6.3.3.1: The Darcy-Weisbach Equation for Gas Production Wells

The Darcy-Weisbach Equation for natural gases can be developed by modifying the equation for liquids using a simple change of units. For standard oilfield units:

or,

Note, this equation assumes that the flow rate, $q_g$, is in ft³/day; however, it can easily be re-written in M ft³/day or MM ft³/day by adjusting the equation constant, 2,308.59, to either 2.30859 or 2.30859x10^-3, respectively. In this equation:

• 411.147 is an equation constant
• 5.615 is a unit conversion constant, ft³/bbl
• Equation constant:
  • 2,308.59 (for ft³/day)
  • 2.30859 (for M ft³/day)
  • 2.30859 x10^-3 (for MM ft³/day)
• 144 is a unit conversion constant, in²/ft²
• $+/-$ is the sign convention used in the equation with “$-$ ” for production or “$+$ ” for injection
• $q$ is the flow rate through the tubing, ft³/day, M ft³/day, or MM ft³/day
• $E_{eff}$ is an efficiency (tuning) factor for the tubing section, dimensionless
• $D_{ID}$ is the Inner Diameter ($ID$ ) of the tubing, in
• $g_c$ is the Universal Gravitational Constant, 32.174 lb_m-ft/lb_f-sec²
• $g$ is the Local Acceleration due to gravity, ft/sec².
• $f_{DW}$ is the Darcy-Weisbach Friction Factor, dimensionless
• $\rho$ is the density of the fluid, lb_m/ft³
• $\Delta l$ is the length of the section of tubing along its axis, ft
• $p_1$ and $p_2$ are the pressures at two points in a section of tubing, psi
• $z_1$ and $z_2$ are the elevations at two points in a section of tubing, psi

In Lesson 3, we saw that the density of a real gas could be determined by the Real Gas Law (Equation 3.71):

Due to the strong dependence of the gas density on local pressure and temperature, the solution of Equation 6.15b is always performed using the segmented well approach with iterations on the outlet pressure, $p_2$. Essentially, the segmented well approach explicitly performs a numerical integration of the differential energy balance equation, Equation 6.08 (in terms of ${\left(\frac{dp}{dl}\right)}_{total}$ ). The steps for this simple iteration are:

1. Assume a fixed length, $\Delta l$ , of tubing (usually in the range of 100 – 200 ft)
2. Calculate the temperature (in ºR) halfway up the tubing length $\left(\frac{\Delta z}{2}=\frac{\Delta l}{2}sin\left(\theta \right)\right)$ using the local temperature gradient in ºR/ft (feet in true vertical depth, $z$ )
3. Start with the known inlet pressure, $p_1$
4. Assume an outlet pressure, $p_2$
5. Calculate the average pressure in the length of tubing, $\overline{p}=\frac{p_1+p_2}{2}$
6. Calculate the Z-Factor and gas density from Equation 6.16 with the temperature from Step 2 and the average pressure from Step 5
7. Evaluate the Reynolds Number from Equation 6.04 using this gas density (viscosity may also change with temperature)
8. Calculate the Darcy-Weisbach Friction Factor from the Moody Diagram (Figure 6.06), the Colebrook Formula^[6] (Equation 6.06), or the Swamee-Jain Equation^[7] (Equation 6.07) using the Reynolds Number from Step 7
9. Apply the Darcy-Weisbach Equation, Equation 6.15b, to calculate a new outlet pressure, $p_2$
10. Go to Step 5 and continue calculations until two successive iterations of $p_2$ agree to a specified tolerance
11. When converged, proceed to the next length of tubing and start from Step 10