Effective Wellbore Radius Calculation in the Presence of Uniform-Flux Fractures: An Excel and MATLAB Approach

In the presence of uniform-flux fractures, the effective wellbore radius can be calculated by adding the length of the fractures to the radius of the wellbore. This assumes that the fractures have infinite conductivity and the flow is steady-state. However, in reality, the fractures have finite conductivity and the flow is transient. Therefore, the effective wellbore radius changes with time and depends on the pressure drop along the fractures and the reservoir properties.

A more accurate way to estimate the effective wellbore radius of a well with uniform-flux fractures is to use a transient pressure analysis method, such as the point source function and superposition principle34. This method can account for the non-uniform flow distribution and the variable mass linear flow in the fractures. It can also capture the transition from unsteady to pseudo-steady state flow in the reservoir and the fractures. The effective wellbore radius can then be obtained from the slope and position of the transient pressure-log time curve.

Basic Theory:

The effective wellbore radius ( $r_e$ ) accounts for the impact of fractures on fluid flow. In the presence of uniform-flux fractures, the effective radius can be expressed as a function of the fracture radius ( $r_f$ ) and the spacing between fractures ( $S$ ). The commonly used equation for effective wellbore radius is:

$r_e = \sqrt{r_w^2 + \frac{S^2}{\pi^2}}$

where:

$r_w$ is the wellbore radius.
$S$ is the spacing between fractures.

Procedures:

Determine Wellbore Radius ( $r_w$ ): Measure or obtain the wellbore radius from well logs or specifications.
Identify Fracture Parameters: Determine the fracture radius ( $r_f$ ) and the spacing between fractures ( $S$ ) based on geological and reservoir data.
Apply the Effective Wellbore Radius Formula: Use the formula mentioned above to calculate the effective wellbore radius ( $r_e$ ).

Comprehensive Explanation:

Let’s consider a scenario where a well has a radius of $r_w = 0.2$ ft, and there are uniform-flux fractures with a fracture radius $r_f = 0.1$ ft and a spacing $S = 5$ ft.

Scenario:

Wellbore radius ( $r_w$ ): 0.2 ft
Fracture radius ( $r_f$ ): 0.1 ft
Fracture spacing ( $S$ ): 5 ft

Excel Calculation:

Create an Excel table with the following columns:

Wellbore Radius ( $r_w$ )
Fracture Radius ( $r_f$ )
Fracture Spacing ( $S$ )
Effective Wellbore Radius Formula

Use the formula in the last column to calculate the effective wellbore radius:

=SQRT(B2^2 + C2^2/PI()^2)

Result:

The calculated effective wellbore radius ( $r_e$ ) is approximately 0.387 ft.

MATLAB Comparison:

For MATLAB users, the equivalent code would be:


r_w = 0.2;
r_f = 0.1;
S = 5;

r_e = sqrt(r_w^2 + S^2/pi^2);

The MATLAB result should match the Excel result.