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Water Retention Models for Scale-Variant and Scale-Invariant Drainage of Mass Prefractal Porous Media

^a Biosystems Engineering and Soil Science, Univ. of Tennessee, 2506 E.J. Chapman Dr., Knoxville, TN 37996
^b Dep. of Earth and Planetary Sciences, Univ. of Tennessee, Knoxville, TN 37996-1410

View larger version (27K): [in this window] [in a new window]   FIG. 1. Example realizations for (a) general scale-variant drainage model, (b) simplified case 1, and (c) simplified case 2 in a b = 3, j = 2, D = 1.771... random Sierpinski carpet with Dd = 1.630..., and P = 0.5 (black = solid, white = air-filled pore, and gray = water-filled pore). (b = scale factor; j = the last iteration level; D = mass fractal dimension; Dd = fractal dimension for the drained pore space; P = ratio of the drained pore space to the total pore space in the generator.)

View larger version (22K): [in this window] [in a new window]   FIG. 2. Sensitivity of the general scale-variant drainage model (GM) to the parameters P and Dd. (S = relative saturation; b = scale factor; E = Euclidean dimension; i = iteration level; D = mass fractal dimension; Dd = fractal dimension for the drained pore space; h/hmin = normalized capillary pressure; P = ratio of the drained pore space to the total pore space in the generator.)

View larger version (34K): [in this window] [in a new window]   FIG. 3. Comparison of the different water retention models for the realizations giving the maximum difference in error sum of squares (ESS) between the general scale-variant drainage model (GM) and simplified model 1 (SC1) (a, c, and e) and between GM and SC2 (b, d, and f) for D = 1.630..., D = 1.771..., and D = 1.892... (b = 3, j = 5). (S = relative saturation; D = mass fractal dimension; b = scale factor; j = the last iteration level; h/hmin = normalized capillary pressure; R&S = Rieu and Sposito.)

View larger version (11K): [in this window] [in a new window]   FIG. 4. Mean values of Pi calculated inversely from Eq. [16] as function of suction level. The error bars indicate one standard deviation around the mean. (D = mass fractal dimension; h/hmin = normalized capillary pressure; Pi = probability of drainage.)

View larger version (12K): [in this window] [in a new window]   FIG. 5. Comparison of predicted relationships (lines) by substituting mean parameter estimates from Table 2 into Eq. [3] with the observed values of Nd (circles) calculated inversely from the simulated data. The error bars indicate one standard deviation around the mean. (Nd = number of drained pores; D = mass fractal dimension; b = scale factor.)