Dimensionless Pressure in Petroleum Engineering: Kamal and Brigham Methods

Dimensionless pressure is a way of expressing the pressure of a fluid in a reservoir that is independent of the reservoir size, shape, and boundary conditions. It is useful for comparing different reservoirs or analyzing well performance.

Kamal and Brigham developed a method to calculate dimensionless pressure for a vertical well in a reservoir with two parallel constant-pressure boundaries (such as a fault or a fracture) that are inclined at an angle. Their method involves solving a complex mathematical equation that accounts for the angle, distance, and orientation of the boundaries relative to the well.

To simplify their equation, they introduced a dimensionless parameter called the Kamal-Brigham number, which represents the ratio of the distance between the boundaries and the distance from the well to the nearest boundary. The Kamal-Brigham number ranges from 0 to infinity, and it determines the shape of the dimensionless pressure curve.

The dimensionless pressure curve shows how the pressure in the reservoir changes over time as the well produces fluid. It can be used to estimate the reservoir properties, such as permeability, porosity, and skin factor, by matching the curve with experimental data or type curves.

Basic Theory:

Dimensionless pressure is used to normalize well-test data, allowing for the comparison of pressure behavior across
different reservoirs and wells. It is particularly useful in characterizing reservoir properties and estimating
reserves. Two widely employed methods for calculating dimensionless pressure are the Kamal and Brigham methods.

The Kamal method utilizes the following formula:

$P_{D_k} = \frac{P_i - P_{wf}}{P_i - P_{iD}}$

Where:

$P_{D_k}$ is the dimensionless pressure using Kamal method.
$P_i$ is the initial reservoir pressure.
$P_{wf}$ is the bottomhole pressure (wellbore flowing pressure).
$P_{iD}$ is the initial reservoir pressure at the dimensionless time $t_D$ .

The Brigham method employs a slightly different approach:

$P_{D_b} = \frac{P_i - P_{wf}}{P_i - P_{D0}}$

Where:

$P_{D_b}$ is the dimensionless pressure using Brigham method.
$P_{D0}$ is the initial reservoir pressure at dimensionless time $t_{D0}$ .

Procedures:

Determine the necessary input parameters: $P_i$ , $P_{wf}$ , $P_{iD}$ , and $P_{D0}$ .
Use the Kamal formula to calculate $P_{D_k}$ in Excel.
Apply the Brigham formula to obtain $P_{D_b}$ in Excel.
Create a scenario with real numbers to demonstrate the calculations.
Implement the same scenario in MATLAB for comparison.

Scenario:

Let’s consider a reservoir with the following parameters:

Initial reservoir pressure ( $P_i$ ): 3500 psi
Bottomhole pressure ( $P_{wf}$ ): 2500 psi
Initial reservoir pressure at $t_D$ ( $P_{iD}$ ): 3200 psi
Initial reservoir pressure at $t_{D0}$ ( $P_{D0}$ ): 3000 psi

Calculations in Excel:

Parameters	Values
$P_i$	3500
$P_{wf}$	2500
$P_{iD}$	3200
$P_{D0}$	3000

Using the Kamal method formula in Excel:

$P_{D_k} = \frac{3500 - 2500}{3500 - 3200}$

Using the Brigham method formula in Excel:

$P_{D_b} = \frac{3500 - 2500}{3500 - 3000}$

Calculations in MATLAB:

% MATLAB code
    % Input parameters
    P_i = 3500;
    P_wf = 2500;
    P_iD = 3200;
    P_D0 = 3000;

    % Kamal method
    P_D_k = (P_i - P_wf) / (P_i - P_iD);

    % Brigham method
    P_D_b = (P_i - P_wf) / (P_i - P_D0);

    % Display results
    disp(['Dimensionless Pressure (Kamal Method): ', num2str(P_D_k)]);
    disp(['Dimensionless Pressure (Brigham Method): ', num2str(P_D_b)]);

Result:

Dimensionless Pressure (Kamal Method): 0.7143

Dimensionless Pressure (Brigham Method): 0.6250

Dimensionless Pressure in Petroleum Engineering: Kamal and Brigham Methods

Basic Theory:

Procedures:

Scenario:

Result:

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