Effective Dispersion of a Solute Cloud in a Chemically Heterogeneous Porous Medium: Comparison of Two Ensemble‐Averaging Procedures

Abstract

We consider solute transport in a porous medium for which we assume the retardation factor R(x), resulting from linear chemical adsorption, to be stochastically varying in space. For large times, the evolution of a solute plume developing from a pointlike, instantaneous solute injection is described by its effective velocity and dispersion. We calculate such quantities using perturbation theory and two different averaging procedures. The first and correct procedure calculates the central moments of the cloud for a given aquifer realization and averages over the ensemble afterward. The second method, which is mathematically less demanding, obtains large‐scale transport coefficients from the moments of the ensemble‐averaged concentration distribution. This last approach is often used to replace the correct procedure, tacitly assuming that the two averaging methods lead to the same effective quantities. We show that the results actually differ in one dimension, whereas the difference vanishes in higher dimensions. The effective retardation factor is found to be the ensemble average of the corresponding small‐scale quantity. The effective dispersion coefficient, on the other hand, differs from the retarded small‐scale dispersion coefficient. It is significantly enhanced by the inhomogeneous fluctuations of the disordered medium.