[1] Flow in fractured porous media was first investigated by and by means of the double-porosity model. A direct, exact, and complete numerical solution of the flow in such media is given in this paper for arbitrary distributions of permeabilities in the porous matrix and in the fracture network. The fracture network and the porous matrix are automatically meshed; the flow equations are discretized by means of the finite volume method. This code has been so far applied to incompressible fluids and to statistically homogeneous media which are schematized as spatially periodic media. Some results pertaining to random networks of polygonal fractures are presented and discussed; they show the importance of the percolation threshold of the fracture network and possibly of the porous matrix. Moreover, the influence of the fracture shape can be taken into account by means of the excluded volume.

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The asymptotic behaviors of the permeability of isotropic fracture networks at small and large densities are characterized, and a general heuristic formula is obtained which complies with the limiting behaviors and accurately predicts the permeability of these networks over the whole density range. Theses developments are based on extensive numerical calculations and on theoretical arguments inspired by the examination of the flow distribution in the fractures at large densities. Then, the results are extended to anisotropic networks with a Fisher distribution of the fracture orientations, to polydisperse networks, and to fractured porous media. Finally, guidelines are provided for the practical evaluation of the required parameters from typical field data. A summary of the results is given in Table III.

The permeability of geological formations which contain fractures with a power-law size distribution is addressed numerically by solving the coupled Darcy equations in the fractures and in the surrounding porous medium. Two reduced parameters are introduced which allow for a unified description over a very wide range of the fracture characteristics, including their shape, density, size distribution, and possibly size-dependent permeability. Two general models are proposed for loose and dense fracture networks, and they provide a good representation of the numerical data throughout the investigated parameter range.

The influence of various parameters such as the domain size, the exponent of the power law, the smallest radius, and the fracture shape on the percolation threshold of fracture networks has been numerically studied. For large domains, the adequate percolation parameter is the dimensionless fracture density normalized by the product of the third moment of fracture radii distribution and of the shape factor; for networks of regular polygons, the dimensionless critical density depends only slightly on the parameters of radii distribution and on the shape of fractures; a model is proposed for the percolation threshold for fractures with elongated shapes. In small domains, percolation is analyzed in terms of the dimensionless fracture density normalized by the sum of two reduced moments of the radii distribution; this provides a general description of the network connectivity properties whatever the dominating percolation mechanism; the fracture shape is taken into account by using excluded volume in the definition of dimensionless fracture density.

Two-phase flow in fractured porous media is investigated by means of a direct and complete numerical solution of the generalized Darcy equations in a three-dimensional discrete fracture description. The numerical model applies to arbitrary fracture network geometry, and to arbitrary distributions of permeabilities in the porous matrix and in the fractures. It is used here in order to obtain the steady-state macroscopic relative permeabilities of random fractured media. Results are presented as functions of the mean saturation and are discussed in comparison with simple models.

Fracture network permeability is investigated numerically by using a three-dimensional model of plane polygons uniformly distributed in space with sizes following a power-law distribution. Each network is triangulated via an advancing front technique, and the flow equations are solved in order to obtain detailed pressure and velocity fields. The macroscopic permeability is determined on a scale which significantly exceeds the size of the largest fractures. The influence of the parameters of the fracture size distribution--the power-law exponent and the minimal fracture radius--on the macroscopic permeability is analyzed. Eventually, a general expression is proposed, which is the product of a dimensional measure of the network density, weighted by the individual fracture conductivities, and of a fairly universal function of a dimensionless network density, which accounts for the influences of the fracture shapes and of the parameters of their size distribution. Two analytical formulas are proposed which successfully fit the numerical data over a wide range of network densities.

The percolation threshold of fracture networks is investigated by extensive direct numerical simulations. The fractures are randomly located and oriented in three-dimensional space. A very wide range of regular, irregular, and random fracture shapes is considered, in monodisperse or polydisperse networks containing fractures with different shapes and/or sizes. The results are rationalized in terms of a dimensionless density. A simple model involving a new shape factor is proposed, which accounts very efficiently for the influence of the fracture shape. It applies with very good accuracy in monodisperse or moderately polydisperse networks, and provides a good first estimation in other situations. A polydispersity index is shown to control the need for a correction, and the corrective term is modelled for the investigated size distributions.