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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.

The true Darcian interblock conductivity mean for vertical unsaturated flow, Kv, is derived from Darcy's law, q/Ks = Kv (dH/dx), and the assumption that flow is constant between grid centers. It can be expressed as Kv = (Δx Kx −Δp Kh)/(Δx −Δp), where Kx is the gravity flow mean, Kh is the capillary flow mean, and H =×− p (total head = elevation ‐matric suction). Plotting contours of Kx, Kh, and Kv in the plane of adjacent grid center conductivities, (kr1, kr2), shows that a component of Kx behaves like a simple weighted mean, Kw = w kr2 + (1 − w) krl, where w is a nonlinear, monotonic function of the grid separation, Δx. Results from a mass‐conservative model of infiltration into dry sand are compared for substituting: (1) the arithmetic mean, Ka = (krl + kr2)/2, for Kv, (2) Kw for Kv, (3) Ka for Kx in Kv (called Km), and (4) Kw for Kx in Kv (called Kmw), over a range of the ratio of Δx to the conductivity displacement pressure, pk, from 0.01 to 1.6. The simulation for Δx/pk = 0.002 is taken to be the exact solution. Kw and Kmw reduce the gross cumulative net inflow error generated with Ka and Km from 3.84 to 8.84 times. Kmw performs consistently better in terms of gross water content distribution error by up to a factor of three.

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