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Transport in Porous Media

, Volume 72, Issue 1, pp 97–120 | Cite as

Numerical modeling of multiphase first-contact miscible flows. Part 2. Front-tracking/streamline simulation

  • Ruben JuanesEmail author
  • Knut-Andreas Lie
Article

Abstract

In this paper we complete the description and application of a computational framework for the numerical simulation of first-contact miscible gas injection processes. The method is based on the front-tracking algorithm, in which numerical solutions to one- dimensional problems are constructed in the form of traveling discontinuities. The efficiency of the front-tracking method relies on the availability of the analytical Riemann solver described in Part 1 and a strategy for simplifying the wave structure for Riemann problems of small amplitude. Several representative examples are used to illustrate the excellent behavior of the front-tracking method. The front-tracking method is extended to simulate higher-dimensional processes through the use of streamlines. The paper presents a validation exercise for a quarter five-spot homogeneous problem, and an application of this computational framework for the simulation of miscible flooding in three-dimensional, highly heterogeneous formations. In this case, we demonstrate that a miscible water-alternating-gas injection scheme is more effective than waterflooding or gas injection alone.

Keywords

Porous media Miscible displacement Water-alternating-gas Riemann problem Front-tracking Streamline simulation 

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References

  1. Aarnes J.E., Kippe V., Lie K.-A. (2005). Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels. Adv. Water Resour. 28: 257–271 CrossRefGoogle Scholar
  2. Batycky R.P., Blunt M.J., Thiele M.R. (1997). A 3D field-scale streamline-based reservoir simulator. SPE Resery. Eng. 11(4): 246–254 Google Scholar
  3. Blunt M and Christie M. (1993). How to predict viscous fingering in three component flow. Transp. Porous Media 12: 207–236 CrossRefGoogle Scholar
  4. Blunt M., Christie M. (1994). Theory of viscous fingering in two phase, three component flow. SPE Advanced Technology Series 2(2): 52–60 Google Scholar
  5. Bratvedt F., Bratvedt K., Buchholz C.F., Gimse T., Holden H., Holden L., Risebro N.H. (1993). Frontline and Frontsim, two full scale, two-phase, black oil reservoir simulators based on front tracking. Surv. Math. Ind. 3: 185–215 Google Scholar
  6. Bressan A., LeFloch P. (1997). Uniqueness of weak solutions to systems of conservation laws. Arch. Rational Mech. Anal. 140(4): 301–317 CrossRefGoogle Scholar
  7. Chen W.H., Durlofsky L.J., Engquist B., Osher S. (1993). Minimization of grid orientation effects through the use of higher-order finite difference methods. SPE Advanced Technology Series 1: 43–52 Google Scholar
  8. Christie M.A., Blunt M.J. (2001). Tenth SPE comparative solution project: A comparison of upscaling techniques. SPE Reserv. Eval. Eng. 4(4): 308–317. url:www.spe.org/csp Google Scholar
  9. Ewing R.E., Russell T.F., Wheeler M.F. (1984). Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics. Comput. Methods Appl. Mech. Eng. 47(1–2): 73–92 CrossRefGoogle Scholar
  10. Gimse T., Risebro N.H. (1992). Solution of the Cauchy problem for a conservation law with a discontinuous flux function. SIAM J. Math. Anal. 23(3): 635–648 CrossRefGoogle Scholar
  11. Gimse T., Risebro N.H. (1993). A note on reservoir simulation for heterogeneous porous media. Transp. Porous Media 10(3): 257–270 CrossRefGoogle Scholar
  12. Glimm, J., Lindquist, B., McBryan O.A., Plohr B., Yaniv S.: Front tracking for petroleum reservoir simulation. In: SPE Reservoir Simulation Symposium. San Francisco, CA (1983) (SPE 12238)Google Scholar
  13. Haugse V., Karlsen K.H., Lie K.-A., Natvig J.R. (2001). Numerical solution of the polymer system by front tracking. Transp. Porous Media 44: 63–83 CrossRefGoogle Scholar
  14. Holden H., Risebro N.H. (1993). A method of fractional steps for scalar conservation laws without the CFL condition. Math. Comp. 60(201): 221–232 CrossRefGoogle Scholar
  15. Holden H., Risebro N.H. (2002). Front Tracking for Hyperbolic Conservation Laws, vol. 152 of Applied Mathematical Sciences. Springer, New York Google Scholar
  16. Isaacson E.L. (1980). Global solution of a Riemann problem for a non-strictly hyperbolic system of conservation laws arising in enhanced oil recovery. Technical report, The Rockefeller University, New York Google Scholar
  17. Jimenez, E., Sabir, K., Datta-Gupta, A., King, M.J.: Spatial error and convergence in streamline simulation. In: SPE Reservoir Simulation Symposium. Houston, TX (2005) (SPE 92873)Google Scholar
  18. Johansen T., Winther R. (1988). The solution of the Riemann problem for a hyperbolic system of conservation laws modeling polymer flooding. SIAM J. Math. Anal. 19(3): 541–566 CrossRefGoogle Scholar
  19. Juanes R., Blunt M.J. (2006). Analytical solutions to multiphase first-contact miscible models with viscous fingering. Transp. Porous Media 64(3): 339–373 CrossRefGoogle Scholar
  20. Juanes R., Lie K.-A. (2007). Numerical modeling of multiphase first-contact miscible flows. Part 1. Analytical Riemann solver. Transp Porous Media 67(3): 375–393. doi:10.1007/s11242–006–9031–1 CrossRefGoogle Scholar
  21. Juanes, R., Lie, K.-A., Kippe, V.: A front-tracking method for hyperbolic three-phase models. In: European Conference on the Mathematics of Oil Recovery, ECMOR IX, vol. 2, paper B025. Cannes, France (2004)Google Scholar
  22. King M.J., Datta-Gupta A. (1998). Streamline simulation: a current perspective. In Situ 22(1): 91–140 Google Scholar
  23. King M.J., Osako I., Datta-Gupta A. (2005). A predictor–corrector formulation for rigorous streamline simulation. Int. J. Numer. Meth. Fluids 47: 739–758 CrossRefGoogle Scholar
  24. Lie K.-A., Juanes R. (2005). A front-tracking method for the simulation of three-phase flow in porous media. Comput. Geosci. 9(1): 29–59 CrossRefGoogle Scholar
  25. Osako I., Datta-Gupta A., King M.J. (2004). Timestep selection during streamline simulation through transverse flux correction. Soc. Pet. Eng. J. 9(4): 450–464 Google Scholar
  26. Pollock D.W. (1988). Semianalytical computation of path lines for finite difference models. Ground Water 26: 743–750 CrossRefGoogle Scholar
  27. Risebro N.H. (1993). A front-tracking alternative to the random choice method. Proc. Amer. Math. Soc 117(4): 1125–1129 CrossRefGoogle Scholar
  28. Risebro N.H., Tveito A. (1991). Front tracking applied to a nonstrictly hyperbolic system of conservation laws. SIAM J. Sci. Stat. Comput. 12(6): 1401–1419 CrossRefGoogle Scholar
  29. Risebro N.H., Tveito A. (1992). A front tracking for conservation laws in one dimension. J. Comput. Phys. 101(1): 130–139 CrossRefGoogle Scholar
  30. Russell T.F., Wheeler M.F. (1983). Finite element and finite difference methods for continuous flows in porous media. In: Ewing, R.E. (eds) The Mathematics of Reservoir Simulation, pp 35–106. SIAM, Philadelphia, PA Google Scholar
  31. Schlumberger: Eclipse technical description, v. 2003A (2003)Google Scholar
  32. Shubin G.R., Bell J.B. (1984). An analysis of the grid orientation effect in numerical simulation of miscible displacements. Comput. Methods Appl. Mech. Eng. 47: 47–71 CrossRefGoogle Scholar
  33. Tveito A., Winther R. (1995). The solution of nonstrictly hyperbolic conservation laws may be hard to compute. SIAM J. Sci. Comput. 16(2): 320–329 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of Applied MathematicsSINTEF ICTOsloNorway

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