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

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. The probabilistic description for the state variable of the nonlinear stochastic unconfined flow process becomes a mixed Lagrangian-Eulerian 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. Similarly, the two-dimensional Boussinesq equation is converted to a nonlinear ODE using the Lie Group theory. Next, these derived ODEs are ensemble averaged with second order cumulant expansion theory and converted into a linear, deterministic partial differential equation (PDE). This new equation is called the FPE, and describes the evolution of the probability density function (PDF) of the hydraulic head. Once a solution of this FPE is obtained, one can then obtain the ensemble average and ensemble variance behavior of the hydraulic head through an expectation operation.;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 in both one-dimension and two-dimensions. This indicates that the derived FPE, as a stochastic model of the ensemble behavior of unconfined groundwater flow, can provide encouraging results in estimating the time-space behavior of the mean and variance of the hydraulic head in heterogeneous aquifers. Modeling of the hydraulic head variance, as shown here, will provide a measure of confidence around the ensemble mean behavior of the hydraulic head.