Flow Through Porous Media Summary

Given below is a schematic for the evolution of the conceptual development of the momentum conservation statement for flow through porous media. The filter-like picture shown in the schemtaic represents the filtration experiments carried out by Henry Philibert Gaspard Darcy in Dijon, France in the middle of the 19th century, when he was working there as the "Dean of the School of Bridges and Roadways".

The evolution essentially traces only the concepts originated through experiments, in other words, the modeling of flow through porous media done at the global scale (to put it crudely, what is seen in the resolution level of the human eye).

Figure 1: Evolution of Global Flow Modeling in Porous Media

The above schematic doesn't give a complete picture of the modeling challenges involved in the differential scale or the macroscopic or pore level modeling of the porous medium. Also, the schematic captures only the evolution of the global momentum conservation statement for a particular restricted class of porous media. This is partly due to the nature of the earlier experiments done (in the 19th and early 20th centuries) to understand the flow through porous media, which involved mostly low permeable media.

Ideally, if one were to perform a volume integration of the equation shown inside the orange box of Fig. 1, it should lead to a result similar to that of the, experimentally verifiable, global momentum conservation equation inside the yellow box, just above the orange box. A more general representation of the macroscopic model (equation inside the orange box in Fig. 1) is given in Fig. 2 below.

The following schematic is a general steady state differential momentum conservation statement valid on a 'porous-continuum', a continuous space whose differential 'point' is in principle equivalent to a [[Representative Elemental Volume (REV)|PM Homogeneity and REV]] of the actual porous medium, which is being modeled.

The in the equation represents the volumetric porosity of the porous medium (total pore volume divided by the total volume of the porous medium); and (vector) are the static pressure and velocity in the pore level, is the effective viscosity, a quantity believed to be a function of the porous medium. The and are the dynamic viscosity and density of the fluid that flow through the porous medium while and are the Permeability ( ) and Form Coefficient]( ), hydraulic properties of the porous medium.

The extensive experiments and testing of low permeable media and the eventual form of the global momentum conservation (equations in yellow boxes in Fig. 1) perhaps have lead some fluid mechanist (mostly experimentalists) to believe that there is NO turbulence in porous media flows because, the smallness of the pores (an artifact of low permeable media) cannot sustain the cascading nature of flows in several scales, a signature of turbulence. This view is not true, and turbulence in porous media is one of the 'hot fields' of current porous medium research. But to discuss that will take a separate post.

Some more references of interest

• Darcy, H. P. G., 1856, Les Fontaines Publiques de la Ville de Dijon, Victor Dalmont, Paris. [ take a look at the English translation of the paper; the crucial data from Appendix D in Excel format of the above paper that led to the Darcy Law; Courtesy: an excellent page on Darcy maintained by Prof. Glenn Brown ]
• Dupuit, A. J. E. J., 1863, etudes Theoriques et Pratiques sur le Mouvement des aux dans les Canaux Decouverts et a Travers les Terrains Permeables, Victor Dalmont, Paris.
• Forchheimer, P., 1901, Wasserbewegung durch Boden, Zeitschrift des Vereines Deutscher Ingenieure, Vol. 45, pp. 1736-1741 and pp. 1781-1788.
• Hazen, A, 1893, "Some Physical Properties of Sand and Gravels with Special Reference to their use in Filtration," Twenty-fourth Annual Report, Massachusetts State Board of Health, p. 541.
• Newton, Sir. I., 1687 (First ed., 1713, Second ed. and 1726, Third ed.), Philosophiae Naturalis Principia Mathematica. 1) translated by A. Motte, Prometheus Books, New York, 1995; originally published as: Newton's Principia. First American ed., New York, 1848. 2) Cajori, F., Newton's Principia (Motte's translation revised). Uty. of Cal. press, Berkeley, CA, 1946.
• Stanek, V. and Szekely, J., 1974, "Three-dimensional flow of fluids through nonuniform packed beds," AIChE Journal, Vol. 20, pp. 974-980.

A good starting point for getting a historical perspective of flow through porous media would be

• Lage, J. L., 1998, "The Fundamental Theory of Flow through Permeable Media: from Darcy to Turbulence," Transport Phenomena in Porous Media, D. B. Ingham and I. Pop, Eds., Pergamon, New York, pp. 1-30.

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