Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

  2. Aziz, K., Settari, A.: Petroleum Reservoir Simulation, vol. 476. Applied Science Publishers, England (1979)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  5. Ballio, F., Guadagnini, A.: Convergence assessment of numerical monte carlo simulations in groundwater hydrology. Water Resour. Res., 40(4) (2004)

  6. Batycky, R.P.: A Three-Dimensional Two-Phase Field Scale Streamline Simulator. Phd thesis, Stanford University. Doctoral dissertation (1997)

  7. Botev, Z.I., Grotowski, J.F., Kroese, D.P.: Kernel density estimation via diffusion. Ann. Stat. 38, 2916–2957 (2010)

    Article  Google Scholar 

  8. Buckley, S.E., Leverett, M.C.: Mechanism of fluid displacement in sands. Trans. AIME 146(1), 107–116 (1942)

    Article  Google Scholar 

  9. Charrier, J.: Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50, 216–246 (2012)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  11. Cline, D.B.H., Hart, J.D.: Kernel estimation of densities with discontinuities or discontinuous derivatives. Statistics 22, 69–84 (1991)

    Article  Google Scholar 

  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)

  13. Dagan, G.: Flow and Transport in Porous Formations. Springer-Verlag, Berlin (1989)

    Book  Google Scholar 

  14. Dagan, G., Neuman, S.: Subsurface Flow and Transport: A Stochastic Approach. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  15. Deutsch, C.V.: Geostatistical Reservoir Modeling. Oxford University Press, Oxford (2002)

    Google Scholar 

  16. Deutsch, C.V., Journel, A.G.: GSLIB: Geostatistical Software Library and User’s Guide. Oxford University Press, Oxford (1998)

    Google Scholar 

  17. Foo, J., Wan, X., Karniadakis, G.: The multi-element probabilistic collocation method: Error analysis and simulation. J. Comput. Phys. 227, 9572–9595 (2008)

    Article  Google Scholar 

  18. Gelhar, L.W.: Stochastic Subsurface Hydrology. Prentice-Hall, Upper Saddle River (1986)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

  24. Jarman, K.D., Tartakovsky, A.M.: Divergence of solutions to solute transport moment equations. Geophys. Res. Lett. 35(15) (2008)

  25. Le Maitre, O., Knio, H., Najm, H., Ghanem, R.: Uncertainty propagation using wiener-haar expansions. J. Comput. Phys. 197, 28–57 (2004)

    Article  Google Scholar 

  26. Li, L., Tchelepi, H.A.: Conditional statistical moment equations for dynamic data integration in heterogeneous reservoirs. In: SPE Reservoir Simulation Symposium. Houston (2005)

  27. Liao, Q., Zhang, D.: Probabilistic collocation method for strongly nonlinear problems: 1. Transform by location. Water Resour. Res. 49, 7911–7928 (2013)

    Article  Google Scholar 

  28. Liao, Q., Zhang, D.: Probabilistic collocation method for strongly nonlinear problems: 2. Transform by displacement. Water Resour. Res. 50, 8736–8759 (2014)

    Article  Google Scholar 

  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)

  30. Loève, M.: Probability Theory, vol. 45. Springer-Verlag, Berlin (1977)

    Google Scholar 

  31. Mallison, B.T., Gerritsen, M.G., Matringe, S.F.: Improved mappings for streamline-based simulation. SPE J. 11(3), 294–302 (2006)

    Article  Google Scholar 

  32. Mariethoz, G., Caers, J.: Multiple-point Geostatistics: Stochastic Modeling with Training Images. Wiley-Blackwell, New York (2014)

    Book  Google Scholar 

  33. Matringe, S.F., Gerritsen, M.G.: On accurate tracing of streamlines. In: SPE Annual Technical Conference and Exhibition. Houston (2004)

  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)

  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)

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  38. Muskat, M., Wyckoff, R.: Theoretical analysis of waterflooding networks. Trans. AIME 107, 62–77 (1934)

    Article  Google Scholar 

  39. Peaceman, D.W.: Fundamentals of numerical reservoir simulation. Elsevier, Amsterdam (1977)

    Google Scholar 

  40. Pollock, D.: Semianalytical computation of path lines for finite-difference models. Ground Water 26, 743–750 (1988)

    Article  Google Scholar 

  41. Pope, S.B.: PDF methods for turbulent reactive flows. Progress Energy Combust. Sci. 11, 119–192 (1985)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  47. Wied, D., Weissbach, R.: Consistency of the kernel density estimator: a survey. Stat. Papers 53, 1–21 (2012)

    Article  Google Scholar 

  48. Zhang, D.: Stochastic Methods for Flow in Porous Media: Coping with Uncertainties. Academic Press, Cambridge (2002)

    Google Scholar 

  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)

    Article  Google Scholar 

  50. Zhang, D., Tchelepi, H.A.: Stochastic analysis of immiscible two-phase flow in heterogeneous media. SPE J. 4(4), 380–388 (1999)

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Institute for Computational & Mathematical Engineering, Stanford University, Stanford, CA, USA

    Fayadhoi Ibrahima

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

    Hamdi A. Tchelepi

  3. Institute of Fluid Dynamics, ETH Zürich, Zürich, Switzerland

    Daniel W. Meyer

Authors
  1. Fayadhoi Ibrahima
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hamdi A. Tchelepi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel W. Meyer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fayadhoi Ibrahima.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ibrahima, F., Tchelepi, H.A. & Meyer, D.W. An efficient distribution method for nonlinear two-phase flow in highly heterogeneous multidimensional stochastic porous media. Comput Geosci 22, 389–412 (2018). https://doi.org/10.1007/s10596-017-9698-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-017-9698-0

Keywords

Search

Navigation