The upscaling of one-dimensional unsaturated soil water flow model under infiltration and evapotranspiration boundary conditions

A new stochastic modeling for one-dimensional unsaturated flow is proposed with major focus on its probabilistic structure. The newly developed model has the form of the Fokker-Planck equation, and its validity as a model for the probabilistic evolution of the nonlinear stochastic unsaturated flow process is investigated under a stochastic soil-related parameter. This model is based on a parabolic type of stochastic partial differential equation (SPDE), and has the advantage of providing the probabilistic solution in the form of probability density function, from which one can obtain the ensemble average behavior of the flow system. It is shown that the proposed model can be used in various applications such as infiltration, soil evaporation, and plant transpiration.; The developed model is based on the cumulant expansion theory. As a point scale soil water flow conservation equation, the Richards equation that is a parabolic type of highly nonlinear partial differential equation, is converted into a simplified ordinary differential equation (ODE) using a depth-integrated scheme. Then, this still nonlinear ODE is further converted into a linear PDE using the stochastic Liouville equation. When compared with Monte Carlo simulations, this model can produce a good agreement in some soil water flow applications such as infiltration, soil evaporation, and plant transpiration conditions. The comparison results with Monte Carlo simulations also shows that this upscaling model can reproduce well the vertically varied soil water wetting front depth.; Overall, the ensemble averaging approach using the cumulant expansion method shows good promise for the stochastic modeling of nonlinear hydrologic processes.