Levine described pressure dependence of cleat width as the sum of the initial width, a term to account for stress, and a term to capture matrix deformation.42 A similar sum was presented to describe permeability variation with pressure. During depletion, stress increases, thereby decreasing cleat width and permeability, while matrix shrinkage increases cleat width and permeability. The two effects compete with one another, and dominance depends on coal properties and reservoir pressure change. A more complete mathematical description of stress and matrix deformation physics begins with an equation for the ratio of porosity at a given pressure to initial porosity. The ratio of permeability at that pressure to initial permeability is, from equation (6.6), simply the cube of the porosity ratio.
Similar to the Levine sum for cleat width, Palmer and Mansoori described the ratio of porosity at a given pressure to initial porosity as43
ϕ cm c0 K Bp Bpi
— = 1 + —– (ϕi ϕi p – pi ) + —– ϕi
(
— – 1M 1+)(
——— – ——— Bp 1+ Bpi)
(6.24)where
cm = 1/M, psia–1,
c0 = volumetric strain coefficient, psia–1, B = reciprocal of Langmuir pressure, psia–1, M = constrained axial modulus, psia, and
K = bulk modulus, psia.
The constrained axial modulus, M, is given by
(1– v)E
M = —–—————
(1+ v)(1– 2v)
Using the definition of cm and the relation
K 1 1+ v -— = — M 3 1–
(
——– v)
the Palmer-Mansoori porosity relation can be written as
ϕ (1+ v)(1– 2v) c0 2(1– 2v) pi p
— = 1 + ——————– (ϕi (1– v)Eϕi p – pi) + — ————– ϕi 3(1– v)
(
———– – ———– pi + pL p + pL)
(6.25)The middle term of the sum on the right-hand side of this equation describes porosity changes due to stress effects, while the last term of the sum expresses the influence of matrix shrinkage on porosity. Inspection of the equation reveals the stress term is controlled by Poisson’s ratio and Young’s modulus. The sorption term is controlled by Poisson’s ratio, the volumetric strain coefficient, and the Langmuir pressure constant. The volumetric strain coefficient, c0, is empirically determined, either from laboratory or field data, and resembles Levine’s maximum matrix strain coefficient. Palmer and Mansoori obtained Poisson’s ratio from laboratory or field data and Young’s modulus from laboratory tests. The stress term in equation (6.25) is negative during depletion, while the sorption term is always positive. Which term dominates depends on rock mechanics properties as well as position on the sorption isotherm. From the porosity ratio of equation (6.25), Palmer and Mansoori calculated the permeability ratio with equation (6.6).
The pressure at which porosity and permeability begin to increase is defined as the rebound pressure. Rebound pressure is determined by setting the derivative with respect to pressure of equation (6.25) to zero, and is
2c0EPL 1/2
pc =
[
————]
– pL (6.26)3(1+ v)
where
pc = rebound pressure, psia.
Coefficients determined by Palmer and Mansoori from matching San Juan coal well performance are given in table 6–8.
Table 6–8. Palmer–Mansoori coefficients—San Juan Basin, Fruitland coal initial porosity = 0.001–0.005 fraction
initial pressure = 1,100 psia
c0 = 0.013 psia-1
Young’s modulus = (1.24–4.45)E+5 psia
Poisson’s ratio = 0.39
pL = 625 psia
Source: Palmer, I., and Mansoori, J. 1996.
Porosity of this high-volatile bituminous coal varied from 0.001 to 0.005, much smaller than the porosity of 0.02 to 0.03 obtained from an interference test in these coals.44 However, it was commensurate with those determined from history matching exercises with reservoir simulators (Mavor, ϕ = 0.0045; Ramurthy, ϕ = 0.0025).45 Young’s modulus varied from 124,000 to 445,000 psia (855 to 3068 MPa), values smaller than those reported by Bell and Jones for gassy coals.46 The extreme sensitivity of the Palmer-Mansoori permeability ratio to rock properties is demonstrated by plotting curves resulting from the four combinations of San Juan parameters in figure 6–8.
The permeability ratio calculated with the smallest values of porosity and Young’s modulus, 0.001 and 124,000 psia (855 MPa), gives physically impossible negative values, but the other three curves are credible.
Fig. 6–8. Palmer-Mansoori permeability ratio—San Juan coal
Rebound pressures, calculated from equation (6.26) using the two values of Young’s modulus, were 682 and 65 psia. For the more elastic coal (E = 445,000 psia), stress effects dominate the first half of depletion, from 1,100 psia initial pressure down to 682 psia, then matrix shrinkage effects dominate the second half. For the stiffer coal (E = 124,000 psia), stress effects dominate into deep depletion, with matrix shrinkage effects dominating stress effects only at reservoir pressures less than 65 psia.
The ICM model of Shi and Durucan for coal porosity and permeability describes stress variation during depletion as the sum of mechanical and matrix deformation terms.47 The mechanical stress term assumes a uniaxial strain regime dominated by horizontal stress changes, while the matrix deformation term is similar to those of Levine and Palmer and Mansoori.48 The permeability ratio is calculated from an equation similar to equation (6.7). Mathematically,
v Eεl p pi
σ – σi = – —–— (p – pi) + ———–
(
——— – ———)
(6.27)1– v 3(1+ v) p + pε pi + pε
where
εl = matrix shrinkage coefficient, psia–1, and pε = matrix deformation Langmuir pressure, psia.
And the permeability ratio is given by
— = exp [ –3ck f (σ – σi)] (6.28)
ki
From equation (6.6), the porosity ratio is simply the cube root of equation (6.28). The Shi and Durucan rebound pressure, the pressure at which matrix deformation effects begin to dominate stress effects, is given by
Eεlpε 1/2
prb =
[
———]
– pε (6.29)3v where
prb = rebound pressure, psia.
Recovery pressure, defined by Shi and Durucan as the pressure at which permeability is equal to initial permeability, is calculated from
Eεlpε
prc = ————— – pε (6.30)
3v(pi + pε) where
prc = recovery pressure, psia.
Shi and Durucan identified five different scenarios of the stress behavior during pressure depletion that depend on the relative magnitudes of rock properties and initial pressure. They noted their model predicts larger recovery and rebound pressures than does the Palmer-Mansoori model. Shi and Durucan employed San Juan data from Palmer and Mansoori and Mavor and Vaughn to determine the coefficients in equations (6.27) and (6.28).49 Coefficients for the Valencia Canyon 32-1 well are collected in table 6–9, and the resulting permeability and porosity ratios plotted in figure 6–9.
Table 6–9. ICM model stress-permeability coefficients—San Juan Basin
Young’s modulus = 421,000 psia
Poisson’s ratio = 0.35
Initial pressure = 957 psia
Max vol strain = 0.01266
Half strain pressure = 625 psia
Cleat compressibility = 0.002 psia-1
Sources: Shi, J. Q., and Durucan, S. 2003. Part I. Paper 0341; and Shi, J. Q., and Durucan, S. 2003. Part II.
Paper 0342.
As seen in this figure, the ICM model predicts steadily increasing permeability and porosity ratios throughout depletion. Both rebound and recovery pressures are above initial pressure.
Average San Juan coal parameters developed by Palmer and Mansoori and Shi and Durucan are collected in table 6–10, and the resulting permeability and porosity ratios plotted in figures 6–10 and 6–11, respectively.50
Table 6–10. Physical constants for Palmer-Mansoori and ICM comparison
Initial porosity = 0.003 fraction
Initial pressure = 1,000 psia
c0 = 0.013 psia-1
Young’s modulus = 400,000 psia
Poisson’s ratio = 0.37
pL = 625 psia
Max vol strain = 0.01266
Cleat compressibility = 0.00146 psia-1
Sources: Shi, J. Q., and Durucan, S. 2003. Part 1. Paper 0341; and Shi, J. Q., and Durucan, S. 2003. Part 2.
Paper 0342; and Palmer, I., and Mansoori, J. 1996.
Fig. 6–10. Comparison of Palmer-Mansoori and ICM permeability ratios
Fig. 6–11. Comparison of Palmer-Mansoori and ICM porosity ratios
From the initial pressure of 1,000 psia, the Palmer-Mansoori theory predicts a shallow decline in both permeability and porosity until the rebound pressure of 623 psia is reached. At rebound pressure, permeability has decreased by 12% and porosity has declined by 4%. At complete depletion, zero psia, permeability has doubled, while porosity has increased by 25% over the initial value. If methane were to be injected into the coal deposit, doubling the initial pressure, this theory yields a steady increase in both permeability and porosity. At 2,000 psia, porosity has increased by 30%, and permeability has doubled.
In contrast, the ICM theory predicts a steady increase in both porosity and permeability throughout depletion of the coal. At zero psia, porosity has increased by a factor of 4.7, and permeability by a factor of 105. Injecting methane would cause both porosity and permeability to decrease slightly before increasing. At rebound pressure, 1,064 psia, both have decreased by a fraction of a percent. Recovery pressure, the pressure at which porosity and permeability return to their initial values, is 1,130 psia. When reservoir pressure reaches 2,000 psia, porosity has increased by one-third and permeability by a factor of 2.3.
Unfortunately, very little field data currently exist to test theories of porosity and permeability changes due to matrix deformation from stress and sorption. Field evidence of increasing coal permeability with depletion has been reported only in the San Juan Basin. Mavor and Vaughn reported permeability increases in some Valencia Canyon wells by a factor of three to seven.51 From a set of four pressure buildup tests on a single dry coal well, Zahner reported exponentially increasing permeability with depletion.52 A reduction in reservoir pressure of 1,200 psia increased permeability by a factor of 2.
Nomenclature
a Cleat spacing A Flow area b Cleat width
B Reciprocal of Langmuir pressure, psia–1 cf Cleat compressibility, psia–1
cm Palmer-Mansoori coefficient = 1/M, psia–1
c0 Palmer-Mansoori volumetric strain coefficient, psia–1 d Depth, ft
E Young’s modulus, psia k Cleat permeability K Bulk modulus, psia ki Initial permeability krg Gas relative permeability
krg Gas relative permeability at irreducible water saturation krw Water relative permeability
l Bedding plane height
M Constrained axial modulus, psia n Number of cleats
n΄ Empirical coefficient, equation (6.20) p Pressure, psia
pc Palmer-Mansoori rebound pressure, psia, equation (6.28) pi Initial pressure, psia
prb Shi and Durucan rebound pressure, psia, equation (6.29) prc Shi and Durucan recovery pressure, psia, equation (6.30) p50 Levine pressure, where strain is one-half maximum strain, psia Δp/L Pressure gradient
q Flow rate
qt Flow rate through n cleats S Overburden load, psia Sgc Critical gas saturation Siw Irreducible water saturation Sw Water saturation
Sw* Normalized water saturation, equation (6.22) Greek
ε Strain or matrix strain
εmax Levine maximum matrix strain coefficient λ Empirical coefficient, equation (6.20) μ Fluid viscosity
ν Poisson’s ratio σ Stress
σh Hydrostatic stress, psia σhi Initial hydrostatic stress, psia ϕ Cleat porosity
References
1. Gash, B. W., Volz, R. F., Potter, G., and Corgan, J. M. 1993. The effects of cleat orientation and confining pressure on cleat porosity, permeability and relative permeability in coal. Paper 9321 in Proceedings of the 1993 International Coalbed Methane Symposium. Tuscaloosa: University of Alabama.
2. Mavor, M. J., and Robinson, J. R. 1993. Analysis of Coal Gas Reservoir Interference and Cavity Well Tests. Paper SPE 25860.
Presented at the Joint Rocky Mountain Regional and Low Permeability Reservoirs Symposium, Denver, Colorado, April 26–28;
and Gas Research Institute. 1996. A Guide to Coalbed Methane Reservoir Engineering. GRI-94/0397. Chicago: Gas Research Institute.
3. Koenig, R. A., and Stubbs, P. B. 1986. Interference Testing of a Coalbed Methane Reservoir. Paper SPE 15225. Presented at the SPE Unconventional Gas Technology Symposium, Louisville, Kentucky, May 18–21.
4. Young, G. B. C., McElhiney, J. E., Dhir, R., Mavor, M. J., and Anbouba, I. K. A. 1991. Coalbed Methane Production Potential of the Rock Springs Formation, Great Divide Basin, Sweetwater County, Wyoming. Paper SPE 21487. Presented at the SPE Gas Technology Symposium, Houston, Texas, January 23–25.
5. Gash, B. W., et al. 1993.
6. Gas Research Institute. 1996; and Bell, G. J., Seccombe, J. C., Rakop, K. C., and Jones, A. H. 1985. Laboratory Characterization of Deeply Buried Coal Seams in the Western U.S. Paper SPE 14445. Presented at the SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, September 22–25.
7. Reiss, L. H. 1980. The Reservoir Engineering Aspects of Fractured Reservoirs. Paris: Gulf Publishing Company.
8. Ibid.
9. Cardott, B. J. 1998. Coal as gas-source rock and reservoir, Hartshorne Formation, Oklahoma. In The Hartshorne Play in Southeastern Oklahoma: Regional and Detailed Sandstone Reservoir Analysis and Coalbed-Methane Resources. Andrews, R. D., Cardott, B. J., and Storm, T. Oklahoma Geological Survey Special Publication 98-7. Norman: University of Oklahoma.
10. Reiss, L. H. 1980; and McKee, C. R., Bumb, A. C., and Koenig, R. A. 1988. Stress-dependent permeability and porosity of coal and other geologic formations. Society of Petroleum Engineers Formation Evaluation. V. 3 (no. 1). p. 81.
11. McKee, C. R., et al. 1988.
12. Shi, J. Q., and Durucan, S. 2003. Changes in permeability of coalbeds during primary recovery. Part 1. Model formulation and analysis. Paper 0341 in Proceedings of the 2003 International Coalbed Methane Symposium. Tuscaloosa: University of Alabama.
13. Palmer, I., and Mansoori, J. 1996. How Permeability Depends on Stress and Pore Pressure in Coalbeds: A New Model. Paper SPE 36737. Presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, October 6–9.
14. Shi, J. Q., and Durucan, S. 2003. Changes in permeability of coalbeds during primary recovery. Part 2. Model validation and field application. Paper 0342 in Proceedings of the 2003 International Coalbed Methane Symposium. Tuscaloosa: University of Alabama.
15. Shi, J. Q., and Durucan, S. 2003. Part 1. Paper 0341.
16. Ibid.
17. Reznik, A. A., Dabbous, M. K., Fulton, P. F., and Taber, J. J. 1974. Air-water relative permeability studies of Pittsburgh and Pocohontas coals. Society of Petroleum Engineers Journal. V. 6 (December). p. 556; and Puri, R., Evanoff, J. C., and Brugler, M. L.
1991. Measurement of Coal Cleat Porosity and Relative Permeability Characteristics. Paper SPE 21491. Presented at the SPE Gas Technology Symposium, Houston, Texas, January 23–25.
18. Bustin, R. M., and Clarkson, C. R. 1999. Free gas storage in matrix porosity: A potentially significant coalbed resource in low rank coals. Paper 9956 in Proceedings of the 1999 International Coalbed Methane Symposium. Tuscaloosa: University of Alabama;
and Gash, B. W. 1991. Measurement of “Rock Properties” in Coal for Coalbed Methane Production. Paper SPE 22909. Presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas, October 6–9.
19. Levine, J. R. 1993. Coalification: The evolution of coal as source rock and reservoir rock for oil and gas. In Hydrocarbons from Coal, AAPG Studies in Geology #38. Law, B. E., and Rice, D. D., eds. Tulsa: American Association of Petroleum Geologists. p. 39.
20. Gash, B. W. 1991.
21. Hyman, L., Ohen, H. A., Amaefule, J. O., and Danehjou, D. 1991. Simultaneous determination of capillary pressure and relative permeability in coal-bed methane reservoirs. Paper 9118 in Proceedings of the 1991 International Coalbed Methane Symposium.
Tuscaloosa: University of Alabama.
22. Gash, B. W. 1991.
23. Testa, S. M., and Pratt, T. J. 2003. Sample preparation for coal and shale gas resource assessment. Paper 0356 in Proceedings of the 2003 International Coalbed Methane Symposium. Tuscaloosa: University of Alabama.
24. Gash, B. W. 1991; Paterson, L. 1990. Laboratory measurement of coal permeability. In Methane Drainage from Coal. Paterson, L., ed. Syndal, Victoria: CSIRO Division of Geomechanics; and Gash, B. W., et al. 1993.
25. Gash, B. W., et al. 1993.
26. Johnson, E. F., Bossler, D. P., and Naumann, V. O. 1959. Calculation of relative permeability from displacement experiment.
Transactions. AIME. V. 216. p. 370.
27. Gash, B. W., et al. 1993.
28. Mavor, M. J., and Robinson, J. R. 1993; Burdine, N. T. 1953. Relative permeability calculations from pore size distribution data.
Transactions. AIME. V. 198. p. 71; Brooks, R. H., and Corey, A. T. 1966. Properties of porous media affecting fluid flow. Journal of the Irrigation and Drainage Division, Proceedings of the ASCE. V. 92 (June). p. 61; and Gash, B. W. 1991.
29. Mavor, M. J., and Robinson, J. R. 1993; and Gash, B. W. 1991.
30. Gash, B. W. 1991; and Conway, M. W., Mavor, M . J., Saulsberry, J., Barree, R. B., and Schraufnagel, R. A. 1995. Multi-Phase Flow Properties for Coalbed Methane Wells: A Laboratory and Field Study. Paper SPE 29576. Presented at the Joint Rocky Mountain Regional Meeting and Low-Permeability Reservoirs Symposium, Denver, Colorado, March 20–22.
31. Ibid.
32. Hower, T. L., Jones, J. E., Goldstein, D. M., and Harbridge, W. 2003. Development of the Wyodak Coalbed Methane Resource in the Powder River Basin. Paper SPE 84428. Presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, October 5–8.
33. Meaney, K., and Paterson, L. 1996. Relative Permeability in Coal. Paper SPE 36986. Presented at the SPE Asia Pacific Oil and Gas Conference, Adelaide, Australia, October 28–31.
34. Ibid.
35. Harpalani, S., and Schraufnagel, R. A. 1990. Influence of Matrix Shrinkage and Compressibility on Gas Production from Coalbed Methane Reservoirs. Paper SPE 20729. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, September 23–26; Harpalani, S. and Chen, G. 1995. Estimation of changes in fracture porosity of coal with gas emission. Fuel.
V. 74 (no. 10). p. 1,491; and Seidle, J. P., and Huitt, L. G. 1995. Experimental Measurement of Coal Matrix Shrinkage Due to Gas Desorption and Implications for Cleat Permeability Increases. Paper SPE 30010. Presented at the International Meeting on Petroleum Engineering, Beijing, China, November 14–17.
36. Levine, J. R. 1996. Model study of the influence of matrix shrinkage on absolute permeability of coal bed reservoirs. Coalbed Methane and Coal Geology, Special Publication No. 109. Gayer, R. A., and Harris, I. A., eds. London: Geological Society. p. 197.
37. Ibid.
38. Harpalani, S., and Schraufnagel, R. A. 1990; Seidle, J. P., and Huitt, L. G. 1995; Levine, J. R. 1996; Moffat, D. H., and Weale, K.
E. 1955. Sorption by coal of methane at high pressures. Fuel. V. 34 (no. 4). p. 449; Gunther, J. 1968. Investigation of the gas-coal bond. Revue de l’Industrie Minérale. V. 47 (October). p. 693. Wubben, P., Seewald, H., and Jurgen, K. 1986. Permeation and sorption behavior of gas and water in coal. Proceedings of the Twelfth Annual Underground Coal Gasification Symposium. August 24–28. DOE/FE/60922-H1. Washington, DC: Department of Energy; Reucroft, P. J., and Patel, H. 1986. Gas-induced swelling in coal. Fuel. V. 65 (no. 6). p. 816; Gray, I. 1987. Reservoir engineering in coal seams. Part 1. The physical process of gas storage and movement in coal seams. SPE 12514. Society of Petroleum Engineers Reservoir Engineering. V. 2 (no. 1). p. 28; Juntgen, H. 1987.
Research for future in situ conversion of coal. Fuel. V. 66 (no. 4). p. 443.
39. Harpalani, S., and Schraufnagel, R. A. 1990.
40. Levine, J. R. 1996.
41. Seidle, J. R., and Huitt, L. G. 1995.
42. Levine, J. R. 1996.
43. Palmer, I., and Mansoori, J. 1996.
44. Mavor, M. J., and Robinson, J. R. 1993.
45. Mavor, M. J. 1994. Coal Gas Openhole Well Performance. Paper SPE 27993. Presented at the University of Tulsa Centennial Petroleum Engineering Symposium, Tulsa, Oklahoma, August 29–31; and Ramurthy, K., Young, G. B. C., Daves, S. B., and Witsell, F. 2003. Case History: Reservoir Analysis of the Fruitland Coals Results in Optimizing Coalbed Methane Completions in the Tiffany Area of the San Juan Basin. Paper SPE 84426. Presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, October 5–8.
46. Bell, G. J., and Jones, A. H. 1989. Variation in mechanical strength with rank of gassy coals. Paper 8924 in Proceedings of the 1989 International Coalbed Methane Symposium. Tuscaloosa: University of Alabama.
47. Shi, J. Q., and Durucan, S. 2003. Part 1. Paper 0341; and Shi, J. Q., and Durucan, S. Part 2. Paper 0342.
48. Levine, J. R. 1996; and Palmer, I., and Mansoori, J. 1996.
49. Palmer, I., and Mansoori, J. 1996; and Mavor, M. J., and Vaughn, J. E. 1997. Increasing absolute permeability in the San Juan Basin Fruitland Formation. Paper 9738 in Proceedings of the 1997 International Coalbed Methane Symposium. Tuscaloosa:
University of Alabama.
50. Palmer, I., and Mansoori, J. 1996; Shi, J. Q., and Durucan, S. 2003. Part 1. Paper 0341; and Shi, J. Q., and Durucan, S. Part 2. Paper 0342.
51. Mavor, M. J., and Vaughn, J. E. 1997.
52. Zahner, B. 1997. Application of Material Balance to Determine Ultimate Recovery of a San Juan Fruitland Coal Well. Paper SPE 38858. Presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, October 5–8.
7
Introduction
For more than half a century, reservoir permeability and pressure and wellbore condition have been determined in conventional wells from the observed pressure response resulting from a change in production rate. More recently, these techniques also have been used in coal wells. Many of the basic principles of conventional well testing can be employed to analyze coal well tests with little or no modification. Coal deposits are often aquifers with little, if any, free gas present. Consequently, initial attempts to characterize the reservoir often involve injection of water into or production of water from the coal. Well tests of producing wells often consider both gas and water flow. Well tests in coals include injection/falloffs, drillstem tests, tank tests, slug tests, diagnostic fracture injection tests, drawdown and buildup tests, interference tests, and micropilot injectivity tests.