27 Apr 2022
27 Apr 2022
Status: a revised version of this preprint was accepted for the journal HESS.

A robust Upwind Mixed Hybrid Finite Element method for transport in variably saturated porous media

Anis Younes1, Hussein Hoteit2, Rainer Helmig3, and Marwan Fahs1 Anis Younes et al.
  • 1Institut Terre et Environnement de Strasbourg, Université de Strasbourg, CNRS, ENGEES, UMR 7063, 67084 Strasbourg, France
  • 2Physical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia
  • 3Institute for Modelling Hydraulic and Environmental Systems, University of Stuttgart, Pfaffenwaldring 61, 15 70569 Stuttgart, Germany

Abstract. The Mixed Finite Element (MFE) method is well adapted for the simulation of fluid flow in heterogeneous porous media. However, when employed for the transport equation, it can generate solutions with strong unphysical oscillations because of the hyperbolic nature of advection. In this work, a robust upwind MFE scheme is proposed to avoid such unphysical oscillations. The new scheme is a combination of the upwind edge/face centred Finite Volume (FV) method with the hybrid formulation of the MFE method. The scheme ensures continuity of both advective and dispersive fluxes between adjacent elements and allows to maintain the time derivative continuous, which permits employment of high order time integration methods via the Method of Lines (MOL).

Numerical simulations are performed in both saturated and unsaturated porous media to investigate the robustness of the new upwind-MFE scheme. Results show that, contrarily to the standard scheme, the upwind-MFE method generates stable solutions without under and overshoots. The simulation of contaminant transport into a variably saturated porous medium highlights the robustness of the proposed upwind scheme when combined with the MOL for solving nonlinear problems.

Anis Younes et al.

Status: final response (author comments only)

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • CC1: 'Comment on hess-2022-153', Thomas Graf, 25 May 2022
    • AC1: 'Reply on CC1', Anis Younes, 06 Aug 2022
  • RC1: 'Comment on hess-2022-153', Anonymous Referee #1, 07 Jul 2022
    • AC2: 'Reply on RC1', Anis Younes, 06 Aug 2022
  • RC2: 'Comment on hess-2022-153', Anonymous Referee #2, 11 Jul 2022
    • AC3: 'Reply on RC2', Anis Younes, 06 Aug 2022

Anis Younes et al.

Anis Younes et al.


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Short summary
Despite its advantages for the simulation of flow in heterogeneous and fractured porous media, the mixed hybrid finite element method has been rarely used for transport as it suffers from strong unphysical oscillations. We develop here a new upwind scheme for the mixed hybrid finite element that can avoid oscillations. Numerical examples confirm the robustness of this new scheme for the simulation of contaminant transport in both saturated and unsaturated conditions.