Kozeny-Carman Relationship in Petroleum Engineering Using Excel

The Kozeny-Carman relationship is a way of estimating the permeability of a porous medium, such as a rock or a soil, based on its porosity and the size and shape of its grains. Permeability is a measure of how easily a fluid can flow through the medium. Porosity is the fraction of the medium that is occupied by pores or empty spaces. The size and shape of the grains affect how the pores are arranged and connected.

The Kozeny-Carman relationship assumes that the porous medium can be modeled as a bundle of tubes with different diameters and lengths, and that the fluid flow is laminar, meaning smooth and steady. The relationship then uses a mathematical formula to calculate the pressure drop of the fluid along the tubes, which is proportional to the permeability of the medium. The formula depends on the following parameters:

The viscosity of the fluid, which is a measure of its resistance to flow
The length of the tubes, which is related to the tortuosity of the medium, or how twisted and curved the flow paths are
The hydraulic radius of the tubes, which is the ratio of the cross-sectional area of the tubes to their perimeter
The porosity of the medium, which is the ratio of the volume of the pores to the total volume of the medium
The sphericity of the grains, which is a measure of how spherical or round they are

The Kozeny-Carman relationship is not exact, but it is useful for estimating the permeability of a porous medium when direct measurements are not available or feasible. It is widely used in the field of fluid dynamics and petroleum engineering.

The Kozeny-Carman relationship is expressed as follows:

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

Where:

k is the permeability of the porous medium,
C is a constant,
$\phi$ is the porosity of the medium.

The specific surface area is defined as $S = \frac{6}{{\phi d}}$ , where $d$ is the diameter of the solid particles.

Procedures:

Define Constants:
- Set up a worksheet in Excel and designate cells for the constant ( $C$ ) and the porosity ( $\phi$ ).
Calculate Specific Surface Area:
- In a new cell, calculate the specific surface area using the formula $S = \frac{6}{{\phi d}}$ .
Apply Kozeny-Carman Relationship:
- In another cell, use the Kozeny-Carman formula to calculate permeability: $k = \frac{C \cdot \phi^3}{{(1-\phi)^2}}$ .

Explanation:

Consider a scenario where the constant $C = 10$ and the porosity $\phi = 0.35$ . The diameter of solid particles is $d = 0.02$ meters.

Set $C = 10$ in a designated cell.
Enter $\phi = 0.35$ in another cell.
Calculate specific surface area: $S = \frac{6}{{\phi d}}$ .
Apply Kozeny-Carman relationship: $k = \frac{C \cdot \phi^3}{{(1-\phi)^2}}$ .

Excel Table:

Constant (C)	Porosity ( $\phi$ )	Diameter ( $d$ )	Specific Surface Area (S)	Permeability (k)
10	0.35	0.02	=6/(B2*C2)	=(A2*B2^3)/((1-B2)^2)

Calculation:

Specific Surface Area ( $S$ ): $\frac{6}{{0.35 \times 0.02}} \approx 85.71 \, \text{m}^2/\text{m}^3$
Permeability ( $k$ ): $\frac{10 \times 0.35^3}{{(1-0.35)^2}} \approx 0.058 \, \text{m}^2$

MATLAB Comparison:

In MATLAB, you can use the same formulas to calculate specific surface area and permeability. Here is a simple script:


% Define constants
C = 10;
phi = 0.35;
d = 0.02;

% Calculate specific surface area
S = 6 / (phi * d);

% Calculate permeability
k = (C * phi^3) / ((1 - phi)^2);

% Display results
fprintf('Specific Surface Area (S): %.2f m^2/m^3\n', S);
fprintf('Permeability (k): %.3f m^2\n', k);

Result:

Specific Surface Area ( $S$ ): 85.71 m²/m³
Permeability ( $k$ ): 0.058 m²