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A Stochastic Model for Colloid Transport and Deposition

^a USDA-ARS, United States Salinity Lab., 450 W. Big Springs Rd., Riverside, CA 92507-4617
^b Faculty of Bioresources, Mie Univ., 1577 Kurimamachiya-cho Tsu, Japan, 514-8507

View larger version (12K): [in this window] [in a new window]   Fig. 1. (a) Plot of the relative flux concentration, < C > /Ci, at a depth of 10 cm as a function of time when kd is stochastic and the value of d from the log-normal probability density function (Eq. [4]) is 0, 0.5, and 1.0. (b, c) Corresponding normalized solid phase colloid concentration, < S > /Nic, and associated variance after 250 min with depth, respectively. Model parameters that were employed in these simulations were D = 0.0313 cm2 min–1, v = 0.313 cm min–1, < kd > = 0.03 min–1, and kr = 0.001 min–1.

View larger version (12K): [in this window] [in a new window]   Fig. 2. (a) Plot of the relative flux concentration, < vC > /( < v > Ci), at a depth of 10 cm as a function of time when v is stochastic and the value of v from the log-normal probability density function (Eq. [4]) is 0, 0.5, and 1.0. (b, c) Corresponding normalized solid phase colloid concentration, < S > /Nic, and associated variance after 250 min with depth, respectively. Model parameters that were employed in these simulations were D = 0.0313 cm2 min–1, < v > = 0.313 cm min–1, kd = 0.03 min–1, and kr = 0.001 min–1.

View larger version (18K): [in this window] [in a new window]   Fig. 3. (a) Plot of the relative flux concentration, < C > /Ci, at a depth of 10 cm as a function of time when kd is stochastic, the values of d1 = 0.1 and d2 = 0.1 from the bimodal log-normal probability density function (Eq. [8]), and f1 is equal to 0, 0.25, 0.5, 0.75, and 1. (b) Corresponding normalized solid phase colloid concentration, < S > /Nic, after 250 min with depth. Other model parameters that were employed in these simulations were D = 0.0313 cm2 min–1, v = 0.313 cm min–1, < kd1 > = 0.015 min–1, < kd2 > = 0.3 min–1, and kr = 0.001 min–1.

View larger version (14K): [in this window] [in a new window]   Fig. 4. (a) Plot of the relative flux concentration, < C > /Ci, at a depth of 10 cm as a function of time when kd and kr are both stochastic parameters and values of dr = –1, –0.5, 0, 0.5, and 1. (b, c) Corresponding normalized solid phase colloid concentration, < S > /Nic, and associated variance after 250 min with depth, respectively. Model parameters that were employed in these simulations were D = 0.0313 cm2 min–1, v = 0.313 cm min–1, < kd > = 0.03 min–1, < kr > = 0.005 min–1, d = 1, and r = 1.

View larger version (17K): [in this window] [in a new window]   Fig. 5. (a) Plot of the relative flux concentration, < vC > /( < v > Ci), at a depth of 10 cm as a function of time when v and kd are both stochastic parameters and values of vd = –1, –0.5, 0, 0.5, and 1. (b, c) Corresponding normalized solid phase colloid concentration, < S > /Nic, and associated variance after 250 min with depth, respectively. Model parameters that were employed in these simulations were D = 0.0313 cm2 min–1, < v > = 0.313 cm min–1, < kd > = 0.03 min–1, kr = 0.001 min–1, v = 1, and d = 1.

View larger version (17K): [in this window] [in a new window]   Fig. 6. (a) Plot of the observed and simulated relative flux concentration, < C > /Ci, at a depth of 12 cm for 3 µm carboxyl modified latex colloids in quartz sands having median grain sizes of 360, 240, and 150 µm. (b) Corresponding observed and simulated normalized solid phase colloid concentration, < S > /Nic, after 250 min with depth. Simulations considered log-normal (Eq. [4]) and bimodal (Eq. [9]) formations for F(kd). (c) Log-normal values of F(kd). Table 1 provides a summary of measured and/or fitted model parameters.