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 ft^{3}/day; however, it can easily be re-written in M ft^{3}/day or MM ft^{3}/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, ft
^{3}/bbl - Equation constant:
- 2,308.59 (for ft
^{3}/day) - 2.30859 (for M ft
^{3}/day) - 2.30859 x10
^{-3}(for MM ft^{3}/day)

- 2,308.59 (for ft
- 144 is a unit conversion constant, in
^{2}/ft^{2} - $+/-$ is the sign convention used in the equation with “$-$ ” for production or “$+$ ” for injection
- $q$ is the flow rate through the tubing, ft
^{3}/day, M ft^{3}/day, or MM ft^{3}/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
, 32.174 lb**Universal Gravitational Constant**_{m}-ft/lb_{f}-sec^{2} - $g$ is the
due to gravity, ft/sec**Local Acceleration**^{2}. - ${f}_{DW}$ is the Darcy-Weisbach Friction Factor, dimensionless
- $\rho $ is the density of the fluid, lb
_{m}/ft^{3} - $\Delta l$ 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:

- Assume a fixed length, $\Delta l$ , of tubing (usually in the range of 100 – 200 ft)
- 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, $z$ )
- Start with the known inlet pressure, ${p}_{1}$
- Assume an outlet pressure, ${p}_{2}$
- Calculate the average pressure in the length of tubing, $\overline{p}=\frac{{p}_{1}+{p}_{2}}{2}$
- Calculate the Z-Factor and gas density from
**Equation 6.16**with the temperature from Step 2 and the average pressure from**Step 5** - Evaluate the Reynolds Number from
**Equation 6.04**using this gas density (viscosity may also change with temperature) - 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** - Apply the Darcy-Weisbach Equation,
**Equation 6.15b**, to calculate a new outlet pressure, ${p}_{2}$ - Go to
**Step 5**and continue calculations until two successive iterations of ${p}_{2}$ agree to a specified tolerance - When converged, proceed to the next length of tubing and start from
**Step 10**