MODELING GAS PRODUCTION

FROM SHALES AND COAL-BEDS

A REPORT SUBMITTED TO THE DEPARTMENT OF ENERGY

RESOURCES ENGINEERING

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE

By

WAQAS ALI

October 2012

iii

I certify that I have read this report and that in my opinion it is fully

adequate, in scope and in quality, as partial fulfillment of the degree

of Master of Science in Petroleum Engineering.

__________________________________

Prof. Khalid Aziz

(Principal Advisor)

v

Abstract

Development of shale gas has transformed the energy outlook of US in the past

decade. Identification of massive global resources and development of technology for

economically producing gas from such sources is making a huge impact on the

energy picture. According to EIA (EIA, 2011) the total gas resources of the world are

estimated to be about 22,600 TCF of which 40% is now contributed by shale plays.

Shales have complex pore networks and distinct scales for fluid flow. There are at

least four types of porosities present in shales — Inorganic porosity, nano-porosity

within organic Kerogen and porosity associated with both natural and artificially

created fractures. Also, significant gas is assumed to be adsorbed in the Kerogen.

Naturally occurring shales have little matrix permeability though. Horizontal drilling

assisted by multi-stage hydraulic fracturing make gas production from shales viable.

This increases near well-bore conductivity and also stimulates the network of natural

fractures.

Modeling fluid transport in shales is challenging. This is due to the combination of

multiple porosities, gas desorption and slippage effects. We present two numerical

schemes to simulate gas production from such systems. In the Single Porosity Model,

we assume that there is only one distinct pore network and the gas desorbs

instantaneously from the shale matrix as the reservoir depletes. In the Dual Porosity

Model we consider two coupled porous networks with desorption in the organic nano-

pores. Langmuir Adsorption Isotherm (Langmuir’s 1916) is used to model the

desorption process. Slippage of gas molecules within nano-pores is incorporated

using Klinkenberg’s Effect, (Klinkenberg 1941).

Single Porosity Model is observed to be less consistent with both pore networks and

desorption process in shales. In Dual Porosity Model, natural fractures permeability

(kf) significantly affects early stage of gas production and matrix-fracture flux

coefficient (Dmf) controls the slow desorbed flux in later half of production. Also,

high Langmuir’s Volume (VL) increase production and delay pressure depletion

within the reservoir.

We also present a scheme to simulate gas recovery from naturally occurring coal

seams. Single porosity, two phase gas-water code with instantaneous desorption is

developed to model initial-dewatering and depletion phase of a typical Coal-bed

Methane well.

The numerical schemes presented here simulate recovery from unconventional gas

reservoirs by considering the physics of flow in complex pore networks and

incorporating other non-darcy effects. Most unconventional wells however are

stimulated by fracturing treatment which is not included rigorously in our current

model. A coupled geomechanical, fracture and flow model is required to simulate

vi

post-fracturing well-performance from these wells. We plan to incorporate these

effects in future.

vii

Acknowledgments

My time here as a graduate student at Stanford has been more inspiring than I ever

thought. It has been exciting and rewarding. I had a chance to learn from the company of

large number of brilliant and creative human beings. I would like to take this opportunity

to express my gratitude towards several of them who have had the greatest impact on my

academic endeavors so far.

First and foremost, I would like to acknowledge my advisor, Prof. Khalid

Aziz, who has guided and motivated me in my studies. He has remained extremely

patient with me during my time at Stanford.

I am indebted to the ERE faculty, my friends, colleagues, class-mates and teachers. I

would especially like to thank Khalid Alnoami, Hai Vo and Dr. Fareed Siddiqui (NED

University) for their support and mentorship; It was due to invaluable discussions with

Khalid Alnoami that this work has been possible.

I acknowledge the support of Reservoir Simulation Consortium (SUPRI-B) and ENI for

financial support. Last but most significantly, I wish to thank my parents for their

unconditional love and support.

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Contents

Abstract ............................................................................................................................... v

Acknowledgments............................................................................................................. vii

Contents ............................................................................................................................. ix

List of Tables ..................................................................................................................... xi

List of Figures ................................................................................................................... xii

1. Chapter 1 Introduction to Unconventional Gas Resources………………….…..1

1.1. Introduction..……….…………………………………………………………… ...1

1.2. Introduction to Shale Gas Resource………………………………………………..2

1.3. Environmental and Economic Consequences ……………………………………...4

1.4. Introduction to Coal-bed Methane Resource ………………………………………5

2. Chapter 2 Characteristics of Shale Gas Reservoirs ………………………………7

2.1. Multiple Porosities in Shale Reservoirs…………………………………………....7

2.2. Ultra-low Permeability in Shale Reservoir…………………………………...........8

2.3. Multiples Porosities in Shales………………………………………….………9

2.4. Water Production from Shales……………………………………………………10

2.5. Discussion………………………………………………………………………...10

3. Chapter 3 Introduction to Single and Multi-component Adsorption ................ 11

3.1. Introduction…………………….............................................................................11

3.2. Adsorption Models for Single Component……………………..........................11

3.2.1. Langmuir's Approach (Langmuir 1916) …….…………...……...………….... 12

3.2.2. The BET Adsorption Models (Brunauer et al.,1938)……….….….……..........14

3.3. Multicomponent Adsorption Models………………………….............................16

4. Chapter 4 Gas In Place Quantification for Shale Gas Reservoirs...................... 17

4.1. Volumetric Calculation for Shale Gas Reservoirs… ………….............................17

4.2. Material Balance for Shale Gas Reservoir.…………………………………...….19

5. Chapter 5 Previous Modeling Efforts-Unconventional Gas Reservoirs ............ 20

5.1. Modeling Gas Slippage/Diffusion……………………………………………….20

x

5.2 Modeling Gas Desorption……………..……………………………………….…21

5.3. Our Methodology for non-Darcy effects……….………………..……..………..22

6. Chapter 6 Numerical Models for Shale Gas and CBM Production…………… 23

6.1. Single Porosity Model with Instantaneous Desorption……………...................... 23

6.2. Dual Porosity Mode with Desorption in nano-Pores.……………………...….….25

6.2.1. Mass Balance in Organic nano-Pores………………………………………….25

6.2.2. Mass Balance in Micro-Fractures……………………………………..……….27

6.3. Modeling CBM Production ……………………………………………………...28

6.4. Code Development ………… …………………………………..………………..28

7. Chapter 7 Model Results …………………………………………………………29

7.1. Single Porosity Model with Instantaneous Desorption………….…..................... 29

7.1.1. Model Validation with Commercial Package.………………......…………......29

7.1.2. Sensitivity to Langmuir's Volume (VL)………………………...……….….….30

7.1.3. Heterogeneity in the Adsorbed Gas Regions.…….…………………….....…...33

7.1.4. Effect of Hydraulic fracturing………….…….……….………...……........…...34

7.2. Dual Porosity Model with Desorption in nano-Pores…………….........................36

7.3. Simulation of CBM Production……………..........................................................38

7.3.1. Two Phase Gas-Water Code Validation…...……………………...…………....38

7.3.2. Comparison of Eclipse®

CBM Model ………………….…...…………………39

7.3.3. Sensitivity to Fractures Permeability (kf) on CBM Production…...…...…….. ..41

8. Summary and Future Work .................................................................................. 42

Nomenclature …………………………………………………………………………….44

References ………………………………………………………………………………..48

Appendices

A. Appendix A Fluid Properties .................................................................................... 51

B. Appendix B Formulation of Jacobian Matrix ........................................................... 54

xi

List of Tables

Table 7-1: Simulation parameters to validate single-phase gas code with Eclipse®

(Schlumberger, 2012)………………………………………………..…………………(29)

Table 7-2: Simulation parameters used for sensitivity of gas production to VL..……...(30)

Table 7-3: Simulation parameters for dealing heterogeneity in adsorbed gas regions...(34)

Table 7-4: Simulation parameters to demonstrate the effect of hydraulic fracturing on

well performance.…………………… ………………………………………...………(35)

Table 7-5: Simulation parameters for sensitivity of gas production to Dmf and Kf ……(36)

Table 7-6: Simulation parameters for validating two phase gas-water code with Eclipse®

…………….…………………………………………………..……………………….(38)

Table 7-7: Simulation parameters for comparing Eclipse® and our CBM

model……..............................................................................................……….(39)

Table 7-8: Simulation parameters used for sensitivity of CBM production parameters to

kf………………………………………………………………………………………..(41)

xii

List of Figures

Figure 1-1: Major US shale gas basins, adapted from "Modern Shale Gas Development in

US - 2009", report published by US DOE……………………………………...……….(2)

Figure 1-2: US natural gas production (past and future projections) from 1990-2035,

adapted from “US EIA, Annual Energy Outlook 2012 Early Release”………………... (3)

Figure 1-3: Global shale gas potential, adapted from US EIA, www.eia.gov……........(3)

Figure 1-4: Major Coal-bed Methane resources in the United States, adapted from USGS

Professional Paper 1625-B……………………………………………………………... (5)

Figure ‎1-5: Typical production behavior of a CBM well, adapted from "Coal-bed

Methane-Potential and Concerns, USGS 2000”…………………………….…………. (6)

Figure ‎1-6: Typical completion setting for CBM well, adapted from "Coal-bed Methane-

potential and concerns, USGS 2000"……………………………...…………………… (6)

Figure 1-1: SEM image of shale sample showing inorganic matrix (light grey) and

organic matter (dark grey), adapted from Ambrose et al., (2010)………..……………. (7)

Figure 2-2: Porosity and Permeability relationships experienced in North American shale

plays, adapted from Wang and Reed, (2009)……………………………...………….. (8)

Figure 2-3: Effect of confining pressure on gas shales permeability, adapted from Wang

and Reed, (2009)………………………………………………………..…………… (9)

Figure 2-4: Schematic of assume production fair-ways in multi-porous gas shales,

adapted from Wang and Reed, (2009)…………………………………...………….... (10)

xiii

Figure 3-1: Five Types of adsorption isotherms for single component, p° is the saturation

pressure (after Yang, 1987)……………………………………………………...……. (12)

Figure 3-2: Typical Langmuir Isotherm, here VL = 200 scf/ton and PL = 500 psia…... (13

Figure 3-3: Sensitivity to VL at fixed PL = 500 psia……………………………...…… (13)

Figure 3-4: Sensitivity to PL at fixed VL = 200 scf/ton……………………..……........ (14)

Figure 3-5: BET Adsorption model for different values of C = 1, 10, 20, 50, 100…... (15)

Figure 3-6: N-layer BET equation for different values of N = 1, 5, 10, 50 and at C = 100

………………………………………………………………………….……………... (15)

Figure 4-1: Various constituents of the total bulk volume of shale gas reservoir (after

Ambrose et al., 2010)………………………………………………………..….….…. (18)

Figure 6-1: Schematic representation of the single porosity model with instantaneous

adsorption…………………………………………………………………..…….…… (24)

Figure 6-2: SEM Image for shale Sample (top) and its dual porosity discrete analog

(bottom), adapted from Ambrose et al., (2010) ……………..………………………...(25)

Figure 6-3: Schematic representation of the organic nano-pores and assumed desorption

process in dual porosity shale reservoir……………………………………….……... (26)

Figure 6-4: Schematic representation of micro-fractures and its flux interaction with

nano-pores in dual porosity shale reservoir…………………………………………… (27)

Figure 7-1: Gas production profile showing model validation with Eclipse®

(Schlumberger, 2012)….……………………………………………..…….…………. (30)

xiv

Figure 7-2: Well performance for various values VL at Pwf = 250 psia………..…….... (31)

Figure 7-3: Pressure map after end of simulation for various values of VL…………... (32)

Figure 7-4: Average gas production data from various shale gas fields in US (Adapted

from Valko and Lee, 2010)…………….……………………………………………... (32)

Figure 7-5: Matrix for VL showing heterogeneous adsorbed gas regions…………....(33)

Figure 7-6: Comparison of base case (VL = 0), heterogeneous case, and mean case….(34)

Figure 7-7: Effect of hydraulic fracturing on well performance……………………… (35)

Figure 7-8: Sensitivity of Qg to Dmf…….……………………………...…………….…(37)

Figure 7-9: Sensitivity of gas production to kf……………...………………………... (37)

Figure 7-10: Comparison of the two phase gas-water code with Eclipse®

……………………………………………………….……..………………………..... (38)

Figure 7-11: Qg comparison of our CBM model with Eclipse®.…………………..…...(40)

Figure 7-12: Sensitivity of CBM production parameters to kf…………...……….…...(40)

Figure 8-1: Models available for simulating gas production from shales………..…… (42)

Figure A-1: Compressibility factor (z), its derivative w.r.t pressure, gas viscosity and its

derivative respectively.………………….………….………………………………...(51)

xv

Figure A-2: Inverse of FVF of gas, its derivative w.r.t pressure, gas compressibility cg,

and its derivative respectively.……………………………………………….……… (52)

Figure ‎A-3: Vg, its derivative w.r.t pressure, Klinkenberg factor and its derivative w.r.t

pressure………………………………………………………………………...….….(52)

Figure ‎A-4: Relative permeability curve used in CBM model……………………….(53)

xvi

1

Chapter 1 Introduction to Unconventional Gas

Resources

1.1. Introduction

Unconventional gas reservoirs have permeability in the range of 10-6

to 10-9

Darcy which

is significantly lower compared to conventional reservoirs (10-3

Darcy scale). Also, in

unconventional gas reservoirs, gas may reside in the interstitial molecular vacancies of

the matrix in an adsorbed state. This ultra-low permeability and presence of adsorbed gas

makes its development challenging, both operationally and economically. However, due

to the rise in the global energy demand and depletion of conventional reservoirs,

unconventional gas resources have gained considerable importance recently.

Unconventional gas reservoirs are classified as: Tight-gas Sands, Gas Shales and Coal-

bed Methane (CBM). Tight-gas Sands are sandstone reservoirs having permeability scale

of 10-3

to 10-6

Darcy. Shales, which are considered to be the source of hydrocarbon

generation for conventional reservoirs have permeability in the range of 10-6

to 10-9

Darcy. CBM is the gas that resides in the naturally occurring coal-beds. In CBM reservoir

gas occurs both in micro-fractures (also termed “cleats” in coals) and in an adsorbed state

within the coal matrix.

Most unconventional gas wells do not produce economically if completed conventionally

(vertical completion without hydraulic fracturing). In order to make gas production

profitable, wells are generally completed horizontally and further assisted by multiple

stages of hydraulic fracturing. This combination enhances the reservoir exposure,

stimulates the near-well bore conductivity and forms a hydraulic connection with little

available natural matrix-permeability. The successful fracturing treatment results in

multi-fold increase of productivity of a well.

Gas in shales is one of the most important of all the unconventional gas resources, this is

due to its enormous resource potential, both in the US and worldwide. In the following

sections we discuss the importance of shale gas resource. Later, we will also introduce

CBM resource.

2

1.2. Introduction to Shale Gas Resource

Two decades ago, gas in shales was considered technically and economically

unrecoverable. The development of Barnett shale is considered a watershed in the shale

gas industry. The two most important factors for this extraordinary success are horizontal

drilling and advancement in fracturing technology. This has been significantly assisted by

high brittleness of Barnett shale (Wang and Reed, 2009). The development of other shale

plays in US is assumed to follow relatively similar trends.

The US has abundant shale gas resource and it is distributed across much of the lower 48

states (US DOE, 2009). According to the Annual Energy Outlook 2012 (early release) the

estimated unproved technically recoverable resources (TRR) for shale gas in the US is

about 482 trillions cubic feet (TCF). The most active shale plays to date in US are

Barnett, Marcellus, Fayetteville, Haynesville, Antrim and New Albany (US DOE, 2009).

Figure 1-1 shows the major shale gas basins in the US.

Figure 1-1: Major US shale gas basins, adapted from "Modern Shale Gas Development in US -

2009", report published by US DOE.

In US, gas production from shales will increase significantly in future. According to US

EIA Annual Energy Outlook (early release) 2012, the gas production in 2012 from shales

will increase from 5.0 TCF (23% of total US dry gas production) to 13.6 TCF in 2035

(49% of total US dry gas production). Figure 1-2 shows past and estimated future gas

production from various gas resources of US.

There is also considerable potential for shale gas exploitation globally. According to EIA

(EIA, 2011) the total gas resources of the world are estimated to be about 22,600 TCF of

3

which 40% is now contributed by shale plays. Also, at the current global consumption

rate of 160 TCF/year, the total gas reserves of the world will last about 140 years. Figure

1-3, shows the global shale gas resource outlook.

Figure 1-2: US natural gas production (past and future projections) from 1990-2035, adapted

from “US EIA, Annual Energy Outlook 2012 Early Release”.

Figure 1-3: Global shale gas potential, adapted from EIA, (2011) www.eia.gov.

4

1.3. Environmental and Economic consequence for Shale Gas Development

In the last five years shale gas developments have revolutionized the energy outlook of

the US and it will continue to play its favorable economic role in the coming decades all

over the globe. Although shale gas development has been rewarding commercially, it has

also caused some environmental concerns. This has instigated many recent studies to

evaluate the exact environmental impact of shale gas development.

Hydraulic fracturing is considered a requisite for wells completed in shales. In this

process, a complex mixture of clays, silica, gels and proppants is injected into the

reservoir at high pressures. If the well is not completed properly or if fluid injection

process is designed or carried out incorrectly, near-by aquifers may be contaminated.

Also, due to the low drainage of shale gas wells, additional development wells have to be

drilled to properly exploit this resource. This may disrupt some natural habitats and cause

additional conservational concerns.

Therefore, shale gas development poses a massive environmental challenge for operating

and service providing companies. It will be interesting to observe how industry develops

the right balance between financial gains and potential environmental hazards connected

with its development.

5

1.4. Introduction to CBM Resource

Gas adsorbed in the coal seams has been commercially produced in many parts of the

world including US, China, Australia and India. The total estimated gas resource of the

world in coal-beds is about 9,000 TCF (Kawata and Fujita, 2001). In the US only, more

than 40,000 wells have been completed in at least 20 different basins which contribute

about 1.7 TCF of gas annually (Jenkins and Boyer, 2008). Figure 1-4 shows major US

CBM basins. According to EIA Energy Outlook 2012 (Early Release), CBM contributes

about 9% of the total US gas production (see Figure 1-2).

Figure 1-4: CBM resources in the United States, adapted from USGS Professional Paper 1625-B.

Productivity of CBM wells depends primarily upon permeability within micro-fractures.

Typical values for cleat-permeability range from a few milli-Darcy to few tens of milli-

Darcy (Jenkins and Boyer, 2008). Generally, the cleats in coal-beds are initially saturated

with water. This water must be produced before gas can start desorbing from the matrix

into these micro-fractures. This initial phase of water production is known as “De-

watering”. In this phase, gas production starts to build-up and reaches a maximum value.

This is followed by a period of relatively stable gas production and then a depletion phase

(see Figure 1-5).

6

Figure ‎1-5: Typical production behavior of a CBM well, adapted from "Coal-bed Methane-

Potential and Concerns, USGS 2000”.

Since most of the CBM resides in an adsorbed state, the quality of the reservoir is

classified by its adsorption capacity (Langmuir’s Volume, VL). Typical value of this

parameter ranges from 100 to 800 scf/ton for anthracite to highly volatile bituminous

coals respectively, (Jenkins and Boyer, 2008).

Some CBM wells are stimulated by a technique called cavitation and others by hydraulic

fracturing. In cavitation, an open-hole completion is repeatedly pressurized and then

rapidly depressurized. This continuous process results in a donut-shaped area of enhance

permeability which increases the productivity of CBM wells.

Figure ‎1-6: Typical completion of CBM wells, adapted from "Coal-bed Methane-potential and

concerns, USGS 2000".

7

Chapter 2 Characteristics of Shale Gas Reservoir

Shales are different from conventional reservoirs in many respects. These reservoirs

usually have multi-porous structure, ultra-low permeability, distinct scales of fluid flow

and potential of adsorbing large quantities of gas. In the ensuing sections, we discuss

various pore networks and attempt to develop an understanding of the flow mechanisms

involved in shales.

2.1. Multiple Porosities in Shale Reservoirs

Shales are attributed with at least four distinct pore scales; Inorganic porosity, organic

porosity, natural micro-fractures and porosity associated with artificially created

fractures.

The inorganic matter such as clay, silica, pyrite and other non-organic minerals which

comprises the bulk of the shale fabric has some inter-granular void spaces within the

matrix. The pores at this level are of micro-scale and are sparsely interconnected (Davies

et al., 1991 and Bustin et al., 2008). Shales also contain organic matter (Kerogen) which

is finely dispersed within the inorganic matrix (see Figure 2-1). This organic matter is

assumed to be formed at the time when oil and gas were generated. It act as a separate

porous medium, has pore size ranging from 5 nm to 1000 nm and can adsorb and store

free gas simultaneously (Wang and Reed, 2009).

Figure 2-1: SEM image of a shale sample showing inorganic matrix (light grey) and organic

matter (dark grey), adapted from Ambrose et al., (2010).

Networks of micro-fractures are critically important to shale gas production. The porosity

in such networks is assumed to be less than 0.5% but they have significant fluid

8

conductivity, which plays a vital role for higher than expected gas rates in shales (Wang

and Reed, 2009). The porosity and permeability at this pore scale may improve

significantly after a successful hydraulic fracturing treatment. The exact reason for this

increase is not well understood at present but it may be due to the opening of fractures as

result of changes in the in-situ stress state of the reservoir.

2.2. Ultra-low Permeability in Shale Reservoirs

One of the challenges in shale resource development is due to its low bulk matrix

permeability. Shale reservoirs generally have permeability in the range of micro to nano-

Darcy (see Figure 2-2) and will not produce at economical rates without stimulation.

Figure 2-2: Porosity and Permeability relationships experienced in North American shale plays,

adapted from Wang and Reed, (2009).

Permeability in shale matrix also varies with confining pressure (or effective stress) in the

reservoir (Soeder, 1988). The depletion of the reservoir causes most of the overburden

stress to be supported by the confining pressure in the grains. This result in the reduction

of permeability both within micro pores and fractures upon pressure depletion (see Figure

2-3).

9

Figure 2-3: Effect of confining pressure on gas shales permeability, adapted from Wang and

Reed, (2009).

2.3. Multiple Porosities in Shale Reservoirs

In this section we outline an assumed flow path for gas flow in shales. The production

mechanism and complex interaction of various pore scales is discussed. This will serve as

a template for the proposed numerical schemes to simulate gas production from shales.

Transport mechanism for the gas stored in the dispersed organic nano-pores (both in free

and adsorbed state) is assumed to be a subtle combination of Darcy and non-Darcy

phenomena. In these organic pores, the average mean free path of the gas molecules is

comparable to the pore-throat size, causing slippage of the gas molecules (Klinkneberg,

1941). Also, desorption of the gas occurs as the pressure in the reservoir depletes. This

combined flux from organic nano-pores is then convected into the inorganic porosity

present in shales. Combination of this flux from two different pore scales is then

transported into the network of micro-fractures.

Micro-fractures in shales tend to be widely dispersed and have little interconnection. In

order to exploit these micro-fractures, artificially induced flow path of high fluid

conductivity is created in the vicinity of well-bore by means of hydraulic fracturing (see

Figure 2-4). This stimulation treatment serves two important purposes: (i) Intersecting

various micro-fractures orthogonally thus completing a flow path from gas-rich organic

matter to the well-bore; (ii) Attempting to change the stress field in the reservoir,

resulting in the opening of micro-fractures. This causes an increase in the fluid

conductivity of the reservoir near the stimulated volume.

10

Figure 2-4: Schematic of assume production fair-ways in multi-porous gas shales, adapted from

Wang and Reed, (2009).

2.4. Water Production from Shale Reservoirs

Water production is a critical factor that affects the performance of any gas well. High

producing water-gas ratios may cause liquid loading of the well and eventually leads to

little or no fluid production. Generally, water produced from shale gas wells is from

surrounding aquifers (Wang and Reed, 2009). Initial gas saturation in organic-rich shales

is high compared to the laboratory data (Boyer et al., 2006). This low initial water

saturation is known as “Sub-irreducible Initial Water Saturation”, (Bennoin and Thomas,

2005). This prevents water production from shales and creates powerful capillary suction

of water (Wang and Reed, 2009). Organic-rich shales such as Barnett shale in North

Texas (Curtis, 2002), Bakken shale in North Dakota (Perrodon, 1983), and Marcellus

shale in Ohio Appalachian Basin (Soeder 1988, Curtis 2002) produced with little water-

gas ratio.

2.5. Discussion

Modeling of gas production from shales is challenging. This is due to the presence of

multiple porosities with added complexity of desorption and slippage effects. To simulate

its production precisely, the physics of these effects must be incorporated in the

numerical model. Also note that, due to Sub-irreducible Initial Water Saturation in

shales, gas production can be modeled assuming single-phase gas flow.

11

Chapter 3 Introduction to Single and Multi-component

Adsorption

3.1. Introduction

Adsorption is a process by which certain porous solids bind large number of molecules to

their surface (Smith et al., 2001). The number of molecules adsorbed on the solid surface

depends on the pressure in the gas phase. At low pressures, few gas molecules cling to

the solid surface. As the pressure increases more gas tends to accumulate in the

interstitial vacancies of porous solids.

The amount of gas adsorbed on the solid surface at a given pressure strongly depends on

the molecular structure of porous solids, properties of the gases involved and their

interaction properties. At some high pressure, all the molecular vacancies are occupied

and solid is said to form “Monolayer” (Smith et al., 2001). Although some solid-gas

combinations form multilayer coverage of molecules on their surface others restrict to

single layer only.

3.2. Adsorption Models for Single Component

An adsorption model is a relationship for the amount of gas adsorbed on solid surface at a

given pressure and temperature. At constant temperature, the amount of gas adsorbed on

the solid surface is a sole function of pressure. This relationship is generally known as an

“Adsorption Isotherm”.

In literature, adsorption models for single component gases are classified into five types.

These models differ from each other in two aspects: (i) The number of layers (single or

multiple); (ii) The extent of pore space (finite or infinite) available for adsorption. Type

(I) is the Langmuir’s Isotherm. This is typical of micro porous solids. At low pressures,

the amount of gas adsorbed on solid surface increases with pressure but reaches a point

where further increase in pressure will not significantly affect the amount of gas

adsorbed. Type (II) and Type (III) allow multilayer coverage of gas molecules on solid

surface and infinite pore space for adsorption (Lin, 2010). Type (IV) and Type (V) are

similar to Type (II) and Type (III) with an additional constraint of finite adsorption

capacity on solid surface (Do, 2008). Typical behavior of these isotherms is shown in

Figure 3-1.

12

Figure 3-1: Five Types of adsorption isotherms for single component adsorption, p° is the

saturation pressure (after Yang, 1987).

3.2.1. Langmuir’s Approach (Langmuir, 1916)

Langmuir adsorption model assumes a dynamic equilibrium between rates of adsorption

and desorption. It also assumes monolayer coverage of gas molecules on solid surface.

Therefore, for a specific solid-gas combination, there exists a maximum for the amount

of gas that can be occupied by the internal surface of adsorbing solid. This parameter will

be termed hereafter as LV (see Figure 3-2). It has units of scf/ton. Langmuir’s model is

defined as:

L

Lg

PP

PVV

(2-1)

Here; gV (scf/ton) is the amount of gas adsorbed at a given gas-phase pressure, P (psia);

LP is the Langmuir’s pressure (psia) at which half of the gas capacity VL remains

adsorbed (see Figure 3-2). Both LV and LP characterize the Langmuir’s isotherm for a

specific solid-gas combination.

13

Figure 3-2: Typical Langmuir Isotherms, here LV = 200 scf/ton and LP = 500 psia.

Figure 3-3 and Figure 3-4 shows sensitivity to LV ( LP fixed at 500 psia) and LP ( LV fixed

at 200 scf/ton) respectively. It can be inferred from these figures that both at high LV and

PL, isotherms have comparatively large gradients, which results in relatively large

desorption rates at low pressures.

Figure 3-3: Sensitivity to LV at fixed LP = 500 psia.

14

Figure 3-4: Sensitivity to LP at fixed LV = 200 scf/ton.

3.2.2. The BET (Brunauer et al., 1938) Adsorption Model

Langmuir’s model assumes mono layer coverage of gas molecules. This assumption is

valid at low pressures and for gases that do not show strong affinity to solid surfaces.

Gases that are intensely adsorbed on solid surfaces such as CO2 on coals, this assumption

does not remain valid (Lin, 2010). BET (Brunauer et al., 1938) equation can be used

here to model up to infinite amount of adsorbed layers.

The BET (Brunauer et al., 1938) equation is given as:

)))(1(1)((

P

PCPP

mCPn

(2-2)

Here, n is the amount of gas adsorbed (moles/ton) at a given pressure, P and Po is the

saturation pressure of the gas adsorbed. Similar to Langmuir’s isotherm, BET equation is

also characterized by two parameters m and C. Here, m is the maximum mono-layer

coverage and C controls the shape of the curve at low pressures. Figure 3-5 shows BET

(Brunauer et al., 1938) curves for different values of C.

15

Figure 3-5: BET Adsorption model for different values of C = 1, 10, 20, 50, 100.

The assumption of infinite adsorbed layers on solid surface is often hypothetical in real

scenarios. A more realistic model is the N-layer BET equation which restricts pore space

to a finite size. The N-layer BET equation is given as:

1

1

)1(1)1(

)1(1

N

rrr

N

r

N

rr

CPPCP

NPPNCP

m

n (2-3)

Here, N is the number of adsorbed layers. N-layer BET equation has three characterizing

parameters, m, C and N. Figure 3-6 shows sensitivity to N.

Figure 3-6: N-layer BET equation for different values of N = 1, 5, 10, 50 and at C = 100.

16

3.3. Multicomponent Adsorption Models

Generally, there are two approaches to modeling multicomponent adsorption. First is the

“Extended Langmuir’s Isotherm” (Langmuir, 1916) which is an extension to Langmuir’s

single component model. It is given as:

cN

j

jLj

iLiLi

gi

pP

pPVV

1

1

(2-4)

Here, LiP and LiV are the Langmuir’s parameters for single component adsorption and ip

is the partial pressure of the component i.

A thermodynamically more consistent approach to model multi-component adsorption is

the “Gibbs Approach”(Lin, 2010). This approach is analogous to predicting “Vapor-

Liquid Equilibrium”. The fundamental assumption of the Gibb’s Approach is the equality

of chemical potential i for component ‘i’ in the adsorbed and the gas phase.

),....,,(),....,,(cc NiiNii xxPTyyPT (2-5)

Here Π is the spreading pressure which accounts for the reduction of surface tension at

the solid-gas interface upon adsorption. The complete development of the “Vapor

Adsorption equilibrium (VAE)” can be found elsewhere (Lin, 2010). Their final equation

reduces to:

iiii xPy (2-6)

Here, iy and ix are the gas and adsorbed phase composition, i is the fugacity

coefficient in the gas phase, i is the activity coefficient accounting for the non-ideality

of the adsorbed mixture (Lin, 2010). This model is also known as “Real Adsorbed

Solution (RAS)’’ model in literature.

17

Chapter 4 Gas in Place Quantification for Shale Gas

Reservoirs

One of the critical calculations in Reservoir Engineering is to quantify how much

hydrocarbon initially resides in oil and gas reservoir. Inefficient estimate of initial gas in

place (IGIP) may lead to unviable development decisions for operators.

The two most important means to quantify IGIP for hydrocarbon reservoirs are

“Volumetrics” and “Material Balance”. Volumetric (or static) calculation uses petro-

physical, fluid and geological data to evaluate IGIP. Material balance (or Dynamic

balance) employs petro-physical, fluid and real time production data to quantify IGIP.

Usually, the closeness of these independent estimates provides assurance to make

development decisions.

In the following sections we discuss these two methods for unconventional gas

reservoirs. The equations presented here are incorporated in the numerical code to

calculate recovery parameters.

4.1. Volumetric Calculation for Shale Gas Reservoirs

Potential of adsorbing large quantities of gas in its organic nano-pores distinguish shales

from conventional reservoirs. Recent work by Ambrose et al., (2010) gives a detail

account to quantify this adsorbed gas accurately. These authors argue that some of the

space in these nano-pores is occupied by the adsorbed phase and thus reduces the space

available for free gas. Figure 4-1 shows a schematic of various constituents that

comprises the total bulk volume in shales.

In shale gas industryIGIP is usually quoted on unit mass basis-scf/ton (Ambrose et al.,

2010). Thus the total gas content in shales is given by:

) (scf/ton+V+V+V=VV swsogft (4-1)

Here, tV is the total gas-in-place; fV is the adsorbed gas corrected free gas in place; gV

is the adsorbed gas; soV and swV are the amount of gas dissolved in oil and water

respectively; each term is given on unit mass basis of shales.

The amount of gas adsorbed gV is described by Langmuir’s Isotherm as:

L

Lg

PP

PVV

(4-2)

18

Figure 4-1: Various constituents of the total bulk volume of shale gas reservoir (after Ambrose et

al., 2010).

Also, adsorbed-phase corrected free gas-in-place, fV is defined as (Ambrose et al., 2010):

L

L

ss

w

g

fPP

PVS

BV

)10(318.1)1(0368.32 6

(4-3)

Here, s is the adsorbed gas phase density. Also, the amount of gas dissolved in oil and

water can be estimated as (Ambrose et al., 2010):

o

soo

soB

RSV

6146.5

0368.32 (4-4)

w

sww

swB

RSV

6146.5

0368.32 (4-5)

19

4.2. Material Balance for Shale Gas Reservoirs

One of the classical tools employed by reservoir engineers to estimate IGIP is the

material balance equation. To apply material balance for unconventional gas reservoirs,

desorption phenomena should be considered. The generalized material balance equation

incorporating desorption term is given by (Moghadam, et al., 2009):

depwipgpgi VVVBGGGB )( (4-6)

Here, wipV is the net water encroachment, epV is the fluid and rock expansion term

(significant for over-pressured gas reservoirs) and dV is effect of desorption. These

terms are defined as:

)(615.5 wpewip BWWV (4-7)

)( PPcScScS

GBV iowiwwit

gi

gi

ep (4-8)

)( PPPP

PV

PP

PV

S

GVBV i

L

L

L

iL

gi

fgib

d

(4-9)

20

Chapter 5 Previous Modeling Efforts for

Unconventional Gas Reservoirs

In this section we review available literature and basic features of the available

commercial packages to simulate gas production from shale gas and CBM reservoirs. We

discuss various types of approaches to incorporate non-Darcy effects (both gas desorption

and slippage) in the organic nano-pores of shales. We then briefly outline our

methodology to include these non-darcy effects in our numerical model.

5.1. Gas Slippage and Diffusion

Numerous studies have been conducted to study gas desorption, slippage and molecular

diffusive mechanism which are assume to occur during the recovery of gas from shales.

Most of the previous studies attempts to modify Darcy’s equation using an apparent

permeability concept to model these effects.

Organic pores in shales have pore sizes in the range of 10-300 nm (Cipolla et al., 2011).

At this small scale, average mean free path of gas molecules is comparable to the pore-

throat size and gas molecules tend to slip in these pores. Ertekin et al. (1986) used

Klinkenberg’s apparent permeability concept (Klinkenberg, 1941) to model production

from tight-gas sands and shales. The apparent gas permeability relationship (after

Klinkenberg, 1941) is given below:

)1(

P

bkk kl

kl (5-1)

Here, bkl, is the gas-slippage factor. In the original work by Klinkenberg (1941) bkl is

assumed constant. Ertekin et al. (1986) modified its definition and proposed pressure

dependence on bkl. Their modified definition is:

k

DPcb

gg

kl

(5-2)

Here, D is the slip coefficient, is the conversion factor, cg is the gas compressibility and

μg is the gas viscosity.

Clarkson et al. (2011) also studied the gas-slippage effect and applied this concept to

analyze pressure transient tests for shale gas and CBM wells. Javadpour (2009) discussed

the importance of Knudsen diffusion in the nano-pores of shales. His work presumes that

the transport mechanism in nano-pores is a combination of viscous and Knudsen

diffusive forces. Javadpour (2009) has formulated an apparent permeability factor which

can be incorporated in reservoir simulators as an additional transmissibility multiplier to

study Knudsen’s diffusion effects on gas production from shales, this is given as:

avgavg

g

app

rF

M

RT

RT

Mrk

8)~

8(

)10(3

2 25.0

23 (5-3)

21

Here, ‘r’ is the radius of pore-throat, M~

is the molecular mass of gas. Also, in nano-

pores the no-slip conditions on the pore surface does not remain valid and the gas

molecules tend to slip in these pores, F is a dimensionless coefficient to account for this

slip-velocity in tubes, refer to Javadpour (2009) for details.

One of the major causes of higher than expected gas-rates from shales is the presence of

natural-fractures in the system. Ozkan et al., (2010) attempted to model the transfer of

flux from the shale matrix to the fracture networks. They used spherical matrix grid

blocks. This is due to their assumption that pressure on the surface of grid block is

uniform and hence flow within matrix will be spherical. Also, this work incorporates

Fick’s first law of diffusion and stress dependent natural-fractures permeability to study

gas flow in shales.

5.2. Gas Desorption

Currently available commercial packages have some desorption modeling capabilities,

however these packages differ in their approaches to model desorption. Eclipse®

(Schulumberger, 2012) uses its black-oil Coal-bed Methane module to incorporate the

desorption process. This assumes that the diffusive flux from matrix into the fractures is

governed by the following relation:

sbcgmf CCDQ

(5-4)

Here, gmfQ is the gas-flux, Ψ is matrix-fractures diffusivity, cD is the diffusion

coefficient, bC is the bulk gas concentration and sC is the gas concentration on the

surface of coal. This requires two-cells for each grid block but the flow equations are

solved in natural fractures only. In the matrix only gas concentration is tracked (Eclipse®

Schulumberger, 2012).

In IMEX (CMG®, 2012) gas desorption is modeled using the method of Seldle and Arrie

(1990). This assumes a pseudo-steady state flow regime and modifies the total

compressibility of the reservoir. Their definition of total system compressibility is given

as:

sggwwrt ccScScc (5-5)

Here, sc is the additional compressibility due to gas adsorbed within the rock matrix.

Their definition of the sorption-compressibility is:

2)1( PPb

PVc

Lg

LL

s

(5-6)

Here, bg is the inverse of formation volume factor (FVF) of gas. Seldle and Arrie (1990)

method further assumes an immobile oil phase in the reservoir and the adsorbed gas is

assumed to be dissolved in it. Also, to use conventional black oil simulators for shale gas

22

and CBM production it assumes that the behavior of Langmuir’s isotherms is similar to

the solution gas-oil ratio. The introduction of this hypothetical oil phase requires increase

in porosity and modification of fluid saturations. Gas-water relative permeability curve

and fluid properties must also be adjusted.

5.3. Our Methodology for Non-Darcy Effects

In order to generalize the conventional reservoir simulation approach to model desorption

we propose the use of an additional “accumulation term” in the mass balance equation of

a given grid block. This term will be described by an appropriate isotherm which is then

coupled with the grid block pressure. This treatment has following advantages: (1) It

preserves the material balance, honoring both adsorbed and free gas, (2) It obviates the

need for an immobile oil phase and does not require any modification of rock and fluid

properties. (3) We expect that for black-oil systems, there is no Fick’s diffusion and

transport is primarily governed by viscous forces (not due to concentration gradients). In

our black-oil two phase gas-water model we assume no concentration gradients.

We have also included the gas-slippage effects in organic pores by using the Klinkenberg

factor (Klinkenberg, 1941). This factor has been included in the definition of

transmissibility (see Appendix B for details). In the next two chapters we present

formulation and results of our proposed schemes to model gas production from shales

and CBM reservoirs.

23

Chapter 6 Numerical Models for Shale Gas and CBM

Production

In this chapter we propose two models of increasing complexity along with their

assumptions to simulate gas production from shales. These are i) single porosity model

with instantaneous desorption into pores; and a ii) dual porosity model with desorption

into nano-pores, coupled with transfer of flux from nano-pores into fractures.

To model CBM production, we will use a single porosity approach where convective

flow is only in the natural fractures with instantaneous desorption to simulate different

phases of production for typical CBM wells.

6.1. Single Porosity Model with Instantaneous Desorption

Simplest numerical approach for modeling shale gas production is based on the

assumption of single porosity in the reservoir. The gas in the adsorbed state is then

accounted for using an additional accumulation term. Langmuir’s adsorption isotherm

(Langmuir’s, 1916) is used here to model the desorption process (assuming no hysteresis

in desorption). This implies that as the pressure in the reservoir depletes, desorbed gas

transfers instantaneously into the pore space with no time lag. However, this is an

oversimplification of the actual desorption processes that occurs in the organic nano-

pores of shales. Also, we assume that the slippage of gas molecules in pores can be

modeled using the Klinkenberg model (Klinkenberg, 1941).

Now we develop the general mass balance equations in finite volume form (Cartesian

grid) for a system with fns fracture (see Figure 6-1).

t

MMmm

n

ig

n

ig

f

ffs

isgm

w

ig

N

l

igl

snct

,

1

,

,

,,,

1

,

21

(6-1)

Here, iglm , represents the mass influx from grid block l to grid block i, w

igm ,is the well

flow term, isgm ,, is the flux transfer from matrix to fracture system and n

igM , is the total

gas that resides in the considered grid block at time step n. Also, Nct represent the number

of connections of a given grid block ‘i’ with its surrounding grid blocks. In order to

incorporate adsorbed gas, n

igM ,is written as:

n

gads

n

gpv

n

ig MMM , (6-2)

24

Figure 6-1: Schematic representation of the single porosity model with instantaneous desorption.

Here, n

gpvM and n

gadsM represent the mass of gas in rock pores and in the adsorbed state

respectively, these are treated here as:

gisc

n

gpv bVM (6-3)

gimsc

n

ads VVM )1( (6-4)

Here, gV represents the amount of gas adsorbed at a given pressure within the shale

matrix (scf/ton), this is described by an appropriate isotherm and Vi is the grid block

volume; in a single porosity system, there is no fracture system, i.e. sn = 0, hence the

residual equation for fully implicit scheme simplifies to:

n

g

ngm

n

g

ng

in

g

N

l

n

li

n

iligi VVbbt

VQPPR

ct

111

1

1

,

1

),()1()( (6-5)

The Eq. 6-5 when written for each grid block results in gN coupled system of non-linear

algebraic equations which can be solved using Newton’s method. The unknown for this

system is a pressure-vector P

containing pressure for each grid block as:

Ng

Ng

P

P

P

P

P

P

1

3

2

1

(6-6)

25

6.2. Dual Porosity Model with Desorption in nano-Pores

6.2.1. Mass-Balance in Organic nano-Pores

In dual porosity systems we assume that shales consist of two distinct pore systems:

organic nano-pores and micro-fractures. The organic nano-pores are widely dispersed

within the inorganic fabric of shales (see Figure 6-2, top). Therefore, we can assume that

no flux occurs between the grid blocks representing organic porosity (see Figure 6-2,

bottom).

Secondly, the simplification made in the single porosity model about instantaneous

desorption (as discussed earlier) can be refined by assuming that desorption now occurs

in the organic porosity of shale, hence incorporated in its mass balance (see Figure 6-3).

Then the combined flux from organic porosity (both free and desorbed gas) is convected

into micro-fractures using a transfer function.

Figure 6-2: SEM image for shale sample (top) and its dual porosity discrete analog (bottom),

adapted from Ambrose et al., (2010), (dark grey, light grey and red for organic nano-

pores, inorganic fabric and micro-fractures respectively).

26

Using the general mass balance formulation, Eq. 6-1, assuming no-flux interaction

between nano-porous grid blocks and considering one fracture system only, i.e. sn = 1,

mass balance for organic porosity simplifies to:

t

MMn

ig

n

ign

fgm

,

1

,1

, (6-7)

The residual equation for fully implicit scheme for nano-porous organic grid blocks is

then given by:

n

g

ngfmm

n

g

ngm

in

fi

n

im

n

mfim VVbbt

VPPR

11111)1()( (6-8)

Here, transfer function, mf is defined as:

ggmfmf bD ~ (6-9)

And matrix-fracture flux coefficient Dmf is given as:

fm

mf

mfl

kD

, (6-10)

Figure 6-3: Schematic representation of the organic nano-pores and assumed desorption process

in dual porosity shale reservoir.

(i)

Pm(i)

Pf(i)

Desorbed gas from

shale matrix

27

6.2.2. Mass-Balance in Micro-Fractures

Micro-fractures in shales are presumed to be connected (at least sparsely) hence we

assume fluid communication with its adjacent grid blocks (see Figure 6-4). We further

assume that no gas is adsorbed in the matrix adjacent to micro-fractures.

Similarly, applying the general mass balance equation, Eq. 6-1, to micro-fracture system

gives:

)(1,

,,,

1

,

n

g

n

gf

ji

ifgm

w

ig

N

l

igl bbt

Vmm

ct

(6-11)

The residual equation for fully implicit scheme for micro-fractures grid block is then

given as:

n

g

ngf

in

fi

n

immfg

N

l

n

lfi

n

fi

n

ilgfif bbt

VPPQPPR

ct

111

1

1

,

11

),()()(

(6-12)

Figure 6-4: Schematic representation of micro-fractures and its flux interaction with nano-pores

in dual porosity shale reservoir.

This finite-volume discretization (Eq. 6-8 and Eq. 6-12) for a dual porosity system results

in 2Ng coupled non-linear algebraic equations which can be solved using Newton’s

method. The unknowns, mP and fP , are stacked in one vector as:

28

fNg

mNg

m

f

m

P

P

P

P

P

P

2

1

1

(6-13)

6.3. Modeling CBM Production

Naturally occurring coal seams are generally completely saturated with water and most of

the gas resides within the coal matrix in an adsorbed state. To model the initial de-

watering phase in CBM production, two phase flow in the reservoir must be considered.

Here, we assume that CBM has only one predominant porosity system, natural fractures,

and the gas desorbs instantaneously from the coal-matrix. The Langmuir’s Isotherm

(Langmuir’s 1916) is used here (with no hysteresis) to model the desorption process.

Applying general mass balance formulation, Eq. 6-1, for both gas and water phase,

results in the following residual equations for fully implicit scheme.

n

g

ngfm

n

g

ng

n

gf

i

N

l

n

li

n

iilgfig VVbbSt

VPPR

ct

111

1

1

,

1

),()1()( (6-14)

n

w

nw

n

wf

iN

l

n

li

n

iilwfiw bbSt

VPPR

ct

11

1

1

,

1

),()( (6-15)

When Eq. 6-15 is written for each grid block it also results in 2Ng coupled non-linear

algebraic equations and can be solved using Newton’s method. The two unknowns for

each grid block lP and gS are stacked in one vector as follows:

gNg

Ng

g

g

S

P

P

S

P

P

2

1

1

(6-16)

6.4. Code Development

In order to demonstrate performance of above presented numerical schemes, separate

code is developed for each of the three scenarios in MATLAB®

programming

environment.. In the next chapter we give results of these schemes under different

reservoir and operating conditions.

29

Chapter 7 Model Results

In this chapter we demonstrate the performance of numerical schemes presented in

Chapter 6 to simulate gas production from shales and CBM. Parameters used here for

simulation are typical of those that are commonly encountered under gas-field operating

conditions. Sensitivity studies presented here will aid to gain insight on key parameters

that govern production from shale gas and CBM reservoirs.

7.1. Single Porosity System with Instantaneous Desorption

7.1.1. Comparison with a Commercial Package

Firstly, we compare results of our single-phase gas code with Eclipse®

(Schlumberger,

2012). Here, we assume no adsorbed gas on shale matrix. This is to show the robustness

of the numerical code, prior to the incorporation of desorption. Here, we assume a 2-D

reservoir and well completed in the center. Gas production is simulated with a constant

bottom-hole pressure constraint. Details of other parameters used in this problem are

shown in Table 7-1. Figure 7-1 compares the results of our code with Eclipse®

(Schlumberger, 2012).

Table 7-1: Simulation parameters to validate single-phase gas code with Eclipse

® (Schlumberger,

2012).

Simulation

Parameters

Values

Φ 0.1

k 0.05 mD

Swi 0.0

VL 0.0 scf/ton

Pi 1000 psia

Pwf 300 psia

Nx = Ny 21

Lx = Ly 2100 fts

Lz 250 fts

IGIP 6.4 Bcf

30

Figure 7-1: Gas production profile showing model validation with Eclipse® (Schlumberger,

2012).

7.1.2. Sensitivity to Langmuir Volume ( LV )

In this section we show the results of sensitivity to LV on gas production. Well placement

and constraints are similar to those in the case presented previously. Simulation

parameters used here are shown in Table 7-2:

Table 7-2: Simulation parameters for sensitivity of gas production to VL.

Simulation

Parameters

Values

Φ 0.07

k 500 nano Darcy

Swi 0.0

VL [0; 2; 4; 6; 8; 10 ] Mscf/ton

PL 535.6 psia

Pi 3500 psia

Pwf 250 psia

Nx = Ny 21

Lx, Ly 1500, 1100 fts

IGIP [6; 60; 113; 166; 220; 274] Bcf

31

Figure 7-2 shows production profiles for different values of LV . It can be observed from

this figure that desorption of gas starts instantaneously (no time-lag) as the pressure in the

reservoir decreases. Shapes of these production profiles also remain similar to each other.

Figure 7-3 shows pressure maps at the end of simulation for these cases. These maps

indicate that high values of LV lead to comparably low reservoir depletion.

Figure 7-2: Well performance for various values VL at Pwf = 250 psia.

It has been observed (Valko and Lee, 2010) in actual field production data from shales

that after an initial decline, gas production is sustained for relatively extended period (see

Figure 7-4). This may be due to an additional flux caused by delayed desorption of gas

from the organic nano-pores in shales. This behavior is not observed with single porosity

numerical model (see Figure 7-2). Therefore, a more sophisticated model is required,

which should be consistent with both observed pore-networks and desorption phenomena

in shales.

32

Figure 7-3: Pressure maps (in psia) at the end of simulation for various values of VL.

Figure 7-4: Average gas production data from various shale gas fields in US Adapted from Valko

and Lee, (2010).

VL = 2 Mscf/ton Base Case VL = 4 Mscf/ton

VL = 6 Mscf/ton VL = 8 Mscf/ton VL = 10 Mscf/ton

Qg

(Msc

f/d

ay)

33

7.1.3. Heterogeneity in the Adsorbed Gas Regions

Heterogeneity is an inherent part of any hydrocarbon reservoir. To deal with

heterogeneity in the adsorption capacity in shales, we can assign Langmuir’s properties

(both LV and LP ) to each grid block. This is similar to handling heterogeneity in any

physical property of the reservoir such as, Φ, k, Swi and others. To demonstrate this, we

prepared a test case in which we assign random numbers to LV (ranging from 0-5000

scf/ton), in a 20 by 20, 2-D reservoir (see Figure 7-5). Other simulation parameters are

shown in Table 7-3. Figure 7-6 shows the comparison of Qg for this heterogeneous case

with base and mean case1.

Figure 7-5: Matrix of VL showing heterogeneous adsorbed gas regions.

1 In the ‘mean case’, average VL from the heterogeneous case is assigned to each grid block.

34

Table 7-3: Simulation parameters used to demonstrate heterogeneity in the adsorbed gas regions.

Simulation

Parameters

Values

Φ 0.07

k 500 nano Darcy

Swi 0.0

PL 535.6 psia

Pi 1000 psia

Pwf 250 psia

Nx = Ny 21

Lx 1500 fts

Ly 1100 fts

IGIP ~73 Bcf

Figure 7-6: Comparison of Base case (VL = 0), heterogeneous case, and mean case.

7.1.4. Impact of Hydraulic Fracturing on Shale Gas Wells

In order to demonstrate the importance of hydraulic fracturing on the performance of

shale gas wells, consider a test case with well in the center of a 2-D reservoir. Relevant

reservoir and fracture parameters are given in Table 7-4. Figure 7-7 compares the pre-

fracturing and post-fracturing well performance. It shows about 10 fold productivity

increase of a well after fracturing treatment. This simple case shows the impact of

hydraulic fracturing on the viability of shale gas wells.

35

Table 7-4: Simulation parameters to demonstrate the effect of hydraulic fracturing on well

performance.

Figure 7-7: Effect of Hydraulic fracturing on well-performance.

2 Hydrualic fracture is oriented in the center strip, parallel to the x-axis in a 2D reservoir.

Simulation

Parameters

Pre-fracturing Post-fracturing

Φ 0.07 0.07

K 500 nano Darcy 5000 nano Darcy

Swi 0.0 0.0

Nx = Ny 21 21

Lx 1500 fts 1500 fts

Ly 1100 fts 1100 fts

IGIP ~73 Bcf ~73 Bcf

Lf2 - 1500 fts

tf - 0.2 fts

khf - 10,000 mD

36

7.2. Dual Porosity Model with Desorption in nano-Pores

Here, we discuss sensitivity of gas production to Dmf and kf. We again consider a 2-D

reservoir, well in the center and constant bottom-hole pressure constraint. Other

parameters are shown in Table 7-5.

Table 7-5: Simulation parameters for sensitivity of gas production to mfD and fk

Simulation

Parameters

Values

Φf 0.01

Φm 0.05

Dmf [ 0.01; 0.05; 0.1; 0.5;] mD/ft

kf [1; 10; 100; 1000] mD

VL 200 scf/ton

PL 535.6 psia

Pi 1000 psia

Pwf 300 psia

Nx = Ny 20

Lx = Ly 1000 fts.

IGIP3

[m,f,ads,t] [0.72; 0.145;3.28;4.158] Bcf

Figure 7-8 shows sensitivity of Qg to Dmf. It indicates that high values of Dmf lead to

comparably high Qg in later-half of production. This is due to relatively high influx from

nano-pores which supplies the desorbed gas. Figure 7-9 shows sensitivity of gas

production to kf. It implies that after initial depletion from fractures (where, Qg do differ

significantly) Qg is only slightly affected by kf. Also, notice that at later times, relatively

stable Qg is due to somewhat slow and steady influx of desorbed gas, which replenishes

the produced gas of the organic nano-pores.

3 Here [m,f,ads,t] stands for IGIP in nono-pores, fractures, adsorbed gas and total gas.

37

Figure 7-8: Sensitivity of Qg to Dmf .

Figure 7-9: Sensitivity of gas production to kf .

38

7.3. Simulation of CBM Production

In order to simulate recovery from a typical CBM well we use our single porosity model

with two phase gas-water flow. While all of the gas is adsorbed on the coal matrix,

because of the assumption of instantaneous desorption, only flow in fractures is

considered in the model

7.3.1. Two Phase Gas-Water Code Validation

This is to demonstrate the robustness of our two phase gas-water code with Eclipse®

(Schlumberger, 2012). Again, we assume no adsorbed gas on matrix. Other relevant

parameters are shown in the Table 7-6. Figure 7-10 shows the match of Qg and Qw with

Eclipse® (Schlumberger, 2012).

Table 7-6: Simulation parameters for validating two phase gas-water code with Eclipse®

(Schlumberger, 2012).

Figure 7-10: Comparison of the developed code with Eclipse® (Schlumberger, 2012).

Simulation

Parameters

Values

Φf 0.1

kf 50 mD

Swi 0.5

Pi 1000 psia

Pwf 400 psia

Nx = Ny 20

Lx = Ly 2000 fts

Lz 250 fts

IGIP 2.9 Bcf

39

7.3.2. Comparison of our scheme with Eclipse® CBM Model

Here we compare the performance of our numerical scheme to simulate CBM production

with Eclipse® Black-Oil CBM model. We assume a 3-D reservoir (three layers) and well

placed at top-right corner of the middle layer. Initially, all the pores (or cleats in CBM)

are completely saturated with water. Gas resides only in an adsorbed state and it desorbs

instantaneously upon pressure depletion. Other relevant properties are listed in Table 7-

74. Figure 7-11 compares Qg for the two models. Significant differences in Qg are due to

differences in handling desorption. In Eclipse®

desorption is driven by concentration

differences, while we assume instantaneous desorption according to the Langmuir’s

desorption model. Also, Figure 7-12 shows gravity segregation of gas and water in our

model.

Table 7-7: Simulation parameters for comparing Eclipse® and our CBM models.

4 Note that Eclipse® CBM Model requires two additional parameters Ψ and Dc.

Simulation

Parameters

Values(Our Model) Values(Eclipse®

)

Φf 0.01 0.01

Swi 1.0 1.0

VL 2425 scf/ton 2425 scf/ton

Pi 4000 psia 4000 psia

Pwf 2000 psia 2000 psia.

Nx = Ny 8 8

Nz 3 3

Lx = Ly 150 fts 150 fts

Lz 150 fts 150 fts

Well co-ordinates (8, 8, 2) (8, 8, 2)

IGIP 3.6 Bcf 3.6 Bcf

Dc 0.2(ft2/s) -

Ψ 0.01 mD/ft -

40

Figure 7-11: Qg comparison of our CBM model with Eclipse

®.

Figure 7-12: Pressure and saturation maps in our model.

41

7.3.3. Sensitivity of CBM Production to Fractures Permeability (kf)

Here we use a 3-D reservoir (three vertical layers) and well completed in the center.

Table 7-8 lists relevant parameters used in simulation. Figure 7-10 shows sensitivity of

production parameters to kf. Initially, Qw is high and Qg increases as the gas starts to

desorb from the coal matrix. This is the de-watering phase followed by a depletion phase

for a typical CBM well. Figure 7-13 show that as expected, high values of kf leads to

quicker recovery.

Table 7-8: Simulation parameters for sensitivity of CBM production to kf.

Figure 7-13: Sensitivity of CBM production parameters to kf.

Simulation

Parameters

Values

Φf 0.01

kf [100; 500; 1000] mD

Swi 1.0

VL 100 scf/ton

Pi 1000 psia

Pwf 300 psia

Nx = Ny 9

Nz 3

Lx = Ly 5000 fts

Lz 150 fts

Well co-ordinates (5, 5, 2)

IGIP 6.29 Bcf

42

Chapter 8 Summary and Future Work

Shales have complex pore networks and distinct scales for fluid flow. There are at least

four unique pore scales in shales. The organic matter which is finely dispersed within the

inorganic matrix of shales has pore size in nano-scale. Gas in these pores resides both in

free and in adsorbed states. Also, the inorganic matrix has very little interconnected

porosity. In addition, shales have networks of sparsely connected micro-fractures. The

hydraulic fracturing treatment in shales not only creates a path of high fluid conductivity

but also aids in stimulating these micro-fractures.

Fluid transport in shales is assumed to be a combination of Darcy and non-Darcy

mechanisms. The pressure depletion in the reservoir causes the gas to desorb in nano-

pores. The desorbed gas is then mixed with free gas flow, both within organic and

inorganic porosity. This combined flux is then assumed to be convected into micro-

fractures and eventually into the well via conductive hydraulic fractures.

We have presented two numerical models to simulate gas production from shales —

“Single Porosity Model with Instantaneous Desorption” and a more rigorous “Dual

Porosity Model with Desorption into nano-Pores”. Figure 8-1 compares the important

features of our routine with two available commercial packages.

Figure 8-1: Comparasion of features of models available for simulating gas production from

shales.

43

The numerical models discussed so far are all black oil formulations. Recently, industry

has shown considerable interest in developing shale plays with rich liquid content.

Presence of liquid in the system will result in at least three different phases in the

reservoir: Gas, adsorbed and liquid phase. In our future work, we plan to incorporate

rigorous compositional treatment based on the Gibbs Approach (Lin, 2010).

Note that the numerical results presented here are from our simulator where hydraulic

fractures are not modeled with any rigor. To simulate performance of typical wells in

shales, rigorous treatment of these artificially-induced, highly conductive flow-paths is

essential. The flow properties of these paths differ significantly from that of unstimulated

shales. These fractures are generally stress sensitive and their flow properties change

considerably over the course of the production life of a well. The geomechanical model,

which predicts the stress-strain field in the reservoir as a function of pressure, will be

numerically coupled with the flow model to simulate these effects. We plan to develop an

algorithm and numerical implementation that should accurately predict recovery from

shale as the stress-field varies over the production life. Also, in future we plan to

incorporate all the current and future implementations in AD-GPRS5 programming

environment.

Shales will be a major supplier of energy for future generations because of their vast

resource base and the technology available for their economic exploitation. Shales are

also expected to play an important role in worldwide economic growth. Although, this

will result in substantial financial gains, careless development can also lead to adverse

environmental consequences. Efficient engineering will be required to preserve a subtle

balance.

5 AD-GPRS: Automatic Differentiation-General Purpose Research Simulator is developed internally by researchers of

ERE department of Stanford University.

44

Nomenclature

bkl = Gas Slippage Factor.

Bcf = Billions cubic feet

Bj = Formation volume factor (scf/ft3) of fluid j; where j = {o,g,w} for oil, gas and water,

respectively

bj = Inverse of formation volume factor (ft3/scf) of fluid j; where j = {o,g,w} for oil, gas

and water respectively

C = Shape factor in BET adsorption model

Cb = Bulk gas concentration (Moles/ft3)

Cs = Surface Gas concentration (moles/ft3)

cj = Compressibility of specie ‘j’, (1/psia); where j = {o,g,w,r,t,s} for oil, gas, water, rock

bulk and sorption respectively

D = Slip coefficient in Klinkerberg’s Factor (mD-psi/cp)

Dmf = Matrix-fracture flux coefficient

Dc = Gas diffusion co-efficient in Eclipse® CBM model (ft

2/s)

fns =Number of fracture system

F = Dimension less coefficient to correct slip velocity in tubes in Javadpour (2009) model

k = Absolute permeability in millidarcies

kapp = Apparent gas permeability in millidarcies

khf = Permeabilitity of hydraulic fractures

kkl = Klinkernberg factor

G = Initial gas in place (scf) or IGIP.

Gp = Total gas produced (scf)

Δlm,f = Hypothetical length between matrix grid block and fracture

L(x,y,z) = Dimension of reservoir (fts)

Lf = Length of fracture (fts)

M = 103, MM = 10

6

= Molecular mass of gas (lb/mole)

m = Maximum monolayer coverage adsorbed on solid in BET adsorption model

(moles/ton)

45

mgl,i = Mass influx from grid block l to grid block i

mg,iw= Mass flow rate for injection or production well in a grid block i

Mg,in = Total gas in the grid block at time step n (lb)

Mgpvn= Mass of gas in pore volume (lb)

Mgadsn= Mass of gas in the adsorbed state (lb)

n = Number of moles adsorbed on solid (moles/ton)

nc = Number of components in multicomponent systems

ns = Number of fracture systems

N = Number of adsorbed layers in N-layer BET equations

Nct = Number of fluid connections of a given grid block ‘i’ with surrounding grid blocks.

Nj = Number of grid block in j-direction where j = {x, y, z}

Ng = Total number of grid blocks

pi = Partial pressureP = Gas phase pressure (psia)

P° = Saturation pressure of gas (psia)

Pfi = Pressure in fracture (psia)

Pi = Initial reservoir pressure (psia)

PL = Langmuir’s pressure constant (psia)

Pmi = Pressure in matrix (psia)

Pr = Reduced pressure

Pwf = Well flowing pressure (psia) gQ = Gas rate at standard conditions (scf/day)

gmfQ = Gas flux from matrix to fractures (scf/day)

r = Radius of pore throats (in)

R = Universal gas constant, 10.37 Psia-ft3/mole-R

Rso = Solution gas-oil ratio (scf/bbl)

Rsw = Solution gas-water ratio (scf/bbl)

Rfi = Residual term for fracture grid block

Rgi = Residual term for gas mass balance

Rmi = Residual term for matrix grid block

Rwi = Residual term for water mass balance

scf = Standards cubic feet ft3

Sj = Fluid saturation of jth

fluid; where j = {o,g,w} for oil, gas and water, respectively

46

Swi = Initial water saturation in reservoir

T = Temperature in R or °F

Tcf = Trillion standard cubic feet

Δt = time step (days)

tf = Thickness of fracture (fts)

Vf = Adsorbed gas corrected free-gas in place (scf/ton)

Vg = Amount of gas adsorbed at gas pressure P on solid surface (scf/ton)

Vi,j =Volume of grid block (i, j) (ft3)

VL = Langmuir’s volume constant (scf/ton)

Vt = Total gas content per-unit mass basis (scf/ton)

Vso= Gas content dissolved in oil (scf/ton)

Vsw = Gas content dissolved in water (scf/ton)

𝜟Vd = Net gas desorbed (ft3)

𝜟Vep = Net rock and fluid expansion (ft3)

𝜟Vwip = Net water encroached (bbl)

Vsw = Gas content dissolved in Water (scf/ton)

We = Total water encroached from aquifers into the reservoir (bbl)

Wp = Total water produced from the reservoir (bbl)

xi = Adsorbed phase composition

yi = Gas phase composition

z = Gas compressibility factor

T = 60 °F and P = 14.7 psia are used here as standard conditions

α = unit coversion factor, 0.006328

= Chemical potential for the component ‘j’

= Viscosity (1/cp) ;where j = {o,g,w} for oil, gas and water respectively

= Inverse of viscosity (1/cp) ;where j = {o,g,w} for oil, gas and water respectively

ρb = Bulk density of rock (gm/cm3)

ρm = Matrix density of grain (gm/cm3)

ρs = Density of adsorbed gas (gm/cm3)

ρsc = Density of gas at standard conditions (gm/cm3)

ρavg = Average density

47

Π = Spreading pressure, psia

φi = Fugacity coefficient for component i

Φj = Porosity; where j = {m,f}, for matrix and fracture porosity, respectively

δi = Activity coefficient

τm,f = Mass-flux rate from matrix to fracture

γmf = Transfer function from matrix to fracture

Γxy(a,b) = Transmissibility in the x system ( x = {m,f} ) of the y phase at (a,b) interface

(mD/ft/cp)

Ψ = Matrix fracture diffusivityin Eclipse® CBM model

48

References

“Annual Energy Outlook Early Release, 2012”, US DOE, 2012.

“Coal-Bed Methane: Potential and Concerns, Oct 2000, USGS Fact Sheet fs-123-00.

“Modern Shale Gas Development in the United States”, April 2009, report prepared for

US DOE, Office of Fossil Energy and National Energy Technology Laboratory, by

Ground Water Prevention Council and All-Consulting.

“World Shale Gas Resources: An Initial Assessment of 14 regions Outside the United

States”, April 2011, report published by US DOE, Washington DC, 20585.“Eclipse®

Reference Manual and Eclipse-100 Black Oil Simulator”, Schlumberger, 2012.

Ambrose, R.J., Hartman, R.C., Campos, M.D., Akktulu, and Y.I., Sondergeld, C.H.,

2010, “New Pore-scale Considerations for Shale Gas-In-Place Calculations”, presented

at 2010 SPE Annual Technical Conference and Exhibition, feb 23-25, Pittsburgh,

Pennsylvania, USA, SPE 131772.

Aziz, K., Durlofsky, L., and Tchelepi, H., Winter 2011, “Notes for ENERGY 223:

Reservoir Simulation”, Stanford University.

Benoin, B.D., and Thomas, F.B., 2005, “Formation Damages Issues Impacting the

Productivity of Low Permeability, Low Initial Water Saturation Gas Producing

Formations”, Trans. AIME, v. 127, pp. 240-247.

Boyer, C., Keischnick, J., Lewis, R.E., Waters, G., 2006, “Producing Gas from its

Source”, Oil Field Review, v. 18, no. 3, pp. 36-49.

Bustin, R.M., Bustin, A.M.M, Cui, X., Ross, D.J.K, and Pathi, V.S.M., 2008, “Impact

of Shale Properties on Pore Structure and Storage Characteristics”, SPE 119892.

Brunauer S., Emmett P.H., and Teller E. 1938. “Adsorption of Gases in Multi

molecular Layers”. Journal of the American Chemical Society 60: 309 – 319.

49

Curtis, J.B., 2002, “Fractured Shale-Gas Systems”, AAPG Bulletin, v. 86, no. 11, pp.

1921-1938.

Dallege, T.A., Barker, C.E., “Coal-bed Methane Gas-In-Place Resource Estimates

Using Sorption Isotherms and Burial History Reconstruction: An Example from the

Ferron Sandstone Member of the Mancos Shale, Utah”, Chapter-L of Geologic

Assessment of Coal in the Colorado Plateau: Arizona, Colorado, New Mexico, and

Utah, U.S. Geological Survey Professional Paper 1625–B

Davies, D, Bryant, W.R., Vessel, R.K., Burkett, P.J., 1991, “Porosity, Permeability and

Micro-fabric of Devonian Shale” in Bennett, R.H., Bryant, W.R., Hulbert, M.H., eds.,

“Micro-Structure of fine grain sediments: from Mudstone to Shale: New York,

Springer-Verlag, pp. 109-119.

Do D.D. 2008. “Adsorption Analysis: Equilibria and Kinetics”. Imperial College Press.

Dranchuk, P.M., Purvis, R.A., Purvis, Robinson, D.B., 1973, “Computer Calculation of

Natural-Gas Compressibility using Standing and Katz Correlation”, presented at

Annual Technical Meeting May 8-12, 1973, Edmonton, Canada, Petroleum Society of

Canada.

Jenkins, C.D., and Boyer, C.M., SPE, 2008, “Coalbed- and Shale-Gas Reservoirs”,

Distinguished Author Series Article, JPT, Feb, 2008, pp 92-99, SPE 103514-MS.

Kawata, Y. and Fujita, K. 2001, “Some Predictions of Possible Unconventional

Hydrocarbons Availability Until 2100”, presented at SPE Asia Pacific Oil and Gas

Conference, Jakarta, Indonesia, 17-19 April, SPE 68755-MS.

Klinkenberg, L.J., 1941, “Permeability of Porous Media to Liquid and Gases”, Drilling

and Productions Practices, API, pp. 200-213.

Langmuir I. 1916. “The Constitution and Fundamental Properties of Solids and Liquids.

Part I. Solids”. Journal of the American Chemical Society 38: 2221–2295.

Lee, A.L. and Eakin, B.E., 1964, “Gas-Phase Viscosity of Hydrocarbon Mixtures”. SPE

Journal, volume 4, no. 3, 1964, pp. 247-249.

50

Lin, W., 2010, “Gas Sorption and the Consequent Volumetric and Permeability Change

of Coal”, Ph.D. Dissertation, Stanford University, California.

Moghadam, S., Jeje, O., Mattar, 2009 “Advanced Gas Material Balance, in Simplified

Format”, presented at Canadian International Petroleum Conference (CIPC), Paper

2009-149.

Perrodon, A., 1983, “Dynamics of Oil and Gas Accumulation”, Elf Aquitaine, pp. 368.

Smith, J.M., Van Ness, H.C., M.M., 2001, “Introduction to Chemical Engineering

Thermodynamics”, 6th edition, pp. 581-585.

Soeder, D.J., 1988, “Porosity and Permeability of Eastern Devonian Gas Shale: SPE

Formation Evaluation March, pp. 116-124.

Valko, P.P., Lee, W.J.,, 1988, “A Better Way to Forecast Production from

Unconventional Gas Wells”, presented at 2010 SPE Annual Technical Conference and

Exhibition, Sep 19-22, Florence, Italy, USA, SPE 134231-MS.

Wang, F.P., and Reed, R.M., 2009, “Pore Networks and Fluid flow in Gas Shales”,

presented at 2009 SPE Annual Technical Conference and Exhibition, Oct 4-7, New

Orleans, Louisiana, USA, SPE 124253.

Yang R.T. 1987. “Gas Separation by Adsorption Processes”. Butterworths Publishers.

51

Appendix A

Fluid Properties

Here, we will show the relevant fluid properties used in the simulation.

A. For compressibility factor (z) we used the Dranchuk correlation (see Dranchuk et

al., 1973).

B. For gas viscosity (μg) ‘Lee Correlation’ is used. (see Lee, 1964).

Figure ‎A-1: Compressibility factor (z), its derivative w.r.t pressure, inverse of gas viscosity and

its derivative, respectively.

52

Figure ‎A-2: Inverse of FVF of gas, its derivative w.r.t pressure, gas compressibility cg, and its

derivative respectively.

Figure ‎A-3: Vg, its derivative w.r.t pressure, Klinkenberg factor and its derivative w.r.t pressure.

53

Figure ‎A-4: Relative permeability curve used in CBM model.

54

Appendix B

Detail formulation of the Jacobian Matrix.

In this section we formulate the elements of Jacobian matrix for the models discussed

in Chapter 6. Here, we have used 2-D Cartesian grid for all the cases.

B.1. Single Porosity Numerical Model.

The residual equation formulated for a single porosity model (as derived in Chap-6, Eq.

6-5) for a Cartesian-grid is given as;

n

g

ngm

n

g

ng

ji

jiscjijijig

jijijig

ji

VVbbt

V

qPPPPR

11,

),(,,1,)

2

1,(

1,,)

2

1,(

,

)1(

)()(

(B.1-1)

The derivative of the residual equation w.r.t to Pi,j is given as;

ji

g

m

ji

gji

ji

jisc

jiji

ji

jig

jigjiji

ji

jig

jigji

ji

P

V

P

b

t

V

P

q

PPP

PPPP

R

,,

,

,

),(,

,1,

,

),2

1(

),2

1(

1,,

,

)2

1,(

)2

1,(

,

,

)1(

)()(

(B.1-2)

Here, ‘Γ’ is the transmissibility at the interface of two grid-blocks. For single phase flow

it is defined as

)()(~)( PbPPkl

kAggklg

(B.1-

3)

Where kkl is the ‘Klinkenberg factor’ (Klinkenberg, 1941), which is defined as;

abs

gg

klk

Dck

1 (B.1-4)

Here, D is the Diffusion coefficient, in this study we have used D = 0.2 (mD-psi/cp).

55

The derivative of the desorption term in Eq. B.1-2 for Langmuir’s Isotherm is given as;

2)( L

LLg

PP

PV

P

V

(B.1-5)

Similarly, the derivative of residual w.r.t Pi±1,j is given as;

)(,1,

,

),2

1(

),2

1(

,1

,

jiji

ji

jig

jig

ji

jiPP

PP

R

(B.1-6)

Other derivative terms can be derived similarly. The above set of equation will result in a

Jacobian Matrix having penta-diagonal structure.

B.2. Dual Porosity Numerical Model.

The residual formulation for Dual porosity model (as derived in Chap-6) for both matrix

and micro-fractures grid-block are given as;

n

g

ngfmm

n

g

ngm

ji

jifjimmfjim VVbbt

VPPR

11,

),(),(,)1()( (B.2-1)

n

g

ngf

ji

jifjimmf

jifjifjigfjifjifjigfjif

bbt

VPP

PPPPR

1,

),(),(

),1(),(),2

1()1,(),()

2

1,(,

)(

)()(

(B.2-

2)

The derivative of Rm(i,j) w.r.t. Pm(i,j) is given as;

1

),(

1

),(

,

),(),(

),(),(

),()1()(

n

jim

g

fmm

n

jim

g

m

ji

jifjim

jim

mf

mf

jim

jim

P

V

P

b

t

VPP

PP

R

(B.2-3)

The derivative of Rm(i,j) w.r.t. Pm(i,j±1) and Pm(i±1,j) both vanishes i.e. ;

0

;0

)1,(

,

),1(

),(

jim

jim

jim

jim

P

R

P

R

(B.2-4)

Also, the derivative of Rm(i,j) w.r.t Pf(i,j) is given by;

56

)( ),(),(

),(),(

),(

jifjim

jif

mf

mf

jif

jimPP

PP

R

(B.2-

5)

The derivative of Rm(i,j) w.r.t. Pf(i,j±1) and Pf(i±1,j) also vanishes i.e. ;

(B.2-6)

The derivative of Rf(i,j) w.r.t Pf(i,j) is given as;

1

),(

,

),(

),(),(

,

),1(),(

,

),1(

),1()1,(),(

,

)2

1,(

)2

1,(

),(

),(

)(

)()(

n

jif

gji

jif

sc

jifjim

jfi

mf

mf

jifjif

ji

jigf

jigfjifjif

ji

jigf

jigfjif

jif

P

b

t

V

P

qPP

P

PPP

PPPP

R

(B.2-7)

Similarly, the derivative of Rf(i,j) w.r.t. Pf(i±1,j) and Pf(i,j±1) are formulated as;

)( ),1(),(

),(

),2

1(

),2

1(

),1(

),(

jifjif

jif

jif

jifjif

jifPP

PP

R

(B.2-8)

)( )1,(),(

),(

)2

1,(

)2

1,(

)1,(

),(

jifjif

jif

jif

jifjif

jifPP

PP

R (B.2-9)

Also, the derivative of Rf(i,j) w.r.t to Pm(i,j) is given as;

)( ),(),(

,),(

),(

jifjim

jfi

mf

mf

jim

jifPP

PP

R

(B.2-

10)

The derivative of Rf(i,j) w.r.t to Pm(i,j±1) and Pm(i±1,j) vanishes i.e.;

0

;0

)1,(

,

),1(

),(

jim

jif

jim

jif

P

R

P

R

(B.2-11)

0

;0

)1,(

),(

),1(

),(

jif

jim

jif

jim

P

R

P

R

57

The above set of equations results in a block (2×2) Penta-diagonal structure.

B.3. Two Phase Gas-Water CBM Model.

The residual equations for CBM model as derived in (chap-6 Eq. 6-14 and Eq.6-15) are

given as;

n

g

ngfm

n

g

ngwf

ji

jijijigjijijigjig

VVbbSt

V

PPPPR

11,

,1,),2

1(1,,)

2

1,(,

)1()1(

)()(

(B.3-1)

n

w

nwwf

ji

jijijiwjijijiwjiw bbSt

VPPPPR

1,

,1,),2

1(1,,)

2

1,(,

)()( (B.3-2)

Here, g and w is defined as;

)()(~)( PbPSkl

kAggwrgg

(B.3-3)

)()(~)( PbPSkl

kAwwwrww

(B.3-4)

The derivative of residual ),( jigR w.r.t. jiP, is given as;

ji

g

fm

ji

g

wf

ji

ji

gsc

jiji

ji

jig

jigjiji

ji

jig

jig

ji

jig

P

V

P

bS

t

V

P

q

PPP

PPPP

R

,,

,

,

,1,

,

),2

1(

),2

1(1,,

,

)2

1,(

)2

1,(

,

),(

)1()1(

)()(

(B.3-5)

The derivative of ),( jigR w.r.t jiP ,1 is given as;

)(,1,

,1

),2

1(

),2

1(,1

),(

jiji

ji

jig

jig

ji

jig

PPPP

R

(B.3-6)

and the derivative of ),( jigR w.r.t. 1, jiP is given as;

58

)(1,,

,1

)2

1,(

)2

1,(

1,

),(

jiji

ji

jig

jig

ji

jigPP

PP

R (B.3-

7)

Similarly the derivative of ),( jigR w.r.t.

),( jiwS is given as;

gf

ji

jiw

gsc

jiji

jiw

jig

jiji

jiw

jig

jiw

jig

bt

V

S

q

PPS

PPSS

R

,

,

,1,

,

),2

1(

1,,

,

)2

1,(

),(

),()()(

(B.3-8)

and the derivative of ),( jigR w.r.t.

),1( jiwS

and )1,( jiwS is given as;

)(,1,

),1(

),2

1(

),1(

),(

jiji

jiw

jig

jiw

jig

PPSS

R

(B.3-9)

)(1,,

)1,(

)2

1,(

)1,(

),(

jiji

jiw

jig

jiw

jig

PPSS

R (B.3-10)

Similarly, the derivative of residual ),( jiwR w.r.t. jiP, is given as;

ji

w

wf

ji

ji

wsc

jiji

ji

jiw

jiwjiji

ji

jiw

jiw

ji

jiw

P

bS

t

V

P

q

PPP

PPPP

R

,

,

,

,1,

,

),2

1(

),2

1(1,,

,

)2

1,(

)2

1,(

,

),()()(

(B.3-11)

and the derivative of ),( jiwR w.r.t. jiP ,1 and 1, jiP is given as

)(,1,

,

),2

1(

),2

1(

,1

),(

jiji

ji

jiw

jiw

ji

jiwPP

PP

R

(B.3-12)

)(1,,

,

)2

1,(

)2

1,(

1,

),(

jiji

ji

jiw

jiw

ji

jiwPP

PP

R (B.3-13)

Also, the derivative of ),( jiwR w.r.t.

),( jiwS is given as;

59

wf

ji

jiw

wsc

jiji

jiw

jiw

jiji

jiw

jiw

jiw

jiw

bt

V

S

q

PPS

PPSS

R

,

,

,1,

,

),2

1(

1,,

,

)2

1,(

),(

),()()(

(B.3-14)

and the derivative of ),( jiwR w.r.t.

),1( jiwS

and )1,( jiwS is given as;

)(,1,

),1(

),2

1(

),1(

),(

jiji

jiw

jiw

jiw

jiwPP

SS

R

(B.3-15)

)(1,,

)1,(

)2

1,(

)1,(

),(

jiji

jiw

jiw

jiw

jiwPP

SS

R (B.3-16)