Ensemble average and ensemble variance behavior of unsteady, one-dimensional groundwater flow in unconfined, heterogeneous aquifers: an exact second-order model

Abstract

A new stochastic model for unconfined groundwater flow is proposed. The developed evolution equation for the probabilistic behavior of unconfined groundwater flow results from random variations in hydraulic conductivity, and the probabilistic description for the state variable of the nonlinear stochastic unconfined flow process becomes a mixed Eulerian–Lagrangian–Fokker–Planck equation (FPE). Furthermore, the FPE is a deterministic, linear partial differential equation (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 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 the ensemble behavior of unconfined groundwater flow, can express the spatial variability of the unconfined groundwater flow process in heterogeneous aquifers adequately. Modeling of the hydraulic head variance, as shown here, will provide a measure of confidence around the ensemble mean behavior of the hydraulic head.