Two‐dimensional stochastic analysis and optimal estimation in aquifers: Random recharge

This paper investigates two‐dimensional aquifer flow under naturally variable recharge and its application to optimal estimation in groundwater. The adopted framework is a stochastic one in which the governing stochastic equation is solved quasi‐analytically using the Galerkin finite element method and matrix exponentials. The continuous‐time stochastic solution relates head perturbations to random initial bead and a convolution of the pertinent stochastic recharge process. Based on the quasi‐analytical solution, continuous time autocorrelation matrices for aquifer head are developed conforming to (1) white noise recharge fluctuations in time and (2) fully correlated recharge fluctuations in time. From a stochastic point of view, transient aquifer flow in its conventional meaning is only inferred when recharge fluctuations are fully correlated in time, provided that the mean flow approaches steady state. Application to optimal state feedback estimation is demonstrated by adopting Kalman filtering and forecasting to a numerical experiment consisting of two‐dimensional aquifer flow instigated by variable leakage. Filtering results demonstrate that under scarce measurements, statistical conditioning on available measurements can result in an estimated aquifer hydraulic response that captures the actual variability that may exist under natural field conditions. Forecasted aquifer heads, however, are sufficient for a relatively short period; they converge asymptotically to their respective average values.