The symmetry properties of the effective diffusivity tensor in anisotropic porous media

The effect of anisotropic grain structures on the average transport properties resulting from flow through porous media is derived using methods that involve ensemble averaging the basic conservation equations and a multiple-scale analysis. The Lagrangian or "moments" effective diffusivity tensor, defined as the long-time limit of the time rate of change of the mean-squared displacement of a tracer particle, is equivalent to the symmetric part of the Eulerian diffusivity, defined as the mass flux induced by a linear concentration gradient in the limit of long times. In an anisotropic medium there may be a component of the mass flux perpendicular to the imposed concentration gradient, resulting in an effective diffusivity tensor that is not symmetric. Such an effect is not detected by the moments or Lagrangian approach.