Advertisement

Computational Geosciences

, Volume 22, Issue 1, pp 389–412 | Cite as

An efficient distribution method for nonlinear two-phase flow in highly heterogeneous multidimensional stochastic porous media

  • Fayadhoi Ibrahima
  • Hamdi A. Tchelepi
  • Daniel W. Meyer
Original Paper

Abstract

In the context of stochastic two-phase flow in porous media, we introduce a novel and efficient method to estimate the probability distribution of the wetting saturation field under uncertain rock properties in highly heterogeneous porous systems, where streamline patterns are dominated by permeability heterogeneity, and for slow displacement processes (viscosity ratio close to unity). Our method, referred to as the frozen streamline distribution method (FROST), is based on a physical understanding of the stochastic problem. Indeed, we identify key random fields that guide the wetting saturation variability, namely fluid particle times of flight and injection times. By comparing saturation statistics against full-physics Monte Carlo simulations, we illustrate how this simple, yet accurate FROST method performs under the preliminary approximation of frozen streamlines. Further, we inspect the performance of an accelerated FROST variant that relies on a simplification about injection time statistics. Finally, we introduce how quantiles of saturation can be efficiently computed within the FROST framework, hence leading to robust uncertainty assessment.

Keywords

Stochastic Buckley-Leverett Saturation Uncertainty propagation Distribution functions Two-phase flow Porous systems Heterogeneity Streamlines 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

Daniel Meyer is very thankful to Florian Müller who kindly provided the streamline code that was applied in this work. Fayadhoi Ibrahima is thankful to Per Pettersson for his constant support in the early stage of the manuscript and to Matthias Cremon for providing SGEMS simulations. The authors are also grateful to the SUPRI-B research group at Stanford University for their financial support.

References

  1. 1.
    Abgrall, R.: A simple, flexible and generic deterministic approach to uncertainty quantifications in non linear problems: application to fluid flow problems. Rapport de recherche. INRIA (2007)Google Scholar
  2. 2.
    Aziz, K., Settari, A.: Petroleum Reservoir Simulation, vol. 476. Applied Science Publishers, England (1979)Google Scholar
  3. 3.
    Babuska, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45, 1005–1034 (2007)CrossRefGoogle Scholar
  4. 4.
    Bäck, J., Nobile, F., Tamellini, L., Tempone, R.: Stochastic spectral galerkin and collocation methods for pdes with random coefficients: a numerical comparison. Spectral High Order Methods Partial Diff Equa. 76, 43–62 (2011)CrossRefGoogle Scholar
  5. 5.
    Ballio, F., Guadagnini, A.: Convergence assessment of numerical monte carlo simulations in groundwater hydrology. Water Resour. Res., 40(4) (2004)Google Scholar
  6. 6.
    Batycky, R.P.: A Three-Dimensional Two-Phase Field Scale Streamline Simulator. Phd thesis, Stanford University. Doctoral dissertation (1997)Google Scholar
  7. 7.
    Botev, Z.I., Grotowski, J.F., Kroese, D.P.: Kernel density estimation via diffusion. Ann. Stat. 38, 2916–2957 (2010)CrossRefGoogle Scholar
  8. 8.
    Buckley, S.E., Leverett, M.C.: Mechanism of fluid displacement in sands. Trans. AIME 146(1), 107–116 (1942)CrossRefGoogle Scholar
  9. 9.
    Charrier, J.: Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50, 216–246 (2012)CrossRefGoogle Scholar
  10. 10.
    Chen, Y., Oliver, D.S., Zhang, D.: Data assimilation for nonlinear problems by ensemble Kalman filter with reparameterization. J. Pet. Sci. Eng. 66, 1–14 (2009)CrossRefGoogle Scholar
  11. 11.
    Cline, D.B.H., Hart, J.D.: Kernel estimation of densities with discontinuities or discontinuous derivatives. Statistics 22, 69–84 (1991)CrossRefGoogle Scholar
  12. 12.
    Cvetkovic, V., Dagan, G.: Transport of kinetically sorbing solute by steady random velocity in heterogeneous porous formations. J. Fluid Mech. 265, 189–215 (1994)Google Scholar
  13. 13.
    Dagan, G.: Flow and Transport in Porous Formations. Springer-Verlag, Berlin (1989)CrossRefGoogle Scholar
  14. 14.
    Dagan, G., Neuman, S.: Subsurface Flow and Transport: A Stochastic Approach. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  15. 15.
    Deutsch, C.V.: Geostatistical Reservoir Modeling. Oxford University Press, Oxford (2002)Google Scholar
  16. 16.
    Deutsch, C.V., Journel, A.G.: GSLIB: Geostatistical Software Library and User’s Guide. Oxford University Press, Oxford (1998)Google Scholar
  17. 17.
    Foo, J., Wan, X., Karniadakis, G.: The multi-element probabilistic collocation method: Error analysis and simulation. J. Comput. Phys. 227, 9572–9595 (2008)CrossRefGoogle Scholar
  18. 18.
    Gelhar, L.W.: Stochastic Subsurface Hydrology. Prentice-Hall, Upper Saddle River (1986)Google Scholar
  19. 19.
    Gu, Y., Oliver, D.S.: The ensemble Kalman filter for continuous updating of reservoir simulation models. J. Energy Resour. Technol. 128, 79–87 (2006)CrossRefGoogle Scholar
  20. 20.
    Hewett, T.A., Yamada, T.: Theory for the semi-analytical calculation of oil recovery and effective relative permeabilities using streamtubes. Adv. Water Resour. 20(5), 279–292 (1997)CrossRefGoogle Scholar
  21. 21.
    Ibrahima, F., Meyer, D.W., Tchelepi, H.: Distribution functions of saturation for stochastic nonlinear two-phase flow. Transp. Porous Media 109, 81–107 (2015)CrossRefGoogle Scholar
  22. 22.
    Ibrahima, F., Tchelepi, H.A., Meyer, D.W.: Uncertainty quantification of two-phase flow in heterogeneous reservoirs using a streamline-based Pdf formulation. In: ECMOR XV-15th European Conference on the Mathematics of Oil Recovery. Amsterdam (2016)Google Scholar
  23. 23.
    Jarman, K.D., Russell, T.F.: Moment equations for stochastic immiscible flow. Technical Report 181. Center for Computational Mathematics, University of Colorado at Denver, vol. 181 (2002)Google Scholar
  24. 24.
    Jarman, K.D., Tartakovsky, A.M.: Divergence of solutions to solute transport moment equations. Geophys. Res. Lett. 35(15) (2008)Google Scholar
  25. 25.
    Le Maitre, O., Knio, H., Najm, H., Ghanem, R.: Uncertainty propagation using wiener-haar expansions. J. Comput. Phys. 197, 28–57 (2004)CrossRefGoogle Scholar
  26. 26.
    Li, L., Tchelepi, H.A.: Conditional statistical moment equations for dynamic data integration in heterogeneous reservoirs. In: SPE Reservoir Simulation Symposium. Houston (2005)Google Scholar
  27. 27.
    Liao, Q., Zhang, D.: Probabilistic collocation method for strongly nonlinear problems: 1. Transform by location. Water Resour. Res. 49, 7911–7928 (2013)CrossRefGoogle Scholar
  28. 28.
    Liao, Q., Zhang, D.: Probabilistic collocation method for strongly nonlinear problems: 2. Transform by displacement. Water Resour. Res. 50, 8736–8759 (2014)CrossRefGoogle Scholar
  29. 29.
    Liu, G., Lu, Z., Zhang, D.: Stochastic uncertainty analysis for solute transport in randomly heterogeneous media using a Karhunen-Loève-based moment equation approach. Water Resour. Res. 43(7) (2007)Google Scholar
  30. 30.
    Loève, M.: Probability Theory, vol. 45. Springer-Verlag, Berlin (1977)Google Scholar
  31. 31.
    Mallison, B.T., Gerritsen, M.G., Matringe, S.F.: Improved mappings for streamline-based simulation. SPE J. 11(3), 294–302 (2006)CrossRefGoogle Scholar
  32. 32.
    Mariethoz, G., Caers, J.: Multiple-point Geostatistics: Stochastic Modeling with Training Images. Wiley-Blackwell, New York (2014)CrossRefGoogle Scholar
  33. 33.
    Matringe, S.F., Gerritsen, M.G.: On accurate tracing of streamlines. In: SPE Annual Technical Conference and Exhibition. Houston (2004)Google Scholar
  34. 34.
    Meyer, D.W., Jenny, P., Tchelepi, H.A.: A joint velocity-concentration PDF method for tracer flow in heterogeneous porous media. Water Resour. Res. 46(12) (2010)Google Scholar
  35. 35.
    Meyer, D.W., Tchelepi, H.A.: Particle-based transport model with Markovian velocity process for tracer dispersion in highly heterogeneous porous media. Water Resour. Res. 46(11) (2010)Google Scholar
  36. 36.
    Meyer, D.W., Tchelepi, H.A., Jenny, P.: A fast simulation method for uncertainty quantification of subsurface flow and transport. Water Resour. Res. 49(5), 2359–2379 (2013)CrossRefGoogle Scholar
  37. 37.
    Müller, F., Jenny, P., Meyer, D.W.: Multilevel Monte Carlo for two phase flow and Buckley-Leverett transport in random heterogeneous porous media. J. Comput. Phys. 250, 685–702 (2013)CrossRefGoogle Scholar
  38. 38.
    Muskat, M., Wyckoff, R.: Theoretical analysis of waterflooding networks. Trans. AIME 107, 62–77 (1934)CrossRefGoogle Scholar
  39. 39.
    Peaceman, D.W.: Fundamentals of numerical reservoir simulation. Elsevier, Amsterdam (1977)Google Scholar
  40. 40.
    Pollock, D.: Semianalytical computation of path lines for finite-difference models. Ground Water 26, 743–750 (1988)CrossRefGoogle Scholar
  41. 41.
    Pope, S.B.: PDF methods for turbulent reactive flows. Progress Energy Combust. Sci. 11, 119–192 (1985)CrossRefGoogle Scholar
  42. 42.
    Shahvali, M., Mallison, B., Wei, K., Gross, H.: An alternative to streamlines for flow diagnostics on structured and unstructured grids. SPE J. 17(3), 768–778 (2012)CrossRefGoogle Scholar
  43. 43.
    Tartakovsky, D.M., Broyda, S.: PDF equations for advective-reactive transport in heterogeneous porous media with uncertain properties. J. Contam. Hydrol. 120, 129–140 (2011)CrossRefGoogle Scholar
  44. 44.
    Thiele, M.R., Batycky, R.P., Blunt, M.J., Orr, F.M.: Modeling flow in heterogeneous media using streamtubes and streamlines. SPE 10, 5–12 (1996)Google Scholar
  45. 45.
    Venturi, D., Tartakovsky, D.M., Tartakovsky, A.M., Karniadakis, G.E.: Exact PDF equations and closure approximations for advective-reactive transport. J. Comput. Phys. 243, 323–343 (2013)CrossRefGoogle Scholar
  46. 46.
    Wang, P., Tartakovsky, D.M., Jarman, K.D., Tartakovsky, A.M.: CDF solutions of Buckley-Leverett equation with uncertain parameters. Multiscale Model. Simul. 11(1), 118–133 (2013)CrossRefGoogle Scholar
  47. 47.
    Wied, D., Weissbach, R.: Consistency of the kernel density estimator: a survey. Stat. Papers 53, 1–21 (2012)CrossRefGoogle Scholar
  48. 48.
    Zhang, D.: Stochastic Methods for Flow in Porous Media: Coping with Uncertainties. Academic Press, Cambridge (2002)Google Scholar
  49. 49.
    Zhang, D., Li, L., Tchelepi, H.A.: Stochastic formulation for uncertainty analysis of two-phase flow in heterogeneous reservoirs. SPE J. 5(1), 60–70 (2000)CrossRefGoogle Scholar
  50. 50.
    Zhang, D., Tchelepi, H.A.: Stochastic analysis of immiscible two-phase flow in heterogeneous media. SPE J. 4(4), 380–388 (1999)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute for Computational & Mathematical EngineeringStanford UniversityStanfordUSA
  2. 2.Department of Energy Resources EngineeringStanford UniversityStanfordUSA
  3. 3.Institute of Fluid DynamicsETH ZürichZürichSwitzerland

Personalised recommendations