On the estimation of parameters of log-unsaturated conductivity covariance from solute transport data

Abstract

An indirect approach to the estimation of parameters of log-unsaturated conductivity (log K) covariance may be based on an inverse problem methodology, employing measurements of the response of the transport domain. In the present investigation, we analyzed the feasibility of estimating parameters of the log K covariance by inversion of an approximate solution of the transport problem under unsaturated flow conditions, employing the spatial moments of the plume of a tracer solute, at various travel times, as input data. The study focused on the question of circumstances under which the resulting inverse problem is well posed. Results of the analyses suggest that inclusion of prior estimates of the covariance parameters in the estimation criterion enhances the likelihood of uniqueness and stability of the inverse solution. The sufficient condition for uniqueness and stability, the convexity of the estimation criterion in the parameter space, however, cannot be guaranteed in all cases. In recognition of the importance of prior information on the uniqueness and stability of the inverse solution, a theoretical framework for the determination of prior estimates of the parameters from the small and large time behavior of the plume is described in this paper. An example involving simulated transport of a pulse of a tracer solute by a quasi-steady-state, unsaturated flow was used for illustration purposes. Results of the analyses of this example show that the inclusion of a prior estimate of a single parameter, the anisotropy ratio, e (that may be obtained from the small-time behavior of the plume) in the estimation criterion, may significantly reduce the uncertainty about the covariance parameters, as compared with the case in which no prior information on the parameters was considered. Controlled field-scale experiments and Monte-Carlo simulations are required, in order to test the findings of this study further before practical tools for the solution of the estimation problem can be provided.