Minimal a priori assignment in a direct method for determining phenomenological coefficients uniquely

Abstract

We identify the coefficients of the transport equation in N dimensions grad c.grad h+c Delta h=d delta h/ delta t+f by solving a differential system of the form grad c+ca=b. The assignment of c at one point only yields a unique solution, found by integration along arbitrary paths. This arbitrariness guarantees a good control of the error, notwithstanding the ill-posedness of the problem. For N=2, the hypotheses allowing for this identification are satisfied when one knows two stationary potentials with non-overlapping equipotential lines and a third non-stationary one-this last needed only for determining d. The theory is applied to a numerical synthetic example, for various grid sizes or for noisy data. Notwithstanding the minimal a priori information required for the coefficients, we are able to compute these at a large number of nodes with good precision. For the sake of completeness, we give other results on identification.

Transmissivity identification through multi-directional aquifer stimulation

“…The idea of using hydraulic head measurements from a variety of steady statē ow conditions has long been suggested [e.g., Scarascia and Ponzini, 1972;Ponzini and Lozej, 1982;Ponzini and Crosta, 1988;Giudici et al, 1995a;and Parravicini et al, 1995]. These authors have shown soundly that the transmissivity may be determined uniquely provided that head measurements under independent¯ow conditions are available.…”

The differential system method for parameter identification; unconfined aquifer case

“…For confined aquifers in the stationary case, if two sets of data are given, say h 1 and h 2 , the full rank condition means that the gradients ∇h 1 , ∇h 2 of the potentials do not vanish and their equipotential lines do not overlap anywhere. This is fully discuss in [6]. Since linear independence of ∇h 1 , ∇h 2 at a point (x, y) is equivalent to linear independence of h 1 ∇h 1 , h 2 ∇h 2 , the same physical condition implies the full rank condition for unconfined aquifers in the stationary case.…”

“…Now, let us rewrite equation (6) recalling definition (4) and writing down the dependence upon x explicitly. We have the following system for the first two components of u…”

“…See the lemma in section 4 of [6]. The solution at a point x = (x, y) is found by choosing an appropriate path joining x with the initial point, and integrating (10) along it.…”

“…For the confined aquifer case, Parravicini et al [6] and Giudici et al [5] proposed a direct method based on the solution of a Cauchy problem that allows for the determination of both transmissivity T and storativity S, when the potentials and source terms are given for three different flow conditions, at least one of them transient. Vázquez et al [8] developing further the method show the advantages of using more than three flow conditions.…”