Laboratory Experiments and Monte Carlo Simulations to Validate a Stochastic Theory of  Tracer-  and Density-Dependent Macrodispersion
Creators: name: Koch, Manfred
name: Starke, Bettina
name: Koch, Manfred
Publisher: name: XVI International Conference on Computational Methods in Water Resources
Year: 2006

description: 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.