Modelling Water Flow and Solute Transport in Heterogeneous Unsaturated Porous Media

New results concerning flow velocity and solute spreading in an unbounded three dimensional partially saturated heterogeneous porous formation are derived. Assuming that the effective water content is a uniformly distributed constant, and dealing with the recent results of Severino and Santini (2005) on mean vertical steady flows, first order approximation of the velocity covariance, and concurrently of the resultant macrodispersion coefficients are calculated. Generally, the velocity covariance is expressed via two quadratures. These quadratures are further reduced after adopting specific (i.e. exponential) shape for the required (cross)correlation functions. Two particular formation structures which are relevant for the applications, and lead to significant simplifications of the computational aspect are also considered. It is shown that the rate at which the Fickian regime is approached is an intrinsic medium property, whereas the value of the macrodispersion coefficients is also influenced by the mean flow conditions as well as the (cross)variances σ² of the input parameters. For a medium of given anisotropy structure, the velocity variances reduce as the medium becomes drier (in mean), and it increases with σ². In order to emphasize the intrinsic nature of the velocity autocorrelation, it is shown the good agreement between our analytical results and the velocity autocorrelation as determined by Russo (1995a) when accounting for groundwater flow normal to the formation bedding. In a similar manner, the intrinsic character of attainment the Fickian regime is demonstrated by comparing the scaled longitudinal macrodispersion coefficients D₁₁(t)/D₁₁(∞) as well as the lateral displacement variance X₂₂(t)/X₂₂(∞)=X₃₃(t)/X₃₃(∞) with the same quantities derived by Russo (1995a) in the case of groundwater flow normal to the formation bedding.