BSEE has outlined the minimum RTM and RTM plan requirements in the regulations. Since the 2016 Rule, the RTM planning requirements have been designed to be “flexible, performance-based measures that better reflect BSEE’s intention that operators use RTM as a tool to improve their own ability to prevent well control incidents while providing BSEE with sufficient access to RTM information to evaluate system improvements.” 2016 final rule preamble, 81 FR 25897; see also 2019 final rule preamble, 84 FR 21942 (“The 2016 WCR’s RTM requirements were themselves largely performance-based, relying primarily on the operator’s development of an RTM plan tailored to its operations but built off of core principles. The revisions implemented here do not reflect a sea change in philosophy…”). With respect to enforcement, the final rule preamble states: “This regulation requires that operators develop and implement RTM plans, and specifically requires that those plans be made available to BSEE upon request. If BSEE has any concerns with an operator’s RTM operations, then BSEE may undertake inspections and enforcement actions to ensure compliance with the regulations. BSEE has additional options such as routine onsite inspections or verifications through the permitting process to ensure that RTM plans are implemented in compliance with the regulations.” 84 FR 21943. The regulations since 2016 have required that operators both “develop and implement” RTM plans. 30 CFR 250.724(c) (emphasis added). These principles are unchanged in this rulemaking.
Velocity of sound in materials in mathematical form is as (Streeter 1962; White 2011)
$$c=\sqrt {\frac{{dp}}{{d\rho }}} ,$$
(15)
where c, P, and ρ are sound velocity, pressure, and density, respectively.
Bulk modulus of elasticity is also defined as (Streeter 1962; White 2011)
$$B= - \frac{{dp}}{{{\raise0.7ex\hbox{${dv}$} \!\mathord{\left/ {\vphantom {{dv} v}}\right.\kern-0pt}\!\lower0.7ex\hbox{$v$}}}}.$$
(16)
B and v are bulk modulus of elasticity and volume of fluid subjected to dp, respectively. P is pressure, as above.
$$\frac{{dv}}{v}= - \frac{{d\rho }}{\rho },$$
(17)
$$\therefore\,B\,=\rho \frac{{dp}}{{d\rho }}~.$$
(18)
Combine equations (15) and (18); from Equation (18), one can write \(\frac{{dp}}{{d\rho }}=\frac{B}{\rho }\) and then from Eq. (15):
$$c=\sqrt {\frac{B}{\rho }} .$$
(19)
The process of ‘flow through choke’ may be considered to be isentropic; so, for gases can be written as
$$p{\rho ^{ - k}}={\text{constan}},$$
(20)
where k is specific heat ratio (k):
$$k=\frac{{{C_p}}}{{{C_v}}}= - \frac{{{\raise0.7ex\hbox{${dp}$} \!\mathord{\left/ {\vphantom {{dp} p}}\right.\kern-0pt}\!\lower0.7ex\hbox{$p$}}}}{{{\raise0.7ex\hbox{${dV}$} \!\mathord{\left/ {\vphantom {{dV} V}}\right.\kern-0pt}\!\lower0.7ex\hbox{$V$}}}}.$$
(21)
Take derivative of Equation (20) with respect to ρ:
$$\frac{{dp}}{{d\rho }}=\frac{{kp}}{\rho }.$$
(22)
Insert Equation (22) and in Equation (15):
$$c=\sqrt {\frac{{kp}}{\rho }} .$$
(23)
By definition, density of gases can be calculated from the following equation (Guo and Ghalambor 2005; Tiab 2000):
$$\rho =\frac{{PM}}{{ZRT}},$$
(24)
where P, M, Z, R, and T are gas pressure, (psia); gas molecular weight, (lbm/lbmole); gas compressibility factor, (dimensionless); universal gas constant, (\(1545.4\frac{{ft.lbf}}{{{\text{lbmole}}.^\circ R}}\)); and gas temperature, (˚R), respectively.
Combine Equations (23) and (24):
$$c=\sqrt {\frac{{ZkRT}}{M}} .$$
(25)
For unit consistency use the conversion factor (\(32.174\frac{{lbm.ft}}{{lbf.{S^2}}}\)):
$$c=\sqrt {\frac{{32.174 \times ZkRT}}{M}} .$$
(26)
Note that by above units, the unit of ‘c’ would be ‘ft/s’.
Let us define gas gravity (G) as (Guo and Ghalambor 2005; Tiab 2000):
$$G=\frac{{{\text{density~of~gas}}}}{{{\text{density~of~air}}}}=\frac{{{\text{molecular~weight~of~gas}}~}}{{{\text{molecular~weight~of~air}}}}=\frac{{{M_{{\text{gas}}}}}}{{{M_{{\text{air}}}}}}.$$
(27)
But molecular weight of air is around 28.97 lbm/lbmole; so,
$${M_{{\text{gas}}}}=28.97G.$$
(28)
Insert data; Equation (28) and universal gas constant (\(1545.4\frac{{ft.lbf}}{{{\text{lbmole}}.^\circ R}}\)) in Equation (26):
$$c=\sqrt {\frac{{32.174 \times 1545.4 \times ZkT}}{{28.97G}}} ,$$
(29)
$$c=41.4284\sqrt {\frac{{ZkT}}{G}} .$$
(30)
Recall that c, z, k, T, and G are sound velocity in gases, (ft/s); gas compressibility, dimensionless; gas specific heat ratio, dimensionless; gas temperature, (˚R); and gas gravity, dimensionless, respectively.
By definition at choking condition, the gas velocity would be equal to sound velocity; so, it means that the actual gas flow rate through the bean at choking condition can ideally be calculated by following equation:
$${{\text{Q}}_{{\text{acfd}}}}=24 \times 3600 \times 41.4284 \times \frac{\pi }{4} \times \frac{{{d^2}}}{{144}} \times \sqrt {\frac{{ZkT}}{G}} ,$$
(31)
$${{\text{Q}}_{{\text{acfd}}}}=19523 \times {d^2}\sqrt {\frac{{ZkT}}{G}} ,$$
(32)
where ‘d’ is choke diameter in ‘inches’.
From equation of state of real gases, it can be written as
$$\frac{{{P_s}{Q_s}}}{{{P_a}{Q_a}}}=\frac{{{Z_s}R{T_s}}}{{{Z_a}R{T_a}}},$$
(33)
where indices ‘s’ and ‘a’ refer to ‘standard’ and ‘actual’ conditions, respectively. By considering the standard conditions as follows, Equation (33) becomes Equation (34):
Standard conditions: P = 14.7 psia, Z = 1.00, T = 520 ˚R.
$${Q_s}=35.3741\frac{{{P_a}}}{{{Z_a}{T_a}}}{Q_a}.$$
(34)
Insert Equation (32) in Equation (34):
$${{\text{Q}}_{{\text{scfd}}}}=690,597 \times \sqrt {\frac{1}{z}} \times \sqrt {\frac{1}{T}} \times \sqrt {\frac{k}{G}} \times P{d^2}.$$
(35)
Equation (35) is the basic equation for estimating single-phase dry gas flow rate through a choke, provided the choking conditions prevail (ratio of pressures of upstream to downstream of choke to be less than about 0.55).
In investigating an equation for calculating fluid flow through any constraint, a phenomenon which is so-called ‘vena contracta’ should be considered. It is defined as the reduction in the area/diameter of a fluid jet after it emerges from a circular aperture in a pressurized reservoir (White 2011). The combination of this amount plus effect of other factors is called discharge coefficient, Cd (Baker 2000). Discharge coefficient is correlated with the ratio of the orifice (choke) diameter to the pipe diameter (β ratio). Therefore, Equation (35) becomes:
$${{\text{Q}}_{{\text{scfd}}}}=690,598 \times {C_d} \times \sqrt {\frac{1}{z}} \times \sqrt {\frac{1}{T}} \times \sqrt {\frac{k}{G}} \times P{d^2}.$$
(36)
To simplify Equation (36) as much as possible and reduce it to a more familiar gas flow through the choke (\({{\text{Q}}_{{\text{scfd}}}}=CP{d^2}\)), let us assume logic values for variables of Cd, Z, T, k, and G.
Heat capacity ratio (k): Let us consider the average specific heat ratio to be equal to 1.28 (Guo et.al. 2007).
Gas gravity (G): The chemical composition of natural gas varies widely around the world; however, the natural gas gravity could be considered in the range of 0.6–0.7 (Faramawy et.al. 2016) for engineering purposes. Here, it is supposed that the average gas gravity is about 0.65.
Temperature (T): In Equation (36), temperature (T) is upstream the choke flowing temperature. The magnitude of this temperature depends on several factors, such as depth of gas reservoir (or better to say gas reservoir’s temperature), sand face pressure drawdown, gas production rate, and size of production well string. Here, based on experience and assuming well’s producing interval depth would be in the range of 10,000–12,000 ft from surface, the flowing wellhead temperature is considered to be in the range of 150–200 °F with average of 175 °F.
Gas compressibility factor: By assuming gas gravity and temperature to be equal to 0.65 and 175 °F, the gas compressibility factor is estimated to be around 0.88. It is assumed that the flowing upstream choke pressure would be in the range of 1800–2800 psi.
Discharge coefficient: By employing the concept of discharge coefficient in flow measurement by orifice and assuming the choke diameter size to inside pipe diameter ratio to be around 0.1–0.25, the discharge coefficient is estimated to be around 0.6 (refer to Baker 2000).
Final choke gas flow equation: Based on the above assumptions, Equation (36) will become:
$${{\text{Q}}_{{\text{scfd}}}}=690,597 \times {C_d} \times \sqrt {\frac{1}{{0.88}}} \times \sqrt {\frac{1}{{(175+460)}}} \times \sqrt {\frac{{1.28}}{{0.65}}} \times P{d^2},$$
(37)
$${{\text{Q}}_{{\text{scfd}}}}=40997 \times {C_d} \times P{d^2},$$
(38)
$${{\text{Q}}_{{\text{scfd}}}}=24598 \times P{d^2}.$$
(39)
Variation in assumed values of each parameter of Eq. (36) in the range of 90–110 percent has little effect on the constant of Equation (39). By variation of each parameter in this range, the numerical value of the constant in Equation (39) will vary in the range of 23,368–25,827.
Equation (39) is for the estimation of single-phase dry gas flow rate through a wellhead’s choke provided the gas velocity through the choke device becomes sonic. The condition for sonic velocity through the choke is defined by the following equation (Streeter 1962; White 2011):
$$\frac{{{P_{{\text{downstream}}}}}}{{{P_{{\text{upstream}}}}}} \leq {\left( {\frac{2}{{k+1}}} \right)^{\left( {\frac{k}{{k - 1}}} \right)}} \leq {\left( {\frac{2}{{1.28+1}}} \right)^{\left( {\frac{{1.28}}{{1.28 - 1}}} \right)}} \leq 0.5494.$$
(40)
Note that in practical engineering purposes, the constant in Equation (39) is considered to be between 19,000 and 26,000 depends on the gas stream conditions and specifications.
The following formula for gas mass flow rate under subsonic conditions is adopted from Streeter (1962):
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$$\dot {m}=\rho vA=A\sqrt {2{P_{{\text{upstream}}}}{\rho _{{\text{upstream}}}}\frac{k}{{k - 1}}{{\left( {\frac{{{P_{{\text{downsream}}}}}}{{{P_{{\text{upstream}}}}}}} \right)}^{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 k}}\right.\kern-0pt}\!\lower0.7ex\hbox{$k$}}}}\left[ {1 - {{\left( {\frac{{{P_{{\text{downsream}}}}}}{{{P_{{\text{upstream}}}}}}} \right)}^{{\raise0.7ex\hbox{${\left( {k - 1} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {k - 1} \right)} k}}\right.\kern-0pt}\!\lower0.7ex\hbox{$k$}}}}} \right]} .$$
(41)
Insert Eq. (24) into Eq. (41):
$$\dot {m}=A\sqrt {2P_{{{\text{upstream}}}}^{2}\frac{M}{{ZR{T_{{\text{upstream}}}}}}\frac{k}{{k - 1}}{{\left( {\frac{{{P_{{\text{downsream}}}}}}{{{P_{{\text{upstream}}}}}}} \right)}^{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 k}}\right.\kern-0pt}\!\lower0.7ex\hbox{$k$}}}}\left[ {1 - {{\left( {\frac{{{P_{{\text{downsream}}}}}}{{{P_{{\text{upstream}}}}}}} \right)}^{{\raise0.7ex\hbox{${\left( {k - 1} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {k - 1} \right)} k}}\right.\kern-0pt}\!\lower0.7ex\hbox{$k$}}}}} \right]} .$$
(42)
In English units and for consistency, apply the conversion factor (\(32.174\frac{{lbm.ft}}{{lbf.{S^2}}}\)):
$$\dot {m}=\frac{\pi }{4}{d^2}{P_{{\text{upstream}}}}\sqrt {2 \times 32.174 \times \frac{{0.65 \times 28.97}}{{0.88 \times 1545.4 \times (460+175)}} \times \frac{{1.28}}{{1.28 - 1}} \times {{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 {1.28}}}\right.\kern-0pt}\!\lower0.7ex\hbox{${1.28}$}}}}\left[ {1 - {{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{{\raise0.7ex\hbox{${\left( {1.28 - 1} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {1.28 - 1} \right)} {1.28}}}\right.\kern-0pt}\!\lower0.7ex\hbox{${1.28}$}}}}} \right]} .$$
(43)
In Equation (43), the assumed average values for some variables such as specific heat ratio, upstream choke temperature, gas molecular weight are employed. Simplify Equation (43):
$$\dot {m}=0.0629 \times {d^2}{P_1}\sqrt {{{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{1.5625}}\left[ {1 - {{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{0.21875}}} \right]} ,$$
(44)
where ṁ, P1, P2, and ‘d’ are gas mass flow rate, lbmass/sec; flowing upstream choke pressure, psi; flowing downstream choke pressure, psi; and choke size diameter, inches.
By dividing the gas mass flow rate by its density at standard conditions (gas gravity = 0.65) and introducing the discharge coefficient in the above equation, the final choke gas flow rate under subsonic conditions is obtained:
$${q_{{\text{scfd}}}}=65554 \times {d^2}{P_1}\sqrt {{{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{1.5625}}\left[ {1 - {{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{0.21875}}} \right]} .$$
(45)
In above equation with assuming P2/P1 values equal to 0.6, 0.7, and 0.8, the following equation (45) becomes:
$${q_{{\text{scfd}}}}=14301 \times {d^2}{P_1}~~~~~~~~~~~~~{\text{for}}~\left( {\frac{{{P_2}}}{{{P_1}}}} \right)=0.6,$$
(46)
$${q_{{\text{scfd}}}}=13592 \times {d^2}{P_1}~~~~~~~~~~~~~{\text{for}}~\left( {\frac{{{P_2}}}{{{P_1}}}} \right)=0.7,$$
(47)
$${q_{{\text{scfd}}}}=12019 \times {d^2}{P_1}~~~~~~~~~~~~~{\text{for}}~\left( {\frac{{{P_2}}}{{{P_1}}}} \right)=0.8.$$
(48)
It is not common to use choke for limiting the liquid flow; however, it is interesting to calculate the magnitude of order of pressure drop across a device (e.g., choke) to reach the choking conditions; i.e., reaching sonic velocity. Let us start with Eq. (19):
$$c=\sqrt {\frac{B}{\rho }} .$$
(49)
Bernoulli’s equation can be used for liquids:
$$\frac{{{P_{{\text{upstream}}}}}}{\gamma }+\frac{{V_{{{\text{upstream}}}}^{2}}}{{2g}}=\frac{{{P_{{\text{downstream}}}}}}{\gamma }+\frac{{V_{{{\text{downstream}}}}^{2}}}{{2g}}.$$
(50)
Consider that downstream conditions refer to choke throat. Also, for liquids, one can write \({A_1}{V_1}={A_2}{V_2}\); so, it can be written as
$${V_{{\text{upstream}}}}=\frac{{{A_{{\text{choke}}}}}}{{{A_{{\text{pipe}}}}}}{V_{{\text{downstream}}}}.$$
(51)
Combine Eqs. (50) and (51) and simplify:
$$\frac{{{P_{{\text{upstream}}}} - {P_{{\text{downstream}}}}}}{\gamma }=\frac{{V_{{{\text{downstream}}}}^{2}}}{{2g}}\left[ {1 - {{\left( {\frac{{{A_{{\text{choke}}}}}}{{{A_{{\text{pipe}}}}}}} \right)}^2}} \right],$$
(52)
$$V_{{{\text{downstream}}}}^{2}=\frac{{2\Delta P}}{\rho } \times \frac{1}{{\left[ {1 - {{\left( {\frac{{{A_{{\text{choke}}}}}}{{{A_{{\text{pipe}}}}}}} \right)}^2}} \right]}}.$$
(53)
At choking conditions, \({V_{{\text{downstream}}}}={\text{Sonic~Velosity}}=c\); so, Equation (53) becomes:
$$\Delta P=\frac{{\rho \left[ {1 - {{\left( {\frac{{{A_{{\text{choke}}}}}}{{{A_{{\text{pipe}}}}}}} \right)}^2}} \right]}}{2}{c^2}.$$
(54)
Equation (54) is the general form for calculating the required differential pressure across a choke to reach choking condition. Combine Eqs. (49) and (54):
$$\Delta P=\left( {\frac{B}{2}} \right) \times \left[ {1 - {{\left( {\frac{{{A_{{\text{choke}}}}}}{{{A_{{\text{pipe}}}}}}} \right)}^2}} \right].$$
(55)
Ignore the term \(\left[ {1 - {{\left( {\frac{{{A_{{\text{choke}}}}}}{{{A_{{\text{pipe}}}}}}} \right)}^2}} \right],\)
$$\Delta P=\frac{B}{2}.$$
(56)
If ‘B’ (bulk modulus of elasticity) is in psi then the unit of ΔP will also be in psi. Batzle and Wang (1992) estimated the bulk modulus of elasticity of some crude in the range 217,500–375,000 psi (1500–2500 MPa). It means that to have choking conditions across a choke in single-phase crude oil service, the differential pressure across the choke should be more than hundreds thousands of psi.
For two-phase flow through the choke, let us assume that part of area of choke is occupied by gas stream and in the remaining of this area liquid flows. In mathematical form, one can write:
$${A_{\text{t}}}={A_{\text{g}}}+{A_{\text{l}}},$$
(57)
where ‘At’, ‘Ag’, and ‘Al’ are total choke throat area, assumed area available for gas flow, and assumed area available for liquid flow, respectively.
$$\pi \frac{{d_{{\text{t}}}^{2}}}{4}=\pi \frac{{d_{{\text{g}}}^{2}}}{4}+\pi \frac{{d_{{\text{l}}}^{2}}}{4},$$
(58)
$$d_{{\text{t}}}^{2}=d_{{\text{g}}}^{2}+d_{{\text{l}}}^{2},$$
(59)
where ‘dt’, ‘dg’, and ‘dl’ are in internal choke diameter, imaginary choke diameter available for gas flow, and imaginary choke diameter available for liquid flow, respectively.
Insert appropriate relation in right-hand side’s terms of Eq. (59):
Use Equation (45) for gas flow:
$$d_{{\text{g}}}^{2}=\frac{{{q_{{\text{scfd}}}}}}{{65554 \times {P_1}\sqrt {{{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{1.5625}}\left[ {1 - {{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{0.21875}}} \right]} }}.$$
(60)
For liquid, start from Equation (53):
$$V_{{{\text{downstream}}}}^{2}=\frac{{2\Delta P}}{\rho } \times \frac{1}{{\left[ {1 - {{\left( {\frac{{{A_{{\text{choke}}}}}}{{{A_{{\text{pipe}}}}}}} \right)}^2}} \right]}}.$$
(61)
Ignore the term \(\frac{1}{{\left[ {1 - {{\left( {\frac{{{A_{{\text{choke}}}}}}{{{A_{{\text{pipe}}}}}}} \right)}^2}} \right]}}\) and by considering that the downstream velocity is choke device’s velocity and employing the unit consistency factor (\(32.174\frac{{lbm.ft}}{{lbf.{S^2}}}\)):
$${q_l}=\pi \frac{{{d^2}}}{4}\sqrt {\frac{{2\Delta P}}{\rho }} ,$$
(62)
$${q_l}=\pi \frac{{{{\text{d}}^2}/144}}{4}\sqrt {\frac{{2 \times 144 \times \Delta P}}{{{\text{SpGr}} \times 62.4}} \times 32.174} .$$
(63)
In Eq. (63), liquid flow rate is in ft3/s, d is choke size diameter in inches, ΔP is differential pressure across the choke in psi and SpGr is oil-specific gravity respect to water. Convert the unit of liquid flow rate to barrel per day:
$${q_{{\text{BPD}}}}=1022.7{\text{d}}_{l}^{2}\sqrt {\frac{{\Delta P}}{{{\text{SpGr}}}}} .$$
(64)
As the liquid is not stabilized, let us introduce an average shrinkage factor equal to 0.9 and introduce the discharge coefficient equal to 0.6:
$${q_{BPD}}=552d_{l}^{2}\sqrt {\frac{{\Delta P}}{{SpGr}}} ,$$
(65)
$$d_{l}^{2}=\frac{{{q_{BPD}}}}{{552\sqrt {\frac{{\Delta P}}{{SpGr}}} }}.$$
(66)
Insert Eqs. (66) and (60) in Eq. (59):
$$d_{t}^{2}=\frac{{{q_{BPD}}}}{{552 \times \sqrt {\frac{{\Delta P}}{{SpGr}}} }}+\frac{{{q_{scfd}}}}{{65554 \times {P_1} \times ~\sqrt {{{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{1.5625}}\left[ {1 - {{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{0.21875}}} \right]} }}~,$$
(67)
$$d_{t}^{2}=\frac{{{q_{BPD}}}}{{{P_1}}}\left\{ {\frac{{{P_1}}}{{552 \times \sqrt {\frac{{\Delta P}}{{SpGr}}} }}+\frac{{GOR}}{{65554 \times \sqrt {{{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{1.5625}}\left[ {1 - {{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{0.21875}}} \right]} }}} \right\},$$
(68)
or
$${q_{BPD}}={P_1}{d^2} \times {\left\{ {\frac{{{P_1}}}{{552 \times ~\sqrt {\frac{{\Delta P}}{{SpGr}}} }}+\frac{{{\text{GOR}}}}{{65554 \times \sqrt {{{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{1.5625}}\left[ {1 - {{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{0.21875}}} \right]} }}} \right\}^{ - 1}}.$$
(69)
Insert \(\Delta P={P_1} - {P_2}={P_1}\left( {1 - \frac{{{P_2}}}{{{P_1}}}} \right)\). in Eq. (69):
$${q_{BPD}}={P_1}{d^2} \times {\left\{ {\frac{{\sqrt {{P_1}} }}{{552 \times ~\sqrt {\frac{{\left( {1 - \frac{{{P_2}}}{{{P_1}}}} \right)}}{{SpGr}}} }}+\frac{{{\text{GOR}}}}{{65554 \times \sqrt {{{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{1.5625}}\left[ {1 - {{\left( {\frac{{{P_2}}}{{{P_1}}}} \right)}^{0.21875}}} \right]} }}} \right\}^{ - 1}}.$$
(70)
Equation (70) is general form of choke formula for estimating the liquid flow rate in two-phase fluid flow.
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