4.3.1 Average Temperature and Compressibility Factor Method

If single average values of temperature and compressibility factor over the entire tubing length can be assumed, Eq. (4.8) becomes

$\frac{\overline{z}R\overline{T}}{29{\gamma }_g}\frac{dP}{P}+\left\{\frac{g}{g_c}\cos \theta +\frac{8f_M{Q}_{cs}^2{P}_{sc}^2{\overline{z}}^2{\overline{T}}^2}{{\pi }^2{g}_{ci}^5{T}_{sc}^2{P}^2}\right\}dL=0.$ (4.9)

(4.9)

By separation of variables, Eq. (4.9) can be integrated over the full length of tubing to yield

$P_{wf}^2=Exp(s)P_{hf}^2+\frac{8f_M[Exp(s)-1]Q_{sc}^2{P}_{sc}^2{\overline{z}}^2{\overline{T}}^2}{{\pi }^2{g}_c{D}_i^5{T}_{sc}^2\cos \theta },$ (4.10)

(4.10)

where

$s=\frac{58{\gamma }_{g}gL\cos \theta }{g_{c}R\overline{z}\overline{T}}.$ (4.11)

(4.11)

Eqs. (4.10) and (4.11) take the following forms when U.S. field units (q_sc in Mscf/d), are used (Katz et al., 1959):

$p_{wf}^2=Exp(s)p_{hf}^2+\frac{6.67\times {10}^{-4}[Exp(s)-1]f_M{q}_{sc}^2{\overline{z}}^2{\overline{T}}^2}{d_i^5\cos \theta }$ (4.12)

(4.12)

and

$s=\frac{0.0375{\gamma }_{g}L\cos \theta }{\overline{z}\overline{T}}$ (4.13)

(4.13)

The Darcy–Wiesbach (Moody) friction factor f_M can be found in the conventional manner for a given tubing diameter, wall roughness, and Reynolds number. However, if one assumes fully turbulent flow, which is the case for most gas wells, then a simple empirical relation may be used for typical tubing strings (Katz and Lee, 1990):

$\begin{array}{ccc}{f}_M=\frac{0.01750}{d_i^{0.224}}& \mathrm{for}& d_i\le 4.277\mathrm{in}.\end{array}$ (4.14)

(4.14)

$\begin{array}{ccc}{f}_M=\frac{0.01603}{d_i^{0.164}}& \mathrm{for}& d_i>4.277\mathrm{in}.\end{array}$ (4.15)

(4.15)

Guo and Ghalambor (2002) used the following Nikuradse friction factor correlation for fully turbulent flow in rough pipes:

$f_M={\left[\frac{1}{1.74-2\log \left(\frac{2\epsilon }{d_i}\right)}\right]}^2$ (4.16)

(4.16)

Because the average compressibility factor is a function of pressure itself, a numerical technique such as Newton–Raphson iteration is required to solve Eq. (4.12) for bottom-hole pressure. This computation can be performed automatically with the spreadsheet program Average TZ.xls. Users need to input parameter values in the Input data section and run Macro Solution to get results.

Example Problem 4.2 Suppose that a vertical well produces 2 MMscf/d of 0.71 gas-specific gravity gas through a $2\frac{7}{8}$ in. tubing set to the top of a gas reservoir at a depth of 10,000 ft. At tubing head, the pressure is 800 psia and the temperature is 150°F; the bottom-hole temperature is 200°F. The relative roughness of tubing is about 0.0006. Calculate the pressure profile along the tubing length and plot the results.

Solution Example Problem 4.2 is solved with the spreadsheet program AverageTZ.xls. Table 4.1 shows the appearance of the spreadsheet for the Input data and Result sections. The calculated pressure profile is plotted in Fig. 4.3.

4.3.2 Cullender and Smith Method

Eq. (4.8) can be solved for bottom-hole pressure using a fast numerical algorithm originally developed by Cullender and Smith (Katz et al., 1959). Eq. (4.8) can be rearranged as

$\frac{\frac{P}{zT}dp}{\frac{g}{g_c}\cos \theta {\left(\frac{P}{zT}\right)}^2+\frac{8f_M{Q}_{sc}^2{P}_{sc}^2}{{\pi }^2{g}_c{D}_i^5{T}_{sc}^2}}=-\frac{29{\gamma }_g}{R}dL$ (4.17)

(4.17)

that takes an integration form of

$\underset{P_{hf}}{\overset{P_{wf}}{\int }}\left[\frac{\frac{P}{zT}}{\frac{g}{g_c}\cos \theta {\left(\frac{P}{zT}\right)}^2+\frac{8f_M{Q}_{sc}^2{P}_{sc}^2}{{\pi }^2{g}_c{D}_i^5{T}_{sc}^2}}\right]dp=\frac{29{\gamma }_{g}L}{R}.$ (4.18)

(4.18)

In U.S. field units (q_msc in MMscf/d), Eq. (4.18) has the following form:

$\underset{p_{hf}}{\overset{p_{wf}}{\int }}\left[\frac{\frac{p}{zT}}{0.001\cos \theta {\left(\frac{p}{zT}\right)}^2+0.6666\frac{f_M{q}_{msc}^2}{d_i^5}}\right]dp=18.75{\gamma }_{g}L$ (4.19)

(4.19)

If the integrant is denoted with symbol I, that is,

$I=\frac{\frac{p}{zT}}{0.001\cos \theta {\left(\frac{p}{zT}\right)}^2+0.6666\frac{f_M{q}_{sc}^2}{d_i^5}},$ (4.20)

(4.20)

Eq. (4.19) becomes

${\int }_{p_{hf}}^{p_{wf}}Idp=1875{\gamma }_{g}L.$ (4.21)

(4.21)

In the form of numerical integration, Eq. (4.21) can be expressed as

$\begin{array}{l}\frac{(p_{mf}-p_{hf})(I_{mf}-I_{hf})}{2}+\frac{(p_{wf}-p_{mf})(I_{wf}-I_{mf})}{2}\hfill \\ =18.75{\gamma }_{g}L,\hfill \end{array}$ (4.22)

(4.22)

where p_mf is the pressure at the mid-depth. The I_hf, I_mf, and I_wf are integrant is evaluated at p_hf, p_mf, and p_wf, respectively. Assuming the first and second terms in the right-hand side of Eq. (4.22) each represents half of the integration, that is,

$\frac{(p_{mf}-p_{hf})(I_{mf}-I_{hf})}{2}=\frac{1875{\gamma }_{g}L}{2}$ (4.23)

(4.23)

$\frac{(p_{wf}-p_{mf})(I_{wf}-I_{mf})}{2}=\frac{18.75{\gamma }_{g}L}{2},$ (4.24)

(4.24)

the following expressions are obtained:

$p_{mf}=p_{hf}+\frac{18.75{\gamma }_{g}L}{I_{mf}+I_{hf}}$ (4.25)

(4.25)

$p_{wf}=p_{mf}+\frac{18.75{\gamma }_{g}L}{I_{wf}+I_{mf}}$ (4.26)

(4.26)

Because I_mf is a function of pressure p_mf itself, a numerical technique such as Newton–Raphson iteration is required to solve Eq. (4.25) for p_mf. Once p_mf is computed, p_wf can be solved numerically from Eq. (4.26). These computations can be performed automatically with the spreadsheet program Cullender-Smith.xls. Users need to input parameter values in the Input Data section and run Macro Solution to get results.

Example Problem 4.3 Solve the problem in Example Problem 4.2 with the Cullender and Smith Method.

Solution Example Problem 4.3 is solved with the spreadsheet program Cullender-Smith.xls. Table 4.2 shows the appearance of the spreadsheet for the Input data and Result sections. The pressures at depths of 5000 ft and 10,000 ft are 937 psia and 1082 psia, respectively. These results are exactly the same as that given by the Average Temperature and Compressibility Factor Method.

4.3.3 Flow of Impure Gas

The average temperature average z-factor method and the Cullender and Smith method were derived on the basis of pure gas flow. In reality, almost all gas wells produce certain amount of liquids (oil and water) and sometimes solid particles (sand and coal). The volume fractions of these liquids and solids are low but their effect on pressure can be significant due to their densities being much higher than the density of gas.

A gas-oil-water-sand four-phase flow model was proposed by Guo and Ghalambor (2005) to describe the flow impure gas. The model takes a closed (integrated) form, which makes it easy to use. The Guo–Ghalambor model can be expressed as follows:

$\begin{array}{l}b\left(P-P_{top}\right)+\frac{1-2bM}{2}\ln \left|\frac{{\left(P+M\right)}^2+N}{{\left(P_{top}+M\right)}^2+N}\right|\hfill \\ -\frac{M+\frac{b}{c}N-b{M}^2}{\sqrt{N}}\hfill \\ \times \left[{\tan }^{-1}\left(\frac{P+M}{\sqrt{N}}\right)-{\tan }^{-1}\left(\frac{P_{top}+M}{\sqrt{N}}\right)\right]\hfill \\ =aL\left(\cos \theta +d^{2}e\right)\hfill \end{array}$ (4.27)

(4.27)

where the group parameters are defined as

$a=\frac{0.0765{\gamma }_g{q}_g+350{\gamma }_o{q}_o+350{\gamma }_w{q}_w+62.4{\gamma }_s{q}_s}{4.07T_{av}{q}_g},$ (4.28)

(4.28)

$b=\frac{5.615q_o+5.615q_w+q_s}{4.07T_{av}{Q}_g},$ (4.29)

(4.29)

$c=0.00678\frac{T_{av}{Q}_g}{A},$ (4.30)

(4.30)

$d=\frac{0.00166}{A}(5.615q_o+5.615q_w+q_s),$ (4.31)

(4.31)

$e=\frac{f_M}{2g{D}_H},$ (4.32)

(4.32)

$M=\frac{cde}{\cos \theta +d^{2}e},$ (4.33)

(4.33)

$N=\frac{c^{2}e\cos \theta }{{(\cos \theta +d^{2}e)}^2},$ (4.34)

(4.34)

where

The Darcy–Wiesbach friction factor (fM) can be obtained from diagram (Fig. 4.2) or based on f_F obtained from Eq. (4.16). The required relation is f_M=4f_F. The Guo–Ghalambor can also be used for describing mist flow in oil wells (Guo et al., 2008).

Because iterations are required to solve Eq. (4.18) for pressure, a computer spreadsheet program Guo-Ghalambor BHP.xls has been developed.

Example Problem 4.4 For the following data, estimate bottom-hole pressure with the Guo–Ghalambor method:

Solution This example problem is solved with the spreadsheet program Guo-GhalamborBHP.xls. The result is shown in Table 4.2. and Table 4.3.

4.4.1 Flow Regimes

As shown in Fig. 4.4, at least four flow regimes have been identified in gas-liquid two-phase flow. They are bubble, slug, churn, and annular flow. These flow regimes occur as a progression with increasing gas flow rate for a given liquid flow rate. In bubble flow, gas phase is dispersed in the form of small bubbles in a continuous liquid phase. In slug flow, gas bubbles coalesce into larger bubbles that eventually fill the entire pipe cross-section. Between the large bubbles are slugs of liquid that contain smaller bubbles of entrained gas. In churn flow, the larger gas bubbles become unstable and collapse, resulting in a highly turbulent flow pattern with both phases dispersed. In annular flow, gas becomes the continuous phase, with liquid flowing in an annulus, coating the surface of the pipe and with droplets entrained in the gas phase.

4.4.2 Liquid Holdup

In multiphase flow, the amount of the pipe occupied by a phase is often different from its proportion of the total volumetric flow rate. This is due to density difference between phases. The density difference causes dense phase to slip down in an upward flow (i.e., the lighter phase moves faster than the denser phase). Because of this, the in-situ volume fraction of the denser phase will be greater than the input volume fraction of the denser phase (i.e., the denser phase is “held up” in the pipe relative to the lighter phase). Thus, liquid “holdup” is defined as

$y_L=\frac{V_L}{V},$ (4.35)

(4.35)

where

Liquid holdup depends on flow regime, fluid properties, and pipe size and configuration. Its value can be quantitatively determined only through experimental measurements.

4.4.3 TPR Models

Numerous TPR models have been developed for analyzing multiphase flow in vertical pipes. Brown (1977) presents a thorough review of these models. TPR models for multiphase flow wells fall into two categories: (1) homogeneous-flow models and (2) separated-flow models. Homogeneous models treat multiphase as a homogeneous mixture and do not consider the effects of liquid holdup (no-slip assumption). Therefore, these models are less accurate and are usually calibrated with local operating conditions in field applications. The major advantage of these models comes from their mechanistic nature. They can handle gas-oil-water three-phase and gas-oil-water-sand four-phase systems. It is easy to code these mechanistic models in computer programs.

Separated-flow models are more realistic than the homogeneous-flow models. They are usually given in the form of empirical correlations. The effects of liquid holdup (slip) and flow regime are considered. The major disadvantage of the separated-flow models is that it is difficult to code them in computer programs because most correlations are presented in graphic form.

4.4.3.1 Homogeneous-flow models

Numerous homogeneous-flow models have been developed for analyzing the TPR of multiphase wells since the pioneering works of Poettmann and Carpenter (1952). Poettmann–Carpenter’s model uses empirical two-phase friction factor for friction pressure loss calculations without considering the effect of liquid viscosity. The effect of liquid viscosity was considered by later researchers including Cicchitti (1960) and Dukler et al. (1964). A comprehensive review of these models was given by Hasan and Kabir (2002). Guo and Ghalambor (2005) presented work addressing gas-oil-water-sand four-phase flow.

Assuming no slip of liquid phase, Poettmann and Carpenter (1952) presented a simplified gas-oil-water three-phase flow model to compute pressure losses in wellbores by estimating mixture density and friction factor. According to Poettmann and Carpenter, the following equation can be used to calculate pressure traverse in a vertical tubing when the acceleration term is neglected:

$\mathrm{\Delta }p=\left(\overline{\rho }+\overline{k}\mathrm{/}\overline{\rho }\right)\frac{\mathrm{\Delta }h}{144}$ (4.36)

(4.36)

where

Δp=pressure increment, psi

$\overline{\rho }$=average mixture density (specific weight), lb/ft³

Δh=depth increment, ft

and

$\overline{k}=\frac{f_{2F}{q}_o^2{M}^2}{7.4137\times {10}^{10}{D}^5}$ (4.37)

(4.37)

where

f_2F=Fanning friction factor for two-phase flow

q_o=oil production rate, stb/day

M=total mass associated with 1 stb of oil

D=tubing inner diameter, ft

The average mixture density $\overline{\rho }$ can be calculated by

$\overline{\rho }=\frac{{\rho }_1+{\rho }_2}{2}$ (4.38)

(4.38)

where

ρ₁=mixture density at top of tubing segment, lb/ft³

ρ₂=mixture density at bottom of segment, lb/ft³

The mixture density at a given point can be calculated based on mass flow rate and volume flow rate:

$\rho =\frac{M}{V_m}$ (4.39)

(4.39)

where

$M=350.17({\gamma }_0+WOR{\gamma }_w)+GO{R}_{{\rho }_{air}}{\gamma }_g$ (4.40)

(4.40)

$V_m=5.615(B_o+WOR{B}_w)+(GOR-R_s)\left(\frac{14.7}{p}\right)\left(\frac{T}{520}\right)\left(\frac{z}{1.0}\right)$ (4.41)

(4.41)

and where

γ_o=oil-specific gravity, 1 for freshwater

WOR=producing water–oil ratio, bbl/stb

γ_w=water-specific gravity, 1 for freshwater

GOR=producing gas–oil ratio, scf/stb

ρ_air=density of air, lb_m/ft³

γ_g=gas-specific gravity, 1 for air

V_m=volume of mixture associated with 1 stb of oil, ft³

B_o=formation volume factor of oil, rb/stb

B_w=formation volume factor of water, rb/bbl

R_s=solution gas–oil ratio, scf/stb

p=in-situ pressure, psia

T=in situ temperature, °R

z=gas compressibility factor at p and T.

If data from direct measurements are not available, solution gas–oil ratio and formation volume factor of oil can be estimated using the following correlations:

$R_s={\gamma }_g{\left[\frac{p}{18}\frac{{10}^{0.0125\mathrm{API}}}{{10}^{0.00091t}}\right]}^{1.2048}$ (4.42)

(4.42)

$B_o=0.9759+0.00012{\left[R_s{\left(\frac{{\gamma }_g}{{\gamma }_o}\right)}^{0.5}+1.25t\right]}^{1.2}$ (4.43)

(4.43)

where t is in-situ temperature in °F. The two-phase friction factor f_2F can be estimated from a chart recommended by Poettmann and Carpenter (1952). For easy coding in computer programs, Guo and Ghalambor (2002) developed the following correlation to represent the chart:

$f_{2F}={10}^{1.444-2.5\log (D\rho v)},$ (4.44)

(4.44)

where (Dρv) is the numerator of Reynolds number representing inertial force and can be formulated as

$(D\rho v)=\frac{1.4737\times {10}^{-5}M{q}_o}{D}.$ (4.45)

(4.45)

Because the Poettmann–Carpenter model takes a finite-difference form, this model is accurate for only short-depth incremental Δh. For deep wells, this model should be used in a piecewise manner to get accurate results (i.e., the tubing string should be “broken” into small segments and the model is applied to each segment).

Because iterations are required to solve Eq. (4.36) for pressure, a computer spreadsheet program Poettmann-CarpenterBHP.xls has been developed. The program is available from the attached CD.

Example Problem 4.5 For the following given data, calculate bottom-hole pressure:

Tubing head pressure:	500 psia
Tubing head temperature:	100°F
Tubing inner diameter:	1.66 in.
Tubing shoe depth (near bottom hole):	5000 ft
Bottom-hole temperature:	150°F
Liquid production rate:	2000 stb/day
Water cut:	25%
Producing GLR:	1000 scf/stb
Oil gravity:	30 °API
Water-specific gravity:	1.05 1 for freshwater
Gas-specific gravity:	0.65 1 for air

Solution This problem can be solved using the computer program Poettmann-CarpenterBHP.xls. The result is shown in Table 4.4.

Table 4.4

Result Given by Poettmann-CarpenterBHP.xls for Example Problem 4.2

Poettmann–CarpenterBHP.xls
Description: This spreadsheet calculates flowing bottom-hole pressure based on tubing head pressure and tubing flow performance using the Poettmann–Carpenter method.
Instruction: (1) Select a unit system; (2) update parameter values in the Input data section; (3) Click “Solution” button; and (4) view result in the Solution section.
Input Data	U.S. Field Units
Tubing ID:	1.66 in
Wellhead pressure:	500 psia
Liquid production rate:	2000 stb/d
Producing gas–liquid ratio (GLR):	1000 scf/stb
Water cut (WC):	25%
Oil gravity:	30 °API
Water-specific gravity:	1.05 freshwater=1
Gas-specific gravity:	0.65 1 for air
N2 content in gas:	0 mole fraction
CO2 content in gas:	0 mole fraction
H2S content in gas:	0 mole fraction
Formation volume factor for water:	1.0 rb/stb
Wellhead temperature:	100°F
Tubing shoe depth:	5000 ft
Bottom-hole temperature:	150°F
Solution
Oil-specific gravity	=0.88 freshwater=1
Mass associated with 1 stb of oil	=495.66 lb
Solution gas ratio at wellhead	=78.42 scf/stb
Oil formation volume factor at wellhead	=1.04 rb/stb
Volume associated with 1 stb oil @ wellhead	=45.12 cf
Fluid density at wellhead	=10.99 lb/cf
Solution gas–oil ratio at bottom hole	=301.79 scf/stb
Oil formation volume factor at bottom hole	=1.16 rb/stb
Volume associated with 1 stb oil @ bottom hole	=17.66 cf
Fluid density at bottom hole	=28.07 lb/cf
The average fluid density	=19.53 lb/cf
Inertial force (Dρv)	=79.21 lb/day-ft
Friction factor	=0.002
Friction term	=293.12 (lb/cf)2
Error in depth	=0.00 ft
Bottom-hole pressure	=1699 psia

4.4.3.2 Separated-flow models

A number of separated-flow models are available for TPR calculations. Among many others are the Lockhart and Martinelli correlation (1949), the Duns and Ros correlation (1963), and the Hagedorn and Brown method (1965). Based on comprehensive comparisons of these models, Ansari et al. (1994) and Hasan and Kabir (2002) recommended the Hagedorn–Brown method with modifications for near-vertical flow.

The modified Hagedorn–Brown (mH-B) method is an empirical correlation developed on the basis of the original work of Hagedorn and Brown (1965). The modifications include using the no-slip liquid holdup when the original correlation predicts a liquid holdup value less than the no-slip holdup and using the Griffith correlation (Griffith and Wallis, 1961) for the bubble flow regime.

The original Hagedorn–Brown correlation takes the following form:

$\frac{dP}{dz}=\frac{g}{g_c}\overline{\rho }+\frac{2f_F\overline{\rho }{u}_m^2}{g_{c}D}+\overline{\rho }\frac{\mathrm{\Delta }(u_m^2)}{2g_c\mathrm{\Delta }z},$ (4.46)

(4.46)

which can be expressed in U.S. field units as

$144\frac{dp}{dz}=\overline{\rho }+\frac{f_F{M}_t^2}{7.413\times {10}^{10}{D}^5\overline{\rho }}+\overline{\rho }\frac{\mathrm{\Delta }(u_m^2)}{2g_c\mathrm{\Delta }z},$ (4.47)

(4.47)

where

M_t=total mass flow rate, lb_m/d

$\overline{\rho }$=in-situ average density, lb_m/ft³

u_m=mixture velocity, ft/s

and

$\overline{\rho }=y_L{\rho }_L+(1-y_L){\rho }_G,$ (4.48)

(4.48)

$u_m=u_{SL}+u_{SG},$ (4.49)

(4.49)

where

ρ_L=liquid density, lb_m/ft³

ρ_G=in-situ gas density, lb_m/ft³

u_SL=superficial velocity of liquid phase, ft/s

u_SG=superficial velocity of gas phase, ft/s

The superficial velocity of a given phase is defined as the volumetric flow rate of the phase divided by the pipe cross-sectional area for flow. The third term in the right-hand side of Eq. (4.47) represents pressure change due to kinetic energy change, which is in most instances negligible for oil wells.

Obviously, determination of the value of liquid holdup y_L is essential for pressure calculations. The mH-B correlation uses liquid holdup from three charts using the following dimensionless numbers:

Liquid velocity number, N_vL:

$N_{vL}=1.938u_{SL}\sqrt[4]{\frac{{\rho }_L}{\sigma }}$ (4.50)

(4.50)

Gas velocity number, N_vG:

$N_{vG}=1.938u_{SG}\sqrt[4]{\frac{{\rho }_L}{\sigma }}$ (4.51)

(4.51)

Pipe diameter number, N_D:

$N_D=120.872D\sqrt{\frac{{\rho }_L}{\sigma }}$ (4.52)

(4.52)

Liquid viscosity number, N_L:

$N_L=0.15726{\mu }_L\sqrt[4]{\frac{1}{{\rho }_L{\sigma }^3}},$ (4.53)

(4.53)

where

D=conduit inner diameter, ft

σ=liquid–gas interfacial tension, dyne/cm

μ_L=liquid viscosity, cp

μ_G=gas viscosity, cp

The first chart is used for determining parameter (CN_L) based on N_L. We have found that this chart can be replaced by the following correlation with acceptable accuracy:

$(C{N}_L)={10}^Y,$ (4.54)

(4.54)

where

$\begin{array}{ll}Y\hfill & =-2.69851+0.15841X_1-0.55100X_1^2\hfill \\ \hfill & +0.54785X_1^3-0.12195X_1^4\hfill \end{array}$ (4.55)

(4.55)

and

$X_1=\log (N_L)+3.$ (4.56)

(4.56)

Once the value of parameter (CN_L) is determined, it is used for calculating the value of the group $\frac{N_{vL}{p}^{0.1}(C{N}_L)}{N_{vG}^{0.575}{p}_a^{0.1}{N}_D}$, where p is the absolute pressure at the location where pressure gradient is to be calculated, and p_a is atmospheric pressure. The value of this group is then used as an entry in the second chart to determine parameter (y_L/ψ). We have found that the second chart can be represented by the following correlation with good accuracy:

$\begin{array}{ll}\frac{y_L}{\psi }\hfill & =-0.10307+0.61777[\log (X_2)+6]\hfill \\ \hfill & -0.63295{[\log (X_2)+6]}^2+0.29598{[\log (X_2)+6]}^3\hfill \\ \hfill & -0.0401{[\log (X_2)+6]}^4,\hfill \end{array}$ (4.57)

(4.57)

where

$X_2=\frac{N_{vL}{p}^{0.1}(C{N}_L)}{N_{vG}^{0.575}{p}_a^{0.1}{N}_D}.$ (4.58)

(4.58)

According to Hagedorn and Brown (1965), the value of parameter ψ can be determined from the third chart using a value of group $\frac{N_{vG}{N}_L^{0.38}}{N_D^{2.14}}$.

We have found that for $\frac{N_{vG}{N}_L^{0.38}}{N_D^{2.14}}>0.01$ the third chart can be replaced by the following correlation with acceptable accuracy:

$\begin{array}{ll}\psi \hfill & =0.91163-4.82176X_3+1,232.25X_3^2\hfill \\ \hfill & -22,253.6X_3^3+116174.3X_3^4,\hfill \end{array}$ (4.59)

(4.59)

where

$X_3=\frac{N_{vG}{N}_L^{0.38}}{N_D^{2.14}}.$ (4.60)

(4.60)

However, ψ=1.0 should be used for $\frac{N_{vG}{N}_L^{0.38}}{N_D^{2.14}}\le 0.01$.

Finally, the liquid holdup can be calculated by

$y_L=\psi \left(\frac{y_L}{\psi }\right).$ (4.61)

(4.61)

The Reynolds number for multiphase flow can be calculated by

$N_{\mathrm{Re}}=\frac{2.2\times {10}^{-2}{m}_t}{D{\mu }_L^{y_L}{\mu }_G^{(1-y_L)}},$ (4.62)

(4.62)

where m_t is mass flow rate. The modified mH-B method uses the Griffith correlation for the bubble-flow regime. The bubble-flow regime has been observed to exist when

${\lambda }_G<L_B,$ (4.63)

(4.63)

where

${\lambda }_G=\frac{u_{sG}}{u_m}$ (4.64)

(4.64)

and

$L_B=1.071-0.2218\left(\frac{u_m^2}{D}\right),$ (4.65)

(4.65)

which is valid for L_B≥0.13. When the L_B value given by Eq. (4.65) is less than 0.13, L_B=0. 13 should be used.

Neglecting the kinetic energy pressure drop term, the Griffith correlation in U.S. field units can be expressed as

$144\frac{dp}{dz}=\overline{\rho }+\frac{f_F{m}_L^2}{7.413\times {10}^{10}{D}^5{\rho }_L{y}_L^2},$ (4.66)

(4.66)

where m_L is mass flow rate of liquid only. The liquid holdup in Griffith correlation is given by the following expression:

$y_L=1-\frac{1}{2}\left[1+\frac{u_m}{u_s}-\sqrt{{\left(1+\frac{u_m}{u_s}\right)}^2-4\frac{u_{sG}}{u_s}}\right],$ (4.67)

(4.67)

where μ_s=0.8 ft/s. The Reynolds number used to obtain the friction factor is based on the in-situ average liquid velocity, that is,

$N_{\mathrm{Re}}=\frac{2.2\times {10}^{-2}{m}_L}{D_{{\mu }_L}}.$ (4.68)

(4.68)

To speed up calculations, the Hagedorn–Brown correlation has been coded in the spreadsheet program Hagedorn Brown Correlation.xls.

Example Problem 4.6 For the data given below, calculate and plot pressure traverse in the tubing string:

Tubing shoe depth:	9700 ft
Tubing inner diameter:	1.995 in.
Oil gravity:	40 °API
Oil viscosity:	5 cp
Production GLR:	75 scf/bbl
Gas-specific gravity:	0.7 air=1
Flowing tubing head pressure:	100 psia
Flowing tubing head temperature:	80°F
Flowing temperature at tubing shoe:	180°F
Liquid production rate:	758 stb/day
Water cut:	10%
Interfacial tension:	30 dynes/cm
Specific gravity of water:	1.05 H2O=1

Solution This example problem is solved with the spreadsheet program HagedornBrownCorrelation.xls. The result is shown in Table 4.5 and Fig. 4.5.

Table 4.5

Result Given by HagedornBrownCorrelation.xls for Example Problem 4.6

HagedornBrownCorrelation.xls
Description: This spreadsheet calculates flowing pressures in tubing string based on tubing head pressure using the Hagedorn–Brown correlation.
Instruction: (1) Select a unit system; (2) update parameter values in the Input data section; (3) click “Solution” button; and (4) view result in the Solution section and charts.
Input Data	U.S. Field Units	SI Units
Depth (D):	9700 ft
Tubing inner diameter (dti):	1.995 in.
Oil gravity (API):	40 °API
Oil viscosity (μo):	5 cp
Production GLR (GLR):	75 scf/bbl
Gas-specific gravity (γg):	0.7 air=1
Flowing tubing head pressure (phf):	100 psia
Flowing tubing head temperature (thf):	80°F
Flowing temperature at tubing shoe (twf):	180°F
Liquid production rate (qL):	758 stb/day
Water cut (WC):	10%
Interfacial tension (σ):	30 dynes/cm
Specific gravity of water (γw):	1.05 H2O=1
Solution
Depth		Pressure
(ft)	(m)	(psia)	(MPa)
0	0	100	0.68
334	102	183	1.24
669	204	269	1.83
1003	306	358	2.43
1338	408	449	3.06
1672	510	543	3.69
2007	612	638	4.34
2341	714	736	5.01
2676	816	835	5.68
3010	918	936	6.37
3345	1020	1038	7.06
3679	1122	1141	7.76
4014	1224	1246	8.48
4348	1326	1352	9.20
4683	1428	1459	9.93
5017	1530	1567	10.66
5352	1632	1676	11.40
5686	1734	1786	12.15
6021	1836	1897	12.90
6355	1938	2008	13.66
6690	2040	2121	14.43
7024	2142	2234	15.19
7359	2243	2347	15.97
7693	2345	2461	16.74
8028	2447	2576	17.52
8362	2549	2691	18.31
8697	2651	2807	19.10
9031	2753	2923	19.89
9366	2855	3040	20.68
9700	2957	3157	21.48