Up-scaling flow in fractured media: Equivalence between the large scale averaging theory and the continuous time random walk method

In a recent paper, (McNabb, 1978), we set up a method allowing to compute b oth the transient and steady-state exchange terms between the matrix and fr actured regions of a naturally fractured porous medium using the continuous time random walk method (CTRW). In particular, the exchange coefficient al pha parametrizing the large-scale exchange term was computed on physical gr ounds using a time integration of the so-called time correlation function c orresponding to the particle presence in the fractures. On the other hand, the large scale averaging theory (LSAT) as developed by Quintard and Whitak er (Quintard and Whitaker, 1996) gives another definition for this exchange coefficient alpha. It also provides a so-called 'closure problem' allowing to compute alpha from the solution of a well-defined steady state boundary value problem, to be solved over a representative volume of the high resol ution fractured map. The goal of the present paper is to show analytically that both definitions coincide, yielding a unique and well defined value of the alpha coefficient. This provides an unification of two approaches whos e respective backgrounds are very different at the first glance.