MS Thesis Jalali

8/4/2019 MS Thesis Jalali

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A Coalbed Methane Simulator Designed for the Independent Producers

Jalal Jalali

Thesis submitted to the

College of Engineering and Mineral Resourcesat West Virginia University

in partial fulfillment of the requirements

for the degree of

Master of Sciencein

Petroleum & Natural Gas Engineering

Shahab D. Mohaghegh, Ph.D., Chair

James AmmerSam Ameri

Department of Petroleum and Natural Gas Engineering

Morgantown, West Virginia

2004

Keywords: Reservoir Simulation, Coalbed Methane, Desorption, Jacobian Matrix


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UMI Number: 1423995

1423995

2005

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Copyright

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ABSTRACT

A Coalbed Methane Simulator Designed for the

Independent Producers

Jalal Jalali

The purpose of this study is to build a PC-Windows based model for two-phase one-

dimension water & gas system based on the formulation of the model developed by King

and Ertekin in 1985. This model is written in Visual Basic 6.0. The advantage of this

model over the King and Ertekins model is the ability of running the model on personal

computers, graphical abilities for input and output, and solving the matrices using an

internal solver instead of using a commercial solver. The advantage of this model over

other existing models is its ease of to use and being inexpensive that makes this model

attractive especially to the independent producers. Fully implicit, generalized Newton-

Raphson procedure was used to solve the nonlinear equations and Gaussian elimination

was used to solve the Jacobian matrix.


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iii

Acknowledgements

I would like to express my sincere thanks to my advisor and committee chairman, Dr.

Shahab D. Mohaghegh, for his guidance and patience throughout my study that without

his help, this study could not have been done. I would also like to thank him for giving

me the opportunity to come to West Virginia University and study under his supervision.

I would like to express my sincere thanks to my committee members Sam Ameri and

James Ammer for their guidance and patience throughout my study.

I would like to express my sincere thanks to Dr. Razi Gaskari for his guidance and

patience throughout my study in West Virginia University.

Finally, I would like to thank my family who has always supported me and provided

me the situation to come to West Virginia University.


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Table of Contents

Acknowledgements ......................................................................... i

Table of Contents .......................................................................... iv

List of Tables .................................................................................vi

List of Figures...............................................................................vii

Chapter I Introduction.................................................................. 1I-1 Statement of the Problem.......................................................................................... 3

Chapter II Literature Review....................................................... 4II-1 Coal Gas................................................................................................................... 6

II-1-1 Coal Seams as Natural Gas Reservoirs............................................................. 6

II-2 Review of Mathematical Models......................................... 14II-2-1 Empirical Sorption Models................................................................................. 15

II-2-1-1 Aireys First Model ..................................................................................... 15II-2-1-2 Decline Curves ............................................................................................ 16

II-2-1-3 Lindine et al. Model .................................................................................... 17

II-2-1-4 McFall et al. Model ..................................................................................... 18II-2-2 Equilibrium Sorption Models............................................................................. 20

II-2-2-1 Aireys Second Model and Kissells Model ............................................... 20

II-2-2-2 Bumbs Model, McKee and Bumbs Model, and Bumb and McKees Model................................................................................................................................... 21

II-2-2-3 Gorbachev et al.s Model, Karagodin and Krigmans Model, and

Vorozhtsov et al.s Model......................................................................................... 26

II-2-3 Non-Equilibrium Sorption Models................................................................. 28

II-2-4 Unsteady State Models ................................................................................... 31Chapter III Methodology ............................................................ 33

III-1 Macropore Transport Equations in Radial Coordinate System............................ 34

III-1-1 Single-Phase Gas Equation ........................................................................... 35

III-1-2 Single-Phase Conventional Gas Equation..................................................... 40III-1-3 Two-Phase Gas Equation .............................................................................. 40

III-1-4 Two-Phase Water Production........................................................................ 41

III-1-5 Auxiliary Macropore Equations .................................................................... 43III-1-6 Diffusion/Sorption Model ............................................................................. 44

III-1-7 Initial and Boundary Conditions ................................................................... 52

III-1-8 Single-Phase Gas Model ............................................................................... 52

III-1-9 Two-Phase Water and Gas Model................................................................. 53III-2 Macropore Transport Equations in Elliptical Coordinate System........................ 55

III-2-1 Single-Phase Gas Equation ........................................................................... 55

III-2-2 Single-Phase Conventional Gas Equation..................................................... 55III-2-3 Two-Phase Gas Equation .............................................................................. 56

III-2-4 Two-Phase Water Equation........................................................................... 56

III-3 Finite-Difference Calculus ................................................................................... 57III-3-1 Finite-Difference Operators .......................................................................... 57


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III-3-2 Backward-Difference Operator..................................................................... 57

III-3-3 Central-Difference Operator ......................................................................... 58III-4 Explicit and Implicit ............................................................................................. 59

III-5 Grid Systems......................................................................................................... 60

III-5-1 Cylindrical Grid Systems .............................................................................. 60

III-5-2 Elliptical Grid Systems.................................................................................. 62III-6 Finite Difference Approximation ......................................................................... 64

III-6-1 Macropore Transport Equations.................................................................... 64III-6-2 Micropore Transport Equation...................................................................... 72

III-6-3 Initial and Boundary Conditions ................................................................... 75

III-6-3-1 Initial Conditions ....................................................................................... 75III-6-3-2 External Boundary Conditions................................................................... 75

III-6-3-3 Internal Boundary Conditions.................................................................... 75

III-7 Solution Procedure ............................................................................................... 76

III-7-1 Newton-Raphson Procedure.......................................................................... 76

Chapter IV Results & Discussion............................................... 82

IV-1 Enhancements of CBM-SWRM to King & Ertekins Model............................... 82IV-2 Comparison of the Results with Published Data and Commercial Simulators .... 98

IV-2-1 Comparison with Published Data.................................................................. 98

IV-2-2 Comparison with Commercial Models ....................................................... 112IV-3 Sensitivity Analysis.117

IV-3-1 Macropore Porosity..................................................................................... 118

IV-3-2 Rock Permeability....................................................................................... 120IV-3-3 Gas Content................................................................................................. 122

IV-3-4 Sorption Time ............................................................................................. 124

IV-3-5 Drainage Area ............................................................................................. 129

Chapter V Conclusion ............................................................... 131


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List of Tables

TABLE II-1 COMPOSITION OF GAS FROM SEVERAL DOMESTIC SEAMS, MOL% (COMPILED

BY JONES ET AL.) 6TABLE II-1 COMPOSITION OF GAS FROM SEVERAL DOMESTIC SEAMS, MOL% (COMPILED

BY JONES ET AL.) 6TABLE III-1. MICROPORE MATRIX GEOMETRIES, PREFACTORS, AND SHAPE FACTORS

(AFTER BOYER ET AL.) 49TABLE III-2 VALUE OF BULK VOLUME AND GEOMETRIC PREFACTORS USED IN THE

CYLINDRICAL FINITE DIFFERENCE EQUATIONS. 69TABLE III-3 VALUE OF BULK VOLUME AND GEOMETRIC PREFACTORS USED IN THE

ELLIPTICAL FINITE DIFFERENCE EQUATIONS. 69TABLE IV-1 INPUT DATA FOR CBM-SWRM 99TABLE IV-2 INPUT DATA FOR CBM-SWRM FOR HISTORY MATCH 109TABLE IV-3 INPUT DATA USED IN CBM-SWRM AND CMG 113TABLE IV-4 ADDITIONAL INPUT DATA USED IN CMG 116TABLE IV-5 INPUT DATA USED IN CBM-SWRM 117TABLE IV-6 INPUT DATA FOR CBM-SWRM 125


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List of Figures

FIGURE II-1 PLAN VIEW OF THE DUAL POROSITY NATURE OF COAL 7FIGURE II-2 RELATIONSHIP BETWEEN RANK, DEPTH, AND SORPTIVE CAPACITY (AFTER

KIM) 9FIGURE II-3 THREE STAGE GAS TRANSPORT IN COALBED METHANE 10FIGURE II-4 RESIDUAL GAS CONTENT AS A FUNCTION OF DEPTH (AFTER HINSLEY ET AL.)

18FIGURE II-5 GAS CONTENT VERSUS DEPTH AND RANK FOR PICEANCE BASIN COALS

(AFTER MCFALL ET AL.) 19FIGURE III-1. RESERVOIR MODEL FOR COAL SEAMS 33FIGURE III-2. SCHEMATIC REPRESENTATION OF THE DUAL MECHANISM APPROACH TO

SLIP FLOW 37FIGURE II-3. TYPICAL LANGMUIR SORPTION ISOTHERM 46FIGURE II-4. COMPARISON OF THE PSEUDOSTEADY STATE DIFFUSION/SORPTION AND THE

UNSTEADY STATE DIFFUSION/SORPTION MODEL FOR SLAB MATRIX SUBELEMENTS

50FIGURE II-5. COMPARISON OF THE PSEUDOSTEADY STATE DIFFUSION/SORPTION AND THE

UNSTEADY STATE DIFFUSION/SORPTION MODEL FOR SPHERE MATRIXSUBELEMENTS 51

FIGURE III-6 TYPICAL GRID SYSTEM FOR AN UNSTIMULATED, DRAINAGE WELL MODELED

IN CYLINDRICAL COORDINATES 61FIGURE III-7 TYPICAL GRID SYSTEM FOR STIMULATED, VERTICAL DRAINAGE WELL

MODELED IN ELLIPTICAL COORDINATES 62FIGURE III-8 BAND STRUCTURE OF JACOBIAN MATRIX 80FIGURE IV-1 INTERFACE OF CBM-SWRM 84FIGURE IV-2 MENU OF THE CBM-SWRM 85FIGURE IV-3 CBM-SWRM INTERFACE. TWO TABS FOR ENTERING RESERVOIR PROPERTIES

86FIGURE IV-4 GAS PRESSURE DISTRIBUTION SHOWN IN CBM-SWRM DURING SIMULATION

PROCESS 87

FIGURE IV-5 GAS PRODUCTION, GAS DESORPTION, AND WATER PRODUCTION SHOWNGRAPHICALLY IN CBM-SWRM 88

FIGURE IV-6 CUMULATIVE GAS AND WATER PRODUCTION SHOWN IN CBM-SWRM 89FIGURE IV-6 CUMULATIVE GAS AND WATER PRODUCTION SHOWN IN CBM-SWRM 89FIGURE IV-7 PERCENTAGE OF CUMULATIVE GAS PRODUCED VS. OGIP SHOWN IN CBM-

SWRM 90FIGURE IV-8 RESERVE ESTIMATION IN CBM-SWRM 91FIGURE IV-9 BUILDING BATCH FILE IN CBM-SWRM 92FIGURE IV-10 SHAPE OF UNIFORM PROBABILITY DENSITY FUNCTION 93FIGURE IV-11 SHAPE OF UNIFORM PROBABILITY DENSITY FUNCTION 93FIGURE IV-12 SHAPE OF TRIANGULAR PROBABILITY DENSITY FUNCTION 94FIGURE IV-13 SHAPE OF UNIFORM PROBABILITY DENSITY FUNCTION 94FIGURE IV-14 SHAPE OF UNIFORM PROBABILITY DENSITY FUNCTION 94

FIGURE IV-15 PRODUCTION SCHEDULE WINDOW FOR SCHEDULING THE PRODUCTION OFTHE WELL 95FIGURE IV-16 GAS PRODUCTION OF A WELL WHEN IT HAS BEEN SHUT-IN FOUR TIMES AND

WHEN IT PRODUCES CONTINUOUSLY 96FIGURE IV-17 GAS PRODUCTION OF THE WELL WHICH HAS STARTED PRODUCING AFTER

IT HAS BEEN SHUT-IN FOR A FEW DAYS 97FIGURE IV-18 RESULTS FROM KING AND ERTEKINS MODEL 100FIGURE IV-19 RESULTS FROM CBM-SWRM 100FIGURE IV-20 RESULTS FROM KING AND ERTEKINS MODEL, GRAPHS 1 & 7 102


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FIGURE IV-21 RESULTS FROM CBM-SWRM FOR 75-FT FRACTURE WITH INFINITE-

CONDUCTIVITY 102FIGURE IV-22 RESULTS FROM KING AND ERTEKINS MODEL 103FIGURE IV-23 RESULTS FROM CBM-SWRM FOR 75-FRACTURE WITH INFINITE-

CONDUCTIVITY 104FIGURE IV-24 COMPARISON OF THE RECTANGULAR, ELLIPTICAL/CYLINDRICAL MODELS

FOR INFINITE-CONDUCTIVITY FRACTURE 105FIGURE IV-25 RESULTS FROM CBM-SWRM FOR 200-FT FRACTURE WITH INFINITE-

CONDUCTIVITY 106FIGURE IV-26 RESULTS FROM CBM-SWRM FOR 200-FT FRACTURE WITH INFINITE-

CONDUCTIVITY 106FIGURE IV-27 RESULTS FROM KING AND ERTEKINS MODEL FOR 200-FT FRACTURE WITH

INFINITE-CONDUCTIVITY 107FIGURE IV-28 GAS PRODUCTION USING CBM-SWRM 110FIGURE IV-29 WATER PRODUCTION USING CBM-SWRM 110FIGURE IV-30 WATER AND GAS PRODUCTION FROM MARY LEE COAL GROUP AND KING

AND ERTEKINS MODEL 111FIGURE IV-31 GAS FLOW RATE PREDICTED WITH CMG AND CBM-SWRM 114FIGURE IV-32 WATER FLOW RATE PREDICTED WITH CMG AND CBM-SWRM 115FIGURE IV-33 GAS FLOW RATE PLOTS FOR DIFFERENT POROSITIES 118FIGURE IV-34 WATER FLOW RATE PLOTS FOR DIFFERENT POROSITIES 119FIGURE IV-35 GAS FLOW RATE PLOTS FOR PERMEABILITIES 10, 25, AND 40 MD 120FIGURE IV-36 WATER FLOW RATE PLOTS FOR PERMEABILITIES 10, 25, AND 40 MD 121FIGURE IV-37 EFFECT OF COAL GAS CONTENT ON GAS FLOW RATE 122FIGURE IV-38 EFFECT OF COAL GAS CONTENT ON GAS PRODUCTION 123FIGURE IV-39 EFFECT OF SORPTION TIME ON GAS PRODUCTION 126FIGURE IV-40 EFFECT OF SORPTION TIME ON WATER PRODUCTION 127FIGURE IV-41 EFFECT OF DRAINAGE AREA ON THE GAS FLOW RATE 129FIGURE IV-42 EFFECT OF DRAINAGE AREA ON WATER FLOW RATE 130


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Chapter I Introduction

Coalbed Methane (CBM) is becoming a significant portion of the US gas resource

and is gaining importance in Australia, China, Indonesia and Europe. CBM reserves in

the United States are estimated at some 450 Tscf. In Australia, CBM resources exceed

300 Tscf, while China has a resource potential greater than the United States and

Australia combined. Recent advances in well design and production technology offer

scope to significantly increase the proportion of gas contained in coal that can be

commercialized. However, a different mindset to that applied to conventional gas

development is necessary for both producers and customers if the full potential of this

vast resource is to be realized.1

Until few years ago, coal-mining practice was to vent the methane associated with

coal into the atmosphere. Obviously, this was a waste of a valuable natural resource. So it

was important to find ways to produce this gas. Due to complexity of the coal as a

reservoir and different mechanism of gas storage in coal, conventional methods are not

applicable to coalbed methane reservoirs. This makes production prediction from CBM

reservoir quite challenging. Type curve analysis can be applied to CBM production after

the reservoir Pseudo-Steady State. Simulation can also be performed for CBM reservoirs,

but the conventional simulators do not apply for CBM due to dual-porosity system of

coal. Gas is stored in CBM reservoirs due adsorption, therefore the storage mechanism

should be modified in conventional models to account for gas storage in CBM reservoirs.

The software presented in this study has the ability of modeling a CBM reservoir, with a

dual-porosity method in a radial coordinate system.

The purpose of building this software is to build an easy to use and inexpensive

simulator for Coalbed Methane reservoir modeling.The formulation of this model is based on King and Ertekin model developed in

1985. The advantage of this model over King and Ertekin model is the ability of using

this model on personal computers. Also, the interface and graphical representation of

results makes it easier to build a model and observe the results interactively. Being an


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easy to use and inexpensive model, is another advantage of this model over the existing

models especially for independent operators.

The models described in this study are single well models in both cylindrical and

elliptical coordinate systems. The fully implicit, generalized Newton-Raphson procedure

was used to solve the nonlinear equations generated by the model. Gaussian elimination

was used to solve the Jacobian matrix generated by the model.

Unlike King & Ertekins model that had used a commercial mathematical library to

solve the resulting matrices, a new code was written, tested and verified that would solve

the matrices internally without the need for a commercial solver.


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I-1 Statement of the Problem

The purpose of this project was to develop a PC-Windows-Based Model (Single-

Well-Radial-Model) for two-phase water and gas in conventional and unconventional

Reservoirs. The formulation of this simulator is based on the model developed by

Gregory King and Turgay Ertekin in 1985. Conventional and unconventional reservoirs

can be modeled using this simulator.


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Chapter II Literature Review

For many years, CBM was a mining safety problem, and many people were trying to

find ways to vent methane from the mine before mining. Over the last decade, there have

been studies on CBM as an unconventional resource. In addition, decreasing the venting

of CBM to the atmosphere from coal mines may help to reduce global warming.2

The presence of explosive gases, especially methane, in coal seams has motivated

research in the U.S. in the past decade. Chamberlain and Darton conducted early research

for the United States Department of the Interior, beginning in 1907, which outlines the

basic mechanisms of methane entrapment and migration. With the inception of the

United States Bureau of Mines in 1910, a government body with the sole concern of

promoting safe and efficient recovery of underground resources was created. Many

studies were conducted on the control of methane emissions under this bureau. 3

In 1928, Rice suggested the use of vertical drainage boreholes to vent methane from

coal seams prior to mining operations. For the next 40 years, expensive ventilation

systems and reduced mining rates were standard methods of controlling methane

emissions.

In 1964, the Bureau of Mines began an extensive investigation of the coal seam

degasification process. This investigation was prompted by the increased mining activity

in deeper, gassier coal seams. Three major advantages of the degasification process were

increases in mining safety, increases in mining rate, and decreases of mining costs.

Another advantage of the degasification process was realized in the early 1970s,

which was the production of a valuable natural resource-methane. With the addition of

section 107 of the 1978 Natural Gas Policy Act, where unconventional gas producers

receive tax credit to produce natural gas, other factors will enhance future gas production

from coal seams. These are:

1. There are many producible coal seams in the eastern United States close to

pipelines and markets.

2. Most major U.S. coal deposits are thought to have been discovered prior to or

during the industrial revolution. These seams are well characterized and

exploration costs, therefore, would be minimal.


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3. Many major domestic coal seams are shallow (depths 1000 ft), therefore,

drilling costs would be minimal.

4. Drilling, completion, and stimulation technology are already well developed.

5. Coal seam methane is generally sweet, requiring only dewatering, metering, and

compressing facilities at the surface.

Over the past decade, in the United States, coalbed methane (CBM) has become an

increasingly important unconventional resource, which also includes gas shales and tight

gas sands.

Because of different storage and fluid flow mechanisms in coalbed methane

reservoirs from other conventional reservoirs and because of different structural

characteristics of coal from other reservoir rocks, past methods used to forecast the

production of a conventional reservoir cannot be applied to coalbed methane reservoirs.

Therefore, new methods should be built to enable the prediction of gas production from

coalbed methane.


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II-1 Coal Gas

Coal gas is a by-product of the physical and chemical reactions associated with

the coaliffication process (the process by which vegetable matter is converted to coal).

Consequently, this makes coal seam reservoirs different than conventional gas reservoirs,

in that the coal seam is both the source rock and the reservoir rock for the gas. As much

as 46 MSCF of gas can be liberated during the formation of one ton of coal.

Coal gas is primarily composed of methane with small amounts of some other

gases. The compositions of gases from several coal seams are presented in Table II-1.

Samples of gas from virgin coal seams yield calorific values ranging from 900 to 1075

B.T.U./SCF, making them of commercial quality with little processing.

II-1-1 Coal Seams as Natural Gas Reservoirs

II-1-1-1 Pore Structures

Coal seams are generally characterized by a dual porosity nature, in which they

contain both a micropore (primary porosity) and a macropore (fracture porosity) system.

The micropores have a diameter ranging from five to ten angstroms (1 A = 10 -10 m)and

exist in the coal matrix between the seams cleat (uniformly spaced natural fractures).

Due to the dimensions of the pores, the micropore system is inaccessible to water.

Table II-1 Composition of gas from several domestic seams, mol% (compiled by Jones et al.) 3


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The macropore system is made up of the volume occupied by the cleat. The cleat spacing

is very uniform, ranging from a fraction of an inch to several inches. Two types of cleats

are present in coal: the face cleat and the butt cleat. The face cleat is continuous

throughout the seam, while the butt cleat in many cases is discontinuous, ending at an

intersection with the face cleat. Generally, the face and butt cleat intersect at right angles.

The width of the macropores varies on the order of angstroms to microns (1 m = 10-6

m). There does not appear to be any transitory pores between the two systems. The total

effective porosity to water is usually less than two percent, while the effective porosity to

free gas in the same coal may be as high as ten percent. This discrepancy is due to the

inaccessibility of the macropore system to water. Figure 2 is a schematic representation

of the pore structures in coal.

Figure II-1 Plan view of the dual porosity nature of coal4


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II-1-1-2 Gas Storage

Because of the small micropore dimensions, the coal matrix will have a large

internal surface area. Two storage mechanisms are possible for methane storage in the

internal surface of the matrix. The first mechanism of entrapment occurs when the gas

molecule penetrates the surface and resides within the molecular lattice of the coal. This

is an absorption process. The second mechanism occurs when the gas molecule adheres

to the surface. This is an adsorption process. Both storage mechanisms are theorized to

occur in coal.

The adsorption process can be classified by the forces, which cause the gas

molecule to adhere to the surface. If these forces are chemical in nature, such as strong

chemical bonding, then the process is a chemisorption process. If, however, the forces are

physical, such as electrostatic forces or Van der Waals forces, the process is a physical

adsorption process. The major portion of the gas stored in coal exists as a physically

adsorbed, molecular monolayer.

Theoretically, the physical adsorption process is reversible. In coal, however, a

small degree of adsorption and chemisorption occurs along with the physical adsorption.

Consequently, a hysteresis effect is often observed between experimentally determined

adsorption and desorption isotherms.Although no general correlation exists, it has been observed that the sorptive

capacity of coal tends to increase with increasing rank (a measure of the coals chemical

structure) and increasing depth. Figure 3 shows the relationship between rank, depth, and

sorptive capacity. Also, the sorptive capacity tends to decrease with increasing water

content.

Along with the adsorbed gas, free gas exists in both the micropores and

macropores of the coal seam. The relative contribution of the free and adsorbed gas to the

overall storage is dependent on the seam.


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Figure II-2 Relationship between rank, depth, and sorptive capacity (after Kim) 3


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II-1-1-3 Gas Transport

The transport of methane through coal matrix is considered a three stage process.

With water production and corresponding pressure decline, gas is being produced fromthe fractures due to pressure gradient. With producing the gas from the fractures, gas

starts desorbing from the matrix walls into the fractures. With desorbing the gas from the

matrix into the fracture, there will be a concentration gradient inside the matrix which

causes the gas diffuse from inside the matrix to the matrix walls. This process is governed

by concentration gradient. In the last stage, gas moves through the cleat system to the

drainage wells. This process is governed by the pressure gradient. Thus, the cleat acts

both as a sink to the micropore system and as a conduit to the wells. Figure below shows

the three-stage process.

Figure II-3 Three stage Gas Transport in Coalbed Methane 5

Based on Scott and Dullien review on the mechanisms of gas transport in porous

media, if a pore channel has a diameter which is much greater than the mean free path of

the gas molecules, then the gas will flow by Poiseuille flow (forced flow) if it is subjected

to a total pressure gradient, by bulk diffusion if it is subjected to a partial pressure

(concentration) gradient, or by two dimensional surface diffusion if an adsorbed layer is

present.

If the diameter of the pore channel is small compared to the mean free path, then

the gas will flow by Knudsen diffusion (molecular streaming) if it is subjected to either a

total or partial pressure gradient.


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The critical value of pore diameter of the medium in order to have forced flow

(due to pressure gradient) is four times larger than the diameter of the micropores in coal.

It has been concluded that the first stage of gas transport in coalbed methane, through the

micropores is a diffusion process. The diffusion of gas through the micropores can be the

result of three distinct mechanisms which may act individually or simultaneously. These

mechanisms are:

1. Bulk diffusion, where molecule-molecule interactions dominate;

2. Knudsen diffusion, where molecule-surface interactions dominate; and

3. Two-dimensional surface diffusion of the adsorbed gas layer.

The following rules can be used to determine which mechanism dominates in a

particular coal. If the diffusion coefficient is experimentally determined to be inversely

proportional to the flowing pressure, then gas transport through the micropores is

dominated by bulk diffusion. If the diffusion coefficient is determined to be constant with

pressure, then Knudsen diffusion is the dominant mode of transport. Finally, if the mass

flux ratio satisfies the following condition:

( )

( )free

ads

ads

free

MW

MW

Q

Q< II-1

where MW is Molecular Weight and Q is Mass Flow Rate, then surface diffusion

dominates.

Regardless of the flow mechanism which dominates, the micropore

transport of gas obeys Ficks Law of diffusion:

( )mi

g

mi

g

r

CDMW

A

Q

= II-2

The micropore diffusion coefficient, miD , has been experimentally

determined to be between 10-8

and 10-13

cm/sec for most coals.

If the micropore diffusion coefficient is not known experimentally, it canbe approximated by the following formula:

mi

mi

FD

D

1= II-3

where, D = the self-diffusion coefficient of the gas

F = the electrical resistivity of the coal saturated with gas


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mi = the micropore porosity

The self-diffusion coefficient for pure methane is 0.214 cm/sec.

The last stage of the transport of gas through coal, flow through the macropore, is

generally considered to be a laminar process. As such, it obeys Darcys Law for laminar

flow through porous media:

r

Pk

A

qv

g

g

mag

g

==

II-4

Unstimulated, homogeneous coal is relatively impermeable to gas and water. The in-situ

permeability of large coal masses is predominantly due to the cleat. The dependence of

coals permeability on the cleat has two important consequences. The first is that the

presence of both face and butt cleat system leads to the permeability anisotropy observed

on most coals, with the direction of greatest permeability oriented parallel to the

continuous face cleat.

The second major consequence is the permeabilitys dependence on confining

stress.24

Increasing the confining stress (or equivalently, reducing the pore pressure)

causes fracture closure, which in turn reduces the absolute permeability of the coal.

Reznik et al. investigated the relative permeability characteristics of coal. The

typical behavior was reported. With an increase in water phase saturation, the relative

permeability to water increased, while the relative permeability to gas decreased.As water and gas are produced, the coal reservoir pressure is decreasing. Some

publications, have reported that as gas is desorbed from the coal matrix during the

decreasing of pressure, the coal matrix shrinks.

Harpalani et al6

presented that shrinkage of the coal matrix can cause the cleat

permeability to increase especially when pressure drops below the desorption pressure.

According to Harpalanis work, at pressures higher than the desorption pressure the cleat

permeability is influenced by the pore volume compressibility and tends to reduce the

permeability. However, after the pressure drops below the desorption pressure, the

shrinkage effect becomes dominant, resulting in increasing permeability. They also

developed an empirical equation to calculate the changing of permeability with pressure.

20260.076.03.3

pp

k ++= II-5


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Seidle and Huitt measured the shrinkage of coal matrix due to gas desorption and

concluded that matrix shrinkage depends on coal rank and adsorbed gas composition.

They reported that the shrinkage of coal matrix increases the cleat width and

subsequently increases the permeability. The equations for calculating the changing of

porosity and permeability by Seidle and Huitt are written below.

( )

+

+

++=

bp

bp

bp

bpVc

i

i

mm

fifi

f

1110

211 6

II-6

where cm is the matrix shrinkage coefficient in s-ton/scf that can be obtained by either

measurement or calculation. The equation for cm is described as the following.

+

+=

bp

bp

V

pcc

m

p

m

1

expII-7

The relationship between cleat permeability and porosity is described by;

3

=

fi

f

fi

f

k

k

II-8


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II-2 Review of Mathematical Models

Based on the King and Ertekin survey, between 1958 and 1994, over 52

mathematical models describing methane production from coal seams have been

developed. These models differ in physical scale and formulation. Their physical scale

range from core samples to single-well, and finally, full field models. Their formulation

is either empirically or analytically and they are solved using either analytical or

numerical techniques.

The models differ in their formulation and solution methods. The empirical

models are limited by the assumptions and observations used in their development. The

analytical models are limited by the available solution techniques, and numerical methods

are limited by their assumptions, coding and machine constraints.

All of these models could be categorized by the way they treat the gas sorption

process. The gas sorption models include empirically based models, equilibrium

(pressure dependent) sorption models, and non-equilibrium (pressure and time

dependent) sorption models.

The empirically based models are the simplest models. They are based on simple

mathematical descriptions of observable physical phenomena. Aireys first model,

decline curves, Lindines model, and the model of McFall et al. are some examples of

empirically based models.

Empirical models are relatively simple, requiring a few input parameters, but they

have theoretical difficulties for detailed predictions. Equilibrium (pressure dependent)

sorption models are the models which take the adsorption/desorption process into

account. In this approach, the gas adsorbed onto the micropore walls is assumed to be in a

continuous state of equilibrium with the free gas pressure in the macropores. In other

words, the desorption/diffusion process is sufficiently rapid that the kinetics of the

process can be neglected. The validity of the assumption of sufficiently rapid can bedetermined by comparing the rate of diffusion to the rate of laminar (Darcy) flow. If gas

flow is limited by the laminar process, and the diffusion process is significantly faster

than the laminar process, then the equilibrium situation can be assumed. Equilibrium

models are single-porosity, and partial differential equation models that are modified for

coal seams. They can be modified in two ways: 1) including a pressure dependent source


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15

term, or 2) the modification of the storage term.7

These models can be formulated and

solved analytically. The examples for equilibrium models are Aireys second model.

They can also be formulated and solved numerically. The examples for them are

INTERCOMPs first model.

The results of equilibrium sorption models are generally optimistic. They give

high production rates when the bottom-hole pressure is specified or high average

reservoir pressure when production rates are specified. The reason for this is, because of

pressure decline, desorbed gas is assumed to enter the macropore system instantaneously.

This approach does not take time dependency during transport through the micropore

system into account. Non-equilibrium (pressure and time dependent) models take time

dependency into consideration.

Non-equilibrium sorption models are modified conventional dual-porosity

models. These modifications are necessary because in coal seams 1) methane (reservoir

fluid) is highly compressible, 2) gas stored in the primary porosity exists in an adsorbed

state, and 3) gas transport through the micropore system is a diffusion process.

Two approaches have been used to formulate the conventional dual porosity

models for coal seams. Pseudosteady state formulations use a discretized form of Ficks

First Law to describe gas transport through the micropore system, and unsteady state

formulations use Ficks Second Law. Some of the models for each category will be

explained here.

II-2-1 Empirical Sorption Models

II-2-1-1 Aireys First Model

Airey proposed an empirical model for core scale samples. The following

assumptions were used:

1. The initial rate of methane desorption is very large for all coal sizes:

=

t

Gq dd ; 0t II-9


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16

2. The volume of desorbed gas approaches a constant for large times:

dG constant ; t II-10

3. The relative volume of desorbed methane is greater for smaller coal samples.

4. The volume of desorbed methane increases with time, while the desorption rate

decreases with time.

Aireys model has the form:

=

N

A

o

c

d

tGG

exp1

1000II-11

The time constant, A, in the equation is dependent on the size of the coal sample, and to

a lesser extent, on the initial gas pressure and water content. The initial gas content,

o

cG ,

is dependent on the same properties and can be calculated using the correlation of

Ettenger et al.

( )c

d

c

cW

GG

31.01+= II-12

Aireys model was developed using samples from the Sherwood Colliery and was

validated by comparing predicted versus experimental results.

II-2-1-2 Decline Curves

Decline curves have been used for several decades to predict production rate

versus time on a single-well or field scale. Decline curves also have been used for

economic analysis of vertical coal seam degasification wells. This approach is based on

the observation that after a certain time, the production rate of a single production well or

an entire field often declines either exponentially, hyperbolically, or harmonically.There is a unique behavior in methane production from coal seams. This behavior

is the negative decline period. During this period, gas production rate is increasing with

time. This phenomenon is attributed to the desorption of methane increasing the relative

permeability to gas. Both field studies and numerical studies have shown this behavior.


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17

Gilles et al. assumed exponential decline and used data from Pocahontas and

Mary Lee coal seams to perform an economic analysis of the degasification process. The

equation used to predict the production rate was:7

( )DCo

gSCgSC tqq /exp = II-13

where DC/1 is the decline constant (a property of the coal seam) ando

gSCq is the initial

gas production rate.

An equation for total gas production during the positive decline period can be

obtained by integrating equation (II-13) with respect to time:

( )[ ]DCo

gSCDCp tqG /exp1 = II-14a

or

( )gSCogSCDCp qqG = II-14b

For drainage area (well spacing), an equation can be defined:

cc

p

Gh

GA

= II-15

The value of Gp used in equation (II-15) must be corrected by adding the volume

of gas produced during the negative decline period.

Chase also suggested the use of decline curves on the core scale to determine the

amount of desorbable methane from samples of broken coal.

Two areas should be studied before using decline curves with confidence for coal

seams. The first area is to determine which decline model exponential, hyperbolic, or

harmonic best fits coal seam degasification wells. The second area is determining the

correct time to evaluateo

gSCq .

II-2-1-3 Lindine et al. Model

Another empirical based model was proposed by Lindine et al. on the observation

that both the initial gas content and residual gas content are nonlinear functions of depth.

Kim presented a relationship between initial gas content and depth while Hinsley et al.


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18

presented a relationship between residual gas content and depth. These relationships are

shown in the figures II.2 and II.4.

Figure II-4 Residual gas content as a function of depth (after Hinsley et al.)7

The equation Lindine et al. used to predict methane emission into a working mine (this

equation also can be used for single-well and full field scales) was:

( )=

=NK

k

r

ccckp kkkGGh

AG

11000 II-16

Gp is total volume of recoverable methane with no time or economic constraints.

II-2-1-4 McFall et al. Model

McFall et al. used a model similar to Lindine et al. with a different observation.

This model was based on the observation that the initial gas content of coal seam is

dependent on the basin, depth, and rank of the coal. McFall et al. used this model to


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19

estimate the methane resource of the Piceance basin. This relationship is shown in the

figure below for the Piceance basin.

Figure II-5 Gas content versus depth and rank for Piceance basin coals (after McFall et al.) 7

The equations used by McFall et al. were:

cc hA

GG

1000= II-17

where the initial gas content can be estimated by a curve fit of figure :

( ) Mc dmG += ln II-18

in this equation, m is a scaling coefficient and M is the y-intercept in figure . these

parameters are the properties of the basin and the rank of the coal. Kelso et al. used this

model to estimate the methane resource of the San Juan Basin.


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20

II-2-2 Equilibrium Sorption Models

The equilibrium sorption models are partial differential equation models. They are

developed from theoretical considerations. This is the opposite of empirical

considerations. The assumption in these models is that gas desorption from coal surfaces

and diffusion through the micropore (primary porosity) system is sufficiently rapid, so

the equilibrium with the gas phase pressure is continuously maintained. Consequently,

these models are single-porosity reservoir models.

II-2-2-1 Aireys Second Model and Kissells Model

Airey and Kissell independently formulated the following partial differential

equation for gas emission into a stationary mine face (for field scale simulations):7

D

D

D

D

t

p

x

p

=

2

22

22

II-19

The dimensionless variables are defined as:

o

g

g

Dp

pp = II-20

maxx

xxD = , where maxmax xxx II-21

2

maxxH

tpkt

g

o

gma

D

= II-22

The dimensionless initial and boundary conditions are:

( ) 1, =DDD txp ; 11 Dx , 0=Dt II-23

( ) 0, =DDD txp ; 1=Dx , 0>Dt II-24

( ) 0, =DDD txp ; 1=Dx , 0>Dt II-25

The definition of dimensionless time is similar to the definition used in well testing of

conventional gas reservoirs, with a Henrys Law Coefficient, H, replacing the

gma c product.


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21

In the derivations of the equations, some assumptions have been made by Airey

that are listed below:

1. The adsorbed gas obeys Henrys Law over the entire pressure range of interest.

ggEHppV = II-26

2. The free gas obeys Boyles Law over the entire pressure range of interest (i.e.,

ideal gas behavior).

Do

g

g

o

g

gp

p

p==

II-27

3. The macropore (secondary porosity) permeability and gas viscosity are constant

over the entire pressure range of interest. This assumption states that stress

dependent permeability and the Klinkenberg effect are negligible.

Airey provided the solution to Dp , subject to initial and boundary conditions mentioned:

( ) ( ) ( ) ( )[ ] ( )[ ]{ }

=

+++=

0

22112 2/1exp.2/1cos2/112

,N

DD

N

DDD tNxNNtxp

II-28

Although this model was derived by assuming Darcy flow through the macropore

system, Airey suggested using Ficks Law for seams under low hydrostatic pressure.

II-2-2-2 Bumbs Model, McKee and Bumbs Model, and Bumb and McKees

Model

Bumb, McKee and Bumb, and Bumb and McKee developed two equilibrium

desorption models, one for single-phase gas flow and one for multiphase gas-water flow,

for well test analysis of coalbed degasification wells and Devonian shale wells.

For single-phase gas flow, the following partial differential equation has been

developed which describes gas transport to a single drainage well:

( )EgSCgma

g

g

g

ma Vtr

pk

rr

+

=

1 II-29


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22

where the storage term (partial derivative with respect to time) has been altered to include

both free gas ( gma ) and adsorbed gas ( EgSCV ). EV in the equation is an equilibrium

isotherm and was considered to be governed by a Langmuir isotherm of the form:

( )gL

gLgE

pppVpV+

= II-30

In this equation, LV and LP are the Langmuir volume constant and Langmuir pressure

constant, respectively, and are the properties of the coal seam.

The partial differential equation for this model was derived and solved using these

assumptions:

1. The reservoir is horizontal, uniformly thick, and of infinite radial extent.

2. Rock properties are homogeneous and isotropic.

3. Gas transport through the macropore system is single-phase, isothermal, and

obeys Darcys Law.

4. The macropore permeability is constant for the entire pressure range.

5. Free gas obeys the real gas law.

6. Adsorbed gas behavior can be modeled within Langmuir isotherm.

7. Gas desorption and diffusion is sufficiently rapid, so that adsorbed gas remains in

equilibrium with the macropore pressure (i.e., dual-porosity effects are ignored).

The initial and boundary conditions considered in this model were:

( ) ogg ptrp =, ; t II-32

t

p

Z

p

RT

MhrkQ

ggg

g

ma

g

=

2; wrr = ; 0>t II-33

Condition II-33 is a mass sink, which for a constant producing pressure and

temperature is equivalent to constant volumetric production.

Applying the real gas law to equation II-29 and expanding the time derivative

yields:

=

r

p

z

pr

rr

g

g

g

1


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23

( ) tp

pP

Pcc

k

g

gLgma

LgSC

ma

r

g

ma

ma

+++

2

II-34a

or

tpc

krp

zpr

rr

g

T

ma

mag

g

g

=

1 II-34b

Where Tc is the total system compressibility which is highly pressure dependent. The

definition for compressibility becomes:

( )2gLgmaLgSC

ma

r

gTpP

Pccc

+++=

II-35

Equation II-34b can be put into the standard well test form by applying the real gas

pseudo-pressure:

( ) =g

SC

p

p

g

g

g

g pdZ

ppm

II-36

Applying conditions II-31 through II-33, equation II-34b can be solved in terms

of the real gas pseudo-pressure:

( )

= T

ma

gma

i

gma

g

g ctk

rE

M

RT

hk

QPm

42

2

II-37

Equation II-37 is the line source solution commonly used in pressure transient

analysis. Bumb and Bumb and McKee suggested the following iterative method to handle

the nonlinear terms g and Tc .

These properties are evaluated at the weighted average pressure defined by:

g

o

gg ppp 1.09.0 = II-38

Equation (II-37) is then solved for pm , converted to gp and gp is reevaluated. This

process is continued until successive gp s converge whiting an acceptable tolerance.

Bumb and McKee and Bumb validated this model by comparing the results with those of

a numerical model.

For two-phase water and gas, Bumb and McKee and Bumb derived the following

set of coupled partial differential equations:


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t

p

dp

dSpSSc

dp

dS

t

p

dp

dSpSSc

dp

dScS

r

pkkr

rr

g

cgw

wcgwwwr

cgw

wma

w

cgw

wcwwr

cgw

wmawwma

w

w

rwma

+

+

++

=

1

1

II-39

and

( )

( )t

p

dp

dSpSccS

dp

dS

t

V

t

p

dp

dSpSSc

dp

dS

r

pkkr

rr

g

cgw

w

cgwwrgmaw

cgw

mag

E

gSC

w

cgw

w

cgwwwr

cgw

mag

g

g

rgma

+++

+

++

=

11

1

1

II-40

These equations were derived using the same assumptions as in the single-phase model

(except for the single-phase assumption).

The multiphase flow of water and gas can be divided into three steps: saturated

water flow ( 1=wS , 0=gS ), unsaturated water flow ( 1


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25

+=

F

cgwcgw

wir

wirwpp

S

SS

2/1exp1

1II-41

2. The relative permeability to water obeys a power law relationship (no

assumptions concerning gas phase relative permeability are needed since gas isimmobile in this step):

N

wir

wirw

rwS

SSk

=

1II-42

3. Gas phase pressure is constant.

The solution for step II, with the same initial and boundary conditions as single-

phase model is:

( )

= To

rwma

mawio

rwma

wwwcgw c

tkkrE

hkkBqp

44615.5

2

II-43

where the definition of pseudo-pressure becomes:

( )

=

cp

cgw

wir

wirw

cgw pdS

SSp

1 II-44

and the definition of total compressibility becomes:

( )

( )

+

+

=

F

wirw

wir

wirw

cgww

N

wir

wirw

ma

wr

wir

wirw

F

wirw

wir

wirw

N

wir

wirw

wwT

SSS

SSpS

S

SS

Sc

S

SS

SSS

SS

S

SS

cSc

11

1

1

11

1II-45

The weighted average cp used in the linearization of this model was suggested as:

cgw

o

cgwcgw ppp 3.07.0 += II-46

wS is evaluated at cgwp . This model was validated by comparing the results to

those of a numerical model and by field testing the model against data from the

Glovier well no. 1 in Archuleta county, Colorado.


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Step III: Two-Phase Flow ( 1


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27

( )g

p

p g

gma

gpd

pkpm

g

SC

= II-49

results in:

( )( )

( )[ ]go

g

o

ma

gL

LL

magpm

dt

dk

pP

RTPCpm

1

2

2

++=

II-50

Equation II-50 was discretized using finite differences and solved with the fully-

implicit, generalized Newton-Raphson technique. The coordinate system and boundary

conditions used in the solution of equation II-50 was dependent on the problem to be

solved. For example, Gorbachev et al. solved this equation in three-dimensional

rectangular coordinates to simulate gas emission from a coal pillar. The initial and

boundary conditions for this full-field problem were:

o

gg pp = ; max0 xx , max0 yy , max0 zz , 0=t II-51

Mg pp = ; max,0 xx = , max,0 yy = , max0 zz , 0>t II-52

0=

z

pg; max0 xx , max0 yy , max,0 zz = , 0>t II-53

The constant in condition II-52, Mp , was the mine pressure, which was

approximately one atmosphere. Condition II-53 implies an impermeable roof and floor

exist in the seam. Gorbachev et al.s model allowed for horizontal boreholes.

Karagodin and Krigman solved equation II-50 in one-dimensional rectangular

coordinates to simulate gas emission into a stationary mine working. The initial and

boundary conditions for this full-field problem were:

o

gg pp = ; x0 ; 0=t II-54

0=gp ; 0=x ; 0>t II-55

o

gg pp = ; x ; 0>t II-56

Karagodin and Krigman used this model to history match gas emission into the

Komsomolets Colliery to determine in situ coal permeability.


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Vorozhtsov et al. solved equation II-50 in one-dimensional, cylindrical

coordinates to simulate the performance of a single vertical degasification well. The

initial and boundary conditions for this problem were:

o

gg pp = ; ew rrr , 0=t II-57

wfgpp = ; wrr = , 0>t II-58

0=

r

pg; err = , 0>t II-59

Other models discussed in King and Ertekins survey are: Model of Nguyen, Model of

Kamal and Six, the model of Ediz and Ediz and Edwards, Modified black oil models,

modified p/Z analysis, and the deliverability model of Seidle.

II-2-3 Non-Equilibrium Sorption Models

Some of the non-equilibrium sorption models are discussed here.

II-2-3-1 Pseudo-Steady State Models

Pseudo-steady state, non-equilibrium sorption models are similar to the Warren

and Root model for conventional dual-porosity reservoirs. The coupled equations used to

describe gas transport are:8

=++

Z

Sp

tq

T

Tpq

T

Tp

Z

p ggmad

SC

SC

gSC

SC

SC

g

gg 1000II-60

( )dt

dVPFq mi

mid = II-61

( )[ ]migEmimi VpVaD

dtdV = II-62a

or

( )[ ]migEp

mi VpVdt

dV=

1II-62b


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29

In addition, a water phase material balance equation is included in two-phase

models.

II-2-3-2 Fedorov et al. Model

Fedorov et al. formulated a single-phase model for gas production into a single

degasification well (Fedorov et al. state that this model is for flow to a borehole.)

In the derivation of their equations, they assumed that 1) gas transport is single-phase,

and 2) free gas obeys the ideal gas law.

II-2-3-3 U.S. Steel Models

Pavone and Schwerer used the pseudo-steady state, non-equilibrium sorption

formulation to derive a set of equations to describe flow of water and gas through dual

porosity coal seams.

II-2-3-4 Penn State Models

Ertekin et al., King et al., and King developed a single-well model. Their model

was similar to the U.S. steel model. The major difference was in the numerical solution of

the system of differential equations. In these models, an algebraic approximation to

equation II-62b was implemented.

II-2-3-5 COMET Model

The COMET (Coalbed Methane Technology) model is a two-phase, two-

dimensional software package for personal computers which was adapted from the

SUGARWAT model (a Devonian shale simulator which will be discussed in the section

on unsteady state models). This model is similar to U.S steel model (ARRAYS) and Penn


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30

State Model (PSU) model. The most significant contributions of the COMET model are

the user friendly features. These include:

1. PC compatible

2. Automatic PVT generation options

3. Automatic gridding options

4. Interactive graphics

5. Restart capabilities

6. Unconditionally stable, fully-implicit solution technique

Other models discussed in King and Ertekins survey are: COALGAS Model, modified

material balance model of King, the type-curve model of Mohaghegh and Ertekin, and

the model of Durucan et al.


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II-2-4 Unsteady State Models

Unsteady state, non-equilibrium sorption models are similar to the De Swaan

model for conventional dual-porosity reservoirs. The coupled equations used to

describe gas transport are:

=++

Z

Sp

tq

T

Tpq

T

Tp

Z

p ggmad

SC

SC

gSC

SC

SC

g

gg 1000II-63

mimi Rr

mi

mi

bmi

mimi

dr

V

V

DAq =

= II-64

t

V

r

Vr

rr

D mi

mi

miI

mi

mi

I

mi

mi

=

II-65

where I in equation II-65 is a coordinate index equal to zero for slabs, one for cylinders,

and two for spheres.

II-2-4-1 Kovaley and Kuznetsov Model

Kovaley and Kuznetsov formulated a moving boundary value problem to

calculate the rate of methane emission into an advancing longwall face.

II-2-4-2 Kuchuk and Sawyer Model and Serra et al. Model

Kuchuk and Sawyer and Serra et al. developed similar models for testing

Devonian shale wells.

II-2-4-3 Smith and Williams Model

A model for broken coal suggested by Smith and Williams used an unsteady state

non-equilibrium sorption formulation. In this model, the coal sample was assumed to


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32

have a spherical macropore geometry, and the micropore matrix sub-elements were also

assumed to be spherical.

Other models that are discussed in King and Ertekins survey are: INTERCOMPs

Second model, Chase Model, SUGAR, SUGARMD, and SUGARWAT Models, Chen

Model, Ertekin and Sung Model, Model of Saghafi and Saghafi et al., Model of Carlson

and Mercer, and Models of Anbarci and Ertekin.


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Chapter III Methodology

To formulate the equations for water and methane transport through coal seams,

we need to give the reservoir an ideal structure. Figure III-1 is a suitable model for most

coal seam reservoirs.

Figure III-1. Reservoir Model for Coal Seams 5

In this figure, the cubes represent the matrix subelements, which contain the micropores,

while spaces between these blocks represent the cleat system. The blocks can be other

shapes (slabs, cylinders, or spheres).

The transport equations are derived by assigning effective permeability and

porosity values to the cleat system. The fracture system can be treated as a porous

medium, with the transport equations developed accordingly.

Once solved, the macropore transport equations give the pressure distribution, as

a function of time, throughout the coal seam. We can use this distribution to obtain the

external boundary condition for the problem of gas diffusion through the micropore

matrix subelements. Since the macropore pressure varies with both time and space, the

boundary condition for an individual subelement depends on both time and that

subelements location in the coal seam.


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34

The solution of the diffusion problem within each subelement yields a source term

to the gas transport equation. This source term, in turn, affects the macropore pressure

distribution. Obviously, the system of equations describing this process is a complex set

of highly nonlinear, coupled equations.

When the equations for water and methane transport are obtained, they should be

simplified in order to be able to solve them. In this study, a radial coordinate system has

been used for unstimulated wells, and Elliptical coordinate system for stimulated wells.

The formulations in both coordinate systems are described here.

III-1 Macropore Transport Equations in Radial Coordinate System

Performing a mass balance on an elemental volume of the reservoir can derive the

equations governing the unsteady state flow of water and gas through the macropore

system of a coal seam. The general form of the continuity equation (differential mass

balance equation), written for phase p in a porous medium, is:

( ) ( )ppmabma

p

pp StV

Qvr

rr

=+

1III-1

In order to develop the equations for water and gas, two assumptions should be

made: The choice of an appropriate equation of state, and a transport law. Once chosen,

these equations are substituted into equation III-1 to obtain the flow equation for each

individual phase.


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III-1-1 Single-Phase Gas Equation

In this study, the free gas in the macropore system is assumed to behave like a

real gas. The real gas law can be rearranged to obtain the following equation for gas

density:

( )

ZRT

PMW gg = III-2

Substituting equation III-2 into equation III-1:

( )

=++

Z

P

tV

Q

MW

RT

V

Q

MW

RTv

Z

Pr

rr

gma

bma

t

bma

g

g

g

)(

1 III-3

The source/sink term has been separated into an external term, Qg, and an internal term,

Qt. The internal source term represents the matrix/fissure methane transfer. After

substitution of equation III-2 into equation III-1, temperature T, and the molecular weight

(MW), were removed from the derivative operators. We can do this if we make some

assumptions first. Temperature can be removed if we assume that the reservoir

temperature remains constant and doesnt change. For molecular weight, in order to

remove it from the derivative operators, if we assume that, desorption or adsorption is not

selective in the micropores. With these assumptions we can remove these two variables

from the derivative operators.

In the storage term (partial derivative with respect to time), the groupZ

Pgma

represents only free gas storage. It has been assumed that the amount of gas adsorbed

onto the walls of the cleats is negligible compared to the amount of the free gas in the

cleats. Finally, the gas saturation for single-phase system is equal to unity.

The real gas law can be rearranged to obtain more convenient expressions for the

source/sink terms in equation III-3:

gSC

SC

SC

g

g

g qT

TPq

Z

PQ

MW

RT 10001000

)(== III-4


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36

t

SC

SC

t qT

TPQ

MW

RT

and

=)(

III-5

Where the term 1000 is a unit conversion constant. Substituting into equation III-3:

=++

Z

P

tV

q

T

TP

V

q

T

TPv

Z

Pr

rr

gma

bma

t

SC

SC

bma

gSC

SC

SC

g

g 10001 III-6

Coal seams usually have low pressures with relatively low absolute

permeabilities. For these reasons, the gas transport equation must allow for the gas

slippage phenomenon. The correction for gas slippage is usually made by applying the

Klinkenberg equation to obtain an effective permeability to gas. This correction has the

form:

+=

gP

bkk 1 III-7

In this study, an approach to gas slippage has been used. In this approach, the slip

velocity is superimposed onto the Darcy velocity and is substituted directly into the

transport equation. Then it is modeled with Ficks law of diffusion. The advantage of this

procedure is that the gas, which is slipping, can be treated as a real gas. In deriving

equation III-7, ideal gas behavior has been assumed.

Figure III-2a shows the flow of an incompressible liquid across the cross-section

A-A. The circles represent fluid molecules. The large arrow represents the laminar,

Darcy flow while the smaller arrows indicate random molecular flow. Since the densities

on both sides are equal, so the molar concentrations are equal, thus the random molecular

motion has a zero net effect.

Figure III-2b shows the flow of a compressible gas across the same cross-section.

Since the gas is less dense on the left side of the cross-section, the random molecular

motion has a positive net effect in the direction of flow.


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Figure III-2. Schematic Representation of the Dual Mechanism Approach to Slip Flow 4

The equation describing this flow process can be derived by defining the total

velocity as the sum of the Darcy and slip components:

S

g

D

gg vvv

+= III-8

Substituting Darcys law:

g

g

D

g Pr

kv

=

III-9

into equation III-8 yields:

S

g

g

gvP

r

kv

+

=

III-10

The slip velocity is a Knudsen flow process and can be modeled with Ficks law of

diffusion.


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Substituting of Ficks law:

gma

g

S

g CrD

MW

v

=

)(

III-11

into equation III-10 yields:

gma

g

g

g

g Cr

DMW

Pr

kv

=

)(III-12

Substituting the definition of gas density (equation III-2) into equation III-12

results in:

gma

g

g

g

g Cr

DP

ZRTP

r

kv

=

III-13

Cg the molar concentration can be obtained from the real gas law:

ZRT

PC

g

g = III-14

Substituting equation III-14 into equation III-13 yields:

=

Z

P

rD

P

ZP

r

kv

g

ma

g

g

g

g

III-15

Expanding the term,

Z

P

r

g, in equation III-15 leads to:

gmagg

g

g Pr

DcPr

kv

=

III-16

or: gmagg

g Pr

Dck

v

+=

III-17

Factoringg

k

out of the parenthesis yields:

g

gmag

g

g Prk

Dckv

+=

1

III-18


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39

Multiplying the second term in the parenthesis byg

g

P

Pyields:

g

g

g

g

g Pr

P

Pbk

v

+

=

)(

1

III-19

where

=k

DPcPb

maggg

g

)( III-20

Substituting equation III-19 into equation III-6 gives the final form of the single-phase,

gas transport equation:

=++

+

Z

P

tV

q

T

TP

V

q

T

TPP

rZ

PP

Pbk

rrr

gma

bma

t

SC

SC

bma

gSC

SC

SCg

g

g

g

g

1000

)(1

1III-21

In the derivation of equation III-7, Klinkenberg assumed ideal gas behavior. In

this study, if this assumption is made, then cg.Pg is equal to unity and the gas viscosity g

is constant. In the other words, if ideal gas behavior is assumed, then b(Pg) is constant.

Although Klinkenbergs theory predicted a constant slippage factor, his experimental

results indicated that this factor increased slightly with increasing pressure.


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40

III-1-2 Single-Phase Conventional Gas Equation

In this study, single-phase conventional gas can be modeled. Using equation III-2

and substituting into equation III-1:

( )

=+

Z

P

tV

Q

MW

RTv

Z

Pr

rr

gma

bma

g

g

g 1III-22

Using Darcys law:

g

g

D

g Pr

kv

=

Substituting Darcys law into equation above and using equation III-4, leads to:

=+

Z

P

tV

q

T

TP

r

P

Z

Pkr

rr

gma

bma

gSC

SC

SCgg

g

10001III-23

III-1-3 Two-Phase Gas Equation

The two-phase gas transport equation can be developed by substituting the

equation for gas density III-2 into the general continuity equation (equation III-1). This

substitution leads to:

=++

Z

SP

tV

q

T

TP

V

q

T

TPv

Z

Pr

rr

ggma

bma

t

SC

SC

bma

gSC

SC

SC

g

g 10001 III-24

The substitution of the two-phase forms of Darcys law and Ficks law into equation

III-8 yields:

=

Z

SP

rD

P

ZP

rv

gg

ma

g

ggg

III-25

The term,

Z

SP

r

gg, cannot be expanded into a convenient form, therefore, equation

III-25 is the final form of the two-phase gas flux. Substituting this equation into equation

III-24 gives the final form of the two-phase gas transport equation:


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41

=++

+

Z

SP

tV

q

T

TP

V

q

T

TP

Z

SP

rrDP

rZ

Pr

rr

ggma

bma

t

SC

SC

bma

gSC

SC

SCgg

mag

gg 10001

III-26In the derivation of equation III-26, it is assumed that the amount of gas dissolved in

water is negligible compared to the amount of free gas.

III-1-4 Two-Phase Water Production

In this study, water is treated as a slightly compressible fluid, and its transport is

assumed to obey Darcys law for laminar flow through porous media. The equation for

slightly compressible fluid can be derived from the definition of compressibility under

isothermal conditions:

T

w

w

w

wdP

dV

Vc

1= III-27

Definition of water density:

w

ww

VM= III-28

considering a given mass of water, the following relationship is obtained:

w

w

w

wdV

V

Md

2= III-29

Substituting equations III-28 and III-29 into equation III-27 yields,

w

w

w

w dP

d

c

1

= III-30

If we separate the variables and integrate the both sides, we obtain:

=w

wSC

w

SC

w

w

P

P

ww dPdc

1III-31


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To simplify the problem to solve, we can consider cw to be constant. Integrating the

equation above yields:

( )

=

wSC

w

SCwwPPc

ln III-32

If we rearrange equation III-32, we get:

( )[ ]SCwww

wSC

ww PPcPB == exp)(

III-33

Bw is the definition of the formation volume factor for a fluid of constant compressibility.

If we solve this equation for w, we get:

w

wSCw

B = III-34

Equation III-34 is the final form of the equation of state for a slightly

compressible fluid. Although this equation was derived by assuming constant

compressibility, it is also valid for any Bw.

Substituting equation III-34 into equation III-1 results:

=+

w

wma

wSCbma

w

w

w B

S

tV

Qv

Br

rr

11III-35

Mass rate and standard density terms can be replaced by the volumetric rate at standard

conditions, qSTB. Substitution yields:

=+

w

wma

bma

STB

w

w B

S

tV

qv

Br

rr

615.511 III-36

Term 5.615 is the unit conversion constant.

It was assumed earlier that water transport obeys Darcys law. If we make another

assumption that the coordinate axes of the model coincide with the principle permeability

axes of the coal seam, then the component of the water velocity in the

directioni becomes:

rP

kSkv w

ww

maiwrw

wi

=

)(

)(III-37


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43

where: dP www144

= III-38

is a unit conversion constant.

We can define the two-phase water mobility in the directioni as:

)(

)(),(

ww

maiwrw

wwwiP

kSkSP

= III-39

Using equations III-37 through III-39, the water velocity becomes:

=

r

d

r

Pv wwww

144

III-40

Substituting equation III-40 into equation III-37 gives the final form of the water

transport equation:

=+

w

wma

bma

STBww

w

w

B

S

tV

q

r

d

r

P

Br

rr

615.5

144

1III-41

III-1-5 Auxiliary Macropore Equations

For multiphase flow problems, the transport equations do not constitute a properly

posed system. In two-phase systems, there are two nonlinear, partial differential

equations in five dependent variables: Pw, Pg, Sw, Sg, and Vmi. Three additional equations

are required to solve these equations. Two of these are obtained from the physics of the

macropore flow, and the third one is obtained from the micropore gas transport. The

equations which are obtained from the physics of the macropore flow are:

1. The summation of water and gas saturation is equal to unity:

Sg + Sw = 1.0 III-42

2. The difference between water and gas pressures is strictly dependent onthe gas phase saturation:

Pcgw(Sg) = Pg - Pw III-43

In order to solve the model, the macropore equations (III-41 and III-26, III-42 and III-43)

should be reduced to a system of two equations. In the auxiliary equations, if we solve

those problems for Sw and Pw and substitute those into equations III-41 and III-26, we can


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reduce the number of unknowns to three. The final form of the macropore equations is

then written in terms of the dependent variables Pg, Sg, and Vmi. Using this procedure, the

reduced system of macropore equations becomes:

)

=+

w

gma

bma

STBw

cgwg

w

w

B

S

tV

qd

rP

rP

rBr

rr

1615.5

144

1 III-44

=++

+

Z

SP

tV

q

T

TP

V

q

T

TP

Z

SP

rrDP

rZ

Pr

rr

ggma

bma

t

SC

SC

bma

gSC

SC

SCgg

mag

gg 10001

III-45

III-1-6 Diffusion/Sorption Model

The macropore transport equations, III-44 and III-45 are two nonlinear, partial

differential equations with three unknowns. To obtain a properly defined system, a third

equation is needed. This equation is obtained from the micropore gas transport.

The major portion of gas stored in coal exists as a physically adsorbed, liquid-like

molecular monolayer on the walls of the micropores. These pores are located in the

interior of the regular geometric subelements of the idealized reservoir. With decreasing

pressure in the cleats due to production, gas begins to desorb from the surface of the

subelements. This creates a concentration gradient in the interior of the subelements,

which is the cause of diffusion. If it is assumed that desorption rate at these surfaces is

sufficiently rapid (so that equilibrium is maintained between the free gas and adsorbed

gas phases), then the methane concentration on the subelements surface is governed

solely by the equilibrium sorption isotherm. For most coals, isotherms based on

Langmuirs theory for simple, single-layer sorption adequately describe the processes

occurring at these surfaces.

Langmuirs theory is based on the premise that molecule-molecule collisions are

inelastic (that is, gas molecules behave like hard spheres), while molecule-surface


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collisions are elastic. Thus, if a free gas molecule traveling towards a coal surface strikes

an adsorbed gas molecule, it will immediately ricochet away from the surface. If,

however, it strikes the surface, the elastic nature of the collision causes the molecule to

stick to the surface for a certain time period. In this theory, it is the residence time

which is the principle cause of adsorption. The following derivation of Langmuirs

equation follows that of Guwaga.

The rate at which gas molecules strike a coal surface is proportional to the free

gas pressure, Pg, and the fraction of the surface that is not already covered by gas

molecules, (1 - ) (Kinetic theory). The rate of adsorption, therefore, can be written as:

( )= 11 gPKq III-46

The rate of gas desorption is proportional to the fraction of the surface covered

with molecules:

= 2Kq III-47

The constants K1 and K2, in equations III-46 and III-47 are proportionality constants with

the units of SCF/ft/day/psi and SCF/ft/day, respectively.

At equilibrium, the rate of adsorption and desorption must be equal, so:

( ) = 21 1 KPK g III-48

or:

g

g

PKK

PK

12

1

+= III-49

The fraction of the surface covered by molecules at any time is equal to the

adsorbed methane concentration divided by the total sorptive capacity of the surface

(concentration at infinite pressure). Since equation III-49 is written for equilibrium

conditions, the adsorbed methane concentration is the equilibrium concentration, that is:

L

gE

V

PV )(= III-50

Equating the two definitions of (equations III-49 and III-50) and rearranging yields:

gL

gL

gEPP

PVPV

+=)( III-51

where:


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1

2)(K

KpsiaP

L = III-52

In equation III-49, the constant, VL, is the Langmuir volume constant and is a

measure of the total sorptive capacity of the surface. The constant, P L, is the Langmuir

pressure constant and is a measure of the residence time, during which an average gas

molecule adheres to the surface. Typical Langmuir isotherm is presented in figure III-3.

Figure II-3. Typical Langmuir Sorption Isotherm5

In cases that the Langmuir pressure is much greater than initial reservoir pressure

(PL>>Pgo):

LgL PPP + III-53

Substituting equation III-53 into equation III-51 yields:

g

L

L

gEP

P

VPV )( III-54

ggE HPPV =)( III-55

which is Henrys law.


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To determine the methane concentration on the surface of the regular geometric

subelements of the idealized model, we can use equations III-51 and III-55, which are the

equilibrium sorption isotherms.

The equations used in the pseudosteady state formulation are derived by

considering a discretised form of Ficks law and have the form:

[ ]migEmimi VPVaDdt

dV= )( III-56

and:

( )dt

dVPF

V

q mimi

bma

t = III-57

along with the initial condition:

)( ogEomimi PVVV == ; t = 0 III-58

Equation III-56 is a first order, ordinary differential equation and has the analytic

solution:

[ ] ( )[ ] +=t

mimigEmi

o

mimi tdttaDVPVaDVV0

exp)(

III-59

Substituting into equation III-56 yields:

[ ] [ ] ( )[ ]

+=

t

mimigEmi

o

migEmi

mi tdttaDVPVaDVPVaDdt

dV

0

exp)()(

III-60

Substituting equation III-60 into equation III-57 results in:

( ) [ ] [ ] ( )[ ]

+= t

mimigEmimigEmimi

bma

t tdttaDVPVaDVPVaDPFV

q

0

exp)()(

..

III-61

Equation III-61 defines, on a unit volume basis, the pseudosteady state source to

the macropore gas transport equation.

The equations used in the unsteady state formulation have the form:

t

V

r

Vr

rr

Dmi

mi

miI

mi

mi

I

mi

mi

=

III-62

and:


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mimi Rr

mi

mi

bmi

mismi

bma

t

r

V

V

DA

V

q=

= III-63

along with the initial and boundary conditions:

)(),(o

gE

o

mimimi PVVtrV == ; mimi Rr 0 , t = 0 III-64a

)(),( gEmimi PVtrV = ; mimi Rr = , t > 0 III-64b

),( trV mimi is finite ; 0=mir , t > 0 III-64c

In equation III-62, the coordinate index, I, is equal to zero for slabs, one for

cylinders, and two for spheres. If the regular geometric subelements are taken to be

spheres (I = 2), then equation III-62, subject to conditions III-64a through III-64c, has the

formal solution:

( ) ( )

=

+=

0 0

2

22

exp)(sin12

),(n

t

mi

mi

gE

mi

min

mimi

mio

mimimi tdttR

nDPV

R

rnn

Rr

DVtrV

III-65

Substituting into equation III-63 yields:

[ ] [ ] ( )

=

=

1 0

2

22

0

2

22

0

2exp)()(

6

n

t

mi

mi

migE

mi

mi

migE

mi

mi

bma

t tdttR

nDVPV

R

nDVPV

R

D

V

q ..

III-66

Equation III-66 gives the unsteady state source, on a unit volume basis, to the

macropore gas transport equation. This equation can be used to obtain expressions for the

shape factor, a , and geometric prefactor, (PF)mi, in equation III-61.

Consider the first term approximation to equation III-66:

[ ] [ ] ( )

t

mi

mi

migE

mi

mi

migE

mi

mi

bma

t tdttR

DVPV

R

DVPV

R

D

V

q

0

2

2

0

2

2

2exp)()(

6

III-67


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Inspection of equation III-61 reveals that it is identical to equation III-67 if the shape

factor, a, is chosen to be:

2

2

miR

a

= III-68

and the geometric prefactor, (PF)mi, is chosen to be:

(PF)mi = 6 III-69

Similar developments can be done for slabs and cylinder micropore matrix

sub-elements. The results are shown in Table below.

Table III-1. Micropore matrix geometries, prefactors, and shape factors (After Boyer et al.)

Micropore Matrix

Geometry

Characteristic

Dimension

Geometric Prefactor

(PF)mi

Shape Factor a

Slabs Half Thickness, hmi 2 2

2 2

4674.2

=

mimihh

Cylinders Radius, Rmi 4 2

2

4082.27832.5

=

mimiRR

Spheres Radius, Rmi 6 2

2

8696.9

=

mimiRR

Comparisons of the pseudosteady state matrix/fissure transfer (equation III-56)

and the unsteady state matrix/fissure transfer (equation III-66), for different subelements

geometries and simple boundary conditions, are presented in figures III-4 and III-5. As

its shown in the figures, they are in agreement.


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Figure II-4. Comparison of the Pseudosteady State diffusion/sorption and the

unsteady state diffusion/sorption model for slab matrix subelements4


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Figure II-5. Comparison of the Pseudosteady State diffusion/sorption and the

unsteady state diffusion/sorption model for sphere matrix subelements 4

The analytic solution of equation III-62 requires the evaluation of an infinite series. For

small times, this series tends to converge rather slowly. On the other hand, to solve this

equation numerically requires solving NR*NI equations (for three dimension is

NR*NI*NJ*NK equations), in addition to the 2*NI macropore (for three dimension is

2*NI*NJ*NK equations). For this reason, and because of the good agreement between

two approaches shown in figures III-4 and III-5, the Pseudosteady state approach has

been used in the model formulation.


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III-1-7 Initial and Boundary Conditio

MS Thesis Jalali

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