Ensemble Modeling of Nonlinear Hydrologic Processes under Uncertainty Using Symmetry Methods and Its Application to 2-D Unconfined Groundwater Flow in Heterogeneous Aquifers

In this paper, a new probabilistic symmetry analysis approach is proposed to the analysis of nonlinear hydrologic processes and is applied to the analysis of nonlinear two-dimensional groundwater flow subject to random hydraulic conductivity field. Symmetry methods can be used to transform almost any kind of linear or nonlinear partial differential equation (PDE) that represents a hydrologic process in any dimension to an equivalent ordinary differential equation (ODE). Following this methodology, the two-dimensional Boussinessq equation for unconfined groundwater flow in a heterogeneous aquifer is transformed to a system of ODEs through Lie Group symmetry analysis in order to eliminate the spatial derivatives occurring in the governing equation. Meanwhile, it was recently shown that the conservation equations of hydrologic processes under uncertainty, expressed as linear or nonlinear stochastic ODEs or PDEs, have a one-to-one correspondence to a mixed Eulerian-Lagrangian nonlocal form of the Fokker-Planck Equation (FPE) when the underlying process has finite correlation lengths (Kavvas 2003). Under such correspondence it is possible to obtain a solution for the ensemble behavior of a particular hydrologic process in terms of the solution of its corresponding FPE for the PDF of its state variables under appropriate initial and boundary conditions. Then, using this ensemble averaging technique the derived nonlinear stochastic ODEs are then transformed to a mixed Lagrangian-Eulerian FPE. Next, the FPE is solved numerically under the appropriate initial and boundary conditions in order to obtain the mean hydraulic head and associated uncertainty (expressed in terms of the hydraulic head variance and hydraulic head probability density function). Finally, the proposed method is tested against the Monte Carlo simulations. Comparison of the results shows that the proposed method is highly promising for estimating the time-space behavior of the mean and variance of the hydraulic head in heterogeneous aquifers.