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Preparation of grids for simulations of groundwater flow in fractured porous media

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Computing and Visualization in Science

Abstract

This work presents an extension of grid generation techniques for finite-volume discretizations of density-driven flow in fractured porous media, in which fractures are considered as low-dimensional manifolds and are resolved by sides of grid elements. The proposed technique introduces additional degrees of freedom for the unknowns assigned to the fractures and thus allows to reconstruct jumps of the solution over a fracture. Through the concept of degenerated elements, the proposed technique can be used for arbitrary junctions of fractures but is sufficiently simple regarding the implementation and allows for the application of conventional numerical solvers. Numerical experiments presented at the end of the paper demonstrate the applicability of this technique in two and three dimensions for complicated fracture networks.

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References

  1. Alliez, P., Ucelli, G., Gotsman, C., Attene, M.: Recent advances in remeshing of surfaces. In: De Floriani, L., Spagnuolo, M. (eds.) Shape Analysis and Structuring, pp. 53–82. Springer, Berlin (2008)

    Chapter  Google Scholar 

  2. Angot, P., Boyer, F., Hubert, F.: Asymptotic and numerical modelling of flow in fractured porous media. ESAIM: M2AN. Math. Model. Numer. Anal. 43(2), 239–275 (2009). doi:10.1051/m2an/2008052

    Article  MATH  MathSciNet  Google Scholar 

  3. Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia, PA (1993)

    MATH  Google Scholar 

  4. Bastian, P., Birken, K., Johannsen, K., Lang, S., Reichenberger, V., Wieners, C., Wittum, G., Wrobel, C.: Parallel solution of partial differential equations with adaptive multigrid methods on unstructured grids. In: Jäger, W., Krause, E. (eds.) High Performance Computing in Science and Engineering, pp. 506–519. Springer, Berlin (2000)

    Google Scholar 

  5. Bastian, P., Chen, Z., Ewing, R.E., Helmig, R., Jakobs, H., Reichenberger, V.: Numerical simulation of multiphase flow in fractured porous media. In: Chen, Z., et al. (eds.) Numerical Treatment of Multiphase Flows in Porous Media. Proceedings of the International Workshop, Beijing, China, August 2–6, 1999, Lect. Notes Phys., vol. 552, pp. 50–68. Springer, Berlin (2000)

  6. Bear, J.: Hydraulics of Groundwater. Dover, Mineola (1979)

    Google Scholar 

  7. Bear, J., Tsang, C.F., deMarsily, G.: Flow and Contaminant Transport in Fractured Rocks. Academic Press, New York (1993)

  8. Bentley, J.L.: Multidimensional binary search trees used for associative searching. ACM Commun. 18(9), 509–517 (1975). doi:10.1145/361002.361007

    Article  MATH  Google Scholar 

  9. Bockgård, N., Vidstrand, P., Åkesson, M.: Modelling the Interaction Between Engineered and Natural Barriers—An Assessment of a Fractured Bedrock Description in the Wetting Process of Bentonite at Deposition Tunnel Scale. Tech. Rep. Rev. 2011–02-23, Technical Committee of Task 8, Svensk Kärnbränslehantering AB (SKB), Stockholm (2011)

  10. Cai, Z.: On the finite volume element method. Numer. Math. 58(1), 713–735 (1990)

    Article  Google Scholar 

  11. Chew, L.P.: Constrained delaunay triangulations. Algorithmica 4(1), 97–108 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fein, E.: Ein Programmpaket zur Modellierung von Dichteströmungen. GRS, Braunschweig 139 (1999). In German.

  13. Frey, P., Borouchaki, H.: Geometric surface mesh optimization. Comput. Vis. Sci. 1(3), 113–121 (1998)

    Article  MATH  Google Scholar 

  14. Frolkovič, P.: Consistent velocity approximation for density driven flow and transport. In: Van Keer, R., et al. (eds.) Advanced Computational Methods in Engineering Part 2 Contributed papers, pp. 603–611. Shaker Publishing, Maastricht (1998)

  15. Frolkovič, P.: Maximum principle and local mass balance for numerical solutions of transport equation coupled with variable density flow. Acta Mathematica Universitatis Comenianae 1(68), 137–157 (1998)

    Google Scholar 

  16. Frolkovič, P., Knabner, P.: Consistent velocity approximations in finite element or volume discretizations of density driven flow. In: Aldama, A.A., et al. (eds.) Computational Methods in Water Resources XI, pp. 93–100. Computational Mechanics Publication, Southhampten (1996)

    Google Scholar 

  17. Fuchs, A.: Almost regular triangulations of trimmend nurbs-solids. Eng. Comput. 17(1), 55–65 (2001)

    Article  MATH  Google Scholar 

  18. Fumagalli, A., Scotti, A.: Numerical modelling of multiphase subsurface flow in the presence of fractures. Commun. Appl. Ind. Math. 3(1) (2012). doi:10.1685/journal.caim.380

  19. Graf, T., Therrien, R.: Variable-density groundwater flow and solute transport in porous media containing nonuniform discrete fractures. Adv. Water Resour. 28(12), 1351–1367 (2005)

    Article  Google Scholar 

  20. Graf, T., Therrien, R.: Stable-unstable flow of geothermal fluids in fractured rock. Geofluids 9(2), 138–152 (2009)

    Article  Google Scholar 

  21. Grillo, A., Lampe, M., Logashenko, D., Stichel, S., Wittum, G.: Simulation of salinity- and thermohaline-driven flow in fractured porous media. J. Porous Media 15(5), 439–458 (2012). doi:10.1615/JPorMedia.v15.i5.40

    Article  Google Scholar 

  22. Grillo, A., Logashenko, D., Stichel, S., Wittum, G.: Simulation of density-driven flow in fractured porous media. Adv. Water Resour. 33(12), 1494–1507 (2010). doi:10.1016/j.advwatres.2010.08.004

    Article  Google Scholar 

  23. Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations. Springer, New-York (1994)

    Book  MATH  Google Scholar 

  24. Hansen, G., Douglass, R., Zardecki, A.: Mesh Enhancement: Selected Elliptic Methods, Foundations and Applications. Imperial College Pr, London (2005)

    Book  Google Scholar 

  25. Henry, H.R.: Effects of dispersion on salt encroachment in coastal aquifers. In: Sea Water in Coastal Aquifers, pp. 70–84. USGS Water Supply Paper 1613-c (1964)

  26. Hoppe, H.: Progressive meshes. In: Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’96, pp. 99–108. ACM (1996)

  27. Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., Stuetzle, W.: Mesh optimization. In: Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, pp. 19–26. ACM (1993)

  28. Johannsen, K., Kinzelbach, W., Oswald, S., Wittum, G.: The saltpool benchmark problem—numerical simulation of saltwater upconing in a porous medium. Adv. Water Resour. 25(3), 335–348 (2002)

    Article  Google Scholar 

  29. Karimian, S.M., Schneider, G.E.: Pressure-based control-volume finite element method for flow at all speeds. AIAA J. 33(9), 1611–1618 (1995)

    Article  MATH  Google Scholar 

  30. Kröhn, K.P.: Qualifying a computer program for simulating fracture flow. Comput. Vis. Sci. 15(1), 29–37 (2012). doi: 10.1007/s00791-013-0191-6

  31. Lang, S., Wittum, G.: Large-scale density-driven flow simulations using parallel unstructured grid adaptation and local multigrid methods. Concurr. Comput.: Pract. Exper. 17(11), 1415–1440 (2005)

    Article  Google Scholar 

  32. Malkovsky, V.I., Pek, A.A.: Conditions for the onset of thermal convection of a homogeneous fluid in a vertical fault. Petrology 5, 381–387 (1997)

    Google Scholar 

  33. Martinez-Landa, L., Carrera, J.: A methodology to interpret cross-hole tests in a granite block. J. Hydrogeol. 325(1–4), 222–240 (2006)

    Google Scholar 

  34. Maryška, J., Severýn, O., Tauchman, M., Tondr, D.: Modelling of processes in fractured rock using fem/fvm on multidimensional domains. J. Comput. Appl. Math. 215(2), 495–502 (2008). doi:10.1016/j.cam.2006.04.074

    Article  MATH  MathSciNet  Google Scholar 

  35. Moore, M., Wilhelms, J.: Collision detection and response for computer animation. SIGGRAPH Comput. Graph. 22(4), 289–298 (1988). doi:10.1145/378456.378528

    Article  Google Scholar 

  36. Murphy, H.D.: Convective instabilities in vertical fractures and faults. J. Geophys. Res. 84(11), 6121–6130 (1979)

    Article  Google Scholar 

  37. Neumann, S.: Trends, prospects and challenges in quantifying flow and transport through fractured rocks. Hydrogeol. J. 13(1), 124–147 (2005)

    Article  Google Scholar 

  38. Oldenburg, C.M., Pruess, K.: Layered thermohaline convection in hypersaline geothermal systems. Transp. Porous Media 33(1–2), 29–63 (1998)

    Article  Google Scholar 

  39. Reiter, S., Logashenko, D., Stichel, S., Wittum, G., Grillo, A.: Models and simulations of variable-density flow in fractured porous media. Int. J. Comput. Sci. Eng. (2013) (in press)

  40. Reiter, S., Wittum, G.: Promesh—A Flexible Interactive Meshing Software for Unstructured Hybrid Grids in 1, 2 and 3 Dimensions. (in preparation) (2013)

  41. Schneider, A.: Enhancement of d3f und r3t (e-dur). fkz 02 e 10336 (bmwi), final report. Tech. rep., Gesellschaft für Anlagen- und Reaktorsicherheit (GRS) mbH, GRS-292, Braunschweig (2012)

  42. Sharp, J.M.J., Shi, M.: Heterogeneity effects on possible salinity-driven free convection in low-permeability strata. Geofluids 34, 263–274 (2009)

    Article  Google Scholar 

  43. Shewchuk, J.R.: Constrained delaunay tetrahedralizations and provably good boundary recovery. In: Proceedings of the 11th International Meshing Roundtable, pp. 193–204 (2002)

  44. Shikaze, S.G., Sudicky, E.A., Schwartz, F.W.: Density-dependent solute transport in discretely-fractured geologic media: is prediction possible? J. Contam. Hydrogeol. 34(10), 273–291 (1998)

    Google Scholar 

  45. Si, H.: TetGen—A Quality Tetrahedral Mesh Generator and Three-Dimensional Delaunay Triangulator. http://wias-berlin.de/software/tetgen/files/tetgen-manual.pdf

  46. Si, H., Gärtner, K.: Meshing piecewise linear complexes by constrained delaunay tetrahedralizations. In: Proceedings of the 14th International Meshing Roundtable, pp. 147–163. Springer, Berlin (2005)

  47. Simpson, M., Clement, T.: Improving the worthiness of the Henry problem as a benchmark for density-dependent groundwater flow models. Water Resour. Res. 40, W01504 (2004). doi:10.1029/2003WR002199

    Article  Google Scholar 

  48. Sorek, S., Borisov, V., Yakirevich, A.: A two-dimensional areal model for density dependent flow regime. Transp. Porous Media 43(1), 87–105 (2001)

    Article  Google Scholar 

  49. Stichel, S., Logashenko, D., Grillo, A., Reiter, S., Lampe, M., Wittum, G.: Numerical methods for flow in fractured porous media. In: Delgado, J. (ed.) Heat and Mass Transfer in Porous Media, Advanced Structured Materials, vol. 13, pp. 83–113. Springer, Berlin (2012). doi:10.1007/978-3-642-21966-5_4

    Chapter  Google Scholar 

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Acknowledgments

We thank Walter Steininger (PTkA-WTE, Karlsruhe, Germany), Klaus-Peter Kröhn and Anke Schneider (GRS, Braunschweig, Germany), Peter Frolkovič (Slovak University of Technology, Bratislava, Slovakia), Sabine Stichel and Michael Lampe (Goethe University Frankfurt, Germany). This work has been supported by the Goethe-Universität Frankfurt am Main, by the German Ministry of Economy and Technology (BMWi) via grants 02E10568 and 02E10326, as well as by the State of Hesse, Germany, via project “NuSim — Numerische Simulation auf Hochleistungsrechnern”.

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  1. Goethe Center for Scientific Computing (G-CSC), Goethe University Frankfurt am Main, Kettenhofweg 139, 60325 , Frankfurt, Germany

    Sebastian Reiter, Dmitry Logashenko & Gabriel Wittum

  2. Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 , Turin, Italy

    Alfio Grillo

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Communicated by Sabine Attinger.

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Reiter, S., Logashenko, D., Grillo, A. et al. Preparation of grids for simulations of groundwater flow in fractured porous media. Comput. Visual Sci. 15, 209–225 (2012). https://doi.org/10.1007/s00791-013-0210-7

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