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Simulating Scale-Dependent Solute Transport in Soils with the Fractional Advective–Dispersive Equation

^a USDA-ARS Hydrology Lab., Bldg. 007, Rm.104 BARC-WEST, Beltsville, MD 20705 USA
^b Desert Research Institute, 2215 Raggio Parkway, Reno, NV 89512-1095 USA

View larger version (17K): [in a new window]   Fig. 1 Observed dependencies of the solute dispersivity on distance in soils. Data sources: Porro et al., 1993 (open circle); Ellsworth et al., 1996 (open square); Yasuda et al., 1994 (open triangle); Zhang et al, 1994 (open inverted triangle); Toride et al., 1995 (open diamond); Snow et al., 1994 (filled circle); Khan and Jury, 1990 (filled square); Khan and Jury, 1990 (filled triangle); Jury, 1988 (filled inverted triangle)

View larger version (15K): [in a new window]   Fig. 2 Probability distribution functions for symmetric Lévy distributions with (a) linear scaling and (b) probability scaling. The curves differ in values of the distribution parameter

View larger version (18K): [in a new window]   Fig. 3 Comparison of measured (filled circles) and calculated with the fractional advective–dispersive equation (FADE) (solid line) and with the advective–dispersive equation (ADE) (dotted line) Cl- breakthrough curves in (a) unsaturated and (b) saturated sand. Measured values are from Toride et al. (1995)

View larger version (13K): [in a new window]   Fig. 4 Comparison of measured (filled circles) and calculated with the fractional advective–dispersive equation (FADE) (line) and with the advective–dispersive equation (ADE) (dotted line) Cl- breakthrough curves in near-saturated clay soil; water flux was (a) 0.28 cm h-1 and (b) 2.75 cm h-1. Measured values are from Dyson and White (1987)

View larger version (17K): [in a new window]   Fig. 5 Using fractional advective–dispersive equation (FADE) parameters found from the breakthrough curve at 17 cm to simulate breakthrough curves at 11 and 23 cm in unsaturated sand. Lines show the simulations, symbols show measured values from Toride et al. (1995)

View larger version (25K): [in a new window]   Fig. 6 Fitting advective–dispersive equation (ADE) to breakthrough curves generated with the fractional advective–dispersive equation (FADE). Optimized values of the ADE dispersivity Ds, (cm h-1) are shown for three depths x (cm); (a) FADE parameters for unsaturated sand, (b) FADE parameters for the clay structured soil

View larger version (16K): [in a new window]   Fig. 7 Breakthrough curves simulated using asymmetrical Lévy distributions. Values of parameters: 1, = 2, ß = 0; 2, = 1.3, ß = 0; 3, = 1.3, ß = 0.9; 4, = 1.3, ß = -0.9; = (Dfs|cos (/2)|)1/