Numerical upscaling of the permeability of a randomly cracked porous medium

The equivalent permeability of a randomly cracked porous material is studied using a finite element program in which a four-nodes zero-thickness element is implemented for modelling the cracks. The numerical simulations are performed for geometries with different cracks densities and for different values of matrix permeability and cracks conductivity, but the cracks length are taken equal to one. The method used for determination of the equivalent permeability resulted in a perfectly symmetric equivalent permeability tensor for each case. Based on the obtained results a simple relation is presented for the equivalent permeability of a randomly cracked porous material as a function of the matrix permeability and the cracks density and conductivity. This relation is then generalized for the cracks of any length using a linear transformation

Une transformation du problème d'élasticité linéaire en vue d'application au problème de l'inclusion et aux fonctions de Green

International audienceNous présentons une transformation simple du problème de structure élastique linéaire. Le problème transformé se présente sous forme d'un nouveau problème de structure élastique linéaire avec un comportement, une géométrie et des données de forces et de déplacements imposés différents. Le problème transformé peut être plus simple à traiter ou correspondre à des cas de solutions analytiques connues. A l'aide de cette transformation, on peut ramener le problème de l'inclusion ellipsoïdale en un problème d'inclusion sphérique, étendre les résultats connus pour le tenseur d'Eshelby pour les cas de matrice isotrope ou d'isotropie transverse, à des cas plus larges, et enfin, établir l'expression analytique de la fonction de Green du milieu infini pour des cas de comportements orthotropes plus riches que l'isotropie transverse, ainsi que des cas non orthotropes

Three-dimensional flow in fractured porous media: A potential solution based on singular integral equations

International audienceGoverning equations for flow in three-dimensional heterogeneous and anisotropic porous media containing fractures or cracks with infinite transverse permeability are described. Fractures are modeled as zero thickness curve surfaces with the possibility of multiple intersections. It is assumed that flow obeys to an anisotropic Darcy's law in the porous matrix and to a Poiseuille type law in fractures. The mass exchange relations at fractures intersections are carefully investigated as to establish a complete mathematical formulation for the flow problem in a fractured porous body. A general potential solution, based on singular integral equations, is established for steady state flow in an infinite fractured body with uniform and isotropic matrix permeability. The main unknown variable in the equations is the pressure field on the crack surfaces, reducing thus from three to two the dimension of the numerical problem. A general transformation lemma is then given that allows extending the solution to matrices with anisotropic permeability. The results lead to a simple and efficient numerical method for modeling flow in three-dimensional fractured porous bodies

Ellipsoidal anisotropy in linear elasticity: Approximation models and analytical solutions

International audienceThe concept of ellipsoidal anisotropy, first introduced in linear elasticity by Saint Venant, has reappeared in recent years in diverse applications from the phenomenological to micromechanical modeling of materials. In this concept, indicator surfaces, which represent the variation of some elastic parameters in different directions of the material, are ellipsoidal. This concept recovers different models according to the elastic parameters that have ellipsoidal indicator surfaces. An interesting feature of some models introduced by Saint Venant is the formation of analytical solutions for basic problems of linear elasticity. This paper has two main objectives. First, an accurate definition of the variety of anisotropy called ellipsoidal is provided, which corresponds to a family of materials that depends on 12 independent parameters, including varieties of orthotropic and non-orthotropic materials. An explicit nondegenerate Green function solution is established for these materials. Then, it is shown that the ellipsoidal model recovers a variety of phenomenological and theoretical models used in recent years, specifically for geomaterials and damaged or micro-cracked materials. These models can be used to approximate the elastic parameters of any anisotropic material with different fitting qualities. A method to optimize the parameters will be given

DETERMINATION OF ROCK MASS STRENGTH PROPERTIES BY HOMOGENIZATION

International audienceA method for determining fractured rock mass properties is presented here on the basis of homogenisation approach. The rock mass is considered to be a heterogeneous medium composed of intact rock and of fractures. Its constitutive model is studied numerically using Finite Element Method and assimilating the fractures to joint elements (Coste1). The method has been applied to a granite formation in France. Geological data on different families of fractures have been used for the statistical representation of the fractures. A mesh-generating tool for the medium with high density of fractures has been developed. The mechanical behaviour of the rock mass (elasticity, ultimate strength and hardening law) has been determined assuming linear elasticity and Mohr-Coulomb strength criterion both for the intact rock and the fractures. Evolution of the mechanical strength in different directions has been determined as a function of the mean stress, thanks to various numerical simulations. The mechanical strength appears to be anisotropic due to the preferential orientation of the fractures. The numerical results allowed us to determine an oriented strength criterion for the homogenized rock mass. A 2D constitutive law for the homogenized medium has been deduced from numerical data. A 3D extension of this model is also presented

Définition de la perméabilité équivalente des massifs fracturés par des méthodes d'homogénéisation

International audienceIn the methods used for determination of the hydraulic behaviour at large scale of fractured rock masses, based on the simulation of flow in a fracture network, the mean flux and mean pressure gradient in the network are not rigorously determined. A method is given for deriving these quantities, in a heterogeneous permeable block, from pressure and flux values on the boundary of the block. A block conductivity tensor is then defined based on the condition of linear variation of the pressure on the boundary of the block. It is shown that this conductivity tensor is symmetric and positive definite. An example of application to a model of fractured medium is given

Numerical modelling of the effect of weathering on the progressive failure of underground limestone mines

International audienceThe observations show that the collapse of underground limestone mines results from a progressive failure due to gradual weathering of the rockmass. The following stages can be considered for the limestone weathering and degradation process in underground mines: condensation of the water on the roof of the gallery, infiltration of water in the porous rock, migration of the air CO2 molecules in the rock pore water by convection and molecular diffusion, dissolution of limestone by CO2 rich water and consequently, reduction of the strength properties of rock. Considering this process, a set of equations governing different hydrochemo-mechanical aspects of the weathering phenomenon and progressive failure occurring in these mines is presented. Then the feasibility of numerical modelling of this process is studied and a simple example of application is presented

Ellipsoidal anisotropy in elasticity for rocks and rock masses

International audienceOne of the interesting features with the ellipsoidal models of anisotropy presented in this paper is their acceptance of analytical solutions for some of the basic elasticity problems. It was shown by Pouya (2000) and Pouya and Zaoui (2006) that many closed-form solutions for basic problems involving linear isotropic materials could be extended by linear transformation to cover a variety of "ellipsoidal" materials. This paper will describe two main varieties of ellipsoidal elastic models and show how well they fit the in situ data for sedimentary rocks; numerical homogenization results for several varieties of fractured rock masses will also be provided