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Estimating permeability based on Kozeny-Carman equation
The Kozeny-Carman equation is typically used to calculate the pressure drop of fluids when crossing a medium that typically includes consolidated grains of some sort. Certain single phase permeability models can be derived based on this equation.
Estimating permeability
The problem of predicting permeability is one of selecting a model expressing k in terms of other, measurable rock properties. Historically, the first approaches were based on a tube-like model of rock pore space known as the Kozeny-Carman relationship.^{[1]}^{[2]}^{[3]}^{[4]} The derivation of this "equivalent channel model" has been reworked by Paterson^{[5]} and Walsh and Brace.^{[6]} The model assumes that flow through a porous medium can be represented by flow through a bundle of tubes of different radii. Within each tube, the flow rate is low enough that flow is laminar rather than turbulent. A tube is assigned:
- Shape factor f, a dimensionless number between 1.7 and 3
- Length L_{a} that is greater than the sample length L
The assumption is that each flow path forms a twisted, tortuous, yet independent route from one end of the rock to the other. The tortuosity is defined as τ=(L_{a}/L)^{2}. From considerations of flow through tubes, the resulting equation is
where the hydraulic radius, r_{h}, is defined as the reciprocal of Σ_{p}, the ratio of pore surface area to pore volume. The pore surface area normalized by a volume is often called the specific surface area. The form of Eq. 1 depends on which volume is used to normalize the pore surface area. If specific surface area is instead expressed as Σ_{r}, the ratio of pore surface area to rock volume, then Eq. 1 becomes
If specific surface area is defined as the ratio of pore surface area to grain volume, Σ_{g}, the expression is
Thus, the functional dependence of k on Φ, which differs among Eqs. 1, 2, and 3, depends on the definition of specific surface area.
Paterson^{[5]} and Walsh and Brace^{[6]} establish a relationship between electrical properties and tortuosity, determining that formation factor F=(L_{a}/L)^{2}/Φ=τ/Φ. They note that this expression differs from earlier incorrect formulations. With it, tortuosity can be eliminated from Eq. 1 to obtain
Different approaches to porous media theory apply the concept of tortuosity in different ways.^{[7]} For our purpose, tortuosity is represented by electrical formation factor, as in Eq. 4, or by porosity raised to an exponent.
Many models that relate k to a pore dimension r are derived, either in spirit or in rigor, from the Kozeny-Carman relationship, which recognizes explicitly the dependence of k on r^{2}.
Nomenclature
f | = | shape factor |
k | = | permeability |
r_{h} | = | hydraulic radius |
Σ_{p} | = | ratio of pore surface area to pore volume |
τ | = | tortuosity |
Φ | = | porosity |
References
- ↑ Carman, P.C. 1956. Flow of Gases Through Porous Media. New York City: Academic Press Inc.
- ↑ Amyx, J.W., Bass, D.M. Jr., and Whiting, R.L. 1960. Petroleum Reservoir Engineering. New York City: McGraw-Hill Book Co.
- ↑ Hearst, J.R., Nelson, P.H., and Paillet, F.L. 2000. Well Logging for Physical Properties. New York City: John Wiley & Sons.
- ↑ Timur, A. 1968. An Investigation Of Permeability, Porosity, & Residual Water Saturation Relationships For Sandstone Reservoirs. The Log Analyst IX (4). SPWLA-1968-vIXn4a2.
- ↑ ^{5.0} ^{5.1} Paterson, M.S. 1983. The equivalent channel model for permeability and resistivity in fluid-saturated rock—A re-appraisal. Mech. Mater. 2 (4): 345-352. http://dx.doi.org/http://dx.doi.org/10.1016/0167-6636(83)90025-X
- ↑ ^{6.0} ^{6.1} Walsh, J.B. and Brace, W.F. 1984. The effect of pressure on porosity and the transport properties of rock. Journal of Geophysical Research: Solid Earth 89 (B11): 9425-9431. http://dx.doi.org/10.1029/JB089iB11p09425
- ↑ Clennell, M.B. 1997. Tortuosity: A Guide Through the Maze. In Developments in Petrophysics, ed. M.A. Lovell and P.K. Harvey, 299. Geological Society Special Publication No. 122.
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