Permeability is a property of a porous material that describes how easily fluids can flow through it. Permeability depends on the size, shape, and connectivity of the pores in the material.
Average permeability is a way of representing the overall permeability of a material that has different layers or regions with different permeabilities. Average permeability is useful for simplifying calculations and modeling fluid flow in complex systems.
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.
Layered beds are a common structure of porous materials that consist of horizontal layers with different permeabilities. Layered beds are often found in natural reservoirs, such as oil and gas fields.
Average permeability for linear flow layered 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 layered beds is lower than the arithmetic mean (the simple average) of the layer permeabilities.
Basic Theory:
The Darcy’s law governs fluid flow through porous media and is defined as:
Q = -kA(∆P/L)
Where:
- Q is the volumetric flow rate,
- k is the permeability,
- A is the cross-sectional area,
- ∆P is the pressure drop across the rock, and
- L is the distance of fluid flow.
For layered beds, the average permeability (k_avg) for linear flow can be calculated using the harmonic mean of individual layer permeabilities:
1/k_avg = h1/k1 + h2/k2 + … + hn/kn
Where:
- k_avg is the average permeability,
- hi is the thickness of the ith layer, and
- ki is the permeability of the ith layer.
Procedures in Excel:
- Data Entry:
- Create an Excel spreadsheet with columns for layer thickness (h) and permeability (k).
- Input the values for each layer in separate rows.
- Calculations:
- In a new column, calculate 1/k for each layer using the formula 1/ki.
- Sum the reciprocals to obtain 1/k_avg using the formula mentioned earlier.
- Final Result:
- Compute k_avg by taking the reciprocal of the summed value.
Comprehensive Explanation:
Consider a reservoir with two layers:
| Layer | Thickness (ft) | Permeability (mD) |
|---|---|---|
| 1 | 50 | 100 |
| 2 | 30 | 150 |
1/k1 = 1/100
1/k2 = 1/150
1/k_avg = 50/100 + 30/150
k_avg = 1 / (50/100 + 30/150)
Result: k_avg = 120 mD
MATLAB Comparison:
For MATLAB, you can use a script to calculate k_avg using the same formula. The comparison between the Excel and MATLAB results ensures accuracy and reliability in the calculations.
% MATLAB Script
h = [50; 30];
k = [100; 150];
k_avg = 1 / sum(h ./ k);
disp(['MATLAB Result: k_avg = ', num2str(k_avg), ' mD']);
