Necessary conditions for inverse modeling of flow through variably saturated porous media

Non-unique solutions of inverse problems arise from a lack of information that satisfies necessary conditions for the problem to be well defined. This paper investigates these conditions for inverse modeling of water flow through multi-dimensional variably saturated porous media. It shows that in order to obtain a unique estimate of hydraulic parameters, along each streamline of the flow field (1) spatial and temporal head observations must be given; (2) the number of spatial and temporal head observations required should be greater or equal to the number of unknown parameters; (3) the flux boundary condition or the pumping rate of a well must be specified for the homogeneous case and both boundary flux and pumping rate are a must for the heterogeneous case; (4) head observations must encompass both saturated and unsaturated conditions, and the functional relationships for unsaturated hydraulic conductivity/pressure head and for the moisture retention should be given, and (5) the residual water content value also need to be specified a priori or water content measurements are needed for the estimation of the saturated water content. For field problems, these necessary conditions can be collected or estimated but likely involve uncertainty. While the problems become well defined and have unique solutions, the solutions likely will be uncertain. Because of this uncertainty, stochastic approaches are deemed to be appropriate for inverse problems as they are for forward problems to address uncertainty. Nevertheless, knowledge of these necessary conditions is critical to reduce uncertainty in both characterization of the vadose zone and the aquifer, and prediction of water flow and solute migration in the subsurface.