Volume 56, Number 2, March-April 2022
|Page(s)||565 - 592|
|Published online||28 February 2022|
Model adaptation for non-linear elliptic equations in mixed form: existence of solutions and numerical strategies
Department of Mathematics, Politecnico di Milano, p.za Leonardo da Vinci 32, Milano 20133, Italy
2 IFP Energies nouvelles, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison, France
* Corresponding author: firstname.lastname@example.org
Accepted: 1 February 2022
Depending on the physical and geometrical properties of a given porous medium, fluid flow can behave differently, going from a slow Darcian regime to more complicated Brinkman or even Forchheimer regimes for high velocity. The main problem is to determine where in the medium one regime is more adequate than others. In order to determine the low-speed and high-speed regions, this work proposes an adaptive strategy which is based on selecting the appropriate constitutive law linking velocity and pressure according to a threshold criterion on the magnitude of the fluid velocity itself. Both theoretical and numerical aspects are considered and investigated, showing the potentiality of the proposed approach. From the analytical viewpoint, we show existence of weak solutions to such model under reasonable hypotheses on the constitutive laws. To this end, we use a variational approach identifying solutions with minimizers of an underlying energy functional. From the numerical viewpoint, we propose a one-dimensional algorithm which tracks the transition zone between the low- and high-speed regions. By running numerical experiments using this algorithm, we illustrate some interesting behaviors of our adaptive model on academic cases and on small networks of intersecting fractures.
Mathematics Subject Classification: 76S05 / 35A01 / 35A15 / 65N50 / 65N55
Key words: Adaptive constitutive law / elliptic equation / fractured porous media / non-linearity / variational formulation
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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