Stochastic analysis of unsaturated steady flows above the water table

Steady flow takes place into a three-dimensional partially saturated porous medium where, due to their spatial variability, the saturated conductivity Ks, and the relative conductivity Kr are modeled as random space functions (RSF)s. As a consequence, the flow variables (FVs), i.e., pressure-head and specific flux, are also RSFs. The focus of the present paper consists into quantifying the uncertainty of the FVs above the water table. The simple expressions (most of which in closed form) of the second-order moments pertaining to the FVs allow one to follow the transitional behavior from the zone close to the water table (where the FVs are nonstationary), till to their far-field limit (where the FVs become stationary RSFs). In particular, it is shown how the stationary limits (and the distance from the water table at which stationarity is attained) depend upon the statistical structure of the RSFs Ks, Kr, and the infiltrating rate. The mean pressure head hWi has been also computed, and it is expressed as hWi5W0ð11wÞ, being w a characteristic heterogeneity function which modifies the zero-order approximation W0 of the pressure head (valid for a vadose zone of uniform soil properties) to account for the spatial variability of Ks and Kr. Two asymptotic limits, i.e., close (near field) and away (far field) from the water table, are derived into a very general manner, whereas the transitional behavior of w between the near/far field can be determined after specifying the shape of the various input soil properties. Besides the theoretical interest, results of the present paper are useful for practical purposes, as well. Indeed, the model is tested against to real data, and in particular it is shown how it is possible for the specific case study to grasp the behavior of the FVs within an environment (i.e., the vadose zone close to the water table) which is generally very difficult to access by direct inspection.