The dimensional character of permeability: Dimensionless groups that govern Darcy's flow in anisotropic porous media

Abstract

The dimensional character of permeability in anisotropic porous media, that is, its dimension or dimensional equation, is an information that allows setting the dimensionless groups that govern the solution of the flow equation in terms of hydraulic potential patterns. However, employing the dimensional basis {L, M, T} (length, mass, time), the dimensionless groups containing the anisotropic permeability do not behave as independent monomials that rule the solutions. In this work, the contributions appearing in the literature on the dimensional character of permeability are discussed and a new approach based on discriminated and general dimensional analysis is presented. This approach leads to the emergence of a new and accurate dimensionless group, normalknormalxnormalknormalynormallnormaly∗2normallnormalx∗2$\frac{{{{\rm{k}}_{\rm{x}}}}}{{{{\rm{k}}_{\rm{y}}}}}\frac{{{\rm{l}}_{\rm{y}}^{{\rm{*}}2}}}{{{\rm{l}}_{\rm{x}}^{{\rm{*}}2}}}$, a ratio of permeabilities corrected by the squared value of an aspect factor, being normallnormalx*${\mathrm{l}}_{\mathrm{x}}^{\ast}$and normallnormaly∗${\rm{l}}_{\rm{y}}^{\rm{*}}$ two arbitrary lengths of the domain in the directions that are indicated in their subscripts. Specific values of this lengths, which we name ‘hidden characteristic lengths’, are also discussed in this article. To check the validity of this dimensionless group, numerical simulations of two illustrative 2‐D seepage scenarios have been solved.

A Universal Graphical Solution to Calculating Seepage in Excavation of Anisotropic Soils and Non-Limited Scenarios

“…A general description of the procedure can be found in [25]. The contribution of this approach to Darcy's flow in anisotropic porous media was described in [26]. The dimensionless groups obtained are summarized in Table 1.…”