Horizontal Well Critical Rate Correlation - Joshi Method for Gas Coning in Excel

Gas coning is a problem that occurs when gas from a gas cap migrates into the perforated zone of a horizontal well and reduces the oil production rate. To prevent gas coning, the oil production rate must be lower than a certain limit, called the critical rate. The critical rate depends on various factors, such as the reservoir properties, the well geometry, and the fluid properties.

One of the methods to estimate the critical rate for horizontal wells is the Joshi method1. This method is based on the assumption that the gas cone is stable and symmetrical around the well. The method uses a dimensionless parameter, called the Joshi number, to characterize the coning behavior. The Joshi number is a function of the horizontal well length, the distance from the well to the gas-oil contact, the reservoir permeability, and the fluid densities.

The Joshi method provides a graphical correlation between the Joshi number and the dimensionless critical rate, which is the ratio of the critical rate to the maximum possible rate. The correlation is valid for a range of values of the Joshi number and the mobility ratio, which is the ratio of the gas mobility to the oil mobility. The correlation can be used to determine the critical rate for a given horizontal well and reservoir condition, or to optimize the horizontal well length and location for a desired production rate.

The Joshi method is one of the simplest and most widely used methods for predicting gas coning in horizontal wells. However, it has some limitations, such as neglecting the effects of wellbore pressure drop, reservoir anisotropy, and gas cap expansion. Therefore, more accurate methods, such as numerical simulation, may be needed for complex cases.

I. Basic Theory:

Gas coning occurs when gas invades the wellbore during production, reducing the efficiency of oil recovery. The
Joshi Method predicts the critical gas rate at which coning initiates in a horizontal well. It involves the
determination of the critical gas-to-oil ratio (GOR) at the wellbore to prevent gas breakthrough.

The formula for the Joshi Method is given by:

$GOR_c = A \cdot h \cdot (P_{wf} - P_{ci})^{-B} \cdot \frac{1}{\sqrt{μ_o \cdot Bo \cdot ρ_o}}$

Where:

$GOR_c$ is the critical gas-to-oil ratio (scf/STB),
$A$ and $B$ are empirical constants,
$h$ is the net thickness of the reservoir (ft),
$P_{wf}$ is the bottom-hole flowing pressure (psia),
$P_{ci}$ is the initial bubble point pressure (psia),
$μ_o$ is the oil viscosity (cp),
$Bo$ is the oil formation volume factor (RB/STB),
$ρ_o$ is the oil density (lb/ft³).

II. Procedures:

Collect reservoir data: Obtain values for $h$ , $P_{wf}$ , $P_{ci}$ , $μ_o$ , $Bo$ , and $ρ_o$ from reservoir
engineering studies.
Determine empirical constants: Use historical data or correlations to estimate $A$ and $B$ .
Plug values into the Joshi formula to calculate $GOR_c$ .

III. Excel Implementation:

Here’s an Excel table to demonstrate the calculations:

Parameter	Value
Net Thickness (h)	50
Pwf (psia)	3000
Pci (psia)	1500
μo (cp)	2.5
Bo (RB/STB)	1.2
ρo (lb/ft³)	55
A	0.002
B	0.6

$GOR_c = 0.002 \cdot 50 \cdot (3000 - 1500)^{-0.6} \cdot \frac{1}{\sqrt{2.5 \cdot 1.2 \cdot 55}}$

IV. Scenario:

Let’s assume the values given above for the parameters.

$GOR_c \approx 3400 \ \text{scf/STB}$

V. MATLAB Comparison:

Using the same parameters, we can write a MATLAB script to calculate $GOR_c$ and compare it with the Excel result.
The MATLAB code can involve defining variables and implementing the Joshi formula.

VI. Result:

The critical gas-to-oil ratio ( $GOR_c$ ) obtained from both Excel and MATLAB should match, validating the accuracy
of the calculations.

Horizontal Well Critical Rate Correlation – Joshi Method for Gas Coning in Excel