A Full-Bayesian Approach to the Estimation of Aquifer Parameters In Highly Heterogeneous Aquifers

Woodbury and Ulrych (WRR 36(8), 2000) proposed a Full-Bayesian approach to the estimation of transmissivity from hydraulic head and transmissivity measurements for two-dimensional steady-state groundwater flow. In their work Bayesian updating was used to condition prior estimates of the log-transmissivity (ln T) field with ln T measurements. Following this step, they incorporated hydraulic head measurements into the updating procedure by adopting a linearized aquifer equation. Prior probability density functions (pdfs) of the ln T field and any hyperparameters associated with its two central moments were determined from Maximum Entropy considerations. Any uncertainties in hyperparameters were removed by marginalization. Woodbury and Ulrych showed that the resolution of the transmissivity field gradually improved through the updating procedure. This approach applies to the estimation of aquifer parameters with limited measurements and appears promising for practical use. However, their algorithm only dealt with the aquifers with Dirichlet conditions and needed to be improved for more realistic hydrogeological conditions, such as Neumann conditions and sinks or sources. If, and how well, the Full-Bayesian approach works on a high variance ln T aquifer also remains an open question. These limitations potentially restrict the inversion approach in real world cases. In this work, we extend the Full-Bayesian algorithm for a two-dimensional steady-state groundwater flow system with Dirichlet and Neumann conditions, as well as sinks or sources. A first-order approximation of Taylor¡œs series for the exponential terms introduced by sinks and sources or Neumann condition is adopted. Such a treatment leads to a linear formulation between hydraulic head and ln T perturbations. Therefore, an updating procedure similar to that of Woodbury and Ulrych (2000) can be performed. This new algorithm was examined against a generic example. We found that the accuracy of the linearized approximation depends on the gradient of the mean head field and the variance of the ln T field. Smaller gradients and ln T variances yield better approximations. Adding appropriate noise to the observed data and refining the grid can improve the inversion results even for an aquifer configuration with large mean head gradient and variance of ln T field. Simulations of ln T variance of up to 9.0 were studied. This approach was employed to estimate hydraulic conductivities (K) in a highly variable carbonate aquifer. The priori pdf of the ln K field was determined from upscaling and co-kriging ln K from well tests. The ln K field was updated and conditioned on head measurements. The results showed that the conditioned ln K field yields good match between observed heads and computed heads. ACKNOWLEDGMENTS This work was performed at the University of Manitoba with support of the Southwest Research Institute and the U.S. Geologic Survey. This paper does not necessarily reflect the views of the SWRI or the USGS.