Approccio Geostatistico per la Soluzione dei Problemi Inversi nelle Acque Sotterranee: Applicazioni e Sviluppi

Abstract

The great interest in environmental issues has drawn the community to an attention to the quality of groundwater. Scientific efforts in groundwater flow studies have primarily focused on the flow and transport behavior and on the identification of the corresponding parameters. After the nineties, increasing attention has been focused on the problem of recovery the release history of a pollutant. The knowledge of the pollution injection function provides information about the future pollution spread and allows better planning of remediation actions [Liu and Ball, 1999; Snodgrass and Kitanidis, 1997; Skaggs and Kabala, 1994; Butera and Tanda, 2003]. Moreover, from a legal and regulatory point of view, it is also important to determine the release time and duration and the highest values of concentration of the injected solution: an available release history can be a useful tool for sharing the costs of remediation of a polluted area among the responsible parties. The Mathematical modeling is the basis of the studies of the evolution of the pollution. Setting up a model is a complex and a difficult task, because the main problem is the evaluation of the aquifer parameters. Usually the scientists have few field data, and with that they have to model wide areas; this implies the introduction of errors (due to the large approximations) into the modeling. These parameters, usually, are estimated from scarce data because they are difficult and costly to obtain. The accuracy of the estimate depends on the number of the measurements, their locations in the studied area, the observation error and the sensitivity of the observed quantity to the real field.Both of the problems - release history and parameters identification - are represented by illposed problems especially inverse problems. The literature regarding the inverse problem is wide and regards several branches of sciences and mathematics. During the last 40 years several methods were developed to solve inverse problems, for instance the Tikhonov regularization, the minimum relative entropy theory, the adjoint state method and the geostatistical method. The methodology applied in this work is the quasi-linear geostatistical approach proposed by Kitanidis [1995]. This approach was chosen because it is a statistical method so that it is possible to evaluate the unknown function and the related uncertainty to it at the same time. It has been widely applied during the last 10 years by several authors with good results [Snodgrass and Kitanidis, 1997; Michalak and Kitanidis, 2002, 2003; Butera and Tanda, 2003; Boano et al., 2005]. This work presents applications and improvements of the quasi-linear geostatistical approach: The first application concerns the recovery of the release history of the pollutants; it consists in the evaluation of the release function of a pollutant starting from concentration measurements. A brief literature review on this topic is presented and the geostatistical approach proposed by Snodgrass and Kitanidis [1997] and subsequent developments are summarized. A new improvement (evaluation of the transfer function) about the possibility to apply the methodology to non uniform flow cases (pumping well, heterogeneous hydraulic conductivity fields, etc.) is described. This new improvement enables the evaluation of the transfer function using a numerical model; this allows extension of the geostatistical approach to any case without using rough simplifications as the use of a 1-D or 2-D homogeneous model. The second application of the geostatistical approach presented in this work is the estimation of the hydraulic parameters. Starting from field measurements, for instance of the transmissivity, it is possible using an interpolator to evaluate the whole transmissivity field of the study area. However, these kind of measurements are expensive and with few monitoring points the resulting transmissivity field is not reliable. Therefore, head measurements are frequently used,because they are easier and cheapless expensive to evaluate. So in the last 30 years several methods [for instance Kitanidis and Vomovoris, 1983; Rubin and Dagan, 1987a; Giudici et al., 1988; etc] regarding parameter identification were developed. In this work the quasi-linear methodology is applied to parameter estimation following the work presented by Kitanidis [1995]. This method is a very efficient procedure, but for strongly nonlinear cases it requires some add ons. It is based on heads measured in specific points of the study area, then a forward problem is performed with an initial value of transmissivity. The following step is to correct the initial transmissivity field as far as the forward problem represents correctly the heads measured. The first proposed improvement is the updating of unknowns from an iteration to the next one. This procedure allows to choose the correct parameter in the Gauss-Newton iterations and to speed up the process. Then the theory regarding the conditional realizations, proposed in Kitanidis [1995], is tested to reproduce the highly nonlinear transmissivity field. Moreover considering the possibility to apply the estimation of hydraulic parameters to a very well defined grid it has been decided to summarize and test the methodology proposed by Nowak et al. [2003] that allows to speed up the matrix multiplication in order to decrease the computation time. The work is structured in two part; the first presents a general introduction on inverse problems and describes the quasi-linear geostatistical approach. The second one proposes few improvements to the methodology and analyzes several cases.