Deficiency of Approximate Interblock Conductivities for Simulation of Horizontal Unsaturated Flow

Abstract

Determination of appropriate horizontal interblock conductivities approximations (k app ) is a critical step for numerical solution of unsaturated flow by finite difference method. This study characterized the horizontal effective interblock conductivity (k H ) and the classical k app by contour map. The deficiencies of the k app were determined by both statistical method and wetting front prediction. Mathematical expressions of the k H were redeveloped for three constitutive hydraulic functions. A large database including ten different k app formulas, three hydraulic functions, and a large range of soil texture is used to determine the deficiencies of the k app for simulation of horizontal unsaturated flow. Some k app may be either higher or lower than the k H since they depend on both the expression forms of the k app and the specific hydraulic functions. For the soil described by the Gardner function, the horizontal wetting front generated from the k H is almost equivalent to the analytical solution, but those predicted from the k app with certain errors. For the soils described by the Brooks-Corey function and the van Genuchten function, the best k app for approximating the k H were also determined, but with conversions for the soils described by the van Genuchten function. For any two k app that hold compatible medians, the k app (>k H ) better tracks the wetting front and makes mass conservation than the k app (<k H ) does. The ranking list of k app with respect to the quality of approximation k H is ultimately determined from the analysis of mass balance of the simulation based on the large database. Ten k app can be divided into four clusters. The logarithm mean and the arithmetic mean calculated from saturation can produce the best simulation results with relative mass balance error averagely less than 2%. Simulations conducted with harmonic mean calculated from hydraulic conductivity are mostly associated with the largest mass balance errors. Such a mean can therefore not be used to simulate horizontal water front advance, whatever the soil type and whatever the model of hydrodynamic function. It is also suggested that use of the classical k app in a numerical process may generate improper results, thus application of the k H is convincingly recommended on the prediction of horizontal wetting front. Relative interblock conductivity approximation k AM Keywords

On equivalent hydraulic conductivity for oscillation–free solutions of Richard’s equation

“…Then, we demonstrate that the integrated formulation remains free from oscillation in the horizontal direction. This result corroborates the conclusion of Pei et al (2012). The criterion of Forsyth and Kropinski (1997) shows that the upstream method would be the unique unconditional monotonic formulation.…”

“…Tables 4 and 5 for the integral mean, one can deduce that monotonicity is always accounted for horizontal flow processes. This fact corroborates the experimental conclusion of Pei et al (2012). For vertical discretizations, a limitation has to be added.…”

“…Previous studies consider the relation between a formulation of equivalent conductivity and the accuracy of the numerical solution. In fact, studies have been found whose authors recommended the use of the geometric mean (e.g., Haverkamp and Vauclin, 1979), arithmetic mean (van Dam and Feddes, 2000), harmonic mean (Oldenburg and Pruess, 1993), upstream mean (Oldenburg and Pruess, 1993), integrated mean (Pei et al, 2012), or more complex averages, such as the Darcian mean (Warrick, 1991;Baker et al, 1995) or optimized algorithm (Szymkiewicz, 2009). A review on averaging approaches for the computation of inter-nodal permeabilities is given in the chapter 4 of the book by Szymkiewicz (2013).…”