Dispersion in heterogeneous porous media: 1. Local volume averaging and large‐scale averaging

Abstract

The problem of diffusion and dispersion in porous media is considered in the absence of either adsorption or chemical reaction. For this situation, local volume averaging has been used to obtain both a convection‐dispersion equation and a method of closure that provides for a direct theoretical prediction of the dispersion tensor. This approach is associated with three length scales, lβ, the pore diameter, ro, the radius of the averaging volume, and L the macroscopic length scale, and makes use of a representative unit cell to determine the dispersion tensor. Previous work suggests that extremely complex unit cells must be used to capture the observed dispersion phenomena, and complex unit cells naturally introduce another length scale, i.e., the length scale associated with the local heterogeneities. To deal with this problem, we propose the use of large‐scale averaging to account for the influence of the local heterogeneities. This leads to a large‐scale convection‐dispersion equation that contains additional terms involving time derivatives which arise as a result of the presence of heterogeneities and which do not appear in the conventional local volume average convection‐dispersion equation. The first new term contains the second derivative with respect to time and is likely to be unimportant for time scales appropriate for groundwater transport processes; however, it may be important for small‐scale laboratory experiments. The second new term contains a mixed space‐time derivative which may be important for many cases of practical interest. A closure scheme is presented that allows for the theoretical determination of all the coefficients that appear in the large‐scale convective‐convection‐dispersion equation.