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Algebraic flux correction finite element method with semi-implicit time stepping for solute transport in fractured porous media

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Abstract

This work is concerned with the numerical modeling of the Darcy flow and solute transport in fractured porous media for which the fractures are modeled as interfaces of codimension one. The hybrid-dimensional flow and transport problems are discretizaed by a lumped piece-wise linear finite element method, combined with the algebraic correction of the convective fluxes. The resulting transport discretization can be interpreted as a conservative finite volume scheme that satisfies the discrete maximum principle, while introducing a very limited amount of numerical diffusion. In the context of fractured porous media flow the CFL number may vary by several order of magnitude, which makes explicit time stepping unfeasible. To cope with this difficulty we propose an adaptive semi-implicit time stepping strategy that reduces to the low order linear implicit discretization in the high CFL regions that include, but may not be limited to the fracture network. The performance of the fully explicit and semi-implicit variants of the method are investigated through the numerical experiment.

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Data Availability Statements

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This project was funded by the National Plan for Science, Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia, award number 14-MAT739-02.

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Authors and Affiliations

  1. Université Côte d’Azur, Inria, CNRS, Laboratoire J.A. Dieudonné, team Coffee, Nice, France

    Konstantin Brenner & Roland Masson

  2. Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia

    Nejmeddine Chorfi

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  1. Konstantin Brenner
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Correspondence to Konstantin Brenner.

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Appendices

Appendix A

A.1: Case 4.1: Approximate solution for Λf = 20

Fig. 20
figure 20

Left: approximate and exact solution along the fracture at t = Tf. Right: zoom view of left figure around x = 0.5. Fracture length is rescaled

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A.2: Case 4.1: Approximate solution for Λf = 2000

Fig. 21
figure 21

Approximate solution at t = Tf using the following discretizations: o1-semi-implicit scheme (left), o1-o2-semi-implicit scheme with minmod limiter (center) and o1-o2-semi-implicit scheme with superbee limiter (right)

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Fig. 22
figure 22

Left: approximate and exact solution along the fracture at t = Tf. Right: zoom view of left figure around x = 0.5. Fracture length is rescaled

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Appendix B: Case 4.1: L 1 relative space-time error

Table 5 Relative L1 space-time error and the experimental order of convergence for explicit schemes (case Λf = 20)
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Table 6 Relative L1 space-time error and the experimental order of convergence for semi-implicit schemes (case Λf)
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Table 7 Relative L1 space-time error and the experimental order of convergence for semi-implicit schemes (case Λf)
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Appendix C: Case 4.2: Pressure and concentration error

Table 8 Error and the experimental order of convergence for pressure problem
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Table 9 Relative L1 space-time error and the experimental order of convergence for transport problem
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Brenner, K., Chorfi, N. & Masson, R. Algebraic flux correction finite element method with semi-implicit time stepping for solute transport in fractured porous media. Comput Geosci 27, 103–126 (2023). https://doi.org/10.1007/s10596-022-10178-y

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