Transport in Porous Media

, Volume 67, Issue 3, pp 375–393 | Cite as

Numerical modeling of multiphase first-contact miscible flows. Part 1. Analytical Riemann solver

  • Ruben JuanesEmail author
  • Knut-Andreas Lie
Original Paper


In this series of two papers, we present a front-tracking method for the numerical simulation of first-contact miscible gas injection processes. The method is developed for constructing very accurate (or even exact) solutions to one-dimensional initial-boundary-value problems in the form of a set of evolving discontinuities. The evolution of the discontinuities is given by analytical solutions to Riemann problems. In this paper, we present the mathematical model of the problem and the complete Riemann solver, that is, the analytical solution to the one-dimensional problem with piecewise constant initial data separated by a single discontinuity, for any left and right states. The Riemann solver presented here is the building block for the front-tracking/streamline method described and applied in the second paper.


Porous media First-contact miscible displacement Water-alternating-gas Shocks Riemann problem Analytical solution Front-tracking 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ancona F., Marson A. (2001). A note on the Riemann problem for general n ×  n conservation laws. J. Math. Anal. Appl. 260, 279–293CrossRefGoogle Scholar
  2. Blunt M., Christie M. (1993). How to predict viscous fingering in three component flow. Transp. Porous Media 12, 207–236CrossRefGoogle Scholar
  3. Blunt M., Christie M. (1994). Theory of viscous fingering in two phase, three component flows. SPE Adv. Technol. Ser. 2(2):52–60Google Scholar
  4. Harten A. (1983). High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49(3):357–393CrossRefGoogle Scholar
  5. Harten A., Engquist B., Osher S., Chakravarthy S. (1987). Uniformly high-order accurate essentially nonoscillatory schemes, III. J. Comput. Phys. 71, 231–241CrossRefGoogle Scholar
  6. Helfferich F.G. (1981). Theory of multicomponent, multiphase displacement in porous media. Soc. Pet. Eng. J. 21(1):51–62 Petrol. Trans. AIME, 271.Google Scholar
  7. Hirasaki G.J. (1981). Application of the theory of multicomponent, multiphase displacement to three-component, two-phase surfactant flooding. Soc. Pet. Eng. J. 21(2):191–204 Petrol. Trans. AIME, 271.Google Scholar
  8. 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 YorkGoogle Scholar
  9. Jessen K., Michelsen M.L., Stenby E.H. (1998). Global approach for calculation of the minimum miscibility pressure. Fluid Phase Equilib. 153, 251–263CrossRefGoogle Scholar
  10. Jessen K., Stenby E.H., Orr F.M. (2004). Interplay of phase behavior and numerical dispersion in finite-difference compositional simulation. Soc. Pet. Eng. J. 9(2):193–201Google Scholar
  11. Johansen T., Tveito A., Winther R. (1989). A Riemann solver for a two-phase multicomponent process. SIAM J. Sci. Comput. 10(5):846–879CrossRefGoogle Scholar
  12. 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):541–566CrossRefGoogle Scholar
  13. Johansen T., Winther R. (1989). The Riemann problem for multicomponent polymer flooding. SIAM J. Math. Anal. 20(4):908–929CrossRefGoogle Scholar
  14. Juanes R. (2005). Determination of the wave structure of the three-phase flow Riemann problem. Transp. Porous Media 60(2):135–139CrossRefGoogle Scholar
  15. Juanes, R., Al-Shuraiqi, H.S., Muggeridge, A.H., Grattoni, C.A., Blunt, M.J.: Experimental and numerical validation of an analytical model of viscous fingering in two-phase, three-component flows. J. Fluid Mech. (In Review) (2005)Google Scholar
  16. Juanes R., Blunt M.J. (2006a). Analytical solutions to multiphase first-contact miscible models with viscous fingering. Transp. Porous Media 64(3):339–373CrossRefGoogle Scholar
  17. Juanes, R., Blunt, M.J.: Impact of viscous fingering on the prediction of optimum WAG ratio. In: SPE/DOE Symposium on Improved Oil Recovery. Tulsa, OK. (SPE 99721) (2006b)Google Scholar
  18. Juanes, R., Lie, K.-A.: Numerical modeling of multiphase first-contact miscible flow. Part 2. Front-tracking/streamline simulation. Transp. Porous Media (In Review) (2006)Google Scholar
  19. 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
  20. Koval E.J. (1963). A method for predicting the performance of unstable miscible displacements in heterogeneous media. Soc. Pet. Eng. J. Petrol. Trans. AIME, 219, 145–150Google Scholar
  21. Kurganov A., Noelle S., Petrova G. (2001). Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations. SIAM J. Sci. Comp. 23, 707–740CrossRefGoogle Scholar
  22. LaForce, T., Johns, R.T.: Analytical solutions for surfactant-enhanced remediation of nonaqueous phase liquids. Water Resour. Res. 41 (2005a) Art. No. W10420, doi:10.1029/2004WR003862Google Scholar
  23. LaForce T., Johns R.T. (2005b). Composition routes for three-phase partially miscible flow in ternary systems. Soc. Pet. Eng. J. 10(2):161–174Google Scholar
  24. Lake L.W. (1989). Enhanced Oil Recovery. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  25. Lax P.D. (1957). Hyperbolic systems of conservation laws, II. Commun. Pure Appl. Math. 10, 537–566Google Scholar
  26. Lie K.-A., Juanes R. (2005). A front-tracking method for the simulation of three-phase flow in porous media. Comput. Geosci. 9(1):29–59CrossRefGoogle Scholar
  27. Liu T.-P. (1974). The Riemann problem for general 2 ×  2 conservation laws. Trans. Am. Math. Soc. 199, 89–112CrossRefGoogle Scholar
  28. Oleinik, O.A.: Discontinuous solutions of nonlinear differential equations. Usp. Mat. Nauk. (N.S.) 12, 3–73 (1957) English transl. in Am. Math. Soc. Trans. Ser. 2, 26, 95–172Google Scholar
  29. Orr Jr., F.M.: Theory of Gas Injection Processes. Stanford University (2005)Google Scholar
  30. Pope G.A. (1980). The application of fractional flow theory to enhanced oil recovery. Soc. Pet. Eng. J. 20(3):191–205 Petrol. Trans. AIME 269.Google Scholar
  31. Risebro N.H., Tveito A. (1991). Front tracking applied to a nonstrictly hyperbolic system of conservation laws. SIAM J. Sci. Stat. Comput. 12(6):1401–1419CrossRefGoogle Scholar
  32. Stalkup Jr., F.I.: Miscible Displacement, vol. 8 of SPE Monograph Series. Dallas, TX: Society of Petroleum Engineers (1983)Google Scholar
  33. Todd M.R., Longstaff W.J. (1972). The development, testing and application of a numerical simulator for predicting miscible flood performance. J. Pet. Technol. 7, 874–882Google Scholar
  34. Wang Y., Orr Jr. F.M. (1997). Analytical calculation of the minimum miscibility pressure. Fluid Phase Equilib. 139, 101–124CrossRefGoogle Scholar
  35. Zauderer E. (1983). Partial Differential Equations of Applied Mathematics. John Wiley & Sons, New YorkGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

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

Personalised recommendations