Multiphase fluid flow measurement

Multiphase fluid flow, such as a mixture of oil, water and gas, is measured by a system which includes one or two densitometers for measuring the multiphase fluid density flowing through a flowline or conduit and a flow restriction, pump, expander or heat exchanger interposed in the flowline between the densitometers to effect a pressure or temperature change in the multiphase fluid. Densities, pressures and temperatures are measured on the upstream and/or downstream sides of the flow restriction, pump, expander or heat exchanger and the measured values of density, pressure and temperature are used to obtain the flow rates of the respective phases in the flowstream.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention pertains to measurement of multiphase fluid (liquid 
and gas) flow using one or more densitometers, a pump, flow restriction or 
heat exchanger to effect a change in pressure and temperature of the fluid 
flow and pressure and temperature sensors for measuring the flowstream 
both upstream and downstream of the restriction, pump or heat exchanger. 
2. Background 
Various methods have been developed for measuring multiphase fluid flow 
where widely varying ratios of gas to liquid are experienced and wherein a 
high degree of measurement accuracy may or may not be important. U.S. 
patent application Ser. No. 08/131,813, filed Oct. 5, 1993, and U.S. 
patent application Ser. No. 08/179,411, filed Jan. 10, 1994, both by M. M. 
Kolpak and both assigned to the assignee of the present invention, 
describe various types of flowmeters for measuring multiphase fluid flow, 
particularly mixtures of oil, water and gas which result from the 
production of hydrocarbons from subterranean wells. U.S. Pat. No. 
5,099,697, issued Mar. 31, 1992 to J. Agar, also describes a multiphase 
fluid flow measurement system adapted to measure flows of oil, water and 
gas. 
However, there has been a continuing need to develop flow measurement 
systems and methods which may be easily adapted to existing flow lines 
leading from subterranean wells, flow lines leading to or from separation 
devices for separating gas from liquid and similar applications where a 
high degree of accuracy is not required, where a relatively high 
percentage of the flow is gas and wherein minimal interruption of the 
fluid flow system is required or is desirable. 
Many types of known flowmeter systems do not perform well where a high 
gas-to-liquid ratio or gas fraction of the total volumetric flow exists or 
wherein a substantial number of fluid parameters, including fluid density, 
must be premeasured or otherwise known in order to perform the measurement 
method. The devices and methods described herein overcome some of the 
deficiencies of existing multiphase fluid flow measurement systems and 
methods. 
SUMMARY OF THE INVENTION 
The present invention provides an improved system and method for measuring 
multiphase (liquid and gas) fluid flow including flow wherein a relatively 
high gas content or so-called gas fraction of the total volume is 
frequently encountered. In accordance with one important aspect of the 
present invention, an improved multiphase fluid flow measurement system is 
provided wherein one or two fluid densitometers are adapted to measure the 
density of the multiphase fluid flowstream and a flow restriction or other 
pressure or temperature changing device is interposed in the flow line 
between the densitometers and further wherein the fluid pressures and 
temperatures are measured on opposite sides of the fluid-pressure or 
temperature changing device. 
In accordance with another important aspect of the present invention, a 
fluid flow measurement system and method is provided wherein the 
determination of the liquid and gas fractions and flow rates take into 
account the temperature changes, gas flashing and liquid shrinkage 
(compressibility). 
In accordance with yet another aspect of the present invention, multiphase 
fluid flow measurement systems and methods are provided wherein multiphase 
fluid flow from a subterranean well may be easily and accurately 
determined by employing two densitometers spaced apart on opposite sides 
of a flow restriction, such as a wellhead choke, and together with 
pressure and temperature sensors disposed for sensing pressure and 
temperature on opposite sides of the choke. An inexpensive and easily 
adapted flow measurement system may thus be employed. 
The present invention provides an inexpensive way to monitor multiphase 
fluid flow, particularly from subterranean wells and from treatment and 
separation facilities for handling the fluid flow from such wells. The 
system may be easily retrofitted to existing facilities which require 
multiphase fluid flow measurement and the cost of the system is expected 
to be less than other types of multiphase fluid flow measurement systems 
and where accuracies of flow measurement in the range of eighty-five 
percent to ninety percent are acceptable. 
Those skilled in the art will recognize the advantages and superior 
features of the systems and methods described above as well as other 
superior aspects of the invention upon reading the detailed description 
which follows in conjunction with the drawing.

DESCRIPTION OF PREFERRED EMBODIMENTS 
In the description which follows, like parts are marked throughout the 
specification and drawing with the same reference numerals. The drawing 
figures are in schematic form in the interest of clarity and conciseness. 
Referring to FIG. 1, there is illustrated a schematic of a flow measurement 
system which is adapted to measure multiphase fluid flow with a relatively 
high gas fraction as a percent of the total volumetric flow rate, 
nominally about twenty five percent (25%) or more. FIG. 1 illustrates a 
wellhead 10 which is operable to produce hydrocarbon fluids such as 
mixtures of liquid and gas and wherein the liquid phase of the fluid may 
comprise oil and water. The wellhead 10 is connected to a flowline 12 
having a first densitometer 14 interposed therein and which may be of a 
type which does not require intrusion into the flowpath of the fluid 
flowing through the flowline 12. A suitable volumetric flowmeter 16 is 
also interposed in the flowline 12 and a suitable flow restriction such as 
a venturi, orifice or oil field choke device 18 is also interposed in the 
flowline 12. The flowmeter 16 and flow restriction 18 are interposed in 
the flowline between the first densitometer 14 and a second densitometer 
20 which also may be of a type which is non-invasive into the flowpath of 
the fluid flowing through the flowline 12. The densitometers 14 and 20 may 
be of the so-called gamma-ray type such as commercially available from 
Texas Nuclear, Inc. of Austin, Tex. as their SGD Series. The flow 
measuring system of FIG. 1 includes pressure and temperature sensors 
disposed for sensing the pressures and temperatures of the fluids flowing 
through the flowline 12 on opposite sides of the flow restriction 18. It 
is assumed that the flowmeter 16 does not cause a significant pressure or 
temperature change of fluid flowing therethrough although the flowmeter 
could have a suitable flow restriction or pressure change device 
incorporated therein in addition to the volumetric flow measuring 
mechanism. The flow restriction 18 can be a valve or similar device also. 
The flow restriction 18 causes a pressure decrease and a corresponding 
expansion of the fluid flowing through the flowline 12. The expansion 
causes the downstream densitometer 20 to register a lower density of the 
fluid flowing therethrough than the upstream densitometer 14. 
Alternatively, in accordance with the present invention, the flow 
restriction 18 could be a pump or turbine. In all cases, the method of 
determining the liquid and gas flow rates for the multiphase fluid flowing 
through the system of FIG. 1 may be carried out whether a pressure 
increase or decrease occurs across the flow restriction 18. 
The liquid fraction, f.sub.L may be determined by a suitable computational 
device such as a microprocessor which has been programmed to make certain 
calculations based on the equations given below. The volumetric liquid 
fraction in the total flow is determined from the following equation. 
EQU f.sub.L =((T.sub.2 /T.sub.1)*(P.sub.1 /P.sub.2)-1)/(d.sub.1 /d.sub.2 
-1)-1(1) 
where, 
(T.sub.1,T.sub.2)=measured temperature upstream and downstream of the 
restriction 18 
(P.sub.1,P.sub.2)=measured pressure upstream and downstream of the 
restriction 18, and 
(d.sub.1,d.sub.2)=measured density at upstream and downstream of the 
restriction 18. 
Liquid and gas volumetric flow rates, (Q.sub.1L, Q.sub.1G), are determined 
from 
EQU Q.sub.1L =f.sub.L *Q (2) 
EQU Q.sub.1G =(1-f.sub.L)*Q (3) 
where Q is the combined flow rate measured by the flowmeter 16. The 
subscripts 1 and 2 denote the flow conditions of the multiphase fluid 
upstream and downstream of the flow restriction 18 or other element 
interposed between the densitometers as indicated by the pressure and 
temperature sensor symbols in the drawing figures. 
The gas laws hold that for a volume of gas, V, 
EQU PV/T=constant (7) 
The change in gas volume (dV) which results from the flow restriction or 
pressure changing element 18 may be expressed as follows in accordance 
with the gas laws. 
EQU dV=V.sub.1 {(T.sub.2 /T.sub.1)(P.sub.1 /P.sub.2)-1} (8) 
If both sides of equation (8) are divided by time, the analogous 
relationship for volumetric rate is provided as: 
EQU dQ=Q.sub.1G {(T.sub.2 /T.sub.1)(P.sub.1 /P.sub.2)-1} (9) 
where Q.sub.1G is the upstream gas flow rate and dQ also equals the 
difference between upstream and downstream flow rates 
EQU dQ=Q.sub.2 -Q.sub.1 (10) 
Equating the mass flow rate upstream and downstream yields 
EQU d.sub.1 Q.sub.1 =d.sub.2 Q.sub.2 (11) 
were d.sub.1 and d.sub.2 are upstream and downstream densities measured by 
the densitometers 14 and 20. 
The upstream flow rate is composed of gas and liquid rates 
EQU Q.sub.1 =Q.sub.1L +Q.sub.1G (12) 
and the liquid fraction, f.sub.L, is defined as 
EQU f.sub.L =Q.sub.1L /(Q.sub.1L +Q.sub.1g) (13) 
Combining equations (7) through (13) yields equations (1) through (3). 
The arrangement of FIG. 1 is not only useful for measuring multiphase fluid 
flow from a well with relatively high gas fraction but the system of FIG. 
1 is also easily adapted to be installed on the gas flowlines from 
multiphase fluid separators, for example. In multiphase fluid handling 
facilities where so-called liquid carry-over into the gas discharge 
flowline from a separator vessel is a frequent occurrence due to 
inadequate separator size, slug flow conditions or the like, the elements 
of a flowmeter, a temperature and pressure sensor and a throttling valve 
are usually present. 
Accordingly, in order to be able to measure both gas and liquid flow, 
all-that is required for such an installation is the additional 
installation of the densitometers 14 and 20 arranged as illustrated in 
FIG. 1 on opposite sides of the flowmeter 16 and the flow restriction 18 
and the addition of a pressure and temperature sensor. Densitometers, such 
as the above-mentioned commercially available gamma ray type, may be 
merely strapped onto the exterior of a conventional, cylindrical pipe 
flowline. The addition of a pressure and temperature sensor to the 
flowline is also normally an easy adaptation. Accordingly, the liquid and 
gas volumetric flow rates and mass flow rates, as well as the liquid and 
gas fractions of the total flow may be easily determined using a flow 
measurement system as shown in FIG. 1 and carrying out the calculations of 
equations (1) through (3) based on the measurements of temperatures and 
pressures on opposite sides of a flow restriction, the total volumetric 
flow rate upstream of a flow restriction and the density of the multiphase 
fluid flowstream on opposite sides of the flow restriction. The method 
described above in conjunction with the system of FIG. 1 is useful for 
fluid flowstreams with relatively high gas to liquid volumetric ratios but 
does not account for gas coming out of solution or liquid "flashing" to 
gas as a result of the pressure or temperature changes. 
FIGS. 2 through 4 illustrate, in schematic form, flow measurement systems 
which may be suitable for many flow measurement applications of multiphase 
fluid flow wherein low cost and easy installation of the system are 
important, the gas fraction may be more or less than twenty five percent 
(25%) of the total fluid flow, on a volumetric basis, and the accuracy of 
the system is a secondary consideration. The systems illustrated in FIGS. 
2 through 4 are capable of providing multiphase fluid flow measurement 
using methodology described herein. The system of FIG. 2 includes the 
densitometers 14 and 20 disposed on opposite sides of a flow restriction 
18 which may be one of several devices as described above in conjunction 
with the system of FIG. 1. Pressures and temperatures are measured in the 
flowline 12 on both sides of the restriction 18 with respect to fluid 
flow. 
FIG. 3 illustrates a system similar to FIG. 2 which includes the 
densitometers 14 and 20 associated with the flow line 12 and a pump 24 
interposed in the flowline between the densitometers. Pressures and 
temperatures are measured on opposite sides of the pump 24 with respect to 
flow through the flowline 12. The "pump" 24 may operate to increase the 
pressure so that pressure P.sub.2 is greater than P.sub.1 or the pump 24 
may comprise an expansion device, such as a turbine, wherein the pressure 
P.sub.2 is lower than P.sub.1. In any case, the method of the invention 
set forth herein is applicable. 
Still further, the flow measurement system of FIG. 4 includes the 
densitometers 14 and 20, the pressure and temperature sensors located 
where indicated and a heat exchanger 26 interposed in the flowline 12 
between the pressure and temperature sensing devices and the 
densitometers. The temperature of the multiphase fluid flowing through the 
flowline 12 and the heat exchanger 26, as well as the pressures, may be 
raised or lowered by suitable heat exchange. Those skilled in the art will 
recognize that the systems illustrated in FIGS. 2 and 4 may be 
characterized by a relatively lengthy section of flowline between the 
densitometers 14 and 20 and the pressure and temperature sensors wherein 
pressure in the flowline 12 will decrease between the densitometers and 
the pressure and temperature sensors due to friction flow losses in the 
flowline or the temperature of the fluid flowing therethrough may decrease 
between measurement points, with the system of FIG. 4, due to heat 
exchange between the fluid in the flowline 12 and the exterior of the 
flowline. For practical purposes, the location of the densitometers 14 and 
20 and the pressure and temperature sensors may require to be close enough 
together that actual flow restriction, pumping devices or actual heat 
exchange devices may require to be used to effect a suitable change in the 
pressures and temperatures of the multiphase fluid. 
The systems of FIGS. 2 through 4 may also be adapted to provide data to an 
on-line computer, not shown, programmed to perform the calculations given 
below. The subscripts 1 and 2 denote the upstream and downstream 
densities, pressures and temperatures measured by the elements indicated 
in the drawing figures. The fluid measurement systems of FIGS. 2 through 4 
are also adapted to determine the flow rates of respective liquids in the 
liquid component of the multiphase fluid flowstream, namely water and oil 
as a mixture of two liquids in the liquid phase. 
The measured data are as follows: 
d.sub.m1, d.sub.m2 =aggregate densities of the flow; given by the 
densitometers 
P.sub.1, P.sub.2 =pressures 
T.sub.1, T.sub.2 =temperatures 
fluid property data 
G.sub.w =water specific gravity; (fresh water=1) 
G.sub.o =oil specific gravity; deg API 
G.sub.g =gas specific gravity; (air=1) 
R.sub.s =solution gas-oil ratio; std. cubic ft. gas dissolved per stock 
tank barrel of oil 
R.sub.sw =solution gas-water ratio; std. cubic ft. gas dissolved per stock 
tank barrel of water 
B.sub.o =oil formation volume factor; volume of oil with dissolved gas per 
stock tank barrel of oil 
B.sub.w =water formation volume factor; volume of water with dissolved gas 
per stock tank barrel of water 
where R.sub.s, R.sub.sw, B.sub.o and B.sub.w are correlations which are 
functions of temperature, pressure and gas and oil gravities. Examples of 
such correlations are given in the publication entitled "Critical and 
Sub-Critical Flow of Multiphase Mixtures Through Chokes" by Perkins, T. 
K., Society of Petroleum Engineers Publication No. SPE 20633, September 
1990. 
Pressure changes in the stream caused by the constriction (or pump) cause 
gas to flash from (or dissolve in) the liquids and to change in volume. 
Equations set forth herein account for these effects in determining the 
fluid fractions f.sub.w1, f.sub.o1 and f.sub.g1 which are, respectively, 
the volumetric water fraction, oil fraction and gas fraction upstream of 
the flow restriction or pump. 
A computer may then determine oil, water and gas flow rates by substituting 
EQU P.sub.1, T.sub.1, P.sub.2, f.sub.w, f.sub.o, f.sub.g, G.sub.w, G.sub.o, 
G.sub.g, R.sub.s, R.sub.sw, B.sub.o, B.sub.w (14) 
into a set of equations describing the physics of flow through the flow 
restriction (or pump), and computing the mass flow rate, W, through it. In 
essence, the flow restriction (pump) provides information about the flow 
which is utilized; it causes measurable changes in the pressure and/or 
temperature and/or density, which uniquely determine W. Fluid volumetric 
flow rates are then computed, knowing W and fluid fractions. 
Gas Densities 
Densities of the gas at standard temperature and pressure (STP) and 
conditions at stations 1 and 2 of the systems of FIGS. 1 through 4 are, 
respectively, 
EQU dgSTP=29.G.sub.g.14.7/(10.73.520) (15) 
EQU dg1=29.G.sub.g.P.sub.1 /(10.73.(460+T.sub.1).z.sub.1) (16) 
EQU dg2=29.G.sub.g.P.sub.2 /(10.73.(460+T.sub.2.z.sub.2) (17) 
Liquid Flow Rates 
Oil volume rates at stations 1 and 2 can be thought of as comprising a 
"dead" oil portion (stock tank barrels per day) and an additional amount 
due to dissolved gas. 
The dead oil portion of the oil rates at stations 1 and are given by 
EQU Q.sub.Do1 =Q.sub.o1 /B.sub.o1 (18) 
EQU Q.sub.Do2 =Q.sub.o2 /B.sub.o2 (19) 
where (Q.sub.o1, Q.sub.o2) are the volumetric flow rates of live oil at 
(1,2) and 
B.sub.o1 =oil formation volume factor; volume of oil (with dissolved gas), 
at [T.sub.1,P.sub.1 ], per stock tank barrel of oil 
B.sub.o2 =oil formation volume factor; volume of oil (with dissolved gas), 
at [T.sub.2,P.sub.2 ], per stock tank barrel of oil 
Conservation of mass requires that 
EQU Q.sub.Do1 =Q.sub.Do2 =Q.sub.Do (2O) 
thus equations (18) and (19) yield 
EQU Q.sub.o2 =Q.sub.o1 (B.sub.o2 /B.sub.o1) (21) 
Similar reasoning for "live" water flow yields 
EQU Q.sub.w2 =Q.sub.w1 (B.sub.w2 /B.sub.w1) (22) 
where 
B.sub.w1 =water formation volume factor; volume of water (with dissolved 
gas), at [T.sub.1 P.sub.1 ], per stock tank barrel of water 
B.sub.w2 water formation volume factor; volume of water (with dissolved 
gas), at [T.sub.2,P.sub.2 ], per stock tank barrel of water 
Gas Flow Rates 
Total gas rate passing station 1, when converted to STP, is 
EQU QTOT.sub.g1STP =Q.sub.g1 (d.sub.g1 /d.sub.gSTP)+Q.sub.Do R.sub.s1 +Q.sub.Dw 
R.sub.sw1 (23) 
in which the first term is the free (undissolved) gas and the second and 
third terms are dissolved gas in the oil and water, respectively. 
Substituting equation (18) and its analog for water into equation (23) 
yields, 
EQU QTOT.sub.g1STP =Q.sub.g1 (d.sub.g1 /d.sub.gSTP)+Q.sub.o1 (R.sub.s1 
/B.sub.o1)+Q.sub.w1 (R.sub.sw1 /B.sub.w1) (24) 
Conservation of mass requires that 
EQU QTOT.sub.g2STP =QTOT.sub.g1STP (25) 
The free gas (undissolved portion) passing station 2 can therefore be 
expressed as 
EQU Q.sub.g2STP =QTOT.sub.g1STP -Q.sub.Do R.sub.s2 -Q.sub.Dw R.sub.sw2(26) 
in which the first term is the total gas and the second and third are the 
gas in solution in the oil and water, respectively. 
Substituting equation (18) and its analog for water, and equation (24) in 
equation (26) and converting to ambient conditions at station 2, yields 
##EQU1## 
Liquid Densities 
Density of dead oil is given by 
EQU d.sub.Do =62.4*141.5/(131.5+G.sub.o) (28) 
where G.sub.o is the API oil gravity. Equation (28) could be altered to 
account for pressure and temperature effects on density. 
At station 1, oil density equals the quotient of mass rate of oil W.sub.o1 
and volume rate of oil Q.sub.o1. 
EQU W.sub.o1 =d.sub.Do (Q.sub.o1 /B.sub.o1)+d.sub.gSTP (Q.sub.o1 R.sub.s1 
/B.sub.o1) (29) 
where the first term is the mass rate of dead oil and the second term is 
the mass rate of the dissolved gas. 
Dividing Q.sub.o1 into equation (29) yields the oil density 
EQU d.sub.o1 =(d.sub.Do +R.sub.s1 d.sub.gSTP)/B.sub.o1 (30) 
Similar reasoning for water density and for densities at station 2 yield, 
EQU d.sub.w1 =(d.sub.Dw +R.sub.sw1 d.sub.gSTP)/B.sub.w1 (31) 
EQU d.sub.o2 =(d.sub.Do +R.sub.s1 d.sub.gSTP)/B.sub.o1 (32) 
EQU d.sub.w2 =(d.sub.Dw +R.sub.sw2 d.sub.gSTP)/B.sub.w2 (33) 
Fluid Fractions at Station 2 
The total flow rate at station 2 is 
EQU QTOT.sub.2 =Q.sub.o2 +Q.sub.w2 +Q.sub.g2 (34) 
Fluid fractions are 
EQU f.sub.o2 =Q.sub.o2 /QTOT.sub.2 (35) 
EQU f.sub.w2 =Q.sub.w2 /QTOT.sub.2 (36) 
EQU f.sub.g2 =Q.sub.g2 /QTOT.sub.2 (37) 
Substituting equations (21), (22) and (27) in (35), (36) and (37) yields 
EQU f.sub.o2 =f.sub.o1 *(a.sub.1 /a.sub.6) (38) 
EQU f.sub.w2 =f.sub.w1 *(a.sub.2 /a.sub.6) (39) 
EQU f.sub.g2 =f.sub.g1 *((a.sub.4 Q.sub.o1 +a.sub.5 Q.sub.w1 +a.sub.3 
Q.sub.g1)/a.sub.6) (40) 
where 
EQU a.sub.6 =f.sub.o1 (a.sub.1 +a.sub.2)+f.sub.w1 (a.sub.2 +a.sub.5)+f.sub.g1 
(a.sub.3) (41) 
EQU a.sub.1 =B.sub.o2 /B.sub.o1 (42) 
EQU a.sub.2 =B.sub.w2 /B.sub.w1 (43) 
EQU a.sub.3 =d.sub.g1 /d.sub.g2 (44) 
EQU a.sub.4 =(d.sub.gSTP /d.sub.g2)(R.sub.s1 -R.sub.s2)/B.sub.o1(45) 
EQU a.sub.5 =(d.sub.gSTP /d.sub.g2)(R.sub.sw1 -R.sub.sw2)/B.sub.w1(46) 
Solution For Two Densitometers 
Three equations in terms of three unknowns, the volumetric fractions of 
oil, water and gas (f.sub.o1,f.sub.w1,f.sub.g1) are solved. These are 
EQU f.sub.o1 +f.sub.w1 +f.sub.g1 =1 (47) 
EQU d.sub.o1 f.sub.o1 +d.sub.w1 f.sub.w1 +d.sub.g1 f.sub.g1 =d.sub.m1(48) 
EQU d.sub.o2 f.sub.o2 +d.sub.w2 f.sub.w2 +d.sub.g2 f.sub.g2 =d.sub.m2(49) 
the first stating that the sum of fractions must equal unity, the second 
and third that the aggregate densities (d.sub.m1, d.sub.m2) of the fluid 
mixture at stations 1 and 2 equal the sum of component densities weighted 
by their respective fractions. Converting (f.sub.o2,f.sub.w2,f.sub.g2) in 
equation (49) to their equivalents in terms of 
(f.sub.o1,f.sub.w1,f.sub.g1); equations (38) through (46), a solution for 
the three fractions is obtained. 
The fluid fractions so determined may be input to a physical model of the 
constriction or pump, together with properties, pressures and 
temperatures, to determine the total mass flow rate, WTOT, through it. The 
equations in the Perkins reference are such a model, suitable for oil well 
chokes. Other possibilities include, but are not restricted to, equations 
for an orifice, venturi or pump. 
Since WTOT is the product of aggregate flow rate QTOT.sub.1 and measured 
density d.sub.m1 
EQU WTOT=QTOT.sub.1 * d.sub.m1 (50) 
the solution for flow rates of individual components is 
EQU Q.sub.o1 =(d.sub.m1 /WTOT)f.sub.o1 (51) 
EQU Q.sub.w1 =(d.sub.m1 /WTOT)f.sub.w1 (52) 
EQU Q.sub.g1 =(d.sub.m1 /WTOT)f.sub.g1 (53) 
Alternatively to obtaining WTOT from a physical model, measuring QTOT.sub.1 
via a volumetric flowmeter will provide values of QTOT.sub.1 in equation 
(50). 
Solution for Heavy Oil 
When oil and water densities are close or equal, the second densitometer, 
represented by equation (49), will not be needed. Equations (47) and (48) 
can be written 
EQU f.sub.11 +f.sub.g1 =1 (54) 
EQU d.sub.11 f.sub.11 +d.sub.g1 f.sub.g =d.sub.m1 (55) 
where oil and water terms have been replaced by terms representing the 
liquid mixture (subscript 1). Equations (50) and (51) are solved for the 
two unknowns f.sub.11 and f.sub.g1. 
Proceeding analogously to the case for two densitometers, either WTOT or 
QTOT.sub.1 is measured and the solution for flow rates is 
EQU Q.sub.11 =(d.sub.m1 /WTOT)f.sub.11 (56) 
EQU Q.sub.g1 =(d.sub.m1 /WTOT)f.sub.g1 (57) 
The total mass flow rate W through a flow restriction, such as a choke, can 
be computed from the equations given in the aforementioned Perkins 
reference. Alternatively, if total flow rate is measured by a volumetric 
flowmeter to give QTOT, then fluid flow rates of the components of the 
multiphase fluid mixture are computed knowing the total volumetric flow 
rate and the fluid fractions. Moreover, if fluid densities are well known 
and the flow is not in the slug regime, accuracies of the systems 
described and illustrated in conjunction with FIGS. 2 through 4 may be at 
least ninety percent (90%), particularly for gas fractions of twenty five 
percent (25%) or more. 
The present invention provides a relatively inexpensive system and method 
for monitoring multiphase fluid flow such as flow from oil and gas 
production wells, process flowstreams and the like. The systems may be 
readily adapted to existing facilities without major modification to the 
flowlines for process conditions and the total cost of the system 
components is expected to be considerably less than more complicated 
multiphase flow measurement systems in the prior art. When measuring 
mixtures of oil and water, if the densities of these fluids are nearly 
equal, such as in the production of heavy oil, then the equations become 
error prone. In such case, the equations can be recast in terms of liquid 
densities rather than separate oil density and water density and the 
downstream densitometer would not be required. 
Although preferred embodiments of a flow measurement system and methods 
associated therewith have been described in detail herein, those skilled 
in the art will recognize that certain substitutions and modifications may 
be made without departing from the scope and spirit of the appended claims 
.