Content of PetroWiki is intended for personal use only and to supplement, not replace, engineering judgment. SPE disclaims any and all liability for your use of such content. More information

# 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. The derivation of this "equivalent channel model" has been reworked by Paterson and Walsh and Brace. 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 La 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=(La/L)2. From considerations of flow through tubes, the resulting equation is ....................(1)

where the hydraulic radius, rh, is defined as the reciprocal of Σp, 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 Σr, the ratio of pore surface area to rock volume, then Eq. 1 becomes ....................(2)

If specific surface area is defined as the ratio of pore surface area to grain volume, Σg, the expression is ....................(3)

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 and Walsh and Brace establish a relationship between electrical properties and tortuosity, determining that formation factor F=(La/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 ....................(4)

Different approaches to porous media theory apply the concept of tortuosity in different ways. 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 r2.

## Nomenclature

 f = shape factor k = permeability rh = hydraulic radius Σp = ratio of pore surface area to pore volume τ = tortuosity Φ = porosity