Dimensionless Vertical Well Critical Rate Correlations in Excel

Dimensionless vertical well critical rate is a measure of the maximum oil production rate that can be achieved from a vertical well without causing water coning. Water coning is a phenomenon where water from the bottom of the reservoir rises up and displaces the oil near the wellbore, reducing the oil recovery and increasing the water production.

Hoyland, Papatzacos, and Skjaeveland proposed two methods to estimate the dimensionless vertical well critical rate for anisotropic, homogeneous formations with the well completed from the top of the formation. Anisotropic means that the reservoir properties vary in different directions, such as horizontal and vertical permeability.

The first method is based on an analytical solution, which means that they derived a mathematical formula using some simplifying assumptions. One of the assumptions is that the wellbore is infinitely conductive, meaning that the pressure drop along the wellbore is negligible. Another assumption is that the water cone does not affect the pressure distribution in the oil zone, meaning that the oil flow is steady and uniform.

The second method is based on a numerical correlation, which means that they used a computer simulation model to run many scenarios with different reservoir parameters and well rates, and then fitted a curve to the results. The correlation is presented as an equation for the isotropic case, where the reservoir properties are the same in all directions, and as a diagram for the anisotropic case, where the reservoir properties vary in different directions.

Both methods are valid for dimensionless radii between 0.5 and 50, where the dimensionless radius is the ratio of the well radius to the reservoir thickness. The methods show that the dimensionless vertical well critical rate depends on the dimensionless radius, the anisotropy ratio, and the mobility ratio. The anisotropy ratio is the ratio of the horizontal permeability to the vertical permeability. The mobility ratio is the ratio of the water mobility to the oil mobility, where the mobility is the product of the permeability and the relative permeability divided by the viscosity.

The methods also show that the dimensionless vertical well critical rate decreases as the dimensionless radius, the anisotropy ratio, and the mobility ratio increase. This means that the critical rate is lower for wells with larger radius, reservoirs with higher horizontal permeability or lower vertical permeability, and fluids with higher water mobility or lower oil mobility.

Basic Theory:

The dimensionless vertical well critical rate correlations by Hoyland, Papatzacos, and Skjaeveland are widely used in the oil and gas industry. They are based on the dimensionless productivity index, which is the ratio of the productivity index to the skin factor. The dimensionless critical rate (Qc) is then determined by empirical correlations involving various reservoir and fluid properties.

Procedures:

1. Gather Data: Collect the necessary reservoir data such as permeability, porosity, fluid properties, and wellbore dimensions.
2. Calculate Dimensionless Parameters: Calculate dimensionless parameters like the dimensionless productivity index (PI*) using the appropriate equations.
3. Apply Correlations: Utilize the Hoyland, Papatzacos, and Skjaeveland correlations to estimate the dimensionless critical rate.
4. Convert to Real Values: Convert the dimensionless critical rate back to real values using the known reservoir and fluid properties.

Explanation:

Let’s consider a hypothetical scenario with the following reservoir properties:

• Permeability (k) = 100 mD
• Porosity (φ) = 0.15
• Wellbore radius (rw) = 0.1 ft
• Fluid viscosity (μ) = 2 cP
• Formation thickness (h) = 50 ft

Scenario:

Using the given reservoir properties, we want to determine the dimensionless critical rate (Qc).

Excel Implementation:

MATLAB Comparison:

Now, let’s solve the same problem using MATLAB to compare the results. Below is a simple MATLAB script:

        % Given reservoir properties
        k = 100;        % mD
        phi = 0.15;
        rw = 0.1;       % ft
        mu = 2;         % cP
        h = 50;         % ft

        % Dimensionless Productivity Index
        PI_star = (2.5 * k * h) / (phi * mu * rw);

        % Dimensionless Critical Rate
        Qc_star = 0.083 * PI_star^(-0.36);

        % Real Critical Rate
        Qc = Qc_star * (k * h * mu) / rw;

        % Display the result
        fprintf('MATLAB Result: Real Critical Rate (Qc) = %.2f bbl/d\n', Qc);