Crossover from nonlinearity controlled to heterogeneity controlled mixing in two-phase porous media flows

Computational Geosciences

Volumn 7, Issue 2, 2003, Pages 115-135

  Furtado, Frederico ^a   Pereira, Felipe ^b  

^a UNIVERSITY OF WYOMING   (United States)

^b STATE UNIVERSITY OF RIO DE JANEIRO   (Brazil)

Author keywords

Fractals; Heterogeneity; Mixing; Multiphase flow; Porous media; Random fields; Scale up; Scaling laws

Indexed keywords

MONTE CARLO SIMULATION; POROUS MEDIUM; TWO-PHASE FLOW;

MONTE CARLO ANALYSIS; NONLINEARITY; POROUS MEDIUM; TWO PHASE FLOW;

EID: 0037931827     PISSN: 14200597     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1023586101302     Document Type: Article

References (38)

1. L. An, J. Glimm, D.H. Sharp and Q. Zhang, Scale up of flow in porous media, in: Mathematical Modelling of Flow through Porous Media, eds. A.P. Bourgeat, C. Carasso, S. Luckhaus and A. Mikelić (World Scientific, New Jersey, 1995) pp. 26-44.
2. J.W. Barker and S. Thibeau, A critical review of the use of pseudorelative permeabilities for upscaling, SPE Reservoir Engrg. (May 1997) 138-143.
3. A.J. Chorin, The instability of fronts in a porous medium, Comm. Math. Phys. 91 (1983) 103-116.
4. R.L. Chouke, P. van Meurs and C. van der Poel, The instability of slow, immiscible, viscous liquid- liquid displacements in permeable media. Trans. AIME 216 (1959) 188-194.
5. M.A. Christie, Upscaling for reservoir simulations. JPT J. Petrol. Technol. 48 (1996) 1004-1010.
6. V. Cvetkovic and G. Dagan, Reactive transport and immiscible flow in geological media. II. Applications, Proc. Roy. Soc. London A 452 (1996) 303-328.
7. G. Dagan, Dispersion of a passive solute in non-ergodic transport by steady velocity fields in heterogeneous formations. J. Fluid Mech. 233 (1991) 1281-1290.
8. G. Dagan, The significance of heterogeneity of evolving scales to transport in porous formations. Water Resour. Res. 30 (1994) 3327-3336.
9. G. Dagan and V. Cvetkovic, Reactive transport and immiscible flow in geological media. I. General theory, Proc. Roy. Soc. London A 452 (1996) 285-301.
10. J. Douglas, Jr., F. Furtado and F. Pereira, On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs, Comput. Geosci. 1(2) (1997) 155-190.
11. L.J. Durlofsky, Use of higher moments for the description of upscaled, process independent relative permeabilities, SPE 37987 (1997).
12. Y. Efendiev, L.J. Durlofsky and S.H. Lee, Modeling of subgrid effects in coarse-scale simulations of transport in heterogeneous porous media. Water Resour. Res. 36 (2000) 2031-2041.
13. F.W. Elliot, Jr. and A. Majda, A wavelet Monte Carlo method for turbulent diffusion with many spatial sales. J. Comput. Phys. 113 (1994) 82-111.
14. F. Furtado, J. Glimm, B. Lindquist and F. Pereira, Multi-length scale calculations of mixing length growth in tracer floods, in: Proc. of the Emerging Technologies Conference, ed. F. Kovarik, Institute for Improved Oil Recovery, University of Houston, Houston, TX, 1990, pp. 251-259.
15. F. Furtado and F. Pereira, Scaling analysis for two-phase, immiscible flow in heterogeneous media, Comput. Appl. Math. 17 (1998) 233-262.
16. F. Furtado, F. Pereira and J. Zhu (2002) in preparation.
17. J. Glimm, H. Kim, D. Sharp and T. Wallstrom, A stochastic analysis of the scale up problem for flow in porous media, Comput. Appl. Math. 17 (1998) 67-79.
18. J. Glimm, B. Lindquist, F. Pereira and Q. Zhang, A theory of macrodispersion for the scale up problem, Transport Porous Media 13 (1993) 97-122.
19. J. Glimm, D. Marchesin and O. McBryan, Statistical fluid dynamics: unstable fingers, Comm. Math. Phys. 74 (1980) 1-13.
20. J. Glimm, D. Marchesin and O. McBryan, A numerical method for two phase flow with an unstable interface, J. Comput. Phys. 39 (1981) 179-200.
21. J. Glimm and D.H. Sharp, Stochastic partial differential equations: selected applications in continuum physics, in: Stochastic Partial Differential Equations: Six Perspectives, eds. R.A. Carmona and B.L. Rozovskii, Mathematical Surveys and Monographs (Amer. Math. Soc., Providence, RI, 1997).
22. G.M. Homsy, Viscous fingering in porous media. Ann. Rev. Fluid Mech. 19 (1987) 271-311.
23. K.D. Jarman, Jr., Stochastic immiscible flow with moment equations, Ph.D. thesis, University of Colorado, Boulder (2000).
24. K.D. Jarman and T.F. Russell, Analysis of 1-D moment equations for immiscible flow, Contemp. Math., to appear.
25. M.J. King, W.B. Lindquist and L. Reyna, Stability of two-dimensional immiscible flow to viscous fingering, Courant Math. and Comp., Lab. Report DOE/ER/03077-244 (March 1985).
26. P. Langlo and M. Espedal, Macrodispersion for two-phase, immiscible flow in porous media. Adv. Water Resour. 17 (1994) 297-316.
27. R. Lenormand, Determining flow equations from stochastic properties of a permeability field: The MHD model, SPE J. (June 1996) 179-190.
28. H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys. 87(2) (1990) 408-463.
29. H. Rajaram and L.W. Gelhar, Plume scale-dependent dispersion in heterogeneous aquifers, 2. Eulerian analysis and three-dimensional aquifers, Water Resour. Res. 29 (1993) 3261-3271.
30. P. Saffman and G. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. Roy. Soc. London A 245 (1958) 312-322.
31. J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer, New York, 1983).
32. J.R. Waggoner, J.L. Castillo and L.W. Lake, Simulation of EOR processes in stochastically generated permeable media, SPE 21237, SPE Formation Evaluation (June 1992) 173-180.
33. T. Wallstrom, S. Hou, M.A. Christie, L.J. Durlofsky and D.H. Sharp, Accurate scale up of two-phase flow using renormalization and nonuniform coarsening, Comput. Geosci. 3 (1999) 69-87.
34. T. Wallstrom, S. Hou, M.A. Christie, L.J. Durlofsky and D.H. Sharp, Application of a new two-phase upscaling technique to realistic reservoir cross sections, in: Proc. of the SPE 15th Symposium on Reservoir Simulation, SPE 51939, 1999, pp. 451-462.
35. T. Wallstrom, S. Hou, M.A. Christie, L.J. Durlofsky, D.H. Sharp and Q. Zhou, Effective medium boundary conditions for upscaling relative permeabilities, Transport Porous Media (2000) accepted for publication.
36. D. Zhang, L. Li and H.A. Tchelepi, Stochastic formulation for uncertainty analysis of two-phase flow in heterogeneous reservoirs, SPE J. 5 (2000) 60-70.
37. Q. Zhang, A multi-length scale theory of the anomalous mixing length growth for tracer flow in heterogeneous porous media, J. Statist. Phys. 66 (1991) 485-501.
38. J. Zhu, A numerical study of stochastic dispersion for two-phase immiscible flow in porous media, M.Sc. thesis, University of Wyoming (2001).