Upscaling of vertical unsaturated flow model under infiltration condition

A new stochastic model for one-dimensional unsaturated water flow is proposed with focus on its probabilistic structure resulting from random variations in saturated hydraulic conductivity. The newly developed model has the form of a Fokker-Planck equation, and its validity is investigated under different stochastic saturated hydraulic conductivity fields. The point-scale Richards equation for soil water flow, which is a parabolic nonlinear partial differential equation (PDE), is first converted into a simplified ordinary differential equation (ODE) using a depth-integrated scheme. This nonlinear ODE is further converted into a linear PDE using the stochastic Liouville equation. Finally, the upscale conservation equation is obtained using the cumulant expansion method. When compared with Monte Carlo simulations, this model yields good agreements. In particular, this upscale model can reproduce the vertical profile of mean soil-water content very well. Besides, the results from model application show that the proposed model can explain the role change between rainfall intensity (system input) and variability in saturated hydraulic conductivity (system parameter) in affecting the spatial variability of soil-water field as infiltration proceeds.