Two‐scale analytical homogenization of Richards' equation for flows through block inclusions

Abstract

[1] In this paper we propose an analytical form for the upscaled coefficient applicable to the nonlinear Richards' equation. Block inclusions are symmetrically centered in a grid cell, and flow is dominated by capillary forces on the small scale. The nonlinear boundary value problems can be defined in a three-dimensional bounded domain, with a periodic and rapidly oscillating saturated hydraulic conductivity coefficient. The new result is derived by applying a corrector to an analytical approximation of the well-known cell problem obtained by the two-scale asymptotic expansion to the original heterogeneous nonlinear problem. The previously known analytical results for the upscaled coefficient, including the geometric average for the checkerboard geometry, can be regarded as particular cases of this new form. We perform numerical simulations to obtain the error between the fine-scale solution and the upscaled solution, respectively, as well as convergence properties of the approximation, corroborating results in the literature. There is no limitation regarding either the ratio of permeability between the matrix and inclusion or its shape for both computing the upscaled coefficient and obtaining the upscaled equation. This is illustrated by the comparison with known results and by numerically demonstrating convergence properties with a medium having square and circular inclusions in a main matrix with heterogeneity ratio of 1:100 and 1:10. Even though the derivation and applications are presented for single phase and steady state, the results are valid for multiphase and transient flow conditions. We also show that the approximation is independent of the relative saturated hydraulic relationship.

Determining the influence of stones on hydraulic conductivity of saturated soils using numerical method

“…HYDRUS-2D is a computer model that solves numerically highly nonlinear Richards equation for a selected set of soil hydraulic functions. Note that the Richards equation can be linearized using the Kirchhoff transformation with an exponential conductivity function and then analytical solutions can be derived for relatively simple geometries and homogeneous (e.g., Kacimov, 2000;Philip, 1968;Raats, 1971) or heterogeneous (Bakker & Nieber, 2004;Sviercoski et al, 2009;Warrick and Knight, 2002) soils. For example, Warrick and Knight (2002) presented an analytic element solution for a single embedded cylindrical inhomogeneity having a different saturated hydraulic conductivity than the surrounding uniform soil.…”

Homogenization of Transport Processes and Hydration Phenomena in Fresh Concrete

“…In [20], the authors proposed an analytical approximation of the solution to the periodic cell problem (17) with a step function A defined by (18). It can be easily shown that the functions (22) satisfy the equation (17) almost everywhere in Y, see [20,Theorem 2.1]. The proposed solution can be seen as an approximation to the solution of (19) in L 2 (Y).…”

“…The homogenization method is one of the most advanced techniques in upscaling the response of the microstructure of heterogeneous materials, e.g. [19][20][21][22][23][24][25][26][27][28]. In this method, the solution of a fine scale problem is used to examine the local material behaviour at the macroscale.…”

Application and Validation of an Upscaling Method for Unsaturated Water Flow Processes in Heterogeneous Soils

“…Toward upscaling soil water processes in the vadose zone, various methodologies for deriving the macroscopic description of the flow process have been developed based on different physical and mathematical considerations such as the stochastic perturbation method, volume averaging methods, and homogenization methods. The homogenization methods have recently been applied to the Richards equation at the laboratory scale within the strictly capillary‐dominated flow regime in soils composed of several distinct materials (Lewandowska and Laurent, 2001; Lewandowska et al, 2004; Szymkiewicz and Lewandowska, 2006; Sviercoski et al, 2009) and have also been combined with ensemble averaging techniques in a statistical framework to model the water flow problem for continuously changing stochastic fields (Neuweiler and Cirpka, 2005; Neuweiler and Eichel, 2006). A more extensive overview of upscaling hydraulic properties and soil water flow processes has been provided by Vereecken et al (2007).…”