Spatial moments analysis of kinetically sorbing solutes in aquifer with bimodal permeability distribution

Abstract

We analyze the influence of the formation structure upon transport of nonreactive and kinetically sorbing solutes in heterogeneous porous formations. Formation structure is represented through a bimodal distribution of hydraulic conductivity with two scales of spatial variability, which is a simplified, yet realistic, model of spatial variability. Kinetic sorption is modeled using a linear reversible rate expression. We derive new closed‐form solutions of the second ergodic spatial moments of a kinetically sorbing solute. Although these solutions are obtained through a perturbation expansion in term of logconductivity variance, they are not limited to any particular model of spatial variability. We show that the formation structure, as described by the bimodal model, has a significant impact on plume moments of a non‐reactive tracer at moderate contrast between mean hydraulic conductivity of the two hydrofacies, when the two scales of variability have comparable weight. On the contrary, the impact diminishes when bimodality weakens at both low and high contrast. As expected, the large time asymptotic longitudinal macrodispersion coefficient is influenced by the structure only through the variance and integral scale of log‐conductivity, and thus the result for the bimodal model is identical to that of a unimodal Gaussian field sharing the same second‐order statistics. The impact of kinetic sorption depends upon the plume travel time since injection. At early times, sorption kinetics tends to accentuate the difference between the longitudinal macrodispersivity in bimodal and unimodal fields, while at later times the difference is minimal because overall macrodispersivity is dominated by the spreading caused by nonequilibrium sorption. In a bimodal formation transverse macrodispersivity peaks later and declines to zero slower than in a Gaussian unimodal field with the same second‐order statistics. These results demonstrate that even a very simplified facies model, such as the bimodal model relying on a statistical distribution of high‐ and low‐conductivity zones, has important impacts upon global parameters describing the migration of nonreactive and reactive tracers. Consequently, larger effects are expected for quantities such as early arrival times and BTC tailing which are more directly influenced by the spatial patterns of high hydraulic conductivity than global parameters, such as macrodispersivity. Furthermore, these solutions contribute to bridge the gap between stochastic modeling and applications, since bimodal fields can be obtained to honor soft data typically available in applications, for example through conditional simulations using stratigraphic information from well logs.