Average permeability is a way of representing the overall permeability of a material that has different layers or regions with different permeabilities. Permeability is a property of a porous material that describes how easily fluids can flow through it.
Linear flow is a type of fluid flow that occurs when the pressure difference between the inlet and outlet of a material is constant. Linear flow is often assumed for convenience and approximation.
Series beds are a common structure of porous materials that consist of vertical layers with different permeabilities. Series beds are often found in natural reservoirs, such as oil and gas fields.
Average permeability for linear flow series beds is calculated by using the harmonic mean of the individual layer permeabilities. The harmonic mean is a type of average that gives more weight to smaller values. This means that the average permeability for linear flow series beds is lower than the simple average of the layer permeabilities.
Basic Theory:
Linear flow through series beds involves multiple rock formations with distinct permeabilities. The total flow across these beds is the sum of individual flows. The average permeability (
) is calculated using Darcy’s law:
![\[Q = \frac{kA\Delta P}{\mu L}\]](https://excelcalculations.refrigeratordiagrams.com/wp-content/ql-cache/quicklatex.com-ecd0babf6d0e85d9e941e30261176bad_l3.svg "Rendered by QuickLaTeX.com")
Where:
is the total flow rate,
is the permeability,
is the cross-sectional area,
is the pressure drop,
is the fluid viscosity, and
is the length of the flow path.
The average permeability for series beds can be expressed as:
![\[k_{avg} = \frac{\sum_{i=1}^{n}k_iL_i}{\sum_{i=1}^{n}L_i}\]](https://excelcalculations.refrigeratordiagrams.com/wp-content/ql-cache/quicklatex.com-88f473b8986120ad5e845ba80dab3375_l3.svg "Rendered by QuickLaTeX.com")
Procedures:
- Data Collection: Gather permeability () and length () data for each bed in the series.
- Excel Formulas:
- Enter the data into an Excel table.
- Use the SUM function to calculate the sum of
and 
. - Apply the average permeability formula.
- Scenario Example: Let’s consider a series of three beds with the following data:
Bed Permeability (mD) Length (ft) 1 100 500 2 150 700 3 80 600
Excel Calculation:
Result:
MATLAB Comparison:
-
- MATLAB Code:
% Input data
permeability = [100, 150, 80]; % in mD
length = [500, 700, 600]; % in ft
% Calculate average permeability
k_avg_matlab = sum(permeability .* length) / sum(length);
disp([‘MATLAB Result: k_avg = ‘, num2str(k_avg_matlab), ‘ mD’]);
-
- MATLAB Result:
