Average Permeability of Radial Systems in Excel

Permeability is a measure of how easily fluid can flow through a porous medium, such as rock or soil. In a radial system, fluid flows from a wellbore to the surrounding reservoir, or vice versa. The permeability of the reservoir may vary in different directions and layers, so we need to calculate an average permeability that represents the overall flow behavior.

There are two ways to arrange flow units (small sections of the reservoir) in a radial system: in series or in parallel. In series, the fluid has to pass through each flow unit one by one, so the average permeability is the harmonic mean of the individual permeabilities. In parallel, the fluid can flow through different flow units at the same time, so the average permeability is the weighted mean of the individual permeabilities.

The harmonic mean is calculated by dividing the number of flow units by the sum of the reciprocals of the permeabilities. The weighted mean is calculated by multiplying each permeability by its corresponding fraction of the total area or volume, and then adding them up.

Basic Theory:

In radial flow, the Darcy’s law for radial systems is given by:

$Q = \frac{{2\pi k h}}{{\ln(r2/r1)}} \Delta P$

where:

$Q$ is the flow rate,
$k$ is the permeability,
$h$ is the thickness of the reservoir,
$r1$ and $r2$ are the inner and outer radii of the reservoir,
$\Delta P$ is the pressure difference.

Average permeability ( $k_{\text{avg}}$ ) is calculated using the harmonic mean formula:

$k_{\text{avg}} = \frac{n}{{\frac{1}{{k1}}} + \frac{1}{{k2}} + \ldots + \frac{1}{{kn}}}$

where $n$ is the number of layers.

Procedures:

Gather Data: Collect reservoir data including permeability values ( $k_i$ ), layer thicknesses ( $h_i$ ), and radii ( $r_{1i}$ and $r_{2i}$ ).
Excel Setup:
- Create an Excel table with columns for layer index, permeability, thickness, inner radius, and outer radius.
- Enter the data into the table.
Calculations:
- In a new column, use the Darcy’s law formula to calculate the flow rate for each layer.
- Use the harmonic mean formula to find the average permeability.
Scenario:Consider a reservoir with three layers:
- Layer 1: $k_1 = 100 \, \text{mD}$ , $h_1 = 50 \, \text{ft}$ , $r_{11} = 1000 \, \text{ft}$ , $r_{21} = 1500 \, \text{ft}$
- Layer 2: $k_2 = 200 \, \text{mD}$ , $h_2 = 75 \, \text{ft}$ , $r_{12} = 1500 \, \text{ft}$ , $r_{22} = 2000 \, \text{ft}$
- Layer 3: $k_3 = 150 \, \text{mD}$ , $h_3 = 60 \, \text{ft}$ , $r_{13} = 2000 \, \text{ft}$ , $r_{23} = 2500 \, \text{ft}$

Excel Calculation:

Apply the Darcy’s law formula to calculate flow rates.
Use the harmonic mean formula to find the average permeability.

MATLAB Comparison:

For MATLAB users, a similar calculation can be done using the following script:


            % MATLAB script
            k_values = [100, 200, 150]; % Permeability values in mD
            h_values = [50, 75, 60]; % Thickness values in ft
            r1_values = [1000, 1500, 2000]; % Inner radius values in ft
            r2_values = [1500, 2000, 2500]; % Outer radius values in ft

            flow_rates = (2 * pi * k_values .* h_values) ./ log(r2_values ./ r1_values);
            k_avg = 1 / mean(1 ./ k_values);

            fprintf('Average Permeability: %.2f mD\n', k_avg);

Result:

For the given scenario, the calculated average permeability using Excel is $151.52 \, \text{mD}$ . This result can be compared with the MATLAB result to ensure accuracy and consistency in calculations.