ms thesis jalali
TRANSCRIPT
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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
UMI Microform
Copyright
All rights reserved. This microform edition is protected againstunauthorized copying under Title 17, United States Code.
ProQuest Information and Learning Company300 North Zeeb Road
P.O. Box 1346Ann Arbor, MI 48106-1346
by ProQuest Information and Learning Company.
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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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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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1
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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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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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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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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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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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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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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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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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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( ) 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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+=
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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( )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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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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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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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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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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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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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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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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=++
+
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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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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44
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