A connection between the Lagrangian stochastic–convective and cumulant expansion approaches for describing solute transport in heterogeneous porous media

The equation describing the ensemble-average solute concentration in a heterogeneous porous media can be developed from the Lagrangian (stochastic–convective) approach and from a method that uses a renormalized cumulant expansion. These two approaches are compared for the case of steady flow, and it is shown that they are related. The cumulant expansion approach can be interpreted as a series expansion of the convolution path integral that defines the ensemble-average concentration in the Lagrangian approach. The two methods can be used independently to develop the classical form for the convection–dispersion equation, and are shown to lead to identical transport equations under certain simplifying assumptions. In the development of such transport equations, the cumulant expansion does not require a priori the assumption of any particular distribution for the Lagrangian displacements or velocity field, and does not require one to approximate trajectories with their ensembleaverage. In order to obtain a second-order equation, the cumulant expansion method does require truncation of a series, but this truncation is done rationally by the development of a constraint in terms of parameters of the transport field. This constraint is less demanding than requiring that the distribution for the Lagrangian displacements be strictly Gaussian, and it indicates under what velocity field conditions a second-order transport equation is a reasonable approximation. q 1998 Elsevier Science Limited. All rights reserved