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Simulation of anisotropic heterogeneous near-well flow using MPFA methods on flexible grids

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Abstract

Control-volume multipoint flux approximations (MPFA) are discussed for the simulation of complex near-well flow using geometrically flexible grids. Due to the strong non-linearity of the near-well flow, a linear model will, in general, be inefficient. Instead, a model accounting for the logarithmic pressure behavior in the well vicinity is advocated. This involves a non-uniform refinement of the grid in the radial direction. The model accounts for both near-well anisotropies and heterogeneities. For a full simulation involving multiple wells, this single-well approach can easily be coupled with the reservoir model. Numerical simulations demonstrate the convergence behavior of this model using various MPFA schemes under different near-well conditions for single-phase flow regimes. Two-phase simulations support the results of the single-phase simulations.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Aavatsmark, I.: Interpretation of a two-point flux stencil for skew parallelogram grids. Comput. Geosci. 11(3), 199–206 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Aavatsmark, I., Eigestad, G.T., Klausen, R.A.: Numerical convergence of the MPFA for general quadrilateral grids in two and three dimensions. In: Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., Shaskov, M., (eds.) Compatible Spatial Discretizations. IMA Vol. Math. Appl., vol. 142, pp. 1–21. Springer, New York (2006)

    Chapter  Google Scholar 

  6. Aavatsmark, I., Eigestad, G.T., Mallison, B.T., Nordbotten, J.M.: A compact multipoint flux approximation method with improved robustness. Numer. Methods Partial Differ. Equ. 24, 1329–1360 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brennier, Y., Jaffré, J.: Upstream differencing for multiphase flow in reservoir simulation. SIAM J. Numer. Anal. 28(3), 685–696 (1991)

    Article  MathSciNet  Google Scholar 

  8. Cao, Y., Helmig, R., Wohlmuth, B.I.: Geometrical interpretation of the multi-point flux approximation L-method. Int. J. Numer. Methods Fluids 60(11), 1173–1199 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media. Computational Science & Engineering. SIAM, Philadelphia (2006). ISBN 0-89871-606-3

    Google Scholar 

  11. Ding, Y., Jeannin, L.: A new methodology for singularity modelling in flow simulations in reservoir engineering. Comput. Geosci. 5, 93–119 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ding, Y., Jeannin, L.: New numerical schemes for near-well modeling using flexible grids. SPE J. 9(1), 109–121 (2004). SPE 87679

    Google Scholar 

  13. Driscoll, T.A.: Algorithm 843: improvements to the Schwarz–Christoffel toolbox for Matlab. ACM Trans. Math. Softw. 31(2), 239–251 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. 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  MATH  MathSciNet  Google Scholar 

  15. 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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Friis, H.A., Edwards, M.G., Mykkeltveit, J.: Symmetric positive definite flux-continuous full-tensor finite-volume schemes on unstructured cell-centred triangular grids. SIAM J. Sci. Comput. 31(2), 1192–1220 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Keilegavlen, E., Aavatsmark, I.: Monotonicity for MPFA-methods on triangular grids. Comput. Geosci. (2009, submitted)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Lambers, J.V., Gerritsen, M.G., Mallison, B.T.: Accurate local upscaling with variable compact multipoint transmissibility calculations. Comput. Geosci. 12(3), 399–416 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mlacnik, M.J., Heinemann, Z.E.: Using well windows in full-field reservoir simulation. SPE Reserv. Evalu. Eng. 6, 275–285 (2003) SPE 85709

    Google Scholar 

  23. Mundal, S., DiPietro, D.A., Aavatsmark, I.: Compact-stencil MPFA method for heterogeneous highly anisotropic second-order elliptic problems. In: Eymard, R., Hérard, J.-M. (eds.) Finite Volumes for Complex Applications V, pp. 905–918. Wiley, New York (2008)

    Google Scholar 

  24. 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  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Peaceman, D.W.: Interpretation of well-block pressures in numerical reservoir simulation. SPE J. 18(3), 183–194 (1978) SPE 6893

    Google Scholar 

  27. Peaceman, D.W.: Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability. SPE J. 23(3), 531–543 (1983). SPE 10528

    Google Scholar 

  28. Peaceman, D.W.: Further discussion of productivity of a horizontal well. In: SPERE, pp. 437–438 (1990). SPE 20799

  29. Peaceman, D.W.: Further discussion of productivity of a horizontal well. In: SPERE, pp. 149–150 (1991). SPE 21611

  30. Pedrosa, O.A. Jr., Aziz, K.: Use of a hybrid grid in reservoir simulation. SPERE 2, 611–621 (1986); Trans., AIME, 281. SPE 13507

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  32. Potsepaev, R., Farmer, C.L., Fitzpatrick, A.J.: Multipoint flux approximation via upscaling. In: SPE Reservoir Simulation Symposium (2009). SPE 118979

  33. Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)

    MATH  Google Scholar 

  34. Wolfsteiner, C., Durlofsky, L.J.: Near-well radial upscaling for the accurate modeling of nonconventional wells. In: SPE Western Regional/AAPG Pacific Section Joint Meeting, Anchorage, Alaska (2002). SPE 76779

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Authors and Affiliations

  1. Department of Mathematics, Centre for Integrated Petroleum Research (CIPR), University of Bergen, P.O. Box 7800, 5020, Bergen, Norway

    Sissel Slettemark Mundal, Eirik Keilegavlen & Ivar Aavatsmark

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  1. Sissel Slettemark Mundal
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  2. Eirik Keilegavlen
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  3. Ivar Aavatsmark
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Correspondence to Sissel Slettemark Mundal.

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Mundal, S.S., Keilegavlen, E. & Aavatsmark, I. Simulation of anisotropic heterogeneous near-well flow using MPFA methods on flexible grids. Comput Geosci 14, 509–525 (2010). https://doi.org/10.1007/s10596-009-9167-5

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  • DOI: https://doi.org/10.1007/s10596-009-9167-5

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