Upscaled flow and transport properties for heterogeneous unsaturated media

Abstract

To represent a heterogeneous unsaturated medium by its homogeneous equivalent, stochastic theory‐based analytical formulas and numerical Monte Carlo simulations are used to obtain upscaled (effective) flow and transport properties. The Monte Carlo experiments simulate steady state flow and transient transport of nonreactive solute in two‐dimensional heterogeneous media. Constitutive relations for unsaturated media at the mesh‐size scale are based on the van Genuchten‐Mualem relationships. A unit‐mean‐gradient approach is used to derive upscaled properties for flow parallel and perpendicular to bedding. Upscaled moisture retention and unsaturated conductivities (K) and longitudinal dispersivities are obtained by simulating steady gravity drainage conditions for a series of applied infiltration rates, characteristic of relatively dry conditions in coarse‐textured sediments. Macrodispersivities are calculated on the basis of spatial moments of the ensemble‐mean plume. Results show that the calculated effective unsaturated K based on the analytical formulas compare well with those based on Monte Carlo simulations. The macroscopic anisotropy (ratio of K parallel to bedding to K perpendicular to bedding) increases with increasing tension, although the increase is rather mild. The longitudinal macrodispersivity for the equivalent homogeneous medium also increases with increasing tension. A comparison of numerical results with stochastic solutions suggests that the computed dispersivities are of the same order of magnitude at low tensions. At higher tensions the dispersivity estimates deviate significantly. Nonetheless, both numerical and analytical results show that the longitudinal dispersivities for flow parallel to bedding are higher than those for flow perpendicular to bedding. In addition, our results suggest that the Fickian regime is reached much earlier for cases with flow perpendicular to bedding than for those with flow parallel to bedding.