Lattice BGK simulations of macrodispersion in heterogeneous porous media

Abstract

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.