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We propose a method for calculating the effective permeability of two-dimensional self-affine permeability fields based on generalizing the one-dimensional concept of a bottleneck. We test the method on fracture faults where the local permeability field is given by the cube of the aperture field. The method remains accurate even when there is substantial mechanical overlap between the two fracture surfaces. The computational efficiency of the method is comparable to calculating a simple average and is more than two orders of magnitude faster than solving the Reynolds equations using a finite-difference scheme.

[1] The miscible displacement of a Newtonian or shear-thinning fluid by another one of same rheological properties has been studied optically in a flat transparent model fracture with a random distribution of identical cylindrical obstacles on one of the walls. At the local scale, the concentration variation on individual pixels satisfies a Gaussian convection-dispersion relation with local transit time (x, y) and dispersivity ld(x, y). The variation of ld with the Peclet number Pe shows that it results from a combination of geometrical and Taylor dispersion, respectively dominant at low and high Pe values. Using shear-thinning solutions instead of a Newtonian fluid enhances the velocity contrasts (and therefore geometrical dispersion) and reduces Taylor dispersion. At the global scale, the front geometry is studied from the isoconcentration lines c = 0.5 (equivalent to lines of constant (x, y) value): beyond a transition travel time, their width in the direction parallel to the flow reaches a constant limit varying linearly with Log(Pe) with a slope increasing with the shear-thinning character of the fluid. These characteristics are compared to previous observations on other model fractures with a self-affine roughness displaying channelization effects.

[1] We use the extended Darcy's law, which also accounts for the Brinkman correction, to study macrodispersion in a two-dimensional (2-D) porous medium. The former is necessary when permeability changes fast at a relatively small scale, and in general, it is a more complete description of flow in a heterogeneous medium. Lattice-gas methods are ideally suited to simulate such flows. Simulations using a lattice BGK method and a small-fluctuation approach are described for an isotropic, exponentially decaying correlation function of the permeability field. The analytical results contain the additional parameter Kl/λ2 (where Kl and λ are the typical permeability and velocity variation length, respectively), the sensitivity to which was studied. As expected, the contribution of the Brinkman effect is insignificant for typical field values of this parameter, in which case, the classical results are recovered. At larger values, for example, for heterogeneous media of a small correlation length, and possibly in laboratory applications, the Brinkman correction leads to a decrease in macrodispersivity, reflecting the smoothing effect of the Brinkman correction on the velocity field. Nonetheless, for practical values of the parameters, this reduction is no larger than 50% of the classical expression. The small fluctuation theory was found to be in good agreement with the simulations, provided that it was consistently applied (namely, by not mixing first-order with second-order expansions). The results also show that lattice-gas simulations can be usefully employed to study macrodispersion in heterogeneous porous media.