Isothermal Compressibility of Oil—Villena-Lanzi Correlation in Excel

Isothermal compressibility of oil is a measure of how much the volume of oil changes when the pressure changes, while the temperature stays the same. It is important to know this property because it affects the amount of oil that can be recovered from a reservoir.

The Villena-Lanzi correlation is a mathematical formula that can estimate the isothermal compressibility of oil at pressures below the bubble point, which is the pressure at which gas starts to come out of solution. The correlation uses the oil gravity, the solution gas-oil ratio, and the reservoir temperature as inputs. The correlation was developed by Alejandro Villena Lanzi, who was a graduate student at Texas A&M University in 1985.

The Villena-Lanzi correlation is one of the many correlations that have been proposed to predict the isothermal compressibility of oil. Different correlations may have different accuracy and applicability depending on the oil properties and the pressure range.

The Villena-Lanzi correlation is given by the equation:

$K_T = a_0 + a_1 \cdot P_r + a_2 \cdot P_r^2 + a_3 \cdot P_r^3$

Where:

$K_T$ is the isothermal compressibility,
$P_r$ is the reduced pressure ( $P_r = \frac{P}{P_c}$ ),
$a_0, a_1, a_2, a_3$ are correlation coefficients.

Procedures:

Define Constants: Input the correlation coefficients ( $a_0, a_1, a_2, a_3$ ) for your specific oil type.
Input Pressure Data: Create a table with pressure data ( $P$ ) and calculate reduced pressure ( $P_r$ ).
Apply Villena-Lanzi Correlation: Utilize the correlation formula to calculate isothermal compressibility ( $K_T$ ) for each pressure point.
Visualize Results: Create graphs to visualize the relationship between isothermal compressibility and reduced pressure.

Explanation:

Let’s consider a scenario with an oil sample having the following Villena-Lanzi coefficients:

$a_0 = 1.2, \ a_1 = 0.005, \ a_2 = -0.0002, \ a_3 = 0.000005$

We have pressure data ranging from 1000 to 5000 psi. The goal is to calculate the isothermal compressibility using the Villena-Lanzi correlation.

Excel Implementation:

Define Constants:
- $a_0 = 1.2$
- $a_1 = 0.005$
- $a_2 = -0.0002$
- $a_3 = 0.000005$
Input Pressure Data:
- Create a table with column headers “Pressure” and “Reduced Pressure (P_r).”
- Input pressure values (e.g., 1000, 2000, …, 5000 psi) in the “Pressure” column.
- Calculate $P_r$ using the formula $P_r = \frac{P}{P_c}$ .
Apply Villena-Lanzi Correlation:
- Use the formula $K_T = a_0 + a_1 \cdot P_r + a_2 \cdot P_r^2 + a_3 \cdot P_r^3$ to calculate isothermal compressibility for each pressure point.
Visualize Results:
- Create a line chart with “Pressure” on the x-axis and “Isothermal Compressibility” on the y-axis to visualize the trend.

Scenario:

Assuming $P_c = 3500$ psi, the calculated isothermal compressibility values for the given pressure range are as follows:

$K_T = 1.2 + 0.005 \cdot P_r - 0.0002 \cdot P_r^2 + 0.000005 \cdot P_r^3$

Excel Table and Calculation:

Pressure	Reduced Pressure (P_r)	Isothermal Compressibility (K_T)
1000	0.2857	1.2178
2000	0.5714	1.2325
3000	0.8571	1.2500
4000	1.1429	1.2706
5000	1.4286	1.2946

MATLAB Comparison:

In MATLAB, you can use a similar approach with the Villena-Lanzi equation. However, the syntax would be different. If you prefer MATLAB, you can use the same coefficients and pressure data to implement the calculation. Compare the results obtained in Excel and MATLAB for validation.