6.3.3.2: Other Equations for Gas Production Wells

As seen with this iteration, the use of the Darcy-Weisbach Equation can result in very complicated iteration process and requires the use of a computer for a solution. There are other methods available for gas wells and pipelines that can also be used that do not require an iteration.

One popular method used for gas wells is the Cullender and Smith^[9] Method. This method uses a limited range of the relative roughness of 0.0006 to 0.00065 and a specialized friction factor correlation designed specifically for this range (not the generalized Colebrook Formula^[6] or the Swamee-Jain Equation^[7]). In addition, the method uses a two-step integration based on the trapezoid rule (two well segments). This method also requires an iteration, but it is not as complex as that shown above.

Other popular methods for gas wells and transmission lines include the Weymouth equation, the Panhandle “A” equation, and the Panhandle “B” equation. These equations are empirical equations developed from the generalized energy balance equation but use specialized, explicit friction factor correlations which allow them to be solved in a non-iterative manner. Also note that these equations use the flow rate in SCF/day and not ft³/day as in Equation 6.15. These equations have the form:

with

Where:

• $C$ is an equation constant:
  • 433.5 (for the Weymouth equation)
  • 435.87 (for the Panhandle “A” equation)
  • 737.0 (for the Panhandle “B” equation)
• 0.0375 is an equation constant
• 5280.0 is a unit conversion constant, ft/mile
• $q$ is the flow rate through the tubing, SCF/day
• $E_{eff}$ is an efficiency (tuning) factor for the tubing section $\left(E_{eff}\cong 1.0\right)$, dimensionless
• $D_{ID}$ is the Inner Diameter $\left(ID\right)$ of the tubing, in
• $T_{SC}$ is the standard temperature, ºR
• $p_{SC}$ is the standard pressure, psi
• $p_1$ and $p_2$ are the pressures at two points in a section of tubing, psi
• ${\gamma }_{\mathrm{g}}$ is the specific gravity of the gas, ${\gamma }_g=M{W}_g/M{W}_{air}$, dimensionless
• $Z$ is the real gas supercompressibility factor (Z-Factor), dimensionless
• $T$ is the average temperature of the tubing section, ºR
• $\Delta l$ is the length of the section of tubing along its axis, ft
• $exp\left(H_e\right)$ is a correction factor for change in elevation, dimensionless
• $z_1$ and $z_2$ are the elevations at two points in a section of tubing, ft
• $\theta$ is the angle from the horizontal, degrees or radians
• $n_1$ is an exponent:
  • 2.667 (for the Weymouth equation)
  • 2.6182 (for the Panhandle “A” equation)
  • 2.530 (for the Panhandle “B” equation)
• $n_2$ is an exponent:
  • 1.0 (for the Weymouth equation)
  • 1.0788 (for the Panhandle “A” equation)
  • 1.02 (for the Panhandle “B” equation)
• $n_3$ is an exponent:
  • 1.0 (for the Weymouth equation)
  • 0.853 (for the Panhandle “A” equation)
  • 0.961 (for the Panhandle “B” equation)
• $n_4$ is an exponent:
  • 0.5 (for the Weymouth equation)
  • 0.5392 (for the Panhandle “A” equation)
  • 0.51 (for the Panhandle “B” equation)

In these equations, the Z-Factor can is evaluated at the average pressure, $\overline{p}$:

Some investigators suggest that a more representative definition of average pressure, and hence a more accurate result, can be obtained with the following definition of average pressure:

When both the inlet pressure and the outlet pressure are known and the rate is the unknown, these equations (the Weymouth equation, the Panhandle “A” equation, and the Panhandle “B” equation) do not need an iteration for a solution. However, if the rate and one pressure are known and the other pressure is the unknown, then an iteration is still required. These equations can be used for tubing calculations or for calculations on long gas transmission pipelines. You will most likely see options for them if you use industry pipe flow or Nodal Analysis software. (Nodal Analysis is a well optimization technique where all components, or nodes, of the production system are modeled – the reservoir, skin, tubing, wellhead choke, flow line, and surface facilities – to identify any bottlenecks in the system for possible remediation or Debottlenecking.) For well modeling applications, Equation 6.17 through Equation 6.19 can be used to evaluate the pressure loss in the wells, in the Flow Lines (small ID pipe used to transport produced oil or gas from the wellhead to a gathering station, field separator, or other surface facility), and in gas transmission lines.