Effective Compressibility in Undersaturated Oil Reservoirs: Hawkins Formula and Excel Implementation

Effective compressibility is a measure of how much the volume of a fluid changes when the pressure changes. In an undersaturated oil reservoir, the oil is not in contact with any gas, so the only sources of compressibility are the oil itself and the rock that contains it. The effective compressibility of the reservoir is a weighted average of the oil compressibility and the rock compressibility, depending on how much oil and rock there are in the reservoir.

The Hawkins method is a way of calculating the effective compressibility of an undersaturated oil reservoir using the initial oil formation volume factor and the change in pressure. The oil formation volume factor is the ratio of the volume of oil at reservoir conditions to the volume of oil at standard conditions. The Hawkins method assumes that the oil formation volume factor is constant for small changes in pressure, and that the rock compressibility is negligible compared to the oil compressibility.

Basic Theory

Effective compressibility (c_e) is a key parameter in reservoir engineering, representing the ability of a porous medium to compress when subjected to changes in pressure. In undersaturated oil reservoirs, the Hawkins formula is commonly employed:

c_e = c_r + (1 – S_w) * c_w

Where:

• c_e is the effective compressibility,
• c_r is the rock compressibility,
• S_w is the water saturation, and
• c_w is the water compressibility.

Procedures

1. Input Parameters:
   • Obtain rock compressibility (c_r),
   • Determine water compressibility (c_w), and
   • Estimate water saturation (S_w).
2. Apply Hawkins Formula:
   • Use the Hawkins formula to calculate effective compressibility (c_e).
3. Excel Implementation:
   • Input parameters into Excel cells,
   • Use Excel formulas to apply the Hawkins formula.

Excel Implementation

Let’s consider the following scenario:

• c_r = 3 * 10^-6 psi^-1,
• c_w = 2 * 10^-6 psi^-1,
• S_w = 0.25.

Excel Formulas

1. Input the parameters in Excel:
   • Cell A1: c_r = 3 * 10^-6,
   • Cell A2: c_w = 2 * 10^-6,
   • Cell A3: S_w = 0.25.
2. Apply the Hawkins formula in Excel (Cell A4):

   =A1 + (1 - A3) * A2

Scenario Calculation

Using the given scenario values:

ce = 3 * 10-6 + (1 - 0.25) * 2 * 10-6

ce = 3 * 10-6 + 0.75 * 2 * 10-6

ce = 3 * 10-6 + 1.5 * 10-6

ce = 4.5 * 10-6 psi-1

Therefore, in this scenario, the effective compressibility (c_e) is 4.5 * 10^-6 psi^-1.

MATLAB Comparison

For MATLAB users, the equivalent code for the Hawkins formula would be:

    % Input parameters
    c_r = 3e-6;
    c_w = 2e-6;
    S_w = 0.25;

    % Hawkins formula
    c_e = c_r + (1 - S_w) * c_w;

    disp(['Effective Compressibility (MATLAB): ', num2str(c_e), ' psi^(-1)']);

Compare the MATLAB result with the Excel result for validation.