6.3.2.1: The Darcy-Weisbach Equation for Single-Segment Oil Production Wells

The Darcy-Weisbach Equation is one of the most common equations for modeling single-phase liquid flow through pipes and tubing. The Darcy-Weisbach Equation is developed by ignoring the acceleration term in Equation 6.02 and replacing the derivative with a finite-difference approximation:

Where the angle, $\theta$, is measured from the horizontal and is 90º $\left(\frac{\pi }{2}radians\right)$ for true vertical wells and 0º $\left(0radians\right)$ for horizontal wells. Solving this equation for the velocity:

Substituting for the velocity term, $v\left(\frac{ft}{\sec }\right)=\frac{q}{A}=\frac{\left(4\right)144\left(\frac{i{n}^2}{f{t}^2}\right)}{\pi D_{ID}^2\left(i{n}^2\right)}\frac{5.615\left(\frac{f{t}^3}{bbl}\right)q\left(\frac{bbl}{day}\right)}{24\left(\frac{hr}{day}\right)60\left(\frac{\min }{hr}\right)60\left(\frac{\sec }{\min }\right)}$:

or, after evaluating the constants and rearranging:

Noting that the term $\Delta lsin\left(\theta \right)=\Delta z$ (the change in elevation over the length of the tubing, $\Delta l$):

This is the theoretically derived Darcy-Weisbach Equation for flow through pipe/tubing in oilfield units. This equation relates the flow rate, $q\left(bbls/day\right)$, to a given pressure drop $\Delta p=p_1-p_2\left(\text{psi}\right)$. In practice, we include a dimensionless efficiency factor, $E_{eff}$, which is approximately equal to one $\left(E_{eff}\cong 1.0\right)$. This efficiency factor is used to tune the equation to actual field measurements.

This version of the Darcy-Weisbach Equation is the version most often used in industry software. In this equation:

• 411.147 is an equation constant
• 144 is a unit conversion constant, in²/ft²
• $q$ is the flow rate through the tubing, bbl/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 ($ID$) 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. The local acceleration due to gravity varies from location to location but is approximately 32.174 ft/sec². The ratio of $\frac{g}{g_c}$ is approximately 1.0 lb_f/lb_m
• $f_{DW}$ 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

We can use this equation in two ways. The first way to use Equation 6.12, is to specify the flow rate and calculate the pressure drop along the section of the pipe/tubing. This calculation is called a Pressure Traverse calculation and is illustrated in Figure 6.08 for a vertical well. In this figure, two tubing diameters are considered, and multiple production rates are plotted for each tubing size. The pressure traverse calculation is used by production engineers to help select the appropriate tubing size for the anticipated well production rates during the completion design phase of the well.

Alternatively, if we know one pressure and the flow rate, then we can calculate the other pressure. This is normally done by specifying the Well Head Pressure, $p_{wh}$, and calculating the flowing bottom-hole pressure, $p_{wf}$, for multiple production rates. This is called Tubing Performance calculation and is illustrated in Figure 6.09 for a well head pressure of $p_{wh}=100\text{ psi}$ and the same two tubing sizes plotted in Figure 6.08: $D_{ID}\text{=1}\text{.995 in}$ in and $D_{ID}\text{=2}\text{.993 in}$.