Computational Geosciences

, Volume 21, Issue 4, pp 807–832 | Cite as

Intrusive uncertainty quantification for hyperbolic-elliptic systems governing two-phase flow in heterogeneous porous media

Original Paper


We model random locations of spatial interfaces of heterogeneities in porous media by means of the hybrid stochastic Galerkin (HSG) approach. This approach extends the concept of the generalized polynomial chaos (PC) expansion for a multi-element decomposition of the multidimensional stochastic space. In this way, the physically two-dimensional hyperbolic-elliptic fractional flow formulation for two-phase flow in heterogeneous porous media is transformed from a random partial differential equation into a deterministic system for the coefficients of the PC expansion of the primary unknown saturation, total velocity, and global pressure. The hyperbolic part is discretized based on a central-upwind finite volume scheme along with a mixed finite element method for the elliptic part. The latter partly uses the tensor product structure of the saddle point system together with appropriate preconditioning to speed up the computations. Since we use the sequential implicit pressure explicit saturation (IMPES) approach, the elliptic part is solved in every time step. The proposed method is particularly well-suited for parallel computations and allows for the consideration of a huge variety of complex flow problems. We illustrate the power of the method by means of several striking numerical examples of different complexities including their numerical convergence analysis.


Hybrid stochastic Galerkin Finite volume method Mixed finite element method Two-phase flow Porous media 

Mathematics Subject Classification (2010)

76S05 65M08 65N30 60H35 65M75 


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The authors would like to thank the German Research Foundation (DFG) for the financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart. Moreover, we thank the referees for the constructive criticism.


  1. 1.
    Alpert, B.K.: A class of bases in L 2 for the sparse representation of integral operators. SIAM J. Math. Anal. 24(1), 246–262 (1993). doi: 10.1137/0524016 CrossRefGoogle Scholar
  2. 2.
    Andreianov, B., Karlsen, K.H., Risebro, N.H.: On vanishing viscosity approximation of conservation laws with discontinuous flux. Netw. Heterog. Media 5(3), 617–633 (2010). doi: 10.3934/nhm.2010.5.617 CrossRefGoogle Scholar
  3. 3.
    Barth, A., Bürger, R., Kröker, I., Rohde, C.: Computational uncertainty quantification for a clarifier-thickener model with several random perturbations: a hybrid stochastic Galerkin approach. Comput. Chem. Eng. 89, 11–26 (2016). doi: 10.1016/j.compchemeng.2016.02.016. CrossRefGoogle Scholar
  4. 4.
    Barth, A., Schwab, C., Zollinger, N.: Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients. Numer. Math. 119(1), 123–161 (2011). doi: 10.1007/s00211-011-0377-0 CrossRefGoogle Scholar
  5. 5.
    Bear, J.: Dynamics of Fluids in Porous Media. No. Bd. 1 in Environmental Science Series. American Elsevier Pub. Co (1972).
  6. 6.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numerica 14(1), 1–137 (2005)CrossRefGoogle Scholar
  7. 7.
    Berres, S., Bürger, R., Karlsen, K.H.: Central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions Proceedings of the 10th International Congress on Computational and Applied Mathematics (ICCAM-2002), vol. 164/165, pp. 53–80 (2004). doi: 10.1016/S0377-0427(03)00496-5
  8. 8.
    Bourgeat, A., Jurak, M., Smaï, F.: Two-phase, partially miscible flow and transport modeling in porous media; application to gas migration in a nuclear waste repository. Comput. Geosci. 13(1), 29–42 (2008). doi: 10.1007/s10596-008-9102-1 CrossRefGoogle Scholar
  9. 9.
    Bürger, R., Kröker, I., Rohde, C.: A hybrid stochastic Galerkin method for uncertainty quantification applied to a conservation law modelling a clarifier-thickener unit. ZAMM Z. Angew. Math. Mech. 77(10), 793–817 (2014). doi: 10.1002/zamm.201200174 CrossRefGoogle Scholar
  10. 10.
    Cameron, R.H., Martin, W.T.: The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. Ann. Math. 48, 385–392 (1947)CrossRefGoogle Scholar
  11. 11.
    Class, H., Ebigbo, A., Helmig, R., Dahle, H.K., Nordbotten, J.M., Celia, M.A., Audigane, P., Darcis, M., Ennis-King, J., Fan, Y., Flemisch, B., Gasda, S.E., Jin, M., Krug, S., Labregere, D., Naderi Beni, A., Pawar, R.J., Sbai, A., Thomas, S.G., Trenty, L., Wei, L.: A benchmark study on problems related to CO2 storage in geologic formations. Comput. Geosci. 13(4), 409–434 (2009). doi: 10.1007/s10596-009-9146-x CrossRefGoogle Scholar
  12. 12.
    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, Grundlehren Der Mathematischen Wissenschaften, 3rd edn., vol. 325. Berlin Springer, Dordrecht (2010). doi: 10.1007/978-3-642-04048-1
  13. 13.
    Eiermann, M., Ernst, O.G., Ullmann, E.: Computational aspects of the stochastic finite element method. Comput. Vis. Sci. 10(1), 3–15 (2007)CrossRefGoogle Scholar
  14. 14.
    Ernst, O.G., Powell, C.E., Silvester, D.J., Ullmann, E.: Efficient solvers for a linear stochastic Galerkin mixed formulation of diffusion problems with random data. SIAM J. Sci. Comput. 31(2), 1424–1447 (2009)CrossRefGoogle Scholar
  15. 15.
    Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: a Spectral Approach. Springer, New York (1991)CrossRefGoogle Scholar
  16. 16.
    Gilman, A., Beckie, R.: Flow of coal-bed methane to a gallery. Transport Porous Med. 41(1), 1–16 (2000). doi: 10.1023/A:1006754108197
  17. 17.
    Kissling, F., Karlsen, K.: On the singular limit of a two-phase flow equation with heterogeneities and dynamic capillary pressure. ZAMM Z. Angew. Math. Mech. 94(7-8), 678–689 (2014). doi: 10.1002/zamm.201200141 CrossRefGoogle Scholar
  18. 18.
    Köppel, M., Kröker, I., Rohde, C.: Finite volumes for complex applications VII—methods and theoretical aspects: FVCA 7, Berlin, June 2014 Stochastic Modeling for Heterogeneous Two-Phase Flow, pp. 353–361. Springer International Publishing, Cham (2014). doi: 10.1007/978-3-319-05684-5_34
  19. 19.
    Kröker, I., Nowak, W., Rohde, C.: A stochastically and spatially adaptive parallel scheme for uncertain and nonlinear two-phase flow problems. Comput. Geosci. 19(2), 269–284 (2015). doi: 10.1007/s10596-014-9464-5 CrossRefGoogle Scholar
  20. 20.
    Kurganov, A., Petrova, G.: Central-upwind schemes on triangular grids for hyperbolic systems of conservation laws. Numerical Methods for Partial Differential Equations 21(3), 536–552 (2005). doi: 10.1002/num.20049 CrossRefGoogle Scholar
  21. 21.
    Le Maître, O.P., Najm, H.N., Ghanem, R.G., Knio, O.M.: Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 197(2), 502–531 (2004). doi: 10.1016/ CrossRefGoogle Scholar
  22. 22.
    Matthies, H.G., Keese, A.: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg. 194(12-16), 1295–1331 (2005). doi: 10.1016/j.cma.2004.05.027 CrossRefGoogle Scholar
  23. 23.
    Mishra, S., Schwab, C.: Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data. Math. Comp. 81(280), 1979–2018 (2012). doi: 10.1090/S0025-5718-2012-02574-9 CrossRefGoogle Scholar
  24. 24.
    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). doi: 10.1016/ CrossRefGoogle Scholar
  25. 25.
    Nielsen, D.R., Th. Van Genuchten, M., Biggar, J.W.: Water flow and solute transport processes in the unsaturated zone. Water Resour. Res. 22(9S), 89S–108S (1986). doi: 10.1029/WR022i09Sp0089S CrossRefGoogle Scholar
  26. 26.
    Oladyshkin, S., Class, H., Helmig, R., Nowak, W.: A concept for data-driven uncertainty quantification and its application to carbon dioxide storage in geological formations. Adv. Water Resour. 34(11), 1508–1518 (2011). doi: 10.1016/j.advwatres.2011.08.005. CrossRefGoogle Scholar
  27. 27.
    Oostrom, M., Hofstee, C., Walker, R., Dane, J.: Movement and remediation of trichloroethylene in a saturated heterogeneous porous medium: 1. Spill behavior and initial dissolution. J. Contam. Hydrol. 37(1–2), 159–178 (1999). doi: 10.1016/S0169-7722(98)00153-3.
  28. 28.
    Panov, E.Y.: Existence of strong traces for generalized solutions of multidimensional scalar conservation laws. J. Hyperbolic Differ. Equ. 2(4), 885–908 (2005). doi: 10.1142/S0219891605000658 CrossRefGoogle Scholar
  29. 29.
    Pettersson, P., Iaccarino, G., Nordström, J.: An intrusive hybrid method for discontinuous two-phase flow under uncertainty. Comput. Fluids J. 86, 228–239 (2013). doi: 10.1016/j.compfluid.2013.07.009.
  30. 30.
    Pettersson, P., Tchelepi, H. A.: Stochastic Galerkin framework with locally reduced bases for nonlinear two-phase transport in heterogeneous formations. Comput. Method. Appl. M. 310, 367–387 (2016). doi: 10.1016/j.cma.2016.07.013.
  31. 31.
    Poëtte, G., Després, B., Lucor, D.: Uncertainty quantification for systems of conservation laws. J. Comput. Phys. 228(7), 2443–2467 (2009). doi: 10.1016/ CrossRefGoogle Scholar
  32. 32.
    Schmidt, A., Siebert, K.G.: Design of Adaptive Finite Element Software. Lecture Notes in Computational Science and Engineering, vol. 42. Springer, Berlin (2005)Google Scholar
  33. 33.
    Shewchuk, J.R.: Triangle: engineering a 2D quality mesh generator and Delaunay triangulator. In: Lin, M. C., Manocha, D. (eds.) Applied Computational Geometry: Towards Geometric Engineering, Lecture Notes in Computer Science, vol. 1148, pp. 203–222. Springer. From the First ACM Workshop on Applied Computational Geometry (1996)Google Scholar
  34. 34.
    Traverso, L., Phillips, T., Yang, Y.: Efficient stochastic FEM for flow in heterogeneous porous media. Part 1: random Gaussian conductivity coefficients. Int. J. Numer. Meth. Fl. 74(5), 359–385 (2014)CrossRefGoogle Scholar
  35. 35.
    Tryoen, J., Le Maître, O., Ndjinga, M., Ern, A.: Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems. J. Comput. Phys. 229(18), 6485–6511 (2010). doi: 10.1016/ CrossRefGoogle Scholar
  36. 36.
    Tryoen, J., Maître, O.L., Ern, A.: Adaptive anisotropic spectral stochastic methods for uncertain scalar conservation laws. SIAM J. Sci. Comput. 34(5), A2459–A2481 (2012). doi: 10.1137/120863927  10.1137/120863927 CrossRefGoogle Scholar
  37. 37.
    Tveito, A., Winther, R.: The solution of nonstrictly hyperbolic conservation laws may be hard to compute. SIAM J. Sci. Comput. 16(2), 320–329 (1995). doi: 10.1137/0916021 CrossRefGoogle Scholar
  38. 38.
    Uzawa, H.: Iterative methods for concave programming. In: Arrow, K., Hurwicz, L., Uzawa, H. (eds.) Studies in Linear and Nonlinear Programming, pp. 154–165. Stanford University Press (1958)Google Scholar
  39. 39.
    Wan, X., Karniadakis, G.E.: An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209(2), 617–642 (2005). doi: 10.1016/
  40. 40.
    Wiener, N.: The homogeneous chaos. Amer. J. Math. 60(4), 897–936 (1938). doi: 10.2307/2371268 CrossRefGoogle Scholar
  41. 41.
    Xiu, D., Karniadakis, G.E.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002). doi: 10.1137/S1064827501387826 CrossRefGoogle Scholar
  42. 42.
    Xiu, D., Karniadakis, G.E.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187(1), 137–167 (2003). doi: 10.1016/S0021-9991(03)00092-5 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.IANSUniverstität StuttgartStuttgartGermany
  2. 2.IANSUniverstität StuttgartStuttgartGermany

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