Example 6.1. Hartshorne coal, Arkoma Basin—cleat permeability and porosity
Cleat properties of the Hartshorne coal of the Arkoma Basin were discussed by Cardott.9 The No. 31-8 L. W.
Stiles well (Sec. 31 T9N R25E) had an average permeability of 3.1 md, an average helium porosity of 2.3%, cleat spacing of 3 to 5 mm, and a cleat aperture of 0.2 mm at surface conditions. Using a permeability of 3 md and a cleat spacing of 4 mm in equation (6.5),
3 md = 1.0555(10)5ϕ342 mm2 Solving for porosity,
ϕ = 0.012
Thus, in-situ porosity is about one-half the reported helium porosity. This is not surprising, as helium can penetrate micropores inaccessible to methane, and laboratory conditions typically allow a rock sample to relax, increasing porosity. In-situ cleat aperture can now be calculated using equation (6.2).
0.012 = ——————b 500(4 mm) Solving for cleat aperture,
b = 24 µm
This in-situ estimate of cleat width is almost an order of magnitude less than cleat width of the sample at surface conditions, again indicative of sample relaxation.
Table 6–1. Coal cleat compressibilities
Basin Seam Cleat compressibility, psia-1
Appalachiana Pittsburgh 1.87E–03
San Juanb Menefee 1.34E–03
Piceancec Cameo 7.76E–04
Piceanced Cameo 1.80E–03
Warriore na 1.87E–03
Warriorf Marylee/Blue Creek 5.79E–04
San Juang na 9.24E–04
San Juanh na 9.61E–04
San Juani Fruitland 2.00E–03
San Juanj Fruitland 1.83E–03
Warriork Marylee/Blue Creek 2.50E–03
San Juanl Fruitland 1.02E–03
Sydneym Bulli 3.00E–04
Average 1.37E–03
na = not available
Sources: aMcKee, C. R., et al. 1988. bIbid. cIbid. dIbid. eMcKee, C. R., Bumb, A. C., and Koenig, R. A. 1987. Stress-dependent permeability and porosity of coal. Paper 8742 in Proceedings of the 1987 International Coalbed Methane Symposium. Tuscaloosa: University of Alabama. fSeidle, J. P., Jeansonne, M. W., and Erickson, D. J. 1992. Application of Matchstick Geometry to Stress-Dependent Permeability in Coals. Paper SPE 24361. Presented at the SPE Rocky Mountain Regional Meeting, Casper, Wyoming, May 18–21.
gMcKee, C. R., et al. 1987. hSeidle, J. P., et al. 1992. iShi, J. Q., and Durucan, S. 2003. Part 1. Paper 0341; and Shi, J. Q., and Durucan, S. 2003. Part 2. Paper 0342. jIbid.
kZuber, M. D., Sawyer, W. K., Schraufnagel, R. A., and Kuuskraa, V. A. 1987. The Use of Simulation and History Matching to Determine Critical Coalbed Methane Reservoir Properties. Paper SPE 16420. Presented at the Low Permeability Reservoirs Symposium, Denver, Colorado, May 18–19. lGash, B. W., et al. 1993. mSpencer, S. J., Somers, M L., Pinczewki, W. V., and Doig, I. D. 1987. Numerical Simulation of Gas Drainage from Coal Seams. Paper SPE 16857. Presented at the SPE Annual Technical Conference and Exposition, Dallas, Texas, September 27–30.
Average cleat compressibility of the values reported in table 6–1 is 1.46(10)–3 psia–1. This important rock property certainly depends on coal age, rank, purity, and other coal properties, but such effects have not been reported in the literature.
A hydrostatic stress regime was assumed by McKee to apply to in-situ coals.11 With vertical and hydrostatic stresses equal, the variation of permeability with depth can be written
—– = exp [ –3ck f(σv – σvi ) ] (6.8)
ki
However, in-situ coal deposits are confined laterally and are therefore more closely approximated by a uniaxial strain regime. Permeability variation with seam depth in a basin or permeability behavior over depletion of a single seam can be obtained by recasting equation (6.7), assuming permeability depends on mean stress or only horizontal stress across the cleats. The constitutive equations for uniaxial strain can be written as follows, assuming a subscript 1 denotes the vertical direction
ε1 = —– (σ1 1 – v(σ2 + σ3)) E
ε2 = ε3 = 0 where
ε = strain,
E = Young’s modulus, σ = stress, and v = Poisson’s ratio.
Furthermore, in uniaxial strain,
σ2 = σ3
and
σ1 = σv where
σv= vertical, lithostatic stress.
As the lateral strain is defined to be zero,
ε2 = 0 = —– (σ1 2 – v(σ3 + σ1)) E
Solving for the lateral stress,
0 = σ2 – vσ2 – vσ1
σ2 (1 – v) = vσ1
σ2 = —–— v σ1 (6.9)
1– v
Mean stress in a uniaxial strain regime is
σm = — (σ1 1 + σ2 + σ3) 3
1 v v
σm = — 3 1–
(
σ1 + ——– σv 1– 1 + ——– v σ1)
1 2v
σm = — 3 1–
(
σ1 + ——– σv 1)
As the 1 direction coincides with the local vertical direction, 1 1+ v σm = — 3 1–
(
——– v)
σvMean stress in a hydrostatic stress regime is
σm = — (σ1 h + σh + σh) 3
σm = σh Equating the two mean stress expressions,
1 1+ v
σh = — 3 1–
(
——– v)
σv (6.10)This expression allows transformation of laboratory results taken in a hydrostatic stress regime to in-situ, uniaxial strain conditions. Substitution of equation (6.10) into equation (6.7) gives
k 1+ v
—– = exp ki 1–
[
–cf(
——— v)
(σv – σvi)]
(6.11)Alternatively, a permeability-stress relation can be derived assuming only horizontal stress affects cleats. The Imperial College model (ICM) of Shi and Durucan noted that as both face and butt cleats are usually nearly vertical, horizontal stresses across the cleats rather than vertical and horizontal stresses parallel to the cleats should control cleat permeability.12 Assuming a uniaxial strain regime and neglecting horizontal stress parallel to the cleat, horizontal stress across the cleat is related to vertical stress by equation (6.9). Employing equation (6.9) in equation (6.7) gives
k v
—– = exp ki 1–
[
–3cf(
——– v)
(σv – σvi)]
(6.12)Assumption of three different stress regimes (hydrostatic, uniaxial strain with mean stress, and uniaxial strain with horizontal stress only) leads to three equations for permeability as a function of vertical stress: equations (6.8), (6.11), and (6.12). These three equations lead to three relations for permeability decrease with depth of burial in a given basin. Vertical stress is defined here as the difference between overburden load and pore pressure,
σv = S – p where
S = overburden load, and p = pore pressure.
Assuming a lithostatic gradient of 1 psi/ft and a normally pressured basin with a pressure gradient of 0.433 psi/ft,
σv = (1 – 0.433)d = 0.567d (6.13)
where
d = depth, ft.
Assuming permeability is governed by mean stress in uniaxial strain, use of (6.13) in equation (6.11) yields a relation between permeability and depth,
k 1+ v
—– = exp ki 1–
[
–cf(
——– v)
0.567(d – di)]
(6.14)Assuming permeability is controlled by horizontal stresses in uniaxial strain, equations (6.13) and (6.12) give
k v
—– = exp ki 1–
[
–3cf(
——– v)
0.567(d – di)]
(6.15)Lastly, assuming a hydrostatic stress regime, the permeability and depth relation, from equations (6.13) and (6.12), is
—– = exp [ –3ck f 0.567(d – di)] (6.16)
ki
As noted above, average coal cleat compressibility is 1.46(10)–3 psia–1. Palmer and Mansoori reported values of Poisson’s ratio for San Juan Basin coal ranging between 0.20 and 0.39.13 The lower value was obtained from laboratory measurements on coal core, the larger value from scaling laboratory values to field scale. Shi and Durucan obtained a Poisson’s ratio of 0.35 from matching San Juan Basin field data.14 Using a cleat compressibility of 1.46(10)–3 psia–1, a Poisson’s ratio of 0.35, and a base permeability of 180 md, permeability as a function of depth as predicted by equations (6.14), (6.15), and (6.16) is plotted in figure 6–3.
Fig. 6–3. Coal permeability vs. depth—data and theory
Also plotted in figure 6–3 are permeability-depth data reported in the literature and listed in table 6–2.
Table 6–2. Coal permeability-depth data
Depth, ft Perm., md Basin Coal Rank
3,118 22.2 San Juana Fruitland na
1,184.5 3.1 Arkomab Hartshorne lo-vol. bit
1,565 20 San Juanc Fruitland na
3,087 3.5 San Juand Fruitland med. vol. bit
2,704 19 San Juane Fruitland na
1,865 17.2 San Juanf Fruitland na
589 632 Powderg Canyon subbit. C
1,033 25.1 Warriorh ML/BC na
2,790 0.73 San Juani Fruitland na
2,430 42.4 San Juanj Fruitland na
1,202 12 Warriork na na
3,201 19 San Juanl Fruitland na
2,322 25 Warriorm Black Crk na
3,157 12.5 GGRBn Rock Spgs hi-vol. B
Table 6–2. Cont.
Depth, ft Perm., md Basin Coal Rank
3,065 65 San Juano Fruitland na
3,164 24.6 San Juanp Fruitland na
3,181 22.5 San Juanq Fruitland na
3,227 19 San Juanr Fruitland na
2,989 0.9 San Juans Fruitland na
3,034 2.1 San Juant Fruitland na
400 750 Powderu Wyodak subbit.
2,650 1.5 San Juanv Fruitland na
2,677 4.5 San Juanw Fruitland na
3,094 6.9 San Juanx Fruitland na
3,104 6.7 San Juany Fruitland na
2,987 3.4 San Juanz Fruitland na
2,500 13 Uintaaa Ferron hi-vol. B
2,609 1.09 San Juanbb Fruitland hi-vol. B
2,706 4.38 San Juancc Fruitland hi-vol. B
3,003 0.37 San Juandd Fruitland na
482 44 Warrioree Pratt na
580 60 Warriorff Pratt na
50 375.5 Warriorgg na na
79 888.0 Warriorhh na na
100 116.7 Warriorii na na
100 183.4 Piceancejj na na
148 158.5 Warriorkk na na
407 80.1 Warriorll na na
568 34.8 Warriormm na na
1,000 78.6 San Juannn na na
1,119 6.4 Warrioroo na na
1,235 13.5 Warriorpp na na
1,256 3.4 Warriorqq na na
2,304 32.1 San Juanrr na na
2,790 0.43 San Juanss na na
2,811 3.9 San Juantt na na
3,092 18.1 Piceanceuu na na
4,110 0.22 Piceancevv na na
6,357 2.5 Warriorww na na
6,475 0.007 Piceancexx na na
na = not available
Sources: aMavor, M. J., and Robinson, J. R. 1993. bCardott, B. J. 1998. cCox, D. O., Young, G. B. C., and Bell, M. J. 1995. Well Testing in Coalbed Methane (CBM) Wells: An Environmental Remediation Case History. Paper SPE 30578. Presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas, October 22–25. dJones, A. H., Ahmed, U., Bush, D. D., Holland, M. T., Kelkar, S. M., Rakop, K. C., Bowman, K. C., and Bell, G. J. 1984. Methane Production Characteristics for a Deeply Buried Coalbed Reservoir in the San Juan Basin. Paper SPE 12876. Presented at the Unconventional Gas Recovery Symposium, Pittsburgh, Pennsylvania, May 13–15. eZahner, B. 1997. fMavor, M. J., and Vaughn, J. E. 1997. gMavor, M. J., Russell, B., and Pratt, T. J. 2003. Powder River Basin Ft. Union Coal Reservoir Properties and Production Decline Analysis. Paper SPE 84427.
Presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, October 5–8. hKoenig, R. A., and Stubbs, P. B. 1986. iIbid. jIbid. kIbid. lIbid. mReeves, S. R., Lambert, S. W., and Zuber, M. D. 1987. A field derived inflow performance relationship for coalbed gas wells in the Black Warrior Basin. Paper 8744 in Proceedings of the 1987 International Coalbed Methane Symposium. Tuscaloosa: University of Alabama. nYoung, G. B. C., et al. 1991. oMavor, M. J. 1994. pIbid. qIbid. rIbid. sIbid. tIbid. uHower, T. L., et al.
2003. vBell, G. J., et al. 1985. wRamurthy, K., Marjerisson, D. M., and Daves, S. B. 2002. Diagnostic Fracture Injection Test in Coals to Determine Pore Pressure and Permeability.
Paper SPE 75701. Presented at the SPE Gas Technology Symposium, Calgary, Alberta. April 30–May 2. xVo, D. T., Ohaeri, C. U., and Montoya, G. L. 1991. Pressure Buildup Analysis of Early-Time Data from Coalbed Methane Wells in the Rincon Unit, San Juan Basin, New Mexico. Paper SPE 22684. Presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas, October 6–9. yIbid. zIbid. aaBurns, T. D., and Lamarre, R. A. 1997. Drunkards Wash Project: Coalbed methane production from Ferron coals in east- central Utah. Paper 9709 in Proceedings of the 1997 International Coalbed Methane Symposium. Tuscaloosa: University of Alabama. bbRamurthy, K., et al. 2003. ccIbid. ddPuri, R., Volz, R., and Duhrkopf, D. 1995. A micro-pilot approach to coalbed methane reservoir assessment. Paper 9556 in Proceedings of the 1995 International Coalbed Methane Symposium. Tuscaloosa: University of Alabama. eeYoung, G. B. C., et al. 1991. ffIbid. ggMcKee, C. R., et al. 1988. hhIbid. iiIbid. jjIbid. kkIbid. llIbid. mmIbid. nnIbid. ooIbid. ppIbid.
qqIbid. rrIbid. ssIbid. ttIbid. uuIbid. vvIbid. wwIbid. xxIbid.
All data are from single-phase tests and are therefore clear of any relative permeability effects. Reported permeabilities vary by a factor of 3,000, from 0.00653 md to 888 md, with a median value of 18 md. Depths range from 50 to 6,500 ft (15 to 1,980 m), with no data from very deep coals. All data are from North American coals of Pennsylvanian, Cretaceous, and Paleocene age. Permeabilities of coals from other areas and other geological ages may differ significantly from these reported data. The reported data exhibit such a high degree of scatter that attempts to fit a curve through them yields only a soft result, bordering on nonunique. By inspection, the best fit curve is perhaps the ICM model of Shi and Durucan, which assumes permeability is controlled by horizontal stresses.15
Inspection of figure 6–3 reveals permeability variation within a basin is greater than permeability variation between basins. Contributing to the variation are a host of coal properties, such as ash and moisture, degree of cleating, and rank. In addition, local stresses and tectonics will affect permeability of a coal deposit. This geological heterogeneity makes the use of a permeability-depth relation to predict sweet spots during exploration or to define a maximum coal depth during development hazardous.
As gas and water are withdrawn from a coal deposit, reservoir pressure declines and stress increases.
This increase in stress decreases cleat width, reducing permeability. Assuming the vertical lithostatic load is unchanging during depletion of a coal, the change in stress is given by
σv – σvi = S – p – (S – pi) = pi – p
Assuming a hydrostatic stress regime, equation (6.8), permeability decrease during depletion is
— = exp [–3ck f(pi –p)] (6.17)
ki
The mean stress approximation, equation (6.11), yields
k 1+ v
— = exp ki 1–
[
–cf(
——– v)
(pi – p)]
(6.18)Assuming a uniaxial stress regime with permeability controlled by horizontal stresses, the ICM model of Shi and Durucan, equation (6.12) gives16
k v
— = exp ki 1–
[
–3cf(
——– v)
(pi – p)]
(6.19)Employing the values of cleat compressibility and Poisson’s ratio discussed above in equations (6.17), (6.18), and (6.19) yields the permeability-depletion pressure plots shown in figure 6–4.
Inspection of this figure shows the importance of choosing the correct conceptual model, as permeabilities for a given pressure can vary by more than an order of magnitude, especially in deep depletion. The harshest permeability decline is seen under the assumption of hydrostatic stress, while the mildest permeability decline is observed under the assumption that only horizontal stresses control permeability. Permeability based upon the mean stress assumption falls between those of the other two conceptual models. Determination of which conceptual model describes a given coal seam is difficult. Even after incorporating all available geologic and engineering data, identification of the correct conceptual model is often difficult and nonunique.
Fig. 6–4. Predicted coal permeability decline with depletion