Some Analytical and Approximate Darcian Means

Abstract

In the finite‐difference modeling of unsaturated infiltration by Richards’equation, standard interblock conductivity means like the arithmetic, geometric, and harmonic have been shown to produce flows different from true Darcian by up to orders of magnitude (Warrick 1991; Baker 1995a, 1998). This may have significant impact on models used for contaminant transport of nuclear and other hazardous waste. Previously, it has been impractical to calculate Darcian means in a running model. This paper introduces the analytic solution for Darcian means for the case of an exponential relative conductivity relation, and one possible approximation for general use, a piecewise Brooks‐Corey Darcian (PBCD) mean. It compares the PBCD mean to the arithmetic mean in a modern mass‐conservative numerical model (Celia et al. 1990) for vertical space steps from 0.8 to 105 cm, using soil parameters from Haverkamp et al. (1977). The results demonstrate that even in a model with mass balance good to one part in 108, non‐Darcian flow errors are alive and well, costing up to 50% of the total mass infiltrated with the arithmetic mean and large model space steps. The new PBCD mean reduces that figure to 0.5% for this example soil.

Deficiency of Approximate Interblock Conductivities for Simulation of Horizontal Unsaturated Flow

“…Equations 7-10 are analytical solutions of k H and q H with explicit forms for the GE and BC functions, which are similar to the expressions of k H as a function of conductivities presented by Baker et al (1999). The k H and q H based on the vG function (Eqs.…”

“…A number of estimation methods have been reviewed (Haverkamp and Vauclin 1979;Schnabel and Richie 1984;Warrick 1991;Ross 1990;Zaidel and Russo 1992;Oldenburg and Pruess 1993;Srivastava and Guzman-Guzman 1995;Desbarats 1995;Romano et al 1998;Baker et al 1999;Gastó et al 2002;Brunone et al 2003;Belfort and Lehmann 2005;Baker 1995Baker , 2000Baker , 2006. By assuming constant flux between adjacent blocks, Warrick (1991) first suggested an effective interblock conductivity which was later denoted as k H for the horizontal flow and k V for the vertical flow by Baker (1995).…”

“…Desbarats (1995) developed a methodology to calculate k V for flow in heterogeneous media and represented k V as a function of alpha parameter for the constitutive Gardner relationship (Gardner 1958). Baker et al (1999) and Baker (2000Baker ( , 2006 further developed a piecewise as well as an integral approximation to k V using the Brooks-Corey (BC) function (Brooks and Corey 1966). They discussed the general validity of k V in unsaturated flow models.…”

“…Baker et al (1999) represented the k H as a function of dimensionless conductivities at two adjacent nodes for the Gardner and the BC functions. It is commonly considered that, on one hand, the horizontal flow is more straightforward to be solved than the vertical flow as the former lacks a gravity item; on the other hand, it is easier to get an explicit expression of k H than that of k V (Warrick 1991) because the k H is not a function of space step ( x).…”

A novel numerical method of high‐order accuracy for flow in unsaturated porous media

“…The exact solution for Equation (26) was obtained in References [54,55] on the basis of the van Genuchten model for the water content Equation (4) and the Brooks-Corey model for the hydraulic conductivity Equation (7). The following soil hydraulic parameters were used for this test case, assuming a homogeneous soil:˛D 1 cm, Á D 1 and n D 1, K s D 1 cm/h,  s D 0.4, and  r D 0.004.…”