EP1409966A1

EP1409966A1 - Improved method and system for measuring multiphase flow using multiple pressure differentials

Info

Publication number: EP1409966A1
Application number: EP00965242A
Authority: EP
Inventors: James R. Fincke
Original assignee: Bechtel BWXT Idaho LLC
Current assignee: Bechtel BWXT Idaho LLC
Priority date: 1999-09-22
Filing date: 2000-09-21
Publication date: 2004-04-21
Also published as: WO2001022041A9; NO20023858D0; EP1409966A4; CA2399536A1; NO20023858L; AU7598600A; WO2001022041A1

Abstract

An improved method and system for measuring a multi-phase flow in a pressure flow meter. An extended venturi (114) is used and pressure of the multi-phase flow is measured at three or more positions in the venturi, which define two or more pressure differentials in the flow conduit. The differential pressures are then used to calculate the mass flow of the gas phases, the total mass flow, and the liquid phase. The method for determining the mass flow of the high void fraction fluid flow and the gas flow includes certain steps. The first step is calculating a gas density for the gas flow (210). The next two steps are finding a normalized gas mass flow rate through the venturi and computing a gas mass flow rate (220). The following step is estimating the gas velocity in the venturi tube throat (230). The next step is calculating the pressure drop experienced by the gas phase due to work performed by the gas phase in accelerating the liquid phase between the upstream pressure measuring point (142) and the pressure measuring point in the venturi throat (240, 250).

Description

IMPROVED METHOD AND SYSTEM FOR MEASURING MULTIPHASE FLOW USING MULTIPLE PRESSURE DIFFERENTIALS

CONTRACTUAL ORIGIN OF THE INVENTION

This invention was made with United States Government support under Contract No. DE-AC07-94ID 13223, now Contract No. DE-AC07-99ID 13727 awarded by the United States Department of Energy. The United States Government has certain rights in the invention.

RELATED APPLICATIONS

This application claims priority from two United States continuation-in-part patent applications, S/N 09/401,375, filed September 22, 1999 and S/N 09/400,946, filed September 22, 1999.

BACKGROUND OF THE INVENTION

Field of the Invention The present invention relates to a flow meter for measuring the flow of very high void fraction multi-phase fluid streams. More particularly, the present invention relates to an apparatus and method in which multiple pressure differentials are used to determine mass flow rates of gas and liquid phases of a predominantly gas fluid stream to thereby determine the mass flow rate of each phase.

State of the Art

There are many situations where it is desirable to monitor multi-phase fluid streams prior to separation. For example, in oil well or gas well management, it is important to know the relative quantities of gas and liquid in a multi-phase fluid stream, to thereby enable determination of the amount of gas, etc. actually obtained. This is of critical importance in situations, such as off-shore drilling, in which it is common for the production lines of several different companies to be tied into a common distribution line to carry the fuel back to shore. In the prior art, a common method for metering a gas is to separate out the liquid phase, but a separation system in not desirable for fiscal reasons. When multiple production lines feed into a common distribution line, it is important to know the flow rates from each production line to thereby provide an accurate accounting for the production facilities.

In recent years, the metering of multi-phase fluid streams prior to separation has achieved increased attention. Significant progress has been made in the metering of multi-phase fluids by first homogenizing the flow in a mixer then metering the pseudo single phase fluid in a venturi in concert with a gamma densitometer or similar device. This approach relies on the successful creation of a homogenous mixture with equal phase velocities, which behaves as if it were a single phase fluid with mixture density p=αp +(1 -α)p where is the volume fraction of the gas phase, and p_g is the gas phase density and p, is the liquid phase density. This technique works well for flows which after homogenizing the continuous phase is a liquid phase. While the upper limit of applicability of this approach is ill defined, it is generally agreed that for void fractions greater than about ninety to ninety-five percent (90-95%) a homogenous mixture is very difficult to create or sustain. The characteristic unhomogenized flow in this void fraction range is that of an annular or ring shaped flow configuration. The gas phase flows in the center of the channel and the liquid phase adheres to and travels along the sidewall of the conduit as a thick film. Depending on the relative flow rates of each phase, significant amounts of the denser liquid phase may also become entrained in the gas phase and be conveyed as dispersed droplets. Nonetheless, a liquid film is always present on the wall of the conduit. While the liquid generally occupies less than five percent (5%) of the cross- sectional volume of the flow channel, the mass flow rate of the liquid may be comparable to or even several times greater than that of the gas phase due to its greater density. The fact that the phases are partially or fully separated, and consequently have phase velocities which are significantly different (slip), complicates the metering problem. The presence of the liquid phase distorts the gas mass flow rate measurements and causes conventional meters, such as orifice plates or venturi meters, to overestimate the flow rate of the gas phase. For example the gas mass flow can be estimated using the standard equation

where m_g is the gas mass flow rate, A is the area of the throat, ΔP is the measured

pressure differential, p_g the gas density at flow conditions, C_c the discharge

coefficient, and Y is the expansion factor. In test samples using void fractions

ranging from 0.997 to 0.95, the error in the measured gas mass flow rate ranges

from 7% to 30%. It is important to note that the presence of the liquid phase

increases the pressure drop in the venturi and results in over-predicting the true

gas mass flow rate. The pressure drop is caused by the interaction between the

gas and liquid phases. Liquid droplet acceleration by the gas, irreversible drag

force work done by the gas phase in accelerating the liquid film and wall losses

determine the magnitude of the observed pressure drop. In addition, the flow is

complicated by the continuous entrainment of liquid into the gas, the redeposition

of liquid from the gas into the liquid film along the venturi length, and also by the

presence of surface waves on the surface of the annular or ringed liquid phase

film. The surface waves on the liquid create a roughened surface over which the

gas must flow increasing the momentum loss due to the addition of drag at the

liquid/gas interface. Other simple solutions have been proposed to solve the overestimation of

gas mass flow rate under multi-phase conditions. For example, Murdock, ignores

any interaction (momentum exchange) between the gas and liquid phases and

proposed to calculate the gas mass flow if the ratio of gas to liquid mass flow is

known in advance. See Murdock, J. W. (1962). Two Phase Flow Measurement

with Orifices, ASME Journal of Basic Engineering, December, 419-433.

Unfortunately this method still has up to a 20% error rate or more.

Another example of a multi-phase measurement device in the prior art, is

U.S. Patent No. 5,461,930, to Farchi, et al, which appears to teach the use of a

water cut meter and a volumetric flow meter for measuring the gas and liquid

phases. This invention is complicated and must use a positive displacement

device to measure the liquid and gas flow rates so it can avoid the problem of slip

between the gas and liquid phases. This system does not appear to be effective

for liquid fractions below (5% - 10%). As mentioned earlier, other such prior art

systems such as U.S. Patent No. 5,400,657 to Kolpak, et al, are only effective for

multi -phase fluid flows where the gas fraction is 25% of the volume and the

liquid is 75% of the volume.

Other volumetric measuring devices such as U.S. Patent No. 4,231,262 to

Boll, et al, measure a flow of solids in a gas stream. For example, coal dust in a

nitrogen stream may be measured. Although these types of devices use pressure

measuring structures, they are not able to address the problem of measuring a

liquid fraction in a multi-phase flow where the liquid phase is less than 10% or

even 5% of the overall volume. Measuring a liquid and gas is significantly different from measuring a gas with a solid particulate. The mass of the liquid is

significant and not uniform throughout the gas. Incorrectly measuring the liquid

throws off the overall measurements significantly. Furthermore, such devices

which have two pressure measuring points on the venturi throat, do not take into

account the fact that a pressure drop is caused by the interaction between the gas

and liquid phases and must be calculated for accordingly.

While past attempts at metering multi-phase fluid streams have produced

acceptable results below the ninety to ninety five percent (90-95%) void fraction

range, they have not provided satisfactory metering for the very high void multi-

phase flows which have less than five to ten (5-10%) non-gas phase by volume.

When discussing large amounts of natural gas or other fuel, even a few percent

difference in the amount of non-gas phase can mean substantial differences in the

value of a production facility. For example, if there are two wells which produce

equal amounts of natural gas per day. The first well produces, by volume, 1%

liquid and the second well produces 5% liquid. If a conventional mass flow rate

meter is relied upon to determine the amount of gas produced, the second well

will erroneously appear to produce as much as 20 - 30 % more gas than the first

well. Suppose further that the liquid produced is a light hydrocarbon liquid (e.g.

a gas condensate such as butane or propane) which is valuable in addition to the

natural gas produced. Conventional meters will provide no information about the

amount of liquid produced. Then if the amount of liquid produced is equally

divided between the two wells, the value of the production from the first well will

be overestimated while the production from the second well will be underestimated. To properly value the gas and liquid production from both wells,

a method of more accurately determining the mass flow rate of both the gas and

liquid phases is required.

The prior art, however, has been generally incapable of accurately

metering the very high void multi -phase fluid streams. In light of the problems

of the prior art, there is a need for an apparatus and method that is less complex

and provides increased accuracy for very high void multi-phase fluid streams.

Such an apparatus and method should be physically rugged, simple to use, and

less expensive than current technology.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide an improved apparatus

and method for metering very high void multi-phase fluid streams.

It is another object of the present invention to provide an apparatus and

method which increases the accuracy of metering with respect to both the gas

phase and the liquid phase when measuring very high void multi-phase fluid

streams.

It is still another object of the present invention to provide such an

apparatus and method which does not require homogenization or separation of the

multi-phase fluid in order to determine flow rate for each of the phases.

The above and other objects of the invention are realized in a specific

apparatus for metering the phases of a multiple phase fluid. The flow meter

includes a cross-sectional area change in the flow conduit such as a venturi with an elongate passage. Disposed along the elongate passage is a converging

section, an extended throat section, and a diffuser. The flow meter also includes

a plurality of pressure monitoring sites which are used to monitor pressure

changes which occur as the multi-phase fluid passes through the elongate passage

and venturi. These pressure changes, in turn, can be processed to provide

information as to the respective flow rates of the phases of the multi -phase fluid.

By determining the flow rates of the components of the multi-phase fluid, the

amount of natural gas, etc., can be accurately determined and accounting

improved.

In accordance with another aspect of the present invention a method for

determining the mass flow of the high void fraction fluid flow and the gas flow

includes a number of steps. The first step is calculating a gas density for the gas

flow. The next two steps are finding the normalized gas mass flow rate through

the venturi and then computing the actual gas mass flow rate. The following step

is estimating the gas velocity in the venturi tube throat. The next step is

calculating the additional pressure drop experienced by the gas phase due to work

performed by the gas phase in accelerating the liquid phase between the upstream

pressure measuring point and the pressure measuring point at the end of the

venturi contraction or throat. Yet another step is estimating the liquid velocity in

the venturi throat using the calculated pressure drop experienced by the gas-phase

due to work performed by the gas phase. Then, the friction loss is computed

between the liquid phase and the conduit wall in the venturi tube using the liquid

velocity. Finally, the total mass flow rate based on measured pressure in the venturi throat is calculated, and the liquid mass flow rate is calculated by

subtracting the total mass flow rate and the gas mass flow rate.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other objects, features and advantages of the invention will

become apparent from a consideration of the following detailed description

presented in connection with the accompanying drawings in which:

FIG. 1 shows a side, cross-sectional view of a differential pressure flow

meter with pressure measuring ports;

FIG. 2 shows a side, cross-sectional view of a differential pressure flow

meter with an round shoulder;

FIG. 3 is a flow chart showing the steps required to calculate the mass

flow in a multiphase flow.

DETAILED DESCRIPTION

Reference will now be made to the drawings in which the various

elements of the present invention will be given numeral designations and in

which the invention will be discussed so as to enable one skilled in the art to

make and use the invention. It is to be understood that the following description

is only exemplary of the principles of the present invention, and should not be

viewed as narrowing the pending claims.

Turning now to FIG. 1 , there is shown another differential pressure flow

meter, generally indicated at 110. The differential pressure flow meter 110 includes a venturi 114 formed by a sidewall 118 which defines a fluid flow

passage 122. The fluid flow passage 122 is segmented into an inlet section 126, a

converging section 130, an extended throat section 134, a diffuser section 138

and an outlet section 140.

The geometry and conduit diameter of the flow obstruction will vary

depending on the particular application. The conduit may be larger or smaller

depending on the specific flow rate, pressure, temperature and other similar

factors. One important characteristic of the flow meter is that the preferred

contraction ratio in the conduit should be between .4 and .75. The contraction

ratio is defined as the ratio of the throat diameter 134 to the upstream conduit

diameter 122. It is also important that the length of the throat is at least ten times

the diameter of the throat. Of course, other throat lengths may be used.

An example of one possible set of conduit measurements will now be

given, but it should be realized that the actual geometry will depend on the

volume and size of the specific application. In one embodiment of the invention,

the inlet section 126 has a diameter of about 3.8 cm adjacent the opening 142 at

the upstream, proximal end 114a of the venturi 114. The converging section 130

tapers inwardly from the inlet section 126 at an angle of about ten degrees (10°)

until it connects with the extended throat section 134, which has a diameter of

about 2.5 cm. The extended throat section 134 remains substantially the same

diameter throughout its length and may be about 30 cm long to provide ample

length to determine acceleration differences between the various phases. At the

end of the extended throat section 134b, the diffuser section 138 tapers outwardly at an angle of about three degrees (3°) until the diameter of the outlet section

passage 140 is substantially the same as that at the inlet section 126 (i.e. 3 cm). It

should be realized that many other specific geometric configurations could be

defined which have characteristics similar to the example above.

In order to monitor the pressure differentials caused by the changes in

fluid velocity, the differential pressure flow meter shown in FIG. 1 utilizes up to

four different measurement points. Each pair of pressure measurement points

defines a pressure differential. Only two pressure differential measurements are

required to determine the gas and liquid flow rates. The preferred pressure

differentials are ΔP₃ and ΔP₂. Pressure differential number three (ΔP₃) is defined

as the pressure change between points 150 and 154. Pressure differential number

two (ΔP₂) is between points 154 and 158. The pressure differential ΔP₂is

important because it is used for the calculation of the pressure drop experienced

by the gas phase due to the work performed by the gas phase in accelerating the

liquid phase.

It should also be apparent based on this disclosure that the combination of

pressure differentials ΔP₃ and ΔP₀ or ΔP₂ and ΔP₀ may be used instead. Each of

these combinations work equally well, with the exception that the numerical

constants in the algorithm change. It is also important that an absolute pressure

and temperature measurement will be provided at the venturi inlet 142.

Now the pressure ports will be described more specifically. A first

pressure measuring port 150 is disposed to measure the pressure in the inlet

section 142. The first pressure measuring port 150 is connected to a pressure monitoring means, such as a pressure transducer 151, to provide a pressure

reading.

A second pressure measuring port 154 is provided at the entrance of the

extended throat section 134. The second pressure measuring port 154 is

disposed adjacent the upstream, proximal end 134a of the extended throat section

134. A pressure transducer 151 is also coupled to the second pressure measuring

port 154.

Distally from the second pressure measuring port 154, but still within the

extended throat section 134, is a third pressure monitoring port 158. Preferably,

the third pressure monitoring port 158 is disposed adjacent the distal end 134b of

the extended throat section 134, and adjacent the beginning 138a of the diffuser

section 138.

The respective pressure measuring ports 150, 154, and 158 are disposed in

communication with a flow processor 153 or similar mechanism through the

pressure monitoring means or pressure transducers 151, 155, and 159. The flow

processor 153 enables the acquisition of the measured pressure differentials, and

thus fluid flow rates in accordance with the present invention. Further, an

accurate determination of the relative acceleration of the two phases can also be

obtained by comparing the pressure drop between the inlet section 126 (through

measuring port 150) and the distal end 134b of the extended throat section 134

(through measuring port 158), as indicated at ΔP₀

In an alternative embodiment of the invention, a fourth pressure

measuring port 161 is disposed at the end of the extended throat 134b. A fifth pressure measuring port 162 is disposed in the outlet section 140 adjacent to the

distal end 138b of the diffuser section 138. Both of these pressure measuring

ports are coupled to pressure monitoring means or pressure transducer 163. The

fourth and fifth monitoring ports allow a pressure differential ΔP] to be measured.

The pressure differential (ΔPj) between the extended throat section 134 and the

distal end 138b of the diffuser section 138 can also be analyzed.

It should also be realized that different angles and lengths can be used for

the venturi constriction and the extended throat of the venturi tube. In fact, the

converging section of the venturi is not required to gradually taper. FIG. 2 shows

a converging section 172 as formed by an annular shoulder in a venturi tube 170

to reduce the cross-sectional area of the inlet section. The preferred size of the

radius of curvature for an annular shoulder 172 is about 0.652 cm. The

converging section can also be formed by placing a solid object in the conduit

which occupies part but not all of the conduit cross-section.

It is vital that the correct method be used in the current invention to

estimate the gas and fluid mass flow. Otherwise errors in the range of 20% or

more will be introduced into the measurements, as in the prior art. Reliable

metering of high void fraction multi-phase flows over a wide range of conditions

(liquid loading, pressure, temperature, and gas and liquid composition) without

prior knowledge of the liquid and gas mass flow rates requires a different

approach than the simple modification of the single phase meter readings as done

in the prior art. Conceptually, the method of metering a fluid flow described here

is to impose an acceleration or pressure drop on the flow field via a structure or venturi constriction and then observe the pressure response of the device across

two pressure differentials as described above. Because the multi-phase pressure

response differs significantly from that of a single-phase fluid, the measured

pressure differentials are a unique function of the mass flow rates of each phase.

As described above, the gas and liquid phases are strongly coupled. When

the gas phase accelerates in the converging section of the nozzle, the denser

liquid phase velocity appreciably lags that of the lighter gas phase. In the

extended throat region, the liquid phase continues to accelerate, ultimately

approaching its equilibrium velocity with respect to the gas phase. Even at

equilibrium, significant velocity differences or slip will exist between the gas and

liquid phases. A method for accurately calculating the gas and liquid

mass flows in an extended venturi tube will now be described. (A derivation of

the method is shown later.) This method uses the four values which are

determined though testing. These values are: ΔP₃ which is the measured pressure

differential across the venturi contraction, ΔP₂ which is the measured pressure

differential across the extended venturi throat, P which is the absolute pressure

upstream from the venturi (psi), and T which is the temperature of the upstream

flow. These measured values are used with a number of predefined constants

which will be defined as they are used. Alternatively, the pressure differentials

ΔP₃ and ΔP₀ or the pressure differentials ΔP₀ and ΔP₂ may be used.

First, the gas density for the gas flow must be calculated based on the

current gas well pressure and temperature. This is done using the following equation which uses English units. Any other consistent set of units may also be

used with appropriate modifications to the equations.

Equation 1

f P + 14.7V 60 + 459.67^ rho g„w„ - rho_r 14.7 J T+ 459.67 where

rho_g is the density of natural gas (i.e. a mixture methane and other

hydrocarbon and non-hydrocarbon gases) at standard temperature (60° F) and

pressure (1 atmosphere) for a specific well;

P is the pressure upstream from the venturi in pounds per square

inch (psi); and

T is the temperature upstream from the venturi in degrees

Fahrenheit.

The value of rho_g will be different for various natural gas compositions

and must be supplied by the well operator. At the standard temperature (60° F)

and pressure (1 atmosphere) the value of rho_g for pure methane is 0.044 lb/ft³.

The second step is finding a normalized gas mass flow rate based on the

square root of a pressure difference across the contraction multiplied by a first

predetermined coefficient, and the square root of a measured pressure differential

across a venturi throat. The normalized gas mass flow rate is found using the

following equation:

Equation 2

mgm = A + B^ P_Z + C^AP₂ where

A, B, and C are experimentally determined constants required to

calculate gas mass flow rate;

ΔP₃ is the measured pressure differential across a venturi

contraction; and

ΔP₂ is the measured pressure differential across a venturi throat.

The preferred values for the constants in the equation above are as follows: A is -

0.0018104, B is 0.008104 and C is -0.0026832 when pressure is in pounds per

square inch (psi), density in lbs/ft³ and mass flow rate in thousands of mass

lbs/minute. Of course, these numbers are determined experimentally and may

change depending on the geometry of the venturi, the fluids used, and the system

of units used.

Calculating the normalized gas mass flow rate is important because it

allows the meter to be applied to the wells or situations where the pressure or

meter diameter for the liquids present are different than the conditions under

which the meter was originally calibrated. This means that the meter does not

need to be calibrated under conditions identical to those present in a particular

application and that the meter may be sized to match the production rate from a

particular well.

The functional form of Equation 2 is arrived at by derivation from the

conservation of mass and energy followed by a simplifying approximation. Other

functional forms of Equation 2 can be used with equivalent results. The functional form of Equation 2 is consistent with the conservation laws and

provides a good representation of the calibration data.

The third step is computing a gas mass flow rate using the normalized gas

mass flow rate, the gas density, and a contraction ratio of the venturi tube. The

equation for calculating the gas mass flow rate from these quantities is

Equation 3

where mgm is the normalized gas mass flow rate;

A_j is the venturi throat area;

β is the contraction ratio of the throat area; and

rho_gw is the gas density at current well conditions.

The fourth step is estimating the gas velocity in the venturi tube throat.

The equation for estimating the gas velocity is:

Equation 4

m„ u„

⁸ rho„ ^■ A,

where m_g is the gas mass flow rate;

rho_g is the density of the gas phase for a specific well; and

A_t is the venturi throat area. The fifth step is calculating the pressure drop experienced by the gas

phase due to work performed by the gas phase in accelerating the liquid phase

between an upstream pressure measuring point and a pressure measuring point in

the distal end of the venturi throat. The pressure drop is calculated as follows:

Equation 5

P_gl = ^ - - r _0gw - u_g ² - l - β⁴)

where ΔP₃ is the measured pressure differential across a venturi

contraction;

rho_gw is gas density at well conditions;

u_g is the gas velocity in the venturi throat; and

β is the contraction ratio of the throat area to the upstream area.

It is important to note that the calculations outlined in steps two and five

are important because they allow for estimating the mass flow of each phase.

Step six is estimating the liquid velocity (u,) in the venturi throat using the

calculated pressure drop experienced by the gas phase due to work performed by

the gas phase. This is performed as follows

Equation 6

2(Δ ₃ - AP_gr u, = ^{l ~} "i rho [(\+ β ) + gcM where

ΔP₃ is the measured pressure differential across a venturi

contraction; ΔP_gl3 is the pressure drop experienced by the gas-phase due to

work performed by the gas phase on the liquid phase;

rho, is the liquid density; and

gcfw is a constant which characterizes wall friction. The preferred

value for gcfw is defined as 0.062. This value may be adjusted depending on

different venturi geometries or different fluids.

The seventh step is computing the friction between the liquid phase and a

wall in the venturi which is performed:

Equation 7

1 , f = gcβv- -- rho u_! where

gcfw is a constant which characterizes wall friction;

rhoj is the liquid density; and

U_! is the liquid velocity in the venturi throat.

The eighth step is calculating the total mass flow rate based on the

measured pressure in the venturi throat, the calculated friction and the gas

velocity. The equation for this is:

Equation 8

where

ΔP₃ is the measured pressure differential across a venturi

contraction; β is the contraction ratio of the throat diameter to the upstream

diameter; and

u_g is the gas velocity in the venturi throat.

The liquid mass flow rate can now be calculated as the difference between

the total and gas mass flow rates.

Equation 9

m, = (m_t - m_g) wherein

m_t is the total mass flow rate; and

m_g is the gas mass flow rate.

Calculating the gas mass flow rate, total mass flow rate, and liquid mass

flow rate using the method outlined above is much more accurate than the prior

art. The accuracy of method outlined above is within ±4% for the gas phase,

±5% for the liquid phase, and ±4% for the total mass flow. This accuracy can

even be increased using measured calibrations for a specific installation to

benchmark the readings.

FIG. 3 shows a summary of the method used to accurately calculate the

mass flow through the elongated venturi. The method for determining the mass

flow of the high void fraction fluid flow and the gas flow includes steps which

were described with Equations 1-9. Referring to FIG. 3, the first step is

calculating a gas density for the gas flow 210. The next two steps are finding a

normalized gas mass flow rate through the venturi 220 and computing a gas mass

flow rate 230. The following step is estimating the gas velocity in the venturi

tube throat 240. The next step is calculating the pressure drop experienced by the gas-phase due to work performed by the gas phase in accelerating the liquid

phase between the upstream pressure measuring point and the pressure measuring

point in the venturi throat 250. Yet another step is estimating the liquid velocity

260 in the venturi throat using the calculated pressure drop experienced by the

gas-phase due to work performed by the gas phase. Then the friction is computed

270 between the liquid phase and a wall in the venturi tube. Finally, the total

mass flow rate based on measured pressure in the venturi throat is calculated 280

and the liquid mass flow rate is determined 290.

Theoretical Gas Mass Flow Rate

Now a discussion of the theoretical derivations will be outlined which

produced the method described above. The theoretical derivation is based on the

physical laws describing the conservation of mass and energy for both the gas and

liquid phases. The conservation of mass and energy equations for each phase are

shown below where the subscript 1 denotes the upstream condition measured at

142 by pressure tap 150 in FIG. 1, and the subscript 2 denotes the venturi throat

entrance measured at 134a by pressure tap 154. ΔP_gl3 is the pressure drop

experienced by the gas phase due to work done by the gas phase in accelerating

the liquid phase between the pressure measuring location at the beginning of the

elongated throat and the pressure measuring location at the end of the throat. It is

assumed that only the liquid phase is in contact with the wall, f_w is the wall

friction coefficient and G_c is a geometry factor which accounts for the

acceleration of the fluid in the venturi contraction and the surface area of the

contraction. Equations 10

m_g = p_gu_gλA_x = ₂p_{g g2}A₂

m_l - (1 - a_l)p_lu_nA₁ = (1 - a₂)p_lu_l2A₂

, + -^ = p₂ + -_Pluf₂ - AP_gl3 + G_cf_w -p_lUf₂

In Equations 10, α is void fraction, p_g is density of a gas at standard

temperature, u_g is the gas velocity, A, is the conduit area upstream of the venturi,

A₂ is the conduit area in the venturi throat, and P, and P₂ are the pressures at

locations 142 (tap 150) and 134a (tap 154) in the conduit.

The gas phase energy equation can be rewritten using the equation for the

gas phase mass flow rate, where D is the diameter of the upstream piping, d is the

throat diameter, β=d/D is the contraction ratio, and ΔP₃ = P₂ - P,_ is the pressure

drop across the contraction.

Equation 11

With the approximation that α, and α₂= 1, the modified orifice equation results.

Equation 12

For single-phase flow ΔP_gl3 is equal to zero and the equation is solved

directly for the mass flow rate m_g. In practice, the single-phase result is modified

by the addition of an empirical constant C_c which accounts for the true discharge

characteristics (non-ideal one-dimensional behavior and friction losses) of the

nozzle and Y which takes compressibility effects into account.

Equation 13

As shown in the introduction, if the Equation 13 above is used under

multiphase conditions, the mass flow rate of the gas phase can be significantly

overestimated. Under multiphase conditions the mass flow rate of the gas phase

is given by:

Equation 14

where α₂A₂ represents the cross sectional area occupied by the gas phase. When

ΔP₃ is large with respect to ΔP_gl3 the quantity under the radical can be

approximated by

Equation 15

where C_gI3 is a constant that is determined experimentally. Empirically it has

been found that ΔP_gl3can be replaced by a function of ΔP₂, the pressure drop in

the extended throat, with appropriate choice of constants. The mass flow rate of

gas under both single phase and multiphase conditions now becomes

Equation 16

where it has been assumed that α₂=αι = l. The constants C_2φ and C₂ have been

determined empirically and the validity of the equation has been tested over a

wide range of conditions. It is important to note that this method can be used not

only with natural gas production but other gas and liquid phase compositions. In

addition, it is also important to recognize that Equations 10 - 16 are used to derive

calculation steps in the calculation method.

We have assumed that α₂=α_!~ l, making Equation 16 above only

approximate. The statistical fitting procedure used to determine the constants

C_2φand C₂ implicitly determines a weighted mean value of α. Because α does not

appear explicitly and is unknown, there is an uncertainty of ±1-2% over the void

fraction range 0.95< α<1.0, implicit in the equation. If α or (1- α) is

independently measured, the observed measurement uncertainties can be

significantly reduced. The uncertainty can also be significantly reduced if, at

installation, the actual flow rates are accurately known. If this measurement is

available then the meter reading can be adjusted to reflect the true value and the

uncertainty in the gas phase mass flow rate measurement can be reduced to less than 0.5% of reading if the gas and liquid flow rates change by less than 50% or

so over time. The repeatability of the measurement is essentially the random

uncertainty in the pressure measurements, less than about 0.5% of reading.

Total and Liquid Mass Flow Rate

If the ratio of liquid to gas flow rate is known a priori with certainty then

the mass flow rate of the liquid phase can be directly obtained from

m,=m_g(m/m _known. Note that because the liquid mass flow rate is only a fraction

(0-30%) of the gas mass flow rate the uncertainty in the measurement is

magnified. For instance, if m,/m_g= 0.01, a 1% error in m_g is magnified to become

a 100%) of reading error for the liquid phase. An additional fixed error of 1% in

the ratio m,/m_g results in a 200% of reading total error for the liquid phase. This

approach, of course, assumes that the m,/m_g ratio remains constant over time.

Unfortunately, without accurate independent knowledge of α or (1-α) the

liquid mass flow rate cannot be obtained directly from one-dimensional theory.

The velocity of the liquid phase can, however, be estimated directly as now

described. Once the mass flow rate of the gas phase is determined the ΔP_gl3 term

can be estimated from the gas phase energy equation:

Equation 17

Equation 17 allows us to derive Equation 5 in the calculation method. Rearranging the liquid phase energy equation yields Equation 18

and using the expression for the mass flow rate of liquid results in:

Equation 19

(i - a_y

With the assumption that ^■ β < < 1 the liquid velocity u₁₂can be

{\ -

estimated. If (1-α) is known then the liquid mass flow rate could be estimated directly from m,=(l-α₂)pu₁₂A. Unfortunately, (1-α) cannot be accurately estimated directly from the differential pressure data; it must be independently measured to pursue this approach.

If we consider the gas and liquid phases together but allow their velocities to differ, the total mass flow rate can be written as:

Equation 20

where the density term in brackets is the effective density, p_s!ip and S=u_g/u, which

is ratio of the gas velocity to the liquid velocity or slip. Since m_t is constant

throughout the venturi, it allows us to write the pressure drop ΔP₃ as Equation 21

The second term on the right hand side is the friction loss assuming that only the

liquid phase is in contact with the wall. The equation can be rearranged to yield

the total mass flow rate

Equation 22

The total mass flow rate m_t can then be obtained directly from ΔP₃ once u_g is

estimated from the measured value of m_g, u_g =171^^ and the liquid velocity is

calculated by solving equation 19 for u₁₂. The total mass flow rate using this

method is a measurement with an uncertainty of ±4% of the actual measured

flow. In principle, (since the total mass flow rate is the sum of the gas and liquid

mass flow rates) the liquid mass flow rate can now be obtained directly from

mι=m,-m_g. The liquid mass flow rate can then be obtained within ±5% of the total

mass flow rate.

As previously noted in the discussion of the measurement of the gas mass

flow rate, if the flow rates of each phase are accurately known at the time of

installation, measurement performance over a reasonable range of mass flow rates

can be significantly enhanced. The uncertainty in the gas mass flow rate measurement can be reduced to <0.5% of reading by benchmarking even if the

gas and/or liquid mass flow rates change by ±50%. Similarly, the uncertainty in

the total mass flow rate can be reduced by <2% of reading for the same ±50%

changes in gas and/or liquid mass flow rates. The corresponding improvement in

accuracy of the liquid phase measurement is also significant. Because the liquid

mass flow rate measurement is dependent on both the gas phase and total mass

flow rate measurements, the uncertainty is also sensitive to changes in both gas

and liquid mass flow rate. If the liquid mass flow rate measurement is

benchmarked at an initial value, the data indicate that the accuracy attainable is

±20% of reading for changes in gas mass flow rate in the range of <±15% and/or

changes in liquid mass flow rate in the range of <±25%. The uncertainty in the

liquid mass flow rate quoted in terms of percent of total mass flow rate becomes

±1%.

Measurement uncertainties can be significantly reduced if flow rates are

accurately known at time of meter installation or periodically measured by

separation and separate metering during the service life of the meter and the well.

Because the liquid phase is generally only a small fraction of the total mass flow

rate the uncertainty in its measurement is inherently high. If the void fraction α is

accurately and independently measured, the liquid mass flow rate can be

calculated directly from m, - (1 - α) l,\\_a A where the u_/2 the liquid velocity is

obtained as described above from equation 19. The void fraction may be

accurately and independently measured using a gamma ray attenuation

densitometer or through ultrasonic film thickness measurements. This approach has been shown to significantly reduce the uncertainty in the liquid mass flow

rate measurement.

Claims

I CLAIM:

1. A method for determining total mass flow rate of a high void

fraction flow having a liquid phase and a gas phase, the method comprising:

a) passing a high void fraction liquid and gas flow through a flow

channel having a variable diameter;

b) detecting a first pressure differential value resulting from a

change in the flow channel diameter;

c) detecting a second pressure differential value resulting from

work performed by the gas phase in accelerating the liquid phase;

d) processing the first and second pressure differential values to

determine the total mass flow rate of the high void fraction flow having a liquid

phase and a gas phase.

2. The method according to claim 1, further comprising the step of

processing the total mass flow rate to calculate a liquid mass flow rate.

3. The method according to claim 1, wherein the flow channel

having a variable diameter is an extended throat venturi having a converging

section and a throat.

4. The method according to claim 3, wherein the step of processing

the first and second pressure differential values to determine the total mass flow rate of a high void fraction flow having a liquid phase and a gas phase further

comprises:

(a) calculating a gas density for a gas flow;

(b) calculating a normalized gas mass flow rate based on measured

pressure differences across the converging section and the throat;

(c) computing a gas mass flow rate in the throat using the

normalized gas mass flow rate, a gas density, venturi throat geometry, and a

contraction ratio of the venturi;

(d) estimating gas velocity in the throat using the gas mass flow

rate;

(e) calculating a pressure drop experienced by the gas phase due to

work performed by the gas phase in accelerating the liquid phase between an

upstream pressure measuring point prior to the venturi and a pressure measuring

point in the throat;

(f) estimating liquid velocity in the throat using the calculated

pressure drop experienced by the gas phase due to work performed by the gas

phase;

(g) computing a friction value between the liquid phase and a wall

in the venturi using the liquid velocity;

(h) calculating the total mass flow rate based on a measured

pressure difference in the throat, the friction value and the gas velocity.

5. The method of claim 4, further comprising the step of calculating a

liquid mass flow rate as the difference between the total mass flow rate and the

gas mass flow rate.

6. The method of claim 4, wherein the step of calculating a gas

density for a gas flow further comprises the step of calculating the gas density for

the gas flow using the equation:

⁽ P 14.7V 60 + 459.67^ rho^ = rho_k

V 14.7 ; T+ 459.67 J

wherein

rho_g is a methane density at standard temperature (60° F) and

pressure (1 atmosphere) for a specific well;

P is a pressure upstream from the venturi tube; and

T is a temperature upstream from the venturi tube.

7. The method of claim 4 wherein the step of calculating a

normalized gas mass (mgm) flow rate further comprises finding the normalized

gas mass flow rate using the following equation:

mgm = A + B^ ^~P ^r C^AP,

where

A, B, and C are experimentally determined constants required to

calculate gas mass flow rate; ΔP₃ is a measured pressure differential across the converging

section; and

ΔP₂ is a measured pressure differential across the throat.

8. The method of claim 7 wherein the constant A is -0.0018104, B is

0.008104 and C is -0.0026832 when pressure is in pounds per square inch (psi),

mass flow rate is in thousands of mass lbs/minute, and density is in lbs/ft³ and

area is in inches².

9. The method of claim 4 wherein the step of computing a gas mass

flow rate further comprises computing a gas mass flow rate (mg) using the

following equation

wherein mgm is the normalized gas mass flow rate;

A_t is an area of the venturi throat in inches; and

β is a contraction ratio of the throat area to an upstream area.

10. The method of claim 4 wherein the step of estimating a gas

velocity further comprises using the following equation:

m „ =

⁸ rho ^■ A_t wherein m_g is the gas mass flow rate;

rho_g is a gas density for a specific well; and

A_t is an area of the venturi throat.

11. The method of claim 4 wherein the step of calculating a pressure

drop experienced by a gas phase due to work performed by the gas phase further

comprises using the following equation:

AP_gl3 = AP - - rh_0gw - u_g ² - (\ - p*)

wherein ΔP₃ is a measured pressure differential across the converging

section;

rho_gw is gas density at well conditions;

u_g is a gas velocity in the throat; and

β is a contraction ratio of the throat area to an upstream area.

12. The method of claim 4 wherein the step of estimating liquid

velocity in throat is performed using the following equation:

wherein

ΔP₃ is a measured pressure differential across the converging

section; ΔP_gl3 is the pressure drop experienced by the gas-phase due to

work performed by the gas phase on the liquid phase;

rhθ] is a liquid density; and

gcfw is a constant which characterizes wall friction.

13. The method as in claim 12 wherein the gcfw which represents wall

friction is defined as 0.062.

14. The method as in claim 4 wherein the step of computing friction

between the liquid phase and a wall in the venturi further comprises computing

friction using the following equation:

1 f = _g - -- rho u, 2^<

wherein

gcfw is a constant which characterizes wall friction;

rho, is a liquid density; and

U) is the liquid velocity in the throat.

15. The method as in claim 4 wherein the step of calculating the total

mass flow rate further comprises calculating the total mass flow rate using the

following equation:

m, = 2(Δ ₃ -. - /) • A _Λ,

' (i - ⁴) - ^ ' wherein

ΔP₃ is a measured pressure differential across the converging

section;

β is a contraction ratio of the throat area to an upstream area; and

u_g is the gas velocity in the throat.

16. The method as in claim 5, wherein the step of calculating the

gas mass flow rate further comprises using the following equation:

m, = (m_t - m )

wherein

m_t is the total mass flow rate; and

m_g is the gas mass flow rate.

17. A method for finding a gas mass flow rate and a liquid mass flow

rate of a combined flow having a liquid phase and gas phase, using an extended

throat venturi having a converging section and a throat, comprising the steps of:

(a) calculating a normalized gas mass flow rate based on measured

pressure differences across the converging section and throat;

(b) computing a gas mass flow rate in the throat using the

normalized gas mass flow rate, venturi throat geometry, a gas density, and a

contraction ratio of the venturi; (c) estimating gas velocity in the throat using the gas mass flow

rate;

(d) calculating a pressure drop experienced by the gas phase due to

work performed by the gas phase in accelerating the liquid phase between an

upstream pressure measuring point prior to the venturi and a pressure measuring

point in the throat;

(e) estimating liquid velocity in the throat using the calculated

pressure drop from step (d);

(f) measuring a void fraction in the throat using a measurement

device; and

(g) calculating the liquid mass flow rate directly from one minus

the void fraction multiplied by a liquid density value, the liquid velocity, and a

throat area value.

18. The method for finding gas and liquid mass flow rates as in claim

17 wherein step (f) further comprises the step of measuring the void fraction in

the throat using a gamma ray attenuation densitometer.

19. The method for finding a gas and liquid mass flow rates as in

claim 17 wherein step (f) further comprises the step of measuring the void

fraction in the throat using an ultrasonic film thickness measuring device.

20. The method for finding a gas and liquid mass flow rate as in claim

the throat using a microwave attenuation densitometer or microwave resonant

cavity densitometer.

21. A differential pressure flow meter for measuring a flow rate of a

high void multi-phase flow to determine mass flow rates of a gas and a liquid

phase through a flow channel, the apparatus comprising:

(a) a flow channel having a variable diameter;

(b) a pressure monitor, disposed in communication with at least

three locations along the flow channel, including locations having different

diameters, for determining at least two pressure differentials within the flow

channel as the high void multi-phase flow passes through the flow channel; and

(c) a processor, disposed in communication with the pressure

monitor, for calculating a pressure drop experienced by the gas phase due to work

performed by the gas phase in accelerating the liquid phase, such that the pressure

drop is used to calculate the mass flow rates for the gas and liquid phases.

22. The differential pressure flow meter of claim 21, wherein the flow

channel having a variable diameter has a first and second segment, the second

segment having a smaller diameter than the first segment.

23. The differential pressure flow meter of claim 21 , wherein the flow

channel is a venturi tube, having a venturi inlet, a venturi outlet, and an extended

throat having a throat inlet coupled to the venturi inlet and a throat outlet coupled

to the venturi outlet.

24. The differential pressure flow meter of claim 22, wherein the at

least two pressure differentials are used by the processor to calculate the pressure

drop experienced by the gas phase due to work performed by the gas phase in

accelerating the liquid phase.

25. The differential pressure flow meter of claim 23, wherein the at

drop experienced by the gas phase due to work performed by the gas phase in

accelerating the liquid phase.

26. The differential pressure flow meter of claim 23, wherein at least

one pressure measuring point is disposed before the venturi inlet, at least one

pressure measuring point is disposed in the throat inlet, and at least one pressure

measuring point is disposed in the throat outlet .

27. The differential pressure flow meter of claim 23, wherein the

pressure monitor further comprises: a first pressure means for determining a first pressure differential

between a location before the venturi inlet and a location in the throat inlet;

a second pressure means for determining a second pressure

differential between a location adjacent the throat inlet and a location adjacent the

throat outlet; and

a calculation means for selectively combining the first and second

pressure differentials to determine a flow rate of a high void fraction liquid and

gas flow, which passes through the venturi tube.

28. The differential pressure flow meter as in claim 21 wherein the

processor uses the pressure drop calculated to further estimate liquid velocity and

friction.

29. The differential pressure flow meter as in claim 28 wherein the

processor calculates the total mass flow rate using the liquid velocity, friction, gas

mass flow rate and at least two pressure differentials.

30. The differential pressure flow meter as in claim 28 wherein the

processor calculates a liquid mass flow rate as a difference between the total mass

flow rate and the gas mass flow rate.