A New Method for the Identification of Distributed Transmissivities

Abstract

For two-dimensional groundwater flow in an isotropic confined aquifer, it has been shown elsewhere that two independent steady state sets of data, i.e., piezometric heads and source terms corresponding to different steady state flow conditions, and the value of transmissivity at one point sutfice to determine transmissivity uniquely in a connected domain. The data are independent if the hydraulic gradients are not parallel anywhere over the domain. Here transmissivity is numerically determined by integration of suitably approximated functions of the data along polygonal lines connecting the nodes of a lattice; integration starts from the node where transmissivity is given. The choice of the integration path is based on the results of the stability analysis and allows us to minimize the effects of the approximations on the data. Since the approximated solution is computed along internode segments, the internode transmissivities are immediately calculated without introducing arbitrary averages of the node transmissivities. The internode transmissivities are the quantities necessary to set up a management model within a conservative finite differences scheme. The applicability of this technique to real cases is tested with two synthetic examples. The first one was set up by ourselves, whereas the second one has been taken from the literature. The internode transmissivities identified with our procedure are compared with the synthetic reference ones. The ultimate check is performed of evaluating new head fields on the basis of the identified and reference internode transmissivities. The fit is good. The relative error for the identified internode transmissivities is very low when error-free data are used, and it varies by an amount approximately constant over the entire aquifer when an error on the initial value of transmissivity is introduced. The errors on the piezometric heads bear more relevance, but nonetheless, the affected results are still good. GIUDICI ET AL.: IDENTIFICATION OF DISTRIBUTED TRANSMISSIVITIESIn management modeling, we have to solve the forward problem, which consists of finding a solution of (2) for h, when the transmissivity and the source term are given together with appropriate boundary conditions on h along the boundary of the aquifer.Measuring the transmissivity is a formidable task, and even if the transmissivity is obtained with experimental field procedures, for instance, by means of pumping tests, it is still a debatable question how these values should be introduced in a mathematical model [Dykaar and Kitanidis, 1993]. A better approach to finding the transmissivity to be used in a forward problem is that of solving an inverse problem.Mathematically, the inverse problem can be stated as follows: given h (x, y) and f(x, y), find r(x, y) such that (2) holds. In this case, (2) is considered as a first-order linear partial differential equation, whose solution is not unique in general. The lack of uniqueness is a very important point. In fact, the solution of the inverse problem is used t...

Discrete stability of the Differential System Method evaluated with geostatistical techniques

“…The plot shows that the results of the inversion with limited data sets tend to the results obtained from the complete data sets when the number of data points increases: recall that this corresponds to a decrease of the We see from ®gure 5 for the complete data set, i.e., without any interpolation error, that we have a relative error less than 0.5 for almost 50% of the nodes. This result strongly depends upon the heterogeneity of the reference transmissivity ®eld; in fact applications of the DSM to examples characterised by more regular ®elds give better results Giudici et al, 1995).…”

“…The approximation of the vectors a and b at the nodes of a regular grid with the procedure described in Giudici et al (1995) and sketched in section 2.2 requires the knowledge of the piezometric head at each node of the grid for both hydraulic states (p 1 and p 2). For real applications the points where the piezometric head is measured are usually few and sparse.…”

“…The fourth section is devoted to a presentation of the synthetic aquifer and the data sets that are used for numerical experiments, the results of which are presented in the ®fth section and their implications are discussed in the sixth and last section. The Differential System Method We simply recall the fundamentals of the DSM, referring the reader to published papers for details Giudici et al, 1995).…”

“…The initial node where the value of T 0 has to be assigned for the DSM has been chosen according to a modi®cation of the rule proposed in Giudici et al (1995). We consider each node together with its eight closest neighbours and we compute a weighted average of the values of jaj approximated at the nine considered nodes and of the values of ja A j evaluated at the twelve internode segments connecting the nine nodes.…”

“…The starting node is chosen where this average is the least. This choice improves the effectiveness of the rule proposed in Giudici et al (1995), especially for such a heterogeneous system as our synthetic aquifer. The value of transmissivity, T 0 , assigned at the initial node is the same as the reference true value at that node.…”