c

Gas sorption and transport in coals: A poroelastic medium approach

Jun Yi a, I.Yücel Akkutlu b,

⁎, C.Özgen Karacan c

, C.R. Clarkson d

a Chongqing University, Chongqing, China b University of Oklahoma, Norman, USA

CDC, NIOSH, Pittsburgh Research Laboratory, Pittsburgh, USA d University of Calgary, Calgary, Alberta, Canada

⁎ Corresponding author. 100 East Boyd, Sarkeys Energy Center, Room T-311, Norman,
 

OK 73079-1003, USA. Tel.: +1 405 325 8141; fax: +1 405 325 7477.
 

E-mail address: akkutlu@ou.edu (I.Y. Akkutlu).
 

abstract

In this paper, single-component gas sorption and transient diffusion processes are described within coal

matrix exhibiting bimodal pore structure. The coal matrix is treated as a poroelastic medium manifesting

swelling and shrinkage effects due to the sorption of gas under effective overburden stress. Gas transport is

considered Fickian with molecular (bulk) and surface diffusion processes simultaneously taking place in the

macro- and micropores of coal, respectively. The numerical formulation is intended to be explicit in nature to

investigate the influences of sorption phenomena on the macropore volumes and on the overall gas transport

for the cases of gas uptake by and release from coal.

Results of the study show the presence of hysteresis during a sorption–desorption cycle of the gas. It is also

found that the overall gas transport takes place at a rate significantly less than that in the macropores only.

Thus the existence of a retardation effect in the overall gas transport is concluded. This retardation effect is

primarily due to the micropore resistances, in particular gas adsorption, and is independent of the changes in

the macropore volumes. It is shown that macroporosity of the coal matrix may change during gas transport

due to combined effects of pressure and sorption-induced swelling or shrinkage of the coal. It is estimated

that the macroporosity variation is non-uniform in space and time, as it is expected in reality, and typically

taking values less than ±10 percent of the initial porosity.

1. Introduction

As an unconventional natural gas resource, coalbed methane

receives worldwide attention. Coal seam gas is produced using

methods and technologies adapted from the conventional oil and

gas industry. Both primary (pressure) depletion and enhanced coalbed

methane recovery (ECBM) methods have been tested for different

pilot scales. In the ECBM process, non-hydrocarbon gases (e.g., flue

gas, nitrogen or carbon dioxide) are injected to improve recovery of

CH4, either by stripping (e.g. with nitrogen) or displacement (e.g. with

CO2). In the case of displacement with CO2 injection, one additional

benefit is to simultaneously sequester CO2, if the geology and

hydrology of the coalbed permit long term storage of injected gas.

Unlike conventional reservoirs, gas storage, flow and transport

processes in coal seams are quite complex, mainly due to the extremely

intricate and heterogeneous nature of the coal. Coals exhibit multi-scale

heterogeneity often characterized by a distinctive matrix structure

involving macropores and micropores (Gan et al.,1972; Unsworth et al.,

1989), which can be treated with a bimodal (bidisperse) pore size

distribution. The matrix-pore structure, specifically the relative abundance of micro-/meso-/macropores volume, has been demonstrated to

be a function of coal organic matter composition and thermal maturity

(ex. Gan et al., 1972; Clarkson and Bustin, 1999a).

Coal can be viewed as a solid, organic microporous material hosting

micropores, interconnected macropores and uniformly distributed

fractures, i.e., cleats. Most gas storage occurs within the microporous

matrix, which has a large internal surface area; mass transfer is primarily

by diffusion. On the other hand, large fractures and cleats penetrating

the coal matrix have very limited gas storage capacity but they are

important for their role in Darcian flow of gas and water, and thus in

production and injection operations. Hence, at moderate to high coalbed

pressures a significant amount of natural gas resides on the surfaces of

micropores in an adsorbed state (Thimons and Kissel,1973), as well as in

intra-molecular voids within complex macromolecular structure of the

solid material in an absorbed/dissolved state (Karacan, 2003; Larsen,

2004; Romanov et al., 2006; Medek et al., 2006).

Gas adsorption and dissolution may cause the coal matrix to swell

and/or shrink. This may change the specific surface areas and total

macropore volume of the coal matrix (Romanov et al., 2006), and thus

gas transport by diffusion in the coal matrix (Cui et al., 2004; Yi et al.,

2008).Also,the change in the matrix volumemay vary the cleat aperture

width which in turn may affect the Darcian flow to the boreholes by

changing the effective permeability of coal. Therefore, during primary
production and ECBM operations, adsorption and dissolution processes

are anticipated to add a complex and dynamic nature to production and

injection operations. Thus, depending on the combined effects of gas

sorption and the effective stress variations, the absolute permeability of

coal may change during primary depletion or greenhouse gas

sequestration/enhanced recovery (ECBM) operations (Palmer and

Mansoori, 1998; Shi and Durucan, 2005).

Based on the premise that the injection and production operations are controlled by flow through cleats and the interconnected

fractures, a series of experimental (Harpalani and Schraufnagel,

1990; Seidle et al., 1992), theoretical (Palmer and Mansoori, 1998; Cui

and Bustin, 2005; Shi and Durucan, 2005; Mavor and Gunter, 2006)

and field (Gierhart et al., 2007; Clarkson et al., 2008) studies were

reported in literature to investigate the overall influences of the

adsorbed gas and effective stress on cleat permeability, particularly

around wellbore. Although, these pioneering studies improved our

understanding of the possible causes of permeability changes around

the borehole, they did not attempt to separate out the effects of

absorption and adsorption and ignored the effect of dissolved gas in

the coal matrix on permeability evolution. Consequently, they may

have simply attributed the swelling effect entirely to adsorption. In

addition, they did not consider the effect of macropore changes in the

coal matrix on diffusive transport.

Smith and Williams (1984), Clarkson and Bustin (1999b), Shi and

Durucan (2003), Cui et al. (2004), and Yi et al. (2008) among others,

previously applied some form of a bidisperse model to describe CH4,

N2, or CO2 transport through the coal matrix. Smith and Williams

(1984) applied Ruckenstein et al.'s (1971) analytical model assuming

linear adsorption of CH4 in both micro- and macropores, whereas the

latter authors used numerical methods to quantify diffusion through

micro-/macropores, assuming non-linear adsorption in the micropores. None of these studies considered storage due to absorption.

The purpose of this paper is to develop a theoretical framework

suitable for a fundamental investigation of single-component gas

sorption and transport in the coal matrix, which shows poroelastic

material behavior. In this paper, a bidisperse coal structure viewed as a

microporous matrix, which contains pores on the order of a few

molecular diameters, penetrated throughout by interconnected macropores (Fig. 1) was considered. It was conceptualized that the microporous matrix retains the bulk of the gas, while the macropores have

relatively negligible gas sorption capacity compared to micropores. In

addition, since the matrix exhibits a large internal surface area for gas

sorption and a relatively strong affinity for gas, it is expected that an

additional diffusive mass transport in the adsorbed phase develops in

the direction of free gas mass fluxes. Thus, itis assumed that gas sorption

rates and flow in and out of coal are controlled by combined pore and

surface diffusive mass fluxes. Furthermore, the sorbed and free gas

mass-transport processes are expected to vary spatially and temporally

and they can be mapped by mathematical models. The so-called “dualsorption” approach is considered for the modeling of adsorption and

dissolution. The approach invokes the existence of two thermodynamically distinct gas populations, namely, molecules dissolved in the solid

coal material by an ordinary dissolution mechanism (obeying Henry's

law) and the molecules residing in the preexisting openings, i.e.,

micropores, in the matrix (obeying Langmuir type of isotherm) with

rapid exchange between these two populations (Rangarajan et al.,1984).

An early development and illustration of the applicability of the dual

sorption model are a result of extensive investigations by Koros et al.

(1977), Chan et al. (1978), Erb and Paul (1981) on sorption and transport

of gases in glassy polymers. An exhaustive review by Paul (1979)

furnishes detailed information pertaining to different aspects of dual

sorption model. Later, the dual sorption model has been applied to

describe solvent (Green and Selby, 1994,) and organic vapor sorption

behaviors in coals (Shimuzu et al., 1998).

Fig. 1. Schematic of bidisperse coal pore structure showing macro- and micropores containing gas in free, adsorbed and dissolved states.

2. Sorption and Transport in Poroelastic Coal

In this section, the governing equations are described in onedimension and in scaled and dimensionless forms. It is shown that the

description can be achieved using a nonlinear second order transientdiffusive gas mass conservation equation coupled with two auxiliary

equations — those that describe the dynamics of solid/pore volume

change and the equilibrium sorption isotherm.

For the solution of the governing equations describing the

fundamentals of gas sorption and transport in coal, the conceptual

coal matrix block is subjected to initial and boundary conditions so

that gas uptake and release experiments can be performed numerically to investigate the influence of adsorbed/dissolved gas on the

microporous matrix and macropore volumes. The results are

presented using the estimated fractional uptake and release rates,

the macroporosity variations and, as an indication of overall gas

transport, the apparent diffusion coefficient.

With Eq. (1) the continuum description is considered valid and,

hence, at a given time and location, gas is distributed at adsorbed and

dissolved states in the microporous coal matrix and as free gas in the

macropores. Gas mass balance in one-dimensional bidisperse coal

contains the following transient and diffusive terms:

( ) A A [ ] 1 A AC ð/CÞ þ ð1−/ÞCμ 1⁄4 xn/Dp At At xn Ax Ax ( ) 1 A n ACμ þ x ð1−/ÞDs ð1Þ xn Ax Ax

In this equation, n is the shape factor for the coal body (where n= 0

for linear, n= 1 for cylindrical and n=2 for spherical coordinates). C is
the free gas concentration in the macropores (moles per unit

macropore volume), whereas Cμ represents concentration of the gas

sorbed by the microporous solid (moles per unit microporous matrix

volume). The first terms on both sides of the governing equation

include a macroporosity φ describing gas mass accumulation and

diffusive mass flux in the macropore void space, respectively. The

second terms, including (1−φ), represent the accumulation and

transport of the sorbed gas in the microporous matrix.

It should be noted that only the total amount of sorbed gas enters

in the mass balance given in Eq. (1) and that its transport through the

microporous matrix is considered to obey Fick's law. Thus, using

Deborah Number, De=α/θ introduced by Vrentas et al. (1975) and

Vrentas and Duda (1977), with the Eq. (1), it is implicitly assumed that

the macromolecular structural relaxation time α is either much larger

(DeNN1) or much smaller (Debb1) than characteristic diffusion time,

θ. When De is in the order of one, on the other hand, the microporous

coal matrix changes phase from glassy structure to rubbery structure

in the presence of a sharp moving boundary. Consequently, a diffusion

process that starts out as Fickian may turn into a so-called Case II, or

anomalous diffusion, with a shock front (Thomas and Windle, 1982;

Neogi 1983; Hui et al., 1987). However, Mazumder and Bruining

(2007) recently showed that, in practice, the latter could only be a

transitional effect for the gas–coal systems. This type of anomalous

behavior is therefore not considered in this work.

In addition, in the development of a mathematical description, the

microporous matrix is considered to be in equilibrium with the gas

and its equilibrium sorption dynamics is described with the

Langmuir–Henry dual-mode isotherm (Green and Selby, 1994;

Shimuzu et al., 1998). If Cμs represents the complete monolayer

coverage in the solid–gas interface and k′d the coefficient of linear

absorption, then the isotherm can be described as:

CμsbVC Cμ 1⁄4 þ kVdC ð2Þ 1 þ bVC

Here b′=bRT and b is often referred to as the Langmuir constant.

When the total gas pressure p is low (i.e., bpbb1), Eq. (2) reduces to a

form similar to Henry's law isotherm, which states that the sorbed gas

concentration in the microporous solid increases linearly with the gas

pressure. The dual-mode isotherm model is an effective approach to

match laboratory experiments, particularly for glassy polymers. This

approach has also been used to investigate organic vapor solvent–coal

interactions (Medek et al., 2006) with the assumptions in the

transport equation describing organic vapor–coal interactions, such

as the two modes are in equilibrium and take place simultaneously;

gas molecules adsorbed under Langmuir mode are immobilized; and

diffusion occurs only in the dissolved mode and the diffusion

coefficient is independent of concentration.

Fig. 2 shows a comparison of the dual mode equilibrium sorption

isotherm (Eq. (2)) with the Langmuir adsorption isotherm versus

pressure, i.e., the free gas macropore concentration C. The values of

pressure in this figure are obtained using the ideal gas law. The dualmode isotherm implies that the gas sorption increases infinitely with

pressure. Thus, although for the pressure ranges of relevance for CBM

and CO2-ECBM scenarios this model yields sorption isotherms that are

typical, the consequence of applying this relationship uncritically may

yield highly unrealistic sorption values for coals. Numerical experiments reported in this work are in between 0–5 MPa, i.e., the lower

end of the pressure domain shown above.

Fig. 2. Comparison of the Langmuir–Henry “dual-mode” sorption isotherm, i.e., Eq. (2),

with the Langmuir adsorption isotherm. The pressures are obtained using ideal gas law.

In order to include macropore-volume strain changes into the

overall transport model, linear elasticity for strain changes in

isothermal coal is considered. For this purpose, Palmer and Mansoori's

(1998) poroelastic model is modified to describe the macroporevolume strain increments under the influence of sorption and

constant overburden stress:

" # bV d/ 1⁄4 a1 þ a2 þ a3 dC ð3Þ 2 ð1 þ bVCÞ

The coefficients, ai, in Eq. (3) represent the stress and sorptioninduced volumetric strains:

[ ( ) ] ( ) ( ) RT K K K a1 1⁄4 − −1 þ f γ RT ; a2 1⁄4 −1 kVd; a3 1⁄4 −1 el M M M M

Eq. (3) is slightly different from the form originally proposed in

that it now explicitly includes the effect of dissolved gas in solid

material. It is developed based on the common assumption (Cui and

Bustin, 2005) that the shape of the volumetric strain-sorbed gas

concentration curve is a Langmuir–Henry isotherm of the form given

in Eq. (2). εl represents a dimensionless adjustment parameter for the

strain and adsorbed gas concentration. The terms on the right side of

Eq. (3) are referred to as the macropore compression (a1) effect,

microporous shrinkage/swelling effects due to dissolution (a2) and

adsorption (a3), respectively.

Inserting Eqs. (2) and (3) and expanding the terms in Eq. (1), and by

applying the chain rule of differentiation, the following form of Eq. (1)

is obtained for gas sorption and transport in coal particles exhibiting

volumetric strain. In this development concentration-independent

diffusion coefficients are assumed.

[ ] ( ) () ACμ ( ) A/ AC /Dp A n AC / þ ð1−/Þ þ C−Cμ 1⁄4 x ð4Þ AC AC At xn Ax Ax ( ) ( ) ð1−/ÞDs A ACμ AC xn

xn Ax AC Ax þ

( ) ( )2 ACμ A/ AC

þ Dp−Ds AC AC Ax

where ∂φ/∂C and ∂Cμ/∂C can be described explicitly using Eqs. (2) and

(3):

A/ bV ACμ CμsbV 1⁄4 a1 þ a2 þ a3 and 1⁄4 þ kVd: AC 2 AC 2 ð1 þ bVCÞ ð1 þ bVCÞ

The second term on the right hand side of Eq. (4) can further be

expanded to yield:

[ ] [ ] () ACμ ( ) A/ AC ACμ 1 A AC xn

AC AC At AC xn Ax Ax / þ ð1−/Þ þ C−Cμ 1⁄4 /Dp þ ð1−/ÞDs

"( ) # ( ) ACμ A/ A2Cμ AC

þ Dp−Ds þ ð1−/ÞDs AC AC AC2 Ax

2
Dividing each term by the coefficient of free gas accumulation (i.e.,

the terms inside the brackets on the left hand side) gives the gas mass

balance in the following simplified form:

( ) ( )2 AC 1 A AC AC 1⁄4 D C x þ f ð Þ S C ð5Þ n ð Þ At xn Ax Ax Ax

In Eq. (5), the only dependent variable is the free gas concentration

C(x,t) in the macropores.

Eq. (5) contains two nonlinear coefficients: D and f s, which are

described as:

[ ] 1 ACμ Dð Þ1⁄4 C /Dp þ ð1−/ÞDs ð6aÞ Δretard AC

and,

"( ) # 1 ACμ A/ A2Cμ f S C 1⁄4 þ ð1−/ÞDs ð Þ Dp−Ds ; ð6bÞ Δretard AC AC AC2

ACμ ( ) A/ Δretard 1⁄4 / þ ð1−/Þ þ C−Cμ : ð6cÞ AC AC

As an immediate consequence of the formulation of the conceptual

model, these equations suggest that diffusive gas transport in bimodal

coal structure is always under the influence of a retardation effect (Eq.

(6c)) during equilibrium sorption phenomena. Although the latter

varies with the effects of free gas amount on the adsorbed phase and

on the macroporosity, it takes values larger than unity during gas

sorption and desorption, therefore, the effective gas transport is

expected to be at a lower rate compared to gas transport in the

macropores (unipore structure) only.

By employing Eqs. (2) and (3) in Eqs. (6a) and (6c), apparent

diffusion coefficient, D , in the coal and the retardation effect can be

described simultaneously as:

" ! # 1 CμsbV Dð Þ C 1⁄4 /Dp þ ð1−/Þ þ kVd Ds ð7aÞ Δ 2 retard ð1 þ bVCÞ

" # CμsbV Δretard 1⁄4 / þ ð1−/Þ 2 þ kVd

ð1 þ bVCÞ

( ) " # CμsbVC bV þ C− −kVd C a1 þ a2 þ a3 ð7bÞ 1 þ 2 bVC ð1 þ bVCÞ

In summary, Eqs. (5) and (6b) together with Eqs. (7a), (7b) describe

the transient gas sorption and transport in poroelastic coal with

bidisperse pore structure in terms of the free and sorbed gas

concentration.

2.1. Scaling and Nondimensionalization

Prior to defining the initial and boundary conditions and to

obtaining numerical solutions, Eq. (5) was transformed to the

following dimensionless form.

( ) ( )2 Ac 1 A Ac Ac ð Þ n 1⁄4 D c r þ fSð Þc ð8Þ Aτ rn Ar Ar Ar

The following nondimensional variables are introduced in Eq. (8):

x Dp t C r 1⁄4 ; τ 1⁄4 ; c 1⁄4 L L n Co þ Cμo

where L corresponds to a characteristic coal sample length defined as

the half-length of a symmetric coal slab in linear geometry (i.e., the

shape factor n equal to zero). Co and Cμo are the initially available free

and sorbed gas concentrations, respectively. The dimensionless

coefficients of Eq. (8) are

" () ! # 1 1−/ λμs D c 1⁄4 e 2 ð Þ 1 þ þ kVd ð9aÞ Δ / ð1 þ λcÞ

 !! ! 1 λ fS c 1⁄4 1−e λμs ð Þ þ kV aV aV a3 ð9bÞ d 1 þ 2 þ Δ 1⁄2 2 2 ð1 þ λcÞ ð1 þ λcÞ

() ! 1−/ 2λμsλ

þe / ð1 þ λcÞ 3 

where

() " # 1−/ λμs Δ 1⁄4 1 þ þ kVd / ð1 þ λcÞ

2

( ) " # 1 λμs c λ

þ c− −kVd c a1V þ aV

2 þ a3 ð9cÞ / 1 þ λ 2 c ð1 þ λcÞ

where the dimensionless quantities are:

( ) ( ) e 1⁄4 Ds ; aVi 1⁄4 ai Co þ Cμo ; λ 1⁄4 bV Co þ Cμo ; λμs 1⁄4 CμsbV Dp

Eq. (8) is a second order nonlinear partial differential equation,

numerical approximation of which could be obtained using an implicit

finite difference scheme and Newton iteration. Time integration of the

ordinary differential equations resulting from the discretization in

space is performed by a solver, which is based on an implicit linear

multi-step method that chooses the time steps dynamically during the

computations (Higham and Higham, 2005).

2.2. Initial/Boundary Value Problems

Gas sorption and transport within a coal particle that experiences

strain due to sorption/desorption, overburden stress and gas dissolution in the matrix are investigated based on solutions to Eq. (8) using

the initial and boundary conditions given in Table 1. This table

indicates that initially gas is either absent (in the case of uptake) or

distributed uniformly (in the case of release) as free gas in macropores

and in sorbed states in the microporous matrix as dictated by the dualmode sorption isotherm. The outer boundary condition (at r= 1) is

described as a source/sink for the gas. Although, the conditions are

defined for the free gas concentration in the macropores, amounts of

the adsorbed dissolved gas at any particular time and location can be

easily predicted at each time step.

Table 1

Initial/boundary value problems

Gas uptake: Gas release:

τ= 0; c(r,0)= 0 τ= 0; c(r,0)= 1

r= 0; ∂c/∂r= 0 r= 0; ∂c/∂r= 0

r= 1; c= 1 r= 1; c= 0

The properties of gas–coal system are given in Table 2. The

investigation considers the sorption and transport behavior comparatively for CH4, N2 and CO2 separately. As shown in Table 2b, we

consider that these components have preferential and graded sorption

tendencies with the coal such that CO2 has the largest, CH4 the

intermediate and N2 the smallest sorption capacity. Henry's constant
k′

d values estimated for these components are 0.54, 0.28, and 0.008,

respectively (Medek et al., 2006).

Table 2

Properties of gas and coal materials

Parameter Unit Value

A — Coal Properties

Temperature, T °K 293.0

Initial macroporosity, φ frac. 0.02

K/M ratio frac. 0.54

Parameter, f frac. 0.50

Grain compressibility, γ 1/MPa 1.3E-10

Initial free gas conc., Co moles/cc 2.0E-03

Parameter Unit Value

N2 CH4 CO2

B — GAS PROPERTIES

Cμs mole/cc 2.0E-3 2.5E-3 3.0E-3

b′ cc/mole 5.0E+2 8.0E+2 12.0E+2

k′d frac. 8.0E-2 28.0E-2 54.0E-2

εl frac. 9.0E-3 12.3E-3 24.6E-3

ε=Ds/Dp frac. 1.0E-3 1.0E-3 1.0E-3

3. Results and Discussion

3.1. Fractional Gas Uptake/Release Rates

Fig. 3 shows free CH4 and CO2 fractions in the coal sample as a

function of dimensionless time during gas sorption and desorption

stages. Hysteresis observed between sorption and desorption curves

points to a complex interplay of the processes describing solid–gas

interactions under different initial conditions. The adsorption hysteresis has a long history (Cohan, 1938); however, the existing discussion

in the literature involves its association with the occurance of capillary

condensation, a phenomenon which is not considered in our model.

Desorption curves show that almost all of the gas in adsorbed and

dissolved phases is eventually being desorbed under the influence of

the existing concentration gradients across the coal sample at longer

times.

Fig. 3. Fractional gas uptake and release for CH4 and CO2. (Blue lines are the fraction of

gas desorbed, whereas green lines are the fraction of gas adsorbed.)

We note thatthe x-coordinate in Fig. 3 is dimensionless time τ, which

is normalized using Dp, defined as macropore diffusivity in the manuscript. Based on the previous experimental studies to date, the latter is

expected to take larger values for CO2. Hence, if we were to convert the xcoordinate to dimensionaltime t using lower macropore diffusivity value

for CH4 in Fig. 3a and higher value for CO2 in Fig. 3b, then the real time

scale for methane would stretch significantly. According to Clarkson and

Bustin (1999b) and Cui et al. (2004), we anticipate that this stretching

will cause the CH4 sorption to take place in a time scale ten to hundred

times longer than the times scale for CO2.

Fig. 4. Estimated macroporosity versus free gas concentration at r=0.5 during gas

release from coal. Initial macroporosity is 2%. Porosity variations are due to overburden

stress only, i.e., a1 ≠0, a2 = 0, a3 = 0.

Fig. 5. Estimated macroporosity versus free gas concentration at r=0.5 during gas

release from coal. Initial macroporosity is 2%. Porosity variations are due to a1 ≠0, a2 = 0,

a3 ≠0.
3.2. Influence of Sorption on Coal Macroporosity

As described previously in Eq. (3), changes in coal macroporosity

during gas sorption and transport are due to three independent effects.

Figs. 4–6 show these effects separately during gas desorption from coal.

Fig. 4 shows the influence of only a constant overburden stress on

the macroporosity as a linear pore-volume compression effect. Using

this analysis dφ/dc is estimated to be a positive value and changing

linearly as a function of free gas concentration, or with pore pressure.

Furthermore, when free gas concentration in the coal drops to zero,

macroporosity decreases from its initial value of 2% to a lower value of

1.7%. The convergence to the same porosity value is expected for all

gases since the intrinsic properties of the gases and their interactions

with coal during sorption are excluded at this stage. The macroporosity change is only due to the gas concentration, thus pore

pressure, change.

In Fig. 5, the influence of adsorption is introduced in addition to the

previous analysis, i.e, a2 = 0, whereas a1 ≠0 and a3 ≠0. Fig. 5 shows the

adsorption isotherm-related deviations from the linearity previously

illustrated in Fig. 4. Depending on the adsorption capacity of coal, each

component shows a different effect on macroporosity changes during

the gas release. As lower concentrations are reached, desorptionrelated coal shrinkage effect becomes somewhat more dominant in

N2. However, in the case of CO2 and CH4 a decrease in macropore

porosity is experienced during desorption when pressure (or

concentration) decreases, followed by an increasing period during

which porosity re-bounds back to values higher than initial porosities

showing more shrinkage taking place. Thus, the porosity increase due

to shrinkage effect overcomes porosity decrease due to increasing

compression effect during desorption. The final porosities depend on

the interplay of these two and the adsorption affinities of the gases.

Fig. 6. Estimated macroporosity versus free gas concentration at r=0.5 during gas

release from coal. Porosity variations due to varying a1, a2, a3 coefficients.

In the next step of the analysis, the influence of gas absorption (or

dissolution) is investigated during the gas desorption experiments,

i.e., a2 ≠0 is also taken. Fig. 6 shows the combined effects of sorption

(adsorption and absorption) processes and overburden stress on

macroporosity. This figure clearly shows that previous observations in

Fig. 5 are amplified when the effect of gas dissolution in the coal

matrix is included. Dissolved gas apparently causes matrix to shrink

more, compared to adsorption only, during a gas-release process. A rebound effect is observed where macroporosity first decreases due to

pressure effects, then increases as a result of matrix-shrinkage; in the

case of CH4 and CO2, the macroporosity at low concentrations actually

exceeds the initial macroporosity. For example, when the dimensionless CH4 amount drops to ∼0.15, the decrease of porosity due to

increasing overburden stress is balanced by increased shrinkage of the

coal matrix and a re-bound effect around 1.9% porosity is observed.

Fig. 7. Estimated macroporosity versus free gas concentration at r=0.5 during gas

uptake by coal. Solid lines with full model; dashed lines with gas adsorption only.

Fig. 8. Dimensionless apparent diffusion coefficient D(c) versus time during CO2 release

from bidisperse coal. The coefficient is normalized using the macropore pore diffusion

coefficient, Dp.
The presence of a gas dissolution mechanism creates additional

shrinkage/swelling effects on macroporosity for N2 and CO2 too.

Fig. 7 shows variations in macroporosity during gas sorption

(uptake) experiments. The analysis of gas sorption on macropore

porosity was performed considering the effects of only adsorption

and, adsorption and absorption combined. We now observe the

interplay of changes between a linear macropore volume expansion

against the overburden stress and a nonlinear macropore decrease

due to coal matrix swelling. In the case of N2, at early stage when its

concentration in coal is relatively low, the sorption related coalswelling effect dominates the change in macroporosity and the

macroporosity becomes smaller; at higher pressures, the macroporosity increases due to compression effects. This is not strictly the case,

however, for CH4 and, more importantly, for CO2. The latter gas

component has the potential to swell the solid material to an extent

that macropore expansion due to gas uptake causes only leveling

effect at large times. This is shown in Fig. 7 as a constant

macroporosity value roughly equal to 1.5%.

Fig. 7 also illustrates the importance of absorption in microporous

matrix during gas sorption in coal. Clearly, when absorption is

included, the swelling of the coal matrix increases significantly.

Especially, in the case of CO2, as the adsorbed and dissolved gas

amount in coal increases, porosity decreases even more. This can be a

major concern during a CO2-ECBM process.

3.3. Gas Transport in Bidisperse-Poroelastic Coal

The apparent diffusion coefficient values are calculated using Eq.

(9a) and are shown in Fig. 8 during a CO2 desorption exercise. For

comparison with the poroelastic approach, the figure also shows gas

transport in rigid coal bodies with a fixed macroporosity value, 2%. The

diffusion coefficient shown in this graph is in dimensionless form,

normalized using macropore diffusion coefficient, Dp. Hence, in the

absence of gas transport in the microporous solid, i.e, the sorption

processes and sorbed gas diffusion, its value is significantly larger and

equal to 1.0, (unipore diffusion). The estimated values during gas

release in bidisperse coals, on the other hand, point to a drastic

deviation from this unity, values varying as low as 0.01 (Fig. 8). It drops

significantly during an early transition period, when free gas in the

macropores is rapidly released and depleted; and later, under the

influence of microporous solid resistances (sorption and surface

diffusion), it stabilizes to a constant low value prior to the complete

gas recovery.

Fig. 8 shows that the stabilized value of the apparent diffusivity

with respect to the macropore diffusivity is as low as 0.01. Yi et al.

(2008) presented a comparative investigation of the overall transport

using rigid bidisperse and unipore coal models. They showed that

retardation due to sorption in microporous solid is mainly responsible

for this significant drop in the stabilized effective diffusivity. Based on

the parameters describing their gas–coal system, they also estimated

an additional 20% drop in the stabilized effective diffusivity solely due

to the presence of surface diffusion in the microporous solid.

Fig. 8 also shows the difference in gas transport in rigid and

poroelastic coal bodies. It is clear that changing macropore and solid

volumes cause an improvement on gas release, although the effect is

almost negligible when compared with the diffusional and sorption

resistances in microporous solids.

4. Conclusions

The theoretical framework considers macropores as the places

where rapid mass transport takes place, whereas the microporous

matrix as poroelastic solid coal bodies retaining majority of the gas in

sorbed (physically adsorbed and dissolved) states. Behavior of the

sorbed gas is then limited by the transport and equilibrium sorption in

the solid. The approach introduces a bimodal pore structure to the

framework in a simple mathematical form that defines gas mass

accumulation and diffusive mass flux terms for each mode. This gives

an ability and significant flexibility to investigate gas–solid interactions for a given initial- and boundary-value problem.

It is theoretically observed that gas sorption and transport

processes and their influence on solid and macropore volumes are

at a fundamental level closely related to the affinity of coal material to

the natural gas component. Based on the conceptual model assumptions inherent to the numerical experiments performed, the following

conclusions are made:

•	 Keeping the overburden stress constant, coal with a large sorption

capacity shrinks during the gas release, consequently its macropore

volume increases;

•	 Similarly, coal with a large sorption capacity swells during the gas

uptake, yielding lower macroporosity values.

•	 As the sorption capacity decreases, or when the coal body consists of

a spatially nonuniform material content with varying preferential

affinity for the gas, it becomes a difficult task to determine the

impact of gas release and uptake on the macroporosity at a given

location in time.

•	 The importance of absorption in microporous matrix during the gas

sorption in coal is illustrated explicitly. Clearly, when absorption is

included, the swelling of the coal matrix increases. Especially, in the

case of CO2, macroporosity decreases even more as the adsorbed and

dissolved gas amount in coal increases. This is proven to be a major

concern during a CO2-ECBM process in pilot field studies.

•	 The dynamic nature of poroelastic coal pore structure, however, is

anticipated to a have a negligible influence on the overall gas mass

transport during the gas production and sequestration operations.

However, the impact of matrix-shrinkage/swelling on cleat porosity

and, hence, fracture transport is well documented.

Nomenclature
 

ai quantities defined in Eq. (3)
 

c nondimensional macropore concentration
 

cμ nondimensional micropore concentration
 

b Langmuir isotherm constant (cm3

/mol)
 

C macropore concentration (mol/cm3

)
 

C 3 o initial macropore concentration (mol/cm )
 

C 3 μο sorbed concentration in equilibrium with Co (mol/cm )
 

Cμ microporous solid-phase concentration (mol/cm3

)
 

CμS maximum sorbed-phase concentration in Langmuir isotherm (mol/cm3

)

D apparent diffusion coefficient (cm2

/s)

De Deborah Number

Ds micropore (solid) diffusion coefficient (cm2

/s)

Dp macropore diffusion coefficient (cm2

/s)

k′d Henry's constant

K bulk modulus (Pa)

M constrained axial modulus, (Pa)

n shape factor

R gas constant (J/K-mol)

T temperature (K)

r dimensionless radial coordinate

t real time coordinate

x dimensional radial coordinate

Greek symbols

α macromolecular structural relaxation time

θ characteristic diffusion time in micropores

τ nondimensional time

γ grain compressibility (1/Pa)

λ dimensionless Langmuir isotherm nonlinearity parameters

φ macroporosity
ε ratio of sorbed phase to macropore flux

εl dimensionless adjustment parameter

δ1 nondimensional macropore capacity

δ2 nondimensional microporous solid phase capacity

Acknowledgement

This research was partially funded by Natural Science Foundation

of Chongqing, CNSFC, China, and by the Natural Science and

Engineering Research Council of Canada, NSERC. Here, we gratefully

acknowledge their financial contributions.

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