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 h eterogeneous formation with three-dimensional, anisotropic structure, first-order approximations of the principal and off-diagonal component s of the velocity covariance tenser 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 veloci ty covariance tensors were employed for modeling transport in the vado se zone for the general case in which J is inclined to the principal a xes of the formation heterogeneity at arbitrary angles, by the use of a Lagrangian formulation. Results of this study suggest that under uns aturated flow, for given statistics of the formation properties and gi ven water saturation theta, macrodispersion will diminish in formation s of coarser texture when J does not coincide with the principal axes of the heterogeneous formation. In this case and in formations of coar ser texture, the principal axes associated with the principal componen ts of the dimensionless macrodispersion tenser, D'(ij), may be deflect ed in a direction opposite to that of J relative to the principal of J relative to the principal axes of the formation heterogeneity. Coarse r texture may compensate for the enhanced lateral dispersion and the e nhanced deflection of the principal axes of the principal components o f D'(ij), due to an increase in the statistical anisotropy of the hete rogeneous formation. For a formation of given statistics and texture a nd for a given theta, the longitudinal component of D'(ij) decreases w hile its transverse components increase when J is more inclined to the longitudinal axis of the formation heterogeneity. The latter situatio n is pertinent to periods of redistribution between successive water a pplications, in which increasing capillary pressure gradients combined with spatial heterogeneity in the formation properties may increase t he transverse components of J.