Richards Equation Inversion

Fluid flow in the vadose zone is governed by Richards equation and parameterized by hydraulic conductivity, which is a non-linear function of pressure head. Investigations in the vadose zone typically require identifications of distributed hydraulic properties. This is increasingly being done through a data-driven approach, using changes in saturation or pressure head to infer hydraulic parameters. Hydrogeophysics allows many more proxy measurements to be taken to help characterize these sites spatially, as well as through time. As such, the number of distributed hydraulic parameters to be estimated in a Richards equation inversion will continue to grow. In this paper, we will describe an efficient technique to invert for hydraulic properties in 3D given a realistic data set. Our algorithm does not store the Jacobian, but rather computes the effect on a vector allowing the size of the problem that we can invert to become much larger. We show our algorithm in practice for a large 3D inversion of saturated hydraulic conductivity using saturation data through time. The code to run our examples is open source, and the algorithm presented allows this inversion process to run on modest computational resources.