A robust Upwind Mixed Hybrid Finite Element method for transport in variably saturated porous media
- 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
- 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: open (until 30 Jun 2022)
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CC1: 'Comment on hess-2022-153', Thomas Graf, 25 May 2022
reply
This manuscript introduces a new upwind Mixed Finite Element (MFE) method to solve for solute transport under saturated and unsaturated conditions. The new scheme is shown to avoid unphysical oscillations that is otherwise potentially caused by advection.
The manuscript is clearly written, and the mathematical context is sound and complete. The two illustrative results are useful and demonstrate very well the capabilities of the new method.
There are some minor comments that I ask the authors to address:
1. There are a number of filling words (especially "indeed") that need to be removed. It is also uncommong to use "the" in e.g. line 129 when generally addressing water content etc.
2. Description of boundary conditions of the two examples in section 4. is not complete. Both figures 4 and 7 should show all BCs for both flow and transport. As is, this is not the case and must be changed. Also, the text does not give the full description of all BCs for both flow and transport, which also has to be completed. A figure that shows the meshes for both examples is missing and would be very helpful. I understand it is an unstructured mesh which explains the fact that result in Fig. 5 is not symmetric. This should be mentioned. For simplicity, it would have been more efficient to use only the upper half of the domain due to symmetry reasons. The authors should give a reason why they did not do so.
Anis Younes et al.
Anis Younes et al.
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