which can be expressed as follows:

k⇀ k⇀ 1 (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 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 (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

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 radiusr . 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:

133

k⇀ k⇀ 1 (21)

where,

b (22)

D.

(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

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, andL ,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

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:

r

0.14 2 Δx Δy                                                                                            k k

0.28⁄

⁄⁄

⁄ .

⁄⁄

⁄⁄                                    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 T q .

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 3.2.1. 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)

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 T q (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)

137

considering the basic DPM with adsorption and non-Darcy permeability change. That is,

bm 0 and bf 0; (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 vdxdy ⋅ γ p v dxdy Tvdxdy (37)

Using Green’s Theorem (Divergence Theory and Integration by parts in multi-dimension),

we obtain that:

⋅ γ p v dxdy γ p ⋅ n v ds γ p ⋅ vdxdy (38)

γφ vdxdy γ p ⋅ n v ds γ p ⋅ vdxdy Tv dxdy(39)

If we just consider the solution on the domain , the weak form for the governing Equation

(31) is:

138

γφ vdxdy γ p ⋅ vdxdy Tvdxdy (40)

Similar process to governing Equation (31), we obtain:

γφ vdxdy γ p ⋅ 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, 7.6 105 Pa; is the gas pressure at the outlet (x=L),

1.0 105 Pa; is the air injection rate, 1.0 10-6 kg/s; is the gas dynamic viscosity,

1.84 10-5 Pa.s; is the length from the inlet (x=0) to the outlet, 10 m; xis the task

location along the gas transport path, m; k is the intrisic permeability for welded tuff

atYucca Mountain, 5.0 10-19 m2; is the compressibility factor, 1.8 10-5 Pa-1m-1. Fig.

8 has presented the match results between model in this paper and analytical solution. As

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 0.065 /K ft , which

correspond to initial reservoir pressure and reservoir temperature of 20 and 353.15

. 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

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, bf 0. Gas viscosity change is represented in the Equation (29). If gas viscosity

141

change is not considered, then bm 0, bf 0, 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)

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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

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 1.0 10-4 mD to 1.0 10-5

mD to 1.0 10-6 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.

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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 2005. The authors also wish to acknowledge for the consultant

support from Baker Hughes and Hess Company.

145

NOMENCLATURE

α =Tangential momentum accommodation coefficient

b =Slip coefficient

b =Non-darcy coefficient

b =Non-darcy coefficient for the fracture system

C =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 =Gas molar mass kg mol⁄

M =Mass accumulation (k g⁄ m . s )

p =Gas pressure in the fracture (Pa)

p =Gas pressure in the matrix (Pa)

p =Bottomhole pressure (Pa)

p =Langmuir pressure of CH4 (Pa)

p =Average pressure in the fracture (Pa)

Q =Source term (k g⁄ m . s )

146

q =Adsorption volume per unit mass of shale (m kg⁄ )

r =Effective pore radius (m)

r =Wellbore radius (m)

r =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 =Density (k g m⁄ )

β =Fluid phase

ϕ =Matrix porosity (%)

ϕ =Fracture porosity (%)

ρ =Density of shale core (k g m⁄ ))

ρ =Gas density in the bedrock (k g m⁄ ))

μ =Gas viscosity (Pa·s)

σ =Crossflow coefficient

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Table 1. Parameters used in the simulation model. Parameter Value Unit Definition

5463 reservoir depth 0.54 ⁄ reservoir pressure gradient

0.059 ⁄ reservoir temperature gradient

k 1.00E-04 matrix initial permeability k 10 fracture initial permeability

0.05 Dimensionless matrix porosity 0.001 Dimensionless fracture porosity

R 8.314 ∗ ⁄ ∗ gas constant z 1 Dimensionless gas compressibility factor

10.4 initial reservoir pressure 3.45 bottom hole pressure 0.016 ⁄ mole weight of CH4 0.0224 ⁄ 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

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).

151

Figure 3. Idealization of the heterogeneous porous medium as DPM.

Matrix Continuum(desorption, viscous flow, Knudsen flow, slip

flow)

Fracture Continuum (viscous flow, Knudsen flow)

Transfer function

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.

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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.

0.01 0.1 10.0

0.2

0.4

0.6

0.8

1.0

eff/

Kn

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., 2005

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).

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0 2 4 6 8 100

1000000

2000000

3000000

4000000

5000000

6000000

Ga

s P

ress

ure

(P

a)

Distance (m)

Results of the model in this paper Analytical Solution by Yu-Shu Wu (1988)

Figure 8. Comparison between analytical solution and numerical simulation in this paper.

2( , , )f wp x y t p

10

p

n

10

p

n

10

p

n

10

p

n

Figure 9. Real reservoir and simplified 2-D reservoir simulation model.

154

1 10 100 10000

10

20

30

40

50

Pro

duc

tion

rate

(x1

03 m3/d

)

time (days)

Without adsorption With adsorption

1 10 100 10000

200

400

600

800

1000

Cu

mu

lativ

e p

rodu

ctio

n (

x103

m3 )

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.

1 10 100 10000.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

mat

rix p

erm

eabi

lity

(mD

)

production time (days)

1 10 100 100099.9

100.0

100.1

100.2

100.3

100.4

100.5

100.6

100.7

100.8

frac

ture

per

mea

bilit

y (1

0-2m

D)

production time (days)

Figure 11. Matrix and fracture permeability change with time. (a) matrix permeability vs. time; (b) fracture permeability vs. time.

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1 10 100 10000.1

1

10

Considering Non-Darcy Flow Without considering Non-Darcy Flow

prod

uctio

n ra

te (

x10

3 m3 /d

)

production time (days)

1 10 100 10000

200

400

600

800

1000

Considering Non-Darcy Flow Without considering Non-Darcy Flow

Cum

mul

ativ

e pr

oduc

tion

(x10

3m

3)

production time (days)

Figure 12. Effect of Non-Darcy flow on gas production. (a) production rate vs. time; (b) cumulative production vs. time.

1 10 100 10000

10

20

30

40

50

Without considering gas viscosity change Considering gas viscosity change

prod

uctio

n ra

te (

x103 m

3 /d)

production time (days)

200 400 600 800 1000 12000

400

800

1200

1600

2000

Without considering gas viscosity change Considering gas viscosity changeC

umul

ativ

e pr

oduc

tion

(x10

3 m3 )

production time (days)

Figure 13. Effect of gas viscosity change on gas production. (a) production rate vs. time; (b) cumulative production vs. time.

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1 10 100 10000

10

20

30

40

50

Considering viscosity change Considering viscosity change and pore

radius change

prod

uctio

n ra

te (

x103

m3 /d

)

production time (days)

0 200 400 600 800 1000 12000

400

800

1200

1600

2000

2400

Considering viscosity change Considering viscosity change and

pore radius change

Cum

ulat

ive p

rodu

ctio

n (x

103 m

3)

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.

Basic model Adsorption Adsorption+Non-Darcy Adsorption+Non-Darcy+Viscosity change Adsorption+Non-Darcy+Viscosity change+radius change

1 10 100 10000

10

20

30

40

50

60

0 200 400 600 800 1000 12000

400

800

1200

1600

2000

2400

Cum

ulat

ive

prod

uctio

n (x

103 m

3 )

Production time (days)

pro

duct

ion

rate

(x1

03 m3 /d

)

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.

157

1 10 100 10000

10

20

30

40

50

produ

ctio

n ra

te (

x103

m3 /d

)

production time (days)

pi=0.2 MPa pi=2.0 MPa pi=20 MPa

0 200 400 600 800 1000 12000

200

400

600

800

1000

pi=0.2 MPa pi=2.0 MPa pi=20 MPa

cum

ulat

ive

prod

uctio

n (

x10

3 m3)

production time (days)

Figure 16. Effect of initial reservoir pressure on gas production. (a) production rate vs. time; (b) cumulative production vs. time.

1 10 100 10000

10

20

30

40

50

kmi

=1E-4 md

kmi

=1E-5 md

kmi

=1E-6 md

prod

uct

ion

rate

(m

3 /d)

production time (days)

0 200 400 600 800 1000 12000

200

400

600

800

1000

kmi

=1E-4 md

kmi

=1E-5 md

kmi

=1E-6 md

Cu

mul

ativ

e pr

oduc

tion

(x10

3m

3 )

production time (days)

Figure 17. Effect of matrix permeability on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time.

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1 10 100 10000

10

20

30

40

50

60

70

kfi=0.1 md

kfi=1.0 md

kfi=10 md

prod

uctio

n ra

te (

x103 m

3/d

)

production time (days)

0 200 400 600 800 1000 12000

200

400

600

800

1000

kfi=1.0 md

kfi=10 md

kfi=100 md

Cum

ulat

ive

prod

uctio

n (x

103 m

3)

production time (days)

Figure 18. Effect of fracture permeability on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time.

1 10 100 10000

10

20

30

40

50

60

70

m=1.0%

m=10%

m=20%

pro

duct

ion

rate

(x1

03 m3/d

)

production time (days)

0 200 400 600 800 1000 12000

400

800

1200

1600

m=1%

m=10%

m=20%

Cum

ulat

ive

prod

uctio

n (x

103 m

3 )

production time (days)

Figure 19. Effect of matrix porosity on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time.

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1 10 100 10000

10

20

30

40

50

f=0.01%

f=0.1%

f=1.0%

pro

duct

ion

rate

(x1

03 m3/d

)

production time (days)

0 200 400 600 800 1000 12000

100

200

300

400

500

600

700

f=0.01%

f=0.1%

f=1.0%

Cum

ulat

ive

prod

uctio

n (x

103 m

3 )

production time (days)

Figure 20. Effect of fracture porosity on gas production. (a) production rate vs. production time; (b) cumulative production vs. production time.

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

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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 2005. 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.

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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

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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

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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

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

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 multi-

stage 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.

2.1. Viscous flow

This is the region where Knudsen number (The ratio of gas mean free path to pore

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)

whereJ 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 .

2.2. 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)

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whereJ is the mass flow caused by Knudsen diffusion (kg/(m2·s)), M is the gas molar

mass (kg/mol), D is the diffusion coefficient of the bedrock (m2/s), C is the gas mole

concentration (mol/ m3).

2.3. 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⇀ k⇀ 1 (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

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, m3/kg; p is the Langmuir

pressure, ; ρ is the density of shale core, kg m⁄ .

170

3. 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 0.8.

3.2. 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,

171

   (8)

or

  1 (9)

Using the form of conventional Darcy flow equation, the fracture apparent

permeability can be expressed as:

k⇀ k⇀ 1 (10)

where

b (11)

D.

(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

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.

4.2. 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:

⋅ ρ v⇀ Q (13)

In the fracture:

⋅ 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:

M ϕ S ρ M ϕ S ρ 1 ∑ (15)

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, kmi 1 bf⁄ , where bf is shown in Equation (11).

3) Gas production from Multi-stage hydraulic fracturing horizontal well

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)

where is the bottom-hole pressure, and re is the effective radius and can be

expressed as:

0.208                                                                                                                                                      when                ,

0.28⁄

⁄⁄

⁄ .

⁄⁄

⁄⁄                    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:

175

γφ ⋅ γ p T  (22)

Equation in the fracture system:

γφ ⋅ γ p T Q (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, andL ,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

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.

5.1. 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

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.

5.2. 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 2431.49 MMSCF to 2759.48

MMSCF, which is 12% increase.

5.3. 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

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 1606.47 MMSCF to 3342.32 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).

5.4. 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 multi-

stage 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

179

from 100 ft to 500 ft, the cumulative gas production changes from 1747.79 MMSCF to

2991.15 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.

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 2005. The authors also wish to acknowledge for the consultant support from

Baker Hughes and Hess Company.

NOMENCLATURE

▽p = pressure gradient, Pa/m

p = fracture pressure, Pa

pm

K

=matrix pressure, Pa

= permeability, m2

kmi = intrinsic matrix permeability, m2

kfi = initial fracture permeability, m2

J = mass flow rate, kg m s⁄

J = mass flow caused by Knudsen diffusion, kg/(m2·s)

C = gas mole concentration, mol/ m3

ν = velocity, m/s

σ = crossflow coefficient between the matrix and fracture systems

181

= Langmuir volume, m3/kg

= Langmuir pressure, Pa

μ = gas viscosity, Pa·s

rw = wellbore radius, m

re = effective radius, m

q = mass production rate, m3/s

qv = volume production rate, kg/s

∅ = fracture porosity, %

∅ = matrix porosity, %

= bottom hole pressure, Pa

= average pressure in the fracture system, Pa

Γ = Outer boundary domain

Γ = Inner boundary domain

t = time, s

x, y, z = cartesian coordinate

L ,L ,L = fracture spacing in the x, y, z directions

ρ = density of shale core, kg/m3

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

182

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Table 1. Knudsen’s Permeability Correction factor for Tight Porous Media (Guo et al., 2015).

Model Correlation factor

Klinkenberg 4 ⁄

Heid et al. 11.419 .

Jones and Owens 12.639 .

Sampath and Keighin 13.851 ⁄ .

Florence et al. ⁄ .

Civan 0.0094 ⁄ .

Table 2. Parameters used in the synthetic model for production performance study.

Parameters Barnett Shale case Unit Model dimensions (LxWxH)

2500x2000x300 (762x609.6x91.44) ft (m)

Initial reservoir pressure 2400 (1.65x107) psi (Pa)

Bottom-hole pressure 500 (3.45x106) psi (Pa)

Production period 20 (6.31x108) year (s)

Reservoir temperature 200 (93.3) oF (oC) Gas viscosity 0.02 (0.00002) cP (Pa*s)

Bulk density 158 (2530.92) lb/ft3 (kg/m3) Reservoir top depth 6800 (2072.6) ft (m) Langmuir pressure 650 (4.48 x106) psi (Pa) Langmuir volume 100 (0.0028) SCF/ton (m3/kg)

Fracture conductivity 9 (1.22x10-14) md-ft (m2-m)

Matrix permeability 0.00035 (6.125x10-19) md (m2) Matrix porosity 0.06 fraction Horizontal wellbore length 1000 (304.8) ft (m)

186

Figure 1. Diagram of dual porosity model.

Figure 2. Synthetic model for production performance study.

187

0

2

4

6

8

10

12

14

16

0.01mD-0.1mD

0.001mD-0.01mD

100nD-1000nD

10nD-100nD

1nD-10nD

Co

unt

s in

the

lite

ratu

re

Figure 3. Frequency distribution diagram of shale matrix permeability.

0 5 10 15 200

500

1000

1500

2000

2500

3000

matrix permeability=10 nD matrix permeability=200 nD matrix permeability=500 nD matrix permeability=800 nD matrix permeability=1000 nD

Note: 1 nD=10-9 Darcy=10-6 mDCu

mu

lativ

e g

as

pro

duct

ion

(M

MS

CF

)

Time (years)

0 5 10 15 20

0.4

0.8

1.2

1.6

Ga

s p

rodu

ctio

n r

ate

(M

MS

CF

/da

y)

Time (years)

matrix permeability=10 nD matrix permeability=200 nD matrix permeability=500 nD matrix permeability=800 nD matrix permeability=1000 nD

Note: 1 nD=10-9 Darcy=10-6 mD

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.

188

<2% 2%-4% 4.1%-6% 6.1%-8% 8.1%-10% >10%0

1

2

3

4

5

6

Co

un

ts in

the

lite

ratu

re

Figure 5. Frequency distribution diagram of shale matrix porosity.

0 5 10 15 200

500

1000

1500

2000

2500

3000

matrix porosity = 2% matrix porosity = 4% matrix porosity = 6% matrix porosity = 8% matrix porosity = 10%

Cu

mu

lativ

e g

as

pro

du

ctio

n (

MM

SC

F)

Time (years)

0 5 10 15 200.0

0.4

0.8

1.2

1.6

2.0

Ga

s p

rod

uct

ion

ra

te (

MM

SC

F/d

ay)

Time (years)

matrix porosity = 2% matrix porosity = 4% matrix porosity = 6% matrix porosity = 8% matrix porosity = 10%

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.

189

0 5 10 15 200

500

1000

1500

2000

2500

3000

3500

4000

Time (years)

fracturing spacing =100 ft fracturing spacing =200 ft fracturing spacing =300 ft fracturing spacing =400 ft

Cu

mu

lativ

e g

as

pro

du

ctio

n (

MM

SC

F)

0 5 10 15 200.0

0.5

1.0

1.5

2.0

2.5

3.0

Ga

s p

rod

uct

ion

ra

te (

MM

SC

F/d

ay)

Time (years)

fracturing spacing =100 ft fracturing spacing =200 ft fracturing spacing =300 ft fracturing spacing =400 ft

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.

(a) (b)

0 5 10 15 200

400

800

1200

1600

2000

2400

2800

3200

Time (years)

fracture half-length =100 ft fracture half-length =200 ft fracture half-length =300 ft fracture half-length =400 ft fracture half-length =500 ft

Cu

mu

lativ

e g

as

pro

du

ctio

n (

MM

SC

F)

0 5 10 15 200.0

0.4

0.8

1.2

1.6

2.0

Ga

s p

rod

uct

ion

ra

te (

yea

rs)

Time (years)

fracture half-length =100 ft fracture half-length =200 ft fracture half-length =300 ft fracture half-length =400 ft fracture half-length =500 ft

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.

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SECTION

3. CONCLUSIONS

Shale gas is one kind of unconventional resources which is produced from organic

tight shale formations. It has becoming an increasing important and hot research topic in

recent decades with the decreasing and shortage of conventional energy resources. Due to

its ultra-low permeability and large portion of nano pores, there exists significant difference

between fluid flow in the shale reservoirs and conventional reservoirs. During recently

years, many apparent permeability models have been proposed to characterize gas flow in

shale nano pores. And also, many models have been studied to simulate gas production

from shale gas reservoirs.

This research work aims to conduct a comprehensive study of gas production from

shale strata. This work will mainly carry out research on gas flow mechanisms from pore

scale to reservoir scale. Apparent permeability expression was developed to characterize

gas flow in nano pores. Well test model has been constructed to understand gas flow

process in the reservoirs. Simulation work have been conducted for shale gas production

with vertical well and multi-stage hydraulic fracturing horizontal well. The main findings

of this study are as follows:

Obvious discrepancy between apparent permeability and intrinsic permeability

has been observed through the experiment of gas flow through nano

membranes.

Relationship between this discrepancy and pore throat diameter has been

analyzed: the smaller the nano pores, the more apparent the discrepancy is.

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Under different pore size condition, there exists different gas flow mechanisms

which depends on the Knudsen number (the ratio of gas mean free path to the

pore throat diameter). Typical gas flow mechanisms include: viscous flow, slip

flow, Knudsen diffusion, etc.

Utilizing the advection-diffusion model and considering the slip flow together

with the Knudsen diffusion, a new apparent permeability formula for gas flow

in nano pores has been derived.

Simulation results were verified against the experimental data for gas flow

through nano membranes and published data with different gases (oxygen,

argon) and different PTDs (235 nm, 220 nm). Results show the proposed model

matches experimental data very closely.

Through comparison against experimental data, the Knudsen/Hagen-Poiseuille

analytical solution, and existing data available in the literature, the model

proposed in this study yielded a more reliable solution.

The well test model for describing fluid flow through fractured vertical well in

shale gas reservoir has been constructed and the solutions, transient pressure

and rate decline curves are analyzed. The flow regimes have been detected:

early afterflow period, afterflow transition period, bilinear flow period,

formation linear flow period, pseudo radial flow of natural fractures, channeling

period from matrix to fracture, formation pseudo radial flow period, boundary

response period.

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

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the trough during this period because big value of VL indicates more gas will

be desorbed during the interporosity flow period.

The number of fractures (MF) mainly affects the early flow regimes of the

fractured horizontal 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.

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.

The numerical simulation for shale gas reservoir with vertical well production

has been constructed, which considers adsorption, non-Darcy permeability

change, gas viscosity change and pore radius increase due to gas desorption.

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.

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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 range of corresponding parameters.

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.

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VITA

Chaohua Guo received his Bachelor of Science degree in petroleum engineering

from China University of Petroleum in China in July 2011. He worked as a post-graduate

at the China University of Petroleum from Sep 2011 to Aug 2012. He joined Dr. Mingzhen

Wei’s research group at the Petroleum Engineering program of Missouri University of

Science and Technology (formerly University of Missouri – Rolla) in August 2012 and

focused on gas flow in nano pores, shale gas simulation, and tight oil simulation. In Dec

2015, he received his Ph.D. degree in Petroleum Engineering from Missouri University of

Science and Technology.