Relation between fractional flow models and fractal or long-range 2-D permeability fields
Résumé
Fractional flow models introduced by Barker (1988) have been increasingly popular as means of interpreting nonclassical drawdown curves obtained from well tests. Fractional flow models are intrinsically isotropic scaling models depending to first order on two exponents n and dw expressing the dimension of the structure available to flow and the flow slowdown, respectively. We study the fractional flow induced either by geometrically scaling structures such as Sierpinski- and percolation-like fractal media or by hydraulically scaling media such as long-range continuous correlated media. First, percolation and Sierpinski structures have two well-separated dw values in the range [2.6, 3] and [1.9, 2.5], respectively. The bottlenecks, characteristic of percolation, induce a more anomalous transport (larger dw values) than the impervious zones present at all scales of Sierpinskis. Second, the realization-based values of n and dw depend both on global and on local characteristics like the fractal dimension and the permeability around the well, respectively. Finally, solving the inverse problem on anomalous transient well test responses consists in comparing the (n, dw) realization-based values with field data. Indeed, well tests performed from a unique pumping well must be taken as realization-based results. For the site of Ploemeur (Brittany, France), from which n and dw have been previously determined (Le Borgne et al., 2004), the only consistent model is given by the continuous multifractals. However, the values obtained from continuous multifractals cover most of the (n, dw) plane, and realization-based results are not selective in terms of model. So this should be replaced by the comparison of (n, dw) values averaged over different pumping well locations, which however requires a significantly larger quantity of field tests.
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