Stochastic analysis of flow and transport in unsaturated heterogeneous porous formation: Effects of variability in water saturation

First-order analysis, based on a stochastic continuum presentation of the flow and a general Lagrangian description of the transport, was used to investigate the effect of the variability in water saturation on the flow and transport in a heterogeneous, partially saturated porous formation. Results of the analyses show that for a given mean water saturation, the variability in water saturation contributes to the variability in the velocity and, concurrently, enhances solute spreading. This applies especially in formations in which (1) the length scale of the heterogeneity in the direction perpendicular to the mean flow is large compared with its counterpart in the direction parallel to the mean flow; (2) the macroscopic capillary length scale is small compared with the length scale of the heterogeneity in the direction parallel to the mean flow; (3) the variability in the soil parameter log alpha is not small compared with the variability in log K-s; (4) the fluctuations in log alpha are negatively correlated with those of log K-s; (5) mean water saturation is relatively small; and (6) the transverse components of the mean head gradient are not small compared with its longitudinal component. Another finding of the present study suggests that the longitudinal components of the velocity and the displacement covariance tensors are much more sensitive to the variability in water saturation than the respective transverse components. For the special case in which the mean flow is aligned along the vertical axis only, the aforementioned transverse components are independent of the spatial variability of water saturation and are influenced only by its mean value. Results of the analysis of the concentration spatial moments suggest that the average cloud centroid will be delayed compared with the mean velocity of the fluid. This stems from the fact that the fluid velocity in the variably saturated formation is not divergence-free and its fluctuations are negatively correlated with those of water saturation.