Increasing the temperature of a coal decreases the amount of gas it can sorb at a given pressure. Furthermore, the curvature of a sorption isotherm changes with temperature, becoming more concave downward as the coal warms. In practice, coal gas reservoir engineering sometimes requires coalbed gas contents or sorption isotherms at temperatures different than those at which available data were acquired. One such example is the inadvertent measurement of sorption isotherms at off-reservoir temperatures. Another possibility is that the only available gas contents for an exploration prospect are from samples at a cool, shallow depth and must be extrapolated to a hotter, deeper portion of the basin.
The effect of temperature on sorption capacity of a Dietz 3 coal sample from the Powder River Basin was studied by Bustin and Downey.27 In-situ, not daf, gas contents for a sample at equilibrium moisture were measured at six temperatures, and the results are reproduced here as figure 5–10. Theoretically, at infinite pressure, a coal surface will be saturated with gas, regardless of coal temperature. As seen in figure 5–10, however, at pressures normally encountered in coal gas exploration, reservoir temperature strongly affects gas sorption potential of the coal deposit.
An approximate relation for the effect of temperature on gas content can be developed from experimental work. Boxho et al. reported that for bituminous coal starting at a temperature of 23°C, sorption capacity drops 0.8%/°C.28 Methane sorption data from Hofer et al., measured on dry samples of high-volatile A bituminous Pittsburgh and low-volatile bituminous Pocahontas coals, indicated gas content decreased an average of 2.2%/°C.29 Gas contents reported by Bustin and Downey fall at 1%–3%/°C. Thus, as a first approximation, gas content drops about 1%/°C (0.6%/°F). Thus, for a temperature increase of 10°F, gas content drops 6%.
Fig. 5–10. Effect of temperature on sorption isotherms—Dietz 3 coal, Powder River Basin30
Semiempirical treatments of thermal influences on sorption typically address the Langmuir volume or pressure constants. Ettinger et al. found an exponential relation between coal temperature and the Langmuir volume constant.31 They developed an equation whereby the Langmuir volume constant of dry coal at a given temperature could be estimated from that at 30°C using empirical temperature coefficients in the exponential arguments. Yee et al. discussed the effect of temperature on the Langmuir pressure constant but neglected the Langmuir volume constant.32 Heats of sorption required by the Yee formulation were measured by Hofer et al.
for two dry Appalachian coals.33
Theoretical treatment of the effect of temperature on coalbed gas content requires a different conceptual model of sorption. In contrast to the Langmuir theory of sorption, in which the sorbed gas is assumed to be a monolayer coating the surface of the coal, Dubinin theory assumes the sorbed gas fills the micropores of the coal.34 Based upon this theory, two equations have been developed that account for the effect of temperature on sorbed gas volumes. Both the Dubinin-Radushkevich equation,
RT p 2
W = W0 exp
[
–(
—— 1n ——βE p) ]
(5.4)and the Dubinin-Astakhov equation,
RT ps n W = W0 exp
[
–(
—— 1n ——βE p) ]
where
W = sorbed gas volume, cm3/g, W0 = volume of micropores, cm3/g,
R = universal gas constant, psi-ft3/lb-mole-°R, T = absolute temperature, °R,
β = sorbate affinity coefficient, dimensionless,
ps = saturation vapor pressure, atm, p = pressure, atm, and
n = a small integer, typically between 1 and 4,
exhibit a declining exponential relationship between sorbed gas volume and temperature. As the temperature of many coal seams is above the critical temperatures of gases sorbed on them, the concept of vapor pressure is ill-defined. There are two relations for the saturation vapor pressure at temperatures exceeding the critical temperature. The first is the Dubinin equation35
T 2
Ps = pc
(
—— Tc)
(5.5)where
pc = critical pressure, atm, and Tc = critical temperature, °R.
The second equation is the reduced Kirchoff equation36
Tnbp T – Tc
ps = pc exp
[
ln pc(
———— Tc – Tnbp)
——— T]
(5.6)where
Tnbp = normal boiling point temperature, °R.
Note that equation (5.6) requires critical pressure be in units of atmospheres. At the time of this writing, validity of the above expressions for saturation vapor pressure for coal gas sorption physics has not been addressed.
The above relations enable correction of isotherms and coalbed gas contents to another appropriate temperature. Experimental sorption data are used to evaluate the two constants of the Dubinin-Radushkevich equation (W0 and the product βE) or the three constants of the Dubinin-Astakhov equation (W0, the product βE, and the exponent n), followed by calculation of gas contents at the desired temperature and pressures. The following example applies the Dubinin-Radushkevich equation to the Powder River Basin coals studied by Bustin and Downey.37
Example 5.1. Temperature effects on sorption isotherms—Dietz 3 coal, Powder River Basin
As noted above, Bustin and Downey reported sorption capacity of the Dietz 3 coal of the Powder River Basin at selected temperatures.38 Using data taken at 20°C, sorption capacity at 10°C and 50°C can be estimated and compared with the reported isotherms. Saturation vapor pressures are calculated with both the Dubinin and reduced Kirchoff equations using methane properties39
Tnbp = 201°R Tc = 344°R pc = 673 psia = 45.78 atm
At 20°C (528°R), saturation vapor pressure from the Dubinin relation, equation (5.5), is
528°R 2
ps = 45.78 atm
(
———–)
= 107.86 atm 344°RSaturation vapor pressure from the reduced Kirchoff equation is
201°R 528 – 344°R ps = 45.78 atm exp
[
ln 45.78 atm(
————–—— 344 – 201°R)
—–————— 528°R]
ps = 45.78 atm
Gas contents calculated from Bustin and Downey’s isotherm at 20°C were fit to the Dubinin-Radushkevich equation using both saturation pressures to determine the two constants in equation (5.4). Results are summarized in table 5–2, and gas contents plotted in figure 5–11. In spite of the Dubinin and reduced Kirchoff saturation vapor pressures differing by a factor of almost three, both appear to work well for this coal at this temperature.
Table 5–2. Calculated and measured gas contents—20°C
20 deg C = 68 deg F = 528 deg R Dubinin Reduced Kirchoff
ps = saturation vapor pressure, atm ps = 107.86 297.96
Wo = volume of micropores, cc/gm Wo = 6.0 8.5
βE = energy constant, psi-ft3/lb-mole βE = 11,705.5 15,018.3
Pressure, psia Gas content, scf/ton P, pressure, atm Gas content, cc/gm Calculated gas content, cc/gm Calculated gas content, cc/gm
0.0 0.0 0.0 0.00 0.0 0.0
88.3 33.2 6.0 1.04 0.9 1.0
146.0 53.2 9.9 1.66 1.6 1.6
199.5 70.9 13.6 2.21 2.2 2.2
261.0 88.6 17.8 2.77 2.8 2.8
347.4 106.3 23.6 3.32 3.5 3.4
400.0 121.8 27.2 3.80 3.9 3.8
465.7 132.0 31.7 4.12 4.2 4.2
549.3 146.2 37.4 4.56 4.6 4.6
600.5 159.5 40.8 4.98 4.8 4.9
666.7 166.1 45.4 5.19 5.0 5.2
Source: Bustin & Downey data
Fig. 5–13. Comparison of reported and calculated isotherms—50°C42
Gas contents at 10°C were estimated from equation (5.4) using micropore gas volumes and energy constants obtained from fitting the 20°C data. Reported and calculated gas contents are shown in figure 5–12. Gas contents based on either saturation vapor pressure, Dubinin or reduced Kirchoff, are in good agreement with reported values at low pressures. Above 400 psia, calculated gas contents fall below measured values, with the reduced Kirchoff saturation vapor pressure yielding more accurate gas contents than that of Dubinin.
Repeating the exercise for 50°C, calculated and reported gas contents are in figure 5–13. Regardless of the saturation pressure employed, plots of gas contents from the Dubinin-Radushkevich equation show unrealistic flexure at low pressures. Again, the reduced Kirchoff saturation vapor pressure relation gave more accurate results than did that from the Dubinin equation.