In this section we consider correlations used to calculate the pressure drop in gas–liquid two-phase flow in wells. As in single-phase flow, the starting point is the mechanical energy balance given by Equation (7-16). Since the flow properties may change significantly along the pipe (mainly the gas density and velocity) in gas–liquid flow, we must calculate the pressure gradient for a particular location in the pipe;
the overall pressure drop is then obtained with a pressure traverse calculation procedure (Section 7.4.4).
A differential form of the mechanical energy balance equation is
In most two-phase flow correlations, the potential energy pressure gradient is based on the in-situ average density, ,
where
Various definitions of the two-phase average velocity, viscosity, and friction factor are used in the different correlations to calculate the kinetic energy and frictional pressure gradients.
There are many different correlations that have been developed to calculate gas–liquid pressure gradients, ranging from simple empirical models to complex deterministic models. The reader is referred to Brill and Mukherjee (1999) for a detailed treatment of several of these correlations.
Although the deterministic models have aided our understanding of multiphase flow behavior, they have not shown a great improvement in predictive accuracy compared with some of the simpler models.
Table 7-1 from Ansari et al. (1994a and 1994b) compares the relative errors of eight different two-phase flow correlations using several combinations of databases. In this table, the smaller the relative
performance factor, the more accurate is the correlation. One of the consistently best correlations was found to be the empirical Hagedorn and Brown correlation.
Table 7-1. Relative Performance Factorsa
aFrom Ansari et al., 1994.
We consider two of the most commonly used two-phase flow correlations for oil wells: the modified Hagedorn and Brown method (Brown, 1977) and the Beggs and Brill (1973) method with the Payne et al. (1979) correction. The first of these was developed for vertical, upward flow and is recommended only for near-vertical wellbores; the Beggs and Brill correlation can be applied for any wellbore
inclination and flow direction. Finally, we will review the Gray (1974) correlation, which is commonly used for gas wells that are also producing liquid.
7.4.3.1. The Modified Hagedorn and Brown Method
The modified Hagedorn and Brown method (mH-B) is an empirical two-phase flow correlation based on the original work of Hagedorn and Brown (1965). The heart of the Hagedorn-Brown method is a correlation for liquid holdup; the modifications of the original method include using the no-slip holdup when the original empirical correlation predicts a liquid holdup value less than the no-slip holdup and the use of the Griffith correlation (Griffith and Wallis, 1961) for the bubble flow regime.
These correlations are selected based on the flow regime as follows. Bubble flow exists if λg < LB, where
and LB ≥ 0.13. Thus, if the calculated value of LB is less than 0.13, LB is set to 0.13. If the flow regime is
found to be bubble flow, the Griffith correlation is used; otherwise, the original Hagedorn-Brown correlation is used.
7.4.3.2. Flow Regimes Other than Bubble Flow: The Original Hagedorn-Brown Correlation
The form of the mechanical energy balance equation used in the Hagedorn-Brown correlation is
which can be expressed in oilfield units as
where f is the friction factor, is the total mass flow rate (lbm/d), is the in-situ average density [Equation (7-90)] (lbm/ft3), D is the diameter (ft), um is the mixture velocity (ft/sec), and the pressure gradient is in psi/ft. The mixture velocity used in H-B is the sum of the superficial velocities,
To calculate the pressure gradient with Equation (7-93), the liquid holdup is obtained from a
correlation and the friction factor is based on a mixture Reynolds number. The liquid holdup, and hence, the average density, is obtained from a series of charts using the following dimensionless numbers.
Liquid velocity number, Nvl:
Gas velocity number, Nvg:
Pipe diameter number, ND,
Liquid viscosity number, NL:
In field units, these are
where superficial velocities are in ft/sec, density is in lbm/ft3, surface tension in dynes/cm, viscosity in
cp, and diameter in ft. The holdup is obtained from Figures 7-12 through 7-14 or calculated from equations that fit the correlation curves presented on the charts (Brown, 1977). First, CNL is read from Figure 7-12 or calculated by
Figure 7-12. Hagedorn and Brown correlation for CNL. (From Hagedorn and Brown, 1965.) Then the group
is calculated; from Figure 7-13, we get yl/ψ, or it is calculated by
Figure 7-13. Hagedorn and Brown correlation for holdup/ψ. (From Hagedorn and Brown, 1965.) Here p is the absolute pressure at the location where pressure gradient is wanted, and pa is atmospheric pressure. Finally, compute
and read ψ from Figure 7-14 or calculate it with
Figure 7-14. Hagedorn and Brown correlation for ψ. (From Hagedorn and Brown, 1965.) The liquid holdup is then
The mixture density is then calculated from Equation (7-90).
The frictional pressure gradient is based on a Fanning friction factor using a mixture Reynolds number, defined as
or, in terms of mass flow rate and using field units,
where mass flow rate, , is in lbm/day, D is in ft, and viscosities are in cp. The friction factor is then obtained from the Moody diagram (Figure 7-7) or calculated with the Chen equation [Equation (7-35)]
for the calculated Reynolds number and the pipe relative roughness.
The kinetic energy pressure drop will in most instances be negligible; it is calculated from the difference in velocity over a finite distance of pipe, Δz.
7.4.3.3. Bubble Flow: The Griffith Correlation
The Griffith correlation uses a different holdup correlation, bases the frictional pressure gradient on the in-situ average liquid velocity, and neglects the kinetic energy pressure gradient. For this correlation,
where ul is the in-situ average liquid velocity, defined as
For field units, Equation (7-111) is
where ml is the mass flow of the liquid only. The liquid holdup is
where us = 0.8 ft/sec. The Reynolds number used to obtain the friction factor is based on the in-situ average liquid velocity,
or
Example 7-8. Pressure Gradient Calculation Using the Modified Hagedorn and Brown Method Suppose that 2000 bbl/d of oil (ρ = 0.8 g/cm3, μ = 2 cp) and 1 MMSCF/d of gas of the same
composition as in Example 7-5 are flowing in 2 7/8-in. tubing. The surface tubing pressure is 800 psia and the temperature is 175°F. The oil–gas surface tension is 30 dynes/cm, and the pipe relative
roughness is 0.0006. Calculate the pressure gradient at the top of the tubing, neglecting any kinetic energy contribution to the pressure gradient,
Solution
From Example 7-5, we have μg = 0.0131 cp and Z = 0.935. Converting volumetric flow rates to superficial velocities with A = (π/4)(2.259/12)2 = 0.0278 ft2,
The gas superficial velocity can be calculated from the volumetric flow rate at standard conditions with Equation (7-45),
The mixture velocity is
and the input fraction of gas is
First, we check whether the flow regime is bubble flow. Using Equation (7-91),
but LB must be ≥ 0.13, so LB = 0.13. Since λg (0.65) is greater than LB, the flow regime is not bubble flow and we proceed with the Hagedorn-Brown correlation.
We next compute the dimensionless numbers, Nvl, Nvg, ND, and NL. Using Equations (7-99) through (7-102), we find Nvl = 10.28, Nvg = 19.20, ND = 29.35, NL = 9.26 × 10–3. Now, we determine liquid holdup, yl, from Figures 7-12 through 7-14 or Equations 7-103 through 7-107. From Figure 7-12 or Equation 7-103, CNL = 0.0022. Then
and, from Figure 7-13 or Equation 7-105, yl/ψ = 0.46. Finally, we calculate
and from Figure 7-14 or Equation 7-107, ψ = 1.0. Note than ψ will generally be 1.0 for low-viscosity liquids. The liquid holdup is thus 0.46. This is compared with the input liquid fraction, λl, which in this case is 0.35. If yl is less than λl, yl is set to λl.
Next, we calculate the two-phase Reynolds number using Equation (7-110). The mass flow rate is The gas density is calculated from Equation (7-44),
so
and
From Figure 7-7 or Equation (7-35), f = 0.0046. The in-situ average density is Finally, from Equation (7-93),
7.4.3.4. The Beggs and Brill Method
The Beggs and Bill (1973) correlation differs significantly from that of Hagedorn and Brown in that the Beggs and Brill correlation is applicable to any pipe inclination and flow direction. This method is based on the flow regime that would occur if the pipe were horizontal; corrections are then made to account for the change in holdup behavior with inclination. It should be kept in mind that the flow regime determined as part of this correlation is the flow regime that would occur if the pipe were perfectly horizontal and is probably not the actual flow regime that occurs at any other angle. The Beggs and Brill method is the recommended technique for any wellbore that is not near vertical.
The Beggs and Brill method uses the general mechanical energy balance [Equation (7-88)] and the in-situ average density [Equation (7-90)] to calculate the pressure gradient and is based on the following parameters:
The horizontal flow regimes used as correlating parameters in the Beggs-Brill method are segregated, transition, intermittent, and distributed (see Chapter 8 for a discussion of horizontal flow regimes). The flow regime transitions are given by the following.
Segregated flow exists if Transition flow occurs when Intermittent flow exists when Distributed flow occurs if
The same equations are used to calculate the liquid holdup, and hence the average density, for the segregated, intermittent, and distributed flow regimes. These are
with the constraint that ylo ≥ λl and where
where a, b, c, d, e, f, and g depend on the flow regime and are given in Table 7-2. C must be ≥ 0.
Table 7-2. Beggs and Brill Holdup Constants
If the flow regime is transition flow, the liquid holdup is calculated using both the segregated and intermittent equations and interpolated using the following:
where
and
The frictional pressure gradient is calculated from
where and
The no-slip friction factor, fn, is based on the actual pipe relative roughness and the Reynolds number,
for ρm in lbm/ft3, um in ft/s, D in in., and μm in cp, and where The two-phase friction factor, ftp, is then
where
and
Since S is unbounded in the interval 1 < x < 1.2, for this interval,
The kinetic energy contribution to the pressure gradient is accounted for with a parameter Ek as follows:
where
In comparisons with extensive measurements with natural gas and water flowing in inclined schedule 40 2-inch I.D. pipe, Payne et al. [1979] found that the Beggs and Brill correlation underpredicted friction factors and overpredicted liquid holdup. To correct these errors, Payne et al. suggest that the friction factor be calculated incorporating pipe roughness (the original correlation assumed smooth pipe), and found the following holdup corrections improved the correlation. Denoting the liquid holdup calculated by the original correlation as ylo, the corrected liquid holdup is
or
Example 7-9. Pressure Gradient Calculation Using the Beggs and Brill Method Repeat Example 7-8, using the Beggs and Brill method.
Solution
First, we determine the flow regime that would exist if the flow were horizontal. Using Equations (7-130) through (7-135) and the values of um (13.39 ft/s) and λl (0.35) calculated in Example 7-8, we find NFR = 29.6, L1 = 230, L2 = 0.0124, L3 = 0.462, and L4 = 606. Checking the flow regime limits
[Equations (7-136) through (7-139)], we see that Then, using Equations (7-142) and (7-143),
so that, from Equation (7-140),
Applying the Payne correction for upward flow, The in-situ average density is
and the potential energy pressure gradient is
To calculate the frictional pressure gradient, we first compute the input fraction weighted density and viscosity from Equations (7-148) and (7-151):
The Reynolds number from Equation (7-150) is
For the relative roughness of 0.0006, from the Moody diagram or the Chen equation, the no-slip friction factor, fn, is 0.005. Then, using Equations (7-152) through (7-154),
From Equation (7-147), the frictional pressure gradient is
and the overall pressure gradient is
7.4.3.5. The Gray Correlation
The Gray correlation was developed specifically for wet gas wells and is commonly used for gas wells producing free water and/or condensate with the gas. This correlation empirically calculates liquid holdup to compute the potential energy gradient and empirically calculates an effective pipe roughness to determine the frictional pressure gradient.
First, three dimensionless numbers are calculated:
where
The liquid holdup correlation is where
The potential energy pressure gradient is then calculated using the in-situ average density.
To calculate the frictional pressure gradient, the Gray correlation uses an effective pipe roughness to account for liquid along the pipe walls. The effective roughness correlation is
or
where
and k is the absolute roughness of the pipe. The constant in Equation 7-183 is for all variables in consistent units. For oilfield units of dynes/cm for σ, lbm/ft3 for ρ, and ft/sec for um,
The effective roughness is used to calculate the relative roughness by dividing by the pipe diameter. The friction factor is obtained using this relative roughness and a Reynolds number of 107.
Example 7-10. Pressure Gradient Calculation Using the Gray Method
An Appendix C gas well is producing 2 MMscf/day of gas with 50 bbl of water produced per MMscf of gas. The surface tubing pressure is 200 psia and the temperature is 100°F. The gas–water surface tension is 60 dynes/cm, and the pipe relative roughness is 0.0006. Calculate the pressure gradient at the top of the tubing, neglecting any kinetic energy contribution to the pressure gradient, At this location, the water density is 65 lbm/ft3 and the viscosity is 0.6 cp.
Solution
First, we need to determine the Z factor for this gas. ppr = 0.298 (p/ppc = 200/671) and Tpr = 1.49 (T/Tpc
= 560/375). From Figure 4-1, Z = 0.97. The flowing area A = 0.0278 ft2 if 2 7/8-in. tubing is used.
The superficial velocities are calculated as
The gas superficial velocity is calculated from the volumetric flow rate at standard conditions with Equation (7-45),
The mixture velocity is
And the input fraction of liquid is
The gas density is calculated from Equation (7-44),
The input fraction weighted density is calculated from Equation (7-148), From Equations (7-175) and (7-176), we calculate N1 and N2,
We calculate Rv and N3 with Equations (7-178) and (7-177),
With Equations (7-180) and (7-179), f1 and y1 are
The in-situ average density is
The potential energy pressure gradient is
The absolute roughness of the pipe is k = εD = (0.0006)(2.259)/12 ft = 0.000113 ft.
Using Equation (7-184),
For Rv = 0.0037 < 0.007,
The effective relative roughness is ke/D = 0.0134. With Equation (7-37), the Fanning friction factor is 0.0105.
From Equation (7-147), the frictional pressure gradient is
and the overall pressure gradient is