Deliverability Equation for Shallow Gas Reservoirs

The deliverability equation is a mathematical relation that describes how much gas a well can produce under different conditions of pressure and flow rate. It is useful for estimating the well’s productivity and performance.

The deliverability equation can be derived from the theory of gas flow in porous media, which accounts for the effects of pressure, temperature, compressibility, viscosity, and turbulence on the gas movement. However, the theoretical equation is complex and requires many parameters that are difficult to measure or estimate.

Therefore, some empirical equations have been proposed that simplify the deliverability equation by using experimental data from many wells. These equations have the form of a power law, which means that the flow rate is proportional to some power of the pressure difference. The power is called the deliverability exponent, and it depends on the flow characteristics of the well and the reservoir.

One of the most widely used empirical equations is the Rawlins and Schellhardt equation, which was developed in 1935 based on the analysis of tests on more than 500 wells. This equation can be written in terms of pseudopressure, which is a function of pressure that accounts for the variation of gas properties with pressure. The pseudopressure difference between the reservoir and the well is proportional to the flow rate raised to the deliverability exponent.

Another empirical equation is the Houpeurt equation, which was derived from the theoretical equation by assuming a constant deliverability exponent and a linear relationship between pseudopressure and pressure. This equation can also be written in terms of pseudopressure difference and flow rate, but it has a different proportionality constant and an additional term that accounts for the non-Darcy flow effects.

Both equations can be used to determine the deliverability exponent and the flow coefficient, which are the parameters that characterize the well’s deliverability. These parameters can be obtained by plotting the pseudopressure difference against the flow rate on a logarithmic scale and fitting a straight line to the data. The slope of the line is the inverse of the deliverability exponent, and the intercept is related to the flow coefficient.

Once the deliverability exponent and the flow coefficient are known, the deliverability equation can be used to calculate the flow rate for any given pressure difference, or vice versa. It can also be used to calculate the absolute open flow (AOF) potential, which is the maximum flow rate that the well could achieve if there was no pressure drop in the reservoir or the well. The AOF is a common indicator of well productivity and is sometimes used by regulatory agencies to allocate production among wells or to set maximum production rates for individual wells.

Basic Theory:

The deliverability equation for a gas reservoir describes the relationship between the wellhead flowing pressure (Pwf) and the gas flow rate (Q). For shallow gas reservoirs, where the compressibility of gas is significant, the pseudopressure concept is often employed. The deliverability equation can be expressed as:

$Q = \frac{{C \cdot P_p}}{{\mu \cdot B_g \cdot \sqrt{T}}} \left( \frac{{P_{wf}^2 - P_{Res}^2}}{{P_{wf}^2}} \right)$

Where:

$Q$ is the gas flow rate (MMscf/day),
$C$ is a constant,
$P_p$ is the pseudopressure (psia),
$\mu$ is the gas viscosity (cp),
$B_g$ is the gas formation volume factor (RB/scf),
$T$ is the reservoir temperature (Rankine),
$P_{wf}$ is the wellhead flowing pressure (psia),
$P_{Res}$ is the reservoir pressure (psia).

Procedures:

Collect Data: Gather data including reservoir pressure ( $P_{Res}$ ), gas viscosity ( $\mu$ ), gas formation volume factor ( $B_g$ ), reservoir temperature ( $T$ ), and constant $C$ .
Calculate Pseudopressure ( $P_p$ ):
$P_p = \frac{{P_{Res}}}{{1.8}} \left( \frac{{P_{wf}}}{{P_{Res}}} \right)^{1.8}$
Apply Deliverability Equation in Excel: Set up an Excel sheet with input cells for the collected data. Use the deliverability equation to calculate gas flow rate for varying wellhead pressures.
Create a Scenario: Assume a shallow gas reservoir with the following data:
- $P_{Res} = 2000$ psia
- $\mu = 0.02$ cp
- $B_g = 0.003$ RB/scf
- $T = 520$ Rankine
- $C = 2$ MMscf/day/psia

Excel Calculation:

Wellhead Pressure (psia)	Pseudopressure ( $P_p$ )	Gas Flow Rate (MMscf/day)
1500	=P_Res/1.8*(A2/P_Res)^1.8	=($C$2$B$2)/($B$3SQRT($B$4))*(1-(A2/P_Res)^1.8)

Drag the formula down for a range of wellhead pressures to create a deliverability curve.

MATLAB Comparison:

In MATLAB, create a script to perform the same calculations using the deliverability equation. Use a loop to iterate through different wellhead pressures and plot the deliverability curve.


      
      % MATLAB Script
      P_Res = 2000; % psia
      mu = 0.02; % cp
      B_g = 0.003; % RB/scf
      T = 520; % Rankine
      C = 2; % MMscf/day/psia

      % Generate wellhead pressures
      P_wf = 1500:100:2500;

      % Calculate pseudopressure and gas flow rate
      P_p = P_Res/1.8*(P_wf/P_Res).^1.8;
      Q = (C*P_p)./(mu*B_g*sqrt(T)).*(1 - (P_wf/P_Res).^1.8);

      % Plot the deliverability curve
      figure;
      plot(P_wf, Q, '-o');
      xlabel('Wellhead Pressure (psia)');
      ylabel('Gas Flow Rate (MMscf/day)');
      title('Deliverability Curve for Shallow Gas Reservoir');
      grid on;

Results:

Upon completing the calculations in Excel and MATLAB, compare the results graphically. The deliverability curve will illustrate the relationship between wellhead pressure and gas flow rate for the given shallow gas reservoir scenario.