Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Comparison of Three FE-FV Numerical Schemes for Single- and Two-Phase Flow Simulation of Fractured Porous Media

Abstract

We benchmark a family of hybrid finite element–node-centered finite volume discretization methods (FEFV) for single- and two-phase flow/transport through porous media with discrete fracture representations. Special emphasis is placed on a new method we call DFEFVM in which the mesh is split along fracture–matrix interfaces so that discontinuities in concentration or saturation can evolve rather than being suppressed by nodal averaging of these variables. The main objective is to illustrate differences among three discretization schemes suitable for discrete fracture modeling: (a) FEFVM with volumetric finite elements for both fractures and porous rock matrix, (b) FEFVM with lower dimensional finite elements for fractures and volumetric finite elements for the matrix, and (c) DFEFVM with a mesh that is split along material discontinuities. Fracture discontinuities strongly influence single- and multi-phase fluid flow. Continuum methods, when used to model transport across such interfaces, smear out concentration/saturation. We show that the new DFEFVM addresses this problem producing significantly more accurate results. Sealed and open single fractures as well as a realistic fracture geometry are used to conduct tracer and water-flooding numerical experiments. The benchmarking results also reveal the limitations/mesh refinement requirements of FE node-centered FV hybrid methods. We show that the DFEFVM method produces more accurate results even for much coarser meshes.

References

  1. Bastian, P.: Numerical computation of multiphase flows in porous media. Habilitationsschrift, Universität Kiel (1999)

  2. Belayneh M.: Paleostress orientation inferred from surface morphology of joints on the southern margin of the bristol channel basin, UK. In: Cosgrove, J.W., Engelder, T. (eds) The Initiation, Propagation, and Arrest of Joints and Other Fractures, Special Publication 231, pp. 243–255. Geological Society Publishing House, Bath (2004)

  3. Benes, M., Illangasekare, T., Mikyska, J.: On the numerical treatment of sharp texture transport in two-phase flow. In: Proceedings of the Czech-Japanese Seminar in Applied Mathematics, pp. 106–116, Oita (2005)

  4. Brooks R.H., Corey A.T.: Hydraulic properties of porous media. Hydrology Papers, Colorado State University, 24 p. Colorado (1964)

  5. Buckley S., Leverett M.: Mechanism of fluid displacement in sands. TAIME 146, 107–1116 (1942)

  6. Chen Z., Huan G., Li B.: An improved impes method for two-phase flow in porous media. Transp. Porous Media 54(3), 361–376 (2006)

  7. Dietrich P., Helmig R., Sauter H., Hötzl H., Kongeter J., Teutsch G.: Flow and transport in fractured porous media. Springer, Berlin (2005)

  8. Durlofsky L.: A triangle based mixed finite element-finite volume technique for modeling two phase flow through porous media. J. Comput. Phys. 105(2), 252–266 (1998)

  9. Eikemo B., Lie K.A., Dahle H., Eigestad G.: A discontinuous galerkin method for transport in fractured media using unstructured triangular grids. Adv. Water Res. 32(4), 493–506 (2009)

  10. Geiger S., Matthäi S., Niessner J., Helmig R.: Black-oil simulations for three-component, three-phase flow in fractured porous media. SPE J. 6, 338–354 (2009)

  11. Geiger, S., Cortis, A., Birkholzer, J.: Upscaling solute transport in naturally fractured porous media with the continuous time random walk method. Water Resour. Res. 46, W12530, 13p. (2010). doi:10.1029/2010WR009133

  12. Gong B., Karimi-Fard M., Durlofsky L.: Upscaling discrete fracture characterizations to dual-porosity, dual-permeability models for efficient simulation of flow with strong gravitational effects. SPE J. 13(1), 58–67 (2008)

  13. Hoteit H., Firoozabadi A.: An efficient numerical model for incompressible two-phase flow in fractured media. Adv. Water Resour. 31(6), 891–905 (2008)

  14. Hoteit H., Firoozabadi A.: Numerical modeling of two-phase flow in heterogeneous permeable media with different capillarity pressures. Adv. Water Resour. 31(1), 56–73 (2008)

  15. Huber R., Helmig R.: Multi-phase flow in heterogeneous porous media: a classical finite element method versus an implicit pressure-explicit saturation-based mixed finite element-finite volume approach. Int. J. Numer. Methods Fluids 29(8), 899–920 (1999)

  16. Huber R., Helmig R.: Node-centered finite volume discretizations for the numerical simulation of multiphase flow in heterogeneous porous media. Comput. Geosci. 4(2), 141–164 (2000)

  17. Juanes R., Samper J., Molinero J.: A general and efficient formulation of fractures and boundary conditions in the finite element method. Int. J. Numer. Methods Eng. 54(12), 1751–1774 (2002)

  18. Karimi-Fard M., Firoozabadi A.: Numerical simulation of water injection in fractured media using the discrete-fracture model and the galerkin method. SPE J. 6(2), 117–126 (2003)

  19. Kim J., Deo M.: Finite element, discrete-fracture model for multiphase flow in porous media. AIChE J. 46(6), 1120–1130 (2000)

  20. Matthäi S., Geiger S., Roberts S., Paluszny A., Belayneh M., Burri A., Mezentsev A., Lu H., Coumou D., Driesner T., Heinrich C.A.: Numerical simulation of multi-phase fluid flow in structurally complex reservoirs. In: Jolley, S.J., Barr, D., Walsh, J.J., Knipe, R.J. (eds) Structurally Complex Reservoirs, Special Publication 292, pp. 405–429. Geological Society, London (2007)

  21. Matthäi S., Nick H.: Upscaling two phase flow in naturally fractured reservoirs. AAPG Bull. 93(11), 1621–1632 (2009)

  22. Matthäi S., Nick H., Pain C., Neuweiler I.: Simulation of solute transport through fractured rock: a higher-order accurate finite-element finite-volume method permitting large time steps. Transp. Porous Media 83(2), 289–318 (2010)

  23. Mikyska, J.: Numerical model for simulation of behaviour of non-aqueous phase liquids in heterogeneous porous media containing sharp texture transitions. Ph.D. thesis, Czech technical university in Prague (2005)

  24. Monteagudo J.E.P., Firoozabadi A.: Control-volume method for numerical simulation of two-phase immiscible flow in two- and three-dimensional discrete-fractured media. Water Resour. Res. 40, W07405, 20 p. (2004). doi:10.1029/2003WR002996

  25. Nick H., Matthäi S.: A hybrid finite-element finite-volume method with embedded discontinuities for solute transport in heterogeneous media. Vadose Zone J. 10(1), 299–312 (2011)

  26. Niessner J., Helmig R., Jakobs H., Roberts J.: Interface condition and linearization schemes in the Newton iterations for two-phase flow in heterogeneous porous media. Adv. Water Resour. 28(7), 671–687 (2005)

  27. Paluszny A., Zimmerman R.: Numerical simulation of multiple 3D fracture propagation using arbitrary meshes. Comput. Methods Appl. Mech. Eng. 200(9–12), 953–966 (2011)

  28. Paluszny A., Matthäi S., Hohmeyer M.: Hybrid finite element finite volume discretization of complex geologic structures and a new simulation workflow demonstrated on fractured rocks. Geofluids 7, 186–208 (2007)

  29. Rangel-German, E., Kovscek, A.: Matrix–fracture transfer functions for partially and completely immersed fractures. Twenty-Eighth Workshop on Geothermal Reservoir Engineering, pp. 1–9, Stanford University, Stanford, January (2003)

  30. Reichenberger V., Jakobs H., Bastian P., Helmig R.: A mixed-dimensional finite volume method for two-phase flow in fractured porous media. Adv. Water Resour. 29(7), 1020–1036 (2006)

  31. Su G., Nimmo J.: Effect of isolated fractures on accelerated flow in unsaturated porous rock. Water Resour. Res. 39(12), 1326–1337 (2003)

  32. Unsal E., Matthèi S., Blunt M.: Simulation of multiphase flow in fractured reservoirs using a fracture-only model with transfer functions. Comput. Geosci. 14(4), 527–538 (2010)

  33. van Duijn C., Molenaar J., de Neef M.: The effect of capillary forces on immiscible two-phase flow in heterogeneous porous media. Transp. Porous Media 21(71), 193–201 (1995)

  34. van Genuchten M.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44, 892–898 (1980)

  35. Zaretskiy Y., Geiger S., Sorbie K., Foerster M.: Efficient flow and transport simulations in reconstructed 3D pore geometries. Adv. Water Resour. 33(12), 1508–1516 (2010)

Download references

Acknowledgments

We would like to thank the sponsors of the ITF project on Improved Simulation of Flow in Fractured and Faulted Reservoirs, and the Technology Strategy Board (TSB) for supporting this research. Chris C. Pain and Martin J. Blunt, receive our thanks for their feedback and comments.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Author information

Correspondence to H. M. Nick.

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

Nick, H.M., Matthäi, S.K. Comparison of Three FE-FV Numerical Schemes for Single- and Two-Phase Flow Simulation of Fractured Porous Media. Transp Porous Med 90, 421–444 (2011). https://doi.org/10.1007/s11242-011-9793-y

Download citation

Keywords

  • DFM
  • FEM
  • FVM
  • Two phase flow
  • Discrete fracture and matrix model