U.S. patent application number 14/451595 was filed with the patent office on 2016-02-11 for reservoir water volume shift model.
This patent application is currently assigned to Baker Hughes Incorporated. The applicant listed for this patent is Nathan F. Andrews, Christopher J. Freitas, Nicholas J. Mueschke. Invention is credited to Nathan F. Andrews, Christopher J. Freitas, Nicholas J. Mueschke.
Application Number | 20160040532 14/451595 |
Document ID | / |
Family ID | 55267058 |
Filed Date | 2016-02-11 |
United States Patent
Application |
20160040532 |
Kind Code |
A1 |
Freitas; Christopher J. ; et
al. |
February 11, 2016 |
RESERVOIR WATER VOLUME SHIFT MODEL
Abstract
A method of estimating properties of a reservoir fluid in an
underground reservoir includes: calculating the density .rho. of
the mixture containing water; determining component-specific volume
shift parameters c.sub.i; replacing the component specific volume
shift parameter for water with a volume shift parameter c.sub.H2O;
determining a total volume shift c required from the sum of the
component specific volume shifts and the mole fraction of each
component in the mixture; determining a corrected molar volume of
the liquid and vapor by subtracting the total volume shift c from
an equation of state (EOS) calculated molar volume v; and
recalculating density .rho. and compressibility factor Z for the
liquid and vapor phases with the corrected molar volume.
Inventors: |
Freitas; Christopher J.;
(San Antonio, TX) ; Mueschke; Nicholas J.; (San
Antonio, TX) ; Andrews; Nathan F.; (San Antonio,
TX) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Freitas; Christopher J.
Mueschke; Nicholas J.
Andrews; Nathan F. |
San Antonio
San Antonio
San Antonio |
TX
TX
TX |
US
US
US |
|
|
Assignee: |
Baker Hughes Incorporated
Houston
TX
|
Family ID: |
55267058 |
Appl. No.: |
14/451595 |
Filed: |
August 5, 2014 |
Current U.S.
Class: |
702/11 |
Current CPC
Class: |
E21B 47/10 20130101;
E21B 47/06 20130101 |
International
Class: |
E21B 49/08 20060101
E21B049/08 |
Claims
1. An apparatus for estimating conditions of reservoir fluid in an
underground reservoir, the apparatus comprising: a sensor for
measuring one or more measured parameters of that fluid, the
measured parameters including at least one of: temperature and
pressure of the fluid; and a processor, the processor configured
to: receive data representing the one or more measured parameters;
calculate the density .rho. of a mixture containing water;
determine component-specific volume shift parameters c.sub.i;
replace the component specific volume shift parameter for water
with a volume shift parameter c.sub.H2O; determine a total volume
shift c required from the sum of the component specific volume
shifts and the mole fraction of each component in the mixture;
determine a corrected molar volume of the liquid and vapor by
subtracting the total volume shift c from an equation of state
(EOS) calculated molar volume v; and recalculate density .rho. and
compressibility factor Z for the liquid and vapor phases with the
corrected molar volume.
2. The apparatus of claim 1, wherein the density is determined
based by solving: .rho. liq = P RT M liq Z liq ; and ##EQU00021##
.rho. vap = P RT M vap Z vap . ##EQU00021.2##
3. The apparatus of claim 2, wherein determining the
component-specific volume shift parameters c.sub.i; includes
solving: c i = 0.40768 ( RT c P c ) ( 0.29441 - Z RA ) .
##EQU00022##
4. The apparatus of claim 2, wherein determining the
component-specific volume shift parameters c.sub.i; includes
solving: c i = S i b i , S i = 1 - d M e , b i = .OMEGA. b RT c P c
, ##EQU00023## where d=2.258 and e=0.1823 for n-alkane hydrocarbons
(different values of d and e are also provided for aromatic and
naphthenic hydrocarbons).
5. The apparatus of claim 2, wherein replacing the component
specific volume shift parameter for water with a volume shift
parameter c.sub.H2O, includes solving: c H 2 O = b H 2 O S H 2 O ,
S H 2 O = a + b exp ( T r ) + c exp ( P r ) + d exp ( T r ) exp ( P
r ) 1 + f exp ( T r ) + exp ( P r ) + h exp ( T r ) exp ( P r ) , b
H 2 O = .OMEGA. b RT c P c = { 2.114176 .times. 10 - 5 m 3 mol ( S
R K ) 1.898464 .times. 10 - 5 m 3 mol ( P R ) P r = P P c and T r =
T T c ##EQU00024##
6. The apparatus of claim 2, wherein determining a total volume
shift c required from the sum of the component specific volume
shifts and the mole fraction of each component in the mixture;
includes solving: c liq = i N x i c i , c vap = i N y i c i .
##EQU00025##
7. The apparatus of claim 2, wherein determining a corrected molar
volume of the liquid and vapor by subtracting the total volume
shift c from an equation of state (EOS) calculated molar volume v;
includes solving:
v.sub.liq.sup.corrected=v.sub.liq.sup.EOS-c.sub.liq,
v.sub.vap.sup.corrected=v.sub.vap.sup.EOS-c.sub.vap.
8. The apparatus of claim 2, wherein recalculating the density
.rho. and compressibility factor Z for the liquid and vapor phases
with the corrected molar volume; includes solving: .rho. liq
corrected = M liq .upsilon. liq corrected , .rho. vap corrected = M
vap .upsilon. vap corrected , Z liq corrected = 1 .upsilon. liq
corrected , Z vap corrected = 1 .upsilon. vap corrected .
##EQU00026##
9. A method of estimating properties of a reservoir fluid in an
underground reservoir, the apparatus comprising: receiving, at a
computing device, data representing one or more measured parameters
of a mixture containing water in the underground reservoir.
calculating, with the computing device, the density .rho. of the
mixture containing water; determining component-specific volume
shift parameters c.sub.i; replacing the component specific volume
shift parameter for water with a volume shift parameter c.sub.H2O;
determining a total volume shift c required from the sum of the
component specific volume shifts and the mole fraction of each
component in the mixture; determining a corrected molar volume of
the liquid and vapor by subtracting the total volume shift c from
an equation of state (EOS) calculated molar volume v; and
recalculating density .rho. and compressibility factor Z for the
liquid and vapor phases with the corrected molar volume.
10. The method of claim 9, wherein determining the
component-specific volume shift parameters c.sub.i; may include
solving: c i = 0.40768 ( RT c P c ) ( 0.29441 - Z RA ) .
##EQU00027##
11. The method of claim 9, wherein determining the
component-specific volume shift parameters c.sub.i; may include
solving: c i = S i b i , S i = 1 - d M e , b i = .OMEGA. b RT c P c
, ##EQU00028## where d=2.258 and e=0.1823 for n-alkane
hydrocarbons.
12. The method of claim 9, wherein replacing the component specific
volume shift parameter for water with a volume shift parameter
c.sub.H2O, includes solving: c H 2 O = b H 2 O S H 2 O , S H 2 O =
a + b exp ( T r ) + c exp ( P r ) + d exp ( T r ) exp ( P r ) 1 + f
exp ( T r ) + exp ( P r ) + h exp ( T r ) exp ( P r ) , b H 2 O =
.OMEGA. b RT c P c = { 2.114176 .times. 10 - 5 m 3 mol ( S R K )
1.898464 .times. 10 - 5 m 3 mol ( P R ) P r = P P c and T r = T T c
##EQU00029##
13. The method of claim 9, wherein determining a total volume shift
c required from the sum of the component specific volume shifts and
the mole fraction of each component in the mixture; includes
solving: c liq = i N x i c i , c vap = i N y i c i .
##EQU00030##
14. The method of claim 9, wherein determining a corrected molar
volume of the liquid and vapor by subtracting the total volume
shift c from an equation of state (EOS) calculated molar volume v;
includes solving:
v.sub.liq.sup.corrected=v.sub.liq.sup.EOS-c.sub.liq,
v.sub.vap.sup.corrected=v.sub.vap.sup.EOS-c.sub.vap.
15. The method of claim 9, wherein recalculating the density .rho.
and compressibility factor Z for the liquid and vapor phases with
the corrected molar volume; includes solving: .rho. liq corrected =
M liq .upsilon. liq corrected , .rho. vap corrected = M vap
.upsilon. vap corrected , Z liq corrected = 1 .upsilon. liq
corrected , Z vap corrected = 1 .upsilon. vap corrected .
##EQU00031##
Description
TECHNICAL FIELD
[0001] This invention is pertinent to the thermodynamic modeling of
water using cubic equation of state (EOS) models and auxiliary
volume shift density correction methods.
BACKGROUND
[0002] Thermodynamic equations of state (EOS) models relate known
state variables, such as pressure and temperature, to unknown state
variables, such as volume, density, and fugacity, as well as other
fluid parameters. Cubic EOS models are a subclass of the broader
class of general thermodynamic EOS models that relate pressure,
temperature, and volume through a cubic polynomial function. Such
models are used to predict phase behavior and fluid properties in a
broad range of applications, including the petrophysical,
geophysical, refrigeration, aerospace, and chemical process design
industries. Their broad adoption and application is due to the
relative simplicity of the model form, ease of computational
implementation, limited number of fluid property inputs required
for operation, and overall computational robustness.
[0003] The cubic EOS model subclass was first developed by van der
Waals as an extension to the ideal gas law, where both attractive
and repulsive molecular forces are included. Since the initial
formulation of a cubic EOS model by van der Waals, numerous
modifications to the model form have been made in an effort to
increase predictive accuracy and general applicability. Of note are
the Soave-Redlich-Kwong (SRK) and Peng-Robinson (PR) model forms,
which achieved broad adoption and use in industrial applications.
Both the SRK and PR models may expressed in a universal form:
P = RT V - b - a .alpha. V 2 + u 1 bV + u 2 b 2 , ( 1 )
##EQU00001##
where P is pressure, T is temperature, V is volume, R is the
universal gas constant, a and b are attractive and repulsive
parameters specific to each pure substance or mixture, and u.sub.1
and u.sub.2 are model-specific constants. The values and model
forms for a, .alpha., b, u.sub.1, and u.sub.2 for the SRK and PR
model forms are listed in Table 1 below. Expressed in its cubic
polynomial form, the SRK and PR models may be written as:
Z.sup.3-(1+B-u.sub.1B)Z.sup.2+(A+u.sub.2B.sup.2-u.sub.1B-u.sub.1B.sup.2)-
Z-(AB+u.sub.2B.sup.2+u.sub.2B.sup.3)=0, (2)
where Z is the compressibility factor, with
A = ( a .alpha. ) mix P R 2 T 2 , and B = b mix P RT . ( 3 )
##EQU00002##
[0004] The attractive and repulsive terms for mixtures are
calculated using standard mixing rules and are given by:
(a.alpha.).sub.mix=.SIGMA..sub.i.sup.N.SIGMA..sub.j.sup.N[x.sub.ix.sub.j
{square root over
(a.sub.ia.sub.j.alpha..sub.i.alpha..sub.j)}(1-k.sub.ij)], (4)
b.sub.mix=.SIGMA..sub.i.sup.Nx.sub.ib.sub.i, (5)
where x.sub.i is the mole fraction of each component in the phase
of interest (liquid or gas) and k.sub.ij are binary interaction
coefficients.
TABLE-US-00001 TABLE 1 SRK and PR cubic EOS parameters. Model a
.alpha. b .OMEGA..sub.a .OMEGA..sub.b u.sub.1 u.sub.2 SRK .OMEGA. a
R 2 T c 2 P c ##EQU00003## [1 + f.sub..omega.(1 -
T.sup.0.5.sub.r)].sup.2 f.sub..omega. = 0.48 + 1.574.omega. -
0.176.omega..sup.2 .OMEGA. b R T c 2 P c ##EQU00004## 0.42748
0.08664 1 0 PR .OMEGA. a R 2 T c 2 P c ##EQU00005## [1 +
f.sub..omega.(1 - T.sup.0.5.sub.r)].sup.2 f.sub..omega. = 0.37464 +
1.54226.omega. - 0.2699.omega..sup.2 .OMEGA. b R T c 2 P c
##EQU00006## 0.45724 0.07780 2 -1
[0005] Given physical properties for each mixture component, a
cubic EOS may be used to iteratively solve for the compressibility
factor Z for each potential phase (Z.sub.liq and Z.sub.vap for
liquid and vapor phases respectively). This cubic EOS iterative
method is part of a "flash algorithm" and upon convergence of the
iterative portion of the flash algorithm, the liquid-phase density
.rho..sub.liq and the vapor-phase density .rho..sub.vap may be
computed through the use of the converged Z factor values:
.rho. liq = P RT M liq Z liq , ( 6 ) .rho. vap = P RT M vap Z vap ,
( 7 ) ##EQU00007##
where M.sub.liq and M.sub.vap are the average molecular weights of
the liquid and vapor compositions.
SUMMARY
[0006] According to one embodiment, an apparatus for estimating
conditions of reservoir fluid in an underground reservoir that
includes a sensor for measuring one or more measured parameters of
that fluid, the measured parameters including at least one of:
temperature and pressure of the fluid a processor is disclosed. The
processor is configured to: receive data representing the one or
more measured parameters; calculate the density .rho. of a mixture
containing water; determine component-specific volume shift
parameters c.sub.i; replace the component specific volume shift
parameter for water with a volume shift parameter c.sub.H2O;
determine a total volume shift c required from the sum of the
component specific volume shifts and the mole fraction of each
component in the mixture; determine a corrected molar volume of the
liquid and vapor by subtracting the total volume shift c from an
equation of state (EOS) calculated molar volume v; and recalculate
density .rho. and compressibility factor Z for the liquid and vapor
phases with the corrected molar volume.
[0007] According to another embodiment, a method of estimating
properties of a reservoir fluid in an underground reservoir is
disclosed. The method includes: calculating the density .rho. of
the mixture containing water; determining component-specific volume
shift parameters c.sub.i; replacing the component specific volume
shift parameter for water with a volume shift parameter c.sub.H2O;
determining a total volume shift c required from the sum of the
component specific volume shifts and the mole fraction of each
component in the mixture; determining a corrected molar volume of
the liquid and vapor by subtracting the total volume shift c from
an equation of state (EOS) calculated molar volume v; and
recalculating density .rho. and compressibility factor Z for the
liquid and vapor phases with the corrected molar volume.
BRIEF DESCRIPTION OF THE DRAWINGS
[0008] The subject matter, which is regarded as the invention, is
particularly pointed out and distinctly claimed in the claims at
the conclusion of the specification. The foregoing and other
features and advantages of the invention are apparent from the
following detailed description taken in conjunction with the
accompanying drawings, wherein like elements are numbered alike, in
which:
[0009] FIG. 1 shows an example drilling system according to one
embodiment; and
[0010] FIG. 2 is a flow diagram illustrating an example of a method
according to one embodiment.
DETAILED DESCRIPTION
[0011] It has been discovered that two-parameter cubic EOS models,
such as the SRK and PR model forms, generally exhibit poor accuracy
in their predictions of liquid-phase density values. While
gas-phase density predictions from cubic EOS models are generally
accurate (less than 5% error), liquid-phase density errors in
excess of 20% are common. This drawback stems from the fact that
only two of the three critical state variables are directly solved
for through two free model parameters (the attractive coefficient a
and repulsive coefficient b) and are formulated so that only two
critical state variables (critical pressure and critical
temperature) of any component are properly represented. Without any
remaining available model parameters to adjust, the third state
variable of critical compressibility factor becomes a consequential
result for the given EOS model form. Note that density, molar
volume, compressibility factor are all related by:
.rho. = M v = MZ ( 8 ) ##EQU00008##
[0012] The fixed or consequential values of critical
compressibility factor for the SRK and PR models are Z.sub.c=0.333
and Z.sub.c=0.307, respectively. However, the physical value of
Z.sub.c for any specific component or mixture may vary and not be
one of these values. As the true value of Z.sub.c for a component
or mixture departs from the Z.sub.c value predicted by the EOS
model, greater liquid-phase density errors are expected. As such,
the SRK and PR EOS models have a demonstrated difficultly
predicting the liquid phase densities of both water
(Z.sub.c,H2O=0.2295) and heavier hydrocarbons (Z.sub.c,C20=0.1865,
Z.sub.c,C48=0.1882) due to the dramatic difference between the
actual critical compressibility values for these compounds and the
fixed or consequential values generated by application of the EOS
models.
[0013] A broadly-adopted strategy for improving the liquid-phase
density prediction of cubic EOS models is to apply a "volume shift"
model to the baseline results initially calculated by the EOS
model. Volume shift models apply a linear correction factor c to
the molar volume value v calculated by the EOS,
v.sub.corrected=v.sub.EOS-c (9)
In the equation above, molar volume is related to density by
v=M/.rho., and thus the corrected molar volume results in a
directly corrected density value as well.
[0014] The two most commonly used volume shift models are given by
Peneloux and Rauzy (Peneloux) for the SRK EOS and by Jhaveri &
Youngren (Jhaveri) for the PR EOS. The Peneloux model form relates
the component critical parameters for the mixture and the component
Rackett factors, Z.sub.RA, in order to formulate the volume
correction parameter c model form:
c = .SIGMA. 1 N c i z i , ( 10 ) c i = 0.40768 ( RT c P c ) (
0.29441 - Z RA ) , ( 11 ) ##EQU00009##
where c.sub.i is the volume correction per component of a mixture,
and z.sub.i is the mole fraction of a component in a mixture. Thus,
for single-component mixture scenarios, c=c.sub.i.
[0015] In the Jhaveri model, the molecular weight of each component
is correlated to the dimensionless volume correction factor
S.sub.i:
c = .SIGMA. 1 N c i z i , ( 12 ) c i = S i b i , ( 13 ) S i = 1 - d
M e , ( 14 ) b i = .OMEGA. b RT c P c , ( 15 ) ##EQU00010##
where d=2.258 and e=0.1823 for n-alkane hydrocarbons (different
values of d and e are also provided for aromatic and naphthenic
hydrocarbons). Neither one of these two model forms directly
addresses the volume shift requirements for water or other polar
molecules. This limitation is resolved by the embodiments disclosed
herein.
[0016] Numerous higher-order volume shifts have been developed and
reported in the open literature, where the volume correction factor
has been formulated as a function of temperature or as a function
of both temperature and density. However, such models simply
introduce further model form error and solution method concerns.
One or more embodiments disclosed herein may reduce such
errors.
[0017] Referring to FIG. 1, an exemplary embodiment of a downhole
drilling, monitoring, evaluation, exploration and/or production
system 10 disposed in a wellbore 12 is shown. A borehole string 14
is disposed in the wellbore 12, which penetrates at least one earth
formation 16 for performing functions such as extracting matter
from the formation and/or making measurements of properties of the
formation 16 and/or the wellbore 12 downhole. The borehole string
14 is made from, for example, a pipe, multiple pipe sections or
flexible tubing. The system 10 and/or the borehole string 14
include any number of downhole tools 18 for various processes
including drilling, hydrocarbon production, and measuring one or
more physical quantities in or around a borehole. Various
measurement tools 18 may be incorporated into the system 10 to
affect measurement regimes such as wireline measurement
applications or logging-while-drilling (LWD) applications.
[0018] In one embodiment, a parameter measurement system is
included as part of the system 10 and is configured to measure or
estimate various downhole parameters of the formation 16, the
borehole 14, the tool 18 and/or other downhole components. The
illustrated measurement system includes an optical interrogator or
measurement unit 20 connected in operable communication with at
least one optical fiber sensing assembly 22. The measurement unit
20 may be located, for example, at a surface location, a subsea
location and/or a surface location on a marine well platform or a
marine craft. The measurement unit 20 may also be incorporated with
the borehole string 12 or tool 18, or otherwise disposed downhole
as desired.
[0019] In the illustrated embodiment, an optical fiber assembly 22
is operably connected to the measurement unit 20 and is configured
to be disposed downhole. The optical fiber assembly 22 includes at
least one optical fiber core 24 (referred to as a "sensor core" 24)
configured to take a distributed measurement of a downhole
parameter (e.g., temperature, pressure, stress, strain and others).
In one embodiment, the system may optionally include at least one
optical fiber core 26 (referred to as a "system reference core" 26)
configured to generate a reference signal. The sensor core 24
includes one or more sensing locations 28 disposed along a length
of the sensor core, which are configured to reflect and/or scatter
optical interrogation signals transmitted by the measurement unit
20. Examples of sensing locations 28 include fibre Bragg gratings,
Fabry-Perot cavities, partially reflecting mirrors, and locations
of intrinsic scattering such as Rayleigh scattering, Brillouin
scattering and Raman scattering locations. If included, the system
reference core 26 may be disposed in a fixed relationship to the
sensor core 24 and provides a reference optical path having an
effective cavity length that is stable relative to the optical path
cavity length of the sensor core 24.
[0020] In one embodiment, a length of the optical fiber assembly 22
defines a measurement region 30 along which distributed parameter
measurements may be taken. For example, the measurement region 30
extends along a length of the assembly that includes sensor core
sensing locations 28.
[0021] The measurement unit 20 includes, for example, one or more
electromagnetic signal sources 34 such as a tunable light source, a
LED and/or a laser, and one or more signal detectors 36 (e.g.,
photodiodes). Signal processing electronics may also be included in
the measurement unit 20, for combining reflected signals and/or
processing the signals. In one embodiment, a processing unit 38 is
in operable communication with the signal source 34 and the
detector 36 and is configured to control the source 34, receive
reflected signal data from the detector 36 and/or process reflected
signal data.
[0022] In one embodiment, the measurement system is configured as a
coherent optical frequency-domain reflectometry (OFDR) system. In
this embodiment, the source 34 includes a continuously tunable
laser that is used to spectrally interrogate the optical fiber
sensing assembly 22.
[0023] The optical fiber assembly 22 and/or the measurement system
are not limited to the embodiments described herein, and may be
disposed with any suitable carrier. That is, while an optical fiber
assembly 22 is shown, any type of now known or later developed
manners of obtaining information relative a reservoir may be
utilized to measure various information (e.g., temperature,
pressure, salinity and the like) about fluids in a reservoir. Thus,
in one embodiment, the measurement system may not employ any fibers
at all and may communicate data electrically.
[0024] A "carrier" as described herein means any device, device
component, combination of devices, media and/or member that may be
used to convey, house, support or otherwise facilitate the use of
another device, device component, combination of devices, media
and/or member. Exemplary non-limiting carriers include drill
strings of the coiled tube type, of the jointed pipe type and any
combination or portion thereof. Other carrier examples include
casing pipes, wirelines, wireline sondes, slickline sondes, drop
shots, downhole subs, bottom-hole assemblies, and drill
strings.
[0025] In support of the teachings herein, various analysis
components may be used, including a digital and/or an analog
system. Components of the system, such as the measurement unit 20,
the processor 38, the processing assembly 50 and other components
of the system 10, may have components such as a processor, storage
media, memory, input, output, communications link, user interfaces,
software programs, signal processors (digital or analog) and other
such components (such as resistors, capacitors, inductors and
others) to provide for operation and analyses of the apparatus and
methods disclosed herein in any of several manners well appreciated
in the art. It is considered that these teachings may be, but need
not be, implemented in conjunction with a set of computer
executable instructions stored on a computer readable medium,
including memory (ROMs, RAMs), optical (CD-ROMs), or magnetic
(disks, hard drives), or any other type that when executed causes a
computer to implement the method of the present invention. These
instructions may provide for equipment operation, control, data
collection and analysis and other functions deemed relevant by a
system designer, owner, user or other such personnel, in addition
to the functions described in this disclosure.
[0026] Further, various other components may be included and called
upon for providing for aspects of the teachings herein. For
example, a power supply (e.g., at least one of a generator, a
remote supply and a battery), cooling unit, heating unit, motive
force (such as a translational force, propulsional force or a
rotational force), magnet, electromagnet, sensor, electrode,
transmitter, receiver, transceiver, antenna, controller, optical
unit, electrical unit or electromechanical unit may be included in
support of the various aspects discussed herein or in support of
other functions beyond this disclosure.
[0027] One embodiment disclosed here consists of a new custom,
higher-order volume shift methodology that is specific to
correcting the density of water. The volume shift calculated is
reported as a volume shift correction factor c.sub.i, so that it
may be used in conjunction with linear Peneloux or Jhaveri volume
shift model forms.
[0028] As discussed above, cubic equations of state (EOS) models
are known to have deficiencies in calculating liquid-phase
densities. This deficiency is remedied by the inclusion of a molar
volume adjustment factor that is applied to the liquid and vapor
phase molar volume once the flash solver has achieved a converged
solution. In particular, a temperature- and pressure-dependent,
water-specific volume shift model is disclosed that may improve the
accuracy of water density calculations. The model calculates the
dimensionless volume shift for water S.sub.H2O as a function of
reduced pressure P.sub.r and reduced temperature T.sub.r. For the
majority of the pressure-temperature phase space, the dimensional
volume shift is given by
S H 2 O = a + b exp ( T r ) + c exp ( P r ) + d exp ( T r ) exp ( P
r ) 1 + f exp ( T r ) + g exp ( P r ) + h exp ( T r ) exp ( P r ) ,
( 16 ) P r = P P c , ( 17 ) and T r = T T c , ( 18 )
##EQU00011##
where the model parameters a-h are provided in Table 2. The
critical parameters of water used are P.sub.c=22.050 MPa and
T.sub.c=647.14 K. The dimensionless volume shift S.sub.H2O may be
converted to a dimensional volume shift c.sub.H2O by:
c.sub.H2O=b.sub.H2OS.sub.H2O, (19)
[0029] where
b H 2 O = .OMEGA. b RT c P c = { 2.114176 .times. 10 - 5 m 3 mol (
SRK ) 1.898464 .times. 10 - 5 m 3 mol ( PR ) ( 20 )
##EQU00012##
[0030] For relatively high temperature, low pressure conditions,
the dimensionless volume shift is modified to the form:
S H 2 O = S H 2 O * exp [ - k ( T r - T r * ) ] , ( 21 ) S H 2 O *
= a + b exp ( T r * ) + c exp ( P r ) + d exp ( T r * ) exp ( P r )
1 + f exp ( T r * ) + g exp ( P r ) + h exp ( T r * ) exp ( P r ) ,
( 22 ) T r * = b ' [ ln ( P r - d ' a ' ) ] - 1 - c ' , ( 23 ) k =
{ 1000 P r .ltoreq. 1.0 10 m exp ( nP r ) P r > 1.0 . ( 24 )
##EQU00013##
[0031] The model coefficients a'-d' and m-n are provided in Table
2.
[0032] This secondary model form is applicable when
P.sub.r<P.sub.r*, where
P r * = a ' exp ( b ' T r + c ' ) + d ' . ( 25 ) ##EQU00014##
TABLE-US-00002 TABLE 2 Custom water volume shift model parameters
optimized for use with SRK and PR EOS models. Model Parameter SRK
EOS PR EOS a 1.686599 .times. 10.sup.-1 9.906580 .times. 10.sup.-2
b -2.414245 .times. 10.sup.-2 -1.499797 .times. 10.sup.-2 c
5.641665 .times. 10.sup.-3 4.148741 .times. 10.sup.-3 d 2.638684
.times. 10.sup.-3 1.960847 .times. 10.sup.-3 f -3.329183 .times.
10.sup.-1 -3.495340 .times. 10.sup.-1 g 3.916869 .times. 10.sup.-2
4.703695 .times. 10.sup.-2 h -3.927259 .times. 10.sup.-3 -6.664902
.times. 10.sup.-3 m 4.1508 4.1508 n -0.34354 -0.34354 a' 7.356856
.times. 10.sup.2 7.356856 .times. 10.sup.2 b' -6.241360 -6.241360
c' -5.620670 .times. 10.sup.-2 -5.620670 .times. 10.sup.-2 d'
-1.061800 .times. 10.sup.-6 -1.061800 .times. 10.sup.-6
[0033] The model parameters were determined by fitting
un-volume-shifted water density data results from calculations with
either the SRK or PR EOS models to water density data from the
IAPWS EOS model. Water density data was generated using the cubic
EOS models and the IAPWS model over a temperature range from
32.degree. F. to 1000.degree. F. at the following isobars: 15, 200,
500, 1000, 2000, 3198, 5000, 6000, 7000, 8000, 10000, 15000, 20000,
and 25000 psia (2815 total data points). The density data from the
cubic EOS models and IAPWS models was converted to molar volume
data by the equation:
v H 2 O ( P , T ) = M H 2 O .rho. H 2 O ( P , T ) ( 26 )
##EQU00015##
where M.sub.H2O=18.015 g/mol. The difference between the calculated
IAPWS and EOS model volumes yields the required total volume shift
that any volume shift model must reproduce:
c.sub.H2O(P,T)=v.sub.H2O.sup.EOS(P,T)-v.sub.H2O.sup.IAPWS(P,T).
(27)
[0034] The required volume shift was then converted to a
dimensionless volume shift form by:
S ^ H 2 O ( P , T ) = c ^ H 2 O ( P , T ) b H 2 O . ( 28 )
##EQU00016##
[0035] The collection of S.sub.H2O(P,T) data points (one data set
for each cubic EOS model form) was then used to determine the model
parameter values listed in Table 2. A nonlinear regression was
performed to minimize the error (relative error squared) between
S.sup.H2O(P,T) and the functional model form of S.sub.H2O(P,T).
[0036] In one embodiment, the volume shift method for water is
implemented by correcting the original un-volume-shifted density
value calculated by a cubic EOS algorithm, in this case,
specifically the SRK and PR EOS algorithms as used in the method
developed here. The overall volume shift algorithm and the
application of the water volume shift method are shown in FIG. 2.
The method of calculating the required volume shift to correct the
cubic EOS prediction of water density occurs in steps 205-206 for
the SRK EOS and steps 216-217 for the PR EOS.
[0037] The disclosed water volume shift method may provide for
accurately calculating water density and has been validated for a
temperature range of 32-1000.degree. F. (273.2-810.9 K) and a
pressure range of 15-25000 psia (0.1034-172.4 MPa). The method is
not intended for predicting the density of solid phase water or
ice, and thus, may not be valid below the triple point of water
(T.sub.t,H2O=273.16 K). For the purpose of implementing this method
in a two-phase flash solver that is not intended to predict solid
phase formation, the required volume shift for water c.sub.H2O at a
(P,T) state where T<T.sub.t,H2O, the value of water density
should be calculated as if T=T.sub.t,H2O.
[0038] In more detail, at block 202 a PR or SRK flash for the
mixture and the measured pressure and temperature is performed.
This may involve, in one embodiment, iteratively solving and
determining the EOS coefficients based on a measured temperature
and pressure. This may generally include solving one or more
equations (1)-(7) above and result in determining values for
liquid-phase density .rho..sub.liq and vapor-phase density
.rho..sub.vap for at least water (and possibly, other
components).
[0039] Block 203 determines whether the SRK or PR EOS model is to
be used. This may be a user defined determination in one
embodiment. In another embodiment, both may be done. In such a case
it shall be understood that portions of the method shown in FIG. 2
may be repeated.
[0040] Assuming that the SRK EOS model is to be used, processing
continues in block 204. At block 204 the component-specific volume
shift parameters c.sub.i are estimated using the Peneloux volume
shift model. In one embodiment, this may include solving:
c i = 0.40768 ( RT c P c ) ( 0.29441 - Z RA ) , ##EQU00017##
where c.sub.i is the volume correction per component of a
mixture.
[0041] At block 205, S.sub.H2O with a water volume shift model with
SRK specific model parameters is calculated. This may include, in
one embodiment, solving equations (16)-(18) above. In cases,
however, where P.sub.r<P.sub.r*, equations (16)-(18) may be
modified as described in equations (21)-(24) above, where P.sub.r*
is found by equation (25).
[0042] At block 206 c.sub.i for water is replaced with c.sub.H2O
per equations (19)-(20) above.
[0043] Alternatively, assuming that the PR EOS model is to be used,
processing continues in block 214. At block 214 the molecular
weight of each component is correlated to the dimensionless volume
correction factor S.sub.i according to the below:
S i = 1 - d M e ##EQU00018##
where d=2.258 and e=0.1823 for n-alkane hydrocarbons (different
values of d and e are also provided for aromatic and naphthenic
hydrocarbons).
[0044] At block 215, for each component c.sub.i is solved per
Jhaveri where
c i = S i b i , b i = .OMEGA. b RT c P c , ##EQU00019##
[0045] At block 216, S.sub.H2O with a water volume shift model with
PR specific model parameters is calculated. This may include, in
one embodiment, solving equations (16)-(18) above. In cases,
however, where P.sub.r<P.sub.r*, equations (16)-(18) may be
modified as described in equations (21)-(24) above, where P.sub.r*
is found by equation (25).
[0046] At block 217 c.sub.i for water is replaced with c.sub.H2O
per equations (19)-(20) above.
[0047] Regardless of how c.sub.H2O is calculated, at block 208 the
total volume shift to be applied to liquid and vapor phases is
calculated. This may include solving:
c.sub.liq=.SIGMA..sub.i.sup.Nx.sub.ic.sub.i; and
c.sub.vap=.SIGMA..sub.i.sup.Ny.sub.ic.sub.i,
where x and y are liquid and vapor compositional mole fractions,
respectively. Thus, for single-component fluid mixture scenarios,
c=c.sub.i.
[0048] At block 210, the liquid and vapor phase molar volumes are
corrected according to:
v.sub.liq.sup.corrected=v.sub.liq.sup.EOS-c.sub.liq; and
v.sub.vap.sup.corrected=v.sub.vap.sup.EOS-c.sub.vap.
[0049] At block 212, the corrected liquid and vapor phase volumes
are utilized to update density and compressibility factors
according to:
.rho. liq corrected = M liq .upsilon. liq corrected ; ##EQU00020##
.rho. vap corrected = M vap .upsilon. vap corrected ;
##EQU00020.2## Z liq corrected = 1 .upsilon. liq corrected ; and
##EQU00020.3## Z vap corrected = 1 .upsilon. vap corrected .
##EQU00020.4##
[0050] Of course, additional processing could also be performed in
one or more embodiments.
[0051] While the invention has been described with reference to
exemplary embodiments, it will be understood that various changes
may be made and equivalents may be substituted for elements thereof
without departing from the scope of the invention. In addition,
many modifications will be appreciated to adapt a particular
instrument, situation or material to the teachings of the invention
without departing from the essential scope thereof. Therefore, it
is intended that the invention not be limited to the particular
embodiment disclosed as the best mode contemplated for carrying out
this invention, but that the invention will include all embodiments
falling within the scope of the appended claims.
* * * * *