1887
  1. Home
  2. Conferences
  3. Conference Proceedings
  4. ECMOR XVII
  5. Conference paper

Abstract

Summary

A nonlinear two-point flux approximation (NTPFA) finite volume method is applied to the modeling of compressible gas flow in anisotropic reservoirs. Gas compressibility factor and gas density are calculated by the Peng-Robinson equation of state. The governing equations are discretized by NTPFA in space and first-order backward Euler method in time. Newton-Raphson iteration is used as the nonlinear solver during each time step. The NTPFA method employs the harmonic averaging points as auxiliary points during the construction of onesided fluxes. A unique nonlinear flux approximation is obtained by a convex combination of the one-sided fluxes. Since a Newton-Raphson nonlinear solver is used, NTPFA will have a denser discretized coefficient matrix compared to the widely used Two-Point Flux Approximation (TPFA) method on grids that are not K-orthogonal. However, its coefficient matrix is still much sparser than the classical Multi-Point Flux Approximation O (MPFA-O) method. Results of numerical examples demonstrate that the pressure profile and gas production rate of NTPFA is in close agreement with that of MPFA-O for most cases while TPFA is inconsistent since the grid is not K-orthogonal. The MPFA-O method is well known to suffer from monotonicity issues for highly anisotropic reservoirs and our numerical experiments show that MPFA-O can fail to converge during the Newton-Raphson iterations when the permeability anisotropy is very high while NTPFA still enjoys good performance.

© EAGE Publications BV
Loading

Article metrics loading...

/content/papers/10.3997/2214-4609.202035002
2020-09-14
2024-04-27

  • From This Site

    /content/papers/10.3997/2214-4609.202035002
    dcterms_title,dcterms_subject,pub_keyword
    -contentType:Journal -contentType:Contributor -contentType:Concept -contentType:Institution
    10
    5
Loading full text...

Full text loading...

References

  1. Aziz, K., K.Aziz, and A.Settari
    , Petroleum reservoir simulation. 1979: Applied Science Publishers.
     [Google Scholar]
  2. Aavatsmark, I., T.Barkve, Ø.Bøe, and T.Mannseth
    , Discretization on Non-Orthogonal, Quadrilateral Grids for Inhomogeneous, Anisotropic Media. Journal of Computational Physics, 1996. 127(1): p. 2–14.
     [Google Scholar]
  3. Edwards, M.G. and C.F.Rogers
    , Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Computational Geosciences, 1998. 2(4): p. 259–290.
     [Google Scholar]
  4. Chen, Z., G.Huan, and Y.Ma
    , Computational Methods for Multiphase Flows in Porous Media. Computational Science & Engineering. 2006: Society for Industrial and Applied Mathematics. 523.
     [Google Scholar]
  5. Aavatsmark, I., T.Barkve, O.Bøe, and T.Mannseth
    , Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: Derivation of the methods. SIAM Journal on Scientific Computing, 1998. 19(5): p. 1700–1716.
     [Google Scholar]
  6. Aavatsmark, I., T.Barkve, and T.Mannseth
    , Control-Volume Discretization Methods for 3D Quadrilateral Grids in Inhomogeneous, Anisotropic Reservoirs. SPE Journal, 1998. 3(02): p. 146–154.
     [Google Scholar]
  7. Aavatsmark, I.
    , An Introduction to Multipoint Flux Approximations for Quadrilateral Grids. Computational Geosciences, 2002. 6(3): p. 405–432.
     [Google Scholar]
  8. Nordbotten, J.M. and I.Aavatsmark
    , Monotonicity conditions for control volume methods on uniform parallelogram grids in homogeneous media. Computational Geosciences, 2005. 9(1): p. 61–72.
     [Google Scholar]
  9. Mlacnik, M.J. and L.J.Durlofsky
    , Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients. Journal of Computational Physics, 2006. 216(1): p. 337–361.
     [Google Scholar]
  10. Keilegavlen, E. and I.Aavatsmark
    , Monotonicity for MPFA methods on triangular grids. Computational Geosciences, 2011. 15(1): p. 3–16.
     [Google Scholar]
  11. Zhang, W. and M.Al Kobaisi
    , A two-step finite volume method to discretize heterogeneous and anisotropic pressure equation on general grids. Advances in Water Resources, 2017. 108: p. 231–248.
     [Google Scholar]
  12. Nordbotten, J.M. and G.T.Eigestad
    , Discretization on quadrilateral grids with improved monotonicity properties. Journal of Computational Physics, 2005. 203(2): p. 744–760.
     [Google Scholar]
  13. Aavatsmark, I., G.T.Eigestad, B.T.Mallison, and J.M.Nordbotten
    , A compact multipoint flux approximation method with improved robustness. Numerical Methods for Partial Differential Equations, 2008. 24(5): p. 1329–1360.
     [Google Scholar]
  14. Chen, Q.-Y., J.Wan, Y.Yang, and R.T.Mifflin
    , Enriched multi-point flux approximation for general grids. Journal of Computational Physics, 2008. 227(3): p. 1701–1721.
     [Google Scholar]
  15. Edwards, M.G. and H.Zheng
    , A quasi-positive family of continuous Darcy-flux finite-volume schemes with full pressure support. Journal of Computational Physics, 2008. 227(22): p. 9333–9364.
     [Google Scholar]
  16. , Double-families of quasi-positive Darcy-flux approximations with highly anisotropic tensors on structured and unstructured grids. Journal of Computational Physics, 2010. 229(3): p. 594–625.
     [Google Scholar]
  17. Friis, H.A. and M.G.Edwards
    , A family of MPFA finite-volume schemes with full pressure support for the general tensor pressure equation on cell-centered triangular grids. Journal of Computational Physics, 2011. 230(1): p. 205–231.
     [Google Scholar]
  18. Zhang, W. and M.Al Kobaisi
    , A simplified enhanced MPFA formulation for the elliptic equation on general grids. Computational Geosciences, 2017. 21(4): p. 621–643.
     [Google Scholar]
  19. Edwards, M.G.
    , Symmetric Flux Continuous Positive Definite Approximation of the Elliptic Full Tensor Pressure Equation in Local Conservation Form, in SPE Reservoir Simulation Symposium. 1995, Society of Petroleum Engineers: San Antonio, Texas. p. 10.
     [Google Scholar]
  20. Nordbotten, J.M., I.Aavatsmark, and G.T.Eigestad
    , Monotonicity of control volume methods. Numerische Mathematik, 2007. 106(2): p. 255–288.
     [Google Scholar]
  21. Le Potier, C.
    , Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles non structurés. Comptes Rendus Mathematique, 2005. 341(12): p. 787–792.
     [Google Scholar]
  22. Lipnikov, K., M.Shashkov, D.Svyatskiy, and Y.Vassilevski
    , Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. Journal of Computational Physics, 2007. 227(1): p. 492–512.
     [Google Scholar]
  23. Lipnikov, K., D.Svyatskiy, and Y.Vassilevski
    , Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes. Journal of Computational Physics, 2009. 228(3): p. 703–716.
     [Google Scholar]
  24. , A monotone finite volume method for advection-diffusion equations on unstructured polygonal meshes. Journal of Computational Physics, 2010. 229(11): p. 4017–4032.
     [Google Scholar]
  25. Danilov, A. and Y.V.Vassilevski
    , A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes. Russian Journal of Numerical Analysis and Mathematical Modelling, 2009. 24(3): p. 207–227.
     [Google Scholar]
  26. Yuan, G. and Z.Sheng
    , Monotone finite volume schemes for diffusion equations on polygonal meshes. Journal of Computational Physics, 2008. 227(12): p. 6288–6312.
     [Google Scholar]
  27. Queiroz, L., M.Souza, F.Contreras, P.Lyra, and D.Carvalho
    , On the accuracy of a nonlinear finite volume method for the solution of diffusion problems using different interpolations strategies. International ournal for Numerical Methods in Fluids, 2014. 74(4): p. 270–291.
     [Google Scholar]
  28. Wu, J. and Z.Gao
    , Interpolation-based second-order monotone finite volume schemes for anisotropic diffusion equations on general grids. Journal of Computational Physics, 2014. 275: p. 569–588.
     [Google Scholar]
  29. Gao, Z. and J.Wu
    , A second-order positivity-preserving finite volume scheme for diffusion equations on general meshes. SIAM Journal on Scientific Computing, 2015. 37(1): p. A420–A438.
     [Google Scholar]
  30. Schneider, M., B.Flemisch, and R.Helmig
    , Monotone nonlinear finite - volume method for nonisothermal two - phase two - component flow in porous media. International Journal for Numerical Methods in Fluids, 2017. 84(6): p. 352–381.
     [Google Scholar]
  31. Agélas, L., R.Eymard, and R.Herbin
    , A nine point finite volume scheme for the simulation of diffusion in heterogeneous media. Comptes rendus de l’Académie des sciences. Série I, Mathématique, 2009. 347(11–12): p. 673–676.
     [Google Scholar]
  32. Zhang, W. and M.Al Kobaisi
    , Cell-centered nonlinear finite volume methods with improved robustness. SPE Journal, 2019. (In Print)
     [Google Scholar]
  33. Terekhov, K. and Y.Vassilevski
    , Two-phase water flooding simulations on dynamic adaptive octree grids with two-point nonlinear fluxes, in Russian Journal of Numerical Analysis and Mathematical Modelling. 2013. p. 267.
     [Google Scholar]
  34. Nikitin, K., K.Terekhov, and Y.Vassilevski
    , A monotone nonlinear finite volume method for diffusion equations and multiphase flows. Computational Geosciences, 2014. 18(3): p. 311–324.
     [Google Scholar]
  35. Schneider, M., B.Flemisch, R.Helmig, K.Terekhov, and H.Tchelepi
    , Monotone nonlinear finite-volume method for challenging grids. Computational Geosciences, 2018. 22(2): p. 565–586.
     [Google Scholar]
  36. Chen, Z.
    , Reservoir simulation: mathematical techniques in oil recovery. Vol. 77. 2007: Siam.
     [Google Scholar]
  37. Aarnes, J.E., T.Gimse, and K.-A.Lie
    , An Introduction to the Numerics of Flow in Porous Media using Matlab, in Geometric Modelling, Numerical Simulation, and Optimization: Applied Mathematics at SINTEF, G.Hasle, K.-A.Lie, and E.Quak, Editors. 2007, Springer Berlin Heidelberg: Berlin, Heidelberg. p. 265–306.
     [Google Scholar]
  38. Peaceman, D.W.
    , Interpretation of Well-Block Pressures in Numerical Reservoir Simulation(includes associated paper 6988). Society of Petroleum Engineers Journal, 1978. 18(03): p. 183–194.
     [Google Scholar]
  39. Peng, D.-Y. and D.B.Robinson
    , A New Two-Constant Equation of State. Industrial & Engineering Chemistry Fundamentals, 1976. 15(1): p. 59–64.
     [Google Scholar]
  40. Lee, A.L., M.H.Gonzalez, and B.E.Eakin
    , The Viscosity of Natural Gases. Journal of Petroleum Technology, 1966. 18(08): p. 997–1000.
     [Google Scholar]
  41. Lie, K.A., S.Krogstad, I.S.Ligaarden, J.R.Natvig, H.M.Nilsen, and B.Skaflestad
    , Open-source MATLAB implementation of consistent discretisations on complex grids. Computational Geosciences, 2012. 16(2): p. 297–322.
     [Google Scholar]
http://instance.metastore.ingenta.com/content/papers/10.3997/2214-4609.202035002
Loading
Modeling Compressible Gas Flow in Anisotropic Reservoirs Using A Nonlinear Finite Volume Method
Conference Proceedings 2020, 1 (2020); https://doi.org/10.3997/2214-4609.202035002
/content/papers/10.3997/2214-4609.202035002
/content/papers/10.3997/2214-4609.202035002
Loading

Data & Media loading...

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error