Stochastic Analysis of the Velocity Covariance and the Displacement Covariance Tensors in Partially Saturated Heterogeneous Anisotropic Porous Formations

Considering steady state, unsaturated flow in an unbounded domain of heterogeneous formation with three-dimensional, anisotropic structure, first-order approximations of the principal and off-diagonal components of the velocity covariance tensor were derived for the general case in which the separation vector is inclined to the mean gradient vector, J, at arbitrary angles, taking into account the texture of the soil materials constituting the heterogeneous formation. The derived velocity covariance tensors were employed for modeling transport in the vadose zone for the general case in which J is inclined to the principal axes of the formation heterogeneity at arbitrary angles, by the use of a Lagrangian formulation. Results of this study suggest that under unsaturated flow, for given statistics of the formation properties and given water saturation θ, macrodispersion will diminish in formations of coarser texture when J does not coincide with the principal axes of the heterogeneous formation. In this case and in formations of coarser texture, the principal axes associated with the principal components of the dimensionless macrodispersion tensor, D′ij may be deflected in a direction opposite to that of J relative to the principal axes of the formation heterogeneity. Coarser texture may compensate for the enhanced lateral dispersion and the enhanced deflection of the principal axes of the principal components of D′ij, due to an increase in the statistical anisotropy of the heterogeneous formation. For a formation of given statistics and texture and for a given θ, the longitudinal component of D′ij decreases while its transverse components increase when J is more inclined to the longitudinal axis of the formation heterogeneity. The latter situation is pertinent to periods of redistribution between successive water applications, in which increasing capillary pressure gradients combined with spatial heterogeneity in the formation properties may increase the transverse components of J.