Simple closed form formulas for predicting groundwater flow model uncertainty in complex, heterogeneous trending media

This note presents approximate, closed form formulas for predicting groundwater velocity variances caused by unmodeled small-scale heterogeneity in hydraulic conductivity. These formulas rely on a linearization of the groundwater flow equation but do not require the common statistical stationarity assumptions. The formulas are illustrated with a two-dimensional analysis of steady state flows in complex, multidimensional trending media and compared with a first-order numerical analysis and Monte Carlo simulation. This comparison indicates that despite the simplifications, the closed form formulas capture the strongly nonstationary distributions of the velocity variances and match well with the first-order numerical model and the Monte Carlo simulation except in the immediate proximity of prescribed head boundaries and mean conductivity discontinuities. The complex trending examples illustrate that the closed form formulas have many of the capabilities of a full stochastic numerical model while retaining the convenience of analytical results.