Symmetry in Nonlinear Hydrologic Dynamics under Uncertainty: Ensemble Modeling of 2D Boussinesq Equation for Unsteady Flow in Heterogeneous Aquifers

Abstract

In this paper, the probabilistic symmetry analysis approach proposed in the previous paper by Cayar and Kavvas in 2009 is adopted and applied to the analysis of nonlinear two-dimensional (2D) groundwater flow subject to random hydraulic conductivity field. The resulting model has the form of a two-dimensional Fokker-Planck equation (FPE). Following the methodology reported by Cayar and Kavvas in 2009, the 2D Boussinessq equation for unconfined groundwater flow in a heterogeneous aquifer is transformed to a system of ordinary differential equations (ODEs) through Lie group symmetry analysis in order to eliminate the spatial derivatives occurring in the governing equation. Next, these derived stochastic ODEs are ensemble averaged with second-order cumulant expansion and converted into a linear, deterministic partial differential equation. This new equation is called the FPE, and describes the evolution of the probability density function (PDF) of the hydraulic head, thereby, describing the ensemble behavior of unconfined aquifer flow in a heterogeneous medium. 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 the Monte Carlo simulations. The validation results indicate that the FPE technique combined with Lie symmetry analysis can provide encouraging results in estimating the time-space behavior of the mean and variance of the hydraulic head in heterogeneous aquifers.