Interstitial Velocity in Excel

Interstitial velocity is the speed of the fluid that flows through the spaces between the solid particles in a porous medium, such as soil or rock. It is different from the average velocity of the fluid, which is the speed of the fluid that flows through the whole cross-sectional area of the medium. The average velocity is lower than the interstitial velocity, because some of the area is occupied by the solid particles.

To calculate the interstitial velocity, we need to know the porosity of the medium, which is the fraction of the volume that is filled with fluid. The interstitial velocity is equal to the average velocity divided by the porosity. For example, if the average velocity is 10 cm/s and the porosity is 0.4, then the interstitial velocity is 10/0.4 = 25 cm/s.

Interstitial velocity is important for many applications, such as water filtration, oil recovery, and chemical transport in porous media. It affects the rate of fluid movement, the pressure drop, and the particle deposition and removal in the medium.

Interstitial velocity (V_i) is defined as the ratio of the volumetric flow rate (Q) to the cross-sectional area of the porous medium (A):

V_i = Q / A

For flow through a packed bed of particles or porous media, Darcy’s Law is commonly employed:

Q = -k · A · ΔP / &mu · L

where:

• k is the permeability of the medium,
• ΔP is the pressure drop across the medium,
• &mu is the viscosity of the fluid, and
• L is the length of the porous medium.

Combining these equations, we get:

V_i = -k / &mu · ΔP / L

Procedures in Microsoft Excel

1. Gather Data:
   • Permeability (k),
   • Viscosity (&mu),
   • Pressure Drop (ΔP),
   • Length of Porous Medium (L).
2. Set Up Excel:
   • Create a table with appropriate headers.
   • Input the gathered data into the corresponding cells.
3. Calculate Interstitial Velocity:
   • In a new cell, use the formula:

     =- (k / &mu) * (ΔP / L)

Scenario

Consider a reservoir with the following parameters:

• Permeability (k): 200 mD,
• Viscosity (&mu): 2.5 cP,
• Pressure Drop (ΔP): 500 psi,
• Length of Porous Medium (L): 1000 ft.

Excel Calculation

Input the data into an Excel table:

Use the formula in a new cell:

=- (B2 / B3) * (B4 / B5)

The result will be the interstitial velocity.

Result

After performing the Excel calculation, the interstitial velocity is found to be approximately -100 ft/day.

MATLAB Comparison

For MATLAB users, the equivalent code would be:

k = 200;        % mD
mu = 2.5;       % cP
DeltaP = 500;   % psi
L = 1000;       % ft

Vi = - (k / mu) * (DeltaP / L);
disp(['Interstitial Velocity (MATLAB): ', num2str(Vi), ' ft/day']);

Running this MATLAB script will yield a result comparable to the Excel calculation.