Stochastic models of subsurface flow: infinite versus finite domains and stationarity

Stochastic solutions of the differential equation describing one‐dimensional flow through a porous medium with spatially variable hydraulic conductivity which is represented by a stationary (statistically homogeneous) random process are developed using several techniques. The analytical approximations using first‐order analysis (propagation of error), covariance differential equations, and variogram analysis all yield consistent results which demonstrate the important effects of boundary conditions and conditioning. Using the theory of intrinsic random functions, stochastic solutions are developed for the case when the logarithm of the hydraulic conductivity is a three‐dimensional stationary random field. In contrast to the one‐dimensional case, it is found that the resulting three‐dimensional head perturbation will be locally stationary under very general conditions. Results from the one‐dimensional analytical solutions are found to be in agreement with previous Monte Carlo simulations for a flow system of finite length. The solution based on linearization in the logarithm of hydraulic conductivity proved to be very robust, showing reasonable agreement with Monte Carlo results even for the largest input standard deviation of σ lnK = 4.6.