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Nonlinear two-point flux approximation for modeling full-tensor effects in subsurface flow simulations

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Abstract

Subsurface flow models can exhibit strong full-tensor anisotropy due to either permeability or grid nonorthogonality effects. Upscaling procedures, for example, generate full-tensor effects on the coarse scale even for cases in which the underlying fine-scale permeability is isotropic. A multipoint flux approximation (MPFA) is often needed to accurately simulate flow for such systems. In this paper, we present and apply a different approach, nonlinear two-point flux approximation (NTPFA), for modeling systems with full-tensor effects. In NTPFA, transmissibility (which provides interblock connections) is determined from reference global flux and pressure fields for a specific flow problem. These fields can be generated using either fully resolved or approximate global simulations. The use of fully resolved simulations leads to an NTPFA method that corresponds to global upscaling procedures, while the use of approximate simulations gives a method corresponding to recently developed local–global techniques. For both approaches, NTPFA algorithms applicable to both single-scale full-tensor permeability systems and two-scale systems are described. A unified framework is introduced, which enables single-scale and two-scale problems to be viewed in a consistent manner. Extensive numerical results demonstrate that the global and local–global NTPFA techniques provide accurate flow predictions over wide parameter ranges for both single-scale and two-scale systems, though the global procedure is more accurate overall. The applicability of NTPFA to the simulation of two-phase flow in upscaled models is also demonstrated.

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References

  1. Aadland, T., Aavatsmark, I., Dahle, H.K.: Performance of a flux splitting scheme when solving the single-phase pressure equation discretized by MPFA. Comput. Geosci. 8, 325–340 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  4. Chen, Y., Durlofsky, L.J.: Adaptive local–global upscaling for general flow scenarios in heterogeneous formations. Transp. Porous Media 62, 157–185 (2006)

    Article  MathSciNet  Google Scholar 

  5. Chen, Y., Durlofsky, L.J.: Efficient incorporation of global effects in upscaled models of two-phase flow and transport in heterogeneous formations. Multiscale Model. Simul. 5, 445–475 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, Y., Durlofsky, L.J., Gerritsen, M., Wen, X.H.: A coupled local–global upscaling approach for simulating flow in highly heterogeneous formations. Adv. Water Resour. 26, 1041–1060 (2003)

    Article  Google Scholar 

  7. Deutsch, C.V., Journel, A.G.: GSLIB: Geostatistical Software Library and User’s Guide, 2nd edn. Oxford University Press, New York (1998)

    Google Scholar 

  8. Durlofsky, L.J.: Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resour. Res. 27, 699–708 (1991)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Eigestad, G.T., Aadland, T., Aavatsmark, I., Klausen, R.A., Nordbotten, J.M.: Recent advances for MPFA methods. In: Proceedings of the 9th European Conference on the Mathematics of Oil Recovery, Cannes, France, 28 August– 2 September 2004

  12. Eigestad, G.T., Klausen, R.: On the convergence of the multi-point flux approximation O-method: Numerical experiments for discontinuous permeability. Numer. Methods Partial Differ. Equ. 21, 1079–1098 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Holden, L., Nielsen, B.F.: Global upscaling of permeability in heterogeneous reservoirs: the output least squares (OLS) method. Transp. Porous Media 40, 115–143 (2000)

    Article  MathSciNet  Google Scholar 

  14. Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134, 169–189 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jenny, P., Lee, S.H., Tchelepi, H.A.: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187, 47–67 (2003)

    Article  MATH  Google Scholar 

  16. Lambers, J., Gerritsen, M.G., Mallison, B.T.: Accurate local upscaling with variable compact multipoint transmissibility calculations. Comput. Geosci. (2008) doi:10.1007/s10596-007-9068-4

  17. 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, 267–277 (2002)

    Google Scholar 

  18. Mallison, B.T., Chen, Y., Durlofsky, L.J.: Nonlinear two-point flux approximation for modeling subsurface flow with nonorthogonal grids and full-tensor permeability anisotropy. In: Proceedings of the 10th European Conference on the Mathematics of Oil Recovery, Amsterdam, The Netherlands, 4–7 September 2006

  19. 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, 337–361 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Nielsen, B.F., Tveito, A.: An upscaling method for one-phase flow in heterogeneous reservoirs. A weighted output least squares (WOLS) approach. Comput. Geosci. 2, 93–123 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Renard, P., de Marsily, G.: Calculating effective permeability: a review. Adv. Water Resour. 20, 253–278 (1997)

    Article  Google Scholar 

  24. Romeu, R., Noetinger, B.: Calculation of internodal transmissivities in finite difference models of flow in heterogeneous porous media. Water Resour. Res. 31, 943–959 (1995)

    Article  Google Scholar 

  25. Wen, X.H., Chen, Y., Durlofsky, L.J.: Efficient 3D implementation of local–global upscaling for reservoir simulation. SPE J. 11, 443–453 (2006)

    Google Scholar 

  26. Wen, X.H., Durlofsky, L.J., Edwards, M.G.: Use of border regions for improved permeability upscaling. Math. Geol. 35, 521–547 (2003)

    Article  MATH  Google Scholar 

  27. Zhang, P., Pickup, G.E., Christie, M.A.: A new upscaling approach for highly heterogeneous reservoirs. Paper SPE 93339 presented at the SPE Symposium on Reservoir Simulation, Houston, TX, 31 January–2 February (2005)

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Authors and Affiliations

  1. Chevron Energy Technology Company, P.O. Box 6019, San Ramon, CA, 94583-0719, USA

    Yuguang Chen & Bradley T. Mallison

  2. Department of Energy Resources Engineering, Stanford University, Stanford, CA, 94305-2220, USA

    Louis J. Durlofsky

Authors
  1. Yuguang Chen
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  2. Bradley T. Mallison
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  3. Louis J. Durlofsky
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Correspondence to Yuguang Chen.

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Chen, Y., Mallison, B.T. & Durlofsky, L.J. Nonlinear two-point flux approximation for modeling full-tensor effects in subsurface flow simulations. Comput Geosci 12, 317–335 (2008). https://doi.org/10.1007/s10596-007-9067-5

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  • DOI: https://doi.org/10.1007/s10596-007-9067-5

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