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A simplified enhanced MPFA formulation for the elliptic equation on general grids

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Abstract

Multi-point flux approximation (MPFA) has proven to be a powerful tool for discretizing the diffusion equation on general grids with heterogeneous anisotropic permeability tensors and hence, removing the O(1) error introduced by two-point flux approximation (TPFA) for non-K-orthogonal grids. However, it is well known that the classical MPFA-O suffers from monotonicity issues and strong unphysical oscillations can be present for highly anisotropic media. Enriched MPFA (EMPFA) and MPFA with full pressure support (FPS) have been proposed in the literature to reduce the strength of the oscillations. In this work, we present a simplified enhanced MPFA formulation (eMPFA) on general grids in 2D. Similar to the MPFA-O method, our formulation starts with an interaction region formed around the vertex of control volumes. Potential at the vertex is introduced as an auxiliary unknown in addition to potential at the centroids of the control volume faces. The original EMPFA and FPS solve the diffusion equation on a small volume around the vertex to close the local system of equations. To simplify the formulation, we propose an average technique to approximate potential at the vertex. Utilizing the potential value at the vertex, full potential continuity can be imposed on control volume faces to construct a more accurate flux across control volume faces. Extensive numerical experiments are conducted to test the eMPFA formulation. Specifically, we compare eMPFA with MPFA-O and two variants of EMPFA in detail. The results show that our average technique is quite robust even for highly distorted quadrilateral grids with large permeability anisotropy ratios and solutions of our formulation are in excellent agreement with EMPFA using bilinear pressure support. The new scheme also reproduces linear potential solutions and has comparable, and in some cases even better, convergence properties to those of the MPFA-O method. Finally, we generalize the formulation to include structured and unstructured triangular and polygonal grids and present the results.

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References

  1. Aziz, K., Settari, A.: Petroleum reservoir simulation. Chapman & Hall (1979)

  2. Aavatsmark, I., Barkve, T., Bœ, Ø., Mannseth, T.: Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127(1), 2–14 (1996)

    Article  Google Scholar 

  3. Avaatsmark, I., Barkve, T., Bœ, Ø., Mannseth, T.: A class of discretization methods for structured and unstructured grids in anisotropic, inhomogeneous media 5th European Conference on the Mathematics of Oil Recovery (1996)

  4. Aavatsmark, I., Barkve, T., Bœ, O., Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. Part II: Discussion and numerical results. SIAM J. Sci. Comput. 19(5), 1717–1736 (1998)

    Article  Google Scholar 

  5. Aavatsmark, I., Barkve, T., Bœ, O., Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: Derivation of the methods. SIAM J. Sci. Comput. 19(5), 1700–1716 (1998)

    Article  Google Scholar 

  6. Aavatsmark, I., Barkve, T., Mannseth, T.: Control-volume discretization methods for 3D quadrilateral grids in inhomogeneous, anisotropic reservoirs. SPE J. 3(02), 146–154 (1998)

    Article  Google Scholar 

  7. Edwards, M.G., Rogers, C.F.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2(4), 259–290 (1998)

    Article  Google Scholar 

  8. Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6(3–4), 405–432 (2002)

    Article  Google Scholar 

  9. Edwards, M.G.: Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci. 6(3–4), 433–452 (2002)

    Article  Google Scholar 

  10. Edwards, M.G.: M-matrix flux splitting for general full tensor discretization operators on structured and unstructured grids. J. Comput. Phys. 160(1), 1–28 (2000)

    Article  Google Scholar 

  11. Verma, S., Aziz, K.: A control volume scheme for flexible grids in reservoir simulation SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (1997)

  12. Lee, S., Durlofsky, L., Lough, M., Chen, W.: Finite difference simulation of geologically complex reservoirs with tensor permeabilities SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (1997)

  13. Lee, S.H., Tchelepi, H.A., Jenny, P., DeChant, L.J.: Implementation of a flux-continuous finite-difference method for stratigraphic, hexahedron grids. SPE J. 7(03), 267–277 (2002)

    Article  Google Scholar 

  14. Eigestad, G.T., Klausen, R.A.: On the convergence of the multi-point flux approximation O-method: numerical experiments for discontinuous permeability. Numerical Methods for Partial Differential Equations 21(6), 1079–1098 (2005)

    Article  Google Scholar 

  15. Pal, M., Edwards, M.G., Lamb, A.R.: Convergence study of a family of flux-continuous, finite-volume schemes for the general tensor pressure equation. Int. J. Numer. Meth. Fluids 51, 1177–1203 (2006)

    Article  Google Scholar 

  16. Aavatsmark, I., Eigestad, G., Klausen, R.A.: Numerical convergence of the MPFA O-method for general quadrilateral grids in two and three dimensions Compatible spatial discretizations, pp 1–21. Springer (2006)

  17. Aavatsmark, I., Eigestad, G.T., Klausen, R.A., Wheeler, M.F., Yotov, I.: Convergence of a symmetric MPFA method on quadrilateral grids. Comput. Geosci. 11(4), 333–345 (2007)

    Article  Google Scholar 

  18. Klausen, R.A., Winther, R.: Convergence of multipoint flux approximations on quadrilateral grids. Numerical Methods for Partial Differential Equations 22(6), 1438–1454 (2006)

    Article  Google Scholar 

  19. Klausen, R.A., Winther, R.: Robust convergence of multi point flux approximation on rough grids. Numer. Math. 104(3), 317–337 (2006)

    Article  Google Scholar 

  20. Klausen, R.A., Radu, F.A., Eigestad, G.T.: Convergence of MPFA on triangulations and for Richards’ equation. Int. J. Numer. Methods Fluids 58(12), 1327–1351 (2008)

    Article  Google Scholar 

  21. Eigestad, G.T., Aavatsmark, I., Espedal, M.: Symmetry and M-matrix issues for the O-method on an unstructured grid. Comput. Geosci. 6(3–4), 381–404 (2002)

    Article  Google Scholar 

  22. Nordbotten, J.M., Aavatsmark, I.: Monotonicity conditions for control volume methods on uniform parallelogram grids in homogeneous media. Comput. Geosci. 9(1), 61–72 (2005)

    Article  Google Scholar 

  23. Nordbotten, J.M., Aavatsmark, I., Eigestad, G.T.: Monotonicity of control volume methods. Numer. Math. 106(2), 255–288 (2007)

    Article  Google Scholar 

  24. Keilegavlen, E., Aavatsmark, I.: Monotonicity for MPFA methods on triangular grids. Comput. Geosci. 15 (1), 3–16 (2011)

    Article  Google Scholar 

  25. Nordbotten, J.M., Eigestad, G.T.: Discretization on quadrilateral grids with improved monotonicity properties. J. Comput. Phys. 203(2), 744–760 (2005)

    Article  Google Scholar 

  26. Aavatsmark, I., Eigestad, G.T., Mallison, B.T., Nordbotten, J.M.: A compact multipoint flux approximation method with improved robustness. Numerical Methods for Partial Differential Equations 24(5), 1329–1360 (2008)

    Article  Google Scholar 

  27. Mlacnik, M.J., Durlofsky, L.J.: Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients. J. Comput. Phys. 216(1), 337–361 (2006)

    Article  Google Scholar 

  28. Chen, Q.-Y., Wan, J., Yang, Y., Mifflin, R.T.: Enriched multi-point flux approximation for general grids. J. Comput. Phys. 227(3), 1701–1721 (2008)

    Article  Google Scholar 

  29. Edwards, M.G., Zheng, H.: A quasi-positive family of continuous Darcy-flux finite-volume schemes with full pressure support. J. Comput. Phys. 227(22), 9333–9364 (2008)

    Article  Google Scholar 

  30. Edwards, M.G., Zheng, H.: Quasi-positive families of continuous Darcy-flux finite volume schemes on structured and unstructured grids. J. Comput. Appl. Math. 234(7), 2152–2161 (2010)

    Article  Google Scholar 

  31. Edwards, M.G., Zheng, H.: Double-families of quasi-positive Darcy-flux approximations with highly anisotropic tensors on structured and unstructured grids. J. Comput. Phys. 229(3), 594–625 (2010)

    Article  Google Scholar 

  32. Friis, H.A., Edwards, M.G.: A family of MPFA finite-volume schemes with full pressure support for the general tensor pressure equation on cell-centered triangular grids. J. Comput. Phys. 230(1), 205–231 (2011)

    Article  Google Scholar 

  33. Crumpton, P., Shaw, G., Ware, A.: Discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients. J. Comput. Phys. 116(2), 343–358 (1995)

    Article  Google Scholar 

  34. Aavatsmark, I: Multipoint flux approximation methods for quadrilateral grids 9th International forum on reservoir simulation Abu Dhabi (2007)

  35. Aavatsmark, I: Comparison of monotonicity for some multipoint flux approximation methods in Finite Volumes for Complex Applications (2008)

  36. Lipnikov, K., Shashkov, M., Svyatskiy, D., Vassilevski, Y.: Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. J. Comput. Phys. 227(1), 492–512 (2007)

    Article  Google Scholar 

  37. Persson, P.-O., Strang, G.: A simple mesh generator in MATLAB. SIAM Rev. 46(2), 329–345 (2004)

    Article  Google Scholar 

  38. Krogstad, S., Lie, K.A., Møyner, O., Nilsen, H.M., Raynaud, X., Skaflestad, B.: MRST-AD—an open-source framework for rapid prototyping and evaluation of reservoir simulation problems SPE reservoir simulation symposium. Society of Petroleum Engineers (2015)

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Acknowledgments

The authors would like to acknowledge ADNOC’s scholarship for Mr. Wenjuan Zhang.

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Authors and Affiliations

  1. Department of Petroleum Engineering, The Petroleum Institute, P.O. Box 2533, Abu Dhabi, United Arab Emirates

    Wenjuan Zhang & Mohammed Al Kobaisi

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  1. Wenjuan Zhang
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  2. Mohammed Al Kobaisi
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Correspondence to Mohammed Al Kobaisi.

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Zhang, W., Al Kobaisi, M. A simplified enhanced MPFA formulation for the elliptic equation on general grids. Comput Geosci 21, 621–643 (2017). https://doi.org/10.1007/s10596-017-9638-z

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  • DOI: https://doi.org/10.1007/s10596-017-9638-z

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