Dimensionless Time for Interference Testing in Homogeneous Reservoirs - Earlougher Formula in Excel

Dimensionless time is a way of comparing how long a test has been running, regardless of the size or shape of the reservoir, or the type of fluid in it. It is like a percentage of the total time needed to get reliable results. For example, if the dimensionless time is 0.1, it means that the test has been running for 10% of the total time. If the dimensionless time is 1, it means that the test has been running for 100% of the total time. The higher the dimensionless time, the more accurate the test results are. Dimensionless time depends on several factors, such as the reservoir permeability, thickness, and pressure, the fluid viscosity and flow rate, and the distance between the wells. Different tests may have different dimensionless times, even if they run for the same amount of time.

Basic Theory:

Dimensionless time (tD) is a normalized time measure used to analyze transient flow behavior in reservoirs. For interference testing in homogeneous reservoirs, Earlougher introduced a formula to calculate dimensionless time based on the following equation:

$tD = \frac{0.25 \cdot r_0^2 \cdot S}{kt}$

Where:

$tD$ = Dimensionless time
$r_0$ = Distance from the observation well to the center of the interference region
$S$ = Reservoir thickness
$k$ = Reservoir permeability
$t$ = Time

Procedures:

Input Parameters:
- Gather the necessary reservoir data: $r_0$ , $S$ , $k$ , and $t$ .
Apply Earlougher Formula:
- Use the Earlougher formula to calculate dimensionless time ( $tD$ ).
Create Excel Table:
- Build an Excel table to organize the input parameters, calculations, and results.
Graphical Representation:
- Plot dimensionless time against pressure for visualization.

Explanation:

Consider a homogeneous reservoir with the following parameters:

$r_0 = 3000$ ft
$S = 50$ ft
$k = 100$ md
$t = 90$ days

Scenario:

We have an observation well at a distance of $r_0 = 3000$ ft from the center of the interference region. The reservoir has a thickness ( $S$ ) of 50 ft, permeability ( $k$ ) of 100 md, and the interference test has been ongoing for $t = 90$ days.

Excel Calculation:

In cell A1, label the parameters: $r_0$ , $S$ , $k$ , $t$ .
In cells B1 to B4, input the corresponding values: 3000, 50, 100, 90.
In cell A6, label the calculation as “Dimensionless Time (tD)”.
In cell B6, input the formula:
```
=0.25 * B2^2 * B3 / B4
```
The result in cell B6 is the dimensionless time ( $tD$ ).

MATLAB Comparison:

% MATLAB code to calculate dimensionless time
r0 = 3000; % ft
S = 50; % ft
k = 100; % md
t = 90; % days

tD = 0.25 * r0^2 * S / (k * t);
disp(['Dimensionless Time (tD): ' num2str(tD)]);

Result:

The dimensionless time ( $tD$ ) for the given scenario is calculated as follows:

$tD = \frac{0.25 \cdot 3000^2 \cdot 50}{100 \cdot 90} \approx 416.67$

Both Excel and MATLAB should yield similar results, confirming the accuracy of the calculations.

Dimensionless Time for Interference Testing in Homogeneous Reservoirs – Earlougher Formula in Excel