Average Permeability for Parallel-Layered Systems in Excel

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.

Parallel-layered systems are a common structure of porous materials that consist of horizontal layers with different permeabilities. Parallel-layered systems are often found in natural reservoirs, such as oil and gas fields.

Average permeability for parallel-layered systems is calculated by using the weighted average of the individual layer permeabilities. The weighted average is a type of average that gives more weight to larger values. This means that the average permeability for parallel-layered systems is higher than the harmonic mean (another type of average) of the layer permeabilities.

Basic Theory:

In parallel-layered systems, each layer contributes to the overall permeability. The harmonic mean is commonly used to calculate the average permeability ( $K_{avg}$ ) for $N$ layers:

$\frac{1}{K_{avg}} = \frac{1}{K_1} + \frac{1}{K_2} + \frac{1}{K_3} + ... + \frac{1}{K_N}$

Where:

$K_{avg}$ is the average permeability.
$K_1, K_2, ..., K_N$ are the permeabilities of individual layers.

Procedures in Microsoft Excel:

Set Up the Excel Table:
- Create a table with columns for layer number (N), permeability (K), and the reciprocal of permeability (1/K).
Enter Data:
- Input the layer numbers and corresponding permeabilities into the Excel table.
Calculate Reciprocal:
- In a new column, use the formula =1/K to calculate the reciprocal of permeability for each layer.
Sum Reciprocals:
- Use the formula =SUM(1/K) to find the sum of reciprocals.
Calculate Average Permeability:
- Finally, calculate the average permeability using the formula =1/(Sum Reciprocals).

Explanation:

Let’s consider a reservoir with three layers having permeabilities $K_1 = 50 \, \text{mD}$ , $K_2 = 75 \, \text{mD}$ , and $K_3 = 100 \, \text{mD}$ . The average permeability can be calculated using the harmonic mean formula mentioned earlier.

Scenario:

Layer (N)	Permeability (K, mD)	1/K
1	50	=1/B2
2	75	=1/B3
3	100	=1/B4
Total		=SUM(B2:B4)
Average		=1/(Total)

Excel Calculation:

$K_{avg} = 1/((1/50) + (1/75) + (1/100)) \approx 34.615 \, \text{mD}$

MATLAB Comparison:

% MATLAB Code
permeabilities = [50, 75, 100];
average_permeability = 1 / mean(1 ./ permeabilities);
disp(['Average Permeability (MATLAB): ', num2str(average_permeability), ' mD']);

MATLAB Calculation: