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 hete rogeneous porous media can be developed from the Lagrangian (stochastic-con vective) approach and from a method that uses a renormalized cumulant expan sion. 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 int erpreted as a series expansion of the convolution path integral that define s the ensemble-average concentration in the La,Lagrangian approach. The two methods can be used independently to develop the classical form for the co nvection-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 wit h their ensemble-average. In order to obtain a second-order equation, the c umulant expansion method does require truncation of a series, but this trun cation is done rationally by the development of a constraint in terms of pa rameters of the transport field. This constraint is less demanding than req uiring that the distribution for the Lagrangian displacements be strictly G aussian, and it indicates under what velocity field conditions a second-ord er transport equation is a reasonable approximation. (C) 1998 Elsevier Scie nce Limited. All rights reserved.