Plane-Strain Propagation of a Fluid-Driven Crack in a Permeable Rock with Fracture Toughness
Publication: Journal of Engineering Mechanics
Volume 136, Issue 9
Abstract
A solution to the problem of a plane-strain fluid-driven crack propagation in elastic permeable rock with resistance to fracture is presented. The fracture is driven by injection of an incompressible Newtonian fluid at a constant rate. The solution, restricted to the case of zero lag between the fluid front and the fracture tip, evolves from the early-time regime when the fluid flow takes place mostly inside the crack toward the large-time response when most of the injected fluid is leaking from the crack into the surrounding rock. This transition further depends on a time-invariant partitioning between the energy expanded to overcome the rock fracture toughness and the energy dissipated in the viscous fluid flow in the fracture. A numerical approach is used to compute the solution for the normalized crack length and crack opening and net-fluid pressure profiles as a function of two dimensionless parameters: the leak-off/storage evolution parameter and the toughness/viscosity number. Relation of this solution to the various available asymptotic solutions is discussed. Obtained mapping of the solution onto the problem parametric space has a potential to simplify the tasks of design, modeling, and data inversion for hydraulic fracturing treatments and laboratory experiments.
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Acknowledgments
The writers thank Dr. José I. Adachi for insightful discussions on the numerical methods. Acknowledgment is made to the donors of The Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research under Grant No. UNSPECIFIEDACS-PRF 35729-G2.
References
Adachi, J. I. (2001). “Fluid-driven fracture in permeable rock.” Ph.D. thesis, Univ. of Minnesota, Minneapolis.
Adachi, J. I., and Detournay, E. (2002). “Self-similar solution of a plane-strain fracture driven by a power-law fluid.” Int. J. Numer. Analyt. Meth. Geomech., 26, 579–604.
Adachi, J. I., and Detournay, E. (2008). “Plane-strain propagation of a hydraulic fracture in a permeable rock.” Eng. Fract. Mech., 75, 4666–4694.
Adachi, J. I., Siebrits, E., Peirce, A., and Desroches, J. (2007). “Computer simulation of hydraulic fractures.” Int. J. Rock Mech. Min. Sci., 44, 739–757.
Batchelor, G. K. (1967). An introduction to fluid dynamics, Cambridge University Press, Cambridge, U.K.
Bilby, B. A., and Eshelby, J. D. (1968). “Dislocations and the theory of fracture.” Fracture, an advanced treatise, H. Liebowitz, ed., Vol. 2, Academic, New York, 191–311.
Bunger, A. P., Detournay, E., and Garagash, D. I. (2005). “Toughness-dominated hydraulic fracture with leak-off.” Int. J. Fract., 134, 175–190.
Carbonell, R. S., Desroches, J., and Detournay, E. (1999). “A comparison between a semi-analytical and a numerical solution of a two-dimensional hydraulic fracture.” Int. J. Solids Struct., 36, 4869–4888.
Carter, E. (1957). “Optimum fluid characteristics for fracture extension.” Drilling and production practices, G. Howard and C. Fast, eds., American Petroleum Institute, Tulsa, Okla., 261–270.
Constien, V. (1989). “Fracturing fluid and proppant characterization.” Reservoir stimulation, 2nd Ed., M. Economides and K. Nolte, eds., Prentice-Hall, Englewood Cliffs, N.J., Chap. 5.
Crouch, S., and Starfield, A. (1983). Boundary element method in solid mechanics, Unwin Hyman, London.
Desroches, J., et al. (1994). “The crack tip region in hydraulic fracturing.” Proc. R. Soc. London, Ser. A, 447, 39–48.
Detournay, E. (2004). “Propagation regimes of fluid-driven fractures in impermeable rocks.” Int. J. Geomech., 4, 35–45.
Economides, M. J., and Nolte, K. G. (2000). Reservoir simulation, 3rd Ed., Wiley, Chichester, U.K.
Garagash, D., and Detournay, E. (1997). “An analysis of the influence of the pressurization rate on the borehole breakdown pressure.” Int. J. Solids Struct., 34(24), 3099–3118.
Garagash, D., and Detournay, E. (2000). “The tip region of a fluid-driven fracture in an elastic medium.” ASME Trans. J. Appl. Mech., 67, 183–192.
Garagash, D. I. (2006a). “Plane-strain propagation of a fluid-driven fracture during injection and shut-in: Asymptotics of large toughness.” Eng. Fract. Mech., 73, 456–481.
Garagash, D. I. (2006b). “Propagation of a plane-strain fluid-driven fracture with a fluid lag: Early-time solution.” Int. J. Solids Struct., 43, 5811–5835.
Garagash, D. I. (2006c). “Transient solution for a plane-strain fracture driven by a shear-thinning, power-law fluid.” Int. J. Numer. Analyt. Meth. Geomech., 30, 1439–1475.
Garagash, D. I., and Detournay, E. (2005). “Plane strain propagation of a fluid-driven fracture: Small toughness solution.” ASME Trans. J. Appl. Mech., 72, 916–928.
Garagash, D. I., Detournay, E., and Adachi, J. I. (2010). “Multiscale tip asymptotics in hydraulic fracture.” J. Fluid Mech., to be published.
Irvin, G. R. (1957). “Analysis of stresses and strains near the end of a crack traversing a plate.” ASME Trans. J. Appl. Mech, 24, 361–364.
Lenoach, B. (1995). “The crack tip solution for hydraulic fracturing in a permeable solid.” J. Mech. Phys. Solids, 43(7), 1025–1043.
Liskovets, O. A. (1965). “The method of lines.” J. Diff. Eqns., 1, 1308–1323.
Lister, J. R. (1990). “Buoyancy-driven fluid fracture: The effects of material toughness and of low-viscosity precursors.” J. Fluid Mech., 210, 263–280.
Madyarova, M. (2003). “Propagation of a fluid-driven penny-shaped fracture in permeable elastic medium.” Master's thesis, Univ. of Minnesota, Minneapolis.
Mitchell, S. L., Kuske, R., and Peirce, A. P. (2007). “An asymptotic framework for the analysis of hydraulic fractures: The impermeable case.” ASME Trans. J. Appl. Mech., 74, 365–372.
Nilson, R. H., and Griffiths, S. K. (1983). “Numerical analysis of hydraulically-driven fractures.” Comput. Methods Appl. Mech. Eng., 36, 359–370.
Press, W., Teukolsky, S., Vetterling, W., and Flannery, B. (1996). Numerical recipes in Fortran 90, 2nd Ed., Cambridge University Press, Cambridge, U.K.
Rubin, A. (1993). “Tensile fracture of rock at high confining pressure: Implications for dike propagation.” J. Geophys. Res., 98(B9), 15919–15935.
Savitski, A. A., and Detournay, E. (2002). “Propagation of a fluid-driven penny-shaped fracture in an impermeable rock: Asymptotic solutions.” Int. J. Solids Struct., 39, 6311–6337.
Sneddon, I., and Lowengrub, M. (1969). Crack problems in the classical theory of elasticity, Wiley, New York.
Spence, D. A., and Sharp, P. W. (1985). “Self-similar solution for elastohydrodynamic cavity flow.” Proc. R. Soc. London, Ser. A, 400, 289–313.
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© 2010 ASCE.
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Received: Jan 29, 2009
Accepted: Mar 18, 2010
Published online: May 6, 2010
Published in print: Sep 2010
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