The Stochastic Upscaling of One-Dimensional Unconfined Groundwater Flow

A new upscaling model for unconfined groundwater flow is proposed from a point-location-scale equation to a scale of computational grid areas. The developed evolution equation for the probabilistic behavior results from random variations in hydraulic conductivity, and the nonlinear stochastic unconfined flow process becomes a mixed Lagrangian-Eulerian Fokker-Planck equation (FPE). Furthermore, the FPE is a deterministic, linear PDE and has the advantage of providing the probabilistic solution in the form of evolutionary probability density functions. Subsequently, the Boussinesq equation for one-dimensional unconfined groundwater flow is converted into a nonlinear ordinary differential equation (ODE) and a two-point boundary value problem through the Boltzmann transformation. The resulting nonlinear ODE is converted to the FPE by means of master key ensemble average conservation equations. The numerical solutions of the FPE are validated with Monte Carlo simulations under varying stochastic hydraulic conductivity fields. Results from the model application to groundwater flow in heterogeneous unconfined aquifers illustrate that the time-space behavior of the mean and variance of the hydraulic head are in good agreement for both the stochastic model and the Monte Carlo solutions. This indicates that the derived FPE, as a stochastic model of unconfined groundwater flow, can express the spatial variability of the unconfined groundwater flow process in heterogeneous aquifers adequately.