Bureau of Mines Report of Investigations/' 1972 

An In-Situ Diffusion Parameter 

for the Pittsburgh and Pocahontas 

No. 3 Coalbeds) 

UNITED STATES DEPARTMENT OF THE INTERIOR 
UNITED STATES DEPARTMENT OF THE INTERIOR 

Rogers C. B. Morton, Secretary 

BUREAU OF MINES 

Elburt F. Osborn, Director 
This publication has been cataloged as follows: 

Kissell, Fred N 

An in-situ diffusion parameter for the Pittsburgh and 

Pocahontas No. 3 coalbeds, by Fred N. Kissell and R. J. 

Bielicki. [Washington] U.S. Dept. of the Interior, Bureau of 

13 p. illus. (U.S. Rureau of Mines. Report of investigations 

Includes bibliography. 

1. Methane. 2. Mine gases. I. Bielicki, Richard J., jt. auth. 

11. Title. (Series) 

TN23.U7 no. 7668 622.06173 

U.S. Dept. of the Int. Library 
CONTENTS 

Page 

Abstract ................................................................. 1 

Introduction ............................................................. 1 

In-situ diffusion parameter .............................................. 2 

Residual gas-- calculation of in-situ diffusion parameter ................. 3 

Calculation for Pittsburgh coalbed .................................. 5 

Calculation for Pocahontas No . 3 coallbed ............................ 7 

Comparison with laboratory and other field data .......................... 7 

Computation of in-situ fracture spacing .................................. 8 

Using diffusion parameter to find slow step .............................. 9 

Conclusions ............................................................. 11 

References ............................................................... 12 

1 . A pressure curve in Pocahontas No . 3 coalbed ......................... 3 

2 . Equilibrium adsorption isotherms for Pittsburgh and Pocahontas 

No . 3 coals ........................................................ 5 

3 . Emission curves for Pittsburgh and Poc~ahontas No . 3 coals ............ 8 
AN IN-SITU DIFFUSION PARAMETER FOR THE PITTSBURGH 

AND POCAHONTAS NIO. 3 COALBEDS 

by 

Fred N. Kissell and R. J. ~ielicki~ 

In this report the importance of diffusion in controlling the emission of 

methane in mines is examined. An in-situ diffusion parameter is defined and 

calculated for a hypothetical unfractured lump of coal in the Pittsburgh and 

Pocahontas No. 3 coalbeds. The values obtained are similar to those obtained 

with lump coal in laboratory experiments. 

INTRODUCTION 

The flow of methane gas in coalbeds has been a topic of Bureau of Mines 

research for many years. Methane, produced as a byproduct of the coalification process, is stored in the coal, frequently in considerable quantities. 

The gas is released where the coalbed is mined, and under some conditions, may 

present considerable danger to those working in the mine. 

Unfractured "solid" coal has a very low permeability; however, bituminous 

coalbeds in place are highly fractured. The quantity of methane entering a 

typically gassy coal mine is far greater than could be imagined from any 

unfractured coalbed, and it is clear that virtually all of the gas entering 

the mine has come through fractures both in the coal and in the adjacent 

strata. 

Not very much is known about the nature of these fractures, but for coal 

two facts have been definitely established: 

1. The density of fractures varies with the coal rank; friable, lowvolatile coals tend to be much more highly fractured. 

2. The fractures in a given coal are not of uniform size or spacing. It 

has been shown by mercury-intrusion studies that there are fractures of all 

sizes from less than 100 A to several microns in width (u) .3 

IPhysical research scientist. 

"physical science technician. 

3 Underlined numbers in parentheses refer to items in the list of references at 

the end of this report. 
cess based on diffusion and laminar flow appears to be a good starting point 

for further analysis. 

IN-SITU DIFFUSION PARAMETER 

Kissell (13) has obtained some values for coalbed permeability by considering coalbed gas pressures and the rate at which methane flows into the mine. 

These values represent the permeability of the fractures through which the 

D flow is laminar. Although the diffusion parameter - of coal powders and 

a2 

lumps has been measured in the laboratory, no estimate of an in-situ diffusion 

parameter of solid coal underground has ever been made. This means that it 

has been impossible to estimate accurately the rate of the first flow step 

compared with the second. 

This report presents a method of obtaining an in-situ diffusion parameter. 

Although the results are not highly accurate, nevertheless they provide a 
reasonable correlation with laboratory diffusion parameters and also some 

insight into how gas migrates in coalbeds. 

RESIDUAL GAS --CALCULATION OF IN-SITU DIFF'U SION PARAMETER 

The Bureau of Mines has been engaged in a program of measuring methane 

pressures in coalbeds. These pressures are measured by drilling a horizontal 

hole into the working face or a rib and sealing the hole with inflatable 

packers (g-2). A pressure curve for a hole 160 ft deep is shown in figure 1. 

Not only is the pressure at the back of the hole measured, but pressures at 

intermediate distances also are obtained. These pressures represent the pressure of gas in the fractures. If the first flow step (the diffusion step) is fast 

compared with the second step (the fracture flow step), these pressures also 

represent the amount of gas in the coal at any given depth. However, if the 

two steps have a comparable rate or if the first step is slower, the amount 

DISTANCE INTO COAL-BED FROM FACE, ft 

FIGURE 1. - A Pressure Curve in Pocahontas No. 3 Coa lbed. 
lected for the residual gas measurement. Two aspects of this sequence are 

important: 

1. A "pressure history" of the hypothetical lump may be deduced from the 

rate of advance of mining, a related pressure curve such as that in figure 1, 

and an estimate of the original gas content. 

4To obtain the residual gas, a sample of coal is collected at a working face 

that has been continuously active and which penetrates into a virgin area 

of the coalbed. The coal is immediately enclosed in a container, and the 

quantity of gas given off over the next several weeks is measured. Then, 

the coal is crushed to at least 200 mesh, and the gas given off in the 

crushing is measured. The gas given off by the coal while in the container 

plus that from the crushed sample is the residual gas. 

Both steps are performed at 1 atm pressure. After crushing, the coal still 

will retain some methane unless the atmospheric pressure is reduced to 

zero. This retained gas is not included in the "residual gas." 
5 cm3 /g is adsorbed (fig. 2). If the density of coal is assumed to be 1.3, 3 this corresponds to 2.9 X moles/cm of coal. At the face, the absolute 

pressure is 1 atm, and 0.44 X moles/cm3 of coal is adsorbed. It follows 

that in the 2-week period the surface concentration of the lump is reduced by 

2.46 X lom4 moles/cm3 of coal. 

A typical value for residual gas in Pitts'burgh coal is about 4 cm3 /g. 

This means that only about 20 pct of the original gas is lost from the coal, 

or 0.20 x 2.9 x moles/cm3 = 0.58 X lo-' moles/cd. 

Crank (5) - gives solutions to the diffusion equation for various shapes 

under different boundary conditions. For a sphere whose surface concentration 

varies linearly with time [Q(t) = kt] the solu.tion is 

c.2 

~n"+t - a" 

Mt -- -(t -GI+- 'a' 1 e 

4 rr4 D 7rm3 k n= 1 

D 1 

where -= the diffusion parameter (z) , a" 

a = sphere radius (cm), 

D = diffusion coefficient (g ) , 

t = time (sec) , 

Mt = amount sorbed or desorbed (moles) at time t, 

M, amount sorbed or desorbed -- - 

4rra3 unit volume 

3 

and Q (t) k = const = - 

t 

If it is assumed that the hypothetical lump is a sphere of radius a, that 

this sphere has a surface concentration that varies linearly at the rate of 

2.46 X lom4 LEk2 

c or k = 2.03 X 10-lo , and that the amount desorbed is 

2 weeks cd sec 

D 0.58 X lom4 moles/cm3, then - = 1.3 X lom8. Note that this quantity is DIP, a" 

not D. This is not surprising, for the size of the hypothetical lump is just 

as important as the diffusion coefficient, and where in-situ coal is concerned, 

a D alone would be meaningless, simply because the lump size is unknown. It 

should be pointed out that both the residual gas and the pressure curve will 

vary somewhat from location to location in the mine, depending on the rate of 

advance and other factors (12-13). However, the diffusion parameter is not 

very sensitive to minor changes in any of these. 
munication with T. C. ~u~~el

 

equilibrium adsorption isotherms have been obtained 

for dry and moist Pittsburgh 

and Pocahontas No. 3 coals. 

In-situ coal always contains 

- moisture, and moist coal 

adsorbs less gas than dry 

coal. This moisture effect 

was first observed by - E t t inger (i) . 

If the coalbed is 

assumed to contain 1.5 pct 

0 10 20 30 40 50 60 moisture at 120 psia, 

PRESSURE, obs otm 

5Research chemist, PittsFIGURE 2. - Equilibrium Adsorption Isotherms for Pitts- burgh Energy Research 

burgh and Pocahontas No. 3 Coals. Center. 
Calculation for Pocahontas No. 3 Coalbed 

A pressure curve for a deep mine (12) in thePocahontasNo. 3 coalbed is 

shown in figure 1. It is much steeper than the curve for the Pittsburgh coalbed, and the original coalbed pressure is reached in about 100 ft. Thus, the 

pressure around the hypothetical lump drops from 680 psia to 15 psia in just 

1 week. If the coals were similar, the residual gas content would be quite 

high, but the opposite is true. The residual gas is only 1 to 1.5 cm31g, 

whereas Ruppel's adsorption curve gives 13.1 cmaIg for the original coal, 

assuming 1.5 pct moisture. This means that nearly 90 pct of the original gas 

is gone, whereas for the Pittsburgh coal it was only 20 pct. 

Applying the same analysis used previously, k = 11.3 X 10-lo, the amount 

desorbed = 6.14 X loe4, and = 1.05 X This is 100 times larger than 

a2 

D the - for Pittsburgh coal. Whether it is due to a larger D or a smaller a is a2 

immaterial, but later findings indicate that it is probably a combination of 

both. 

COMPARISON WITH LABORATORY AND OTHER FIELD DATA 

Another way to obtain D/a2 for a lump of coal is to pressurize the lump 

with methane in the laboratory. After the coal is equilibrated, the pressure 

is reduced abruptly, and the rate of emission of gas from the coal is measured. 

From the emission rate, D/a2 is calculated. The authors have obtained a number of emission curves for different size lumps of both Pittsburgh and 

Pocahontas coals; these will be published elsewhere. However, a few typical 

results are given in figure 3. 

A simple procedure used by Hofer, Bayer, and Anderson (10) can be used to 

calculate D/a2 from these curves. The equation for diffusion from a sphere is 

applied again except that the boundary conditions are different [constant concentration at t = 0 and surface concentration = 0 for t>0]. A solution valid 

for short times (2) is 

and DIP is found readily by plotting Mt 1% against t1I2 . Using this proceD dure, we obtained - = 1.4 X for 114- to 112-in size range Pittsburgh a2 

D coal and - = 0.5 X for 114- to 112-in size range Pocahontas coal. 

a2 

It is known that D/a2 will vary with lump size only if the lump size is 

smaller than the fissure network of the coal (2) . For Pocahontas coal, lumps 

114 to 112 in in size already are larger tlhan the fissure network as measured 

in the laboratory, and D/a2 does not change if the lump size is increased 

beyond this. Such is not the case for Pittsburgh coal; D/a2 will decrease 

slightly with an increase in lump size, but D/a2 still will be approximately 

lo-. 
coalbed near Amity, Pa., and a 4-in-diam core was brought to the surface. The 

core was enclosed immediately in a vessel, and the emission rate was measured 

(fig. 3). Some of the data are missing owing to the amount of time (150 min) 

required to cut the core and bring it to the surface, but a plot of $/Moo 

against t1i2 still could be made, and D/a2 was calculated to be 4.95 X 

These results may be summarized as follows: For the Pittsburgh coalbed, 

the in-situ D/a2 = 1.3 X Lumps in the laboratory gave D/a2 = 1.4 X lo-', 

and a core from a vertical well gave D/a2 = 4.95 X lo-'. For Pocahontas No. 3 

coal, the in-situ D/a2 = 1.05 X and lumps in the laboratory gave 

D/a2 = 0.5 X lo-=. 

COMPUTATION OF IN-SITU FRACTURE SPACING 

It is possible to compute an in-situ fracture spacing from the D/a2 values obtained. E. D. ~himons' (personal communication) has obtained values for 

GPhysicist, Pittsburgh Mining and Safety Research Center. 
the transient diffusion coefficient D for Pittsburgh and Pocahontas No. 3 coal 

in the laboratory by flowing dry methane through small disks of coal. Thimons 

obtained 1.3 X for Pittsburgh coal and 70 X lo-" for Pocahontas No. 3 

coal. He also found that if the methane was saturated with water vapor (this 

is more realistic than using dry methane) the coefficient for Pittsburgh coal 

was reduced by a factor of 8, and for Pocahontas coal the coefficient was 

reduced by a factor of 14. This gives more realistic values of D = 1.6 X lo-' 

for Pittsburgh coal and D = 5 X for Pocahontas No. 3 coal. Now if 

(% tt,sburgh 

2 

and ( 5 I,. . .ho ,, , a. 

lo-", then 2api ttabur gh = 8 cm 

and Zap, ,., = 4.5 cm, where 2a is the diameter of the hypothetical sphere. 

These values seem large, especially since the laboratory lump emissions 

results discussed previously indicate that the fissure spacing of Pittsburgh 

coal is closer to 2 cm and that of Pocaho~ntas coal is approximately 1 mm. 

However, it must be remembered that coal in the laboratory has been treated 

rather harshly. The confining forces that squeeze the coal underground have 

been removed, and it has been torn from the coalbed by the mining machine. It 

has been sieved, evacuated, and dried in ithe desorption experiment. Just dry-, 

ing a lump of coal will crack it; therefore, it is not surprising that a fissure spacing in the laboratory will be much less than that found underground. 

USING DIFFUSION PARAMETER TO FIND SLOW STEP 

Previously, we pointed out that the flow of gas from a coalbed could be 

viewed as a two-step process, and that the analysis could be considerably 

simplified if one step could be considered much slower than the other. In 

particular, it has been assumed (12-13) that in the Pocahontas No. 3 coalbed 

the second step (laminar flow) is much slower than the first step (the diffusion from lumps). Since a value for ~/a~ is available now, this hypothesis 

can be tested. 

The easiest way to do this is to compare the time required for the hypothetical lump to lose half the original gas by diffusion with a similar time 

required for a hypothetical lump some distance back in a homogeneous coalbed. 

Using the equation valid for short times, 

If M,/& = 0.5 and ~/a~ = then t = 6 hr. This should be the time 

required for the hypothetical lump of Pocahontas No. 3 coal to lose half of 

its original gas by diffusion from the lump. 

The foregoing may be compared with a hypothetical lump in a homogeneous 

coalbed where there is no diffusion step and the flow is entirely laminar and 

follows the Darcy equation, 

8p3 - 1s a? ---- 

ax" ~~at 
It can be seen that diffusion of half of the gas out of the hypothetical 

lump requires 6 hr, whereas a lump in a homogeneous coalbed with only laminar 

flow through cracks will lose half of its gas in 523 hr if that lump is 50 ft 

from the working face. It is clear that the first step (the diffusion step) 

is much faster than the second. The low residual gas content of Pocahontas 

No. 3 coal verifies this. 

In Pittsburgh coal, the residual gas content is high, and so it seems the 

first step may not be the fast one. Repeating the same procedure that was 

used for Pocahontas No. 3 coal, if D/a2 = half the gas has diffused from 

the hypothetical lump in 600 hr. For the - lump in the homogeneous coalbed, K/S 

is assumed to be 50 X darcy (l3), P is 5 atm, and then t, = 150 hr for a 

lump 50 ft from the working face. Here it may be seen the first step is 

slightly slower than the second step. 

These calculations are very approximate. For example, it may be seen 

that for the lump in the homogeneous coalbed, trn depends on the distance from 
the lump to the face. As the lump-to-face distance decreases, then t, 

decreases also. This makes no difference for most of the Pocahontas No. 3 

coalbed, but in the Pittsburgh coalbed where the two steps are comparable, 

consideration of the depth is important. For example, at 100 ft, t, = 600 hr, 

and both steps will proceed at the same rate. At distances greater than 

100 ft, the first step becomes faster than the second step, which is the opposite of what prevailed at less than 100 ft. 

CONCLUSIONS 

An in-situ diffusion parameter has been calculated for the Pittsburgh and 

Pocahontas No. 3 coalbeds using typical values for residual gas and typical 

pressure-depth curves. A two-step process consisting of diffusion and flow 

through fractures out of the coalbed was assumed. It was found that the diffusion step was much faster in the Pocaho~itas No. 3 coalbed, whereas the two 

steps were comparable in the Pittsburgh coalbed. 

The quantity obtained was ~/a~, not I), for if diffusion is taking place 

from some hypothetical solid lump within the coalbed, the size of that lump is 

as important as the diffusion coefficient. 

In-situ fracture spacings were comput:ed to be 8 cm and 4.5 cm for the 

Pittsburgh and Pocahontas No. 3 coalbeds, respectively. These seem large compared with laboratory results, but the breakage of coal that takes place during mining may in part account for this difference. Given the approximate 

nature of the calculations, these results may be regarded as satisfactory. 
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Conf. of Safety in Mines Res., Donezk, U.S.S.R., 1971, 29 pp. 

2. Barrer, R. M., and D. W. Brook. Molecular Diffusion in Chabazite, 

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1059. 

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IN T.-BU.OF MIN ES,PGH.,PA. 17779