A Direct Determination of the Transient Exchange Term of Fractured Media Using a Continuous Time Random Walk Method

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In two recent papers, Nœtinger and Estébenet, 2000; Nœtinger et al., submitted, we set-up a method allowing to compute both the transient and steady-state exchange terms between the matrix and fractured regions of a naturally fractured porous medium using continuous time random walk methods (CTRW). The goal of the present paper is to show that a new version of the CTRW algorithm provides a direct determination of the so called transient exchange function f(t) (or its Laplace transform f(s)) widely used in well test interpretation. It is shown that this function is directly linked with the probability density of the first escape time in the fractured region of a Brownian particle launched initially in the matrix region. This new interpretation allows relating directly the exchange coefficient α with the mean escape time of brownian particles in the matrix. From a practical point of view, these new results allow to derive a simpler version of the CTRW method. In addition, we obtain a considerable speed up of the CTRW method for up-scaling fractured reservoirs.

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