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Relation between fractional flow models and fractal or long‐range 2‐D permeability fields
Author(s) -
de Dreuzy JeanRaynald,
Davy Philippe
Publication year - 2007
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/2006wr005236
Subject(s) - scaling , fractal , sierpinski carpet , percolation (cognitive psychology) , isotropy , mathematics , sierpinski triangle , inverse , fractal dimension , realization (probability) , flow (mathematics) , percolation threshold , statistical physics , mathematical analysis , geometry , physics , statistics , optics , electrical resistivity and conductivity , quantum mechanics , neuroscience , biology
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 d w 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 d w values in the range [2.6, 3] and [1.9, 2.5], respectively. The bottlenecks, characteristic of percolation, induce a more anomalous transport (larger d w values) than the impervious zones present at all scales of Sierpinskis. Second, the realization‐based values of n and d w 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 , d w ) 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 d w 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 , d w ) plane, and realization‐based results are not selective in terms of model. So this should be replaced by the comparison of ( n , d w ) values averaged over different pumping well locations, which however requires a significantly larger quantity of field tests.