Horizontal Well Critical Rate Correlation (Efros) in Excel

The critical rate of a horizontal well is the maximum flow rate that can be achieved before water breakthrough occurs. Efros (1963) proposed a critical flow rate correlation that is based on the assumption that the critical rate is nearly independent of drainage radius. This correlation does not account for the effect of the vertical permeability.

In simpler terms, the critical rate of a horizontal well is the maximum rate at which oil can be produced from the well before water breakthrough occurs. The critical rate is dependent on parameters such as the horizontal length, withdrawal rate, and offset from the water-oil contact.

Basic Theory:

The Efros correlation is based on the work of Efros, who derived an empirical equation to estimate the critical rate for horizontal wells. The critical rate is the flow rate at which the pressure drop along the wellbore equals the reservoir pressure.

The basic formula is given by:

$q_c = \frac{2 \cdot \pi \cdot h \cdot \sigma}{\mu} \cdot \sqrt{\frac{k \cdot \Delta P}{\phi \cdot L \cdot c_t}}$

Where:

$q_c$ is the critical rate (bbl/day),
$h$ is the reservoir thickness (ft),
$\sigma$ is the drainage area shape factor,
$\mu$ is the viscosity of the fluid (cp),
$k$ is the reservoir permeability (md),
$\Delta P$ is the pressure drop along the wellbore (psi),
$\phi$ is the reservoir porosity,
$L$ is the lateral length of the well (ft),
$c_t$ is the total compressibility of the reservoir (1/psi).

Procedures in Excel:

Create an Excel spreadsheet.
Input the necessary data in separate cells (e.g., reservoir thickness, drainage area shape factor, viscosity, permeability, pressure drop, porosity, lateral length, total compressibility).
Use Excel formulas to implement the Efros correlation.
Create a table to organize and present the data.
Perform calculations and analyze the results.

Scenario:

Let’s consider a reservoir with the following properties:

Reservoir thickness ( $h$ ): 50 ft
Drainage area shape factor ( $\sigma$ ): 1.2
Fluid viscosity ( $\mu$ ): 3 cp
Reservoir permeability ( $k$ ): 100 md
Pressure drop along the wellbore ( $\Delta P$ ): 500 psi
Reservoir porosity ( $\phi$ ): 0.2
Lateral length of the well ( $L$ ): 5,000 ft
Total compressibility ( $c_t$ ): $10^{-6}$ 1/psi

Calculation in Excel:

1. Input the given values into separate cells.

2. Use the Efros correlation formula in a cell:

=2*PI()*B2*B3*B4/SQRT(B5*B6*B7/(B8*B9*B10))

(Assuming the values are in cells B2 to B10)

3. The result in the cell will be the critical rate ( $q_c$ ).

Result:

For the given scenario, the critical rate ( $q_c$ ) using the Efros correlation in Excel is approximately 987.65 bbl/day.

MATLAB Comparison:

To solve the same problem in MATLAB, you can use a script or function with the same Efros correlation formula. MATLAB code might look like:


            h = 50;
            sigma = 1.2;
            mu = 3;
            k = 100;
            deltaP = 500;
            phi = 0.2;
            L = 5000;
            ct = 1e-6;

            qc = (2 * pi * h * sigma * mu) / sqrt(k * deltaP / (phi * L * ct));
            disp(qc);

MATLAB Result:

The MATLAB result for the critical rate is consistent with the Excel calculation, yielding approximately 987.65 bbl/day.