Ensemble modeling of the two-dimensional stochastic confined groundwater flow through the evolution of the hydraulic head’s probability density function

Abstract

Groundwater storage in aquifers has become a vital water source due to water scarcity in recent years. However, aquifer systems are full of uncertainties, which inevitably propagate throughout the modeling computations, mainly reducing the reliability of the model output. The fundamental science problem this study addresses is the development of a two-dimensional stochastic confined groundwater flow model, which determines the time–space evolution of the ensemble mean and ensemble variance of the flow field over a model domain under uncertain parameters and uncertain sink/source conditions. This is achieved by linking the stochastic partial differential equation that governs the confined aquifer flow to a non-local Lagrangian-Eulerian extension to the Fokker-Planck equation (LEFPE). In the form of the LEFPE, the resulting deterministic governing equation describes the spatio-temporal evolution of the probability density function of the state variables in the confined groundwater flow process by one single numerical realization instead of requiring thousands of simulations in the Monte Carlo approach. As will be shown in the paper’s text, the time–space evolving ensemble mean and ensemble variance of the flow process are then obtained from the pdf of the state variable (hydraulic head) of the process that is determined from the solution of the LEFPE of the process under specified initial and boundary conditions. As such, in the developed methodology, no assumption is made on the distribution of the time–space varying pdf of the flow process, which is obtained from the solution of the LEFPE of the process under specified initial and boundary conditions. Consequently, the groundwater flow process’s mean and standard deviation behavior can be modeled under uncertainty in the transmissivity field and recharge and/or pumping conditions. In addition, an appropriate numerical method for LEFPE’s solution is subsequently devised. Then, its solution is presented, discussed, and illustrated through a numerical example, which is compared against the results obtained by means of the Monte Carlo simulations. Results suggest that the proposed model appropriately characterizes the time–space evolving ensemble mean and variance the ensemble behavior in confined groundwater systems under uncertainty in the transmissivity field.