Abstract
Effective stress laws and their application are not new, but are often overlooked or applied inappropriately. The complexity of using a proper effective stress law increases when analyzing stress variation in coal as a result of gas production or mining. In this paper, an effective stress law is derived analytically for coalbed methane reservoirs, combining the concepts of matrix shrinkage/swelling and external stress by including the effect of sorbing gas pressure on the elastic response of the reservoir. The proposed law reduces to that of Terzaghi when the compressibility of bulk material is sufficiently greater than the compressibility of the solid grain, and without the strain associated with matrix shrinkage/swelling effect. Moreover, it is shown that the Biot coefficient (α) can have a value larger than unity for self-swelling/dilation materials, such as coal. The proposed stress–strain relationship was validated using experimental results. Overall, the effective stress law for deformation was extended for sorptive materials, providing a new and unique technique to analyze the elastic behavior of coal by reducing three variables, namely, external stress, pore pressure and matrix shrinkage/swelling along with the associated stress, down to one variable, “effective stress”.
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References
Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12:155–164
Biot MA (1955) Theory of elasticity and consolidation for a porous anisotropic solid. J Appl Phys 26:182–185
Bishop AW (1955) The principle of effective stress. Tekniske Ukeblad 39:859–863
Christensen NI, Wang HF (1985) The influence of pore pressure and confining pressure on dynamic elastic properties of Berea Sandstone. Geophysics 50:207–213
Detournay E, Cheng AH-D (1993) Fundamentals of poroelasticity. In: Fairhurst C (ed) Comprehensive rock engineering: principles, practice and projects, vol. II, analysis and design method. Pergamon Press, New York, pp 113–171
Fatt I (1959) The Biot-Willis elastic coefficients for a sandstone. J Appl Mech 26:296–297
Geertsma J (1957) The effect of fluid pressure decline on volumetric changes of porous rocks. Trans AIME 210:331–340
George JG, Barakat MA (2001) The change in effective stress associated with shrinkage from gas desorption in coal. Int J Coal Geol 45:105–113
Harpalani S, Chen GL (1995) Estimation of changes in fracture porosity of coal with gas emission. Fuel 74:1491–1498
Harpalani S, Mitra A (2010) Impact of CO2 injection on flow behavior of coalbed methane reservoirs. Transp Porous Media 82:141–156
Hubbert MK, Rubey WW (1959a) Role of fluid pressure in mechanics of overthrust faulting. Bull Geol Soc Am 70:115–205
Hubbert MK, Rubey WW (1959b) Role of fluid pressure in mechanics of overthrust faulting: a reply. Bull Geol Soc Am 71:617–628
Izadi G, Wang S, Elsworth D, Liu J, Wu Y, Pone D (2011) Permeability evolution of fluid-infiltrated coal containing discrete fractures. Int J Coal Geol 85:202–211
Knaap WV (1959) Nonlinear behavior of elastic porous media. Trans AIME 216:179–187
Lade PV, Boer RD (1997) The concept of effective stress for soil, concrete and rock. Geotechnique 47(1):61–78
Levine JR (1996) Model study of the influence of matrix shrinkage on absolute permeability of coalbed reservoirs. In: Gayer R, Harris I (eds) Coalbed methane and coal geology. Geol Soc Special Pub, London, pp 197–212
Li C (2000) The effective stress study of porous media and it application. PhD dissertation, University of Science and Technology of China
Liu S, Harpalani S (2013a) A new theoretical approach to model sorption-induced coal shrinkage or swelling. AAPG Bull 97(7):1033–1049
Liu S, Harpalani S (2013b) Permeability prediction of coalbed methane reservoirs during primary depletion. Int J Coal Geol 113:1–10
Liu HH, Rutqvist J (2010) A new coal-permeability model: internal swelling stress and fracture–matrix interaction. Transp Porous Media 82:157–171
Ma Q, Harpalani S, Liu S (2011) A simplified permeability model for coalbed methane reservoirs based on matchstick strain and constant volume theory. Int J Coal Geol 85:43–48
Maggs FAP (1946) The adsorption-swelling of several carbonaceous solids. Trans Faraday Soc 42:284–288
Moffat DH, Weale KE (1955) Sorption by coal of methane at high pressures. Fuel 34:449–462
Nur A, Byerlee JD (1971) An exact effective stress law for elastic deformation of rock with fluid. J Geophys Res 76:6414–6419
Nuth M, Laloui L (2008) Effective stress concept in unsaturated soils: clarification and validation of a unified framework. Int J Numer Anal Methods Geomech 32:771–801
Palmer I, Mansoori J (1998), How rermeability depends on stress and pore pressure in coalbeds: a new model. SPE Reserv Eval Eng 539–544
Palmer I, Mavor M, Gunter B (2007) Permeability changes in coal seams during production and injection. In: Proceedings of 2007 international Coalbed methane symposium. Paper 0713
Pan Z, Connell LD (2007) A theoretical model for gas adsorption-induced coal swelling. Int J Coal Geol 69:243–252
Robertson EP (2005), Measurement and modeling of sorption-induced strain and permeability changes in coal. Phd dissertation, Colorado School of Mine, USA
Schiffman RL (1970) The stress components of a porous medium. J Geophys Res 75:4035–4038
Seidle JP, Jeansonne DJ, Erickson DJ (1992) Application of matchstick geometry to stress dependent permeability in coals. In: SPE Rocky maintain regional meeting, Casper, Wyoming, Paper SPE 24361, pp 433–444
Shi JQ, Durucan S (2004) Drawdown induced changes in permeability of coalbeds: a new interpretation of the reservoir response to primary recovery. Transp Porous Media 56:1–16
Skempton AW (1960) Effective stress in soils, concrete and rock. In: Conference on pore pressure and suction in soils, Butterworths, pp 4–16
Skempton AW, Bishop AW (1954) Soils, in building materials, their elasticity and inelasticity. North Holland Publishing Company, Amsterdam, pp 417–482
Suklje L (1969) Rheological aspects of soil mechanics. Wiley Interscience, New York, p 123
Terzaghi K (1943) Theoretical soil mechanics. Wiley, New York
US EIA (2009) Natural gas gross withdrawals and production released on 10/29/2010. US Energy Information Administration
Warpinski NR, Teufel LW (1992) Determination of the effective stress law for permeability and determination in low-permeability rocks. SPE Form Eval 7:123–131
Zhao Y (2009) Development and utilization of coal mine methane in China. 2009. In: International Coalbed methane & Shale gas symposium, Paper 0938. University of Alabama, Tuscaloosa, pp 65–74
Zhao Y, Hu Y, Wei J, Yang D (2003) The experimental approach to effective stress law of coal mass by effect of methane. Transp Porous Media 53:235–244
Zimmerman RW, Somerton WH, King MS (1986) Compressibility of porous rocks. J Geophys Res 91:765–777
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The authors thank Co-Editor Dr. Herbert Einstein and two anonymous reviewers for valuable suggestions that helped improve the manuscript.
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Liu, S., Harpalani, S. Determination of the Effective Stress Law for Deformation in Coalbed Methane Reservoirs. Rock Mech Rock Eng 47, 1809–1820 (2014). https://doi.org/10.1007/s00603-013-0492-6
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DOI: https://doi.org/10.1007/s00603-013-0492-6