Abstract
The equations governing plane steady-state flow in heterogeneous porous body containing cracks are presented first. Then, a general transformation lemma is presented which allows extending a particular solution obtained for a given flow problem to another configuration with different geometry, behaviour and boundary conditions. An existing potential solution in terms of discharges along the cracks, established by Liolios and Exadaktylos (J Solids Struct 43:3960–3982, 2006) for non-intersecting cracks in isotropic matrix, is extended to intersecting cracks in anisotropic matrix. The basic problem of a single straight crack in an infinite body submitted to a pressure gradient at infinity is then investigated and a closed-form solution is presented for the case of void cracks (infinite conductivity), as well as a semi-analytical solution for the case of cracks with Poiseuille type conductivity. These solutions, derived first for an isotropic matrix, are then extended to anisotropic matrices using the general transformation lemma. Finally, using the solution obtained for a single crack, a closed-form estimation of the effective permeability of micro-cracked porous materials with weak crack density is derived from a self-consistent upscaling scheme.
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References
Barthélémy J.-F.: Effective permeability of media with a dense network of long and micro fractures. Transp Porous Media 76, 153–178 (2009)
Bonnet M.: Equations Intégrales et Éléments de Frontière. CNRS Editions/Eyrolles, Paris (1995)
Castany G.: Traité Pratique des Eaux Souterraines. Editions Dunod, Paris (1967)
Dormieux L., Kondo D.: Approche micromécanique du couplage perméabilité-endommagement. C.R. Mecanique 332, 135–140 (2004)
Eshelby J.D.: Determination of the elastic filed of an ellipsoidal inclusion and related problems. Proc. R. Soc. Lond. A241, 376–396 (1957)
Goméz-Hernández J.J., Wen X.H.: Upscaling hydraulic conductivities in heterogeneous media: an overview. J. Hydrol. 183, ix–xxxii (1996)
Guéguen Y., Palciauskas V.: Introduction to Physics of Rocks. Princeton University Press, Princeton (1994)
Kachanov M.: Elastic solid with many cracks and related problems. In: Hutchinson, J.W., Wu, T. (eds) Advances in Applied Mechanics, vol 30, pp. 259–445. Academic Press, New York (1993)
Liolios P.A., Exadaktylos G.E.: A solution of steady-state fluid flow in multiply fractured isotropic porous media. Int. J. Solids Struct. 43, 3960–3982 (2006)
Milton, G.W.: The Theory of Composites, pp. 1454–147. Cambridge University Press, Cambridge (2002)
Mourzenko, V.V., Thovert, J.-F., Adler, P.M.: Macroscopic properties of polydisperse, anisotropic and/or heterogeneous fracture networks. Proceedings of the International Conference on Rock Joints and Jointed Rock Masses, Tucson, Arizona, USA, (2009)
Muskhelishvili N.I.: Singular Integral Equations. P. Noordhoff Ltd., Groningen, Holland (1953)
Pouya A.: Equivalent permeability tensors of a finite heterogeneous block. C.R. Géosci. 337, 581–588 (2005)
Pouya A., Courtois A.: Définition de la perméabilité équivalente des massifs fracturés par des méthodes d’homogénéisation. C.R. Géosci. 334, 975–979 (2002)
Pouya A., Fouché O.: Permeability of 3D discontinuity networks: new tensors from boundary-conditioned homogenization. Adv. Water Resour. 32, 303–314 (2009)
Pouya A., Zaoui A.: A transformation of elastic boundary value problems with application to anisotropic behavior. Int. J. Solids Struct. 43, 4937–4956 (2006)
Renard P., de Marsily G.: Calculating equivalent permeability: a review. Adv. Water Resour. 20(5-6), 253–278 (1997)
Sánchez-Vila X., Girardi G.P., Carrera J.: A synthesis of approaches to upscaling of hydraulic conductivities. Water Resour. Res. 31(4), 867–882 (1995)
Zhao C., Hobbs B.E., Ord A., Hornby P., Mühlhaus H., Peng S.: Theoretical and numerical analyses of pore-fluid flow focused heat transfer around geological faults and large cracks. Comput. Geotech. 35, 357–371 (2008)
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An erratum to this article can be found at http://dx.doi.org/10.1007/s11242-010-9544-5
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Pouya, A., Ghabezloo, S. Flow Around a Crack in a Porous Matrix and Related Problems. Transp Porous Med 84, 511–532 (2010). https://doi.org/10.1007/s11242-009-9517-8
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DOI: https://doi.org/10.1007/s11242-009-9517-8