Some analytical and approximate Darcian means

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 10(8), 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.