Identification of hydrogeological systems via inverse procedures

Abstract

This thesis is concerned with three topics. The first two comprise the development of a new model of salinity transport in a stream connected to a salty aquifer and the derivation of a new method for the identification of transmissivity in an aquifer. A third topic interwoven with the first two is the analysis of ill-posedness often associated with the identification of hydrological systems. The salinity transport model is derived from basic mass conservation equations and from the representation of the stream-aquifer interaction by a convolution integral. In order to restrict model complexity to a level compatible both with the available data and with the objectives of the modelling exercise, several simplifying assumptions are invoked and result in a model linear in the parameters. The identification procedure is performed via recursive instrumental variable techniques. The ill-posedness associated with the deconvolution of the stream-aquifer system is addressed through the choice of a low level of parameterization for the kernel of the convolution. The salinity transport model is tested on a 207 km stretch of the River Murray. The identification of transmissivity in a confined and steady aquifer is carried out via the use of a direct approach based on a weak formulation of the aquifer flow equation. The approach leads to a natural discretization by a Galerkin method via the use of spectral expansions for the piezometric head, the sink/source flow and the transmissivity over subregions of the aquifer. The ill-posedness of the problem is counteracted by inclusion of a smoothing constraint involving the linearized curvature of the unknown transmissivity. The computation of the regularizing parameter is perfomed by generalized cross-validation. The method provides guides to assess the amount of information truly present in the data. In addition, it is amenable to error analysis. Results based on synthetic data are provided. The general question of ill-posedness encountered in inverse problems is addressed in detail. Furthermore, a review of the work reported in recent years in the mathematical literature and relevant to the identification of hydrological systems is presented. Finally, an analysis of the conditioning of the geo statistical approach to aquifer identification is provided.