Eulerian moment equations for 2-D stochastic immiscible flow

Multiscale Modeling and Simulation

Volumn 1, Issue 4, 2003, Pages 598-608

  Jarman, Kenneth D ^a   Russell, Thomas F ^b  

^a PACIFIC NORTHWEST NATIONAL LABORATORY   (United States)

^b UNIVERSITY OF COLORADO   (United States)

Author keywords

Moment equations; Porous media; Random fields; Stochastic

Indexed keywords

DIFFERENTIAL EQUATIONS; MONTE CARLO METHODS; POROUS MATERIALS; SECONDARY RECOVERY;

FRACTIONAL FLOW; IMMISCIBLE FLOW; MOMENT EQUATIONS; RANDOM FIELDS; SECONDARY OIL RECOVERIES; STATISTICAL MOMENTS; STOCHASTIC; TAYLOR EXPANSIONS;

STOCHASTIC SYSTEMS;

EID: 10644286198     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/S1540345902413176     Document Type: Article

References (44)

1. M. B. Allen, G. A. Behie, and J. A. Trangenstein, Multiphase Flow in Porous Media, Lecture Notes in Engrg. 34, Springer, Berlin, 1988.
2. O. Amir and S. Neuman, Gaussian closure of one-dimensional unsaturated flow in randomly heterogeneous soils, Transp. Porous Media, 44 (2001), pp. 355-383.
3. V. Artus, B. Noetinger, and L. Ricard, Dynamics of the water-oil front for two-phase, immiscible flows in heterogeneous porous media. 1-Stratified media, Transp. Porous Media, submitted.
4. S. E. Buckley and M. C. Leverett, Mechanism of fluid displacement in sands, Trans. AIME, 146 (1942), pp. 107-116.
5. V. Cvetkovic and G. Dagan, Reactive transport and immiscible flow in geological media. II. Applications, Proc. Roy. Soc. London Ser. A, 452 (1996), pp. 302-328.
6. V. Cvetkovic, A. Shapiro, and G. Dagan, A solute flux approach to transport in heterogeneous formations 2. Uncertainty analysis, Water Resour. Res., 28 (1992), pp. 1377-1388.
7. G. Dagan, Flow and Transport in Porous Formations, Springer, Heidelberg, Berlin, 1989.
8. G. Dagan and V. Cvetkovic, Reactive transport and immiscible flow in geological media. I. General theory, Proc. Roy. Soc. London Ser. A, 452 (1996), pp. 285-301.
9. G. Dagan, V. Cvetkovic, and A. Shapiro, A solute flux approach to transport in heterogeneous formations 1. The general framework, Water Resour. Res., 28 (1992), pp. 1369-1376.
10. G. Dagan and S. P. Neuman, Nonasymptotic behavior of a common Eulerian approximation for transport in random velocity fields, Water Resour. Res., 27 (1991), pp. 3249-3256.
11. L. P. Dake, Fundamentals of Reservoir Engineering, Developments in Petroleum Science 8, Elsevier, New York, 1986.
12. G. de Marsily, Quantitative Hydrogeology: Groundwater Hydrology for Engineers, Academic Press, New York, 1986.
13. D. W. Dean, An Analysis of the Stochastic Approaches to the Problems of Flow and Transport in Porous Media, Ph.D. thesis, University of Colorado, Boulder, CO, 1997.
14. J. P. Delhomme, Spatial variability and uncertainty in groundwater flow parameters: A geostatistical approach, Water Resour. Res., 15 (1979), pp. 269-280.
15. R. A. Freeze, A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media, Water Resour. Res., 11 (1975), pp. 725-741.
16. F. Furtado and F. Pereira, Crossover from nonlinearity controlled to heterogeneity controlled mixing in two-phase porous media flows, Comput. Geosci., 7 (2003), pp. 115-135.
17. L. W. Gelhar, Stochastic Subsurface Hydrology, Prentice-Hall, Englewood Cliffs, NJ, 1993.
18. W. Graham and D. McLaughlin, Stochastic analysis of nonstationary subsurface solute transport 1. Unconditional moments, Water Resour. Res., 25 (1989), pp. 215-232.
19. A. Gutjahr, B. Bullard, and S. Hatch, General joint conditional simulations using a fast Fourier transform method, Math. Geol., 29 (1997), pp. 361-389.
20. J. O. Hinze, Turbulence, 2nd ed., McGraw-Hill, New York, 1975.
21. K. D. Jarman, Stochastic Immiscible Flow with Moment Equations, Ph.D. thesis, University of Colorado, Boulder, CO, 2000.
22. K. D. Jarman and T. F. Russell, Analysis of 1-D moment equations for immiscible flow, in Fluid Flow and Transport in Porous Media: Mathematical and Numerical Treatment, Z. Chen and R. E. Ewing, eds., AMS, Providence, RI, 2002, pp. 293-304.
23. K. D. Jarman and T. F. Russell, Moment Equations for Stochastic Immiscible Flow, Tech. Report 181, University of Colorado Denver Center for Computational Mathematics, 2002, http://www-math.cudenver.edu/ccm/reports/rep181.ps.gz.
24. P. Langlo and M. S. Espedal, Heterogeneous reservoir models, two-phase immiscible flow in 2-D, in Computational Methods in Water Resources, Vol. 9, Elsevier, New York, 1992, pp. 71-79.
25. P. Langlo and M. S. Espedal, Macrodispersion for two-phase, immiscible flow in porous media, Adv. in Water Resour., 17 (1994), pp. 297-316.
26. R. J. Leveque, CLAWPACK Version 4.0 User's Guide, University of Washington, Seattle, WA, rjl@amath.washington.edu (1999).
27. J. L. Lumley and H. A. Panofsky, The Structure of Atmospheric Turbulence, Monographs and Texts in Physics and Astronomy 12, Wiley, New York, 1964.
28. S. P. Neuman, Eulerian-Lagrangian theory of transport in space-time nonstationary velocity fields: Exact nonlocal formalism by conditional moments and weak approximation, Water Resour. Res., 29 (1993), pp. 633-645.
29. S. P. Neuman and A. Guadagnini, private communication, 2000.
30. S. P. Neuman, C. L. Winter, and C. M. Newman, Stochastic theory of field-scale Fickian dispersion in anisotropic porous media, Water Resour. Res., 23 (1987), pp. 453-466.
31. B. Noetinger, V. Artus, and L. Ricard, Dynamics of the water-oil front for two-phase, immiscible flows in heterogeneous porous media. 2-Isotropic media, Transp. Porous Media, submitted.
32. C. S. Simmons, A stochastic-convective transport representation of dispersion in onedimensional porous media systems, Water Resour. Res., 18 (1982), pp. 1193-1214.
33. C. S. Simmons, T. R. Ginn, and B. D. Wood, Stochastic-convective transport with nonlinear reaction: Mathematical framework, Water Resour. Res., 31 (1995), pp. 2675-2688.
34. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Comprehensive Studies in Math- ematics, Springer, New York, 1983.
35. G. Sposito and D. A. Barry, On the Dagan model of solute transport in groundwater: Foundational aspects, Water Resour. Res., 23 (1987), pp. 1867-1875.
36. D. M. Tartakovsky and S. P. Neuman, Transient flow in bounded randomly heterogeneous domains 1. Exact conditional moment equations and recursive approximations, Water Resour. Res., 34 (1998), pp. 1-12.
37. A. F. B. Tompson and L. W. Gelhar, Numerical simulation of solute transport in threedimensional, randomly heterogeneous porous media, Water Resour. Res., 26 (1990), pp. 2541-2562.
38. D. Zhang, Numerical solutions to statistical moment equations of groundwater flow in nonstationary, bounded, heterogeneous media, Water Resour. Res., 34 (1998), pp. 529-538.
39. D. Zhang, Nonstationary stochastic analysis of transient unsaturated flow in randomly heterogeneous media, Water Resour. Res., 35 (1999), pp. 1127-1141.
40. D. Zhang, Quantification of uncertainty for fluid flow in heterogeneous petroleum reservoirs, Phys. D, 133 (1999), pp. 488-497.
41. D. Zhang, Stochastic Methods for Flow in Porous Media: Coping with Uncertainties, Academic Press, San Diego, 2001.
42. D. Zhang and H. Tchelepi, Stochastic analysis of immiscible two-phase flow in heterogeneous media, Soc. Pet. Eng. J., 4 (1999), pp. 380-388.
43. D. Zhang, H. Tchelepi, and L. Li, Stochastic formulation for uncertainty assessment of twophase flow in heterogeneous reservoirs, in 1999 SPE Reservoir Simulation Symposium, Society of Petroleum Engineers, Houston, TX, 1999, pp. 389-402.
44. D. Zhang and C. L. Winter, Moment-equation approach to single phase fluid flow in heterogeneous reservoirs, Soc. Pet. Eng. J., 4 (1999), pp. 118-127.