Conditioning nonlocal steady-state flow on hydraulic head and conductivity through geostatistical inversion

Nonlocal moment equations allow one to render optimum predictions of flow in randomly heterogeneous media deterministically conditional on measured values of medium properties and to assess the corresponding predictive uncertainty. I present a geostatistical inverse algorithm for steady-state flow that makes it possible to further condition such predictions and assessments on measured values of hydraulic head and (or) flux. My algorithm is based on recursive finite-element approximations of exact first and second conditional moment equations. Computational efficiency is enhanced through the use of a direct sparse matrix solver. Hydraulic conductivity is parameterized via universal kriging based on unknown values at pilot points and (optionally) measured values at other discrete locations. Correlation among parameter estimates (or priors) is considered in the universal kriging equations. Optimum unbiased inverse estimates of natural log hydraulic conductivity, head and flux are obtained by minimizing a calibration criterion, composed of residuals of head or (and) flux and (possibly) log conductivity, using the Levenberg-Marquardt algorithm. Statistical parameters characterizing the natural variability of hydraulic conductivity can also be estimated using this algorithm. I illustrate the method for superimposed mean uniform and convergent flows in a bounded two-dimensional domain under various conditions for a range of parameters. My examples illustrate how conductivity and head data act separately or jointly to reduce parameter estimation errors and model predictive uncertainty. Over-parameterization is seen to create zones of high mean conductivity, in which flux prediction is more uncertain than is in other regions. It is found that a regular distribution of pilot points works better than does an irregular layout and that the number of pilot points should be as close as possible to the number of head data while maintaining parameters reasonably uncorrelated. Head and flux predictions are very satisfactory for cases with either log conductivity variance or integral scale between one and four, though prediction quality deteriorates with either larger variances or shorter integral scales. The method may perform satisfactorily in cases with no conductivity measurements and only a few head data.