Parameters Identification in a Saltwater Intrusion Problem
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
PARAMETERS IDENTIFICATION IN A SALTWATER INTRUSION PROBLEM∗
oG)
Ji LI (
College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing Key Laboratory of Social Economy and Applied Statisties, Chongqing 400067, China E-mail :liji [email protected]
Carole ROSIER† Universite du Littoral Cˆ ote d’Opale, UR 2597, LMPA, Laboratoire de Math´ematiques Pures et Appliqu´ees Joseph Liouville, F-62100 Calais, France, CNRS FR 2037, France E-mail : [email protected] Abstract This article is devoted to the identification, from observations or field measurements, of the hydraulic conductivity K for the saltwater intrusion problem in confined aquifers. The involved PDE model is a coupled system of nonlinear parabolic-elliptic equations completed by boundary and initial conditions. The main unknowns are the saltwater/ freshwater interface depth and the freshwater hydraulic head. The inverse problem is formulated as an optimization problem where the cost function is a least square functional measuring the discrepancy between experimental data and those provided by the model. Considering the exact problem as a constraint for the optimization problem and introducing the Lagrangian associated with the cost function, we prove that the optimality system has at least one solution. Moreover, the first order necessary optimality conditions are established for this optimization problem. Key words
parameters identification; optimization problem; strongly coupled system; nonlinear parabolic equations; seawater intrusion
2010 MR Subject Classification
1
49J20; 37N10; 35K51; 35K55
Introduction
In order to get an optimal exploitation of freshwater and then to control seawater intrusion in coastal aquifers, we need to develop efficient and accurate models to simulate the transport of saltwater front in coastal aquifer. Thus an essential step is to identify the parameters involved in the model. ∗ Received
January 31, 2018; revised October 19, 2019. The first author was supported by the Natural Science Foundation of Chongqing Municipal Education Commission (KJ1706167), and the Program for the introduction of High-Level Talents (1756006, 1752003). † Corresponding author: Carole ROSIER.
1564
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
In this article, we focus on the identification of physical parameters such as the hydraulic conductivity K and the porosity φ. The estimation of these parameters is based on observations or field measurements made on hydraulic heads and on the depth of the freshwater/saltwater interface. We note that concretely, there are only specific observations (in space and in time) corresponding to the number of monitoring wells. We also emphasize that the parameters identification problematic was often addressed as part of the underground hydraulic (cf. [1–6]), but more rarely with regard to saltwater intrusion phenomena. The first studies done by Sun et Yeh (cf. [7]) develop th
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