Abstract
Patterns and fronts arise in most fluid flow situations. Waves, clouds, convection patterns, and turbulence are well-known examples. In porous media the displacement of one fluid by another fluid leads to fronts and patterns that are often fractal (Mandelbrot, 1982; Feder, 1988). The disorder of the porous matrix plays a key role that is not well understood. Depending on the displacement rates, viscosity ratios, miscibility, interfacial tensions, and pore geometry a bewildering variety of displacement fronts arise. Lenormand (Lenormand and Zarcone, 1985a, Lenormand and Zarcone, 1985b; Lenormand, 1985; Lenormand et al., 1988) has studied many of the regimes observed under various conditions during two fluid displacement processes in micromodels of porous media. In our group (Måløy et al., 1985; Feder et al., 1986; Måløy et al., 1987; Furuberg et al., 1988; Oxaal et al., 1987; Måløy et al., 1987; Måløy et al., 1988; Hinrichsen et al., 1989; Oxaal, 1991; Oxaal et al., 1991; Birovljev et al., 1991; Meakin et al., 1992; Frette et al., 1992) we have also studied displacement fronts in two-dimensional micromodels and random bead packs. We have also studied the three-dimensional displacement in transparent models (Frette et al., 1990, Frette et al., 1992). Theoretical simulations are compared in detail to the observed displacement fronts.
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References
Aharony, A., Percolation, in: Directions in Condensed Matter Physics (G. Grinstein and G. Mazenko, eds.), World Scientific, Singapore, pp. 1–50 (1986).
Aharony, A., and Feder, J., eds., Fractals in Physics: Essays in Honour of Benoit B. Mandebrot, North-Holland, Amsterdam (1989).
Bak, P., and Chen, K., Fractal dynamics of earthquakes, in: Fractals and Their Uses in Earth Sciences Chapter (C. C. Barton and P. R. La Pointe, eds.), Geological Society of America (1990a).
Bak, P., and Chen, K., Predicting earthquakes, in: Nonlinear Structures in Physical Systems—Pattern Formation, Chaos and Waves (L. Lam and H. C. Morris, eds.), Springer, Heidelberg (1990b).
Bak, P., and Tang, C., Earthquakes as a self-organized critical phenomena, J. Geophys. Res. 94, 15635–15637 (1989).
Bak, P., Tang, C., and Wiesenfeld, K., Self-organized criticality: An explanation of 1/f noise, Phys. Rev. Lett. 59, 381–384 (1987).
Bak, P., Tang, C., and Wiesenfeld, K., Self-organized criticality, Phys. Rev. A 38, 364–374 (1988).
Bear, J., Dynamics of Fluids in Porous Media, American Elsevier, New York (1972).
Bear, J., and Bachmat, Y., Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic Publishers, Dordrecht, Boston, London (1990).
Bensimon, D., Kadanoff, L. P., Liang, S., Shraiman, B., and Tang, C., Viscous flows in two dimensions, Rev. Mod. Phys. 58, 977–999 (1986).
Birovljev, A., Furuberg, L., Feder, J., Jøssang, T., Måløy, K. J., and Aharony, A., Gravity invasion percolation in two dimensions: Experiment and simulation, Phys. Rev. Lett. 67, 584–587 (1991).
Bonnet, J., and Lenormand, R., Constructing micromodels for the study of multiphase flow in porous media, Rev. Inst. Fr. Pet. 42, 477–480 (1977).
Cancelliere, A., Chang, C., Foti, E., Rothman, D. H., and Succi, S., The permeability of a random medium: Comparison of simulation with theory, Physics of Fluids 2, 2085–2088 (1991).
Carman, P. C., Fluid flow through granular beds, Trans. Inst. Chem. Eng. Lond. 15, 150–166 (1937).
Chandler, R., Koplik, J., Lerman, K., and Willemsen, J. E., Capillary displacement and percolation in porous media, J. Fluid Mech. 119, 249–267 (1982).
Charlaix, E., Hulin, J. P., and Plona, T. J., Experimental study of tracer dispersion in sintered glass porous materials of variable compaction, Phys. Fluids 30, 1690–1698 (1987).
Chen, J. D., and Wilkinson, D., Pore-scale viscous fingering in porous media, Phys. Rev. Lett. 55, 1892–1895 (1985).
Chuoke, R. L., van Meurs, P., and van der Poel, C., The instability of slow, immiscible, viscous liquid-liquid displacements in permeable media, Trans. Metall. Soc. of AIME 216, 188–194 (1959).
Clément, E., Baudet, and Hulin, J. P., Multiple scale structure of nonwetting fluid invasion fronts in 3d model porous media, J. Physique Lett. 46, L1163–L1171 (1985).
Daccord, G., Chemical dissolution of a porous medium by a reactive fluid, Phys. Rev. Lett. 58, 479–482 (1987).
Daccord, G., Nittmann, J., and Stanley, H. E., Fractal viscous fingers: Experimental results, in: On Growth and Form (H. E. Stanley and N. Ostrowsky, eds.), Martinus Nijhoff, Dordrecht, pp. 203–210 (1986).
Darcy, D., Les Fontaines Publiques de la Ville de Dijon, Victor Dalmont, Paris (1856).
de Gennes, P. G., and Guyon, E., Lois générales pour l’injection d’un fluide dans un milieu poreux aléatoire, J. Mec. 17, 403–432 (1978).
DeGregoria, A. J., and Schwartz, L. W., Saffman-Taylor finger width at low interfacial tension, Phys. Rev. Lett. 58, 1742–1744 (1987).
Deutch, J. M., and Meakin, P., Diffusion controlled colloidal growth rates for nonspherical clusters, J. Chem. Phys. 78, 2093–2094 (1983).
Dhar, D., Self-organized critical state of sandpile automaton models, Phys. Rev. Lett. 64, 1613–1616 (1990).
Dullien, A. L., Fluid Transport and Pore Structure, Academic Press, New York (1979).
Engelberts, W. F., and Klinkenberg, L. J., Laboratory experiments on the displacement of oil by water from packs of granular material, Petr. Congr. Proc. Third World, 544–554 (1951).
Essam, J. W., Percolation theory, Rep. Prog. Phys. 43, 833–912 (1980).
Feder, H. J. S., and Feder, J., Self-organized criticality in a stick-slip process, Phys. Rev. Lett. 66, 2669–2672 (1991) Erratum: Phys. Rev. Lett. 67, 282 (1991).
Feder, J., Fractals, Plenum Press, New York (1988).
Feder, J., Hinrichsen, E. L., Måløy, K. J., and Jøssang, T., Geometrical crossover and self-similarity of DLA and viscous fingering clusters, in: Fractals in Physics (A. Aharony and J. Feder, eds.), North-Holland, Amsterdam, pp. 104–11 (1989).
Feder, J., Jøssang, T., Frette, V., and Birovljev, A., Fysisk modell for en horisontal brønn i Tabertformasjonen, Gullfaks sør Fracton a/s (1988) unpublished.
Feder, J., Jøssang, T., Måløy, K.J., and Oxaal, U., Models of viscous fingering, in Fragmentation Form and Flow in Fractured Media (R. Englman and Z. Jaeger, eds.), Ann Israel Phys. Soc. 8, 531–548 (1986).
Frette, V., Feder, J., Jøssang, T., and Meakin, P., Buoyancy driven fluid migration in porous media, Phys. Rev. Lett. 68, 3164–3167 (1992).
Frette, V., Måløy, K. J., Boger, F., Feder, J., Jøssang, T., and Meakin, P., Diffusion-limited-aggregation-like displacement structure in a three-dimensional porous medium, Phys. Rev. A 42, 3432–3437 (1990).
Frisch, U., Hasslacher, B., and Pomeau, Y., Lattice-gas automata for the Navier-Stokes equation, Phys. Rev. Lett. 56, 1505–1507 (1986).
Furuberg, L., Computer Simulations of Percolation Phenomena, Master’s thesis, University of Oslo (1988).
Furuberg, L., Feder, J., Aharony, A., and Jøssang, T., Dynamics of invasion percolation, Phys. Rev. Lett. 61, 2117–2120 (1988).
Grossman, T., and Aharony, A., Structure and perimeters of percolation clusters, J. Phys. A 19, L745–L751 (1986).
Gunstensen, A., Rothman, D. H., Zaleski, S., and Zanetti, G., A lattice-Boltzman model of immiscible fluids, Phys. Rev. A 43, 4320–4327 (1991).
Guyon, E., Nadal, J., and Pomeau, Y., Disorder and Mixing, NATO ASI Series E: Vol. 152, Kluwer Academic Publishers, Dordrecht (1988).
Held, G. A., Solina, D. H., Keane, D. T., Haag, W. J., Horn, R. M., and Grinstein, G., Experimental study of critical-mass fluctuations in an evolving sandpile, Phys. Rev. Lett. 65, 1120–1123 (1990).
Hele-Shaw, H. S., The flow of water, Nature 58, 34–36 (1898).
Hentschel, H. G. E., Deutch, J. M., and Meakin, P., Dynamical scaling and the growth of diffusion-limited aggregates, J. Chem. Phys. 81, 2496–2502 (1984).
Hill, S., Channeling in packed columns, Chemical Engineering Sciences 1, 247–253 (1952).
Hinrichsen, E. L., The large scale off-lattice simulation was performed using a program that is a modification of programs developed by P. Meakin (1988).
Hinrichsen, E. L., Måløy, K. J., Feder, J., and Jøssang, T., Self-similarity and structure of DLA and viscous fingering clusters, J. Phys. A: Math. Gen. 22, L271–L277 (1989).
Homsy, G. M., Viscous fingering in porous media, Ann. Rev. Fluid Mech. 19, 271–311 (1987).
Hong, D. C., and Langer, J. S., Pattern selection and tip perturbations in the Saffman-Taylor problem, Phys. Rev. A 36, 2325–2332 (1987).
Horton, R. E., Erosional development of streams and their drainage basins; hydrophysical approach to quantitative morphology, Geol. Soc. Am. Bull. 56, 275–370 (1945).
Hulin, J. P., Cazabat, A. M., Guyon, E., and Carmona, F., eds., Hydrodynamics of Dispersed Media, North-Holland, Amsterdam (1990).
Jensen, M. H., Libchaber, A., Pelcé, P., and Zocchi, G., Effect of gravity on the Saffman-Taylor meniscus: Theory and experiment, Phys. Rev. A 35, 2221–2227 (1987).
Jullien, R., and Botet, R., Aggregation and Fractal Aggregates, World Scientific, Singapore (1987).
Kadanoff, L. P., Simulating hydrodynamics: A pedestrian model, J. Stat. Phys. 39, 267–283 (1985).
Kadanoff, L. P., Exact solutions for the Saffman-Taylor problem with surface tension, Phys. Rev. Lett. 65, 2986–2988 (1990).
Kadanoff, L. P., Nagel, S. R., Wu, L., and Zhou, S., Scaling and universality in avalanches, Phys. Rev. A 39, 6524–6537 (1989).
Kessler, D. A., and Levine, H., Stability of finger patterns in Hele-Shaw cells, Phys. Rev. A 32, 1930–1933 (1985).
Lee, J., and Stanley, H. E., Phase transition in the multifractal spectrum of diffusion-limited aggregation, Phys. Rev. Lett. 61, 2945–2948 (1985).
Lenormand, R., Différentes méchanismes de déplacements visqueux et capillaries en milieux poreux: Diagramme de phase, C. R. Acad. Sci. Paris Ser. II 301, 247–250 (1985).
Lenormand, R., Touboul, E., and Zarcone, C., Numerical models and experiments on immiscible displacements in porous media, J. Fluid Mech. 189, 165–187 (1988).
Lenormand, R., and Zarcone, C., Invasion percolation in an etched network: Measurement of a fractal dimension, Phys. Rev. Lett. 54, 2226–2229 (1985a).
Lenormand, R., and Zarcone, C., Two-phase flow experiments in a two-dimensional permeable medium, Physico-Chemical Hydrodynamics 6, 497–506 (1985b).
Lutsko, J. F., Boon, J. P., and Somers, J. A., Lattice gas automata simulations of viscous fingering in a porous medium preprint (1991).
Maher, J. V., Development of viscous fingering patterns, Phys. Rev. Lett. 54, 1498–1501 (1985).
Måløy, K.J., Feder, J., and Jøssang, T. Unpublished observations made during the preparation of Feder et al., 1986 (1985).
Måløy, K. J., Feder, J., and Jøssang, T., Viscous fingering fractals in porous media, Phys. Rev. Lett. 55, 2688–2691 (1985b).
Måløy, K. J., Boger, F., Feder, J., and Jøssang, T., Dynamics and structure of viscous fingers in porous media, in: Time-Dependent Effects in Disordered Materials (R. Pynn and T. Riste, eds.), Plenum Press, New York, pp. 111–138 (1987).
Måløy, K. J., Boger, F., Feder, J., Jøssang, T., and Meakin, P., Dynamics of viscous-fingering fractals in porous media, Phys. Rev. A 36, 318–324 (1987b).
Måløy, K. J., Feder, J., Boger, F., and Jøssang, T., Fractal structure of hydrodynamic dispersion in porous media, Phys. Rev. Lett. 61, 2925–2928 (1988).
Mandelbrot, B. B., How long is the coast of Britain? statistical self-similarity and fractal dimension, Science 155, 636–638 (1967).
Mandelbrot, B. B., The Fractal Geometry of Nature, W.H. Freeman, San Francisco (1982).
Mandelbrot, B. B., Self-affine fractals and fractal dimension, Physica Scripta 32, 257–260 (1985).
Mandelbrot, B. B., Self-affine fractal sets, in: Fractals in Physics, North-Holland, Amsterdam, pp. 3–28 (1986).
Mandelbrot, B. B., and Evertz, C. J. G., The potential distribution around growing fractal clusters, Nature 348, 143–145 (1990).
Matsushita, M., Sano, M., Hayakawa, Y., Honjo, H., and Sawada, Y., Fractal structures of zinc metal leaves grown by electrodeposition, Phys. Rev. Lett. 53, 286 (1984).
McLean, J. W., and Saffman, P. G., The effect of surface tension on the shape of fingers in a Hele-Shaw cell, J. Fluid Mech. 102, 455–469 (1981).
Meakin, P., Diffusion-controlled cluster formation in 2–6 dimensional space, Phys. Rev. A 27, 1495–1507 (1983).
Meakin, P., Fractal aggregates and their fractal measures, in: Phase Transitions and Critical Phenomena (C. Domb and J. L. Lebowitz, eds.), Academic Press, New York, pp. 336–489 (1987).
Meakin, P., and Deutch, J. M., Monte Carlo simulations of diffusion controlled growth rates in two and three dimensions, J. Chem. Phys. 80, 2115–2122 (1984).
Meakin, P., Feder, J., Frette, V., and Jøssang, T., Invasion percolation in a destabilizing gradient, Phys. Rev. A 46, 3357–3368 (1992).
Nittmann, J., Daccord, G., and Stanley, H. E., Fractal growth of viscous fingers: Quantitative characterization of a fluid instability phenomenon, Nature 314, 141–144 (1985).
Nittmann, J., and Stanley, H. E., Tip splitting without interfacial tension and dendritic growth patterns arising from molecular anisotropy, Nature 321, 663–668 (1986).
Oxaal, U., Fractal viscous fingering in inhomogeneous porous models, Phys. Rev. A 44, 5038–5051 (1991).
Oxaal, U., Murat, M., Boger, F., Aharony, A., Feder, J., and Jøssang, J., Viscous fingering on percolation clusters, Nature 329, 32–37 (1987).
Oxaal, U., Boger, F., Feder, J., Jøssang, T., Meakin, P., and Aharony, A., Viscous fingering in square-lattice model with two types of bonds, Phys. Rev. A 44, 6564–6576 (1991).
Paterson, L., Diffusion-limited aggregation and two-fluid displacements in porous media, Phys. Rev. Lett. 52, 1621–1624 (1984).
Paterson, L., Radial fingering in a Hele-Shaw cell, J. Fluid Mech. 113, 513–529 (1981).
Pitts, E., Penetration of a fluid into a Hele-Shaw cell: The Saffman-Taylor experiment, J. Fluid Mech. 97, 53–64 (1980).
Rabaud, M., Couder, Y., and Gerard, N., Dynamics and stability of Saffman-Taylor fingers, Phys. Rev. A 37, 935–947 (1988).
Rothman, D. H., Cellular-automaton fluids: A model for flow in porous media, Geophysics 53, 509–518 (1988).
Rothman, D. H., and Keller, J. M., Immiscible cellular-automaton fluids, J. Stat. Phys. 52, 1119–1127 (1988).
Roux, S., and Guyon, E., Temporal development of invasion percolation, J. Phys. A: Math. Gen. 22, 3693–3705 (1989).
Saffman, P. G., A theory of dispersion in a porous medium, J. Fluid Mech. 6, 321–349 (1959).
Saffman, P.G., Dispersion due to molecular diffusion and macroscopic mixing through a network of capillaries, J. Fluid Mech. 7, 194–208 (1960).
Saffman, P. G., and Taylor, G., The penetration of a fluid into a medium or Hele-Shaw cell containing a more viscous liquid, Proc. Soc. London, Ser. A 245, 312–329 (1958).
Sahimi, M., Davis, H. T., and Scriven, L. E., Dispersion on percolation clusters, Chem. Eng. Com. 23, 329 (1983).
Sapoval, B., Rosso, M., and Gouyet, J. F., The fractal nature of a diffusing front and the relation to percolation, J. Phys. Lett. 46, L149–L156 (1985).
Sapoval, B., Rosso, M., Gouyet, J. F., and Colonna, J. F., Dynamics of the creation of fractal objects by diffusion and 1/f noise, Solid State Ionics 18, 21–30 (1986).
Scheidegger, A. E., The Physics of Flow Through Porous Media, University of Toronto Press, Toronto (1974).
Shaw, T. M., Movement of a drying front in a porous material, Better Ceramics Through Chemistry II, Materials Research Society Symposia Proceedings 73, 215–223 (1986).
Shaw, T. M., Drying as an immiscible displacement process with fluid counterflow, Phys. Rev. Lett. 59, 1671–1674 (1987).
Sornette, A., and Sornette, D., Self-organized criticality and earthquakes, Europhys. Lett. 9, 197–202 (1989).
Stauffer, D., Introduction to Percolation Theory, Taylor & Francis, London (1985).
Stauffer, D., and Aharony, A., Introduction to Percolation Theory, 2nd edition, Taylor & Francis, London (1992).
Tang, C., and Bak, P., Mean field theory of self-organized critical phenomena, J. Stat. Phys. 51, 797–802 (1988).
Tryggvason, G., and Aref, H., Numerical experiments on Hele-Shaw flow with a sharp interface, J. Fluid Mech. 136, 1–30 (1983).
VanDamme, H., Obrecht, F., Levitz, P., Gatineau, L., and Laroche, C., Fractal viscous fingering in clay slurries, Nature 320, 731–733 (1986).
Vicsek, T., Fractal Growth Phenomena, World Scientific, Singapore (1989).
Voss, R. F., The fractal dimension of percolation cluster hulls, J. Phys. A 17, L373–L377 (1984).
Wilkinson, D., Percolation model of immiscible displacement in the presence of buoyancy forces, Phys. Rev. A 30, 520–531 (1984).
Wilkinson, D., Percolation effects in immiscible displacement, Phys. Rev. A 34, 1380–1391 (1986).
Wilkinson, D., and Willemsen, J. F., Invasion percolation: A new form of percolation theory, J. Phys. A: Math. Gen. 16, 3365–3376 (1983).
Witten, T. A., and Sander, L. M., Diffusion-limited aggregation, a kinetic critical phenomenon, Phys. Rev. Lett. 47, 1400–1403 (1981).
Wolfram, S., Cellular automaton fluids: I: Basic theory, J. Stat. Phys. 45, 471–526 (1986a).
Wolfram, S., Theory and Applications of Cellular Automata, World Scientific, Singapore (1986b).
Zaitoun, A., Kalaydjian, F., and Jacquin, C., Two-phase flow through porous media: Influence of viscosity ratio on relative permeabilities, in: Fundamentals of Fluid Transport in Porous Media (ed.), IFP Institut Français du Pétrole, Réf. I.F.P. 37 981-1, pp. 83–86 (1990).
Zick, A. A., and Homsy, G. M., Stokes flow through periodic arrays of spheres, J. Fluid Mech. 115, 13–26 (1982).
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Feder, J., Jøssang, T. (1995). Fractal Patterns in Porous Media Flow. In: Barton, C.C., La Pointe, P.R. (eds) Fractals in Petroleum Geology and Earth Processes. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-1815-0_10
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