Understanding Capillary Number in Excel

Capillary number is a dimensionless quantity that compares the effects of viscous forces and surface tension forces in a fluid flow. Viscous forces are related to the resistance of a fluid to flow, while surface tension forces are related to the attraction between molecules at the interface of two fluids. Capillary number tells us how easily one fluid can displace another fluid in a porous medium, such as rock or soil.

The capillary number is usually very small for slow flows in porous media, such as oil and water in a reservoir. This means that the surface tension forces are dominant, and the displacing fluid has difficulty overcoming the capillary pressure of the displaced fluid. The capillary number is larger for fast flows in pipes or channels, such as oil and water in a well. This means that the viscous forces are dominant, and the displacing fluid can easily push the displaced fluid out of the way.

Basic Theory:

Capillary number is defined as the ratio of viscous forces to capillary forces and is represented by the formula:

$Ca = \frac{\mu V}{\sigma}$

Where:

Ca: Capillary number
$\mu$ : Dynamic viscosity of the fluid
V: Characteristic velocity of the fluid
$\sigma$ : Interfacial tension between the fluid phases

Procedures:

Collect Data: Gather the necessary data – dynamic viscosity ( $\mu$ ), characteristic velocity ( $V$ ), and interfacial tension ( $\sigma$ ).
Insert Data into Excel: Create an Excel table with columns for each parameter. Input the values into the respective cells.
Calculate Capillary Number: Use the formula $Ca = \frac{\mu V}{\sigma}$ in a new cell to calculate the Capillary number.

Scenario:

Consider a scenario where we have the following values:

Dynamic viscosity ( $\mu$ ) = 0.01 Pa.s
Characteristic velocity ( $V$ ) = 0.1 m/s
Interfacial tension ( $\sigma$ ) = 0.02 N/m

Excel Calculation:

$\mu$	V	$\sigma$	Ca
0.01	0.1	0.02	=B2*C2/A2

In this scenario, the Capillary number would be calculated as $Ca = \frac{0.01 \times 0.1}{0.02} = 0.05$ .

MATLAB Comparison:

If you prefer MATLAB, you can use the following script:


% Scenario Values
mu = 0.01;    % Pa.s
V = 0.1;      % m/s
sigma = 0.02; % N/m

% Capillary Number Calculation
Ca = (mu * V) / sigma;
disp(['Capillary Number (MATLAB): ' num2str(Ca)]);

Results:

Excel Calculation: $Ca = 0.05$
MATLAB Calculation: $Ca = 0.05$

The Capillary number in this scenario is 0.05, indicating the dominance of viscous forces over capillary forces in the fluid flow.