U.S. patent application number 12/137756 was filed with the patent office on 2009-12-17 for evaluating multiphase fluid flow in a wellbore using temperature and pressure measurements.
This patent application is currently assigned to SCHLUMBERGER TECHNOLOGY CORPORATION. Invention is credited to Joerg Henning Meyer, Bobby Dale Poe, JR..
Application Number | 20090308601 12/137756 |
Document ID | / |
Family ID | 41413713 |
Filed Date | 2009-12-17 |
United States Patent
Application |
20090308601 |
Kind Code |
A1 |
Poe, JR.; Bobby Dale ; et
al. |
December 17, 2009 |
EVALUATING MULTIPHASE FLUID FLOW IN A WELLBORE USING TEMPERATURE
AND PRESSURE MEASUREMENTS
Abstract
A system, method and program product for analyzing multiphase
flow in a wellbore. A system is provided that includes: an input
system for receiving pressure and temperature readings from a pair
of sensors located in the wellbore; a computation system that
utilizes a flow analysis model to generate a set of wellbore fluid
properties, wherein the set of wellbore fluid properties includes
at least one of: a fluid mixture value, a phase velocity value, a
flow rate, a mixture density, a mixture viscosity, a fluid holdup,
and a slip velocity; and a system for outputting the wellbore fluid
properties.
Inventors: |
Poe, JR.; Bobby Dale;
(Houston, TX) ; Meyer; Joerg Henning; (Houston,
TX) |
Correspondence
Address: |
Schlumberger Technology Corporation
P. O. Box 425045
Cambridge
MA
02142
US
|
Assignee: |
SCHLUMBERGER TECHNOLOGY
CORPORATION
Cambridge
MA
|
Family ID: |
41413713 |
Appl. No.: |
12/137756 |
Filed: |
June 12, 2008 |
Current U.S.
Class: |
166/250.01 ;
703/10 |
Current CPC
Class: |
E21B 47/06 20130101;
E21B 47/10 20130101 |
Class at
Publication: |
166/250.01 ;
703/10 |
International
Class: |
E21B 47/06 20060101
E21B047/06 |
Claims
1. A system for analyzing multiphase flow in a wellbore,
comprising: an input system for receiving pressure and temperature
readings from a pair of sensors located in the wellbore; a
computation system that utilizes a flow analysis model to generate
a set of wellbore fluid properties from the pressure and
temperature readings, wherein the set of wellbore fluid properties
includes at least one of: a fluid mixture value, a phase velocity
value, a flow rate, a mixture density, a mixture viscosity, a fluid
holdup, and a slip velocity; and a system for outputting the
wellbore fluid properties.
2. The system of claim 1, wherein the pair of sensors are
permanently mounted in the wellbore.
3. The system of claim 1, further comprising a contribution
analysis system for analyzing pressure and temperature from a
plurality of sensor pairs to provide wellbore fluid properties for
different inflow zones and/or branches of a multiple production
zone well.
4. The system of claim 1, wherein the flow analysis model includes
a single-phase flow model that utilizes three equations to solve
for a fluid superficial velocity (V.sub.si), a Reynolds number
(N.sub.Re), and a Fanning friction factor (f).
5. The system of claim 1, wherein the flow analysis model includes
a two-phase oil-water flow model that utilizes 10 equations to
solve for: oil and water holdups, oil, water, and mixture
superficial velocities, mixture density and dynamic viscosity, a
slip velocity between oil and water phases, a Reynolds number and
Fanning friction factor, and a pressure loss that occurs over a
metering length of a flow conduit.
6. The system of claim 1, wherein the flow analysis model includes
a two-phase gas-oil flow model that utilizes 10 equations to solve
for: oil and gas holdups, superficial velocities, a mixture
superficial velocity, density and viscosity, a gas-liquid slip
velocity, a Reynolds number and a Fanning friction factor.
7. The system of claim 1, wherein the flow analysis model includes
a two-phase water-gas flow model that utilizes 10 equations to
solve for: water and gas holdups, water and gas superficial
velocities, a mixture superficial velocity, density and dynamic
viscosity, a gas-liquid slip velocity, a Reynolds number and a
Fanning friction factor.
8. The system of claim 1, wherein the flow analysis model includes
a three-phase flow model that utilizes 13 equations to solve for:
holdups for oil, water and gas phases, superficial velocities of
oil, water and gas, a mixture superficial velocity, a mixture
density, a viscosity, a Reynolds number, a friction factor, a
water-oil slip velocity and a gas-liquid slip velocity.
9. The system of claim 1, wherein said wellbore is used to produce
formation fluids or to inject fluids into a formation.
10. The system of claim 1, wherein said computation system
calibrates one or more of said wellbore fluid properties using data
obtained from a retrievable production logging device.
11. A method for analyzing multiphase flow in a wellbore,
comprising: obtaining pressure and temperature readings from a pair
of sensors located in the wellbore; utilizing a flow analysis model
to generate a set of wellbore fluid properties from the pressure
and temperature readings, wherein the set of wellbore fluid
properties includes at least one of: a fluid mixture value, a phase
velocity value, a flow rate, a mixture density, a mixture
viscosity, a fluid holdup, and a slip velocity; and outputting the
wellbore fluid properties.
12. The method of claim 11, wherein the pair of sensors are
permanently mounted in the wellbore.
13. The method of claim 11, further comprising analyzing pressure
and temperature from a plurality of sensor pairs to provide
wellbore fluid properties for different inflow zones and/or
branches of a multiple production zone well.
14. The method of claim 11, wherein the flow analysis model
includes a single-phase flow model that utilizes three equations to
solve for a fluid superficial velocity, a Reynolds number, and a
Fanning friction factor.
15. The method of claim 11, wherein the flow analysis model
includes a two-phase oil-water flow model that utilizes 10
equations to solve for: oil and water holdups, oil, water, and
mixture superficial velocities, mixture density and dynamic
viscosity, a slip velocity between oil and water phases, a Reynolds
number and Fanning friction factor, and a pressure loss that occurs
over a metering length of a flow conduit.
16. The method of claim 11, wherein the flow analysis model
includes a two-phase gas-oil flow model that utilizes 10 equations
to solve for: oil and gas holdups, superficial velocities, a
mixture superficial velocity, density and viscosity, a gas-liquid
slip velocity, a Reynolds number and a Fanning friction factor.
17. The method of claim 11, wherein the flow analysis model
includes a two-phase water-gas flow model that utilizes 10
equations to solve for: water and gas holdups, water and gas
superficial velocities, a mixture superficial velocity, density and
dynamic viscosity, a gas-liquid slip velocity, a Reynolds number
and a Fanning friction factor.
18. The method of claim 11, wherein the flow analysis model
includes a three-phase flow model that utilizes 13 equations to
solve for: holdups for oil, water and gas phases, superficial
velocities of oil, water and gas, a mixture superficial velocity, a
mixture density, a viscosity, a Reynolds number, a friction factor,
a water-oil slip velocity and a gas-liquid slip velocity.
19. The method of claim 11, wherein said wellbore is used to
produce formation fluids or to inject fluids into a formation.
20. The system of claim 11, wherein data obtained from a
retrievable production logging device is used to calibrate one or
more of said wellbore fluid properties.
21. A computer readable medium for storing a computer program
product, which when executed by a computer system analyzes
multiphase flow in a wellbore, comprising: program code for
inputting pressure and temperature readings from a pair of sensors
located in the wellbore; program code for implementing a flow
analysis model to generate a set of wellbore fluid properties from
the pressure and temperature readings, wherein the set of wellbore
fluid properties includes at least one of: a fluid mixture value, a
phase velocity value, a flow rate, a mixture density, a mixture
viscosity, a fluid holdup, and a slip velocity; and program code
for outputting the wellbore fluid properties.
22. The computer readable medium of claim 21, wherein the wellbore
fluid properties are outputted in an on-demand manner.
23. The computer readable medium of claim 21, wherein the pair of
sensors are permanently mounted in the wellbore.
24. The computer readable medium of claim 21, further comprising
program code for analyzing pressure and temperature from a
plurality of sensor pairs to provide wellbore fluid properties for
different branches of a multiple production zone well.
25. The computer readable medium of claim 21, wherein the flow
analysis model includes a single-phase flow model that utilizes
three equations to solve for a fluid superficial velocity, a
Reynolds number, and a Fanning friction factor.
26. The computer readable medium of claim 21, wherein the flow
analysis model includes a two-phase oil-water flow model that
utilizes 10 equations to solve for: oil and water holdups, oil,
water, and mixture superficial velocities, mixture density and
dynamic viscosity, a slip velocity between oil and water phases, a
Reynolds number and Fanning friction factor, and a pressure loss
that occurs over a metering length of a flow conduit.
27. The computer readable medium of claim 21, wherein the flow
analysis model includes a two-phase gas-oil flow model that
utilizes 10 equations to solve for: oil and gas holdups,
superficial velocities, a mixture superficial velocity, density and
viscosity, a gas-liquid slip velocity, a Reynolds number and a
Fanning friction factor.
28. The computer readable medium of claim 21, wherein the flow
analysis model includes a two-phase water-gas flow model that
utilizes 10 equations to solve for: water and gas holdups, water
and gas superficial velocities, a mixture superficial velocity,
density and dynamic viscosity, a gas-liquid slip velocity, a
Reynolds number and a Fanning friction factor.
29. The computer readable medium of claim 21, wherein the flow
analysis model includes a three-phase flow model that utilizes 13
equations to solve for: holdups for oil, water and gas phases,
superficial velocities of oil, water and gas, a mixture superficial
velocity, a mixture density, a viscosity, a Reynolds number, a
friction factor, a water-oil slip velocity and a gas-liquid slip
velocity.
30. The computer readable medium of claim 21, further including
program code for calibrating one or more of said wellbore fluid
properties using data obtained from a retrievable production
logging device.
Description
FIELD OF THE INVENTION
[0001] This disclosure relates to evaluating fluid flow in an oil
or gas well, and more particularly relates to a system and method
of evaluating multiphase fluid flow in a wellbore using temperature
and pressure measurements.
BACKGROUND OF THE INVENTION
[0002] Reliable and accurate downhole temperature and pressure
measurements have been available in the petroleum industry for the
past several years. Permanent downhole pressure monitoring
equipment has now been installed in a number of producing basins
around the world, with successful measurement operations exceeding
five or more years at this time. Downhole permanent temperature
measurements have also become more common, with both conventional
or fiber optic thermal measurements currently available for most
reservoir conditions. While continuous pressure and temperature
readings provide an important part of understanding oil and gas
production, quantitative information must typically be obtained
using other types of data.
[0003] For example, the quantitative evaluation of the production
or injection profile in an oil and/or gas well has traditionally
involved the use of production log measurements of flow rate,
pressure, density, and fluid holdup to derive estimates of the
wellbore fluid mixture phase velocities, densities, pressure
distributions, and completed interval inflow or outflow
contributions. Modern production logs can be used in many
situations to obtain the necessary measurements that are required
to perform these quantitative computations. The measurements made
in these cases however are periodic and reflect the wellbore fluid
inflows/outflows at the time that the production log was run.
Unfortunately, the known art does not provide a solution to obtain
continuous or real-time quantitative measurements and evaluations
using downhole pressure and temperature readings obtained from a
plurality of sensors in the wellbore.
SUMMARY OF THE INVENTION
[0004] The present invention relates to a system, method and
program product that provides a computational model and evaluation
technique for using array pressure and temperature measurements
obtained in a flow conduit to evaluate the phase flow rates and
velocities, fluid phase holdup, slip velocities between fluid
phases, and mixture density and viscosity. These values can then be
used, for instance, to quantify the inflow and outflow
contributions of completed zones in a wellbore.
[0005] In one embodiment, there is a system for analyzing
multiphase flow in a wellbore, comprising: an input system for
receiving pressure and temperature readings from a pair of sensors
located in the wellbore; a computation system that utilizes a flow
analysis model to generate a set of wellbore fluid properties from
the pressure and temperature readings, wherein the set of wellbore
fluid properties includes at least one of: a fluid mixture value, a
phase velocity value, a flow rate, a mixture density, a mixture
viscosity, a fluid holdup, and a slip velocity; and a system for
outputting the wellbore fluid properties.
[0006] In a second embodiment, there is a method for analyzing
multiphase flow in a wellbore, comprising: obtaining pressure and
temperature readings from a pair of sensors located in the
wellbore; utilizing a flow analysis model to generate a set of
wellbore fluid properties from the pressure and temperature
readings, wherein the set of wellbore fluid properties includes at
least one of: a fluid mixture value, a phase velocity value, a flow
rate, a mixture density, a mixture viscosity, a fluid holdup, and a
slip velocity; and outputting the wellbore fluid properties.
[0007] In a third embodiment, there is a computer readable medium
for storing a computer program product, which when executed by a
computer system analyzes multiphase flow in a wellbore, comprising:
program code for inputting pressure and temperature readings from a
pair of sensors located in the wellbore; program code for
implementing a flow analysis model to generate a set of wellbore
fluid properties from the pressure and temperature readings,
wherein the set of wellbore fluid properties includes at least one
of: a fluid mixture value, a phase velocity value, a flow rate, a
mixture density, a mixture viscosity, a fluid holdup, and a slip
velocity; and program code for outputting the wellbore fluid
properties.
[0008] An advantage of this invention is the implementation of a
quantitative evaluation methodology for characterizing the
temperature, pressure, wellbore fluid mixture density and
viscosity, and fluid holdup distributions in a wellbore using the
temperature and pressure distributions in the well. This is
achieved by the development and use of a comprehensive multiphase
capillary flow analysis model. The results provide a reliable,
accurate, and continuous characterization of the wellbore fluid
flow properties such as pressure, temperature, mixture density,
mixture viscosity, fluid phase holdup distributions, and completed
zone inflow/outflow contributions.
[0009] This invention is directly applicable in wellbore
environments and conditions in which modern production logging
techniques may not be readily accessible or may not be deployable
as a result of the wellbore geometry, well depth, water depth, or
other operational and economical considerations.
[0010] The illustrative aspects of the present invention are
designed to solve the problems herein described and other problems
not discussed.
BRIEF DESCRIPTION OF THE DRAWINGS
[0011] These and other features of this invention will be more
readily understood from the following detailed description of the
various aspects of the invention taken in conjunction with the
accompanying drawings.
[0012] FIG. 1 depicts a computer system having a multiphase flow
analysis system in accordance with an embodiment of the present
invention.
[0013] FIG. 2 depicts an embodiment of a multiphase flow analysis
system that provides a contribution analysis in accordance with an
embodiment of the present invention.
[0014] FIG. 3 depicts a graph showing a Fanning friction factor
correlated with the Reynolds number and relative roughness for
single-phase flow systems.
[0015] FIG. 4 depicts a graph showing a friction factor showing the
transition between the laminar and turbulent flow regimes in
multiphase flow systems.
[0016] The drawings are merely schematic representations, not
intended to portray specific parameters of the invention. The
drawings are intended to depict only typical embodiments of the
invention, and therefore should not be considered as limiting the
scope of the invention. In the drawings, like numbering represents
like elements.
DETAILED DESCRIPTION OF THE INVENTION
[0017] Referring to the drawings, FIG. 1 depicts an overview of an
illustrative system 11 for implementing aspects of the present
invention. As shown, a computer system 10 is provided that includes
a multiphase flow analysis system 18 for analyzing fluid
characteristics flowing through a wellbore 34. Also provided are at
least two sensors 30, 32 placed within the wellbore to provide
multipoint pressure and temperature readings.
[0018] Multiphase flow analysis system 18 includes a pressure and
temperature input system 20 for obtaining pressure and temperature
readings from each sensor 30, 32 in a continuous, as needed, or
periodic manner. Also included is a computation system 22 that
utilizes a flow analysis model 24 for computing wellbore fluid
properties, including one or more of: (1) the fluid mixture; (2)
phase velocities; (3) flow rates; (4) mixture density; (5) mixture
viscosity; (6) fluid holdups; and (7) estimates of the slip
velocities between the wellbore liquid and gas phases and between
the oil and water phases, if those phases are present in the
system. Wellbore fluid properties 28 may be computed and outputted
by output system 29 in an "on-demand" manner, i.e., continuously,
as needed, periodically, in real-time, etc. It is also possible to
output the wellbore fluid properties when pre-selected system
conditions are reached, such as anomalous incidents or trends,
conditions exceeding thresholds, etc. A description of the flow
analysis model 24 and how the computations may be implemented is
provided below.
[0019] Also included in multiphase flow analysis system 18 is a
contribution analysis system 26 to quantitatively evaluate an oil
or gas well with multiple production or injection zones. For
example, FIG. 2 depicts a well 50 having multiple production zones
that include a main branch 42, a first contribution branch 44, and
a second contribution branch 46. In this case, multipoint
measurements are obtained with sets of sensors configured into a
multipoint measurements array. In particular, the complex
multi-branched well 50 is fitted with three sets of sensors (A, B,
and C). Each set is strategically located to obtain contribution
readings from each different zone in the well. Namely, sensor set C
is located to obtain readings for main branch 42; sensor set B is
located to obtain contribution readings from main branch 42 and
first contribution branch 44; and sensor set A is located to obtain
contribution readings from main branch 42, first contribution
branch 44, and second contribution branch 46.
[0020] A contribution analysis 40 may be obtained for each
contribution branch 44, 46 by subtracting all the downstream fluid
property computations. For instance, by subtracting computation
values obtained from sensor set C from computation values obtained
from sensor set B, a contribution analysis 40 for the first
contribution branch 44 can be obtained. Similarly, by subtracting
computation values obtained from sensor sets B and C from
computation values obtained from sensor set A, a contribution
analysis 40 for the second contribution branch 46 can be obtained.
Contribution analysis 40 for main branch 42 is simply obtained from
sensor set C, which has no additional downstream contributions.
[0021] Note that each sensor set A, B, C may include more than two
sensors in order to provide redundancy. In this example, each set
is shown including four sensors, e.g., set A includes sensors A1,
A2, A3, and A4. This thus allows six different sensor pairs (e.g.,
A1-A2, A1-A3, A1-A4, A2-A3, A2-A4, A3-A4) to be used as a basis
calculating wellbore fluid properties. Any one or more of the
sensor pairs may be used for evaluation purposes. While FIG. 2
depicts a well that has a main branch and first and second
contribution branches, embodiments of the invention may also be
used with a wellbore having only a main branch with different
inflow or outflow zones, such as a cased well having separate
perforated intervals.
[0022] As noted in FIG. 1, computation system 22 provides a flow
analysis model 24 for generating wellbore fluid properties 28 for a
sensor pair 30, 32. An explanation for how such properties may be
obtained from temperature and pressure readings from sensor pair
30, 32 begins with a review of the fundamental governing
relationships that pertain to multiphase fluid flow in a tubular
conduit (e.g., a wellbore). The following notation is used
throughout the discussion.
Variable Description
[0023] D Flow conduit inside diameter [0024] f Fanning friction
factor [0025] f.sub.o Fraction of oil in liquid component of the
system [0026] f.sub.w Fraction of water in liquid component of the
system [0027] g Gravitational acceleration [0028] H.sub.L Liquid
holdup [0029] h.sub.L Elevation at end of flow conduit segment
[0030] h.sub.0 Elevation at start of flow conduit segment [0031] L
Measured length of the flow conduit segment [0032] N.sub.Re
Reynolds number [0033] P.sub.L Pressure at outlet end of flow
conduit segment [0034] P.sub.0 Pressure at start end of flow
conduit segment [0035] q.sub.g Insitu gas volumetric flow rate
[0036] q.sub.L Insitu liquid volumetric flow rate [0037] V Fluid
average velocity in circular conduit [0038] V.sub.m Wellbore fluid
mixture superficial velocity, ft/s [0039] V.sub.sg Gas superficial
velocity, ft/s [0040] V.sub.sgL Gas-liquid slip velocity, ft/s
[0041] V.sub.sL Liquid superficial velocity, ft/s [0042] V.sub.so
Oil superficial velocity, ft/s [0043] V.sub.sow Oil-water slip
velocity, ft/s [0044] V.sub.sw Water superficial velocity, ft/s
[0045] Y.sub.g Gas holdup [0046] Y.sub.L Liquid holdup [0047]
Y.sub.o Oil holdup [0048] Y.sub.w Water holdup
Greek Description
[0048] [0049] .alpha. Wellbore deviation angle from vertical, deg
[0050] .epsilon. Pipe roughness [0051] .lamda..sub.L No-slip liquid
holdup [0052] .mu..sub.g Gas dynamic viscosity, cp [0053]
.mu..sub.L Liquid dynamic viscosity, cp [0054] .mu..sub.m Fluid
mixture dynamic viscosity, cp [0055] .mu..sub.o Oil dynamic
viscosity, cp [0056] .mu..sub.w Water dynamic viscosity, cp [0057]
.upsilon..sub.m Fluid mixture kinematic viscosity, cp-cu ft/lbs
[0058] .rho..sub.g Gas density, lbs/cu ft [0059] .rho..sub.L Liquid
density, lbs/cu ft [0060] .rho..sub.m Fluid mixture density, lbs/cu
ft [0061] .rho..sub.o Oil density, lbs/cu ft [0062] .rho..sub.osg
Oil density, g/cc [0063] .rho..sub.w Water density, lbs/cu ft
[0064] .rho..sub.wsg Water density, g/cc
[0065] One of the fundamental parameters that can be used to
quantify and correlate the level of inertial to viscous forces in a
fluid flowing in a circular conduit is the Reynolds number. This
dimensionless parameter is defined in Eq. 1.
N Re = DV m .upsilon. m ( 1 ) ##EQU00001##
The kinematic viscosity of a fluid mixture appearing in Eq. 1 is
defined as the ratio of the dynamic fluid viscosity to its density.
This relationship is expressed mathematically in Eq. 2.
.upsilon. m = .mu. m .rho. m ( 2 ) ##EQU00002##
The general relationship that describes the pressure loss exhibited
due to fluid flow in a circular tubular conduit is given by
Fanning's equation. Note that gravitational effects have been
included in this expression.
P 0 - P L + .rho. m g ( h 0 - h L ) L = 2 .rho. m fV m 2 D ( 3 )
##EQU00003##
[0066] Substitution of Eqs. 2 and 3 into Eq. 1 results in
expression that can be used to correlate the Reynolds number and
friction factor to the conduit dimensions, the pressure loss over a
given length of conduit, and the physical properties of the fluid
flowing in the conduit. Note that the relationship given in Eq. 4
is explicitly independent of the fluid velocity, except that the
effect of this parameter is implicitly manifested in the fluid flow
problem in the form of the Reynolds number.
N Re f = D 3 2 .rho. m 1 2 [ P 0 - P L + .rho. m g ( h 0 - h L ) ]
1 2 2 L .mu. m = D 3 2 [ P 0 - P L + .rho. m g ( h 0 - h L ) ] 1 2
2 L .rho. m .upsilon. m ( 4 ) ##EQU00004##
[0067] The Fanning friction factor encountered in Eqs. 3 and 4 is a
function of the Reynolds number and the relative roughness of the
conduit in which the flow occurs. The Fanning friction factor is
correlated with the Reynolds number and relative roughness as
presented in FIG. 3. This friction factor correlation is generally
considered to be applicable to single-phase fluid flow. Note that
there is an unstable transition regime for Reynolds numbers in the
range of 2,000 to 3,000. For Reynolds numbers below about 2,000,
laminar flow conditions exist. For Reynolds numbers greater than
approximately 3,000, turbulent flow conditions are generally
considered to prevail.
[0068] Based upon gas-liquid experimental data, the friction factor
that is applicable for multiphase flow generally tends to have a
smooth, continuous transition between the laminar and turbulent
flow regimes. This transition regime behavior is depicted in FIG.
4. This transition regime behavior was found to be valid for the
typical range of pipe relative roughness values that are commonly
found in commercially available oilfield tubular goods
( D .ltoreq. 0.004 ) . ##EQU00005##
[0069] Note that in this case, the transition regime is a smooth
transition that deviates from that of laminar flow at a Reynolds
number of approximately 1,000, characterized by the
inertial-turbulent friction factor values at higher Reynolds
numbers.
[0070] The Fanning friction factor that corresponds to the laminar
flow regime in FIGS. 3 and 4 (N.sub.Re<2000 and
N.sub.Re<1000, respectively) can be described mathematically
with the relationship given in Eq. 5.
f = 16 N Re ( 5 ) ##EQU00006##
The Fanning friction factor that corresponds to the turbulent flow
regime (N.sub.Re>3,000) can be accurately evaluated using the
relationship described in Colebrook, C. F.: "Turbulent Flow in
Pipes, with Particular Reference to the Transition Region Between
the Smooth and Rough Pipe Laws," J. Inst. Civil Engs., London,
(1938-1939).
[0071] The evaluation of this expression requires an iterative
numerical solution procedure and is presented in Eq. 6.
1 f = - 4 log ( 0.269 D + 1.255 N Re f ) ( 6 ) ##EQU00007##
[0072] In addition to the governing fluid flow relationships
presented thus far, a conservation of mass relationship for the
fluids present in the system can also be defined. The fluid mixture
density is generally computed in multiphase flow analyses in the
manner depicted in Eq. 7. However, an alternate form of this
relationship for flow in horizontal circular conduits is described
in Dukler, A. E.: "Gas-Liquid Flow in Pipelines," AGA, API, Vol. I,
Research Results, (May 1969). That expression is given in Eq.
8.
.rho. m = .rho. L Y L + .rho. g ( 1 - Y L ) ( 7 ) .rho. m = .rho. L
( .lamda. L 2 Y L ) + .rho. g ( ( 1 - .lamda. L ) 2 1 - Y L ) ( 8 )
##EQU00008##
[0073] The no-slip liquid holdup is utilized in Dukler's alternate
fluid mixture density relationship. The no-slip liquid holdup is
defined in Eq. 9.
.lamda. L = q L q L + q g ( 9 ) ##EQU00009##
[0074] In a similar manner, the fluid mixture dynamic viscosity can
be evaluated by various means. Hagedorn, A. R. and Brown, K. E.:
"The Effect of Liquid Viscosity in Vertical Two-Phase Flow," JPT,
(Feb. 1964), 203, suggested that the fluid mixture viscosity in a
multiphase flow system should be evaluated in the manner given by
Eq. 10.
.mu. m = .mu. L Y L .mu. g 1 - Y L ( 10 ) ##EQU00010##
[0075] The fluid mixture dynamic viscosity has been more commonly
estimated in previous investigations of multiphase fluid flow using
a holdup-weighted combination of the liquid and gas viscosities,
given by Eq. 11.
.mu..sub.m=.mu..sub.LY.sub.L+.mu..sub.g(1-Y.sub.L) (11)
[0076] A relationship for the fluid mixture dynamic viscosity that
is identical to that given in Eq. 11 has been proposed, except that
the no-slip liquid holdup is the weighting parameter used in those
analyses rather than the slip-adjusted liquid holdup.
.mu..sub.m=.mu..sub.L.lamda..sub.L+.mu..sub.g(1-.lamda..sub.L)
(12)
[0077] Where required, the kinematic viscosity (Eq. 3) can be
evaluated using the fluid mixture density obtained with Eqs. 7 or 8
and dynamic fluid mixture viscosity evaluated with Eqs. 10, 11, or
12. An alternative approach is to evaluate the kinematic viscosity
of the fluid mixture in a manner analogous to that used for the
holdup-weighted mixture density and viscosity. This expression is
given in Eq. 13.
.upsilon. m = .mu. L .rho. L Y L + .mu. g .rho. g ( 1 - Y L ) ( 13
) ##EQU00011##
[0078] Regardless of the particular fluid mixture density and
viscosity relationship used in the analysis, most of the previous
investigations of multiphase flow in pipe have evaluated the liquid
density and dynamic viscosity of oil and water mixtures using the
relationships given in Eqs. 14 and 15. Note that other mixture
viscosity models may be used in the analysis such a medium emulsion
model for oil-water vertical flow systems.
.rho..sub.L=.rho..sub.of.sub.o+.rho..sub.wf.sub.w (14)
.mu..sub.L=.mu..sub.of.sub.o+.mu..sub.wf.sub.w (15)
[0079] Typically when these relationships for computing the liquid
density and dynamic viscosity of oil-water systems are used, the
fraction of oil and water are often evaluated assuming no-slip
conditions. However, a similar analysis could also be performed
using an appropriate slip relationship between the water and the
less dense oil phase in the system without any loss in generality.
In addition, the slip velocity relationship between the oil and
water phases in a two-phase liquid flow system can be reliably
determined using Eq. 16. An illustrative embodiment provided herein
utilizes this relationship (Eq. 16) for the oil-water slip velocity
for wellbore inclinations up to about 70 degrees, but the invention
may also use other applicable oil-water slip velocity correlations.
This disclosure includes but is not limited to the use of only a
single oil-water slip velocity relationship in the invention.
V sow = 0.6569 ( .rho. wsg - .rho. osg ) 0.25 exp [ 0.788 ln ( 1.85
.rho. wsg - .rho. osg ) ( 1 - Y w Y L ) ] ( 1 + 0.04 .alpha. ) ( 16
) ##EQU00012##
[0080] The fundamental definition of the slip velocity between the
oil and water phases in a two-phase oil-water system is given by
Eq. 17. It is noted that the slip velocity between the oil and
water phases is simply the difference between the average
velocities of the oil and water phases. Note that when the
definition of the slip velocity between the oil and water phases
(two-phase relationship) is applied to a three-phase analysis (as
is considered in this invention), the holdups of the oil and water
phases must be normalized by the liquid holdup in the three-phase
system. This normalization of the oil and water phase holdups to
the liquid holdup (oil+water) in a three-phase system is presented
in Eq. 17.
V sow = V so Y L Y o - V sw Y L Y w ( 17 ) ##EQU00013##
[0081] A similar slip velocity relationship exists between the gas
and liquid phases in a multiphase system. An accurate and reliable
correlation for estimating the slip velocity between the gas and
liquid phases in a multiphase system is given in Eq. 18. Other
gas-liquid slip velocity relationships may also be used in the
computational analysis described in this invention disclosure.
While the gas-liquid slip velocity relationship given in Eq. 18
provides an illustrative embodiment, the use of other applicable
gas-liquid slip velocity correlations may also be utilized and fall
within the scope of this invention.
V.sub.sgL=[(0.95-Y.sub.g.sup.2).sup.0.5+0.025](1+0.04.alpha.)
(18)
The fundamental definition of the slip velocity relationship
between the gas and liquid phases in a multiphase system is given
by Eq. 19.
V sgL = V sg Y g - V sL Y L ( 19 ) ##EQU00014##
The liquid mixture superficial velocity in a multiphase system is
the sum of the oil and water superficial velocities.
V.sub.sL=V.sub.so+V.sub.sw (20)
The sum of the holdups of each of the fluid phases must total to 1,
the sum of all of the fluids in the system.
1=Y.sub.o+Y.sub.w+Y.sub.g=Y.sub.L+Y.sub.g (21)
The wellbore mixture fluid superficial velocity is the sum of the
superficial velocities of each of the fluid phases present in the
system.
V.sub.m=V.sub.so+V.sub.sw+V.sub.sg=V.sub.sL+V.sub.sg (22)
The wellbore fluid mixture kinematic viscosity can be evaluated as
the sum of the kinematic viscosities of each of the fluid phases
and their associated fluid holdups.
.upsilon. m = .mu. o Y o + .mu. w Y w + .mu. g Y g .rho. o Y o +
.rho. w Y w + .rho. g Y g ( 23 ) ##EQU00015##
A final governing relationship that may be utilized to resolve the
unknowns in the fluid flow problem is an expression relating the
insitu mixture density directly to the measured pressure and
temperature, and the composition of the fluids in the system. This
relationship can be an equation-of-state, such as the model
proposed by Peng and Robinson. Other equations-of state can also be
used to evaluate the mixture density and fluid mixture viscosity at
the insitu conditions of temperature and pressure, for a given
composition of wellbore fluid.
Evaluation of Single-Phase Flow Metering Parameters
[0082] In single-phase flow metering cases, the evaluation of the
fluid flow parameters involves the solution of three equations for
three unknown parameter values in the problem. In single-phase
flow, the unknown parameters that must be determined in the
analysis are the fluid superficial velocity (V.sub.si), the
Reynolds number (N.sub.Re), and the Fanning friction factor (f).
The i subscript appearing on the phase superficial velocity and
fluid properties in Eq. A-1 represents the individual fluid phase
(oil, gas, or water: i.e. o, g, or w). The definition of the
single-phase Reynolds number in conventional oilfield units is
given in Eq. A-1.
N Re = 124 D V si .gamma. i = 124 D .rho. i Vsi .mu. i ( A - 1 )
##EQU00016##
[0083] The Fanning friction factor for single-phase flow conditions
is determined from FIG. 3. The Fanning friction is a function of
the Reynolds number and the relative pipe roughness.
[0084] For laminar flow conditions (N.sub.Re<2,000), the Fanning
friction factor given in FIG. 3 is defined by the relationship
given in Eq. A-2.
f = 16 N Re ( A - 2 ) ##EQU00017##
Under turbulent flow conditions (N.sub.R>3,000), the Fanning
friction factor can be determined using the non-linear
Colebrook-White relationship given in Eq. A-3.
1 f = - 4 log ( 0.269 D + 1.255 N Re f ) ( A - 3 ) ##EQU00018##
[0085] The final relationship that is used to resolve the unknowns
in the single-phase flow metering problem is the capillary flow
relationship that relates the pressure loss due to frictional and
gravitational effects of flow in the conduit to the Reynolds
number, fluid properties, and relative pipe roughness is given in
Eq. A-4 using conventional oilfield units.
N Re f 1 / 2 = 1722.9 D 3 / 2 .rho. i 1 / 2 [ P 0 - P L + 0.006945
.rho. i L cos .theta. ] 1 / 2 L 1 / 2 .mu. i ( A - 4 )
##EQU00019##
[0086] The solution of these relationships for the three unknowns
in the problem may for example be accomplished using a non-linear
root-solving procedure such as Secant-Newton. The parameter of
variation in the root-solving procedure is the superficial velocity
(V.sub.si). With the single-phase fluid physical properties
(.mu..sub.i, .rho..sub.i) known as a function of the pressure and
temperature, the superficial velocity is used to determine the
Reynolds number as defined in Eq. A-1, the Fanning friction factor
from FIG. 3 (or Eqs. A-2 or A-3), and the basis function
constructed by rearranging the capillary flow relationship given in
Eq. A-4.
[0087] Note that for a single-phase system, the oil-water and
gas-liquid slip velocities are of course equal to zero. The same is
true of the superficial velocities and holdups of the fluid phases
not present in the single-phase system.
Evaluation of Oil-Water Two-Phase Flow Metering Parameters
[0088] The solution of two-phase flow metering computations using
temperature and pressure measurements described in this invention
involve the resolution of a non-linear system of 10 independent
relationships for the 10 unknown parameters in the problem. This is
true in oil-water, oil-gas, and water-gas two-phase flow metering
analyses using distributed temperature and pressure
measurements.
[0089] For an oil-water system, the unknowns that must be resolved
in the analysis are the oil and water holdups, the oil, water, and
mixture superficial velocities, mixture density and dynamic
viscosity, the slip velocity between the oil and water phases, the
Reynolds number and Fanning friction factor, and pressure loss that
occurs over the metering length of the flow conduit. Note that the
gas holdup and superficial velocity are equal to zero for an
oil-water system, as is the gas-liquid slip velocity.
[0090] The first relationship that is used to construct the
multiphase flow metering analysis in oil-water systems is the
holdup relationship given in Eq. B-1.
1=Y.sub.o+Y.sub.w (B-1)
The mixture density in oil-water two-phase flow can be defined by
the expression given in Eq. B-2.
.rho..sub.m=.rho..sub.oY.sub.o+.rho..sub.wY.sub.w (B-2)
The simultaneous solution of Eqs. B-1 and B-2 results in
expressions for the oil and water holdups, expressed in terms of
the unknown mixture density.
Y o = .rho. w - .rho. m .rho. w - .rho. o ( B - 3 ) T w = .rho. m -
.rho. o .rho. w - .rho. o ( B - 4 ) ##EQU00020##
The two-phase oil-water flow mixture dynamic viscosity for
non-emulsion fluid systems may be expressed by the relationship
given in Eq. B-3.
.mu..sub.m=.mu..sub.oY.sub.o+.mu..sub.wY.sub.w (B-5)
In terms of the unknown mixture density, the mixture viscosity is
defined as given in Eq. B-6.
.mu. m = .mu. o ( .rho. w - .rho. m ) + .mu. w ( .rho. m - .rho. o
) .rho. w - .rho. o ( B - 6 ) ##EQU00021##
The mixture superficial velocity of an oil-water two-phase system
is the sum of the superficial velocities of the oil and water
phases.
V.sub.m=V.sub.so+V.sub.sw (B-7)
The superficial mass velocity of a two-phase oil-water flow stream
is best characterized using Eqs. B-2, B-7, and an
equation-of-state. An expression that relates Eqs. B-2 and B-7 to
the mass velocity is given by Eq. B-8.
.rho..sub.mV.sub.m=(.rho..sub.oY.sub.o+.rho..sub.wY.sub.w)(V.sub.so+V.su-
b.sw) (B-8)
Expressions for estimating the oil and water superficial velocities
expressed in terms of the mixture superficial velocity and density
may be obtained using the definition of the mass velocity given in
Eq. B-8, in combination with an independent equation-of-state for
computing the mixture density using the temperature, pressure, and
fluid composition. With the two measurements (differential pressure
and temperature), two parameters may be resolved in the oil-water
two-phase system analysis, the mixture density and the
velocity.
[0091] A slip velocity relationship that is applicable for oil and
water multiphase systems is presented in Eq. B-9, expressed in
terms of conventional oilfield units. This relationship relates the
slip between the oil and water phases to the differences in
densities of the two fluids and the conduit inclination angle.
V sow = 0.6569 ( .rho. w - .rho. o 62.428 ) 0.25 exp [ - 0.788 ln (
115.5 .rho. w - .rho. o ) ( .rho. w - .rho. o .rho. w - .rho. o ) ]
( 1 + 0.04 .alpha. ) ( B - 9 ) ##EQU00022##
The Reynolds number of oil-water two-phase flow in a circular
conduit is given by Eq. B-10.
N Re = 124 .rho. m D V m .mu. m ( B - 10 ) ##EQU00023##
Substitution of the oil and water superficial velocities, holdups
(Eqs. B-3 and B-4), and the oil-water slip relationship into the
definition of the Reynolds number (Eq. B-9), results in an
expression for Reynolds number that represents one component of the
root-solving procedure basis function. The resulting expression can
be used in conjunction with the capillary flow relationship for a
two-phase oil-water system, given in Eq. B-11, to construct a basis
function for a non-linear root-solving procedure with the mixture
density as the variable parameter.
N Re = 1722.9 D 3 / 2 .rho. m 1 / 2 ( .rho. w - .rho. o ) [ P 0 - P
L + 0.006945 .rho. m L cos .theta. ] 1 / 2 Lf [ .mu. o ( .rho. w -
.rho. m ) + .mu. w ( .rho. m - .rho. o ) ] ( B - 11 )
##EQU00024##
Once the system mixture density has been determined with the
root-solving procedure, the oil-water system slip velocity is
evaluated with Eq. B-19, and the mixture velocity is evaluated with
Eq. B-12.
V m = [ .rho. w ( .rho. o - .rho. m ) + .rho. o ( .rho. w - .rho. m
) ] V sow .rho. w 2 - .rho. o 2 ( B - 12 ) ##EQU00025##
The Reynolds number can then be determined with Eq. B-10 and the
Fanning friction factor (f) is obtained with FIG. 4, which shows
the Fanning friction factor for multiphase flow systems (or with
the laminar or Colebrook-White turbulent flow relationships).
[0092] The oil and water phase superficial velocities are
subsequently evaluated using expressions derived from the mixture
and mass velocity relationships (Eqs. B-7 and B-8), and the oil and
water holdups are evaluated with Eqs. B-3 and B-4. The mixture
dynamic viscosity can then be readily evaluated using Eq. B-5 or
B-6.
Evaluation of Oil-Gas Two-Phase Flow Metering Parameters
[0093] Metering analyses using distributed temperature and pressure
measurements in a two-phase oil-gas system involves the
determination of 10 unknown parameter values using 10 independent
relationships, some of which are non-linear and/or piece-wise
continuous. The unknown parameters that must be resolved in an
oil-gas two-phase system analysis are the following: oil and gas
holdups, superficial velocities, the mixture superficial velocity,
density and viscosity, and the gas-liquid slip velocity, Reynolds
number and Fanning friction factor. The water holdup and
superficial velocity in this case are equal to zero, as is the
oil-water slip velocity. Essentially with the two physical
measurements that are being made in this case, the differential
pressure and the temperature, the mixture density and superficial
velocity can be resolved.
[0094] The holdup relationship for a two-phase oil-gas system is
given in Eq. C-1.
1=Y.sub.o+Y.sub.g (C-1)
The mixture density is defined as in Eq. C-2.
.rho..sub.m=.rho..sub.oY.sub.o+.rho..sub.gY.sub.g (C-2)
The solution of these two relationships results in expressions for
the oil and gas holdups in terms of the mixture density.
Y o = .rho. m - .rho. g .rho. o - .rho. g ( C - 3 ) Y g = .rho. o -
.rho. m .rho. o - .rho. g ( C - 4 ) ##EQU00026##
[0095] The mixture viscosity in a two-phase oil-gas system is
generally defined in one of two ways, with the more common
relationship given in Eq. C-5 or with the Hagedorn-Brown model
given in Eq. C-6.
.mu..sub.m=.mu..sub.oY.sub.o+.mu..sub.gY.sub.g (C-5)
.mu..sub.m=.mu..sub.o.sup.Y.sup.o.mu..sub.g.sup.Y.sup.g (C-6)
These expressions can be readily rewritten in terms of the oil and
gas holdups given in Eqs. C-3 and C-4 as functions of the mixture
density.
.mu. m = .mu. o ( .rho. m - .rho. g ) + .mu. g ( .rho. o - .rho. m
) .rho. o - .rho. g ( C - 7 ) .mu. m = .mu. o ( .rho. m - .rho. g
.rho. o - .rho. g ) .mu. g ( .rho. o - .rho. m .rho. o - .rho. g )
( C - 8 ) ##EQU00027##
The mixture superficial velocity of the two-phase system is defined
in Eq. C-9, with the superficial mass velocity being evaluated with
Eq. C-10.
V.sub.m=V.sub.so+V.sub.sg (C-9)
.rho..sub.mV.sub.m=(.rho..sub.oY.sub.o+.rho..sub.gY.sub.g)(V.sub.so+V.su-
b.sg) (C-10)
Solution of Eqs. C-9 and C-10, with substitution of the previously
determined relationships for the holdups (Eqs. C-3 and C-4), the
oil and gas superficial velocities can be expressed in terms of the
mixture density and superficial velocity. The mixture density in
this case is best characterized using an accurate equation-of-state
to determine the densities of the liquid and vapor phases in the
system. The gas-liquid slip velocity relationship is defined for an
oil-gas two-phase system as shown in Eq. C-11.
V sg Y g - V so Y o = [ ( 0.95 - Y g 2 ) 0.5 + 0.025 ] ( 1 + 0.04
.alpha. ) ( C - 11 ) ##EQU00028##
The Reynolds number of a two-phase oil-gas flow is determined using
Eq. C-12.
N Re = 124 .rho. m D V m .mu. m ( C - 12 ) ##EQU00029##
[0096] Substitution of the mixture superficial velocity given by
Eq. C-9 into the Reynolds number relationship (Eq. C-12) yields an
expression that can be used to construct a basis function for a
root-solving procedure to evaluate the unknown parameter values in
the oil-gas two-phase flow metering problem.
[0097] Another expression for the Reynolds number can be obtained
from the capillary flow relationship that describes the pressure
differential in the flow conduit due to frictional and
gravitational effects. This relationship is given in Eq. C-13 and
is used to complete the construction of the root-solving basis
function used in the analysis. Note that the Fanning friction
factor appearing in Eq. C-13 is obtained from FIG. 4 as a function
of the Reynolds number and the relative pipe roughness, or by the
solution of the laminar or turbulent flow relationships that
correspond to the graphical solution of the Fanning friction
factor.
N Re = 1722.9 D 3 / 2 .rho. m 1 / 2 ( .rho. o - .rho. g ) [ P 0 - P
L + 0.006945 .rho. m L cos .theta. ] 1 / 2 Lf [ .mu. o ( .rho. m -
.rho. g ) + .mu. g ( .rho. o - .rho. m ) ] ( C - 13 )
##EQU00030##
[0098] The unknown parameter used as the variable of the
root-solving procedure in this analysis is the mixture density.
Once the mixture density has been determined, the Reynolds number
can be readily evaluated using either Eq. C-12 or C-13. The mixture
superficial velocity is then evaluated with Eq. C-9. The oil and
gas phase holdups may be determined with Eqs. C-3 and C-4, followed
by the mixture dynamic viscosity computed with Eq. C-5.
[0099] A similar solution methodology can be developed for the
alternate mixture viscosity relationship given in Eq. C-6, as that
given when Eq. C-5 is used. The solution methodology developed in
this invention is applicable in general for all oil-gas two-phase
flow cases. Substitution for an alternate dynamic viscosity or
gas-liquid slip velocity relationship is permitted in the
analysis.
Evaluation of Water-Gas Two-Phase Flow Metering Parameters
[0100] In a water-gas two-phase flow metering system developed
using distributed temperature and pressure measurements, the
evaluation of the 10 unknown parameters require the use of 10
independent functional relationships involving those parameters in
order to resolve the multiphase flow metering problem. The unknown
parameter values that must be determined from the metering analysis
are the water and gas holdups, the water and gas superficial
velocities, the mixture superficial velocity, density and dynamic
viscosity, the gas-liquid slip velocity, Reynolds number and
Fanning friction factor. In a manner similar to that described
previously for the other two-phase flow metering analyses, a
non-linear root-solving procedure is required to resolve the
unknowns of the problem. Note that in a two-phase water-gas flow
metering analysis, the oil holdup and superficial velocity are
equal to zero, as well as is the oil-water slip velocity.
[0101] The holdup relationship that is applicable for a two-phase
water-gas metering analysis is given by Eq. D-1.
1=Y.sub.w+Y.sub.g (D-1)
The mixture density of the water-gas two-phase flow is defined by
Eq. D-2.
.rho..sub.m=.rho..sub.wY.sub.w+.rho..sub.gY.sub.g (D-2)
Simultaneous solution of Eqs. D-1 and D-2 results in expressions
for the water and gas holdups, expressed in terms of the fluid
mixture density of the water-gas system.
Y w = .rho. m - .rho. g .rho. w - .rho. g ( D - 3 ) Y g = .rho. w -
.rho. m .rho. w - .rho. g ( D - 4 ) ##EQU00031##
[0102] There are at least two fluid mixture viscosity relationships
that can be used for characterizing the dynamic fluid viscosity in
a water-gas two-phase metering analysis. The more commonly used of
these is the relationship given in Eq. D-5, with an alternate fluid
mixture viscosity relationship proposed by Hagedorn and Brown given
in Eq. D-6.
.mu..sub.m=.mu..sub.wY.sub.w+.mu..sub.gY.sub.g (D-5)
.mu..sub.m=.mu..sub.w.sup.Y.sup.w.mu..sub.g.sup.Y.sup.g (D-6)
[0103] Application of the holdup relationships obtained in Eqs. D-3
and D-4 in the mixture viscosity model given by Eq. D-5, results in
a fluid mixture viscosity relationship that is only a function of
the unknown fluid mixture density.
.mu. m = .mu. w ( .rho. m - .rho. g ) + .mu. g ( .rho. w - .rho. m
) .rho. w - .rho. g ( D - 7 ) ##EQU00032##
[0104] The mixture superficial velocity in a water-gas two-phase
flow metering analysis is the sum of the water and gas phase
superficial velocities.
V.sub.m=V.sub.sw+V.sub.sg (D-8)
[0105] The superficial mass velocity in the water-gas system can be
evaluated as defined in Eq. D-9.
.rho..sub.mV.sub.m=(.rho..sub.wY.sub.w+.rho..sub.gY.sub.g)(V.sub.sw+V.su-
b.sg) (D-9)
Solution of Eqs. D-8 and D-9, with the definitions of the water and
gas holdups previously obtained in Eqs. D-3 and D-4, the
superficial velocity of the water and gas phases can be expressed
in terms of the mixture density and superficial velocity. The
gas-liquid slip velocity can be evaluated using the slip velocity
relationship presented in Eq. D-10.
V sg Y g - V sw Y w = [ ( 0.95 - Y g 2 ) 0.5 + 0.025 ] ( 1 + 0.04
.alpha. ) ( D - 10 ) ##EQU00033##
The fluid mixture superficial velocity is given in Eq. D-8 and the
Reynolds number for a water-gas multiphase flow is defined by the
relationship given in Eq. D-11.
N Re = 124 .rho. m D V m .mu. m ( D - 11 ) ##EQU00034##
Substitution of the results of mixture dynamic viscosity (Eq. D-7)
and superficial velocity (Eq. D-8) in the Reynolds number
relationship results in one component of the basis function for
evaluating the unknowns in the multiphase metering problem.
[0106] The other component of the basis function (alternate
Reynolds number relationship) is obtained from the capillary flow
relationship that relates the pressure differential observed in
flow in a conduit to the frictional and gravitational effects.
N Re = 1722.9 D 3 / 2 .rho. m 1 / 2 ( .rho. w - .rho. g ) [ P 0 - P
L + 0.006945 .rho. m L cos .theta. ] 1 / 2 Lf [ .mu. w ( .rho. m -
.rho. g ) + .mu. g ( .rho. w - .rho. m ) ] ( D - 12 )
##EQU00035##
[0107] With the fluid mixture density obtained from the
root-solving procedure just described, the Reynolds number can then
be determined using either Eq. D-11 or D-12. With the two
independent measurements available (differential pressure and
temperature) two parameters of the problem can be resolved. These
are the mixture density and the superficial velocity. A mixture
density can be derived from the constituitive relationships of the
problem, including the capillary flow relationship and the
differential pressure measurements. The pressure and temperature
also provides a means of computing the mixture density under these
conditions as a function of the fluid composition using an accurate
and robust equation-of-state.
Evaluation of Three-Phase Flow Metering Parameters
[0108] The evaluation of the unknown metering parameters in a
three-phase system (oil, gas, and water) using distributed
temperatures and pressures is by far the most difficult to
implement in a stable numerical solution procedure due to the
complex relationships between the slip velocities, holdups, mixture
viscosities, and superficial velocities of the phases present in
the system. The unknowns in the three-phase metering analysis
include the holdups of all three phases, their superficial
velocities, as well as the mixture superficial velocity, the
mixture density, viscosity, Reynolds number and friction factor, in
addition to the water-oil and gas-liquid slip velocities of the
system. There are a total of 13 unknowns in the three-phase
metering analysis problem. Therefore, a total of 13 independent
relationships are required to properly resolve the unknowns in the
three-phase flow metering analysis using distributed temperature
and pressure measurements.
[0109] As was demonstrated with the two-phase flow problems above,
the holdup relationship is the first fundamental independent
relationship that is used to construct the system of equations in
the analysis. The three-phase holdup relationship is given by Eq.
E-1.
1=Y.sub.o+Y.sub.w+Y.sub.g (E-1)
The fluid mixture density is defined in the three-phase analysis
with Eq. E-2.
.rho..sub.m=.rho..sub.oY.sub.o+.rho..sub.wY.sub.w+.rho..sub.gY.sub.g
(E-2)
The fluid mixture dynamic viscosity is commonly evaluated using the
model presented in Eq. E-3.
.mu..sub.m=.mu..sub.oY.sub.o+.mu..sub.wY.sub.w+.mu..sub.gY.sub.g
(E-3)
An alternate expression for estimating the fluid mixture dynamic
viscosity has been proposed by Hagedorn and Brown. The Hagedorn and
Brown fluid mixture viscosity model is given in Eq. E-4.
.mu. m = ( .mu. o Y o + .mu. w Y w Y o + Y w ) ( Y o + Y w ) .mu. g
Y g ( E - 4 ) ##EQU00036##
One fluid mixture relationship that has been found to characterize
the kinematic viscosity of the three-phase system reasonably well
is given by Eq. E-5. The kinematic viscosity is defined as the
ratio of the dynamic viscosity to the fluid mixture density.
.gamma. m = .mu. m .rho. m = .mu. o Y o + .mu. w Y w + .mu. g Y g
.rho. o Y o + .rho. w Y w + .rho. g Y g ( E - 5 ) ##EQU00037##
The simultaneous solution of Eqs. E-1 through E-5 can be used to
develop expressions for the three fluid phase holdups and the
mixture viscosity, expressed in terms of the unknown mixture
density. The mixture kinematic viscosity is a sum of the kinematic
viscosities of the individual phases, the gas holdup can then be
evaluated as a function of the mixture density and dynamic
viscosity.
[0110] The water holdup may then be evaluated using Eq. E-6 as a
function of the mixture density and viscosity, and the gas holdup.
The oil phase holdup can subsequently be computed from the
fundamental holdup relationship given in Eq. E-1 using the results
of the gas holdup and E-6.
Y w = .rho. m - .rho. o - Y g ( .rho. g - .rho. o ) .rho. w - .rho.
o ( E - 6 ) ##EQU00038##
[0111] The three-phase flow metering analysis solution procedure
next addresses the issue of the fluid phase and mixture superficial
velocities and the two sets of two-phase slip velocity
relationships that are required in the analysis of a three-phase
flow system. The slip velocity relationships that are applicable
for the oil and water phases in a three-phase analysis are given by
Eqs. E-7 and E-8.
V sow = V so ( Y o + Y w ) Y o - V sw ( Y o + Y w ) Y w ( E - 7 ) V
sow = 0.6569 ( .rho. w - .rho. o 62.428 ) 0.25 exp [ - 0.788 ln (
115.5 .rho. w - .rho. o ) ( Y o Y o + Y w ) ] ( 1 + 0.04` .alpha. )
( E - 8 ) ##EQU00039##
The gas-liquid slip velocity relationships that are applicable in
three-phase flow analyses are presented in Eqs. E-9 and E-10.
V sgL = V sg Y g - V so + V sw Y o + Y w ( E - 9 ) V sgL = [ ( 0.95
- Y g 2 ) 0.5 + 0.025 ] ( 1 + 0.04 .alpha. ) ( E - 10 )
##EQU00040##
The three-phase mixture superficial velocity is given by Eq.
E-11.
V.sub.m=V.sub.so+V.sub.sw+V.sub.sg (E-11)
The mass superficial velocity can best be characterized using a
relationship such as the one given in Eq. E-12 and a value of the
mixture density derived from an accurate equation-of-state.
.rho..sub.mV.sub.m=(.rho..sub.oY.sub.o+.rho..sub.wY.sub.w+.rho..sub.gY.s-
ub.g)(V.sub.so+V.sub.sw+V.sub.sg) (E-12)
The solution of Eqs. E-7 through E-12 results in expressions for
the phase and mixture superficial velocities and slip velocities
that are only functions of the previously determined fluid phase
holdups and dynamic viscosity, all of which can be directly related
to the fluid mixture density. One component of the root-solving
procedure basis function is obtained in the form of the Reynolds
number, given by Eq. E-13.
N Re = 124 .rho. m D V m .mu. m ( E - 13 ) ##EQU00041##
[0112] Substitution into Eq. E-13 for the mixture superficial
velocity (Eq. E-11) and dynamic viscosity (Eq. E-3) results in one
component of the basis function of the root-solving procedure used
in the three-phase flow metering analysis. The other component of
the basis function used in the root-solving procedure for
evaluating the mixture density, satisfying all of the conditions
and relationships in the three-phase flow metering analysis, is
obtained from the capillary flow relationship. Rearranged in terms
of the Reynolds number, this relationship is given in Eq. E-14. The
Fanning friction factor in this expression is evaluated using FIG.
4 for the Reynolds number defined by Eq. E-14.
N Re = 1722.9 D 3 / 2 .rho. m 1 / 2 ( .rho. w - .rho. g ) [ P 0 - P
L + 0.006945 .rho. m L cos .theta. ] 1 / 2 Lf [ .mu. o Y o + .mu. w
Y w + .mu. g Y g ] ( E - 14 ) ##EQU00042##
[0113] With the three-phase fluid mixture density resolved with the
non-linear root-solving mixture described herein, the unknown
parameters in the problem are recovered by back-substitution in the
analysis procedure. For instance, the Reynolds number can be
computed directly using the mixture density and Eqs. E-13 or E-14.
The mixture superficial velocity is determined with Eq. E-11 and
the mixture dynamic viscosity is obtained with Eq. E-3. The
oil-water system slip velocity can be evaluated using Eq. E-8 and
the gas-liquid slip velocity can be computed with Eq. E-10. The
water phase superficial velocity can be evaluated using Eq. E-15
and the gas phase superficial velocity can be evaluated with Eq.
E-16. The oil phase superficial velocity can then be determined by
rearranging Eq. E-11.
V sw = Y w [ V m - V sgL Y g - V sow Y o ( Y o + Y w ) 2 ] ( E - 15
) V sg = Y g [ V m + V sgL ( Y o + Y w ) ] ( E - 16 )
##EQU00043##
Example Computational Results
[0114] The results of an example computation of multiphase flow
velocities, holdup, slip velocities, and mixture density and
viscosity for a pressure traverse in a vertical section of wellbore
production tubing using the computational methodology disclosed in
this invention is presented in the following discussion. The fluids
considered in this theoretical example include a 40.degree. API
hydrocarbon liquid (oil) with a density of 45.923 lbs/cu ft and a
dynamic viscosity of 0.487 cp, produced formation water with a
salinity of 40,000 ppm that has a density of 65.762 lbs/cu ft and a
dynamic viscosity of 0.271 cp, and a natural gas mixture that has a
density at downhole wellbore conditions of 2.456 lbs/cu ft and a
dynamic viscosity of 0.014 cp.
[0115] Simulated temperature and pressure measurements are modeled
for two spatial positions in a vertical section of the wellbore for
multiphase flow metering purposes, at wellbore depths of 10,000 and
10,100 ft. The temperature in the wellbore at 10,000 ft of depth
was assumed to be 240.degree. F. and the flowing wellbore pressure
at that depth was assumed to be 1,000 psia. At 10,100 ft of depth,
the corresponding temperature was modeled to be 241.8.degree. F.
and the wellbore flowing pressure was assumed to be 1,025 psia. The
production tubing (flow conduit) in this section of the wellbore in
this example is 2 3/8in OD tubing which has an internal diameter of
1.995 inches and a relative roughness of 0.004.
[0116] An example of the output results obtained using a computer
program consisting of the computational methodology described in
this invention disclosure is presented in the following summary
table. Note that in this synthetic example there is three-phase
flow in the wellbore. In fact, there is upward flow of gas while
there is fallback (downward flow) of the hydrocarbon liquid (oil)
and water phases. The Reynolds number indicates that the flow
conditions are in the transition flow regime range (not quite fully
developed turbulent flow) and the pipe friction is relatively low
due to the moderate Reynolds number and the relatively low relative
roughness of the conduit.
[0117] The gas holdup obtained for these conditions indicates that
gas occupies 36% of the wellbore flow area, with water present in
about 30%, and oil occupying about 34% of the flow area or volume.
The volumetric flow rates obtained in the analysis are presented in
the summary tables as well. Note that the gas volumetric flow rate
includes the free gas present in the flowstream, as well as the
solution gas dissolved in the oil and water phases at downhole
conditions.
TABLE-US-00001 Wellbore Segment Computed Results: Oil holdup =
0.337 Water holdup = 0.298 Gas holdup = 0.364 Oil superficial
velocity = -0.009 ft/s Water superficial velocity = -0.123 ft/s Gas
superficial velocity = 0.062 ft/s Liquid superficial velocity =
-0.131 ft/s Mixture superficial velocity = -0.069 ft/s Oil-water
slip velocity = 0.244 ft/s Gas-liquid slip velocity = 0.378 ft/s
Mixture density = 35.999 lbs/cu ft Mixture dynamic viscosity =
0.250 cp Mixture kinematic viscosity = 0.007 cp-cu ft/lbs Reynolds
number = 2456.4 Fanning friction factor = 0.01238 Oil flow rate =
-2.54 STB/D Water flow rate = -38.84 STB/D Gas flow rate = 5.62
Mscf/D
[0118] Referring again to FIG. 1, it is understood that computer
system 10 may be implemented as any type of computing
infrastructure. Computer system 10 generally includes a processor
12, input/output (I/O) 14, memory 16, and bus 17. The processor 12
may comprise a single processing unit, or be distributed across one
or more processing units in one or more locations, e.g., on a
client and server. Memory 16 may comprise any known type of data
storage, including magnetic media, optical media, random access
memory (RAM), read-only memory (ROM), a data cache, a data object,
etc. Moreover, memory 16 may reside at a single physical location,
comprising one or more types of data storage, or be distributed
across a plurality of physical systems in various forms.
[0119] I/O 14 may comprise any system for exchanging information
to/from an external resource. External devices/resources may
comprise any known type of external device, including sensors 30,
32, a monitor/display, speakers, storage, another computer system,
a hand-held device, keyboard, mouse, wireless system, voice
recognition system, speech output system, printer, facsimile,
pager, etc. Bus 17 provides a communication link between each of
the components in the computer system 10 and likewise may comprise
any known type of transmission link, including electrical, optical,
wireless, etc. Although not shown, additional components, such as
cache memory, communication systems, system software, etc., may be
incorporated into computer system 10.
[0120] Access to computer system 10 may be provided over a network
such as the Internet, a local area network (LAN), a wide area
network (WAN), a virtual private network (VPN), etc. Communication
could occur via a direct hardwired connection (e.g., serial port),
or via an addressable connection that may utilize any combination
of wireline and/or wireless transmission methods. Moreover,
conventional network connectivity, such as Token Ring, Ethernet,
WiFi or other conventional communications standards could be used.
Still yet, connectivity could be provided by conventional TCP/IP
sockets-based protocol. In this instance, an Internet service
provider could be used to establish interconnectivity. Further,
communication could occur in a client-server or server-server
environment.
[0121] It should be appreciated that the teachings of the present
invention could be offered as a business method on a subscription
or fee basis. For example, a computer system 10 comprising a
multiphase flow analysis system 18 could be created, maintained
and/or deployed by a service provider that offers the functions
described herein for customers. That is, a service provider could
offer to provide wellbore fluid property information as described
above.
[0122] It is understood that in addition to being implemented as a
system and method, the features may be provided as a program
product stored on a computer-readable medium, which when executed,
enables computer system 10 to provide a multiphase flow analysis
system 18. To this extent, the computer-readable medium may include
program code, which implements the processes and systems described
herein. It is understood that the term "computer-readable medium"
comprises one or more of any type of physical embodiment of the
program code. In particular, the computer-readable medium can
comprise program code embodied on one or more portable storage
articles of manufacture (e.g., a compact disc, a magnetic disk, a
tape, etc.), on one or more data storage portions of a computing
device, such as memory 16 and/or a storage system, and/or as a data
signal traveling over a network (e.g., during a wired/wireless
electronic distribution of the program product).
[0123] As used herein, it is understood that the terms "program
code" and "computer program code" are synonymous and mean any
expression, in any language, code or notation, of a set of
instructions that cause a computing device having an information
processing capability to perform a particular function either
directly or after any combination of the following: (a) conversion
to another language, code or notation; (b) reproduction in a
different material form; and/or (c) decompression. To this extent,
program code can be embodied as one or more types of program
products, such as an application/software program, component
software/a library of functions, an operating system, a basic I/O
system/driver for a particular computing and/or I/O device, and the
like. Further, it is understood that terms such as "component" and
"system" are synonymous as used herein and represent any
combination of hardware and/or software capable of performing some
function(s).
[0124] The block diagrams in the figures illustrate the
architecture, functionality, and operation of possible
implementations of systems, methods and computer program products
according to various embodiments of the present invention. In this
regard, each block in the block diagrams may represent a module,
segment, or portion of code, which comprises one or more executable
instructions for implementing the specified logical function(s). It
should also be noted that the functions noted in the blocks may
occur out of the order noted in the figures. For example, two
blocks shown in succession may, in fact, be executed substantially
concurrently, or the blocks may sometimes be executed in the
reverse order, depending upon the functionality involved. It will
also be noted that each block of the block diagrams can be
implemented by special purpose hardware-based systems which perform
the specified functions or acts, or combinations of special purpose
hardware and computer instructions.
Calibration Using Data From A Retrievable Production Logging
Device
[0125] Embodiments of the inventive system, method, and program
code can also utilize data obtained from a retrievable production
logging device to calibrate one or more of the generated wellbore
fluid properties. These retrievable production logging devices are
typically deployed in the wellbore on wireline, slickline, or
coiled tubing. In the case of highly deviated or horizontal
wellbores, the production logging devices may be pushed into
position using coiled tubing or stiff wireline cable or may be
pulled into position using a downhole tractor. Examples of the
types of retrievable production logging devices that may be used
include the Production Logging Tool, Memory PS Platform, Gas Holdup
Optical Sensor Tool, and Flow Scanner Tool, all available from
Schlumberger. The calibration process may involve the
identification of or confirmation that one or more sensors that are
providing inaccurate downhole measurements and
elimination/rejection of the data provided by these sensors.
Alternatively, the wellbore fluid property generation process
and/or results may be adjusted to either match or more closely
reflect the data obtained from the retrievable production logging
device.
[0126] Although specific embodiments have been illustrated and
described herein, those of ordinary skill in the art appreciate
that any arrangement which is calculated to achieve the same
purpose may be substituted for the specific embodiments shown and
that the invention has other applications in other environments.
This application is intended to cover any adaptations or variations
of the present invention. The following claims are in no way
intended to limit the scope of the invention to the specific
embodiments described herein.
* * * * *