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Upscaling Non-Darcy Flow

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Abstract

We consider upscaling of non-Darcy flow in heterogeneous porous media. Our approach extends the pressure-based numerical homogenization procedure for linear Darcy flow, due to Durlofsky, to the nonlinear case. The effective coefficients are not constants but rather mildly varying functions of prevailing gradients of pressure. The upscaled model approximates the fine grid model accurately and, in some cases, more accurately than what is expected for Darcy flow; this is due to the non-Darcy effects which suppress heterogeneity. We provide comparisons of alternative approaches as well as consider several variants of numerical realizations of the non-Darcy flow model. Numerical results show effectiveness of the upscaling procedure.

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References

  • Arbogast, T.: An overview of subgrid upscaling for elliptic problems in mixed form. In: Current trends in scientific computing (Xi’an, 2002), Contemp. Math. 329, 21–32 (2003)

  • Barker, J., Thibeau, S.: A criticial review of the use of pseudo relative permeabilities for upscaling. In: SPE 35491 (1996)

  • Bear J.: Dynamics of Fluids in Porous Media. Dover, New York (1972)

    Google Scholar 

  • Bennethum L.S., Giorgi T.: Generalized Forchheimer equation for two-phase flow based on hybrid mixture theory. Transp. Porous Media 26(3), 261–275 (1997)

    Article  Google Scholar 

  • Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic analysis for periodic structures. In: Studies in Mathematics and its Applications, vol. 5. North-Holland Publishing Co., Amsterdam (1978)

  • Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. In: Springer Series in Computational Mathematics, vol. 15. Springer-Verlag, New York (1991)

  • Chen Z.: Expanded mixed finite element methods for linear second-order elliptic problems. I. RAIRO Modél. Math. Anal. Numér. 32(4), 479–499 (1998)

    Google Scholar 

  • Chen Y., Durlofsky L.J.: Adaptive local-global upscaling for general flow scenarios in heterogeneous formations. Transp. Porous Media 62(2), 157–185 (2006)

    Article  Google Scholar 

  • Chen Z., Yue X.: Numerical homogenization of well singularities in the flow transport through heterogeneous porous media. Multiscale Model. Simul. 1(2), 260–303 (2003) (electronic)

    Article  Google Scholar 

  • 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 

  • Chen Z., Huan G., Li B.: A pseudo function approach in reservoir simulation. Int. J. Numer. Anal. Model. 2(suppl.), 58–67 (2005)

    Google Scholar 

  • Douglas Jr., J., Paes-Leme, P.J., Giorgi, T.: Generalized Forchheimer flow in porous media. In: Boundary Value Problems for Partial Differential Equations and Applications, RMA Research Notes in Applied Mathematics, vol. 29, 99–111, Masson, Paris (1993)

  • Dullien F.: Porous Media. Academic Press, San Diego (1972)

    Google Scholar 

  • Durlofsky L.J.: Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resour. Res. 27(5), 699–708 (1991)

    Article  Google Scholar 

  • Durlofsky L.: Upscaling of geological models for reservoir simulation: issues and approaches. Comput. Geosci. 6, 569–571 (2002)

    Article  Google Scholar 

  • Efendiev Y., Durlofsky L.J.: Numerical modeling of subgrid heterogeneity in two phase flow simulations. Water Resour. Res. 38(8), 3.1–3.11 (2002)

    Article  Google Scholar 

  • Efendiev Y., Pankov A.: Numerical homogenization and correctors for nonlinear elliptic equations. SIAM J. Appl. Math. 65(1), 43–68 (2004) (electronic)

    Article  Google Scholar 

  • Efendiev Y., Ginting V., Hou T., Ewing R.: Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys. 220(1), 155–174 (2006)

    Article  Google Scholar 

  • Efendiev Y.R., Hou T.Y., Wu X.H.: Convergence of a nonconforming multiscale finite element method. SIAM J. Numer. Anal. 37(3), 888–910 (2000) (electronic)

    Article  Google Scholar 

  • Ergun S.: Fluid flow through packed columns. Chem. Eng. Prog. 48, 89–94 (1952)

    Google Scholar 

  • Ergun S., Orning A.: Fluid flow through randomly packed columns and fluidized beds. J. Ind. Eng. Chem. 41, 1179–1184 (1949)

    Article  Google Scholar 

  • Ewing R.E., Lazarov R.D., Lyons S.L., Papavassiliou D.V., Pasciak J., Qin G.: Numerical well model for non-Darcy flow through isotropic porous media. Comput. Geosci. 3(3–4), 185–204 (1999a)

    Article  Google Scholar 

  • Ewing R.E., Wang J., Weekes S.L.: On the simulation of multicomponent gas flow in porous media. Appl. Numer. Math. 31(4), 405–427 (1999b)

    Article  Google Scholar 

  • Fabrie P., Langlais M.: Mathematical analysis of miscible displacement in porous medium. SIAM J. Math. Anal. 23(6), 1375–1392 (1992)

    Article  Google Scholar 

  • Frih N., Roberts J.E., Saada A.: Modeling fractures as interfaces: a model for Forchheimer fractures. Comput. Geosci. 12, 91–104 (2008)

    Article  Google Scholar 

  • Forchheimer, P.: Wasserbewegung durch Boden. Zeit. Ver. Deut. Ing. (45), 1781–1788 (1901)

  • Fourar M., Lenormand R., Karimi-Fard M., Horne R.: Inertia effects in high-rate flow through heterogeneous porous media. Transp. Porous Media 60, 353–370 (2005). doi:10.1007/s11242-004-6800-6

    Article  Google Scholar 

  • Geertsma J.: The effects of non-Darcy flow on the behavior of hydraulically fractured gas wells. J. Pet. Tech. 883, 1169 (1976)

    Google Scholar 

  • Holden L., Nielsen B.F.: Global upscaling of permeability in heterogeneous reservoirs; the output least squares (OLS) method. Transp. Porous Media 40(2), 115–143 (2000)

    Article  Google Scholar 

  • Jikov V.V., Kozlov S.M., Oleinik O.A.: Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin (1994)

    Google Scholar 

  • Jones, S.: Using the inertial coefficient beta to characterize heterogeneity in reservoir rock. In: SPE 16949, pp. 165–175 (1987)

  • Lake, L.W.: Enhanced oil recovery. Prentice Hall, Englewood Cliffs (1989)

    Google Scholar 

  • Marusic-Paloka E., Mikelic A.: The derivation of a nonlinear filtration law including the inertia effects via homogenization. Nonlinear Anal. Ser. A Theory Methods 42(1), 97–137 (2000)

    Article  Google Scholar 

  • Narayanaswamy G., Sharma M.M., Pope G.: Effect of heterogeneity on the non-Darcy flow coefficient. SPE Reser. Eval. Eng. 2(3), 296–302 (1999)

    Google Scholar 

  • Park E.J.: Mixed finite element methods for generalized Forchheimer flow in porous media. Numer. Methods Partial Differ. Equ. 21(2), 213–228 (2005)

    Article  Google Scholar 

  • Peaceman D.W.: Fundamentals of numerical reservoir simulation, 1st edn. Elsevier Scientfic Publishing Company, Amsterdam (1977)

    Google Scholar 

  • Peszyńska M.: Mortar adaptivity in mixed methods for flow in porous media. Int. J. Numer. Anal. Model 2(3), 241–282 (2005)

    Google Scholar 

  • Peszyńska, M., Showalter, R.E.: Multiscale elliptic-parabolic systems for flow and transport. Electron. J. Differ. Equ. (147), 30 pp. (electronic) (2007)

  • Peszyńska, M., Jenkins, E., Wheeler, M.F.: Boundary conditions for fully implicit two-phase flow model. In: Feng, X., Schulze, T.P. (eds.) Recent Advances in Numerical Methods for Partial Differential Equations and Applications. Contemporary Mathematics Series, vol. 306, pp. 85–106. American Mathematical Society, Providence (2002a)

  • Peszyńska M., Wheeler M.F., Yotov I.: Mortar upscaling for multiphase flow in porous media. Comput. Geosci. 6, 73–100 (2002b)

    Article  Google Scholar 

  • Peszynska, M., Trykozko, A., Augustson, K.: Computational modeling of inertia effects at porescale. In: Allen, G., Nabrzyski, J., Seidel, E., van Albada, D., Dongarra, J., and Sloot, P. (eds.) ICCS 2009 Proceedings, Springer Lecture Notes in Computer Science (2009, accepted)

  • Raviart R.A., Thomas J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics, vol. 606, 292–315. Springer-Verlag, New York (1977)

  • Reese, J.P., Long, K.R., Kelley, C.T., Gray, W.G., Miller, C.T.: Simulating non-Darcy flow through porous media using sundance. In: Binning, P.J., Engesgaard, P.K., Dahle, H.K., Pinder, G.F., Gray, W.G. (eds.) Proceedings of the XVI International Conference on Computational Methods in Water Resources, Copenhagen. http://proceedings.cmwr-xvi.org (2006)

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

    Article  Google Scholar 

  • Russell T.F., Wheeler M.F. (1983) Finite element and finite difference methods for continuous flows in porous media. In: Ewing, R.E. (ed.) The Mathematics of Reservoir Simulation, pp. 35–106. SIAM, Philadelphia

  • Ruth D., Ma H.: On the derivation of the Forchheimer equation by means of the averaging theorem. Transp. Porous Media 7(3), 255–264 (1992)

    Article  Google Scholar 

  • Weinan E., Engquist B., Li X., Ren W., Vanden-Eijnden E.: Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2(3), 367–450 (2007)

    Google Scholar 

  • Zijl W., Trykozko A.: Numerical homogenization of the absolute permeability tensor around wells. SPE J. 5, 399–408 (2001)

    Google Scholar 

  • Zijl W., Trykozko A.: Numerical homogenization of two-phase flow in porous media. Comput. Geosci. 6(1), 49–71 (2002)

    Article  Google Scholar 

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  1. Mathematics Department, Oregon State University, Corvallis, OR, 97331, USA

    C. R. Garibotti & M. Peszyńska

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  1. C. R. Garibotti
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  2. M. Peszyńska
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Correspondence to M. Peszyńska.

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Garibotti, C.R., Peszyńska, M. Upscaling Non-Darcy Flow. Transp Porous Med 80, 401–430 (2009). https://doi.org/10.1007/s11242-009-9369-2

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