Keywords: mechanics, fractured media, deformation, non-linear elasticity, permeability, hydromechanical coupling.

This paper presents a numerical model and a set of results for the deformation of fractured porous media, their effective mechanical moduli, and demonstrating the influence of the deformation on the flow properties. The three-dimensional model incorporates randomly located discrete fractures, and accounts for their non linear rheology.

The deformations of a fractured porous medium are governed by the Navier elastostatic equation together with Hooke's constitutive law for the matrix material, supplemented with boundary conditions at the fracture surfaces. The behaviour of the fracture is represented by two normal and tangential stiffness coefficients, which relate the fracture closure and tangential displacement to the embedding stress. These coefficients can be evaluated from Hertz contact theory and from the statistical geometrical parameters of the fracture surfaces. They depend on the current deformed state of the fracture, whose rheology is therefore non linear.

The mechanical properties of intact porous media and fractures have been investigated separately and at the microscale in earlier works [1,2]. We combine here these ingredients to address the deformation of fractured porous media. Macroscopic properties such as the effective Lamé coefficients or the Young modulus and Poisson's ratio can be derived, and the impact of the deformation on the flow properties can be quantified.

Two major steps are required in the numerical solution of this problem. First, an unstructured tetrahedral mesh of the fractures and of the porous matrix located in between is constructed; second, the equations are discretized and solved, in a finite volume formulation. The general methodology can be found in reference [3], which also describes the solution of the flow problem in a fractured permeable matrix. Fractures with various plane polygonal shapes have been considered, which range from nearly circular to very elongated quadrilaterals. The degree of fracturation is quantified by a dimensionless fracture density which incorporates a shape factor.

The calculations were conducted in cubic samples with periodicity conditions, submitted to successive increments of the macroscopic strain tensor. Three regimes can be distinguished. At first, the fractures are open and the fractured medium is linearly elastic, with initial effective moduli. Then, when the external stresses increase, the fractures are progressively closed and the problem becomes non linear; the effective properties depend on the type and magnitude of the load. Finally, in an ultimate stage when the external stresses are very large, the fractures are totally closed but their surfaces may still slip tangentially according to a Coulomb-type failure criterion.

Initial effective moduli have been calculated using statistical sets of random realizations. They always decrease as the degree of fracturation increases. The data for small fracture densities are in good agreement with the existing theoretical predictions. Over a wide range of , they can be fairly accurately fitted by using decaying exponential laws, which are much more successful than effective field or self-consistent models from the literature. Furthermore, the influence of the fracture geometry is almost fully accounted for by the shape factor which is involved in the definition of the dimensionless fracture density .

In the non linear regime, marginal instantaneous moduli can be defined for each deformed state. Examples are provided of incremential deformations, which result in a transition from the initial to the ultimate effective properties. In addition, the flow properties of the fractured medium can be evaluated in the successive states. The permeability tensor evolves from its initial value to that of the matrix material. In the intermediate states, some major flow paths are redistributed, and a significant anisotropy is observed, which depends on the type of external load.

Thus, a methodology for the determination of the deformations of fractured porous media, of their effective mechanical moduli, and of the influence of these deformations on the permeability tensor has been put on a firm basis. An extensive set of results relative to the mechanical aspects, and preliminary results on the permeability tensor have been obtained. Further investigation of the permeability is to be conducted, and the modeling of creep and fracturation will be addressed in the near future.

1

J. Poutet, D. Manzoni, F. Hage-Chehade, C.G. Jacquin, M.J. Bouteca, J.-F. Thovert, P.M. Adler, "The effective mechanical properties of random porous media", J. Mech. Phys. Solids (44), 1587-1620, 1996. doi:10.1016/0022-5096(96)00051-8

2

V.V. Mourzenko, O. Galamay, J.-F. Thovert, P.M. Adler, "Fracture deformation and influence on permeability", Phys. Rev. E (56) 3167-3184, 1997. doi:10.1103/PhysRevE.56.3167

3

I.I. Bogdanov, V.V. Mourzenko, J.-F. Thovert, P.M. Adler, "Effective permeability of fractured porous media in steady-state flow", Water Resour. Res. (39), 1029-1044, 2001WR000756, 2003. doi:10.1029/2001WR000756

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