A numerical comparison between two upscaling techniques: non-local inverse based scaling and simplified renormalization

In this paper, we face the problem of upscaling transmissivity from the macroscopic to the megascopic scale; here the macroscopic scale is that of the continuous flow equations, whereas the megascopic scale is that of the flow models on a coarse grid. In this paper, we introduce the non-local inverse based scaling (NIBS) and compare it with the simplified renormalization (SR). The latter is a classical technique that we adapt to compute internode transmissivities for a finite differences flow model in a direct way. NIBS is implemented in three steps: in the first step, the macroscopic transmissivity, together with arbitrarily chosen auxiliary boundary conditions and sources, is used to solve forward problems (FPs) at the macroscopic scale; in the second step, the resulting heads are sampled at the megascopic scale; in the third step, the upscaled internode transmissivities are obtained by solving an inverse problem with the differential system method (DS) for which the heads resulting from the second step are used. NIBS is a non-local technique, because the computation of the internode transmissivities relies upon the whole transmissivity field at the macroscopic scale. We test NIBS against SR in the case of synthetic, isotropic, confined aquifers under the assumptions of two-dimensional (2D) and steady-state flow; the aquifers differ for the degree of heterogeneity, which is represented by a normally distributed uncorrelated component of lnT. For the comparison, the reference heads and fluxes at the megascopic scale are computed from the solution of FPs at the macroscopic scale. These reference values are compared with the heads and the fluxes predicted from models at the megascopic scale using the upscaled parameters of SR and NIBS. For the class of aquifers considered in this paper, the results of SR are better than those of NIBS, which hints that non-local effects can be disregarded at the megascopic scale. The two techniques provide comparable results when the heterogeneity increases, when the megascopic scale is large with respect to the heterogeneity length scale, or when the source terms are relevant.