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Stochastic Analysis of Solute Mass Flux in Gravity-Dominated Flow through Bimodal Heterogeneous Unsaturated Formations

Dep. of Environmental Physics and Irrigation Institute of Soils, Water and Environmental Sciences, Agricultural Research Organization, The Volcani Center, Bet Dagan 50250, Israel

View larger version (28K): [in a new window]   Fig. 1. Longitudinal component of the scaled velocity covariance tensor, u11 = Cu11/U12, as a function of the scaled separation distance, for selected values of mean saturation, S, the inclusions' volume fraction, P*, and the conductivities' ratio, Kr = Kg2/Kg1. Kg1 = 1 cm h–1, 1 = 0.01 cm–1, = 0.25, = 1, = 5, µ = 1, and rfa = 0.

View larger version (29K): [in a new window]   Fig. 2. Longitudinal component of the scaled velocity variance tensor, u11 = Cu11/U12, as a function of the dimensionless mean pressure head, H* = 1H, for selected values of the inclusions' volume fraction, P*, the conductivities' ratio, Kr = Kg2/Kg1, and the cross-correlation coefficient, rfa. Kg1 = 1 cm h–1, 1 = 0.01 cm–1, = 0.25, = 5, µ = 1, and = 1.

View larger version (25K): [in a new window]   Fig. 3. Mean water saturations (a) at which u11(0) reaches its minimum, Sm and (b) at which u11(0) equals its value in saturated formation, Sc, as functions of the conductivities' ratio, Kr = Kg2/Kg1, for selected values of the cross-correlation coefficient, rfa. P* = 0.2, Kg1 = 1 cm h–1, 1 = 0.01 cm–1, = 0.25, = 5, µ = 1, and = 1.

View larger version (32K): [in a new window]   Fig. 4. Scaled travel time covariance, '2TT' = 2TT'U12/yv12, as a function of scaled travel distance, x'1 = tU1/yv1, for different values of the scaled separation distance in the transverse direction, ß' = ß/yv1, for selected values of mean saturation, S, and the conductivities' ratio, Kr = Kg2/Kg1. P* = 0.2, Kg1 = 1 cm h–1, 1 = 0.01 cm–1, = 0.25, = 5, µ = 1, = 1, and rfa = 0. Note that the scaled travel time variance, '2T = 2TU12/yv12 is given by the curve labeled by ß' = 0.

View larger version (32K): [in a new window]   Fig. 5. Scaled travel time variance, '2T = 2TU12/yv12, as a function of scaled travel distance, x'1 = tU1/yv1, for different values of the mean water saturation, S, for selected values of the inclusions' volume fraction, P*, and the conductivities' ratio, Kr = Kg2/Kg1. Kg1 = 1 cm h–1, 1 = 0.01 cm–1, = 0.25, = 5, µ = 1, = 1, and rfa = 0.

View larger version (33K): [in a new window]   Fig. 6. Scaled expected solute discharge, D' = ZD/M0yv1, crossing a control plane (CP) at a scaled vertical distance, = Z/yv1 = 10 from the injection zone, as a function of scaled travel time, t' = tU1/Z, for different values of the mean water saturation, S, for selected values of the inclusions' volume fraction, P*, and the conductivities' ratio, Kr = Kg2/Kg1. Kg1 = 1 cm h–1, 1 = 0.01 cm–1, = 0.25, = 1, = 5, µ = 1, and rfa = 0.

View larger version (34K): [in a new window]   Fig. 7. Scaled variance of solute discharge, '2d = Z22d/2, crossing a CP located at a scaled vertical distance, = Z/Iyv = 10 from the injection zone, as a function of scaled travel time, t' = tU1/Z, for different values of the scaled side length, = L/yv1 of a square injection area for selected values of mean water saturation, S, and the conductivities' ratio, Kr = Kg2/Kg1. P* = 0.2, Kg1 = 1 cm h–1, 1 = 0.01 cm–1, = 0.25, = 1, = 5, µ = 1, and rfa = 0.

View larger version (27K): [in a new window]   Fig. 8. Scaled expected solute discharge, D' = ZD/M0yv1, crossing a control plane (CP) at a scaled vertical distance, = Z/yv1 = 10, from the injection zone, as a function of scaled travel time, t' = tU1/Z, for different values of the mean water saturation, S, for (a) unimodal (the background soil) and (b) bimodal formations. Kg1 = 1 cm h–1, 1 = 0.01 cm–1, = 0.25, and rfa = 0. For the bimodal formation, = 1, = 5, µ = 1, Kr = Kg2/Kg1 = 0.05, and P* = 0.1.

View larger version (28K): [in a new window]   Fig. 9. The probability, Pm, of observing a scaled accumulated mass, m/m0, smaller than or equal to a critical value, mc/M0, crossing a CP at a scaled vertical distance, = Z/yv1 = 10, from the planar L x L square source ( = L/yv1 = 1), as a function of mc/m0. Pm is depicted for different amounts of water equivalent to t' pore volumes that crossed the CP (denoted by the numbers labeling the curves), for the unimodal and the bimodal formations and for two different mean water saturations, (a) S = 1 and (b) S = 0.7. Kg1 = 1 cm h–1, 1 = 0.01 cm–1, = 0.25, and rfa = 0. For the bimodal formation, = 1, = 5, µ = 1, Kr = Kg2/Kg1 = 0.05 and P* = 0.1.