Numerical approximation of head and ̄ux covariances in three dimensions using mixed ®nite elements
نویسندگان
چکیده
A numerical method is developed for accurately approximating head and ̄ux covariances and cross-covariances in ®nite twoand three-dimensional domains using the mixed ®nite element method. The method is useful for determining head and ̄ux covariances for non-stationary ̄ow ®elds, for example those induced by injection or extraction wells, impermeable subsurface barriers, or non-stationary hydraulic conductivity ®elds. Because the numerical approximations to the ̄ux covariances are obtained directly from the solution to the coupled problem rather than having to dierentiate head covariances, the approximations are in general more accurate than those obtained from conventional ®nite dierence or ®nite element methods. Results for uniform ̄ow example problems are consistent with results from previously published ®nite domain analyses and demonstrate that head variances and covariances are quite sensitive to boundary conditions and the size of the bounded domain. Flux variances and covariances are less sensitive to boundary conditions and domain size. Results comparing approximations from lower-order Raviart±Thomas±Nedelec and higher order Brezzi±Douglas±Marini ®nite element spaces indicate that higher order element space improve the estimate of the ̄ux covariances, but do not signi®cantly aect the estimate of the head covariances. Ó 1999 Elsevier Science Ltd. All rights reserved
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