Flow in Porous Media: Permeability and Displacement Patterns
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flow and give an update on its progress. It is also my hope to show how basic research in this field is intimately coupled to real applications. Permeability For many applications, the most important physical property of porous media is the hydraulic permeability, k, which is defined by Darcy's law for single-phase fluid flow: Jf=
VP. (1) V Here, Jf is the volume current density (volume/area-time) of the flowing fluid, TJ is its viscosity, and P is the pressure field that drives the flow. Darcy's law is the analog of Ohm's law for electrical conduction and permeability is the analog of conductivity. For a fluid with 17 = 1 cP (0.01 dyne-s/cm2), if Jf = 1 cm/s for a pressure gradient of 1 atm/cm, then the medium is said to have a permeability k = 1 darcy. Numerically, 1 darcy = 1 /j,m2. This already indicates that k is related to the microstructure of the porous media. How one derives the permeability from the microgeometry and how one measures it experimentally are two main issues of concern. A material such as concrete can have very low permeability (in the microdarcy range) and it is difficult to measure it accurately, but the value will determine whether the concrete can make an effective barrier to contain environmental contaminants. In an oil field, the permeability of the rock formation determines whether the oil can be produced economically, but there are no simple methods to measure the reservoir permeability directly.
Mesoscopic Length Scales and Formation Factor During the last decade, the general approach to understanding permeability was to study the microgeometry of the pores and use it to estimate the permeability.2 One of the best known examples of this approach comes from Johnson, Koplik, and Schwartz,3 who introduced an effective pore size parameter called the A-parameter. They proposed a permeability estimation formula A78F
(2)
where F is the formation factor, defined as the electrical conductivity ratio of the pore fluid and the fluid saturated medium, i.e., F = o-f/o-j. So if one can determine F by an electrical measurement and find A by some other method, one would have an estimate of the permeability. The validity of this general idea was demonstrated by the independent and concurrent work of Katz and Thompson (KT).4 They used mercury injection to determine a characteristic pore diameter €c for 50 rock samples and found that a good estimate of the permeability is k = €2/226F.
(3)
The different numerical prefactors in Equations 2 and 3 can be attributed to the difference in the theoretical definition of A and the experimental definition of €c. Taking into account that €c is the diameter rather than the radius, the ratio €C/2A is (226/32)w = 2.66. This shows a remarkable agreement between Equations 2 and 3 even though they were obtained by very different ideas. We can understand Equations 2 and 3 from Poiseuille's flow in a cylindrical pipe: (4)
where Q is the total flow rate (volume per unit time) and (Jf) is the average flux over the cross-sectional area -nr\. If we identify the pip
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