Abstract
We derive a large scale mixing parameter for a displacement process of one fluid by another immiscible one in a two-dimensional heterogeneous porous medium. The mixing of the displacing fluid saturation due to the heterogeneities of the permeabilities is captured by a dispersive flux term in the large scale homogeneous flow equation. By making use of the stochastic approach we develop a definition of the dispersion coefficient and apply a Eulerian perturbation theory to determine explicit results to second order in the fluctuations of the total velocity. We apply this method to a uniform flow configuration as well as to a radial one. The dispersion coefficient is found to depend on the mean total velocity and can therefore be time varying. The results are compared to numerical multi-realization calculations. We found that the use of single phase flow stochastics cannot capture all phenomena observed in the numerical simulations.
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References
Arfken, G. B. and Weber, H. J.: 1995, Mathematical Methods for Physicists, Academic Press.
Attinger, S., Dentz, M., Kinzelbach, H. and Kinzelbach, W.: 1999, Temporal behaviour of a solute cloud in a chemically heterogeneous porous medium, J. Fluid Mech. 386, 77–104.
Attinger, S., Neuweiler, I. and Kinzelbach, W.: 2001, Macrodispersion in a radially diverging flow field with finite Peclet numbers II: Homogenization theory approach, Water Resour. Res. 37(3), 495–507.
Batycky, R. P.: 1997, A three-dimensional two-phase field scale streamline simulator, Dissertation at Stanford University, Stanford, California.
Batycky, R. P., Blunt, M. J. and Thiele, M. R.: 1997, A 3D field scale streamline simulator with gravity and changing well conditions, SPE Reservoir Eng. 246–254.
Binning, P. and Celia, M. A.: 1999, Practical implementation of the fractional flow approach to multi-phase flow simulation, Adv. Water Resour. 22(5), 461–478.
Blunt, M. J., Barker, J. W., Rubin, B., Mansfield, M., Culverwell, I. D. and Christie, M.: 1994, Predictive theory for viscous fingering in compositional displacement, SPE Reservoir Eng. 36(12) (SPE Paper 25234), 73–80.
Cvetkovic, V. and Dagan, G.:1996, Reactive transport and immiscible flow in geological media. II. Applications, P. Roy. Soc. Lond. A 452, 303–328.
Dagan, G.: 1988, Time dependent macrodispersion for solute transport in anisotropic heterogeneous aquifers, Water Resour. Res. 24, 1491–1500.
Dagan, G.: 1989, Flow and Transport in Porous Formations, Springer.
Dagan, G. and Cvetkovic, V.: 1996, Reactive transport and immiscible flow in geologic media: I. General theory, P. Roy. Soc. Lond. A 452, 285–302.
Dentz, M., Kinzelbach, H., Attinger, S. and Kinzelbach, W.: 2000, Temporal behaviour of a solute cloud in a heterogeneous porous medium: Point-like injection, Water Resour. Res. 36(12), 3591–3604.
Ewing, R. E.: 1997, Aspects of upscaling in simulation of flow in porous media, Adv. Water Resour. 20(5-6), 349–358.
Fiori, A., Indelman, P. and Dagan, G.: 1998, Correlation structure of flow variables for steady flow toward a well with application to highly anisotropic heterogeneous formations, Water Resour. Res. 34(4), 699–708.
Gelhar, L. W.: 1993, Stochastic Subsurface Hydrology, Prentice Hall.
Gelhar, L. W. and Axness, C. L.: 1983, Three-dimensional analysis of macrodispersion in aquifers, Water Resour. Res. 19(1), 161–180.
Hsu, K. C. and Neuman, S. P.: 1997, Second-order expressions for velocity moments in two-and three-dimensional statistically anisotropic media, Water Resour. Res. 33(4), 625–637.
Indelman, P. and Dagan, G.: 1999, Solute transport in divergent radial flow through heterogeneous porous media, J. Fluid Mech. 384, 159–182.
Langlo, P. and Espedal, M. S.: 1995, Macrodispersion for two-phase, immiscible flow in porous media, Adv. Water Resour. 17, 297–316.
Marle, C. M.: 1981, Multiphase Flow in Porous Media, 1st edn., Institut Francais du Petrole, Paris.
McComb, W. D.: 1991, The Physics of Fluid Turbulence, Clarendon Press, Oxford.
Neuweiler, I., Attinger, S. and Kinzelbach, W.: 2001, Macrodispersion in a radially diverging flow field with finite Peclet numbers I: Perturbation theory approach, Water Resour. Res. 37(3), 481–494.
Pruess, K.: 1996, A Fickian diffusion model for the spreading of liquid plumes infiltrating in heterogeneous media, Transport Porous Med. 24(1), 1–33.
Robin, M. J. L., Gutjahr, A. L., Sudicky, E. A. and Wilson, J. L.: 1993, Cross-correlated random field generator with the direct Fourier transform method, Water Resour. Res. 29(7), 2385–2397.
Rubin, Y.: 1990, Stochastic modeling of macrodispersion in heterogeneous porous media, Water Resour. Res. 26(4), 133–141.
Zhang, D. and Tchelepi, H.: 1999, Stochastic analysis of immiscible two-phase flow in heterogeneous media, SPE J. 4(4), 380–388
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Neuweiler, I., Attinger, S., Kinzelbach, W. et al. Large Scale Mixing for Immiscible Displacement in Heterogeneous Porous Media. Transport in Porous Media 51, 287–314 (2003). https://doi.org/10.1023/A:1022370927468
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DOI: https://doi.org/10.1023/A:1022370927468