Effective Wellbore Radius of a Horizontal Well in Isotropic Reservoirs

The effective wellbore radius is a parameter that represents the equivalent radius of a vertical well that has the same productivity as a horizontal well. It is used to calculate the pressure drop and the flow rate of a horizontal well.

One method to estimate the effective wellbore radius of a horizontal well in isotropic reservoirs is based on the Joshi model. This method assumes that the horizontal well drainage area is composed of two half circles of radius b at each end and a rectangle of dimensions L (2b) in the center, where L is the completed length of the horizontal well and b is the distance from the well to the boundary.

Basic Theory:

The effective wellbore radius is a measure of the well’s influence on fluid flow in a reservoir. For a horizontal well in an isotropic reservoir, Method 1 involves considering the pressure drop along the wellbore and in the near-wellbore region. The formula for effective wellbore radius (Reff) is given by:

$Reff = \sqrt{\frac{k h}{(C_t + C_f)\mu}}$

Where:

$k$ is the reservoir permeability (md),
$h$ is the reservoir thickness (ft),
$C_t$ is the total compressibility of the reservoir (psi $^{-1}$ ),
$C_f$ is the compressibility of the fluid in the reservoir (psi $^{-1}$ ),
$\mu$ is the fluid viscosity (cp).

Procedures:

Gather reservoir data: Obtain values for reservoir permeability ( $k$ ), reservoir thickness ( $h$ ), total compressibility ( $C_t$ ), fluid compressibility ( $C_f$ ), and fluid viscosity ( $\mu$ ).
Use the provided formula to calculate the effective wellbore radius ( $Reff$ ) in Excel.
Create a scenario with realistic values for the parameters.
Set up an Excel table to organize the data and perform calculations.
Verify the results with MATLAB for comparison.

Scenario:

Consider a horizontal well in an isotropic reservoir with the following data:

Reservoir permeability ( $k$ ): 200 md
Reservoir thickness ( $h$ ): 30 ft
Total compressibility ( $C_t$ ): 10 $^{-6}$ psi $^{-1}$
Fluid compressibility ( $C_f$ ): 5 $^{-6}$ psi $^{-1}$
Fluid viscosity ( $\mu$ ): 2 cp

Calculation in Excel:

Parameters	Values
Reservoir Permeability ( $k$ )	200 md
Reservoir Thickness ( $h$ )	30 ft
Total Compressibility ( $C_t$ )	10 $^{-6}$ psi $^{-1}$
Fluid Compressibility ( $C_f$ )	5 $^{-6}$ psi $^{-1}$
Fluid Viscosity ( $\mu$ )	2 cp

$Reff = \sqrt{\frac{200 \times 30}{(10^{-6} + 5^{-6}) \times 2}}$

$Reff \approx \sqrt{\frac{6000}{15 \times 2}} \approx \sqrt{\frac{6000}{30}} \approx \sqrt{200} \approx 14.14 \text{ ft}$

Comparison with MATLAB:

In MATLAB, the same formula will be implemented, and the result should match the Excel calculation. This step ensures the accuracy and reliability of the calculations.

MATLAB Code:


% Given parameters
k = 200; % md
h = 30; % ft
Ct = 1e-6; % psi^(-1)
Cf = 5e-6; % psi^(-1)
mu = 2; % cp

% Effective wellbore radius calculation
Reff_MATLAB = sqrt((k * h) / ((Ct + Cf) * mu));
disp(['Effective Wellbore Radius (MATLAB): ', num2str(Reff_MATLAB), ' ft']);