An inverse model for three-dimensional flow in variably saturated porous media

Geostatistical theory has shown promise in dealing with issues of stability , uniqueness, and identity of estimates inherent in inverse problems of sub surface flow. Here the geostatistical method is extended to three-dimension al, unsteady flow in variably saturated porous geological media (the vadose zone) that are modeled using the Richards equation and the van Genuchten-M ualem constitutive equations. The saturated hydraulic conductivity, alpha, and n parameters of this relationship are treated as spatially correlated, statistically independent, stochastic processes for representing heterogene ity of porous media. For given covariance functions of the parameters the a djoint-state sensitivity method is used to calculate first-order approximat ions for covariances of capillary pressure and moisture content and cross c ovariances between capillary pressure, moisture content, and the hydraulic properties. These covariances and cross covariances are then used in a succ essive linear estimator (SLE) to estimate the conditional means of the hete rogeneous hydraulic property fields based on measurements of pressure and m oisture content data. A sequential conditioning approach for our SLE was al so applied to data sets collected at different sampling times during a tran sient infiltration event. This approach has the benefit of reducing the siz e of the matrices and so helps avoid numerical stability problems. On the b asis of our study, pressure and moisture content data sets collected at lat er times of an infiltration event or during steady state flow were found to provide better estimates (smaller mean-square error compared to the true f ield) of the hydrological parameters of the vadose zone than data from very early times.