Cumulative Gas Production Using Tarner's Method in Excel

Tarner’s method is a technique for predicting the cumulative gas production (Gp) from a reservoir as a function of reservoir pressure (P). It is based on the material balance equation (MBE), which relates the initial and remaining volumes of oil, gas, and water in the reservoir, and the instantaneous gas-oil ratio ®, which is the ratio of the gas and oil produced at any given time.

The basic idea of Tarner’s method is to assume a series of reservoir pressure drops from the initial pressure (Pi) to the desired pressure (P), and calculate the corresponding values of Gp and R for each pressure using the MBE and the R equation. The method is iterative, meaning that the calculations are repeated until the desired accuracy is achieved.

The advantage of Tarner’s method is that it can be applied to both undersaturated and saturated reservoirs, and it does not require any information about the reservoir rock or fluid properties, except for the initial values of Pi, Gp, R, and the oil formation volume factor (Bo). The disadvantage is that it can be tedious and time-consuming, especially for large pressure drops or complex reservoirs.

Here is a summary of the steps involved in Tarner’s method:

Assume an initial value of P, such as Pi – 100 psi.
Calculate the value of R using the R equation, which depends on the type of reservoir (undersaturated or saturated).
Calculate the value of Gp using the MBE, which depends on the type of reservoir (undersaturated or saturated).
Compare the calculated value of Gp with the actual value of Gp from the production data. If they are close enough, stop the iteration. If not, go back to step 1 and assume a different value of P, such as P – 100 psi.
Repeat steps 1 to 4 until the desired accuracy is achieved.

Basic Theory

Tarner’s method is based on the concept of hyperbolic decline in gas reservoirs. The hyperbolic decline equation is
given by:

$q_t = \frac{q_i}{(1 + b \cdot d \cdot t)^{1/b}}$

where:

$q_t$ is the gas flow rate at time $t$ ,
$q_i$ is the initial gas flow rate,
$b$ is the hyperbolic decline constant, and
$d$ is the hyperbolic decline exponent.

The cumulative gas production ( $Q_t$ ) at any time $t$ can be calculated by integrating the hyperbolic decline
equation:

$Q_t = \int_{0}^{t} q_t \,dt$

Procedures

Collect Data: Gather gas production data over time, including the initial flow rate ( $q_i$ ),
time ( $t$ ), and the hyperbolic decline parameters ( $b$ and $d$ ).
Calculate Gas Flow Rate ( $q_t$ ): Use the hyperbolic decline equation to calculate the gas flow
rate ( $q_t$ ) at each time step.
Calculate Cumulative Gas Production ( $Q_t$ ): Integrate the gas flow rate over time to obtain
the cumulative gas production at each time step.
Excel Implementation: Utilize Excel formulas to automate the calculations and create a table to
present the results.
MATLAB Comparison: Implement the same calculations in MATLAB to compare results and validate the
accuracy of the Excel implementation.

Scenario

Let’s consider a gas well with the following parameters:

$q_i = 5000 \, \text{SCF/day}$ ,
$b = 0.1$ ,
$d = 0.5$ ,
Time steps ( $t$ ): 0, 1, 2, 3, and 4 years.

Excel Calculation

Create an Excel table with columns for time ( $t$ ), gas flow rate ( $q_t$ ), and cumulative gas production
( $Q_t$ ). Use Excel formulas to implement the hyperbolic decline equation and integrate to find cumulative
production.

Time (years)	Gas Flow Rate ( $q_t$ )	Cumulative Gas Production ( $Q_t$ )
0	$=5000/(1+0.10.50)^1/0.1$	$=0$
1	$=5000/(1+0.10.51)^1/0.1$	$=Integration of $q_t$ up to 1$
2	$=5000/(1+0.10.52)^1/0.1$	$=Integration of $q_t$ up to 2$
3	$=5000/(1+0.10.53)^1/0.1$	$=Integration of $q_t$ up to 3$
4	$=5000/(1+0.10.54)^1/0.1$	$=Integration of $q_t$ up to 4$

MATLAB Calculation

Implement the same hyperbolic decline equation and integrate using MATLAB. Compare the cumulative gas production
values with the Excel results to ensure consistency.

    
      qi = 5000;
      b = 0.1;
      d = 0.5;
      t = [0 1 2 3 4];

      qt = qi ./ (1 + b * d * t).^ (1 / b);
      Qt = cumtrapz(t, qt);

Results

The Excel table will provide the cumulative gas production values at each time step. The MATLAB variables $qt$ and
$Qt$ will contain the gas flow rates and cumulative production, respectively. Compare the results to ensure
accuracy and consistency between the two methods.

Time (years)	Gas Flow Rate ( $q_t$ )	Cumulative Gas Production ( $Q_t$ )
0	$=5000/(1+0.10.50)^1/0.1$	$=0$
1	$=5000/(1+0.10.51)^1/0.1$	$=Integration of \(q_t$ up to 1\)
2	$=5000/(1+0.10.52)^1/0.1$	$=Integration of \(q_t$ up to 2\)
3	$=5000/(1+0.10.53)^1/0.1$	$=Integration of \(q_t$ up to 3\)
4	$=5000/(1+0.10.54)^1/0.1$	$=Integration of \(q_t$ up to 4\)

Cumulative Gas Production Using Tarner’s Method in Excel