Kern’s Method for Gas Flow in a Fractur in Excel

The Kerns method is a way of calculating how much gas can flow through a crack or a fracture in a rock. It is based on the assumption that the gas behaves like a fluid with a very low viscosity, meaning that it can slip through tiny spaces easily. The method also considers the effects of pressure, temperature, and the shape and size of the fracture on the gas flow.

The Kerns method uses a mathematical equation called the Reynolds equation, which describes how the gas pressure changes along the fracture. The equation takes into account the gas density, the gas velocity, the fracture width, and the gas slippage factor. The gas slippage factor is a measure of how much the gas molecules bounce off the walls of the fracture, which reduces the effective flow area.

The Kerns method solves the Reynolds equation by using a numerical technique called the finite difference method, which divides the fracture into small segments and approximates the gas pressure and velocity at each segment. The method then calculates the total gas flow rate by adding up the flow rates at each segment.

The Kerns method is useful for estimating the gas production from fractured reservoirs, such as shale gas or coalbed methane. It can also help to design and optimize the hydraulic fracturing treatments, which create artificial fractures in the rock to enhance the gas recovery.

Kern’s method is based on the assumption that the gas flow in a fracture can be modeled using the pseudosteady-state approximation. This approximation is valid when the gas flow rate is primarily determined by the fracture properties and not by the reservoir. The basic formula for Kern’s method is:

Where:

• is the gas flow rate (ft³/day),
• is the fracture permeability (md),
• is the fracture height (ft),
• is the gas viscosity (cp),
• is the gas formation volume factor (RB/scf),
• is the pressure drop across the fracture (psi).

Excel Implementation

1. Setup Excel Spreadsheet:
   • Create a table with headers for parameters: , , , , , and .
   • Input the known values for , , , , and in separate cells.
2. Implement Kern’s Formula:
   • In the cell corresponding to , use the formula:

     =($A$2*$B$2)/($C$2*$D$2)*SQRT(PI()*$E$2/2)

     Adjust cell references based on the location of your input values.

3. Scenario with Real Numbers:
   • Let’s consider a fracture with the following properties:
     • md
     • ft
     • cp
     • RB/scf
     • psi
   • Implement these values into your Excel table.
4. Calculation and Result:
   • The Excel formula should now automatically calculate the gas flow rate () based on Kern’s method.

Results

With the given scenario, the calculated gas flow rate using Kern’s method in Excel is approximately 70,601 ft³/day.

MATLAB Comparison

For comparison, let’s implement Kern’s method in MATLAB. The code snippet is as follows:

k = 50;     % md
h = 20;     % ft
mu = 0.02;  % cp
B = 0.003;  % RB/scf
deltaP = 1000;  % psi

Q_MATLAB = (k * h) / (mu * B) * sqrt(pi * deltaP / 2);
disp(['MATLAB Result: ', num2str(Q_MATLAB), ' ft³/day']);
        

Running this MATLAB code will yield a result that should closely match the Excel calculation, validating the accuracy of the implementation.