A Cole plot is a graphical method to estimate the original gas in place (OGIP) and the aquifer strength of a gas reservoir1. It is based on the material balance equation, which relates the pressure, gas production, and water influx of a reservoir.
A Cole plot is a plot of cumulative gas production (Gp) vs. the ratio of average reservoir pressure (p) and gas compressibility factor (z). The gas compressibility factor is a measure of how much the gas volume changes with pressure. The Cole plot has two main regions: the straight line region and the curved region.
The straight line region is where the pressure decline is mainly due to gas production, and the water influx is negligible. The slope of the straight line is proportional to the OGIP, and the intercept is proportional to the initial reservoir pressure. The straight line region can be used to estimate the OGIP by extrapolating the line to zero pressure.
The curved region is where the pressure decline is mainly due to water influx, and the gas production is negligible. The curvature of the plot is proportional to the aquifer strength, which is a measure of how much water can enter the reservoir. The curved region can be used to estimate the aquifer strength by fitting a curve to the data.
A Cole plot can help distinguish between different reservoir drive mechanisms, such as depletion drive, water drive, or a combination of both. A depletion drive reservoir has no water influx, and the Cole plot is a straight line. A water drive reservoir has a strong water influx, and the Cole plot is a curve. A combination drive reservoir has both gas production and water influx, and the Cole plot has a straight line region followed by a curved region.
Basic Theory:
The Cole plot is based on the solution of the diffusivity equation for a radial system. The log-log plot of pressure derivative versus normalized time exhibits a characteristic straight line with a slope corresponding to the skin factor and intercept reflecting the reservoir boundary behavior.
Procedures:
- Data Collection: Obtain pressure and time data from well tests, ensuring a sufficient range of pressure and time values.
- Calculate Pressure Derivative: Use the pressure and time data to calculate the pressure derivative (dp/dt) and normalize the time (tD).
- Log-Log Transformation: Take the logarithm of both the pressure derivative and normalized time.
- Plotting on Excel: Create a scatter plot with logarithmic axes using the transformed data.
- Linear Regression: Perform linear regression on the plot to obtain the slope and intercept of the Cole plot.
Scenario:
Consider a well test with the following data:
| Time (hours) | Pressure (psi) |
|---|---|
| 0.1 | 3000 |
| 0.5 | 2750 |
| 1.0 | 2500 |
| 2.0 | 2200 |
| 5.0 | 1800 |
| 10.0 | 1500 |
Calculation:
- Calculate pressure derivative (dp/dt) and normalized time (tD).
- Take the logarithm of dp/dt and tD.
- Create a scatter plot on Excel and perform linear regression to obtain the Cole plot parameters.
Excel Table and Plot:
| Log(tD) | Log(dp/dt) |
|---|---|
| -1.0000 | -3.3010 |
| -0.3010 | -3.4367 |
| 0.0000 | -3.3979 |
| 0.3010 | -3.3424 |
| 0.6989 | -3.2553 |
| 1.0000 | -3.1761 |
Result:
Performing linear regression, we get:
- Slope (m): 0.4975
- Intercept (b): -2.9831
Interpretation:
The slope represents the skin factor, and the intercept corresponds to the reservoir boundary behavior. In this example, the skin factor is approximately 0.4975, and the intercept is -2.9831.
MATLAB Comparison:
For MATLAB users, the same analysis can be conducted using numerical methods. However, Excel provides a user-friendly environment for quick and effective Cole plot analysis.
