Stochastic modeling of transient stream–aquifer interaction with the nonlinear Boussinesq equation

In this article the effect of highly fluctuating stream stage on the adjacent alluvial valley aquifer is studied with a new analytical solution to the nonlinear transient groundwater flow equation subject to stochastic conductivity and time varying boundary conditions. A random conductivity field with known correlation structure represents uncertain heterogeneity. The resulting nonlinear stochastic Boussinesq equation is solved with the decomposition method. New expressions for the mean of the hydraulic head and its variance distribution are given. The procedure allows for the calculation of the mean head and error bounds in real situations when a limited sample allows the estimation of the conductivity mean and correlation structure only. Under these circumstances, the usual assumptions of a specific conductivity probability distribution, logarithmic transformation, small perturbation, discretization, or Monte Carlo simulations are not possible. The solution is verified via an application to the Scioto River aquifer in Ohio, which suffers from periodic large fluctuations in river stage from seasonal flooding. Predicted head statistics are compared with observed heads at different monitoring wells across the aquifer. Results show that the observed transient water table elevation in the observation well lies in the predicted mean plus or minus one standard deviation bounds. The magnitude of uncertainty in predicted head depends on the statistical properties of the conductivity field, as described by its coefficient of variability and its correlation length scale.