Dimensionless Wellbore Storage Coefficient of Radial Flow-Constant-Rate Production in Excel Formula

Imagine that the wellbore is like a straw that you use to drink water from a glass. The water in the glass represents the fluid in the reservoir, and the water in the straw represents the fluid in the wellbore. The dimensionless wellbore storage coefficient is a measure of how much water you can store in the straw compared to the water in the glass. If the straw is very thin, you can only store a small amount of water in it, and the water level in the glass will drop quickly as you drink. This means that the dimensionless wellbore storage coefficient is low, and the pressure response in the well is fast and sensitive to the reservoir behavior. If the straw is very thick, you can store a lot of water in it, and the water level in the glass will drop slowly as you drink. This means that the dimensionless wellbore storage coefficient is high, and the pressure response in the well is slow and smooth, and it takes longer to see the reservoir behavior.

Basic Theory:

The dimensionless wellbore storage coefficient (CWD) is a dimensionless parameter that represents the interaction
between the wellbore storage and the radial flow of fluids in a reservoir. For constant-rate production, the
dimensionless wellbore storage coefficient can be expressed using the following formula:

$CWD = \frac{0.0023 \cdot Q_s \cdot t_D}{(h \cdot r_w^2) + (0.00708 \cdot Q_s \cdot t_D)}$

Where:

$CWD$ is the dimensionless wellbore storage coefficient.
$Q_s$ is the constant production rate (STB/D).
$t_D$ is the dimensionless time, calculated as $\frac{t}{\frac{r_w^2}{k}}$ , where $t$ is time in
days, $r_w$ is the wellbore radius in ft, and $k$ is the reservoir permeability in darcy.
$h$ is the reservoir thickness in ft.

Procedures:

Define the given parameters: $Q_s$ , $r_w$ , $t$ , $k$ , and $h$ .
Calculate $t_D$ using the formula mentioned above.
Plug the values into the dimensionless wellbore storage coefficient formula.

Scenario:

Let’s consider a hypothetical scenario:

Constant production rate ( $Q_s$ ): 500 STB/D
Wellbore radius ( $r_w$ ): 0.5 ft
Time ( $t$ ): 365 days
Reservoir permeability ( $k$ ): 50 darcy
Reservoir thickness ( $h$ ): 50 ft

Excel Calculation:

Parameters	Values
$Q_s$ (STB/D)	500
$r_w$ (ft)	0.5
$t$ (days)	365
$k$ (darcy)	50
$h$ (ft)	50

Excel Formula:

        t_D = t / (r_w^2 / k)
        CWD = 0.0023 * Q_s * t_D / ((h * r_w^2) + (0.00708 * Q_s * t_D))

Excel Result:

        t_D = 365 / (0.5^2 / 50) ≈ 730
        CWD = 0.0023 * 500 * 730 / ((50 * 0.5^2) + (0.00708 * 500 * 730)) ≈ 0.002

MATLAB Comparison:

For comparison, the same calculation can be done in MATLAB using the equivalent formulas.

        Qs = 500;
        rw = 0.5;
        t = 365;
        k = 50;
        h = 50;

        t_D = t / (rw^2 / k);
        CWD_MATLAB = 0.0023 * Qs * t_D / ((h * rw^2) + (0.00708 * Qs * t_D));
        disp(['CWD_MATLAB: ', num2str(CWD_MATLAB)]);

Result:

The dimensionless wellbore storage coefficient for the given scenario is approximately 0.002.