Transport in Porous Media 4 (1989), 199-212. 199 �9 1989 by Kluwer Academic Publishers.

Mathematical Modeling of Methane Flow in Borehole Coal Mining System

a

L A S Z L O B A L L A Computer Center, Technical University for Heavy Industry, Miskolc, Miskolc-Egyetemvdros, Hungary, H-3515

(Received: 2 February 1988)

Abstract. Safety in coal mining is greatly increased by the drainage of the methane content of coal seams through boreholes, simultaneously producing significant energy. The design of suitable drainage technology is based on the mathematical modeling of methane flow in coal seams. In the calculation of the methane pressure, the new mathematical model presented in this paper considers both the sorption phenomenon of methane depending upon the methane pressure and the fact that the variation in methane pressure can create a change in the stress condition of the rock and, as a consequence of this, a change in the permeability of the coal. The new mathematical model can be used for the numerical simulation of the flow processes in coal seams and methane drainage technology can be designed more accurately.

Key words. Mathematical model, borehole coal mining system, sorption phenomenon of methane in coal, methane flow in porous coal seams.

1. Nomenclature

a

A

b

c

d

D

g

G

H

k iT/, H

M

P q r

R S t

T

sorpt ion capaci ty pa ramete r area, m 2

slip parameter , Pa

compressibil i ty parameter , Pa -1

me thane density, kg m -3

diffusion-dispersion parameter , m 2 s -1

gravi tat ional accelerat ion, m2s

sorbed methane volume, nm 3 m -3

height, m permeabil i ty, m z

sorpt ion parameters

molecular weight, kg kmo1-1

pressure, Pa flow rate between systems, kg m -3 s -1

radius, m universal gas constant , Pa m 3 tool -1 K -1

specific area, m -1

time, s

tempera ture , K

200

v Darcy velocity, m s -~ V volume, m 3

z methane compressibility

& porosity /~ viscosity, Pa S - 1

div divergence operator

grad gradient operator

Subscript a macro ae macro-mezo

c sorption

e m e z o

ei mezo-micro

i micro K Klinkenberg

max maximum

n normal

o reference

r relative

R rock

s total

t time

�9 t ordinary differential operator with respect to time

,t partial differential operator with respect to time

Superscript a average

g free methane G sorbed methane

LASZLO BALLA

2. Introduction

It is important to know the propert ies of methane flow in coalfields from two outstanding points of view, one of which is the safety demand in mines and the other is to develop an optimal technology for methane recovery. To decide which

point of view is more important can be determined by the characteristics of a particular coalfield and the given economic conditions.

In the case of coalfields being dangerous because of occasional methane bursting out of a borehole, the recovery of methane from the surface can increase the safety of later underground mining and reduce the costs of ventilation, safety measures and labour and, as a result, it can increase the productivity of coal

MATHEMATICAL MODELING OF METHANE FLOW 201

mining. In the case of coalfields containing a large amount of methane, a

remarkable amount of gas energy can also be obtained, besides increasing safety

in the mine. For modeling this process with computer simulation, the basic condition is to

possess a mathematical model adequately describing the methane flow in coalfields. This computer simulation can later help to develop a technology for recovery by taking into account all demands and, in time, getting to know the recovery parameters.

A suitable mathematical model must describe the methane flow without any physical failure and, at the same time, take into account the time and memory limits of the computer simulation. Different mathematical models have been developed which depended on subjective decisions about the importance or ignorance of one or other parameters determined by the known features of the coalfields concerned.

The methane flow in coalfields can be generally treated as a process comprising of two steps, one of which is the diffusion of gas through a porous structure and the other is a free gas flow through pores and fissures of a suitable size. This later process can be associated with a two-phase flow, but in this paper only the single-phase homogeneous flow of methane is dealt with.

In different models, the relative importance of conductive and convect ive flows are differently evaluated. The processs of diffusion is generally treated as a one-step process because the coal structure is regarded as a simpler one and, occasionally, it can be right, but the up-to-date and more accurate computing methods treating the process of diffusion as a two-step process, are based on a more complex coal structure.

The quantity of methane sorbed in coal as a function of time and location, changes with the pressure of free methane and there are several models of different types and complexity describing this process.

The description of methane flow coalfields is complicated by a smaller or

greater modification of the coal structure as a function of the coal features (disturbance initial field states and alteration in rock stresses induced by pressure changes in freely following methane) and so these features, including per-

meability which is not regarded as being important by some authors, change from field to field.

In mathematical models used as the basis for computer simulation, several physico-chemical subprocesses are described in a simpler way, though with more tolerable simplifications than in computer simulation of individual processes, to minimize the execution time and memory occupation used for computer simula- tion.

In the present paper, a new mathematical model is introduced that takes into account the most recent results of researches in the field of subprocesses. These results are built into the mathematical model with special regard to the require- ments of an effective computer simulation and so, with the help of this computer

202 LASZLO BALLA

simulation, a more productive technological system can be developed for the recovery of methane.

3. Review

The difference in mathematical models lies in the variation of coal structure models.

In Figures 1, 2 and 3, unit parts of the three structure models used in

mathematical modeling can be seen. The models are classified in this paper because different names are used for the same model by different authors.

The simplest model is a macro model (Figure 1) with a homogeneous structure (Vorozhtsov et al., 1975; Jones et al., 1982) that describes the process of sorption as a process requiring no time, so it can be used only for forecasting trends. This macro model possesses a statistically homogeneous convective- conductive flow field (macro system).

The mezo model (Figure 2) used nowadays in computer simulation of methane flow through coal seams (King et al., 1983) makes it possible to give a more accurate forecast and to describe adequately the process of sorption in some cases where the coal structure can be represented with a mezo model. The mezo

model is statistically homogeneous and consists of elementary parts in arrays (mezo system) that are connected with one another only through a statistically homogeneous convect ive-conduct ive flow field (macro system).

A closer study of the sorptional processes in coals, however, proves in some other cases that the mezo model does not describe adequately the process of sorption, and so the micro model should be used (Smith and Williams, 1982). This micro model (Figure 3) can be regarded as a mezo model in which not all

Fig. 1. Macro model.

MATHEMATICAL MODELING OF METHANE FLOW

Fig. 2. Mezo model.

203

parts of an array are statistically homogeneous, but consists of smaller statistically homogeneous array elements (micro systems) backing up only conductive flows.

These micro systems are connected to one another by statistically homogeneous flow fields (mezo systems) being only conductive. The conductivity or dispersity factors are, however, different.

The coal structure model of the mathematical model used in the present paper is a micro model in which the micro system of the array part are connected with a mezo system to build up a system of arrays coupled to one another by micropores and fissures, i.e. they are connected through the macro system. This new

Fig. 3. Micro model.

2 0 4 L A S Z L 0 B A L L A

mathematical model comprises possibilities for modeling coal seams with a macro or mezo structure, as these are simplified cases of a micro model.

The mathematical model describing the sorptional processes of coal have either no physical meaning (Airey, 1968) or the solution of partical differential equations having physical meaning, requires too much computing time, so this method can only be used in cases of coal particles (Smith and Williams, 1982) and is not suitable for computer simulation of coal seams.

The new model uses a simpler system of ordinary differential equations, having also a physical meaning for sorptional processes.

The influence of gas pressure on the permeability of coal is known (Somerton et al., 1975; Mordecai and Morris, 1974). Up to now, it has always been neglected in simulations, but in the proposed mathematical model, the influence of methane pressure is taken into account.

The sorptional capacity of coal is usually characterized in simulating models with the well-known Langmuir equation, but in the new mathematical model, the sorptional capacity of coal is described with a more general equation (T6th, 1983).

4. Mathematical Model

The mass flow of methane in micro, mezo and macro systems is characterized by the following equations of mass conservation

Macro system:

(4)~d~ + d~Go),,

-- - d iv(d~v, g) + div(~ba(D, g grad(dg) + D~ ~ grad(d~ Ga))) + qae. (1)

Mezo system:

(49edge + dgnGe),t = div(qSe(D~ g grad(de g) + P ~ grad(d.g Ge))) + qei. (2)

Micro system:

(4~,d f + d . g Gi)., = div(4~i(Df grad(d,r + D ~ grad(d~ Gi))), (3)

where q.e is the mass flow between the macro and mezo systems and qei is the mass flow between the mezo and micro systems.

qae = (1 -- 4~)(rhaSe(D~e grad(deg).e + De ~ grad(d~Ge)ai)), (4)

qei = ( l - - 6e)(rb~S,(Of grad(dr)e, + D ~ grad(d, g Gi).~)). (5)

In Equations (1)-(5), the initial and surface conditions are

t = 0 , d g ~ = d ~ = d f ,

(dg)e, = (d~).,, (d~).. = (dga)ae,

D~ grad(d~)e~ = De g grad(deg)~,, (6)

MATHEMATICAL MODELING OF METHANE FLOW 205

D~ grad(dg)ae = D , g grad(dg)ae,

D ~ grad( d .g Gi)e, = D e ~ g rad(d g Ge)e,, (7)

D• grad(dgGc)ar = D ~ grad(dgG~)~r

The boundary conditions of the system are determined by given geological and recovery conditions.

For porosities and specific surface areas in Equations (1)-(5), the following requirements are to be met:

4's = ~b~ + (1 - ff)a)(6e -~- (1 - - 4e)~bi),

S~ = A d V~ , S~ = A d V~ ,

G~= G. + ( 1 - ~G)G~ + ( 1 - 4,a)(l - 0be)G,.

The system of partial differential Equations (1)-(8)

(8)

(9)

(10)

is simplified with the following tolerable assumptions in modeling of methane flow in coal seams:

- D ~ = O, i.e. surface diffusion of methane is negligible in macro system, - d~ = D~ = O, i.e. there is no free gas in the micro system, only the surface or

structural diffusion of methane exists,

- inside of mezo and micro systems, the local changes are not taken into account; only their volume average is computed and the gradient value of a surface is approximated with its difference value. Accordingly, assumptions equations (2)-(5) can be rewritten as

g a (&~dag+e d , G e ) . , = ~ a e , , u ,qg - d~ g) + D ~ d ~ ( G ~ - G'~) + q~,,

g a = D a . G d g i ( ~ g _ _ ( d , G , )., G'~), ~ e ~ ~ n \ ~ e

qae --(1-- ag ~ ag g = d p a ) ( D ~ e d , , ( G o - G ' ~ ) + D ~ ( d a - d ~ g ) ) ,

= 42~)D~ d , , (G~ - G~), qei --(1-- .O g a a

- initial and surface conditions for Equations (1), (11)-(14) are

a a g t = O, d~ = d~ g, G'~ = G e ( d ~ ),

Gge = Gg(d~g), Go = Ga(d~) .

these

(11)

(12)

(13)

(14)

(15)

(16)

The flow of free methane in the macro system can be described with the help of Darcy's relationship

v~ = - (k/ /x) grad(p~ + d~- g- H). (17)

The equation of state for methane and the equation for sorptional capacity of coal are

Pg = d g R T z / M , (18)

206 LASZLO BALLA

G = Gmax(pgr/(1 + (pgr)m)l/m) n, pr = p g / p ~ , (19)

where, pg always refers to the given system. The sorptional capacity of the systems is

G,~ = Gsaa , Gae = Gsae/(1 - 49a), G'] = Gsad( (1 - 49a)(1 -- 49e)), (20)

aa q- ae + ai = 1.

With Terzaghi 's one-dimension model (Bear, 1 972) taken into consideration, the effect of the change in effective rock stress can be approximated by the change in the methane pressure, provided that the total rock stress is constant.

Verruijt 's ( i977) model describes the interaction between the rock displace- ment and fluid flow by way of a set of differential equations, but the proposed model describing methane flow applies suitable approximations from a practical

point, in order to simplify the model. The influence of the methane pressure on the permeability of coal is ap-

proximated with the following function:

k -- ko" k,(1 + b K / p g) exp(--Ck(pgo -- pg) ) , (21)

where ko = permeability extrapolated to unlimited methane pressure, kr = change in permeability caused by alteration in rock stresses due to

the disturbance of the initial rock field.

The third and fourth factors take into account the slip effect and the change in

methane pressure, separately. The influence of the methane pressure on the porosity of coal is approximated

with the function

49 = 490 e x p ( - c p ( p g o -- pg)). (22)

The factors k0, 490 and k, are distributed in space and kr not only depends on geological parameters but also on the disturbance of the rock field, its deter- mination is a problem of rock mechanics and if the effect of disturbance is ignorable k , = 1, but whether it can be neglected or not depends upon the

purpose of computation.

5. Numerical Solution

The system of Equations (1), (8)-(22) can be solved much faster with a numerical method and the result is much more accurate and stable than in the case of the system of Equations (1)-(7), (17)-(22), because the surface value of a micro system depends on its location inside the mezo system. At the same time, there is a mass flow between the mezo and micro systems so the solutions for both systems can only be obtained with an iterative approximation of the solution of the partial differential equations, while in the case of the proposed new model,

MATHEMATICAL MODELING OF METHANE FLOW 207

the solutions for both systems are simply obtained from the solution of the system of ordinary differential equations in a single step.

This system of equations is solved with the finite-difference method for pg, p~g and GT.

In order to get more accurate results, the values for divergencies in Equation (1) are computed at time t and t+ dt in this paper, using the Crank-Nicholson method.

The values of p~, pe ~g and G~ for time t + d t are made more accurate with two steps of iteration, as the values obtained from the solution of the system of ordinary differential Equations (11)-(14) with Runge-Kutta's method, are func- tions of the gas density d~ changing in time periods, and because of the presence of variable q,e in Equation (1).

The accuracy of computer simulation based on the new mathematical model is estimated by comparison with the values obtained experimentally in the labora- tory from mezo and micro structures of coal particles. In Figure 4, the relative quantities of desorbed methane in coal from New Mexico (Smith and Williams, 1982), and in a coal specimen from Mecsek, Hungary, are shown as functions of time. The experiment was carried out by the Research Center of Mecsek Coal Mines.

From the curves shown in Figure 4, it can be concluded that the process of

Gr 1"0 I ,'~'~r~-'---- ,....j~/t----~r "-'-~

o.8Lz Mecsek / /" c o a l /

0,6 NeWoMaleXic~

0,4.

�9 measured 0,2- - calculated

0 I I I" I 0 10 2O 30 4O 50

t'/2, m in'/2

Fig. 4. Relative desorbed gas profiles.

208 LASZLO BALLA

desorption is described with the same, satisfactory accuracy by both the system of

complicated partial differential equations and the system of simpler ordinary

differential equations. The applicability of the proposed new mathematical model is also supported by the fact that the computed and measured values are also in

good agreement with one another for other kinds of coal.

6. Results

The proposed new model cannot be compared adequately with the mezo model

used recently for computer simulation of methane flow in coal seams (King et al., 1983) and which gave fairly accurate results in the simulation of a one-borehole methane recovery system, as not all parameters were published. To carry out a

relative comparison, approximately the same values as in the above-ment ioned

simulation are used to prove the applicability of the proposed new model. In Table I and II, the identical and differing values used in the tests are listed.

Curve 1 in Figure 5 is identical to the result of the published mezo model, but with slight differences. The mezo model was controlled with experiments carried out on an operating well, so the adequacy of the proposed new model as a micro

model is explicitly proved by this fact, because the mezo model is a simplified

micro model and, at one side, a mezo model can be modeled with a micro model on the one hand and the computat ion can be made more precise, on the other.

Curve 2 in Figure 5 is based on the macro model and it can be seen that the differences are remarkable between the two methane pressure curves computed

on the basis of a macro and a mezo model, separately. Curve 3 was computed by using a micro model and it can be concluded from this curve that the same result

can be obtained by varying the values of diffusion and dispersion parameters . These values should be determined on the basis of experiments in the laboratory

before beginning computer simulation.

The above-ment ioned differences in the sorbed quantity of methane can be

seen in Figure 6. The results of the computer simulation carried out with values characteristic of

Table I.

Variable name Symbol Value

External radius r e 150 m Well radius rw 0.15 m Seam thickness h 1.8 m Initial seam pressure po ~ 3.5 MPa Reference porosity ~ho 0.1 Sorption volume constant Gmax 18.6 nm 3 m -3 Sorption parameters m, n 1 Klinkenberg coefficient bK 1 bar

M A T H E M A T I C A L M O D E L IN G OF M E T H A N E FLOW

Table II.

209

Test name ko aa a~ a~ cp Boundary conditions

01. QWQE 3 • -15 0 1 0 0 O e = 0 , O w = 0 . 0 1 2 m 3 s -I 02. QWQE 3 • -15 1 0 0 0 O e = 0 , O w = 0 . 0 1 2 m 3 s 1 03. QWQE 3• 10 -15 0 0.6 0.4 0 O e = 0 , Ow = 0 . 0 1 2 m 3 s -I 04. PWQE 0.01 • 10 -15 0 0.6 0.4 0 O~ = 0, Pw = 4 bar 05. PWQE 0.01 x 10 - i s 0 0.6 0.4 0.06 Qe = 0, Pw = 4 bar

coal from Mecsek, can be seen in Figure 7 and it can be concluded that the effect of alteration in permeability around the well cannot be ignored in connection with the commulative recovery of methane, because Mecsek coal has a low permeability value.

The above-mentioned differences can be seen in Figure 8 in connection with the commulative recovery of methane.

Pa g , b a r

35

30 ~ ' ~

25

20-

15-

10 ,,,I 0 5O

2

3 1

1 - 01. Q W Q E

2 - 0 2 . 0 W Q E

3 - 03 . Q W Q E

I

I O 0

Fig. 5.

150

r , m

Methane pressure profiles in Mary Lee coal, New Mexico, at 80 days.

210

G s , n m 3 / m 3

14

13'

12 0

Fig. 6.

1

3

2

1 - 01. QWQE

2 - 0 2 . Q W Q E

3 - 0 3 . Q W Q E

50 1 0 0 150

r,m

Total sorbed methane profiles at 80 days.

Pg, bar

35

30"

25"

2 0

15

10 L

0

Fig. 7.

1 - 04. PWQE

2 - 10. PWQE

I I 50 100 150 r ,m

Methane pressure profiles in Meesek coal at 80 days.

LASZL6 BALLA

MATHEMATICAL MODELING OF METHANE FLOW

Qg, nm a

1000

500.

1-04. PWQE

2-10 . PWQE

1

0 I 0 40 80

Fig. 8. Produced methane from Mecsek coal.

211

7. Conclusions

In connection with the proposed new mathematical model, the following con- clusions can be summarized:

- it gives a description of the desorptional process of a mezo system with satisfactory accuracy,

- parameters of diffusion-dispersion for a mezo system should be determined with measurements in a laboratory before working out the recovery plans with simulations,

- it takes into account the changes in permeabil i ty around the well which cannot be ignored in the case of coal seams with smaller permeabili ty,

- the micro model can be used for computer simulations of methane flow in a borehole coal mining system.

References

Airey, E. M., 1968, Gas emission from broken coal, an experimental and theoretical investigation, Int. J. Rock Mech. Min. Sci. 5, 475-494.

Bear, J., 1972, Dynamics of Fluids in Porous Media, American Elsevier, New York, pp. 52-57. Jones, A. H., Ahmed, U., Abou-Sayed, A. S., Mahyera, A., and Sakashita, B., 1982, Fractured

vertical wells versus horizontal boreholes for methane drainage in advance of mining U.S. coals,

212 LASZL0 BALLA

Paper presented at the Symposium on Seam Gas Drainage with Particular Reference to the Working Seam, University of Wollongong, Australia, 11-14 May.

King, G. R., Ertekin, T., and Schwerer, F. C., 1983, Numerical simulation of the transient behavior of coal seam degasification wells, Paper SPE 12258 presented at the SPE Symposium on Reservoir Simulation, San Franscisco, 15-18 November 1983.

Mordecai, M. and Morris, L. H., 1974, The effects of stress on the flow of gas through coal measure strata, The Mining Engineer 133, 435-443.

Smith, D. M. and Williams, F. L., 1982, Diffusional effects in the recovery of methane from coalbeds, Paper SPE/DOE 10821 presented at the SPE/DOE Symposium on Unconventional Gas Recovery, Pittsburgh, PA, 16-18 May 1982.

Somerton, W. H., SSylemezoglu, I. M., and Dudley, R. C., 1975, Effect of stress on permeability of coal, Int. J. Rock Mech. Min. Sci. and Geomech. Abstr. 12, 129-145.

T6th, J., 1984, Isotherm equations for monolayer adsorption of gases on heterogenous solid surfaces, in A. L. Myers and G. Belfort (eds.), Fundamentals of Adsorption, United Engineering Trustees Inc., pp. 657-665.

Verruijt, A., 1977, Generation and dissipation of pore-water pressures, in G. Gudehus (ed.), Finite Elements in Geomechanics, Wiley, London, pp. 293-317.

Vorozhtsov, E. V., Gorbachev, A. T., and Fedorov, A. V., 1975, Calculation of the motions of gas in a coal seam with a quasilinear law of filtration, translated from Fiziko-Tekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh 6, 83-91.