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. This study develops a novel two-dimensional stochastic confined groundwater flow model. The proposed model is developed by linking the stochastic governing partial differential equations by means of their one-to- one correspondence to the nonlocal 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. Consequently, the ensemble 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 ensemble behavior in confined groundwater systems under uncertainty in the transmissivity field.