Numerical simulations of transport of non-ergodic solute plumes in heterogeneous aquifers

Abstract

Transport of non-ergodic solute plumes by steady-state groundwater ow with a uniform mean velocity, l, were simulated with Monte Carlo approach in a two-dimensional heterogeneous and statistically isotropic aquifer whose transmissivity, T, is log-normally distributed with an exponential covariance. The ensemble averages of the second spatial moments of the plume about its center of mass, hS ii ti, and the plume centroid covariance, R ii t i 1Y 2, were simulated for the variance of Y log T, r 2 Y 0X1, 0.5 and 1.0 and line sources normal or parallel to l of three dimensionless lengths, 1, 5, and 10. For r 2 Y 0X1, all simulated hS ii ti À S ii 0 and R ii t agree well with the ®rst-order theoretical values, where S ii 0 are the initial values of S ii t. For r 2 Y 0X5 and 1.0 and the line sources normal to l, the simulated longitudinal moments, hS 11 ti À S 11 0 and R 11 t, agree well with the ®rst-order theoretical results but the simulated transverse moments hS 22 ti À S 22 0 and R 22 t are signi®cantly larger than the ®rst-order values. For the same two larger values of r 2 Y but the line sources parallel to l, the simulated hS 11 ti À S 11 0 are larger than but the simulated R 11 are smaller than the ®rst-order values, and both simulated hS 22 ti À S 22 0 and R 22 t stay larger than the ®rst-order values. For a ®xed value of r 2 Y , the summations of hS ii ti À S ii 0 and R ii , i.e., X ii i 1Y 2, remain almost the same no matter what kind of source simulated. The simulated X 11 are in good agreement with the ®rst-order theory but the simulated X 22 are signi®cantly larger than the ®rst-order values. The simulated X 22 , however, are in excellent agreement with a previous modeling result and both of them are very close to the values derived using Corrsin's conjecture. It is found that the transverse moments may be signi®cantly underestimated if less accurate hydraulic head solutions are used and that the decreasing of hS 22 ti À S 22 0 with time or a negative effective dispersivity, de®ned as 1 2l dhS 22 i dt , may happen in the case of a line source parallel to l where r 2 Y is small.

Numerical simulations of non-ergodic solute transport in three-dimensional heterogeneous porous media

“…5) and Zhang and Lin (1998) found that their simulated X 22 are significantly higher than the first-order solution and are very close to the values derived using Corrsin's conjecture. Most recently, Dentz et al (2002) carried out numerical simulations of the temporal behavior of a passive solute in a saturated three-dimensional heterogeneous medium and their results are compared to the second-order perturbation theory expressions derived early by the same authors (Dentz et al 2000a, b).…”

“…They (Fig. 3, Zhang and Lin 1998) showed that the simulated transverse centroid variance R 22 and thus the one-particle transverse variance X 22 are very sensitive to the value of e: the simulated R 22 increases as the value of e decreases. While the simulated Z 22 for all values of e they used are above the first-order theoretical curve, the simulated R 22 increases from below to above the first-order theoretical curve.…”

“…The source is represented by a number of particles and the number of particles in each cell is 4. In their two-dimensional simulations Zhang and Lin (1998) tested that the simulation results remain the same when a larger number of particles per cell than 5 are used. The solute plume originated from the square source is advected downstream and the particles consisting of the plume are tracked (local dispersion is neglected) with a semi-analytical method (Pollock 1994).…”

“…The iteration stops before a sufficiently close approximation to the solution is attained if a relatively larger convergence criterion is used. As demonstrated by Zhang and Lin (1998), a very small convergence criterion must be used in order to obtain accurate results for the second spatial moments, especially in transverse directions. They (Fig.…”

“…Rubin 1990; Chin and Wang 1992;Salandin and Fiorotto 1993;Burr et al 1994;Selroons and Cvetkovic 1994;Cvetkovic et al 1996;Dentz et al 2002;Hassan et al 2002) or with a small number (less than 100 to a few hundred) of realizations (e.g. Tompson and Gelhar 1990;Quinodoz and Valocchi 1990;Burr et al 1994) or in two dimensions (Zhang and Lin 1998;Hassan et al 1998Hassan et al , 2002 or addressed only the longitudinal macrodispersion (Quinodoz and Valocchi 1990;Salandin et al 1991;Salandin and Fiorotto 1993), even though our understanding about the transverse dispersion is less satisfactory than the longitudinal dispersion in both ergodic and non-ergodic conditions.…”