Line-Source Solution for Damaged or Stimulated Wells in Excel

The line-source solution is a mathematical model that describes how the pressure and flow rate of a well change over time, depending on the properties of the reservoir and the well. It is based on some simplifying assumptions, such as:

The well is vertical and fully penetrates the reservoir.
The reservoir is homogeneous, isotropic, and infinite in extent.
The fluid is single-phase, incompressible, and has constant viscosity.
The well produces or injects at a constant rate.

The line-source solution can be used to analyze the performance of a well and estimate the reservoir parameters, such as permeability, skin factor, and drainage area. It can also be used to evaluate the effects of well damage or stimulation on the well productivity or injectivity.

Well damage is a reduction in the effective permeability near the wellbore, caused by factors such as drilling mud invasion, scale deposition, or fines migration. Well damage reduces the flow rate and increases the pressure drop in the well, resulting in a negative skin factor.

Well stimulation is a treatment that enhances the effective permeability near the wellbore, by creating fractures, dissolving minerals, or removing damage. Well stimulation increases the flow rate and decreases the pressure drop in the well, resulting in a positive skin factor.

The line-source solution can be modified to account for the presence of well damage or stimulation, by introducing a term called the skin factor in the equation. The skin factor represents the additional pressure drop or gain due to the altered zone near the wellbore. The skin factor can be positive or negative, depending on whether the well is stimulated or damaged, respectively.

The line-source solution can also be extended to account for other factors, such as wellbore storage, partial penetration, multiphase flow, and finite reservoir boundaries. However, these factors complicate the analysis and require more advanced methods and tools.

The Line-Source solution is derived from Darcy’s law and considers the radial flow of fluids in a reservoir. For a vertically fractured well, the inflow performance relationship (IPR) can be expressed using the Line-Source solution.

$q = \frac{{2 \pi k h}}{{\mu B}} \left( \frac{{P_{wf} - P_{wf,0}}}{{\ln \left( \frac{{r_{e,0}}}{{r_w}} \right)}} \right)$

where:

$q$ is the production rate,
$k$ is the reservoir permeability,
$h$ is the reservoir thickness,
$\mu$ is the fluid viscosity,
$B$ is the formation volume factor,
$P_{wf}$ is the wellbore flowing pressure,
$P_{wf,0}$ is the initial wellbore flowing pressure,
$r_{e,0}$ is the initial effective drainage radius, and
$r_w$ is the wellbore radius.

Procedures for Implementation in Excel

Step 1: Define Parameters

Start by defining the relevant parameters in an Excel table. This includes permeability ( $k$ ), thickness ( $h$ ), fluid viscosity ( $\mu$ ), formation volume factor ( $B$ ), wellbore radius ( $r_w$ ), initial wellbore flowing pressure ( $P_{wf,0}$ ), and initial effective drainage radius ( $r_{e,0}$ ).

Step 2: Input Wellbore Flowing Pressure

Input the current wellbore flowing pressure ( $P_{wf}$ ) in a designated cell.

Step 3: Use Excel Formula

Apply the Line-Source formula in a separate cell to calculate the production rate ( $q$ ). Utilize Excel functions such as $\text{LN}$ , $\text{PI}$ , and basic arithmetic operations to implement the formula.

Example

Let’s consider a reservoir with the following parameters:

Permeability ( $k$ ): 100 md,
Thickness ( $h$ ): 20 ft,
Fluid Viscosity ( $\mu$ ): 1.5 cp,
Formation Volume Factor ( $B$ ): 1.2 RB/STB,
Wellbore Radius ( $r_w$ ): 0.25 ft,
Initial Wellbore Flowing Pressure ( $P_{wf,0}$ ): 3000 psi,
Initial Effective Drainage Radius ( $r_{e,0}$ ): 150 ft.

Assuming the current wellbore flowing pressure ( $P_{wf}$ ) is 2500 psi, we can use the Line-Source solution to calculate the production rate.

Excel Calculation

Set up an Excel table with the given parameters.
Input the current wellbore flowing pressure ( $P_{wf}$ ).
Use the Line-Source formula to calculate the production rate ( $q$ ).

Parameter	Value
Permeability ( $k$ )	100 md
Thickness ( $h$ )	20 ft
Fluid Viscosity ( $\mu$ )	1.5 cp
Formation Volume Factor ( $B$ )	1.2 RB/STB
Wellbore Radius ( $r_w$ )	0.25 ft
$P_{wf,0}$	3000 psi
$r_{e,0}$	150 ft
$P_{wf}$	2500 psi

$q = \frac{{2 \pi \times 100 \times 20}}{{1.5 \times 1.2}} \left( \frac{{2500 - 3000}}{{\ln \left( \frac{{150}}{{0.25}} \right)}} \right)$

$q \approx 150 \, \text{STB/D}$