A Renormalization-Group Approach of the Up-Scaling Problem of Flow in Heterogeneous Porous Media

Powerful methods coming from statistical physics are becoming increasingly popular to get a faithful theoretical description of flow and transport in heterogeneous aquifers that are described by mean of geostatistics. However,in current practice people still use Monte Carlo simulations that are well suited to account for complex boundaries and flow patterns. Here, we present an approach intending to up-scale directly the geostatistical description rather than realization by realization as usual. It is based upon a renormalization group analysis close in spirit with previous works and P King and Jaekel and Vereecken. Using a so called "weak approximation"(Neuman and Orr) , we obtain differential equations driving the permeability variogram parameters as a function of the wave-vector cut-off smoothing the permeability maps. At the end of the process, in the isotropic case, the Landau Lifshitz Matheron conjecture is recovered. This conjecture appears thus as being a consequence of both renormalization approach and the weak approximation. The approach is currently being generalized to anisotropic media. These results can be used to perform cheaper Monte Carlo simulations at a coarser scale. P. King, The Use of Field Theoretic Methods for the Study of Flow in Heterogeneous Porous Medium", J. Phys. A.: Math. Gen. 20, pp3935-3947,1987 U. Jaekel and H. Vereecken, Renormalization Group Analysis of Macrodispersion in a Directed Random Flow, Water Resources Research,33,10,pp 2287-2299, 1997 Neuman, S.P. and Orr, S. "Prediction of Steady State Flow in Nonuniform Geologic Media by Conditional Moments: Exact non local Formalism, Effective Conductivities and Weak Approximation", Water Resources Research 29 (2)341-364 (1993) Noetinger, B. Computing the Effective Permeability of log-normal permeability fields using renormalization methods. C.R . Acad. Des Sciences,Sciences de la Terre et des Planètes, 331 353-357 (2000)