Kozeny Equation in Petroleum Engineering

The Kozeny equation is a relation that describes the pressure drop of a fluid flowing through a packed bed of solids, such as sand or gravel. The equation assumes that the fluid flow is very slow and laminar, meaning that there are no turbulence or eddies in the fluid. The equation also assumes that the packed bed can be modeled as a collection of curving tubes that cross the bed, and that the fluid flow in each tube follows Poiseuille’s law, which relates the pressure drop to the tube diameter and length.

The Kozeny equation depends on several factors, such as the height of the bed, the velocity of the fluid, the viscosity of the fluid, the porosity of the bed, the sphericity of the particles, and the diameter of the particles. The porosity is the fraction of the bed volume that is occupied by the fluid, and the sphericity is a measure of how spherical the particles are. The equation also includes an empirical constant that accounts for the tortuosity of the bed, which is the degree of twisting or bending of the tubes.

The Kozeny equation can be used to calculate the pressure drop of a fluid flowing through a porous medium, such as a filter or a soil layer. The equation is useful for applications in fluid dynamics, hydrology, and petroleum engineering. However, the equation is only valid for very low fluid velocities and particle Reynolds numbers, which are dimensionless numbers that indicate the ratio of inertial forces to viscous forces in the fluid. For higher velocities and Reynolds numbers, the pressure drop becomes more dependent on the kinetic energy losses of the fluid, and the Kozeny equation needs to be modified or replaced by other empirical relations, such as the Ergun equation or the Burke-Plummer equation.

Basic Theory of Kozeny Equation:

The Kozeny equation is used to estimate the permeability ( $k$ ) of a porous medium using porosity ( $\phi$ ) and specific surface area ( $S$ ) of the rock. The equation is expressed as:

$k = \frac{\alpha \cdot \phi^3}{(1 - \phi)^2}$

Where:

$k$ is the permeability.
$\phi$ is the porosity.
$\alpha$ is a dimensionless constant related to the shape of the grains.

Procedures for Implementing Kozeny Equation in Excel:

Follow these steps to implement the Kozeny equation in Excel:

Open a new Excel sheet.
Label cells for input parameters: A1 for $\phi$ , B1 for $\alpha$ .
Enter the values for $\phi$ and $\alpha$ in cells A2 and B2, respectively.
In cell C1, label it as “Permeability”.
In cell C2, enter the formula:
```
=B2*A2^3/(1-A2)^2
```
The result in cell C2 will be the estimated permeability.

Example:

Let’s consider a reservoir with a porosity ( $\phi$ ) of 0.25 and a shape factor ( $\alpha$ ) of 10.

Excel Calculation:

Input: $\phi = 0.25$ , $\alpha = 10$
Excel Formula: $k = \frac{10 \cdot 0.25^3}{(1 - 0.25)^2}$
Excel Result: $k \approx 0.578$ (Darcy)

MATLAB Comparison:

For MATLAB, the script would look like:

phi = 0.25;
alpha = 10;
k = (alpha * phi^3) / (1 - phi)^2;
disp(['Permeability from MATLAB: ', num2str(k), ' Darcy']);

Result:

Excel Result: $k \approx 0.578$ Darcy
MATLAB Result: $k \approx 0.578$ Darcy