Laboratory Experiments and Monte Carlo Simulations to Validate a Stochastic Theory of Tracer- and Density-Dependent Macrodispersion

Monte Carlo simulations using the SUTRA density-dependent flow and transport model have been performed, in order to calibrate and validate tank experiments, as well as stochastic theory (Welty and Gelhar, 1991; 1992; Gelhar, 1993) of macrodispersion in both tracer and density-dependent flow within a heterogeneous medium. Objective of this ongoing long-term study is the analysis of the effects of the stochastic parameters of the porous medium on the steady-state macrodispersion quantified, namely, by the transversal dispersivity AT. Numerous tank experiments of hydrodynamically stable stratified flow and transport with saltwater concentrations ranging from c0 = 250 (tracer) to c0 =100000 ppm and three inflow velocities of u = 1, 4 and 8 m/day each were carried out for three stochastic, anisotropically packed sand structures with different mean Kg, variance σ2, and horizontal and vertical correlation lengths λx , λy for the permeability variations. For each flow and transport experiment carried out in one tank pack, a large number of Monte Carlo simulations with stochastic realizations taken from the corresponding statistical family (with predefined Kg, σ2, λx , λy) are simulated under steady-state conditions. From moment analyses and laterals widths of the simulated saltwater plume, variances σD2 of lateral dispersion are calculated as a function of the horizontal distance x from the tank inlet. Using simple square root regression analysis of σD2(x), an expectation value for the transversal dispersivity E(AT) is then computed which should be representative for the particular medium family and the given flow conditions. To validate a theoretical formula proposed by Welty et al. (2003) for the transversal macro- dispersivity AT, the sets of experimental as well as numerical results for AT are subject to a multiple linear regression analysis of the form y = X1 b1 + X2 b2, whereby X1 b1 and X2 b2 denote a density-independent and density-dependent factor, respectively, with X1 and X2 comprising combinations of the known (σ2, λx , λy), and b1, b2 unknown regressors of a combination of the flow factor γ and density gradient and that are to be determined. The regression of the experimentally measured values of AT show that statistically particularly reliable estimates of b1, b2 can be obtained if the flow factor γ is assumed to be constant. For the numerical Monte Carlo simulations, however, the situation is more intricate. Whereas E(AT ) for the numerical tracer cases can be reliably regressed for either a constant or a variance-dependent γ, the density- dependent regressions of E(AT) are more sensitive to the form of γ . Among the reasons for this discrepancy we expect an insufficient number of Monte Carlo realizations for the extremely time-consuming density-dependent cases to achieve an asymptotically stable E(AT) and/or the inadequacy of the classical Fick’s law implemented in the SUTRA code to simulate high-density dependent transport and, finally, an inappropriate numerical representation of the free outlet boundary condition., especially at high concentrations, Interestingly enough, for moderately high concentrations of up to c0 =35000 ppm the individual experimental results obtained for AT are reasonably well mimicked by the numerical simulations. The latter are also used to calculate the sensitivities ∂AT /∂p for the most influential flow, transport and medium parameters.