Transport in Porous Media

, Volume 69, Issue 3, pp 383–409 | Cite as

Efficient integration of stiff kinetics with phase change detection for reactive reservoir processes

  • Morten R. Kristensen
  • Margot G. Gerritsen
  • Per G. Thomsen
  • Michael L. Michelsen
  • Erling H. Stenby
Orginal Paper

Abstract

We propose the use of implicit one-step Explicit Singly Diagonal Implicit Runge–Kutta (ESDIRK) methods for integration of the stiff kinetics in reactive, compositional and thermal processes that are solved using operator-splitting type approaches. To facilitate the algorithmic development we construct a virtual kinetic cell model. The model serves both as a tool for the development and testing of tailored solvers as well as a testbed for studying the interactions between chemical kinetics and phase behavior. As case study, two chemical kinetics models with 6 and 14 components, respectively, are implemented for in situ combustion, a thermal oil recovery process. Through benchmark studies using the 14 component reaction model the new ESDIRK solvers are shown to improve computational speed when compared to the widely used multi-step BDF methods DASSL and LSODE. Phase changes are known to cause convergence problems for the integration method. We propose an algorithm for detection and location of phase changes based on discrete event system theory. Experiments show that the algorithm improves the robustness of the integration process near phase boundaries by lowering the number convergence and error test failures by more than 50% compared to direct integration without the new algorithm.

Keywords

Reactive transport processes Stiff ODE solvers ESDIRK methods Discrete event systems Phase change detection Differential-algebraic equations Multi-scale methods Operator splitting Enhanced oil recovery In situ combustion 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adegbesan, K.O., Donnelly, J.K., Moore, R.G., Bennion, D.W.: Low-temperature-oxidation kinetic parameters for in-situ combustion: numerical simulation. Society of Petroleum Engineers. SPE 12004: Presented at the 58th Annual Technical Conference and Exhibition, San Francisco, CA, October 5–8 (1987)Google Scholar
  2. Alexander R. (2003). Design and implementation of DIRK integrators for stiff systems. Appl. Numer. Math. 46: 1–17 Google Scholar
  3. Aris R. (1989). Elementary Chemical Reactor Analysis. Dover Publication, Inc., Mineola, New York Google Scholar
  4. Bauer I., Bock H.G., Körkel S. and Schlöder J.P. (2000). Numerical methods for optimum experimental design in DAE systems. J. Comput. Appl. Math. 120: 1–25 CrossRefGoogle Scholar
  5. Bijl H., Carpenter M.H., Vatsa V.N. and Kennedy C.A. (2002). Implicit time integration schemes for the unsteady compressible Navier-Stokes equations: laminar flow. J. Comput. Phys. 179: 313–329 CrossRefGoogle Scholar
  6. Castanier, L.M., Brigham, W.E.: In-situ combustion. Society of Petroleum Engineers Handbook (2004)Google Scholar
  7. Clara C., Durandeau M., Quenault G. and Nguyen T.-H. (2000). Laboratory studies for light-oil air injection projects: potential application in handil field. SPE Reservoir Eval. Eng. 3(3): 239–248 Google Scholar
  8. CMG: STARS, Advanced Process and Thermal Reservoir Simulator. 2004 edn.Google Scholar
  9. Coats K.H. (1980). In-situ combustion model. Soc. Petrol. Eng. J. 269: 533–554 Google Scholar
  10. Crookston, R.B., Culham, W.E., Chen, W.H.: A numerical simulation model for thermal recovery processes. Soc. Petrol. Eng. J. 37–58 (1979)Google Scholar
  11. Enright W.H., Jackson K.R., Nørsett S.P. and Thomsen P.G. (1986). Interpolants for Runge-Kutta formulas. ACM Trans. Math. Software 12(3): 193–218 CrossRefGoogle Scholar
  12. Fassihi, M.R., Brigham, W.E., Ramey, H.J. Jr.: Reaction kinetics of in-situ combustion: part 2 - modelling. Soc. Petrol. Eng. J. 408–416 (1984)Google Scholar
  13. Freitag N.P. and Exelby D.R. (2006). A SARA-based model for simulating the pyrolysis reactions that occur in high-temperature EOR processes. J. Can. Petrol. Technol. 45(3): 38–44 Google Scholar
  14. Freitag N.P. and Verkoczy B. (2005). Low-temperature oxidation of oils in terms of SARA fractions: why simple reaction models don’t work. J. Can. Petrol. Technol. 44(2): 54–61 Google Scholar
  15. Friedly J.C. (1991). Extent of reaction in open systems with multiple heterogeneous reactions. AIChE J. 37(5): 687–693 CrossRefGoogle Scholar
  16. Friedly J.C. and Rubin J. (1992). Solute transport with multiple equilibrium-controlled or kinetically controlled chemical reactions. Water Resour. Res. 28(6): 1935–1953 CrossRefGoogle Scholar
  17. Gillham, T.H., Cerveny, B.W., Turek, E.A., Yannimaras, D.V.: Keys to increasing production via air injection in gulf coast light-oil reservoirs. Society of Petroleum Engineers. SPE 38848: Presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, October 5–8 (1997)Google Scholar
  18. Grabowski, J.W., Vinsome, P.K., Lin, R.C., Behie, A., Rubin, B.: A fully implicit general purpose finite-difference thermal model for in-situ combustion and steam. Society of Petroleum Engineers. SPE 8396: Presented at the 54th Annual Fall Technical Conference and Exhibition of the SPE, Las Vegas, Nevada, September 23–26 (1979)Google Scholar
  19. Gustafsson, K.: Control of error and convergence in ODE solvers. PhD thesis, Department of Automatic Control, Lund Institute of Technology (1992)Google Scholar
  20. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II, 2nd revised edn. Springer (1996)Google Scholar
  21. Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I, 2nd revised edn. Springer (1996)Google Scholar
  22. Hascoët, L., Pascual, V.: Tapenade 2.1 User’s Guide. INRIA, France (2004)Google Scholar
  23. Hindmarsh A.C. (1983). ODEPACK, a generalized collection of ODE solvers. In: Stepleman, R.S. (eds) Scientific Computing (IMACS Transactions on Scientific Computing, vol 1), pp 55–64. North- Holland, Amsterdam Google Scholar
  24. Islam, M.R., Chakma, A., Farouq Ali, S.M.: State-of-the-art of in-situ combustion modelling and operations. Society of Petroleum Engineers. SPE 18755: Presented at the SPE California Regional Meeting, Bakersfield, CA, April 5–7 (1989)Google Scholar
  25. Kennedy C.A. and Carpenter M.H. (2003). Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44: 139–181 CrossRefGoogle Scholar
  26. Kräutle, S., Knabner, P.: A new numerical reduction scheme for fully coupled multicomponent transport-reaction problems in porous media. Water Resour. Res. 41(9), W09414, DOI: 10.1029/2004WR003624 (2005)Google Scholar
  27. Kutta W. (1901). Beitrag zur näherungsweisen integration totaler differentialgleichungen. Z. Math. Phys. 46: 435–453 Google Scholar
  28. Kværnø A. (2004). Singly diagonally implicit Runge-Kutta methods with an explicit first stage. BIT Numer. Math. 44: 489–502 CrossRefGoogle Scholar
  29. Moore, R.G., Mehta, S.A., Ursenbach, M.G.: A guide to high pressure air injection (HPAI) based oil recovery. Society of Petroleum Engineers. SPE 75207: Presented at the SPE/DOE Improved Oil Recovery Symposium, Tulsa, Oklahoma, April 13–17 (2002)Google Scholar
  30. Park T. and Barton P.I. (1996). State event location in differential-algebraic models. ACM Trans. Model. Comput. Simul. 6(2): 137–165 CrossRefGoogle Scholar
  31. Petzold, L.R.: DASSL: A differential/algebraic system solver. In: 10th IMACS World Congress on System Simulation and Scientific Computation (1982)Google Scholar
  32. Prats, M.: Thermal Recovery, vol 7 of SPE Monograph Series. Society of Petroleum Engineers (1986)Google Scholar
  33. Prothero A. and Robinson A. (1974). On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comput. 28(125): 145–162 CrossRefGoogle Scholar
  34. Rasmussen, C.P., Krejbjerg, K., Michelsen, M.L., Bjurstrøm, K.E.: Increasing the computational speed of flash calculations with applications for compositional, transient simulations. SPE Reservoir Eval. Eng. 9(1), 32–38 (2006)Google Scholar
  35. Ren, Y., Freitag, N.P., Mahinpey, N.: A simple kinetic model for coke combustion during an in situ combustion (ISC) process. Presented at the 6th Canadian International Petroleum Conference, Calgary, Alberta, June 7–9 (2005)Google Scholar
  36. Rosenbrock H.H. (1963). Some general implicit processes for numerical solution of differential equations. Comp. J. 6(4): 329–330 CrossRefGoogle Scholar
  37. Runge C. (1895). Über die numerische auflösung von differentialgleichungen. Math. Ann. 46: 167–178 CrossRefGoogle Scholar
  38. Sandu A., Verwer J.G., Bloom J.G., Spee E.J., Carmichael G.R. and Potra F.A. (1997a). Benchmarking stiff ODE solvers for atmospheric chemistry problems: II. Rosenbrock solvers. Atmos. Environ. 31(20): 3459–3472 CrossRefGoogle Scholar
  39. Sandu A., Verwer J.G., Loon M.V., Carmichael G.R., Potra F.A., Dabdub D. and Seinfeld J.H. (1997b). Benchmarking stiff ODE solvers for atmospheric chemistry problems: I. Implicit vs. explicit. Atmos. Environ. 31(19): 3151–3166 CrossRefGoogle Scholar
  40. Strang G. (1968). On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5(3): 506–517 CrossRefGoogle Scholar
  41. Thomsen, P.G.: Discontinuities in ODEs – systems with change of state. Technical Report IMM-2006-07, Informatics and Mathematical Modelling, Technical University of Denmark (2006)Google Scholar
  42. Williams R., Burrage K., Cameron I. and Kerr M. (2002). A four-stage index 2 diagonal implicit Runge-Kutta method. Appl. Numer. Anal. 40: 415–432 CrossRefGoogle Scholar
  43. Younis, R., Gerritsen, M.: Multiscale process coupling by adaptive fractional stepping: an in-situ combustion model. Society of Petroleum Engineers. SPE 93458: Presented at the 2006 SPE/DOE Symposium on Improved Oil Recovery, Tulsa, Oklahoma, April 22–26 (2006)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • Morten R. Kristensen
    • 1
  • Margot G. Gerritsen
    • 2
  • Per G. Thomsen
    • 3
  • Michael L. Michelsen
    • 1
  • Erling H. Stenby
    • 1
  1. 1.Department of Chemical EngineeringTechnical University of DenmarkLyngbyDenmark
  2. 2.Department of Energy Resources EngineeringStanford UniversityStanfordUSA
  3. 3.Informatics and Mathematical ModellingTechnical University of DenmarkLyngbyDenmark

Personalised recommendations