The purpose of this paper is to present a stochastic double porosity model. This model is based on a stochastic spatial distribution of the fractures, and can be coupled to a stochastic flow rate between the two porosities: matrix-blocks and fractures. A Geometrical model is presented. The fluid flow through the connected fractures is stated using reciprocal principle. The second porosity is taken into account by identifying the matrix-blocks delimited by the nearby connected fractures. A comparison with analytical solutions for simple double porosity model is presented.
In the 60s some discontinuous models were proposed which took into account the co-existence of a double-porosity: fractures and rock matrices. These models considered a piling-up of cubics or spheres to represent the fractured media (Figure 1.a). Fractures and matrices interact through a flow rate which is a function of the pressure in each one and the geometry of the rock matrix. In the early 70s, other models were proposed based upon the existence of a single fracture surrounded by a rock matrix interacting and a flow rate from the block matrix to the fracture was defined. These models assume that each porosity is a continuum media. But these approaches are a simplification of the geological structure of the aquifers and the physical phenomenon in which they occur. Because of the geometric scale on which the observations are made and in which the phenomena occur, it is not usually possible to use an equivalent porous medium approach, and a discontinuous representation might therefore be chosen. Moreover, data of observed characteristics in the fractured medium are overall very scarce, making it seem preferable to consider each family of fractures in the network as random. Therefore, several stochastic models were developed for flow and transport in fractured media where fractures distributions are known only by their statistics (Figure 1.b). These models assume that the fluid flow occurs only in the fractures and the walls are impervious, therefore the matrix-blocks is ignored. But, at any given point, the high-diffusivity fractures respond to an imposed pressure change much more rapidly than do the low-diffusivity matrix blocs, causing a pressure imbalance between the two.
Unlike in classical fluid theory, for each point in space, not one liquid pressure but two, pi and pa, are introduced. The pi pressure represents the average liquid pressure in the fractures in the neighborhood of the given point, whereas the pressure p2 is the average liquid pressure in the pores [of the matrix blocks] in the neighborhood of the given point. To obtain reliable averages, the scale of the averaging should include a sufficiently large number of blocks. Therefore one must assume not only that any infinitely small volume includes a large number of pores, as is assumed in classical fluid flow theory, but also that it contains a large number of blocks. This assumption allows one to use the analysis of infinitesimals in studying fractured rocks [Barenblatt, Zheltov, and Kochina [1961], p. 853].
