Method Documentation
PVTsim Nova
PVTsim Method Documentation Contents 2
Contents
Introduction 7
Introduction ............................................................................................................................... 7
Pure Component Database 8
Pure Component Database ......................................................................................................... 8 Component Classes ..................................................................................................... 8 Component Properties ............................................................................................... 10 User Defined Components ........................................................................................ 12 Missing Properties ..................................................................................................... 12
Composition Handling 14
Composition Handling ............................................................................................................. 14 Types of fluid analyses .............................................................................................. 14 Handling of pure components heavier than C6 .......................................................... 15 Fluid handling operations .......................................................................................... 16 Mixing ....................................................................................................................... 16 Weaving .................................................................................................................... 16 Recombination ........................................................................................................... 16 Characterization to the same pseudo-components ..................................................... 16
QC of Fluid 18
QC of Fluid .............................................................................................................................. 18 Bottomhole samples .................................................................................................. 18 Separator Samples ..................................................................................................... 22 References ................................................................................................................. 27
Flash Algorithms 28
Flash Algorithms ..................................................................................................................... 28 Flash Options ............................................................................................................. 28 Flash Algorithms ....................................................................................................... 29 K-factor Flash ............................................................................................................ 32 Other Flash Specifications ......................................................................................... 33 Phase Identification ................................................................................................... 33 Components Handled by Flash Options .................................................................... 34 References ................................................................................................................. 35
Phase Envelope and Saturation Point Calculation 36
Phase Envelope and Saturation Point Calculation ................................................................... 36 No aqueous components ............................................................................................ 36 Mixtures with Aqueous Components ........................................................................ 37 Components handled by Phase Envelope Algorithm ................................................ 37 References ................................................................................................................. 38
Equations of State 39
Equations of State .................................................................................................................... 39
PVTsim Method Documentation ContentsIntroduction 3
SRK Equation ............................................................................................................ 39 SRK with Volume Correction ................................................................................... 41 PR/PR78 Equation ..................................................................................................... 42 PR/PR78 with Volume Correction ............................................................................ 43 Classical Mixing Rules .............................................................................................. 43 Temperature Dependent Binary Interaction Parameters ............................................ 44 The Huron and Vidal Mixing Rule ............................................................................ 44 PC-SAFT Equation .................................................................................................... 45 PC-SAFT with Association ....................................................................................... 48 Phase Equilibrium Relations ..................................................................................... 49 References ................................................................................................................. 50
Characterization of Heavy Hydrocarbons 52
Characterization of Heavy Hydrocarbons ................................................................................ 52 Classes of Components .............................................................................................. 52 Properties of C7+-Fractions ........................................................................................ 53 Extrapolation of the Plus Fraction ............................................................................. 54 Estimation of PNA Distribution ................................................................................ 55 Grouping (Lumping) of Pseudo-components ............................................................ 56 Delumping ................................................................................................................. 56 Characterization of Multiple Compositions to the Same Pseudo-Components ......... 57 References ................................................................................................................. 58
Thermal and Volumetric Properties 59
Thermal and Volumetric Properties ......................................................................................... 59 Density ...................................................................................................................... 59 Enthalpy .................................................................................................................... 59 Internal Energy .......................................................................................................... 61 Entropy ...................................................................................................................... 61 Heat Capacity ............................................................................................................ 62 Joule-Thomson Coefficient ....................................................................................... 62 Velocity of sound ...................................................................................................... 62 References ................................................................................................................. 62
Transport Properties 63
Transport Properties ................................................................................................................. 63 Viscosity .................................................................................................................... 63 Thermal Conductivity ................................................................................................ 71 Gas/oil Interfacial Tension ........................................................................................ 76 References ................................................................................................................. 76
PVT Experiments 78
PVT Experiments ..................................................................................................................... 78 Constant Mass Expansion .......................................................................................... 78 Differential Liberation ............................................................................................... 79 Constant Volume Depletion ...................................................................................... 79 Separator Experiments ............................................................................................... 79 Viscosity Experiment ................................................................................................ 80 Swelling Experiment ................................................................................................. 80 Equilibrium Contact Experiment ............................................................................... 80 Multiple Contact Experiment .................................................................................... 80 Slim Tube Experiment ............................................................................................... 81 References ................................................................................................................. 84
Compositional Variation due to Gravity 85
Compositional Variation due to Gravity .................................................................................. 85
PVTsim Method Documentation Contents 4
Isothermal Reservoir ................................................................................................................ 85 Reservoirs with a Temperature Gradient ................................................................................. 86
Prediction of Gas/Oil Contacts .................................................................................. 88 References ................................................................................................................. 88
Regression to Experimental Data 89
Regression to Experimental Data............................................................................................. 89 Experimental data ...................................................................................................... 89 Object Functions and Weight Factors........................................................................ 90 Regression for Plus Compositions ............................................................................. 90 Regression for already characterized compositions ................................................... 92 Regression on fluids characterized to the same pseudo-components ........................ 93 Regression Algorithm ................................................................................................ 93 References ................................................................................................................. 93
Minimum Miscibility Pressure Calculations 94
Minimum Miscibility Pressure Calculations ............................................................................ 94 References ................................................................................................................. 95
Unit Operations 96
Unit Operations ........................................................................................................................ 96 Compressor ................................................................................................................ 96 Expander .................................................................................................................... 99 Cooler ........................................................................................................................ 99 Heater ........................................................................................................................ 99 Pump.......................................................................................................................... 99 Valve ......................................................................................................................... 99 Separator .................................................................................................................. 100 References ............................................................................................................... 100
Modeling of Hydrate Formation 101
Hydrate Formation ................................................................................................................. 101 Types of Hydrates ................................................................................................... 101 Hydrate Model ......................................................................................................... 102 Hydrate P/T Flash Calculations ............................................................................... 104
Calculation of Component Fugacities .................................................................................... 105 Fluid Phases ............................................................................................................. 105 Hydrate Phases ........................................................................................................ 105 Ice ............................................................................................................................ 106 References ............................................................................................................... 106
Modeling of Wax Formation 108
Modeling of Wax Formation ................................................................................................. 108 Vapor-Liquid-Wax Phase Equilibria ....................................................................... 108 Extended C7+ Characterization ................................................................................ 109 Viscosity of Oil-Wax Suspensions .......................................................................... 111 Wax Inhibitors ......................................................................................................... 111 References ............................................................................................................... 111
Asphaltenes 112
Asphaltenes ............................................................................................................................ 112 Cubic Equations of State ......................................................................................... 112 PC-SAFT ................................................................................................................. 113 References ............................................................................................................... 114
PVTsim Method Documentation ContentsIntroduction 5
H2S Simulations 115
H2S Simulations ..................................................................................................................... 115 References ............................................................................................................... 116
Water Phase Properties 117
Water Phase Properties .......................................................................................................... 117 Properties of Pure Water ......................................................................................... 117 Properties of Aqueous Mixture................................................................................ 125 Salt Solubility in Pure Water ................................................................................... 129 Salt Solubility Salt-Inhibitor-Water Systems .......................................................... 132 Viscosity of water-oil Emulsions ............................................................................ 133 References ............................................................................................................... 134
Modeling of Scale Formation 135
Modeling of Scale Formation ................................................................................................ 135 Thermodynamic equilibria ...................................................................................... 136 Amounts of CO2 and H2S in water .......................................................................... 139 Activity coefficients of the ions ............................................................................... 139 Calculation procedure .............................................................................................. 145 References ............................................................................................................... 146
Wax Deposition Module 147
Modeling of wax deposition .................................................................................................. 147 Discretization of the Pipeline into Sections ............................................................. 147 Energy balance ........................................................................................................ 147 Overall heat transfer coefficient .............................................................................. 148 Inside film heat transfer coefficient ......................................................................... 149 Outside Film Heat Transfer Coefficient .................................................................. 150 Pressure drop models ............................................................................................... 150 Single-phase flow .................................................................................................... 151 Two-phase flow ....................................................................................................... 152 Mukherjee and Brill pressure drop model ............................................................... 152 Handling of an aqueous phase in the model ............................................................ 154 Wax deposition ........................................................................................................ 154 Boost pressure ......................................................................................................... 155 Porosity .................................................................................................................... 155 Boundary conditions ................................................................................................ 155 Mass Sources ........................................................................................................... 156 References ............................................................................................................... 156
Clean for Mud 157
Clean for Mud ........................................................................................................................ 157 Cleaning Procedure ................................................................................................. 157 Cleaning with Regression to PVT Data ................................................................... 158 References ............................................................................................................... 158
Black Oil Correlations 159
Black Oil Correlations ........................................................................................................... 159 Bubble-point Pressure ............................................................................................. 159 Saturated Gas/Oil Ratio ........................................................................................... 161 Oil Formation Volume Factor ................................................................................. 162 Dead-Oil Viscosity .................................................................................................. 165 Saturated Oil Viscosity ............................................................................................ 166 Gas Formation Volume Factor ................................................................................ 168 Gas Viscosity ........................................................................................................... 169 Nomenclature .......................................................................................................... 171
PVTsim Method Documentation 6
References ............................................................................................................... 172
STARS 173
VISCTABLE ......................................................................................................................... 173 Introduction ............................................................................................................. 173 Outline of Procedure................................................................................................ 174 Generating Artificial Live Oil Viscosity Data from Dead Oil Viscosity Data ........ 174 Generating Tabulation Viscosity Data Points .......................................................... 174 Calculating Component Viscosity Contributions .................................................... 175 Checking for Monotonicity and Performing Corrections ........................................ 176
Allocation 177
Allocation .............................................................................................................................. 177 References ............................................................................................................... 179
PVTsim Method Documentation Introduction 7
Introduction
Introduction
When installing PVTsim the Method Documentation describing the calculation procedures used in PVTsim. is
copied to the installation directory as a PDF document (PVTdoc.pdf). The Methid Documentation may further be
accessed from the <Help> menu in PVTsim. The <Help> menu also gives access to a Users Manual, which during
installation is copied to the PVTsim installation directory as the PDF document PVThelp.pdf.
PVTsim Method Documentation Pure Component Database 8
Pure Component Database
Pure Component Database
The Pure Component Database contains approximately 100 different pure components and pseudo-components
divided into 6 different component classes
Component Classes
PVTsim distinguishes between the component classes
Water
Hydrate inhibitors
Salts
Other inorganic
Organic defined
Pseudo-components
The program is delivered with a pure component database consisting of the following components:
Short Name Systematic Name Formula Name
Water H2O Water H2O
Hydrate inhibitors MeOH Methanol CH4O
EtOH Ethanol C2H6O
PG Propylene-glycol C6H8O2
DPGME Di-propylene-glycol-methylether C7H16O3
MEG Mono-ethylene-glycol C2H6O2
PGME Propylene-glycol-methylether C7H10O2
DPG Di-propylene-glycol C6H14O3
DEG Di-ethylene-glycol C4H10O3
TEG Tri-ethylene-glycol C6H14O4
PVTsim Method Documentation Pure Component Database 9
Glycerol Glycerol C3H8O3
Salts NaCl Sodium chloride NaCl
KCl Potassium chloride KCl
NaBr Sodium bromide NaBr
CaCl2 Calcium chloride (anhydrous) CaCl2
HCOONa Sodium formate (anhydrous) HCOONa
HCOOK Potassium formate (anhydrous) HCOOK
KBr Potassium bromide KBr
HCOOCs Caesium formate (anhydrous) HCOOCs
CaBr2 Calcium bromide (anhydrous) CaBr2
ZnBr2 Zinc bromide ZnBr2
Other inorganic He Helium-4 He(4)
H2 Hydrogen H2
N2 Nitrogen N2
Ar Argon Ar
O2 Oxygen O2
CO2 Carbon dioxide CO2
H2S Hydrogen sulfide H2S
Organic defined C1 Methane CH4
C2 Ethane C2H6
C3 Propane C3H8
c-C3 Cyclo-propane C3H6
iC4 Iso-butane C4H10
nC4 Normal-butane C4H10
2,2-dim-C3 2,2-Dimethyl-propane C5H12
c-C4 Cyclo-propane C4H8
iC5 2-methyl-butane C5H12
nC5 Normal-pentane C5H12
c-C5 Cyclo-pentane C5H8
2,2-dim-C4 2,2-Dimethyl-butane C6H14
2,3-dim-C4 2,3-Dimethyl-butane C6H14
2-m-C5 2-Methyl-pentane C6H14
3-m-C5 3-Methyl-pentane C6H14
nC6 Normal-hexane C6H14
C6 Hexane --------
m-c-C5 Methyl-cyclo-pentane C6H12
Benzene Benzene C6H6
Napht Naphthalene C10H8
c-C6 Cyclo-hexane C6H12
223-tm-C4 2,2,3-Trimethyl-butane C7H16
3,3-dim-C5 3,3-Dimethyl-butane C7H16
2-m-C6 2-Methyl-hexane C7H16
c13-dm-cC5 Cis-1,3-Dimethyl-cyclo-pentane C7H14
t13-dm-cC5 Trans-1,3-Dimethyl-cyclo-pentane C7H14
3-m-C6 3-Methyl-hexane C7H16
t12-dm-cC5 Trans-1,2-Dimethyl-cyclo-pentane C7H14
nC7 Normal-heptane C7H16
m-c-C6 Methyl-cyclo-hexane C7H14
et-c-C5 Ethyl-cyclo-pentane C7H14
113-tr-cC5 1,1,3-Trimethyl-cyclo-pentane C8H16
Toluene Toluene C7H8
2-m-C7 2-Methyl-heptane C8H18
c-C7 Cyclo-heptane C7H14
3-m-C7 3-Methyl-heptane C8H18
PVTsim Method Documentation Pure Component Database 10
11-dm-cC6 1,1-Dimethyl-cyclo-hexane C8H16
c13-dm-cC6 Cis-1,3-Dimethyl-cyclo-hexane C8H16
t12-dm-cC6 Trans-1,2-Dimethyl-cyclo-hexane C8H16
nC8 Normal-octane C8H18
c12-dm-cC6 Cis-1,2-Dimethyl-cyclo-hexane C8H16
Et-cC6 Ethyl-cyclo-hexane C8H16
et-Benzene Ethyl-Benzene C8H10
p-Xylene Para-xylene C8H10
m-Xylene Meta-xylene C8H10
2-m-C8 2-Methyl-octane C9H20
o-Xylene Ortho-xylene C8H10
1m-3e-cC6 1-Methyl-3-Ethyl-cyclo-hexane C9H18
1m-4e-cC6 1-Methyl-4-Ethyl-cyclo-hexane C9H18
c-C8 Cyclo-octane C8H16
4-m-C8 4-Methyl-octane C9H20
nC9 Normal-nonane C9H20
Mesitylene 1,3,5-Tri-methyl-Benzene C9H12
Ps-Cumene 1,2,4-Tri-methyl-Benzene C9H12
nC10 Normal-decane C10H22
Hemellitol 1,2,3-Tri-methyl-Benzene C9H12
nC11 Normal-undecane C11H24
nC12 Normal-dodecane C12H26
nC13 Normal-tridecane C13H28
1-m-Napht 1-methyl-Naphthalene C11H10
nC14 Normal-tetradecane C14H30
nC15 Normal-pentadecane C15H32
nC16 Normal-hexadecane C16H34
nC17 Normal-heptadecane C17H36
nC18 Normal-octadecane C18H38
nC19 Normal-nonadecane C19H40
nC20 Normal-eicosane C20H42
nC21 Normal-C21 C21H44
… … …
nCn Normal-Cn CnH2n+2
… … …
nC40 Normal-C40 C40H82
The database furthermore contains carbon number fractions from a C7 to C100. Each fraction Cn consists of all
components with a boiling point in the interval from that of nCn-1 + 0.5°C/0.9°F to that of nCn + 0.5°C/0.9°F.
Finally the database contains the components CHCmp_1 to CHCmp_6, which are dummy pseudo-components and
only accessible when working with characterized fluids. The only properties given in the database are the molecular
weight, a and b. The molecular weight will usually have to be modified by the user. All other component
properties must be entered manually.
Component Properties
For each component the database holds the component properties
Name (short, systematic, and formula)
Molecular weight
Liquid density at atmospheric conditions (not needed for gaseous components)
PVTsim Method Documentation Pure Component Database 11
Critical temperature (Tc)
Critical pressure (Pc)
Acentric factor ( )
Normal boiling point (Tb)
Weight average molecular weight (equal to molecular weight unless for pseudo-components)
Critical volume (Vc)
Vapor pressure model (classical or Mathias-Copeman)
Mathias-Copeman coefficients (only available for some components)
Temperature independent and temperature dependent term of the volume shift (or Peneloux) parameter for the
SRK or PR equations
Ideal gas absolute enthalpy at 273.15 K/0°C/32°F (Href)
Coefficients in ideal gas heat capacity (Cp) polynomial
Melting point temperature (Tf)
Melting point depression (Tf)
Enthalpy of melting (Hf)
PNA distribution (only for pseudo-components)
Wax fraction (only for n-paraffins and pseudo-components)
Asphaltene fraction (only for pseudo-components)
Parachor
Hydrate formation indicator (None, I, II, H and combinations)
Hydrate Langmuir constants
Number of ions in aqueous solution (only for salts)
Number of crystal water molecules per salt molecule (only for salts)
Pc of wax forming fractions (only for n-paraffins and pseudo-components)
a and b in the SRK and PR equations
The component properties needed to calculate various physical properties and transport properties will usually be
established as a part of the fluid characterization. It is however also possible to input new components without
entering all component properties and it is possible to input compositions in characterized form.
Tc, Pc, , a, b and molecular weight are required input for all components to perform simulations. What other
component properties are needed depend on the simulation to be performed and may be seen from the below table.
Physical or transport property Component properties needed
Volume Peneloux parameter*1)
Density Peneloux parameter*1)
Z factor Peneloux parameter*1)
Enthalpy (H) Ideal gas CP coefficients, Peneloux parameter*1)
Entropy (S) Ideal gas CP coefficients, Peneloux parameter
*1)
Heat capacity (CP) Ideal gas CP coefficients
Heat capacity (CV) Ideal gas CP coefficients, Peneloux parameter*1)
Kappa (CP/ CV) Ideal gas CP coefficients, Peneloux parameter*1)
Joule-Thomson coefficient Ideal gas CP coefficients, Peneloux parameter*1)
Velocity of sound Peneloux parameter*1)
Viscosity Weight average molecular weight*2)
, Vc*3)
PVTsim Method Documentation Pure Component Database 12
Thermal conductivity
Surface tension Parachor, Peneloux parameter*1)
*1) Only if an equation of state with Peneloux volume correction is used.
*2) Only if corresponding states viscosity model selected.
*3) Only if LBC viscosity model selected.
User Defined Components
User defined components may be added to the database. It is recommended to enter as many component properties
for new components as possible. The following properties must always be entered
Component type
Name
Critical temperature (Tc)
Critical pressure (Pc)
Acentric factor ()
a and b
Molecular weight (M)
For pseudo-components it is highly recommended also to enter the liquid density.
Missing Properties
PVTsim has a <Complete> option for estimating missing component properties for a fluid composition entered in
characterized form. The number of missing properties estimated depends on the properties entered manually. It is
assumed that Tc, Pc, , a, b, and molecular weight have all been entered. Below is shown what other properties
are needed to estimate a given missing property and a reference is given to the section in the Method Documentation
where the property correlation is described.
Property Component properties
needed for estimation
Section where described
Liquid density T independent term of Peneloux
parameter
SRK with Volume Correction. PR
with Volume Correction.
Normal boiling point None Extrapolation of Plus Fraction.
Weight average molecular weight Assumed equal to number average
molecular weight
-
Critical volume None Lohrenz-Bray-Clark (LBC) part of
Viscosity section.
Vapor pressure model Not estimated -
Mathias-Copeman coefficients Not estimated -
T-independent term of SRK or PR
Peneloux parameter for defined components. Liquid
density for pseudo-components
SRK with Volume Correction or PR
with Volume Correction
T-dependent term of SRK or PR
Peneloux parameter
Not estimated for defined
components. Liquid density for
SRK with Volume Correction or PR
with Volume Correction
PVTsim Method Documentation Pure Component Database 13
pseudo-components
Melting point depression
(Tf)
Only for pseudo-components.
Viscosity data for an
uninhibited/inhibited fluid.
Ideal gas absolute enthalpy at 273.15
K/0°C/32°F (Href)
Molecular weight Compositional variation due to
gravity
Ideal gas Cp coefficients Not estimated for defined
components. Liquid density for
pseudo-components
Enthalpy
Melting temperature (Tf) Irrelevant for defined components.
None for pseudo-components
Extended C7+ Characterization
Enthalpy of melting (Hf) Irrelevant for defined components.
None for pseudo-components
Extended C7+ Characterization
PNA distribution Irrelevant for defined components.
Liquid density for pseudo-
components
Estimation of PNA Distribution
Wax fraction Irrelevant for defined components.
None for pseudo-components.
Extended C7+ Characterization
Asphaltene fraction Irrelevant for defined components.
Liquid density for pseudo-
components
Asphaltenes
Parachor Not estimated for defined
components. Liquid density for
pseudo-components
Gas/Oil interfacial tension.
Hydrate former or not Not estimated -
Hydrate Langmuir constants Not estimated -
Number of ions in aqueous solution
(only for salts)
Not estimated -
Number of crystal water molecules
per salt molecule (only for salts)
Not estimated -
Pc of wax forming fraction Irrelevant for defined components.
Liquid density for pseudo-
components
Extended C7+ Characterization
PVTsim Method Documentation Composition Handling 14
Composition Handling
Composition Handling
PVTsim distinguishes between the fluid types
Compositions with Plus fraction
Compositions with No-Plus fraction
Characterized compositions
Compositions with Plus fraction are compositions as reported by PVT laboratories where the last component is a
plus fraction residue. A C20+ fraction for example contains the carbon number fractions from C20 and heavier. For
this type of composition the required input is mole%’s of all components and molecular weights and densities of the
C7+ components (carbon number fractions). It is possible to enter the mole%’s to a higher carbon number than
molecular weights and densities. If the mole%’s are given to C20 and the molecular weights and densities to C10, the
program will interpret the molecular weight and density entered for C10 as properties of the whole C10+ fraction.
Compositions with No-Plus fraction require the same input as compositions with a plus fraction. In this case the
heaviest component is not a residue but an actual component or a boiling point cut. Gas mixtures with only a
marginal content of C7+ components are to be usually classified as No-Plus fraction compositions.
Simulations can only be made on characterized compositions. These are usually generated from a Plus fraction or
No-Plus fraction type of composition. They may alternatively be entered manually.
Types of fluid analyses
A reservoir fluid may either be sampled as a bottom hole sample or as a separator sample. Bottom hole samples are
taken in the bottom of the well and are usually single-phase at sampling conditions and therefore representative for
the reservoir fluid. A separator sample consists of two samples, a separator gas and a separator oil from a well head
separator.
In the laboratoy the samples are flashed to standard conditions before making any analyses. Flashing the oil results
in a gas and a liquid sample that are analyzed separately. The gas will always be analyzed by a gas chromatographic
(GC) analysis. Two alternative types of fluid analyses are used for the liquid. These are a gas chromatographic (GC)
PVTsim Method Documentation Composition Handling 15
analysis and a true boiling point (TBP) analysis. None of these analyses will identify all the chemical species
contained in the fluid but will separate the C7+ fraction into boiling point cuts.
GC analysis
Also oil compositions are often analyzed by GC. It is relatively cheap, very fast, and requires only small sample
volumes. A GC analysis suffers from the problem that heavy ends may be lost in the analysis, especially heavy
aromatics (asphaltenes). The main problem with a GC analysis is however that no information is retained on
molecular weight (M) and density of the cuts above C9. Instead standard molecular weights and densities are
assigned to the heavier fractions. This may results in large uncertainties on the molecular weight and density of the
plus fractions. Because the component quantities measured in a GC analysis are on weight basis, this uncertainty
also transfers to an uncertainty on the mole% of the plus fraction.
A GC composition may for example consist of mole%’s given to C30+ while molecular weight and density are only
given to C7+. In this case one may enter the mole%'s to C30 together with the M and density of the total C7+ fraction,
leaving the M and density fields blank for C8-C30. With this input the program will estimate the molecular weights
and densities of the fractions C7-C30 while honoring the reported composition and matching the input C7+ molecular
weight and density. One may as an alternative input the composition (the mole%’s) lumped back to C7+, which will
often provide equally accurate simulation results as with the detailed GC composition.
TBP Distillation
A TBP distillation requires a larger sample volume, typically 50 – 200 cc and is more time consuming than a GC
analysis. The method separates the components heavier than C6 into fractions bracketed by the boiling points of the
normal alkanes. For instance, the C7 fraction refers to all species with a boiling point between that of nC6 +
0.5C/0.9°F and that of nC7 + 0.5C/0.9°F, regardless of how many carbon atoms these components contain. Each
of the fractions distilled off is weighed and the molecular weight and density are determined experimentally. The
density and molecular weight in combination provide valuable information to the characterization procedure. The
residue from the distillation is also analyzed for amount, M and density.
Whenever possible, it is recommended that input for PVTsim is generated based on a TBP analysis. The accuracy of
the characterization procedure relies on good values for densities and molecular weights of the C7+ fractions.
Parameters such as the Peneloux volume shift for the heavier pseudo-components are estimated based on the input
densities, and consequently the quality of the input directly affects the density predictions of the equation of state
(EOS) model.
Handling of pure components heavier than C6
When the compositional input is based on a GC analysis, there will often be defined components (pure chemical
species) reported, which in the TBP-terminology would belong to a boiling point cut. Such components may be
entered alongside with the boiling point fraction, which then represents the remaining unresolved species within that
boiling point interval. Before the entered composition is taken through the characterization procedure, the pure
species are lumped into their respective boiling point fraction and the properties of that fraction adjusted
accordingly. After the characterization, the pure species and the remaining fraction (pseudo-component) are split
again and the properties adjusted accordingly.
PVTsim Method Documentation Composition Handling 16
Fluid handling operations
Quite often there is a need to mix two or more fluids and continue simulations with the mixed composition. PVTsim
supports a ‘Mixing’, a ‘Weaving’ and a ‘Recombination’ option for combining two or more fluid compositions.
Mixing
PVTsim may be used to mix 2 to 50 fluid compositions. A mixing will not necessarily retain the pseudo-components
of the individual compositions. Averaging the properties of the pseudo-components in the individual compositions
generates new pseudo-components. Mixing may be performed on all types of compositions. For fluids characterized
in PVTsim, mixing is done at the level where the fluid has been characterized but not yet lumped. The mixed not yet
lumped fluid is afterwards lumped to the specified number of components.
Weaving
Weaving will maintain the pseudo-components of the individual compositions and can only be performed for already
characterized compositions. In a weaved fluid all pseudo-components from all the original fluids are maintained in
the resulting weaved fluid. This may lead to several components having the same name, and it is therefore advisable
to tag the component names before weaving in order to avoid confusion later on. The weaving option is useful to
track specific components in a process simulation or for allocation studies.
Recombination
Recombination is a mixing on volumetric basis performed for a given P and T (usually separator conditions).
Recombination can only be performed for two compositions, an oil and a gas composition. The recombination option
is often used to combine a separator gas phase and a separator oil phase to get the feed to the separator. When the
two fluids are recombined, the GOR and liquid density at separator conditions must be input. Alternatively the
saturation point of the recombined fluid can be entered along with the liquid density. When the GOR is specified, the
program determines the number of moles corresponding to the input volumes and mixes the two fluids based on this
molar ratio. When the saturation pressure is specified, the amount of gas tobe added to the oil to yield this saturation
pressure is determined in an iterative manner.
Characterization to the same pseudo-components
The goal of characterizing fluids to the same pseudo-components is to obtain a number of fluids, which are all
represented by the same component set. Numerically this is done in a similar fashion as the mixing operation with
the only difference that the same pseudos logic keeps track of the molar amount of each pseudo-component
contained in each individual fluid.
The characterization to the same pseudo-components option is useful for a number of tasks. In compositional
pipeline simulations where different streams are mixed during the calculations or in compositional reservoir
simulations where zones with different PVT behavior are considered, mixing is straightforward when all fluids have
the same pseudo-components. It is furthermore possible to do regression in combination with the characterization to
the same pseudos, in which case one may put special emphasis on fluids for which PVT data sets are available.
PVTsim Method Documentation Composition Handling 17
Characterization to same pseudo-components is described in more detail in the section on Characterization of Heavy
Hydrocarbons.
PVTsim Method Documentation QC of Fluid 18
QC of Fluid
QC of Fluid High quality PVT simulation results on petroleum reservoir fluids are heavily dependent on representative and
accurate fluid compositions. The characterization procedure in PVTsim (Pedersen et al. (1992) and Krejbjerg and
Pedersen (2006)) generally provides PVT simulations results in good correspondence with experimental data. When
a bad correspondence is seen with experimental PVT data, the reason could be an inaccurate reservoir fluid
composition.
The PVTsim QC module is designed to analyze reservoir fluid compositions for any inconsistencies between
compositional analyses, sampling data and basic PVT data.
Reservoir fluid samples can either be
Bottomhole samples
Separator samples
The approach to QC evaluation is dependent on the sample type. All conducted QC evaluations must pass for the
sample to pass the overall QC evaluation.
Bottomhole samples
The following input is mandatory to conduct a QC on a bottomhole sample. The information should be readily
available in a PVT report
Molar composition of reservoir fluid sample. The composition must be a Plus composition
Either Reservoir Pressure or Bottom Hole Flowing Pressure
Reservoir Temperature
STO Oil Density (Single Stage Flash)
GOR (Single Stage Flash)
Saturation Pressure at Reservoir Temperature
Reservoir Fluid Type
The following additional, optional information can be entered when available from the PVT report
FVF (single stage flash)
PVTsim Method Documentation QC of Fluid 19
The QC evaluation scheme for a bottomhole sample is
No. Evaluation Always
performed
Only when
enough data
1 Single Phase at Sampling Conditions X
2 GOR X
3 STO Oil Density X
4 FVF X
5 Fluid Type X
6 Saturation Pressure at Tres X
7 Ln(mol%) vs. Carbon Number X
8 Possible OBM Contamination X
9 Pus Fraction Mole/Mass X
10 Plus Fraction Density X
11 Plus Fraction Molecular Weight X
In the following the QC evaluations are described in terms of
Simulation method
Accepted deviation between measured and simulated results
Possible key sources in case of failure are listed in the QC report with suggestions on how to correct the sample to
pass the QC.
1 – Single Phase at Sampling Conditions
The saturation pressure must be lower than reservoir pressure and/or bottom hole flowing pressure. This is required
for the sample to be single phase at sampling conditions.
2 - GOR
The GOR from a single stage flash of the bottomhole composition at standard conditions is compared with the input
GOR.
The evaluation fails if the deviation exceeds ± 10%. The same applies if a single phase is detected at standard
conditions.
3 - STO Oil Density (Single Stage Flash)
The bottomhole composition is flashed to standard conditions (typically 1.01 bara/15°C or 14.7 psia/59°F), and the
density of the flashed liquid compared with the input STO Oil density.
The evaluation fails if the deviation exceeds ± 4%.
The evaluation will also fail if a single-phase gas is detected at standard conditions.
4 – FVF (Saturated at Tres to standard Conditions)
FVF is the ratio of the oil volume at the saturation pressure at the reservoir temperature and the oil volume from a
flash to standard conditions (typically 1.01 bara/15°C or 14.7 psia/59°F).
PVTsim Method Documentation QC of Fluid 20
The evaluation fails if the deviation between the reported and the simulated FVF exceeds ± 5%.
5 - Fluid Type
The following should apply
Critical temperature less than reservoir temperature plus 20 K: Fluid Type: Gas or gas condensate.
Critical temperature higher than reservoir temperature minus 20K; Fluid Type: Oil or heavy oil.
Critical temperature within 20K from reservoir temperature; Fluid Type: Near Critical.
The test is only performed on fluids with one simulated critical point.
6 - Saturation Pressure at Tres
The simulated saturation pressure at reservoir temperature is compared with the input saturation pressure.
The evaluation fails if the deviation exceeds ± 15%.
7 – Ln(Mol%) vs. Carbon Number
For most reservoir fluids the logarithm of the mole% of the C7+ fractions versus carbon number will follow an
almost straight line (Pedersen et al., 1992). With a fluid composition to for example C20+ an almost straight line is
to be expected for logarithm of the mole% of C7-C19 versus carbon number. A best fit line should have a coefficient
R2 above 0.80 for the fluid to pass the test.
For heavy oils, the carbon number, at which the logarithmic decay starts, is dependent on the STO API Gravity of
the heavy oil. Based on the findings by Krejbjerg and Pedersen (2006), the following equation can be derived
198.225492.0CNB API
where CNB is the carbon number where the logarithmic decay begins for heavy oils, and API is the API gravity
measured for the heavy oil.
For gases, gas condensates and oils the test is not performed unless the fluid composition is given to at least C20+.
For heavy oils, the fluid analysis must be to at least CNB +
8 - Possible OBM Contamination
For most (clean) reservoir fluids the logarithm of the mole fraction of the C7+ versus carbon number will follow an
almost straight line (Pedersen et al. (1992)). With a fluid composition to for example C20+ an almost straight line is
to be expected for logarithm of the mole% of C7-C19 versus carbon number.
The evaluation is conducted by a calculation of the best-fit straight line through the logarithm of the mole fraction of
the C11+ fractions (except plus component) versus carbon number.
The evaluation will fail if the average deviation of the component mole%’s above the trend line is more than 100%
higher than the average deviation of the component mole%’s below the trend line.
For condensates and oils the test is not performed unless the fluid composition is given to at least C20+. For heavy
oils, the fluid analysis must be to at least CNB + 4.
9 - Plus Fraction Mole/Mass
By extrapolation of the best-fit line in the logarithmic mole fraction vs. carbon number plot, an estimate can be
provided of the plus component amount (C20+ if the compositional analysis ends at C20+). The plus component
amount is calculated from
PVTsim Method Documentation QC of Fluid 21
max
zC
C
iz
where C+ is plus fraction carbon number and Cmax is C80 (C200 for heavy oils). The zi up to Cmax are found from
CnBA)Ln(z i
where Cn is carbon number and the constants A and B are found from the best-fit line of Ln(mol%) vs carbon
number plot.
A deviation of more than -50/+100% from the reported plus component amount will make the evaluation fail.
10 - Plus Component Density
By extrapolation of the best-fit line in the logarithmic mole fraction vs. carbon number plot, an estimate can be
provided of the plus component density. The plus component density is calculated from
max
max
C
C
C
C
i
ii
ii
Mwz
Mwz
where C+ is plus fraction carbon number and Cmax is C80 (C200 for heavy oils). The densities of the carbon
number fractions contained in the plus fraction are found from
Ln(Cn)DCi
where Cn is carbon number and the constants C and D are found from a best-fit line of density versus ln(carbon
number) for the carbon number fractions except the plus fraction. A best fit should have a coefficient R2 above 0.85
for the fluid to pass the test.
A deviation of more than ± 5% from the reported plus density will make the test fail.
11 - Plus Component Molecular Weight
By extrapolation of the best-fit line in the logarithmic mole fraction vs. carbon number plot, an estimate can be
provided of the plus component molecular weight. The plus component molecular weight is calculated from
max
max
C
C
C
C
i
ii
z
Mwz
Mw
where C+ is plus fraction carbon number and Cmax is C80 (C200 for heavy oils). The molecular weights of the
carbon number fractions contained in the plus fraction are calculated from
4 -Cn 41i Mw
PVTsim Method Documentation QC of Fluid 22
where Cn is carbon number. A deviation of more than ± 25% from the reported plus molecular weight will make the
evaluation fail.
Separator Samples
A separator sample is taken from a separator operating at elevated P and T. The separator oil is in the PVT
laboratory flashed to standard (typically 1.01 bara/15°C or 14.7 psia/59°F) in a single stage.
The following input is mandatory if a QC evaluation is to be conducted for a separator sample. The information
should be readily available in a PVT report.
Molar composition of separator gas and oil. The oil must be a Plus composition and the gas either a plus or
a No Plus composition
Molar composition of recombined fluid
Separator Pressure
Separator Temperature
STO Oil Density
Separator GOR
Reservoir Temperature
Reservoir Fluid Type (Oil must be chosen if the STO API Gravity is above 25 °API and Heavy Oil must be
chosen if the STO API Gravity is below 25 °API).
The following additional information can optionally be entered
Separator Gas Opening Pressure and Temperature
PVTsim Method Documentation QC of Fluid 23
The QC evaluation scheme for a separator sample is
No. QC Evaluation Always
performed
Only when
enough data
1 Separator GOR X
2 STO Oil Density X
3 FVF Separator Oil X
4 Separator Conditions X
5 Gas Opening Pressure X
6 K-Factor Plot X
7 Fluid Type X
8 Mass Balance Check X
9 Separator Oil Saturation Pressure X
10 Separator Gas Saturation Temperature X
11 Ln (Mole%) vs. Carbon Number X
12 Plus Fraction Mole/Mass X
13 Plus Fraction Density X
14 Plus Fraction Molecular Weight X
15 Hoffmann Plot (*) X
(*) Not considered in overall evaluation
In the following the QC evaluations are described in terms of
Simulation method
Accepted deviation between measured and simulated results
Possible key sources in case of failure are listed in the QC report with suggestions on how to correct the sample to
pass the QC.
1 - Separator GOR
The recombined fluid is flashed to separator conditions. The gas from this flash is flashed to standard conditions
(typically 1.01 bara/15 ⁰C and 14.7 psia/59 ⁰F) and so is the oil. The separator GOR is the volume of the separator
gas at standard conditions divided by the oil from the flash of the separator oil to standard conditions.
The evaluation will fail if the simulated separator GOR deviates by more than ± 10% from the reported separator
GOR.
2 - STO Oil Density
The recombined separator sample is flashed to at standard conditions (typically 1.01 bara/15°C or 14.7 psia/59°F),
and the density of the flashed liquid compared with the input STO Oil density.
The evaluation will fail if the deviation exceeds ± 4%.
3 – FVF Separator Oil
FVF Separator Oil is the ratio of the oil volume at the separator conditions and the oil volume from a flash of the
separator oil to standard conditions (typically 1.01 bara/15°C or 14.7 psia/59°F).
PVTsim Method Documentation QC of Fluid 24
The evaluation fails if the deviation between the reported and the simulated FVF exceeds ± 5%.
4 - Separator Conditions
Phase envelopes for the separator gas and the separator oil should ideally meet at the separator P and T. In the QC
module the deviation between the simulated separator P and T and the reported separator conditions are defined as
22
100Deviation
rep
repsim
rep
repsim
T
TT
P
PP
The evaluation will fail if the deviation exceeds 20%. The test will also fail if the phase envelopes do not intersect.
5 – Gas Opening Pressure
The opening pressure of the gas sample at the laboratory can be calculated from a VT flash if the opening
temperature is known. The molar volume of the gas is first calculated by a PT flash of the gas at separator
conditions. The opening pressure is calculated by a VT flash with this molar volume and the opening temperature as
input
The evaluation will fail if the deviation exceeds ± 5%.
6 - K-Factor Plot
The K-factor of component I is determined through
i
i
x
yiK
where yi is the mole fraction of the i’th component in the separator gas, and xi is the mole fraction of the i’th
component in the separator oil.
To check whether the sampled separator compositions were at equilibrium at separator conditions the K-factors of
the sampled compositions may be compared with the K-factors of the compositions from a flash of the recombined
fluid to separator conditions.
The test should ideally yield a straight line (y=x) when plotting the simulated K-factors against the reported K-
factors. Only defined components are included in the test since heavier components are not always contained in both
separator gas and separator liquid analysis. N2 is not included in this evaluation, the reason being that sample
cylinders may be contaminated with N2.
The line coefficient R2 must be above 0.98 to pass the K-Factor Plot – Linearity test.
Furthermore the y-value for x=0 should be in the interval from -0.05 to 0.05 and the y-value for x=1 should be in the
interval from 0.9 to 1.1.
7 - Fluid Type
The following should apply
Critical temperature less than reservoir temperature plus 20 K: Fluid Type: Gas or gas condensate.
Critical temperature higher than reservoir temperature minus 20K; Fluid Type: Oil or heavy oil.
Critical temperature within 20K from reservoir temperature; Fluid Type: Near Critical.
The test is only performed on fluids with one simulated critical point.
8 - Mass Balance Check
PVTsim Method Documentation QC of Fluid 25
A recombination of the separator gas and oil according to the separator GOR should give the composition of the
recombined reservoir fluid composition in the PVT report. The mass balance over a separator is given by
iii x1z y
where zi is the mole fraction of component i in the feed to the separator, yi is the mole fraction of the i’th component
in the separator gas, xi is the mole fraction of the i’th component in the separator oil, and β is the vapor fraction.
Watanasiri et al. (1982) rewrites this equation to
1
z
x1
z
y
i
i
i
i
which shows that plotting yi/zi against xi/zi should yield a straight line. The line should be downward sloping as 0 ≤ β
≤ 1. Only defined components are included in the test since heavier components are not always contained in both
separator gas and separator liquid analysis.
The line coefficient R2 must be at least 0.98 to pass the Mass Balance Check.
9 - Separator Oil Saturation Pressure
The saturation pressure of the separator oil at the separator temperature should ideally equal the separator pressure.
The saturation pressure of the oil at separator temperature is calculated and must be within ± 10% of the separator
pressure for the test to pass.
10 - Separator Gas Saturation Temperature
The saturation temperature of the separator gas at the separator pressure should ideally equal the separator
temperature.
The saturation temperature of the gas at separator pressure is calculated and must not deviate by more than -10/+5%
from the separator temperature for the test to pass. The reason for a too low simulated saturation temperature could
be that the gas analysis was not extended to heavy components, which is not a serious problem. A too high simulated
saturation temperature may on the other hand signal liquid carryover in the sampled gas, which is more serious. That
is the reason a 10% too low simulated saturation temperature is accepted whereas it is not accepted that the
saturation temperature is more than 5% too high.
11 – Ln(mol%) vs. Carbon Number (Recombined Fluid)
For most reservoir fluids the logarithm of the mole fraction of C7+ fractions (except the plus component) versus
carbon number will follow an almost straight line (Pedersen et al., 1992). With a fluid composition to for example
C20+ an almost straight line is to be expected for logarithm of the mole% of C7-C19 versus carbon number. A best
fit should have a coefficient R2 above 0.80 for the fluid to pass the test.
For heavy oils, the carbon number, at which the logarithmic decay starts, is dependent on the STO API Gravity of
the heavy oil. Based on the findings by Krejbjerg and Pedersen (2006), the following equation can be derived
198.225492.0CNB API
where CNB is the carbon number where the logarithmic decay begins for heavy oils, and API is the API gravity
measured for the heavy oil.
For gases, gas condensates and oils the test is not performed unless the fluid composition is given to at least C20+.
For heavy oils, the fluid analysis must be to at least CNB + 4.
12 - Plus Fraction Mole/Mass (Recombined Fluid)
By extrapolation of the best-fit line in the logarithmic mole fraction vs. carbon number plot, an estimate can be
provided of the plus component amount (C20+ if the compositional analysis ends at C20+). The plus component
amount is calculated from
PVTsim Method Documentation QC of Fluid 26
max
zC
C
iz
where C+ is plus fraction carbon number and Cmax is C80 (C200 for heavy oils). The zi up to Cmax are found from
CnBA)Ln(z i
where Cn is carbon number and the constants A and B are found from the best-fit line of Ln(mol%) vs carbon
number plot.
A deviation of more than -50/+100% from the reported plus amount will make the test fail.
13 - Plus Component Density
By extrapolation of the best-fit line in the logarithmic mole fraction vs. carbon number plot, an estimate can be
provided of the plus component density. The plus component density is calculated from
max
max
C
C
C
C
i
ii
ii
Mwz
Mwz
where C+ is plus fraction carbon number and Cmax is C80 (C200 for heavy oils). The densities of the carbon
number fractions contained in the plus fraction are found from
Ln(Cn)DCi
where Cn is carbon number and the constants C and D are found from a best-fit line of density versus ln(carbon
number) for the carbon number fractions except the plus fraction. A best fit should have a coefficient R2 above 0.85
for the fluid to pass the test.
A deviation of more than ± 5% from the reported plus density will make the test fail.
14 - Plus Component Molecular Weight
By extrapolation of the best-fit line in the logarithmic mole fraction vs. carbon number plot, an estimate can be
provided of the plus component molecular weight. The plus component molecular weight is calculated from
max
max
C
C
C
C
i
ii
z
Mwz
Mw
where C+ is plus fraction carbon number and Cmax is C80 (C200 for heavy oils). The molecular weights of the
carbon number fractions contained in the plus fraction are calculated from
4 -Cn 41i Mw
where Cn is carbon number. A deviation of more than ± 25% from the reported plus molecular weight will make the
test fail.
15 - Hoffmann Plot
PVTsim Method Documentation QC of Fluid 27
The Hoffmann Plot (Hoffmann et al. (1953)) is an alternative/supplement to the K-factor plot for determining
whether the given separator gas and oil compositions are in equilibrium at separator conditions.
The correlation is given by
sepib TT
11b)
P
PLog(K
,std
sep
i
where Ki is the K-factor of component i, Psep is the separator pressure, Pstd is the standard pressure (typically 1.01
bara/14.7 psia), Tb,i is the normal boiling point of component i, Tsep is the separator temperature and b is a
parameter given by
icib TT ,,
std
ic,
11
)P
PLog(
b
where Pc,i is the critical pressure of component i and Tc,i is the critical temperature of component i. Finally, α and β
are the slope and the intercept of the straight line respectively.
The Hoffmann Plot is included in the QC module because it is an accepted QC standard in the oil industry, Whitson
and Brulé (2000) have shown that the Hoffmann correlation can be derived from the Wilson Equation for
approximate K-factors (Wilson, 1966) when the Edmister correlation (Edmister, 1958) is used to determine the
acentric factor in the Wilson equation. Being an approximate correlation it is less refined than the K-factor plot
evaluations and therefore not assigned any importance in the overall QC evaluation.
References
Hoffmann, A. E., Crump, J. S. and Hocott, C. R., “Equilibrium Constants for a Gas condensate System”, Petroleum
Transactions, AIME 198, 1953, pp. 1-10.
Krejbjerg, K., Pedersen, K.S., “Controlling VLLE Equilibrium With a Cubic EoS in Heavy Oil Modeling”,
Presented at the 7th Canadian International Petroleum Conference, Calgary, Alberta, Canada, June 13-15, 2006.
Pedersen, K.S., Blilie, A.L., Meisingset, K.K., “PVT Calculations on Petroleum Reservoir Fluids Using Measured
and Estimated Compositional Data for the Plus Fraction”, I&EC Research, 31, 1992, pp. 1378-1384.
Watanasiri, S., Brulé, M.R., Starling, K.E., “Correlation of Phase-Separation Data for Coal-Conversion Systems”,
AIChE Journal, 28, 1982, pp. 626-637.
Whitson, C., Brulé, M.R., “Phase Behavior”, SPE Monograph, Volume 20, SPE, 2000, pp. 41-42.
Wilson, G. M., "A Modified Redlich-Kwong Equation of State, Application to General Physical Data Calculation",
Paper No. 15C presented at the 1969 AIChE 65th National Meeting, Cleveland, Ohio, March 4-7, 1969.
PVTsim Method Documentation Flash Algorithms 28
Flash Algorithms
Flash Algorithms
The flash algorithms of PVTsim are the backbone of all equilibrium calculations performed in the various simulation
options. The different flash options are described in the following. A more detailed description can be found in
Michelsen and Mollerup (2004).
The input to a PT flash calculation consists of
Molar composition of feed (z)
Flash specifications (e.g. Pressure (P) and temperature (T))
The flash result consists of
Number of phases
Amounts and molar compositions of each phase
Physical properties and transport properties of each phase.
Flash Options
PVTsim supports the flash options
PT non aqueous (gas and oil)
PT aqueous (gas, oil, and aqueous)
PT multi phase (gas, max. two oils, and aqueous)
PH where H is enthalpy (gas, oil, and aqueous)
PVTsim Method Documentation Flash Algorithms 29
PS where S is entropy (gas, oil, and aqueous)
VT where V is molar volume (gas, oil, and aqueous)
UV where U is internal energy (gas, oil, and aqueous)
HS (gas, oil, and aqueous)
P where is hydrocarbon vapor mole fraction of total hydrocarbon phase(s) (gas, oil, and aqueous)
T (gas, oil, and aqueous)
K - factor (gas and oil)
Split - factor (gas and oil)
Specific PT flash options considering the appropriate solid phases are used in the hydrate, wax, and asphaltene
options.
Flash Algorithms
PVTsim uses the PT flash algorithms of Michelsen (1982a, 1982b). They are based on the principle of Gibbs energy
minimization. In a flash process a mixture will settle in the state at which its Gibbs free energy
N
1iiiμnG
is at a minimum. ni is the number of moles present of component i and i is the chemical potential of component i.
The chemical potential can be regarded as the “escaping tendency” of component i, and the way to escape is to form
an additional phase. Only one phase is formed if the total Gibbs energy increases for all possible trial compositions
of an additional phase. Two or more phases will form, if it is possible to separate the mixture into two phases having
a total Gibbs energy, lower than that of the single phase. With two phases (I and II) present in thermodynamic
equilibrium, each component will have equal chemical potentials in each phase
II
i
I
i μμ
The final number of phases and the phase compositions are determined as those with the lowest total Gibbs energy.
The calculation that determines whether a given mixture at a specified (P,T) separates into two or more phases is
called a stability analysis. The starting point is the Gibbs energy, G0, of the mixture as a single phase
G0 = G(n1, n2, n3,……,nN)
ni stands for the number of moles of type i present in the mixture, and N is the number of different components.
PVTsim Method Documentation Flash Algorithms 30
The situation is considered where the mixture separates into two phases (I and II) of the compositions (n1 -1 , n2 --,
n3 - -3 …., nN-N) and (1 , 2 , 3,……,N) where i is small. The Gibbs energy of phase I may be approximated by a
Taylor series expansion truncated after the first order term
N
1ini
ii01
n
GεGG
The Gibbs energy of the second phase is found to be
GII = G ((1 , 2 , 3,……,N)
The change in Gibbs energy due to the phase split is hence
N
1i0iIIii0iIIi
N
1ii0III ))(μ)((μyε))(μ(μεGGGΔG
where
N
1iiε and yi is the mol fraction of component i in phase II. The sub-indices 0 and II refer to the single
phase and to phase II, respectively. Only one phase is formed if G is greater than zero for all possible trial
compositions of phase II. The chemical potential, i, may be expressed in terms of the fugacity, fi, as follows
)P1nlnzRT(1nμf1nRTμμ ii
0
ii
0
ii
where 0 is a standard state chemical potential, a fugacity coefficient, z a mole fraction, P the pressure, and the
sub-index i stands for component i. The standard state is in this case the pure component i at the temperature and
pressure of the system. The equation for G may then be rewritten to
N
1i0iiIIiii ))1n(zln)1n(y(1ny
εRT
ΔG
where zi is the mole fraction of component i in the total mixture. The stability criterion can now be expressed in
terms of mole fractions and fugacity coefficients. Only one phase exists if
N
1i0iiIIiii 0))ln(zln)ln(y(lny
for all trial compositions of phase II. A minimum in G will at the same time be a stationary point. A stationary point
must satisfy the equation
k)ln(lnz)ln(yln 0iiIIii
where k is independent of component index. Introducing new variables, Yi, given by
ln Yi = ln yi – k
the following equation may be derived
1n Yi = 1n zi + 1n(i)0 – 1n(i)II
PVTsim Method Documentation Flash Algorithms 31
PVTsim uses the following initial estimate (Wilson, 1969) for the ratio Ki between the mole fraction of component i
in the vapor phase and in the liquid phase
)
T
T(15.373exp
P
PK cici
i
where
Ki= yi/xi
and Tci is the critical temperature and Pci the critical pressure of component i. As initial estimates for Yi are used Kizi,
if phase 0 is a liquid and zi/Ki, if phase 0 is a vapor. The fugacity coefficients, (i)II, corresponding to the initial
estimates for Yi are determined based on these fugacity coefficients, new Yi-value are determined, and so on. For a
single-phase mixture this direct substitution calculation will either converge to the trivial solution (i.e. to two
identical phases) or to Yi-values fulfilling the criterion
N
1ii 1Y
which corresponds to a non-negative value of the constant k. A negative value of k would be an indication of the
presence of two or more phases. In the two-phase case the molar composition obtained for phase II is a good starting
point for the calculation of the phase compositions. For two phases in equilibrium, three sets of equations must be
satisfied. These are
1) Materiel balance equations
N1,2,3,...,i,zxβ1βy iii
2) Equilibrium equations
N1,2,3,...,i,xy L
ii
V
ii
3) Summation of mole fractions
N
1iii 0)x(y
In these equations xi, yi and zi are mole fractions in the liquid phase, the vapor phase and the total mixture,
respectively. is the molar fraction of the vapor phase. V
i and L
i are the fugacity coefficients of component i in the
vapor and liquid phases calculated from the equation of state. There are (2N + 1) equations to solve with (2N + 3)
variables, namely (x1, x2, x3,…, xN), (y1, y2, y3,….,yN), , T and P. With T and P specified, the number of variables
equals the number of equations. The equations can be simplified by introducing the equilibrium ratio or K-factor, Ki
= yi/xi. The following expressions may then be derived for xi and yi
N1,2,3,...,i,xKy
N1,2,3,...,i,1Kβ1
zx
iii
i
ii
and for Ki
N1,2,3,...,i,KV
i
L
ii
The above (2N+1) equations may then be reduced to the following (N+1) equations
N1,2,3,...,i,ln
lnKln
V
i
L
ii
PVTsim Method Documentation Flash Algorithms 32
i
N
1iiiiii 01))β(K1)/(1(Kz)x(y
For a given total composition, a given (T, P) and Ki estimated from the stability analysis, an estimate of may be
derived. This will allow new estimates of xi and yi to be derived and the K-factors to be recalculated. A new value of
is calculated and so on. This direct substitution calculation may be repeated until convergence. For more details on
the procedure it is recommended to consult the articles of Michelsen (1982a, 1982b).
For a system consisting of J phases the mass balance equation is
0H
1)(KzN
1i i
imi
where
1)(Kβ1H1j
1m
m
i
m
i
m is the molar fraction of phase m. m
iK equals the ratio of mole fractions of component i in phase m and phase J.
The phase compositions may subsequently be found from
N1,2,3,...,i,H
zy
J1,2,3,...,mN;1,2,3,...,i,H
Kzy
i
iJ
i
i
m
iim
i
where m
iy and J
iy are the mole fractions of component i in phase m and phase J, respectively.
K-factor Flash
The Flash option and some of the interface options in PVTsim support K-factor and Split-factor flashes. The K-
factor of component i is the mole fraction of component i in the vapor phase (yi) divided by the mole fraction (xi) of
component i in the liquid phase (i.e. Ki=yi/xi). The Split-factor of component i equals the molar amount of
component i in the vapor phase divided by the molar amount of component i in the feed composition. Split-factor are
converted to K-factors and the below N+1 equations solved.
1) Materiel balance equations
N1,2,3,...,i,zxβ1βy iii
2) Summation of mole fractions
N
1i
N
1ii
ii
ii 0)1(K1
)1(Kz)x(y
In the multiphase meter interface in PVTsim full flash calculations are carried out for the individual separator stages.
The total separation is then converted to overall K-factors and these are used to calculate the black oil properties
written out by this interface option.
PVTsim Method Documentation Flash Algorithms 33
Other Flash Specifications
P and T are not always the most convenient flash specifications to use. Some of the processes taking place during oil
and gas production are not at a constant P and T. Passage of a valve may for example be approximated as a constant
enthalpy (H) process and a compression as a constant entropy (S) process. The temperature after a valve may
therefore be simulated by initially performing a PT flash at the conditions at the inlet to the valve. If the enthalpy is
assumed to be the same at the outlet, the temperature at the outlet can be found from a PH flash with P equal to the
outlet pressure and H equal to the enthalpy at the inlet. A PT flash followed by a PS flash may similarly be used to
determine an approximate temperature after a compressor.
To perform a PH or a PS flash, PVTsim starts with a temperature of 300 K/26.85°C/80.33°F. Two object functions
are defined. These are for a two-phase PH flash
N
1iiii1 1)ζ(Kzg
spec2 HHg
where
1Kβ1ς ii
H is total molar enthalpy for the estimated phase compositions, and Hspec is the specified molar enthalpy. At
convergence both g1 and g2 are zero.
Other flash specifications are VT, UV and HS. V is the molar volume and T the absolute temperature. A VT
specification is useful to for example determine the pressure in an offshore pipeline during shutdown. U is the
internal energy. A dynamic flow problem may sometimes more conveniently be expressed in U and V than in P and
T.
Michelsen (1999) has given a detailed description on how to perform flash calculations with other specification
variables than P and T.
Phase Identification
If a PT flash calculation for an oil or gas mixture shows existence of two phases, the phase of the lower density will
in general be assumed to be gas or vapor and the phase of the higher density liquid or oil. In the case of a single-
phase solution it is less obvious whether to consider the single phase to be a gas or a liquid. There exists no generally
accepted definition to distinguish a gas from a liquid. Since the terms gas and oil are very much used in the oil
industry, a criterion is needed for distinguishing between the two types of phases.
The identification criterion used in PVTsim is
Liquid if
The pressure is lower than the critical pressure and the temperature lower than the bubble point temperature.
PVTsim Method Documentation Flash Algorithms 34
The pressure is above the critical pressure and the temperature lower than the critical temperature.
Gas if
The pressure is lower than the critical pressure and the temperature higher than the dew point temperature.
The pressure is above the critical pressure and the temperature higher than the critical temperature.
In the flash options handling water, a phase containing more than 80 mole% total of the components water, hydrate
inhibitors and salts is identified as an aqueous phase.
Components Handled by Flash Options
The non-aqueous PT-flash option handles the following component classes
Other inorganic
Organic defined
Pseudo-components
The PT aqueous and multiflash options handle
Water
Hydrate inhibitors
Other inorganic
Organic defined
Pseudo-components
Salts
The PH, PS, and HS flash options handle
Water
Hydrate inhibitors
Other inorganic
Organic defined
Pseudo-components
Salts
The VT and UV flash options handle
Water
Hydrate inhibitors
Other inorganic
PVTsim Method Documentation Flash Algorithms 35
Organic defined
Pseudo-components
The T and P flash options handle
Water
Hydrate inhibitors
Other inorganic
Organic defined
Pseudo-components
Salts
References
Michelsen, M.L., “The Isothermal Flash Problem. Part I: Stability”, Fluid Phase Equilibria 9, 1982a, 1.
Michelsen, M.L., “The Isothermal Flash Problem. Part II: Phase-Split Calculation”, Fluid Phase Equilibria 9, 1982b,
21.
Michelsen, M.L., “State Based Flash Specification”, Fluid Phase Equilibria 158-161, 1999, pp. 617-626.
Michelsen, M. L. and Mollerup, J., “Thermodynamic Models: Fundamental and Computational Aspects”, Tie-Line
Publication, Holte, Denmark, 2004.
Wilson, G. M., “A Modified Redlich-Kwong Equation of State, Application to General Physical Data Calculation”,
Paper No. 15C presented at the 1969 AIChE 65th
National Meeting, Cleveland, Ohio, March 4-7, 1969.
PVTsim Method Documentation Phase Envelope and Saturation Point Calculation 36
Phase Envelope and Saturation Point Calculation
Phase Envelope and Saturation Point Calculation
No aqueous components
A phase envelope consists of corresponding values of T and P for which a phase fraction of a given mixture equals
a specified value. The phase fraction can either be a mole fraction or a volume fraction. The phase envelope option
in PVTsim (Michelsen, 1980) may be used to construct dew and bubble point lines, i.e. corresponding values of T
and P for which equals 1 or 0, respectively. Also inner lines (0 < < 1) may be constructed.
The construction of the outer phase envelope ( = 1 and = 0) and inner molar lines follows the procedure outlined
below. The first (T, P) value of a phase envelope is calculated by choosing a fairly low pressure (P). The default in
PVTsim is 5 bara/4.93 atm/72.52 psia. An initial estimate of the equilibrium factors (Ki = yi/xi) is obtained from the
following equation
)
T
T5.42(1exp
P
PK cici
i
This relation and the mass balance equation
N
1i
N
1iiiiii 01))β(K1)/(1(Kz)x(y
are solved for T and equal to the specified vapor mole fraction. The correct value of T is subsequently calculated
by solving this equation in conjunction with
V
i
L
ii
ln
lnlnK
where the liquid (L) and vapor (V) phase fugacity coefficients, , are found using the equation of state.
PVTsim Method Documentation Phase Envelope and Saturation Point Calculation 37
An initial estimate of the second point on the phase envelope is calculated using the derivatives of T and Ki with
respect to P calculated in the first point. The correct solution is again found by solving the above equations.
From the third point and on the extrapolation is based on the two latest calculated points and the corresponding
derivatives. This stepwise calculation is continued until the temperature is below the specified lower temperature
limit.
In simulations of PVT experiments, knowledge of the complete phase envelope is not needed but only the saturation
pressure at the temperature of the experiment. A saturation point is also located through a phase envelope
calculation. A critical point may be considered a special type of saturation point, and the critical point is easily
identified as a point where the lnKi changes sign. Some fluids have more than one critical point. The critical point is
furthermore verified by a more direct method as described by Michelsen and Heidemann (1981).
The basic phase envelope option only considers two phases (one gas and one liquid). For many reservoir fluid
mixtures a PT-region exists with 3 phases (1 gas and 2 liquids). This is for example often the case for gas condensate
mixtures at low temperatures. The phase envelope option in PVTsim allows a check to be performed of the possible
existence of a 3-phase region.
For fluids with no aqueous components (i.e. water, hydrate inhibitors or salts) it is possible to obtain other phase
envelope diagrams than the traditional PT-phase envelope diagram. PVTsim allows combinations of the following
properties on the axes of the phase envelope diagram
Pressure (P)
Temperature (T)
Enthalpy (H)
Entropy (S)
Volume (V)
Internal Energy (U)
Mixtures with Aqueous Components
Only the outer lines (=1 and 0) will be located for mixtures containing aqueous components. The phases
considered are (hydrocarbon) gas, (hydrocarbon) liquid and aqueous. The mutual solubility between all phases is
taken into account. The algorithm is described by Lindeloff and Michelsen (2003).
Components handled by Phase Envelope Algorithm
The algorithm handles components belonging to the classes
Other inorganic
Organic defined
Pseudo-components.
PVTsim Method Documentation Phase Envelope and Saturation Point Calculation 38
Water (no inner lines)
Hydrate inhibitors (no inner lines)
The saturation point algorithm used in the saturation point option and the PVT simulations is also based on the phase
envelope algorithm, but does not handle water and hydrate inhibitors.
References
Lindeloff, N. and Michelsen, M.L., “Phase Envelope Calculations for Hydrocarbon-Water Mixtures”, SPE 85971,
SPE Journal, September 2003, pp. 298-303.
Michelsen, M.L., “Calculation of Phase Envelopes and Critical Points for Multicomponent Mixtures”, Fluid Phase
Equilibria 4, 1980, pp. 1-10.
Michelsen, M.L. and Heidemann, R.A., “Calculation of Critical Points from Cubic Two-Constant Equations of
State”, AIChE J. 27, 1981, pp. 521-523.
PVTsim Method Documentation Equations of State 39
Equations of State
Equations of State
The phase equilibrium calculations in PVTsim are based on one of the following equations
Soave-Redlich-Kwong (SRK) (Soave, 1972)
Peng-Robinson (PR) (Peng and Robinson, 1976)
Modified Peng-Robinson (PR78) (Peng and Robinson, 1978)
All equations may be used with or without Peneloux volume correction (Peneloux et al., 1982). A constant or a
temperature dependent Peneloux correction may be used. The temperature dependent volume correction is
determined to comply with the ASTM 1250-80 correlation for volume correction factors for stable oils (Pedersen et
al., 2002).
For selected models like asphaltene and, Flash and PVT simulations, the PC-SAFT equation (Chapman et al. 1988
and 1990) may be used.
SRK Equation
The SRK equation takes the form
b)V(V
a(T)
bV
RTP
where P is the pressure, T the temperature, V the molar volume, R the gas constant and a and b are equation of state
parameters, which for a pure component are determined by imposing the critical conditions
pointcrit.T2
2
T 0))V
P()
V
P((
The following relation is then obtained for parameter a of component i at the critical point
PVTsim Method Documentation Equations of State 40
ci
2
ci
2
aciP
TRΩa
and for parameter b of component i
ci
ci
biP
TRΩb
where
aΩ = 0.42748
bΩ = 0.08664
Tci is the critical temperature of component i and Pci the critical pressure. Values for Tc, Pc and may be seen from
the PVTsim pure component database. All the values except those for salts are taken from Reid et al. (1977). The
values for the salts are chosen to ensure that these components remain in the aqueous phase (Sørensen et al. (2002)
and Pedersen and Milter (2004)).
The temperature dependence of the a-parameter is expressed in the form of a term ai(T), which multiplied with aci
gives the final expression for the a-parameter of the SRK-equation
ai(T) = acii(T)
The parameter is by default obtained from the following expression
20.5
c
iT
T1m1(T)α
where
2
iii 0.176ω1.574ω0.480m
It is seen that i(T) equals 1 at the critical temperature at which temperature ai(T) therefore becomes equal to aci. is
the acentric factor that is defined as follows (Pitzer, 1955)
1Plogω0.7T
Vap
ri10ir
where Vap
riP is the reduced vapor pressure of component i (vapor pressure divided by critical pressure).
PVTsim Method Documentation Equations of State 41
An alternative temperature dependence as suggested by Mathias and Copeman (1983) may be applied
1T,)T1CT1CT1C(1α(T) r
23
r3
2
r2r1
1T,))T(1C(1(T)α r
2
r1
It is seen that the proposed temperature dependence reduces to the default (classical) one for C1 = m and C2 = C3 = 0.
In general the Mathias-Copeman (M&C) expression offers a more flexible temperature dependence than the classical
expression. It can therefore be used to represent more complicated pure component vapor pressure curves than is
possible with the classical expression. M&C is not used default in PVTsim, but is it possible for the user to change
temperature dependence from classical to M&C and to enter M&C coefficients (C1, C2 and C3) when these are not
given in the PVTsim database. The M&C coefficients used in PVTsim are from Dahl (1991).
SRK with Volume Correction
With Peneloux volume correction the SRK equation takes the form
2cbVcV
a
bV
RTP
The SRK molar volume, V~
, and the Peneloux molar volume, V, are related as follows
cVV~
The b parameter in the Peneloux equation b~
is similarly related to the SRK b-parameter as follows
cb~
b
The parameter c can be regarded as a volume translation parameter and is given by the following equation
c = c’ + c’’ (T – 288.15)
where T is the temperature in K. The parameter c’ is the temperature independent volume correction and c’’ the
temperature dependent volume correction. Per default the temperature dependent volume correction c’’ is set to zero
unless for C+ pseudo-components. In general the temperature independent Peneloux volume correction for defined
organics and “other organics” is found from the following expression
RA
c
c Z0.29441P
RT0.40768c'
where ZRA is the Racket compressibility factor
PVTsim Method Documentation Equations of State 42
ZRA = 0.29056 – 0.08775
For some components, e.g. H2O, MEG, DEG, TEG, and CO2, the values have been found from pure component
density data. For heavy oil fractions c is determined in two steps. The liquid density is known at 15°C/59°F from the
composition input. By converting this density () to a molar volume V = M/, the c’ parameter can be found as the
difference between this molar volume and the SRK molar volume for the same temperature. Similarly c’’ is found as
the difference between the molar volume at 80°C/176°F given by the ASTM 1250-80 density correlation and the
Peneloux molar volume for the same temperature, where the Peneloux volume is found assuming c=c’.
PR/PR78 Equation
The PR/PR78 equations both take the form
bVbbVV
a(T)
bV
RTP
where
a(T) = ac (T)
c
2
c
2
acP
TRΩa
20.5
cT
T1m1α(T)
c
cb
P
TRb
and
aΩ = 0.45724
bΩ = 0.07780
The parameter m is for the PR equation found from
20.269922 - 1.54226 0.37464 m
With the PR78 equation m is found from the same correlation if 0.49. Otherwise the below correlation is used
PVTsim Method Documentation Equations of State 43
m = 0.379642 + (1.48503 0.164423 + 0.016662)
The Mathias-Copeman temperature dependence presented in the SRK section may also be applied with both the
Peng-Robinson equation and the Peng-Robinson 78 equation.
PR/PR78 with Volume Correction
With Peneloux volume correction the PR and PR78 equations become
bVcbb2cVcV
a(T)
bV
RTP
where c is a temperature dependent constant as presented in the SRK section. In general the temperature independent
Peneloux volume correction for defined organics and “other organics” is found from
)Z(0.25969P
RT0.50033c' RA
c
c
where ZRA is defined as for the Peneloux modification of the SRK equation. For other components c’ is found as
explained in the SRK section, which also explains how to determine the temperature dependent term c”.
For some components, e.g. H2O, MEG, DEG, TEG, and CO2, the values have been found from pure component
density data. For heavy oil fractions c is determined in two steps. The liquid density is known at 15°C/59°F from the
composition input. By converting this density () to a molar volume V = M/, the c’ parameter can be found as the
difference between this molar volume and the PR molar volume for the same temperature. Similarly c’’ is found as
the difference between the molar volume at 80°C/176°F given by the ASTM 1250-80 density correlation and the
Peneloux molar volume for the same temperature, where the Peneloux volume is found assuming c=c’.
Classical Mixing Rules
The classical mixing rules for a, b and c are
N
1i
N
1jijji azza
N
1iii bzb
N
1iiiczc
where zi and zj are mole fractions, i and j component indices, and
PVTsim Method Documentation Equations of State 44
ijjiij k1aaa
The parameter kij is a binary interaction.
The greater part of the interaction coefficients in the PVTsim database has been found in Knapp et al. (1982).
The option exists to calculate interaction parameters from critical volumes using the following equation (Chueh and
Prausnitz, 1967)
n
3
1
cj3
1
ci
3
1
cj3
1
ci
ij
VV
VV21k
In PVTsim the exponent n is user-specified with a default value of 1.
Temperature Dependent Binary Interaction Parameters
PVTsim supports temperature dependent kij’s. The following temperature dependence is used
kij =kij_A+CNj (kij_B+kij_C (T-T0))
where kij_A, kij_B and kij_C are user input, T0 is a reference temperature of 288.15 K and CNj the carbon number of
component j.
The Huron and Vidal Mixing Rule
For binary pairs of components of which at least one is polar, the classical mixing rule is often insufficient for the a-
parameter. In PVTsim the mixing rule suggested by Huron and Vidal (H&V) (1979) is default used for most
interactions with water, fluid inhibitors and salts. The H&V a-parameter mixing rule takes the form
N
1i
E
i
i
iλ
G
b
azba
where is specific for the selected equation of state. For SRK and PR the values for are
12
12ln
22
1λ:PR
ln2λ:SRK
PVTsim Method Documentation Equations of State 45
EG is the excess Gibbs energy at infinite pressure. EG is found using a modified NRTL mixing rule
N
liN
1kkikikk
N
1jjijijjji
i
E
ταexpzb
ταexpzbτ
zRT
G
where ji is a non-randomness parameter, i.e. a parameter taking into account that the mole fraction of molecules of
type i around a molecule of type j may deviate from the overall mole fraction of molecules of type i in the mixture.
When ji is zero, the mixture is completely random. The parameter is defined by the following expression
RT
ggτ
iiji
ji
where gij is an energy parameter characteristic of the j-i interaction. In PVTsim the g-parameters are temperature
dependent and given by the expression (Pedersen et al., 2001)
gji – gii = (gji – gii)’ + T (gji – gii)”
The parameter b entering into the expression for EG is the b-parameter of the equation of state. The classical mixing
rule is used for the b-parameter.
The local composition of a binary pair that can be described using the classical mixing rule, will not deviate from the
overall composition, i.e. ji should be chosen equal to zero. By further selecting the following expressions for the
interaction energy parameters
λb
ag
i
iii
ij
0.5
jjii
ji
ji
ji k1ggbb
bb2g
the H&V mixing rule reduces to the classical one. When the H&V mixing rule is used, the latter expressions are
therefore used for gij and gii of binary pairs not requiring the advanced mixing rule. This gives a continuous
description of both hydrocarbons and aqueous components.
PC-SAFT Equation
The PC-SAFT equation of state was first introduced in PVTsim in the asphaltene module as the result of an
Asphaltene JIP carried out with industry sponsors. PC-SAFT stands for Perturbed Chain Statistical Association Fluid
Theory (Chapman et al. (1988 and 1990) and (Gross and Sadowski (2001)).
The PC-SAFT model expresses the compressibility factor as a deviation from the ideal gas compressibility factor of
1.0
disphc ZZ1Z
PVTsim Method Documentation Equations of State 46
Zhc
is the hard-chain contribution to the compressibility factor accounting for repulsive molecular interactions and
Zdisp
is an attractive (dispersive) term.
Each molecule is represented through 3 parameters
Number of segments: m
Segment diameter:
Segment energy:
The number of segments is 1 for methane. For heavier hydrocarbons it is a little lower than the number of
hydrocarbon segments.
PC-SAFT sees a pure fluid as consisting of equal-sized hard-spheres or segments. These hard-spheres are then
combined to hard-chain molecules. The hard-chain molecules interact with each other.
The hard chain term to the PC-SAFT compressibility factor is expressed as
ρ
lng
g
ρ1)(mxZmZ
hsii
hsii
i
N
1i
ihshc
where N is the number of components, xi the mole fraction of component i and
i
N
1i
imxm
Zhs
is the hard sphere contribution to Zhc
, which term is expressed as
330
323
32
230
21
3
3hs
)(1
33
)(1
3
1Z
where
nii
N
1i
in (T)dmxρ6
π
The parameter n may take the values 0, 1, 2, and 3. The term packing fraction is used for 3. The temperature
dependent diameter, d, is expressed through
kT
3ε0.12exp1σ(T)d i
ii
The term is the total number density of molecules
N
1i
3
iii
3
dmxπ
6ζρ
while hs
iig in is the molar radial pair distribution function for two segments of component i in the hard sphere system.
The radial pair distribution function takes the general for segments of component i and j
3
3
2
2
2
i
i
3
2
i
i
3
hs
ij)(1
2
d
d
)(1
3
d
d
1
1g
j
j
j
j
d
d
d
d
PVTsim Method Documentation Equations of State 47
The radial pair distribution function is a measure of the probability of finding a particle in a given distance from a
fixed particle in the fluid. The density derivative of the radial distribution function may be found from
4
3
3
2
2
3
3
2
2
2
i
i
3
3
32
2
3
2
i
i
2
3
3
hs
ij
)(1
6
)(1
4
d
d
)(1
6
)(1
3
d
d
1
g
j
j
j
j
d
d
d
d
PC-SAFT uses the following expression for the dispersion contribution to the compressibility factor, Zdisp
322232
3
231
32
3
13disp σεmIζCζ
IζCmπρεσm
ζ
Iζ2πZ
where
233
43
33
233
4
3
233
1)ζ(2ζ1
2ζ12ζ27ζ20ζ)m(1
ζ1
2ζ8ζm1C
3
33
323
33
5
3
3232
12)ζ(2ζ1
4048ζ12ζ2ζ)m(1
ζ1
820ζ4ζmCC
3N
1i
N
1j
ij
jiji
32 σkT
εmmxxεσm
ij
3N
1i
N
1j
2
ij
jiji
322 σkT
εmmxxσεm
ij
j3
6
0j
j1 ζmaI
j3
6
0j
j2 ζmbI
The cross energy term ij equals
)k(1εεε ijjiij
and
)σ(σ2
1σ jiij
where kij is a binary interaction parameter. Finally the terms )m(a j and )m(b j equal
2j1j0jj am
2m
m
1ma
m
1ma)m(a
2j1j0jj bm
2m
m
1mb
m
1mb)m(b
The universal constants for a0j, a1j, a2j, b0j, b1j and b2j are given in the below table.
j a0i a1j a2j b0i b1j b2j
0 0.9105631445 -0.3084016918 -0.0906148351 0.7240946941 -0.5755498075 0.0976883116
1 0.6361281449 0.1860531159 0.4527842806 2.2382791861 0.6995095521 -0.2557574982
2 2.6861347891 -2.5030047259 0.5962700728 -4.0025849485 3.8925673390 -9.1558561530
3 -26.547362491 21.419793629 -1.7241829131 -21.003576815 -17.215471648 20.642075974
4 97.759208784 -65.255885330 -4.1302112531 26.855641363 192.67226447 -38.804430052
PVTsim Method Documentation Equations of State 48
5 -159.59154087 83.318680481 13.776631870 206.55133841 -161.82646165 93.626774077
6 91.297774084 -33.746922930 -8.6728470368 -355.60235612 -165.20769346 -29.666905585
PC-SAFT with Association
For aqueous components an association term is used with PC-SAFT. This term is added to the perturbation in Z so
that the equation takes the form
assocdisphc ZZZZ 1
where Zassoc
is the contribution from association.
To calculate the effect contribution of associaton 2 additional pure components are needed for each associating
component
Association energy: iiBA
ii
Association volume: iiBA
ii
Zassoc
can be derived from the contribution of association to the Helmholtz energy, which can be written on the
following form
i
ii
A
AA
i
i XXn2
1
2
1)ln(
RT
Aassoc
i is the index for components and Ai is the index of association sites on component i. XAi is the fraction of sites of
type A on component i, that is not bonded to other sites
j B
BA
Bj
A
j
ji
j
i
XnVX
)/1(1
1
The assocciation strength,jiBA
, between site A on component i and site B on component j are calculated from the
association energy and the association volume
1exp3
kTg
ji
jiji
BA
ijBA
ijij
hs
ij
BA
The combining rules employed for the cross-association energy and volume are those suggested by Wolbach and
Sandler
2
ijii
ji
BA
jj
BA
iiBA
ij
3
21
jjii
jjiiBA
jj
BA
ii
BA
ijjjiiji
Currently fixed parameters are used for associating components. The parameters and association schemes used are
shown in the below table. (Values are taken from various sources.)
Name m σ (Å) ε/k (K) εAiBj/k (K) κAiBj Scheme
Water 2.1945 2.229 141.66 1804.17 0.2039 4C
PVTsim Method Documentation Equations of State 49
Methanol 1.88238 3.0023 181.77 2738.03 0.054664 2B
Ethanol 2.3827 3.1771 198.24 2653.4 0.032384 2B
MEG 1.90878 3.5914 325.23 2080.03 0.0235 4C
DEG 3.05823 3.6143 310.29 2080.03 0.0235 4C
TEG 3.18092 4.0186 333.17 2080.03 0.0235 4C
PG 2.33917 3.6351 284.62 2080.03 0.0235 4C
Glycerol 1.5728 4.1901 554.73 4364.57 0.0007 2B
DPG 3.2435 3.7575 187.84 4469.34 0.010795 3B
PGME 3.5966 3.2182 154.82 2531.97 0.09821 3B
DPGME 3.1354 3.9782 174.78 3482.77 0.017518 4B
PC-SAFT binary interaction parameters of 0 are used for the components in the above table internally and versus any
other component.
If any other hydrate inhibitors than those appearing in the above tabled are contained in the fluid, an error message is
returned and no calculation is done.
Note that the PC-SAFT parameters for H2O differ from those shown in the Fluid View. Those parameters are
assuming that H2O is not associating and are not used in the calculations.
Phase Equilibrium Relations
In case of two phases, each component will have equal fugacities, fi, in both phases
L
i
V
i ff
The following general thermodynamic relation exists for determination of the fugacity coefficient
V
nV,T,ii lnZdVRT/VnP/1/RTlnj
where ni is the number of moles of type i. The following relation exists for the fugacity coefficient derived from the
SRK equation with classical mixing rules
)Z
Bln(1
b
b)k(1aza2
a
1
B
A
b
b1)(ZB)ln(Zln i
ij
N
1jji
ii
j
For the PR equation the expression for the fugacity coefficient takes the form
1)b(2Z
1)b(2Zln
b
b)k(1aza2
a
1
B2
A
b
b1)(ZB)ln(Zln
0.5
0.5
iij
N
1jji1.5
ii
j
A and B in these expressions are given by
22 TR
PTaA
PVTsim Method Documentation Equations of State 50
RT
bPB
The fugacity coefficient can also be derived for the PC-SAFT equation.
With two phases present, the phase compositions are related to the total composition as follows
1Kβ1
zx
i
ii
1Kβ1
zKy
i
iii
where zi is the mole fraction of component i in the total mixture and is the molar vapor phase fraction.
For details on how to determine the number of phases and on how to determine the amounts of each phase, the P/T
flash section should be consulted.
References
Chapman, W. G., Jackson, G. and Gubbins, K. E., “Phase Equilibria of Associating Fluids. Chain Molecules with
Multiple Bonding Sites”, Mol. Phys 65, 1988, pp. 1057-1079.
Chapman, W. G., Gubbins, K. E., Jackson, G. and Radosz, M., “New Reference Equation of State for Associating
Liquids”, Ind. Eng. Chem. Res. 29, 1990, pp. 1709-1721.
Chueh, P.L., and Prausnitz, J.M., “Vapor-Liquid Equilibrium at High Pressures: Calculation of Partial Molar
Volumes in Non-Polar Liquid Mixtures”, AIChE Journal 13, 1967, pp. 1099-1107.
Dahl, S., “Phase Equilibria for Mixtures Containing Gases and Electrolytes”, Ph.D. thesis, Department of Chemical
Engineering, Technical University of Denmark, 1991.
Gross, J. and Sadowski, G., “Perturbed-Chain SAFT: An Equation of State Based on Pertubation Theory for Chain
Molecules”, Ind. Eng. Chem. Res. 40, 2001, pp. 1244-1260.
Huron, M.J. and Vidal, J., “New Mixing Rules in Simple Equations of State for Representing Vapor-liquid
Equilibria of Strongly Non-Ideal Mixtures”, Fluid Phase Equilibria 3, 1979, p. 255.
Knapp H.R., Doring, R., Oellrich, L., Plocker, U., and Prausnitz, J.M., “Vapor-Liquid Equilibria for Mixtures of
Low Boiling Substances”, Chem. Data. Ser., Vol. VI, 1982, DECHEMA.
Mathias, P.M. and Copeman, T.W., “Extension of the Peng-Robinson Equation of State to Complex Mixtures:
Evaluation of the various Forms of the Local Composition Concept”, Fluid Phase Equilibria 13, 1983, pp. 91-108.
Pedersen, K.S., Milter, J., and Rasmussen, C.P., “Mutual Solubility of Water and Reservoir Fluids at High
Temperatures and Pressures, Experimental and Simulated Phase Equilibrium Data”, Fluid Phase Equilibria 189,
2001, pp. 85-97.
Pedersen, K. S. and Milter, J., “Phase Equilibrium Between Gas Condensate and Brine at HT/HP Conditions”, SPE
90309, presented at the SPE ATCE, Houston, TX, September 26-29, 2004.
Pedersen, K.S., Milter, J. and Sørensen, H., “Cubic Equations of State Applied to HT/HP and Highly Aromatic
Fluids”, SPE 88364, SPE Journal, June 2004, pp. 186-192.
Peneloux, A., Rauzy, E. and Fréze, R., “A Consistent Correlation for Redlich-Kwong-Soave Volumes”, Fluid Phase
Equilibria 8, 1982, pp. 7-23.
PVTsim Method Documentation Equations of State 51
Peng, D.-Y. and Robinson, D.B., “A New Two-Constant Equation of State”, Ind. Eng. Chem. Fundam. 15, 1976, pp.
59-64.
Peng, D.-Y., and Robinson, D.B., “The Characterization of the Heptanes and Heavier Fractions for the GPA Peng-
Robinson Programs”, GPA Research Report RR-28, 1978.
Pitzer, K. S., “Volumetric and Thermodynamic Properties of Fluids. I., Theoretical Basis and Virial Coefficients”, J.
Am. Chem. Soc. 77, 1955, 3427.
Reid, R.C., Prausnitz, J.M. and Sherwood, J. K., “The Properties of Gases and Liquids” McGraw-Hill, New-York
1977.
Soave, G., “Equilibrium Constants From a Modified Redlich-Kwong Equation of State”, Chem. Eng. Sci. 27, 1972,
pp. 1197-1203.
Sørensen, H., Pedersen, K.S. and Christensen, P.L., "Modeling of Gas Solubility in
Brine", Organic Geochemistry 33, 2002, pp. 635-642.
PVTsim Method Documentation Characterization of Heavy Hydrocarbons 52
Characterization of Heavy Hydrocarbons
Characterization of Heavy Hydrocarbons
To use a cubic equation of state as for example the SRK or the PR equations on oil and gas condensate mixtures the
critical temperature, Tc, the critical pressure, Pc, and the acentric factor, , must be known for each component of the
mixture. Naturally occurring oil or gas condensate mixtures may contain thousands of different components. This
number exceeds what is practical in a usual phase equilibrium calculation. Some of the components must be lumped
together and represented as pseudo-components. C7+-characterization consists in representing the hydrocarbons with
seven and more carbon atoms as a reasonable number of pseudo-components and to find the needed equation of state
parameters, Tc, Pc and for these pseudo-components.
Classes of Components
Naturally occurring oil and gas condensate mixtures consist of three classes of components
Defined Components
These are per default N2, CO2, H2S, C1, C2, C3, iC4, nC4, iC5 and C6 in PVTsim. C6 is in PVTsim considered to be
pure nC6.
C7+ Fractions
Each C7+ fraction contains hydrocarbons with boiling points within a given temperature interval. Carbon number
fraction n consists of the components with a boiling point between that of nCn-1 + 0.5C/0.9°F and that of nCn +
0.5C/0.9°F. The C7 fraction for example consists of the components with a boiling point between those of nC6 +
0.5C/0.9°F and nC7 + 0.5C/0.9°F . For the C7+-fractions the density at standard conditions (1 atm/14.969 psia and
15°C/59°F) and the molecular weight must be input.
The Plus Fraction
The plus fraction consists of the components, which are too heavy to be split into individual C7+-fractions. The
average molecular weight and the density must be known.
PVTsim Method Documentation Characterization of Heavy Hydrocarbons 53
Properties of C7+-Fractions
PVTsim supports two different characterization procedures
Standard oil characterization to C80
Heavy oil characterization to C200
Cubic Equations of State
Tc, Pc and are found from empirical correlations in density, , and molecular weight, M
Tc = c1 + c2 1n M + c3 M + c4/M
lnPc = d1 + d2d5
+ d3/M + d4/M2
m = e1 + e2 M + e3 + e4 M2 (standard characterization)
Meρeln(M)eem 4321 (heavy oil characterization)
where m is defined in the Equation of State section and the coefficients are given in the tables below.
Standard characterization - SRK (Pedersen et al., 1989b and 1992)
Sub-index/
Coefficient
1 2 3 4 5
c 1.6312 x 102
8.6052 x 10 4.3475 x 10-1
-1.8774 x 103
-
d -1.3408 x 10-1
2.5019 2.0846 x 102 -3.9872 x 10
3 1.0
e 7.4310 x 10-1
4.8122 x 10-3
9.6707 x 10-3
-3.7184 x 10-6
-
Standard characterization – PR/PR78 (Pedersen et al., 2002)
Sub-index/
Coefficient
1 2 3 4 5
c 7.3404 x 10 9.7356 x 10 6.1874 x 10-1
-2.0593 x 103
-
d 7.2846 x 10-2
2.1881 1.6391 x 102
-4.0434 x 103
1/4
e 3.7377 x 101
5.4927 x 10-3
1.1793 x 10-2
-4.9305 x 10-6
-
Heavy oil characterization – SRK (Krejbjerg and Pedersen, 2006)
Sub-index/
Coefficient
1 2 3 4 5
c 8.30631 102 1.75228 10 4.55911 10
-2 -1.13484 10
4 -
d 8.0298810-1 1.78396
1.5674010
2 -6.96559 10
3 0.25
e -4.7268010-2
6.0293110-2 1.21051
-5.7667610
-3
Heavy oil characterization – PR/PR78 (Krejbjerg and Pedersen, 2006)
Sub-index/
Coefficient
1 2 3 4 5
c 9.13222102
1.0113410
4.5419410-2
-1.35867104 -
d 1.28155
1.26838
1.67106102
-8.10164103 0.25
e -2.3838010-1
6.1014710-2 1.32349
-6.5206710
-3
M is in g/mole, is in g/cm3, Tc is in K and Pc in atm. The correlations are the same with and without volume
correction.
PVTsim Method Documentation Characterization of Heavy Hydrocarbons 54
PC-SAFT
The PC-SAFT parameters mi , i and i are found from empirical correlations in density, , and molecular weight, M
17ρ
7Mi2
7miC
CM102.82076Cm
i
0.25
7ρ7M
0.25
iim7εii CCρM7.97066C
k
mε
where k is Boltzmann's constant, Mi is the molecular weight and i the density of carbon number fraction i, and
AC7NC7PC77m m)fraction(i-Am)fraction(i-Nm)fraction(i-PC
AC7NC7PC77M M)fraction(i-AM)fraction(i-NM)fraction(i-PC
AC7
AC7
NC7
NC7
PC7
PC7
7M7ρ
ρ
M)fraction(i-A
ρ
M)fraction(i-N
ρ
M)fraction(i-P
CC
PC7AC7PC7NC7PC7PC7m7ε εm)fraction(i-Amε)fraction(i-Nεm)fraction(i-PC
P-fraction(i), N-fraction(i) and A-fraction(i) stand for respectively paraffinic, naphthenic and aromatic fraction of
carbon number fraction i. These fractions (PNA distribution) are found using the procedure of Nes and Westerns
(1951). The sub-index PC7 stands for property of C7 normal paraffin (n-heptane), NC7 for property of C7 naphthene
(methyl-cyclohexane) and AC7 for property of C7 aromatic (benzene). These properties may be seen from the below
table.
mPC7 3.4831
mNC7 2.5303
mAC7 2.4653
MPC7 100.203
MNC7 84.137
MAC7 78.114
PC7 (g/cm3) 0.690
NC7 (g/cm3) 0.783
AC7 (g/cm3) 0.886
PC7 238.40
NC7 278.11
AC7 287.35
The relations applied for the C7 properties ensure that n-heptane, methyl-cyclo-hexane and benzene will have the
PC-SAFT parameters tabulated in literature and shown in the above table.
The parameter i is found to comply with the density of the fraction at atmospheric conditions.
Extrapolation of the Plus Fraction
PVTsim Method Documentation Characterization of Heavy Hydrocarbons 55
Characterization of the plus fraction consists in
Estimation of the molar distribution, i.e. mole fraction versus carbon number.
Estimation of the density distribution, i.e. the density versus carbon number.
Estimation of the molecular weight distribution, i.e. molecular weight versus carbon number.
Calculation of Tc, Pc and of the resulting pseudo-components.
The molar composition of the plus fraction is estimated by assuming a logarithmic relationship between the molar
concentration zN, of a given fraction and the corresponding carbon number, CN, for CN >7
CN = A1 + B1 ln zN
A1 and B1 are determined from the measured mole fraction and the measured molecular weight of the plus fraction.
The densities of the carbon number fractions contained in the plus fraction are estimated by assuming a logarithmic
dependence of against carbon number.
Boiling points are required to estimate ideal gas heat capacity coefficients for the C7+ fractions (see section on
Thermal and Volumetric Properties). The boiling points recommended by Katz and Firoozabadi (1978) are used up
to C45. The following relation is used for heavier components
TB = 97.58 M0.3323
0.04609
where TB is in K and in g/cm3.
Estimation of PNA Distribution
The following procedure is used to estimate the PNA-distribution of the C7+ fractions. The refractive index, n, of
each C7+-fraction is calculated from the density, the normal boiling point and the molecular weight using the
correlations of Riazi and Daubert (1980)
I1
2I1n
I is a characterization factor, which is found from the following correlation
0.91820.02269
B ρT0.3773I
TB is the boiling point in K and the liquid density at atmospheric conditions in g/cm3. Based on the refractive
index, the density and the molecular weight the PNA distribution (in mole%) can be estimated as described by Nes
and Westerns (1951)
PVTsim Method Documentation Characterization of Heavy Hydrocarbons 56
v = 2.51 (n – 1.4750) - + 0.8510
w = - 0.8510 – 1.11 (n – 1.4750)
%A = 430 v + 3660/M for v > 0
%A = 670 v + 3660/M for v < 0
R = 820 w + 10000/M for w > 0
R = 1440 w + 10600/M for w < 0
%N = R- %A
%P = 100 – R
Grouping (Lumping) of Pseudo-components
The extrapolated mixture may consist of more than 200 components and pseudo-components. In the simulation
options PVTsim can handle a maximum of 120 components. The number of components is reduced through a
grouping or lumping. The default number of C7+ components in PVTsim is 12. The Carbon number fractions C7, C8
and C9 will not be lumped when more than five pseudo-components are specified.
Weight Based Lumping
PVTsim default uses a weight based lumping where each lumped pseudo-component contains approximately the
same weight amount and where Tc, Pc and of the individual carbon number fractions and found as weight mean
average values of Tc, Pc and of the individual carbon number fractions. If the k’th pseudo-component contains the
carbon number fractions M to L, its Tc, Pc and will be found from the relations
L
Miii
L
Miciii
ck
Mz
TMz
T
L
Miii
L
Miciii
ck
Mz
PMz
P
L
Miii
L
Miiii
ck
Mz
ωMz
ω
where zi is the mole fraction and Mi the molecular weight of carbon number fraction i. The weight-based procedure
ensures that all hydrocarbon segments of the C7+ fraction are given equal importance.
Delumping
In compositional reservoir simulations it is desirable to use as few components as possible in order to minimize the
computation time. This is accomplished by a component lumping. Not only C7+ components but also some of the
PVTsim Method Documentation Characterization of Heavy Hydrocarbons 57
defined components may have to be lumped. In subsequent process simulations it is often desirable to work with all
the defined components and possibly also an increased number of C7+ pseudo-components. Expansion of a lumped
composition may in PVTsim be accomplished by use of the Delumping Option. A lumped component consisting of
defined components is split into its constituents. The relative molar amounts of the individual components are
assumed to be the same as in the original composition before lumping. The C7+ pseudo-components of the lumped
fluid are possibly split to cover smaller carbon number ranges. To start with the C7+ pseudo-component containing
the largest weight fraction is split into two new pseudo-components of approximately equal weight amounts. Next
the pseudo-component, which now contains the largest weight amount is split into two and so on until the number of
C7+ pseudo-components equals that specified.
It is possible to adjust the gas/oil ratio of the delumped composition to match that of the lumped composition.
Characterization of Multiple Compositions to the Same Pseudo-Components
In process simulations and compositional reservoir simulations it is often advantageous to characterize a number of
different reservoir fluids to a unique set of pseudo-components. This is practical for example when numerous
process streams are let to the same separation plant in which case there is a need for simulating each stream
separately as well as the mixed stream as a whole. If each composition is represented using the same pseudo-
components, the streams can readily be mixed without having to increase the number of components.
Initially the plus fractions of the compositions to be characterized to the same pseudo-components are split into
carbon number fractions. For each C7+ carbon number fraction Tc, Pc and are estimated in the usual manner. Tc’s,
Pc’s and ’s representative for all the compositions are calculated from
NFL
1j
j
i
NFL
1j
j
ci
j
iunique
ci
zjWgt
TzjWgt
T
NFL
1j
j
i
NFL
1j
j
ci
j
iunique
ci
zjWgt
PzjWgt
P
NFL
1j
j
i
NFL
1j
jj
imix
i
zjAmount
ωzjAmount
ω
NFL is the number of compositions to be characterized to the same pseudo-components, j
iz is the mole fraction of
component i in composition number j, and Amount(j) is the weight (molar or weight based) to be assigned to
composition number j.
To decide what carbon number fractions to include in each pseudo-component, a molar composition is calculated,
which is assumed to be reasonably representative for all compositions. In this imaginary composition, component i
enters with a mole fraction of
PVTsim Method Documentation Characterization of Heavy Hydrocarbons 58
NFL
1j
NFL
1j
j
iunique
i
jAmount
zjAmount
z
and a molecular weight of
NFl
1j
j
i
j
i
NFL
1j
j
iunique
i
zjAmount
MzjAmount
M
This composition is now treated like an ordinary composition to be lumped into pseudo-components. The lumping
determines the carbon number ranges to be contained in each pseudo-component, and Tc, Pc and of each pseudo-
component. The properties of the lumped composition are assumed to apply for all the individual compositions. If
the k’th pseudo-component contain the carbon number fractions M to L, the mole fraction of this pseudo-component
in the j’th composition will be
L
Mi
j
i
j
k zz
References
Katz, D.L. and Firoozabadi, A., ”Predicting Phase Behavior of Condensate/Crude-Oil Systems Using Methane
Interaction Coefficients”, J. Pet. Technol. 20, 1978, pp. 1649-1655.
Krejbjerg, K. and Pedersen, K. S., “Controlling VLLE Equilibrium with a Cubic EoS in Heavy Oil Modeling”,
presented at 57th
Annual Technical Meeting of the Petroleum Society (Canadian International Petroleum
Conference), Calgary, Canada, June 13-15, 2006
Lomeland F. and Harstad, O., “Simplifying the Task of Grouping Components in Compositional Reservoir
Simulation”, SPE paper 27581, presented at the European Petroleum Computer Conference in Aberdeen, U.K., 15-
17 March, 1997.
Nes, K. and Westerns, H.A., van, ”Aspects of the Constitution of Mineral Oils”, Elsevier, New York, 1951.
Pedersen, K.S., Thomassen, P. and Fredenslund, Aa., ”Thermodynamics of Petroleum Mixtures Containing Heavy
Hydrocarbons. 3. Efficient Flash Calculation Procedures Using the SRK Equation of State”, Ind. Eng. Chem.
Process Des. Dev. 24, 1985, pp. 948-954.
Pedersen, K.S. , Fredenslund, Aa. and Thomassen, P., ”Properties of Oils and Natural Gases”, Gulf Publishing Inc.,
Houston, 1989a.
Pedersen, K.S., Thomassen, P. and Fredenslund, Aa., Advances in Thermodynamics 1, 1989b, 137.
Pedersen, K.S., Blilie, A. and Meisingset, K.K., "PVT Calculations of Petroleum Reservoir Fluids Using Measured
and Estimated Compositional Data for the Plus Fraction", Ind. Eng. Chem. Res. 31, 1992, pp. 924-932.
Pedersen, K.S., Milter, J. and Sørensen, H., “Cubic Equations of State Applied to HT/HP and Highly Aromatic
Fluids”, SPE 88362, SPE Journal, June 2004, pp. 186-192.
Riazi, M.R. and Daubert, T.E., ”Prediction of the Composition of Petroleum Fractions”, Ind. Eng. Chem. Process
Des. Dev. 19, 1980, pp. 289-294.
PVTsim Method Documentation Thermal and Volumetric Properties 59
Thermal and Volumetric Properties
Thermal and Volumetric Properties
Density
The phase densities are calculated using the selected equation of state, i.e. either
SRK
SRK-Peneloux
SRK-Peneloux(T)
PR
PR-Peneloux
PR-Peneloux(T)
PR78
PR78-Peneloux
PR78-Peneloux(T)
where (T) means that the Peneloux volume translation parameter is temperature dependent.
Enthalpy
The enthalpy, H, is calculated as the sum of two contributions, the ideal gas enthalpy and residual enthalpy, Hres
N
li
resid
ii HHzH
PVTsim Method Documentation Thermal and Volumetric Properties 60
where N is the number of components, zi is the mole fraction of component i in the phase considered and id
iH is the
molar ideal gas enthalpy of component i.
T
T
id
pi
id
iref
dTCH
Tref is a reference temperature (273.15 K (= 0°C/32°F) in PVTsim). id
piC is the molar ideal gas heat capacity of
component i, which is calculated from a third degree polynomial in temperature
3
i4,
2
i3,i2,i1,
id
pi TCTCTCCC
The default values used in PVTsim for the coefficients C1-C4 of the lighter petroleum mixture constituents are those
recommended by Reid et al. (1977).
For C7+ hydrocarbon fractions C1-C4 are for heat capacities in Btu/lb calculated from the following correlations
(Kesler and Lee, 1976)
C1 = -0.33886 + 0.02827 K – 0.26105 CF + 0.59332 CF
C2 = -(0.9291 – 1.1543 K + 0.0368 K2) 10
-4 + CF(4.56 - 9.48)10
-4
C3 = -1.6658 · 10-7
+ CF(0.536 – 0.6828)10-7
C4 = 0
where
CF = ((12.8 – K)(10-K)/(10))2
and K is the Watson characterization factor defined as
/SGTK 1/3
B
TB is the normal boiling point in °R and SG the specific gravity, which is approximately equal to the liquid density in
g/cm3.
The acentric factors, are calculated from (Kesler and Lee, 1976)
)0.8T(for
0.43577T13.4721lnTT
15.6875-15.2518
0.169347T-1.28862lnTT
6.096495.92714Pln
ω Br6
BrBr
Br
6
BrBr
Br
BR
0.8)T(forT
0.01063K1.408T8.3590.007465K0.1352K7.904ω Br
Br
Br
2
PVTsim Method Documentation Thermal and Volumetric Properties 61
PBr is atmospheric pressure divided by Pc and TBr is TB/Tc.
For hydrocarbons with a molecular weight above 300, is replaced by 1.0 if < 1. Acentric factors below 0.1 are
replaced by = 0.1.
The residual term of H is derived from the equation of state using the following general thermodynamic relation
T
lnRTH 2res
where is the fugacity coefficient of the mixture and the derivative is for a constant composition.
Internal Energy
The internal energy, U, is calculated as U = H – PV. Where H is the enthalpy, P the pressure and V the molar
volume.
Entropy
The entropy is calculated as the sum of two contributions, the ideal gas entropy and residual entropy
N
1i
resid
ii SSzS
The ideal gas term at the temperature T is calculated from
T
T
i
ref
id
piid
i
ref
zlnRP
PlnRdT
T
CS
Pref is a reference pressure (1 atm/14.696 psia in PVTsim). id
piC is the molar ideal gas heat capacity of component i,
which is calculated as outlined in the Enthalpy section.
The residual term is calculated from
lnRT
HS
resres
PVTsim Method Documentation Thermal and Volumetric Properties 62
Heat Capacity
The heat capacity at constant pressure is calculated from
P
PT
HC
and the heat capacity at constant volume from
VP
PVT
P
T
VTCC
where the derivatives are evaluated using the equation of state. H is the enthalpy, T the temperature, P the pressure
and V the molar volume.
Joule-Thomson Coefficient
The Joule-Thomson coefficient is defined as the pressure derivative of the temperature for constant enthalpy. It is
derived as follows
TpH
jTP
H
C
1
P
Tμ
Velocity of sound
The velocity of sound is derived as
PVV
P
S
sonicV
T
T
P
C
C
MW
V
V
P
MW
Vu
where M is the molecular weight and the derivatives are evaluated using the equation of state.
References
Kesler, M.G. and Lee, B.I., ”Improve Prediction of Enthalpy of Fractions”, Hydrocarbon Processing 55, 1976, pp.
153-158.
Reid, R.C., Prausnitz, J. M. and Sherwood, J.K., ”The Properties of Gases and Liquids”. McGraw-Hill, New-York
1977.
PVTsim Method Documentation Transport Properties 63
Transport Properties
Transport Properties
Viscosity Corresponding States Method
The viscosity calculations in PVTsim are default based on the corresponding states principle in the form suggested
by Pedersen et al. (1984, 1987) and Lindeloff et al. (2004).
The idea behind the corresponding states principle is that the relation between the reduced viscosity
1/22/3
c
1/6-
c
rMPT
ηη
and the reduced pressure (P/Pc) and temperature (T/Tc) is the same for a group of substances that is
rrr T,Pfη
If the function f is known for one component (a reference component) within the group it is possible to calculate the
viscosity at any (P,T) for any other component within the group. The viscosity of component x at (P,T) is for
example found as follows
cx
co
cx
co
o
1/2
o
x
2/3
co
cx
1/6
co
cx
rrxT
TT,
P
PPη
M
M
P
P
T
TT,Pη
where o refers to the reference component.
In PVTsim methane is used as reference component unless at conditions where methane is in solid form at the
reference conditions. The methane viscosity model of McCarty (1974) is used. The deviations from the simple
corresponding states principle is expressed in terms of a parameter, , giving the following expression for the
viscosity of a mixture (Pedersen et al., 1984)
PVTsim Method Documentation Transport Properties 64
oooomix
1/2
omix
2/3
comixc,
1/6
comixc,mix T,Pη)/α(α/MM/PP/TTTP,η
where
mixmixc,
oco
oαT
αTTP
; mixmixc,
oco
oαT
αTTT
The critical temperature and the critical molar volume for unlike pairs of molecules (i and j) are found using the
below formulas
cjcicij TTT
31/3
cj
1/3
cicij VV8
1V
The critical molar volume of component i may be related to the critical temperature and the critical pressure as
follows
ci
cicici
P
TRZV
where Zci is the compressibility factor of component i at the critical point. Assuming that Zc is a constant
independent of component, the expression for Vcij may be rewritten to
31/3
cj
cj
1/3
ci
cicij
P
T
P
Tconstant
8
1V
The critical temperature of a mixture is found from
N
1i
N
1jcijji
N
1i
N
1jcijcijji
mixc,
Vzz
VTzz
T
where zi and zj are mole fractions of components i and j, respectively and N the number of components. This
expression may be rewritten to
PVTsim Method Documentation Transport Properties 65
N
1i
31/3
cj
cj
1/3
ci
ciN
1jji
N
1i
N
1j
1/2
cjci
31/3
cj
cj
1/3
ci
ciji
mixc,
P
T
P
Tzz
TTP
T
P
Tzz
T
For the critical pressure of a mixture, Pc,mix, the following relation is used
Pc,mix = constant Tc,mix / Vc,mix
where Vc,mix is found as follows
N
1i
N
1icijjimixc, VzzV
The following expression may now be derived for Pc,mix
2
N
1i
N
1j
31/3
cj
cj
1/3
ci
ciji
N
1i
1/2
cjci
3
N
1j
1/3
cj
cj
1/3
ci
ciji
mixc,
P
T
P
Tzz
TTP
T
P
Tzz8
P
The applied mixing rules are those recommended by Murad and Gubbins (1977).
The mixture molecular weight is found as follows
n
CSP) (2nd2.303
n
CSP) (2nd2.303
w4
mix MMMCSP)(1st101.304M
where and wM and nM are the weight average and number average molecular weights, respectively
N
1jii
N
1i
2
ii
w
Mz
Mz
M
N
1jin,i
N
1iiw,in,i
w
Mz
MMz
M
N
1iiin MzM
N
1iin,in MzM
PVTsim Method Documentation Transport Properties 66
The expressions in parentheses are those used for fluid mixtures containing lumped components. Mn,i is the number
average molecular weight and Mw,i the weight average molecular weight of the lumped component.
In the expression for the mixture molecular weight (1st CSP) and (2nd CSP) are tuning parameters, which are 1.0 by
default.
The parameter of the mixture is found from the expression
0.5173
mix
1.847
r
3
mix Mρ107.3781.000α
The reduced density r is defined as
co
mixc,
co
mixc,
coo
rρ
P
PP,
T
TTρ
ρ
The reference viscosity correlation is based on the methane viscosity model of Hanley et al. (1975)
Tρ,Δη'ρTηTηTρ,η' 1o
where '
10 Δηandη,η are functions defined in the above reference. The methane density is found using the BWR-
equation in the form suggested by McCarty (1974). In the dense liquid region this expression is mainly governed by
the term ’(,T)
1.0
T
j
T
jjθρ
T
jjρexpT/jjexpTρ,Δη'
2
765
0.5
3/2
32
0.1
41
In the work of Hanley (1975) the coefficients j1 – j7 have the following values (viscosities in P)
j1 = -10.3506
j2 = 17.5716
j3 = -3019.39
j4 = 188.730
j5 = 0.0429036
j6 = 145.290
j7 = 6127.68
θ is given by
PVTsim Method Documentation Transport Properties 67
c
c
ρ
ρρθ
The presented viscosity calculation method presents problems when methane is in a solid form at its reference state.
This is the case when the methane reference temperature is below 91 K. For methane reference temperatures above
75 K the term ’(,T) is replaced by (Pedersen and Fredenslund, 1987)
1.0
T
k
T
kkθρ
T
kkρexp/TkkexpTρ,'Δη'
2
765
0.5
3/2
32
0.1
41
with
k1 = -9.74602
k2 = 18.0834
k3= -4126.66
k4 = 44.6055
k5 = 0.9676544
k6 = 81.8134
k7= 15649.9
Continuity between viscosities above and below the freezing point of methane is secured by introducing ” as a
fourth term in the viscosity expression
Tρ,'Δη'FTρ,Δη'FρTηTηTρ,η 2110
2
1HTANF1
2
HTAN1F2
ΔTexpΔTexp
ΔTexpΔTexpHTAN
with
5
TTΔT F
where TF is the freezing point of methane.
PVTsim Method Documentation Transport Properties 68
When the methane reference temperature is below 75 K there is the need for a different reference model. Lindeloff et
el. (2004) have proposed to use a correlation proposed by Rønningsen (1993) for use on stable oils
T
M6.215
T
371.8M0.011010.07995ηlog10
T is the temperature in K and M is the average molecular weight. For T > 564.49 K, the sign in front of 0.01101 is
changed from – to +. As the correlation in a PVTsim context is not always used on stable oils, it is necessary to have
a procedure for evaluating a representative average molecular weight, M, also applicable to live oils.
1.5M
Mfor
CSP)(3rdVisfac3
1.5MM
n
w
CSP)(4thVisfac4
n
1.5M
Mfor
MCSP)(3rdVisfac3
MMM
n
w
CSP)(4thVisfac4
n
w
n
where (3rd CSP) and (4th CSP) are tuning parameters, which are 1.0 by default. nM is the number average
molecular weight, wM the weight average molecular weight, and
0.9738M
T0.2252 Visfac3
n
0.1170-Visfac30.5354 Visfac4
where T is in K.
Stable oils will usually have 1.5M
M
n
w
for which type of oils M using default viscosity correction factors will be
equal to .Mn The correlation of Rønningsen applies to systems at atmospheric pressure. In order to capture pressure
effects on the reference fluid, the following pressure dependence is used
0.8226
1P0.00384
0
0.8226
eηη
for viscosities in cP. 0 is the viscosity at the actual temperature and atmospheric pressure and P is the actual
pressure in atm.
For methane reference temperatures > 75 K the classical corresponding states (CSP) model is used. For reference
temperatures < 50 K the heavy oil model is used. The 50 K < T < 75 K the viscosity is calculated as
Heavy2CSP1 ηFηFη
where F1and F2 are defined above, and T in this case is
PVTsim Method Documentation Transport Properties 69
5
70TΔT
for the temperature T in K.
Lohrenz-Bray-Clark (LBC) Method
The viscosity may in PVTsim alternatively be calculated using the Lohrenz-Bray-Clark correlation (1964). Gas and
oil viscosities are related to a fourth-degree polynomial in the reduced density, r = /c.
4
r5
3
r4
2
r3r21
1/44* ρaρaρaρaa10ξηη
where
a1 = 0.10230
a2 = 0.023364
a3 = 0.058533
a4 = -0.040758
a5 = 0.0093324
*η is the low-pressure gas mixture viscosity. is the viscosity-reducing parameter, which for a mixture is given by
the following expression:
2/3N
1icii
1/2N
1iii
1/6N
1icii PzMzTzξ
where N is the number of components in the mixture and zi the mole fraction of component i.
The critical density, c, is calculated from the critical volume
1
N
1icii
1
cc VzVρ
For C7+ fractions the critical volume in ft3/lb mole is found from
Vc = 21.573 + 0.015122 M – 27.656 + 0.070615 M
In this expression, M is the molecular weight and the liquid density in g/cm3. For defined components literature
values are used for the critical volumes.
PVTsim Method Documentation Transport Properties 70
If the composition has been entered in characterized form and densities are not available, the critical volume is
calculated from a correlation of Riedel (1954)
1
c
c
cc 7.0)0.26(α3.72
P
RTV
c
b
c
c
b
c
T
T1
lnPT
T
1.00.9076α
If the normal boiling point is not available, the critical volume is calculated from the following correlation (Reid et
al., 1977)
c
cc
P
)RT0.0928(0.2918V
The dilute gas mixture viscosity * is determined from (Herning and Zippener, 1936)
N
1i
1/2
ii
N
1i
1/2
i
*
ii*
MWz
MWηz
η
The following expressions (Stiel and Thodos, 1961) are used for the dilute gas viscosity of the individual
components, *
iη
1.5T,Tξ
11034η ri
0.94
ri
i
5*
i
1.5T,1.67T4.58ξ
11017.78η ri
5/8
ri
5*
i
where i is given by
2/3
ci
1/2
i
1/6
ci
iPM
Tξ
When performing tuning on the LBC viscosity model either the critical volumes, the coefficients a1-a5 or both may
be selected as tuning parameters. The ability to tune the coefficients makes the LBC model extremely flexible, but if
no data are available the CSP model generally provides better predictions.
For fluids containing solid wax particles, a non-Newtonian viscosity model may be applied as is described in the
Wax section.
PVTsim Method Documentation Transport Properties 71
Emulsion viscosities are dealt with in the section on Water Phase Properties.
Thermal Conductivity
Corresponding States Method
The thermal conductivity is defined as the proportionality constant, , in the following relation (Fourier’s law)
dx
dTλq
where q is the heat flow per unit area and (dT/dx) is the temperature gradient in the direction of the heat flow.
The thermal conductivity is in PVTsim calculated using a corresponding states principle (Christensen and
Fredenslund (1980) and Pedersen and Fredenslund (1987)).
According to the corresponding states theory, the thermal conductivity can be found from the expression
rrr T,Pf
where f is the same function for a group of substances obeying the corresponding states principle. For the reduced
thermal conductivity, r, the following equation is used
1/22/3
c
1/6
c
rMPT
TP,λTP,λ
Using simple corresponding states theory, the thermal conductivity of component x at the temperature T and the
pressure P may be found from the following equation
ooo
1/2
ox
2/3
cocx
1/6
cocxx T,Pλ/MMP/PT/TTP,λ
where Po = PPco/Pcx and To = TTco/Tcx and o is the thermal conductivity of the reference substance at the
temperature To and pressure Po. As is the case for viscosity, methane is used as reference substance. However some
corrections must be introduced as compared with the simple corresponding states principle. The thermal conductivity
of polyatomic substances (Hanley (1976)) can be separated into two contributions, one due to transport of
translational energy and one due to transport of internal energy
= tr + int
PVTsim Method Documentation Transport Properties 72
PVTsim uses the modification of Christensen and Fredenslund (1980), which only applies the corresponding states
theory to the translational term. A term int,mix is used to correct for the deviations from the simple corresponding
states model. The final expression for calculation of the thermal conductivity of a mixture at the temperature, T, and
the pressure, P, is the following
(T)λTλP,Tλα/α
M/MP/PT/TTP,λ
mixint,ooint,oooomix
1/2
omix
2/3
comixc,
1/6
comixc,mix
where
oco
mixmixc,
o
coco
mixmixc,
oαP
αPP/Pand
αT
αTT/T
The mixture molecular weight Mmix is found from Chapman-Enskog theory as described by Murad and Gubbins
(1976)
4/3
mixc,
1/3
mixc,
2
21/3
cj
cj
1/3
ci
ci
N
1i
N
1j
1/4
cjci
1/2
ji
ji
mix PT
P
T
P
T
T/TM
1
M
1zz
16
1M
where z are mole fractions and i and j component indices. The internal energy contributions to the thermal
conductivity, int,o (reference substance) and int,mix (mixture) are both given by
3
r
2
rrr
r
id
piint
ρ0.029725ρ0.030182ρ0.0534321ρf
/Mρf2.5RC1.18653ηλ
is the gas viscosity at the actual temperature and a pressure of 1 atm, id
PC the ideal gas heat capacity at the
temperature T. R is the gas constant. The -parameter is found from the following expression (Pedersen and
Fredenslund (1987))
1.086
i
2.043
rii Mρ0.00060041α
where
co
ci
co
ci
coo
riρ
P
PP,
T
TTρ
ρ
PVTsim Method Documentation Transport Properties 73
α mix is found using the mixing rule
N
1i
N
1j
0.5
jijimix ααzzα
which ensures that components having small -values, i.e. small molecules, are attributed more importance than
those having larger -values. Smaller molecules are more mobile than larger ones and contribute relatively more to
the transfer of energy than do the larger ones.
The calculation of the thermal conductivity of the reference substance, methane, is based on a model of Hanley et al.
(1975), which has the form
Tρ,ΔλTρ,Δλ'ρTλTλTρ,λ c1o
In the dense liquid region the major contribution to this expression comes from '(,T), which has the same
functional form as the expression for '(,T) in the viscosity section. The coefficients ji – j7 have the following
values (for thermal conductivities in mW/(mK)
j1 = 7.0403639907
j2 = 12.319512908
j3= -8.8525979933 102
j4= 72.835897919
j5= 0.74421462902
j6= -2.9706914540
j7= 2.2209758501 103
As for viscosities a ”low temperature term” (Pedersen and Fredenslund (1987) is used. The final expression for the
thermal conductivity of methane is then the following
Tρ,ΔλTρ,'Δλ'FTρ,Δλ'FρTλTρ,λ c210
F1 and F2 are defined in the viscosity section. The following expression is used for "(,T),
1.0
T
l
T
llθρ
T
llρexp/TllexpTρ,'Δλ'
2
765
0.5
3/2
32
0.1
41
where
l1= -8.55109
l2= 12.5539
l3= -1020.85
PVTsim Method Documentation Transport Properties 74
l4= 238.394
l5= 1.31563
l6= -72.5759
l7= 1411.60
LBC method
The thermal conductivity may in PVTsim alternatively be calculated using the LBC method, which is a modified
Lohrenz-Bray-Clark type expression. The thermal conductivity is derived from two contributions
The translatoric thermal conductivity
The internal thermal conductivity
The total thermal conductivity may therefore be expressed as follows
InternalicTranslatorTotal λλλ
In the LBC method, the gas and oil translatoric conductivities are related to a fourth-degree polynomial in the
reduced density, r = /c
The translatoric thermal conductivity is expressed as a function of temperature, pressure and reduced density.
4
5
3
r4
2
r321
CC
1icTranslator 5r32 ρaρaρaρaaPTCλ
where
C1 = 2.30528
C2 = -0.59394
C3 = 0.06928
a1 = 270.28341
a2 = -148.95858
a3 = 408.63577
a4 = -127.74598
a5 = 13.52979
The critical density, c, is calculated from the critical volume
1
N
1icii
1
cc VzVρ
For C7+ fractions the critical volume in ft3/lb mole is found from
Vc = 21.573 + 0.015122 M – 27.656 + 0.070615 M
PVTsim Method Documentation Transport Properties 75
In this expression, M is the molecular weight and the liquid density in g/cm3. For defined components literature
values are used for the critical volumes.
If the composition has been entered in characterized form and densities are not available, the critical volume is
calculated from a correlation of Riedel (1954)
1
c
c
cc 7.0)0.26(α3.72
P
RTV
c
b
c
c
b
c
T
T1
lnPT
T
1.00.9076α
If the normal boiling point is not available, the critical volume is calculated from the following correlation (Reid et
al., 1977)
c
cc
P
)RT0.0928(0.2918V
The internal thermal conductivity is determined by the following equation
w
'
rvInternal
M
ρCξ1.1865λ
where
3
r
2
rr
'
r 0.029725ρ0.30182ρ0.053432ρ1ρ
and
N
1i
1/2
ii
N
1i
1/2
ii
MWz
MWz i
The following expressions are used for i of the individual components
1.5T,Tξ
11034 ri
0.94
ri
i
5
i
1.5T,1.67T4.58ξ
11017.78 ri
5/8
ri
i
5
i
where i is given by
2/3
ci
1/2
i
1/6
ci
iPM
Tξ
PVTsim Method Documentation Transport Properties 76
The data material used in the determination of the parameters consisted of 8660 data sets, with the temperature and
pressure ranging from 223.15 K to 473.15 K, and 1 atm. to 1000 atm. For the entire data material the average
deviation was 16%, and the maximum deviation for a single point was 98% but for pressures equal or above 25 atm.
and temperatures equal or below 448.15 K the maximum single point deviation was 54%. For low pressure the
accuracy of the translatoric thermal conductivity is not important, because the thermal conductivity is determined by
the ideal gas thermal conductivity, which is kept unchanged.
Gas/oil Interfacial Tension
The interfacial tension between an oil and a gas phase is in PVTsim calculated using the procedure of Weinaug and
Katz (1943). The interfacial tension (in dyn/cm = 1 mN/m) is expressed in terms of the Parachors [P] of the
individual components
N
1iiiviiL
1/4 yPρxPρσ
Lρ and Vρ are the molar densities in mole/cm3 (the density divided by the molecular weight) of the oil and gas
phases, respectively and xi and yi are the mole fractions of component i in the oil and gas phases. The Parachors of
the defined components have fixed values. The Parachor of a C7+ component is calculated from the following
expression
ii M2.3459.3P
where Mi is the molecular weight of the component. The phase densities are calculated using the equation of state.
References
Christensen, P.L. and Fredenslund Aa., ”A Corresponding States Model for the Thermal Conductivity of Gases and
Liquids”, Chem. Eng. Sci. 35, 1980, pp. 871-875.
Hanley, H.J.M., McCarty, R.D. and Haynes, W.M., ”Equation for the Viscosity and Thermal Conductivity
Coefficients of Methane”, Cryogenics 15, 1975, pp. 413-417.
Hanley, H.J.M., ”Prediction of the Viscosity and Thermal Conductivity Coefficients of Mixtures”, Cryogenics 16,
1976, pp. 643-651.
Herning, F. and Zippener, L., ”Calculation of the Viscosity of Technical Gas Mixtures from the Viscosity of the
Individual Gases”, Gas u. Wasserfach 79, 1936, pp. 69-73.
Lindeloff, N., Pedersen, K.S., Rønningsen, H.P. and Milter, J., “The corresponding States Viscosity Model Applied
to Heavy Oil Systems”, Journal of Canadian Petroleum Technology 43, 2004, pp. 47-53.
Lohrenz, J., Bray, B.G. and Clark, C.R., ”Calculating Viscosities of Reservoir Fluids from Their Compositions”, J.
Pet. Technol., Oct. 1964, pp. 1171-1176.
McCarty, R.D., ”A Modified Benedict-Webb-Rubin Equation of State for Methane Using Recent Experimental
Data”, Cryogenics 14, 1974, pp. 276-280.
Murad, S. and Gubbins, K.E., ”Corresponding States Correlation for Thermal Conductivities of Dense Fluids, Chem.
Eng. Sci. 32, 1977, pp. 499-505.
PVTsim Method Documentation Transport Properties 77
Pedersen, K.S., Fredenslund, Aa., Christensen, P.L. and Thomassen, P., ”Viscosity of Crude Oils”, Chem. Eng. Sci.
39, 1984, pp. 1011-1016.
Pedersen, K.S. and Fredenslund, Aa., ”An Improved Corresponding States Model for the Prediction of Oil and Gas
Viscosities and Thermal Conductivities”, Chem. Eng. Sci. 42, 1987, pp. 182-186.
Reid, R. C. and Sherwood, T. K., "The Properties of Gases and Liquids", 2nd ed. Chap 2, McGraw-Hill, New York,
1966.
Rønningsen, H.P., "Prediction of Viscosity and Surface Tension of North Sea Petroleum Fluids by Using the
Average Molecular Weight", Energy & Fuels 7, 1993, pp. 565-573.
Reidel L., “A New Universal Vapor Pressure Equation. I. The Extension of the Theories of the Corresponding
States”, Chem. Ing. Tech., 26, 1954, pp. 83-89
Stiel, L. I. and Thodos, G., ”The Viscosity of Non-Polar Gases at Normal Pressures”, AIChE J. 7, 1961, pp. 611-615.
Weinaug, C.F. and Katz, D.L., “Surface Tensions of Methane-Propane Mixtures”, Ind. Eng. Chem. 35, 1943, pp.
239-246.
PVTsim Method Documentation PVT Experiments 78
PVT Experiments
PVT Experiments
PVTsim may be used to simulate the most commonly performed PVT-experiments. A description of these
experiments has been given by Pedersen et al. (1984, 1989) ans by Pedersen and Christensen (2006).
PVT experiments are carried out with reference to standard conditions that may be specified in PVTsim. Default
values are default 1 atm/14.696 psia and 15°C/59°F. The results tabulated in a simulation of a PVT experiment are
explained in the following.
Constant Mass Expansion
The reservoir fluid is kept in a cell at reservoir conditions. The pressure is reduced in steps at constant temperature
and the change in volume is measured. The saturation point volume, Vsat, is used as a reference value and the
volumetric results presented are relative volumes, i.e., the volumes divided by Vsat.
Oil Mixtures
For oil systems the primary output for each pressure stage comprises
Relative volume
V/Vb where V is the actual volume and Vb is bubble point or saturation point volume.
Compressibility (only for pressures above the saturation point)
T
oP
V
V
1c
Y factor (only for pressures above the saturation point)
1V
VP
PPY
sat
t
sat
Vt is the total volume of gas and liquid.
Gas Condensate Mixtures
PVTsim Method Documentation PVT Experiments 79
For gas condensate systems the primary output for each pressure stage comprises
Rel Vol V/Vd (Vd is dew point or saturation point volume)
Liq Vol Liquid vol% of Vd.
Z Factor (only above saturation point)
Differential Liberation
This experiment is only carried out for oil mixtures. The reservoir fluid is kept in a cell at the reservoir temperature.
The experiment is usually started at the saturation pressure. The pressure is reduced stepwise and all the liberated gas
is displaced and flashed to standard conditions. This procedure is repeated 6-10 times. The end point is measured at
standard conditions.
The primary output for each pressure stage comprises:
Oil FVF Oil formation volume factor (Bo) defined as the oil volume at the actual
pressure divided by the residual oil volume at standard conditions
Rsd Solution gas/oil ratio, which is the total standard volume of gas liberated
from the oil in the stages to follow, divided by the residual oil volume. The
volume of the liquid condensing when flashing the gas to standard
conditions is converted to an equivalent gas volume.
Gas FVF Gas formation volume factor defined as the volume of the gas at the actual
conditions divided by the volume of the same gas at standard conditions. The
volume of the liquid condensing when flashing the gas to standard
conditions is converted to an equivalent gas volume.
Gas Gravity Molecular weight of the gas divided by the molecular weight of atmospheric
air (=28.964).
Constant Volume Depletion
This experiment is performed for gas condensates and volatile oils.
The reservoir fluid is kept in a cell at reservoir temperature and saturation point pressure. The pressure is reduced in
steps, and at each level as much gas is removed that the volume of the remaining gas and oil mixture equals the
saturation point volume.
For each pressure stage the primary output consists of
Liq vol Liquid volume% of dew point volume
%Prod Cumulative mole% of initial mixture removed
Z factor gas
Viscosity Viscosity of the gas in the cell
Separator Experiments
Separators in Series
A separator experiment is customarily started at the saturation pressure at the reservoir temperature. The volume and
the density are recorded. Subsequently a series of PT flash separations is performed. The gas phase from each
separator stage is flashed to standard conditions. The liquid phase is let to a new separator in which a new PT flash
separation takes place, and so on. The last separator is at atmospheric conditions.
The primary output consists of
PVTsim Method Documentation PVT Experiments 80
GOR Volume of gas from the actual stage at standard conditions divided by the volume of
the oil from the last stage (atmospheric conditions)
Gas Gravity Molecular weight of the gas divided by the molecular weight of air (28.964)
FVF Oil formation volume factor, which is the oil volume at the actual stage divided by the
oil volume from the last stage.
Sometimes the separator GOR is seen reported as the standard volume of gas divided by the separator oil volume (oil
volume at actual stage). The latter GOR can be converted into that reported by PVTsim by dividing it by FVF.
Viscosity Experiment
A viscosity experiment is performed at the reservoir temperature. The pressure is reduced in steps as in a differential
liberation experiment. At each step the gas and oil viscosities are recorded.
Swelling Experiment
When gas is injected into a reservoir containing undersaturated oil, the gas may dissolve in the oil. The volume of
the oil increases, which is called swelling. A swelling test experiment may simulate this process. The cell initially
contains reservoir oil. A known molar amount of a gas is added at a constant temperature. The saturation pressure of
the swollen mixture and the volume at the saturation point divided by the volume of the original reservoir oil are
recorded. More gas is added. The new saturation pressure and saturation point volume are recorded and so on. The
primary output consists of:
Mole% Cumulative mole% of gas added
GOR Std. volume of gas added per volume of original reservoir fluid
Sat P Saturation pressure after gas injection
Swollen volume Volume of the mixture per volume original reservoir fluid
Density Density of swollen mixture at saturation point
It is further indicated in the output whether the saturation point is a bubble point (Pb) or a dew point (Pd).
Equilibrium Contact Experiment
When gas is injected into a reservoir containing undersaturated oil, the gas may either dissolve in the oil or split the
reservoir fluid into two equilibrium phases – a gas and an oil. An Equilibrium Contact experiment may simulate this
process. The cell initially contains a known amount of reservoir oil. A user-specified amount of a gas is added at a
constant pressure and temperature. The amount of gas is specified as a molar ratio between gas and oil.
The output consists of amount and properties of gas and oil arising from equilibrating the mixture at the specified
PT-conditions.
This process is continued for a 1 stage.
Multiple Contact Experiment
When gas is injected into a reservoir containing undersaturated oil, the gas may either dissolve in the oil or split the
reservoir fluid into two equilibrium phases – a gas and an oil. A Multiple Contact experiment may simulate this
process. The cell initially contains a known amount of reservoir oil. A user-specified amount of a gas is added at a
constant pressure and temperature. The amount of gas is specified as a molar ratio between gas and oil.
PVTsim Method Documentation PVT Experiments 81
The Drive Type may be either
Forward
The gas phase is moved to the subsequent stage and mixed with a known amount of fresh reservoir oil.
Reverse (backward)
The oil phase is moved to the subsequent stage and mixed with a known amount of fresh injection gas
For a forward contact the gas/oil input ratio is per amount oil at the actual stage. For a backward contact it is per
amount of initial oil.
The output consists of amount and properties of gas and oil arising from equilibrating the mixture at the specified
PT-conditions.
This process is continued for a number of stages.
Slim Tube Experiment
As a result of the production from a petroleum field, the reservoir pressure will begin to decrease. In order to
maintain the reservoir pressure at a level, where the recovery of reservoir oil is at an optimum, gas is often injected
into the reservoir.
The optimum pressure level is known as the minimum miscibility pressure (MMP). The MMP may either be
calculated through an MMP calculation (see Minimum Miscibility Pressure Calculation), or it may be estimated
through a series of simulated slim tube experiments conducted at different pressures.
The MMP can often be seen as a distinct bend on a curve of oil recovery versus pressure. This is exemplified by the
figure below.
The slim tube experiment is simulated in PVTsim as shown in the figure below (Metcalfe et al., 1972).
PVTsim Method Documentation PVT Experiments 82
The pressure and temperature in each cell are, in PVTsim, assumed constant throughout a run. The sum of all cell
volumes is assumed to be equal to the total pore volume of the tube.
The input to the slim tube experiment consists of
A reservoir oil composition and an injection gas composition characterized to the same pseudo-components.
Temperature (constant) and a maximum of 8 pressure stages.
Number of cells.
Number of time steps (or ‘gas injections’).
Transport mechanism.
- Moving excess: The cell volume remains constant throughout the simulation, and the excess volume is
transferred to the next cell. If the oil volume exceeds that of the original cell, all gas and the excess volume
of oil are transferred to the next cell. If the oil volume is lower than that of the original cell, only the excess
gas volume is transferred to the next cell.
- Phase mobility: The cell volume remains constant throughout the simulation, and the excess volume is
transferred to the next cell. If two phases are present, gas and liquid are moved according to their relative
phase mobilities, M. These are calculated from relative permeability data (k) and from the phase viscosities
( For a given phase the mobility is defined as η
kM . Each time one unit of gas with a mobility of 2 is
removed from the cell, half a unit of oil with a mobility of 1 is removed from the cell. The relative
permeabilities of the gas and oil phases are determined by interpolating in user input for relative
permeability versus oil saturation.
- Phase viscosities: The cell volume remains constant throughout the simulation, and the excess volume is
transferred to the next cell. If two phases are present, gas and liquid are moved according to their relative
phase viscosity mobilities defined as
η
fraction volumePhaseM visc
The oil and gas volume transferred from one cell to the next one are
Voil(moved) = Vexcess* Mvisc(oil)/(Mvisc(oil)+Mvisc(gas))
Vgas(moved) = Vexcess- Voil(moved)
PVTsim Method Documentation PVT Experiments 83
Corresponding values of oil saturation (volume % oil) and relative gas and oil permeability (only for phase
mobility option).
The two former transport mechanisms are illustrated in the figure below (Pedersen et al. 1989).
Moving excess Phase mobility
The simulation scheme followed to simulate the slim tube experiment, in PVTsim, is outlined below.
1. The cells are loaded with reservoir oil (1 mole per cell).
2. The oil volume at standard conditions, which defines a recovery of 100%, is calculated.
3. Calculation of number of moles of injection gas.
4. Injection of gas into cell 1. The amount to be injected into cell 1 in each time step equals the number of moles
of injection gas from 3. divided by the number of time steps. The gas and the oil are assumed to mix perfectly
and to reach phase equilibrium instantaneously.
5. A flash calculation is carried out for cell 1 at the specified pressure and temperature in order to determine the
phase split and the phase compositions.
6. Excess hydrocarbon fluid is transferred to cell 2 according to the selected transport mechanism.
7. A flash calculation is carried out for cell 2, and the excess volume transferred to cell 3, etc.
8. The excess hydrocarbon fluid from the last cell is flashed to standard conditions. The oil volume at standard
conditions is added to the oil volume produced in previous 'time steps'. Recovery after a given time equals
cumulative oil volume at standard conditions divided by oil volume calculated in 2.
9. If there is more gas to inject, continue from 4. Otherwise continue with next pressure stage. Stop when all
pressure stages are covered.
The output consists of
Recovery table and plot of % recovered oil as a function of pressure.
For each cell
- Volume % oil.
- Viscosity of gas and oil.
- Density of gas and oil.
- K-factors for each component, if both gas and oil are present.
PVTsim Method Documentation PVT Experiments 84
- Total, gas, and oil composition.
Composition, density and viscosity of cumulative produced oil.
Pressure range within which MMP is found. Only input pressures are considered. MMP is reached when
two almost identical near-critical phases are present in a cell.
References
Metcalfe, R.S., Fussel, D.D., Shelton, J.L., (1972), "A Multicell Equilibrium Separation Model for the Study of
Multiple Contact Miscibility in Rich-Gas Drives", Paper presented at the SPE-AIME 47th
Annual Meeting in San
Antonio, Tx, Oct. 8 – 11.
Pedersen, K.S., Thomassen, P. and Fredenslund, Aa., ”Thermodynamics of Petroleum Mixtures Containing Heavy
Hydrocarbons. 3. Efficient Flash Calculation Procedures Using the SRK Equation of State”, Ind. Eng. Chem.
Process Des. Dev. 24, 1985, pp. 948-954.
Pedersen, K.S., Fredenslund Aa. and Thomassen, P., ”Properties of Oils and Natural Gases”, Gulf Publishing
Company, Houston, 1989.
Pedersen, K.S. and Christensen, P.L., ”Phase Behavior of Petroleum Reservoir Fluids”, CRC Taylor & Francis, Boca
Raton, 2006.
PVTsim Method Documentation Compositional Variation due to Gravity 85
Compositional Variation due to Gravity
Compositional Variation due to Gravity
Hydrocarbon reservoirs show variations in the composition in the direction from the top to the bottom of the
reservoir. The mole fractions of the lighter components decrease, whereas the mole fractions of the heavier
components increase. This is at least partly explained by the fact that gravity forces introduce a compositional
gradient.
The Depth Gradient option of PVTsim considers
Isothermal reservoirs
Reservoirs with a vertical temperature gradient.
With isothermal reservoirs the compositional variation with depth is assumed only to originate from gravitational
forces. For non-isothermal reservoirs both gravitational forces and vertical heat flux are accounted for.
Isothermal Reservoir
For an isothermal system the chemical potentials, , of component i located in height h and in height h0 are related as
follows
0
i
0
ii hhgMhμhμ
M stands for molecular weight and g is the gravitational acceleration. The chemical potential is related to the
fugacity through the following relation
ii flnRTμ
PVTsim Method Documentation Compositional Variation due to Gravity 86
where T is the temperature.
The fugacities of component i in height h and in height h0 are therefore related through
RT
hhgMflnfln
0
ih
i
h
i
o
The fugacity of component i is related to the fugacity coefficient of component i as
Pzf iii
which gives the following relation between the fugacity coefficients of component i in height h and in height h0
RT
hhgMPzlnPzln
0
ihh
i
h
i
hh
i
h
i
000
This equation is valid for any component i. For a system with N components there are N such equations. The mole
fractions of the components must sum to 1.0 giving one additional equation
N
1ii 1z
If the pressure0hP and the composition N)1,2,...,i,(z
0h
i are known in the reference height h0, there are N + 1
variables for a given height h, namely N)1,2,...,i,(z0h
i and Ph. A set of N + 1 equations with N + 1 variables may
be solved to give the molar composition and the pressure as a function of height. The equations are solved as
outlined by Schulte (1980).
In general the SRK and PR equations give the same phase equilibrium results with and without the Peneloux volume
correction. This is not true in depth gradient calculations. The fugacity coefficients of component i calculated with
the SRK and SRK-Peneloux equations are interrelated as follows
RT
Pclnln i
PENi,i,SRK
where c is the volume translations term. In a usual phase equilibrium calculation the temperature and pressure are the
same throughout the system and the term on the right hand side of the equation cancels. This is not the case in a
calculation of the compositional variations with depth. The pressure changes with depth and this change is related to
the fluid density for which different results are obtained with the SRK and PR Peneloux equations. The SRK and PR
Peneloux equations are both presented in the Equation of State section.
Reservoirs with a Temperature Gradient
PVTsim Method Documentation Compositional Variation due to Gravity 87
A petroleum reservoir can only be at thermodynamic equilibrium if the temperature is constant with depth. In
petroleum reservoirs the temperature typically increases by of the order of 0.02C/m - 0.011°F/ft from the top to the
bottom of the reservoir. A temperature gradient introduces a flow of heat between locations at different temperature
and it can no longer be assumed that the reservoir is in thermodynamic equilibrium. For relatively thin reservoirs it is
often reasonable to neglect the temperature variation.
The heat flux results in an entropy production in the system. To set up the equations needed to solve for the molar
compositions in a reservoir with a thermal gradient it is necessary to make use of the terminology of irreversible
thermodynamics. To simplify the problem one may assume that the system is at a stationary state, that is, all
component fluxes are zero and the gradient assumed constant in time. Relative to the equilibrium situation addressed
by Schulte (1980), this constitutes a dynamically stabilized system balanced by the gravity and heat flow effects.
An observed compositional gradient in a petroleum reservoir may furthermore be affected by capillary forces, by
convection and by secondary migration of hydrocarbons into the reservoir. None of these effects are considered here.
PVTsim uses a model of Pedersen and Lindeloff (2003) for describing the non-isothermal case. It is essentially the
same as that proposed by Haase (1971). The approach can be summarized as follows
N1,i;T
ΔT
M
H~
M
H~
M)hg(hM)PzRTln()PzRTln(i
i
i
0
i
h0h0
i
h0
i
hh
i
h
i
Relative to the isothermal expression by Schulte, an additional term including the effect of the temperature gradient
ΔT has been added. The term furthermore contains average molecular weight, M, component molecular weight Mi
and mixture and component partial molar enthalpies, H~
and .H~
i
A proper determination of partial molar enthalpies is the key to obtaining reasonable predictions with the model. In
typical process simulations it is appropriate to work with enthalpy differences since the overall composition is
normally constant, and the reference state therefore the same in all cases. This assumption cannot be applied to the
present problem. Instead, absolute enthalpies with a unique reference state must be used.
In PVTsim enthalpies are normally calculated relative to the enthalpy of an ideal gas at 273.15 K/0°C/32°F and the
same composition. Absolute enthalpies, being the sum of an ideal gas contribution and a residual term are obtained
as follows
ig
273.15K
ig
273.15K
igresig
273.15K
PVTsimabs H)H(HHHHH
PVTsim by default uses the following expressions for the ideal gas enthalpy of component i at 273.15 K (Pedersen
and Hjermstad, 2006)
i
ig
i,273.15M67.86342,1
R
H
where R
Hig
i,273.15 is in Kelvin.
The ideal gas enthalpy of component i at 273.15 K make up the tuning parameters when tuning to match
experimental data for the compositional variation with depth. The values may vary freely depending on the number
of data points available.
PVTsim Method Documentation Compositional Variation due to Gravity 88
Prediction of Gas/Oil Contacts
Assume an oil of a given composition at a reference depth. Moving upwards in the reservoir the concentration of
lighter components increases, causing the bubble point of the oil to increase and the reservoir pressure to decrease.
At a certain depth the reservoir pressure and the bubble point pressure of the oil may coincide. This is the depth of
the gas/oil contact in the reservoir. This depth is determined and written out in PVTsim.
References
Haase, R., Borgmann, H.-W., Dücker, K. H. and Lee, W. P., "Thermodiffusion im kritischen Verdampfungsgebiet
Binärer Systeme", Z. Naturforch. 26 a, 1971, pp. 1224-1227.
Schulte, A.M., ”Compositional Variations within a Hydrocarbon Column due to Gravity”, paper SPE 9235 presented
at the 1980 SPE Annual Technical Conference and Exhibition Dallas, Sept. 21-24, 1980.
Pedersen, K.S. and Lindeloff, N., “Simulations of Compositional Gradients in Hydrocarbon Reservoirs Under the
Influence of a Temperature Gradient”, SPE Paper 84364, presented at the SPE ATCE in Denver, 5-8 October, 2003.
Pedersen, K. S. and Hjermstad, H. P., “Modeling of Large Hydrocarbon Compositional Gradient” presented at 2006
SPE Abu Dhabi International Petroleum Exhibition and Conference, November 5-8, 2006 in Abu Dhabi, UAE.
PVTsim Method Documentation Regression to Experimental Data 89
Regression to Experimental Data
Regression to Experimental Data
PVTsim is basically a predictive tool. No experimental PVT-data are needed to perform the C7+-characterization and
once the C7+-characterization is completed, all the simulations can be readily performed. When a particularly good
match of the experimental PVT-data is needed or heavy lumping is a requirement, the simulation results can be
improved using the regression module.
Experimental data
The two tables below show the type of PVT-data to which regression may be performed.
Oil mixtures.
Sat. Point CME Dif. Dep. Separator Viscosity Swelling CVD MMP
Saturation
Point
*) x x x x x x x
MMP x
Bo x x
GOR (Rs) x x x
Rel. volume x x
Compressibility x
Y-Factor x
Oil density x x x x
Z factor Gas x x
Two phase Z factor x
Liquid volume %
Gas Gravity x x
Bg x
Mole % removed x
Oil viscosity x x
Gas viscosity x x x *)
May also be the critical point.
PVTsim Method Documentation Regression to Experimental Data 90
Gas condensate mixtures
Sat. points CME CVD Separator Viscosity MMP
Saturation
Point
*) x x x x x
MMP x
Z factor gas x x
Two phase
Z factor
x
Rel volume x
Liq vol% x x
Bo x
GOR x
Gas density x
Oil density x
Gas gravity x
Mole% removed x
Oil viscosity x
Gas viscosity x x *)
May also be critical point.
Object Functions and Weight Factors
The object function to be minimized during a regression calculation is defined as
NOBS
1j
2
j
j
w
rOBJ
where NOBS is the number of experimental observations used in the regression, wi is the weight factor for the j’th
observation, and rj is the jth
residual
j
jj
jOBS
CALCOBSr
where OBS stands for the observed and CALC for calculated. For liquid dropout curves from a constant mass
expansion and constant volume depletion experiment, a constant is added to all OBS and CALC-values. This
constant equals the maximum liquid dropout divided by 3 and is added to reduce the weight assigned to data points
with small liquid dropout relative to data points with larger liquid dropouts. The weight factor, wj, and the user
specified weight, WOBS to be assigned to the j’th observation are interrelated as follows
2
jw
1WOBS
Regression for Plus Compositions PVT Data
PVTsim Method Documentation Regression to Experimental Data 91
If the user has allowed the plus molecular weight to be adjusted, an initial regression calculation is performed where
the plus molecular weights are adjusted to give the best possible match of the measured saturation points. The
molecular weight of the plus fraction is used as regression parameter because there is usually an experimental
uncertainty of 5-10% on the experimental determination of this quantity. Furthermore even small changes in the
molecular weight of the plus fraction may have a major influence on the calculated saturation point. When
modifying the molecular weight of the plus fraction, the weight composition is kept constant while the molar
composition is recalculated. The weight composition is the one actually measured and is accordingly kept constant.
Secondly a regression is performed where the coefficients in the Tc, Pc and m correlations presented in the
Characterization of Heavy Hydrocarbons section are treated as regression parameters. The default number of
regression parameters is
NPAR = 1 + ln (NDAT)
Where NDAT is the number of experimental data points not considering viscosity data. The maximum number of
regression parameters is 10. The NPAR regression parameters are selected in the following order (Christensen,
1999):
Coefficient c2 in Tc correlation.
Coefficient d2 in Pc correlation.
Peneloux volume shift parameter.
Coefficient c3 in Tc correlation.
Coefficient d3 in Pc correlation.
Coefficient e2 in m correlation.
Coefficient e3 in m correlation.
Coefficient c4 in Tc correlation.
Coefficient d4 in Pc correlation.
Coefficient e4 in m correlation.
In each iteration the parameters c1, d1 and e1 are recalculated to give the same Tc, Pc and m of a component with a
molecular weight of 94 and a density of 0.745 g/cm3 as is obtained with the standard coefficients. This is done to
ensure that Tc, Pc and m of the lower C7+ fractions are assigned properties, which are physically meaningful. The
user therefore has no control of the parameters c1, d1 and e1 in the regression input menu.
The user may modify the default selection of regression parameters, but the number of regression parameters must
not exceed the number of experimental data points.
Regression to Viscosity Data
The regression parameters depend on applied viscosity correlation. The below parameters are defined in the
Transport Property section. With the corresponding states model the assumed mixture molecular weight is found
from the following equation when methane is used as reference component
n,W
Corfac2VISC2
n,W
Corfac2VISC2
w,Wmixw, MMMVISC1Corfac1M
PVTsim Method Documentation Regression to Experimental Data 92
VISC1 = 1.304 x 10-4
and VISC2 = 2.303. Corfac1 and Corfac2 are by default 1.0 but can be modified by regression
to viscosity data (1st and 2
nd CSP viscosity correction factors).
When the stable oil viscosity correlation is used as reference the average molecular weight is found from
Corfac4VISC4
n
wn
MCorfac3VISC3
MMM
Corfac3 and Corfac4 are 1.0 by default, but may be regressed on (3rd
and 4th
CSP viscosity correction factors).
With the LBC viscosity correlation three regression options exist. The default one is to let the regression determine a
unique correction factor to be multiplied with the critical volumes of the pseudo-components. It is further possible to
determine optimum values of the five coefficients a1 – a5 in the LBC correlation. A third option is to combine the Vc
and a1 – a5 regression.
The optimum viscosity correction factors and/or the optimum values of a1 – a5 may be viewed in the Char Options
menu accessed from the composition input menu.
Regression for already characterized compositions
The following component properties may be specified as regression parameters:
Tc
Pc
VPEN (volume shift parameter)
Vc
aΩ
bΩ
kij (binary interaction parameter)
kij A, kij B, kij C (parameters in expression for T-dependent binary interaction parameters)
The mentioned properties are all defined in the Equation of State section. A maximum of 15 regression parameters
may be specified. The number of experimental data points must be at least as high as the number of regression
parameters. One regression parameter may consist of for example Tc of one specific component or it may consist of
the Tc’s of a number of consecutive components in the component list. In the latter case the Tc’s of all these
components will be adjusted equally.
The critical volume only affects the viscosities and only if the LBC correlation has been specified (see Transport
Property section)
With the LBC viscosity model it is further possible to regress on the coefficients a1 – a5.
PVTsim Method Documentation Regression to Experimental Data 93
For the binary interaction parameters it is possible to specify single pairs of components for which the binary
interaction parameters are to be adjusted. Alternatively one may specify a component triangle. The binary interaction
parameters for each component pair contained in this triangle will in that case be adjusted equally.
The user may specify a maximum allowed adjustment for each parameter.
Regression on fluids characterized to the same pseudo-components
It is possible to perform regression on fluids, which have been characterized to the same set of pseudo-components.
Experimental PVT data is not required for all fluids. Consider a regression to the same pseudos in a case where data
is available say for 2 fluids out of 5 fluids to be characterized to the same pseudo-components. In this case the
regression procedure will modify the properties of all 5 fluids while honoring the best possible match of the available
data sets for the two fluids.
Regression Algorithm
The minimization algorithm used in the parameter regression is a Marquardt algorithm (Marquardt, 1963).
References
Christensen, P.L., ”Regression to Experimental PVT Data”, Journal of Canadian Petroleum Technology 38. 1999,
pp. 1-9.
Marquardt, D.W., SIAM J 11 1963, 431-441.
PVTsim Method Documentation Minimum Miscibility Pressure Calculations 94
Minimum Miscibility Pressure Calculations
Minimum Miscibility Pressure Calculations
Injection of gas into oil fields is commonly used to obtain an enhanced recovery. The injected gas influences the
reservoir oil in several ways. It reduces the pressure drop associated with the production, it influences the phase
properties (density, viscosity, etc.) and it influences the gas/oil phase equilibrium. The gas may take up components
from the oil phase (vaporizing mechanism), the oil may take up components from the gas phase (condensing
mechanism) or the oil and the gas may exhibit first contact miscibility. This means that only one phase is formed, no
matter in what proportion the oil and the gas are mixed. If the gas and the oil are not miscible by first contact,
miscibility may take place as a result of multiple contacts between the oil and the gas. A miscible drive is
advantageous, because valuable heavy components will be contained in a phase of a fairly high mobility. The
mobility is inversely proportional to the viscosity and the viscosity decreases when the oil takes up gaseous
components.
The MMP option in PVTsim (Jessen et al., 1996) considers the situation where miscibility may develop somewhere
between the injection well and the gas/oil front (combined drive). Imagine a 1-dimensional reservoir (tube). Gas is
being injected at one end and a series of gas and oil compositions develop along the tube ending with original oil in
some distance from the injection point. The approach used to calculate a combined vaporizing and condensing MMP
is based on the assumption (Wang and Orr, 1998) that all neighboring key tie lines are coplanar and hence have a
point of intersection. That the tie lie in section (1) has a point of intersection (zi, i=1,2,…,N) with the tie line in
section (2) can be expressed as follows
N,...,2,1i;x)1(yx)1(yz)2(
i2)2(
i2)1(
i1)1(
i1i
Starting at the far end with fresh oil the first, tie line is defined as
N,...,2,1i;x)1(yz1j
ioil1j
ioiloili
This tie line is coplanar (has a point of intersection) with the 2nd tie-line (at gas/oil front)
N,...,2,1i;x)1(yx)1(y1j
ioil1j
ioil2j
i22j
i2
This 2nd tie line is coplanar with the 3rd tie line
N,...,2,1i;x)1(yx)1(y2j
i22j
i23j
i33j
i3
PVTsim Method Documentation Minimum Miscibility Pressure Calculations 95
and so on. There are a total of N-1 intersecting tie lines (called key tie lines). The last one defines the point of
intersection between the injection gas and the oil at the injection point
N,...,2,1i;x)1(yx)1(y2Nj
i2N2Nj
i2N1Nj
iinj1Nj
iinj
Each equation is subject to the equilibrium constraints
1N,1j;N,...,2,1i;xy Li
ji
Vi
ji
and the summation of mole fraction condition
N,...,1j;0xyN
1i
ji
ji
-1
The above 2(N2-1) equations may be solved starting with a moderate pressure and gradually increasing the pressure
until one of the tie lines becomes critical. This is when the x-and the y-compositions become identical corresponding
to a tie line of zero length.
The % vaporizing drive is contained in the output. It follows the definition of Johns et al. (2002) for how to quantity
the displacement mechanisms. For each tie line the point is located for which the vapor mole fraction is equal to
0.5. The term d1 is used for the distance from the point on the oil tie line where =0.5 to the point on 2nd tie line
where =0.5. The term d2 is used for the distance from the latter point on the 2nd
tie line to the point on the 3rd tie
line where =0.5. For a 4-component mixture the 3rd tie line is the one passing through the injection gas and for
which number of components the fraction of a combined vaporizing/condensing drive that is vaporizing is given by
21
2m
dd
dV
For a multi-component system the vaporizing fraction is defined as the ratio of the total vaporizing path length to the
entire composition path
2N
1kk
2N
1kvk,
m
d
d
V
where dk,v is non-zero for tie lines for which the displacement mechanism between that tie line and the next one is
vaporizing. This is the case if the tie lines are longer in the direction towards the gas tie line than in the direction
towards the oil tie line.
References
Jessen, K., Michelsen, M.L. and Stenby, E.H.: ”Effective Algorithm for Calculation of Minimum Miscibility
Pressure”, SPE Paper 50632, 1998.
Johns, R.T., Yuan, H. and Dindoruk, B., "Quantification of Displacement Mechanisms in Multicomponent
Gasfloods", SPE 77696 presented at the SPE ATCE in San Antonio, Tx September 29-October 2, 2002.
Wang, Y., and Orr, F.M., ”Calculation of Minimum Miscibility Pressure”, SPE paper 39683, 1998.
PVTsim Method Documentation Unit Operations 96
Unit Operations
Unit Operations
Compressor
PVTsim supports two compressor options:
Compressor with classical isentropic efficiency.
Compression following constant efficiency path (PACE), which is a polytropic compression generalized to
multiple phases.
The two options differ in the way the compression path is corrected for isentropic efficiency.
The isentropic efficiency, , is defined as
dH
dPVη
where V is the molar volume, P the pressure and H the enthalpy. From the general thermodynamics relation
dH = VdP+TdS
where S is the entropy it can be seen that =1 for S=0 and that
VdP
dH
S
meaning that the definition of the efficiency can be rewritten to
PVTsim Method Documentation Unit Operations 97
dH
(dH)
dP
dH
dP
dH
η SS
Neglecting variations in efficiency along the compression path, one arrives at the classical definition of the
efficiency
ΔH
H)(η S
where (H)S is the enthalpy change of a compression following an isentropic path (=reversible adiabatic
compression) and H is the enthalpy change of the real compression (adiabatic but partly irreversible).
The difference between the two compressor options is illustrated in the below figure.
P
Pout
in
S
HP
P
P
P1
2
..
..
H
His
Isentropic
Real
PACE
The dashed line illustrates a compression path following the classical definition of isentropic efficiency. Initially an
isentropic path is followed from inlet pressure Pin to outlet pressure Pout. The corresponding enthalpy change is
(H)S. The outlet enthalpy is determined by dividing the isentropic enthalpy change by the efficiency. The Pout
pressure line is followed to the outlet enthalpy meaning that the efficiency is determined by the slope of the Pout
curve.
Schematic HS-diagram.
PVTsim Method Documentation Unit Operations 98
The dotted line shows a compression path of an almost constant efficiency (polytropic compression). The
compression path is divided into small P-segments each of the size P as illustrated by the dotted line in the figure.
Each segment is simulated as an isentropic compression with the pressure increase P. The corresponding enthalpy
change (H)S is derived. The actual enthalpy change, H=(H)S/, and P determine the conditions in the next
point on the compression path including the volume.
The sequence of calculations is the following
1) Divide the compression into n pressure steps where each step is P =(Pout-Pin)/n.
2) Perform a PT-flash for Tin, Pin. Flash determines Sin and Hin.
3) Perform a PS-flash for P2=Pin +P, Sin. Flash determines isentropic outlet temperature (T2)S and (H2)S from
segment.
4) Determine ininS2
2 Hη
H)(HH
5) Determine T2 and S2 by PH-flash for P2,H2.
6) Perform a PS-flash for P3=P2+P, S2. Flash determines isentropic temperature (T3)S and (H3)S
7) Determine 22S3
3 Hη
H)(HH
8) Determine T3 and S3 by PH flash for P3,H3.
9) Continue from 6. with P4, and so on until Pn-1 (Pn=outlet pressure Pout).
The outlined procedure is applicable to gases as well as mixtures of gases and liquids.
The output for the Path of Constant Efficiency (PACE) option includes maximum and minimum values of the
compressibility functions, X and Y as defined by Schultz (1962)
T
P
P
V
V
PY
1T
V
V
TX
Also given in the output is the HEAD defined as:
fmg
WORKHEAD
where WORK is the total work done by the compressor on the fluid, g the gravitational acceleration and mf the flow
rate of the fluid through the compressor.
As can be seen from the above equation, the unit of HEAD is m or ft depending on selected unit. HEAD therefore
expresses the vertical lift height corresponding to the total work done by the compressor on the fluid.
PVTsim Method Documentation Unit Operations 99
Expander
The input is inlet pressure and temperature and outlet pressure. An efficiency can be specified. It is 1.0 by default.
For an efficiency of 1 the expansion process is assumed to be isentropic (constant entropy (S)). In general the
efficiency is defined as
sΔH
ΔHη
where (H)S is the enthalpy change by an isentropic expansion and H the actual enthalpy change.
Cooler
Input consists of inlet and outlet temperature and pressure. The outlet pressure is entered as a pressure drop, which is
zero by default. The cooling capacity is calculated. It is defined as the enthalpy to be removed from the flowing
stream per time unit.
Heater
Input consists of inlet and outlet temperature and pressure. The outlet pressure is entered as a pressure drop, which is
zero by default. The heating capacity is calculated. It is defined as the enthalpy to be transferred to the flowing
stream per time unit.
Pump
Input consists of inlet temperature and pressure and outlet pressure. A thermal efficiency can be specified, which is
defined through the relation
ΔH
P)V(Vη inout
where Vout is the outlet volume, Vin the inlet volume and the enthalpy change as a result of the pumping.
Valve
The outlet temperature is found by assuming that there is no enthalpy change by the passage of the valve.
PVTsim Method Documentation Unit Operations 100
Separator
Input consists of inlet temperature and pressure for which a PT-flash calculation is performed.
References
Schultz, J. M., "The Polytropic Analysis of Centrifugal Compressors", Journal of Engineering for Power, January
1962, pp. 69-82.
PVTsim Method Documentation Modeling of Hydrate Formation 101
Modeling of Hydrate Formation
Hydrate Formation
Hydrates consist of geometric lattices of water molecules containing cavities occupied by lighter hydrocarbons or
other light gaseous components (for example nitrogen or carbon dioxide). Hydrates may be formed where the
mentioned components are in contact with water at temperatures below approximately 35°C/95°F. Using the hydrate
module in PVTsim it is possible to calculate the conditions at which hydrates may form and in what quantities.
Calculations concerning the effect of the most commonly applied liquid hydrate inhibitors may be performed, and
the inhibiting effect of dissolved salts in the water phase is also accounted for. The hydrate phase equilibrium
calculations considers the phases
Gas
Oil
Aqueous
Ice
Hydrates of structures I, II and H
Solid salts.
The loss of hydrate inhibitors to the hydrocarbon phases is also determined.
Types of Hydrates
PVTsim considers three different types of hydrate lattices, structures I, II and H. Each type of lattice contains a
number of smaller and a number of larger cavities. In a stable hydrate, components (guest molecules) occupy some
of these cavities.
Structures I and II hydrates can only accommodate molecules of a rather modest size and appropriate geometry. The
table below indicates which of the components in the PVTsim component database may enter into the cavities of
hydrate structures I and II. The cavities may contain just one type of molecules or they may contain molecules of
different chemical species.
Component sI - Small sI - Large sII - Small sII - Large
PVTsim Method Documentation Modeling of Hydrate Formation 102
Cavities cavities cavities cavities
N2 + + + +
CO2 + + + +
H2S + + + +
O2 + + + +
Ar + + + +
C1 + + + +
C2 - + - +
C3 - - - +
iC4 - - - +
nC4 - - - +
2,2-dim-C3 - - - +
c-C5 - - - +
c-C6 - - - +
Benzene - - - +
The last four components in the above table are designated structure II heavy hydrate formers (HHF).
The number of cavities available for guest molecules are given as follows:
Number Per Unit Cell sI sII
H2O molecules 46 136
Small cavities 2 16
Large cavities 6 8
Structure H consists of three different cavity sizes. These are in PVTsim modeled as just two cavity sizes, a
small/medium one and a huge one. The huge cavity can accommodate molecules containing from 5 to 8 carbon
atoms. The small/medium sized cavities will usually be accommodated with N2 or C1. The below table gives an
overview of structure H formers considered in PVTsim.
Component Small/Medium Cavities Huge Cavities
Methane + -
Nitrogen + -
Isopentane - +
Neohexane - +
2,3-Dimethylbutane - +
2,2,3-Trimethylbutane - +
3,3-Dimethylpentane - +
Methylcyclopentane - +
1,2-Dimethylcyclohexane - +
Cis-1,2-Dimethylcyclohexane - +
Ethylcyclopentane - +
Cyclooctane - +
Hydrate Model
Hydrates are formed when the hydrate state is energetically favorable as compared to a pure water state (fluid water
or ice). The transformation from a pure water state to a hydrate state can be regarded as consisting of two steps:
pure water () empty hydrate lattice ()
empty hydrate lattice () filled hydrate lattice (H)
PVTsim Method Documentation Modeling of Hydrate Formation 103
where , and H are used to identify each of the three states considered. The - state is purely hypothetical and only
considered to facilitate the hydrate calculations. The energetically favorable state is that of the lowest chemical
potential. The difference between the chemical potential of water in the hydrate state (H) and in a pure water state
() can be expressed as
αββHαH μμμμμμ
The first term on the right hand side βH μμ can be regarded as the stabilizing effect on the hydrate lattice caused
by the adsorption of gas molecules. This latter effect depends on the tendency of the molecules to enter into the
cavities of the hydrate lattice. This tendency is in PVTsim expressed using a simple adsorption model. The
difference between the chemical potential of water in the empty and in the filled hydrate lattice is calculated as
follows
NCAV
li
N
1KKii
βH Y1lnvTRμμ
where i is the number of cavities of type i and YKi denotes the possibility that a cavity i is occupied by a gas
molecule of type K. NCAV is the number of cavities per unit cell in the hydrate lattice and N is the number of
components present, which may enter into a cavity in the hydrate lattice. The probability YKi is calculated using the
Langmuir adsorption theory
N
ljjji
KKiKi
fC1
fCY
where fK is the fugacity of component K. CKi is the temperature dependent adsorption constant specific for the cavity
of type i and for component K. The adsorption constant accounts for the water-hydrate forming component
interactions in the hydrate lattice. The adsorption constant C is calculated from the following expression (Munck et
al., 1988)
/TBexp/TAC KiKiKi
For each component K capable of entering into a cavity of type i, AKi and BKi must be determined from experimental
data. The A and B values used in PVTsim may be seen from the Pure Component database. Most of the structure I
and II hydrate parameters are from Munck et al. (1988) and Rasmussen and Pedersen (2002), and the parameters for
structure H are from Madsen et al. (2000). The hydrate parameters are specific for the selected equation of state
(SRK or PR).
The term μμ is equal to the difference between the chemical potentials of water in the empty hydrate lattice
(the -state) and water in the form of liquid or ice (the -state). An expression for this difference in chemical
potentials can be derived using the following thermodynamic relation
dPRT
ΔVdT
RT
ΔH
RT
Δμd
2
where R is the gas constant and H and V are the changes in molar enthalpy and molar volume associated with the
transition. The following expression may be obtained for the difference between the chemical potentials of water in
the - and -states at the temperature, T, and the pressure, P
P
P0
T
T0 20
00αβ
dPTR
ΔVdT
RT
ΔH
RT
P,TΔμ
RT
PT,Δμ
RT
μμ
where T0, P0 indicates a reference state at which is known. In this equation it has been assumed that H is
independent of pressure. The temperature dependence of the second term has been approximated by the average
temperature
PVTsim Method Documentation Modeling of Hydrate Formation 104
2
TTT 0
If the reference pressure, P0, is chosen to be equal to be zero, the above equation can be rewritten to
P
P0
T
T0 20
00αβ
dPTR
ΔVdT
RT
ΔH
RT
P,TΔμ
RT
PT,Δμ
RT
μμ
H is calculated from the difference, CP, in the molar heat capacities of the - and the -states
T
T0 pdTΔCTΔH
The constants needed in the calculation of for the transition at a given temperature and pressure are taken
from Erickson (1983) (structure I and II) and from Mehta and Sloan (1994) (structure H) and shown below.
Property Unit Structure I Structure II Structure H
0Δμ (liq) J/mole 1264 883 1187.33
0ΔH (liq) J/mole -4858 -5201 -5162.43
0ΔH (ice) J/mole 1151 808 846.57
0ΔV (liq) cm3/mole 4.6 5.0 5.45
0ΔV (ice) cm3/mole 3.0 3.4 3.85
pΔC (liq) J/mole/K -39.16 -39.16 -39.16
Using the procedure outlined above, the difference in chemical potentials μμ H between water in a hydrate state
(H) and in a pure water state () may now be calculated.
A hydrate phase equilibrium curve represents the T, P values for which
0μμ αH
At those conditions the hydrate state and the liquid or solid water states are equally favorable. To the left of the
hydrate curve
0μμ αH
and some of the water will at equilibrium be in a hydrate form. Whether this is a structure I or a structure II hydrate
depends on which of the two structures has the lower chemical potential in the presence of the actual gas
components as potential guest molecules. To the right of the hydrate curve
0μμ αH
i.e. at equilibrium at those conditions no hydrate can exist and the water will be in the form of either liquid or ice.
Hydrate P/T Flash Calculations
Flash calculations are in PVTsim performed using an ”inverse” calculation procedure as outlined below.
PVTsim Method Documentation Modeling of Hydrate Formation 105
1) Initial estimates are established of the fugacity coefficients of all the components in all phases except in the
hydrate phases and in any pure solid phases. This is done by assuming an ideal gas and ideal liquid solution,
neglecting water in the hydrocarbon liquid phase and by assuming that any water phase will be pure water.
2) Based on these fugacity coefficients and the total overall composition (zK, K = 1,2,…..N) a multi phase P/T
flash is performed (Michelsen, 1988). The results of this calculation will be the compositions and amounts
of all phases (except any hydrate and pure solid phases) based on the guessed fugacity coefficients, i.e.: xKj
and j, K = 1,2…,N, j hyd and pure solid. The subscript K is a component index, j a phase index, stands
for phase fraction and N for number of components.
3) Using the selected equation of state and the calculated compositions (xKj), the fugacities of all components
in all the phases except the hydrate and pure solid phases are calculated, i.e. (fKj, K = 1,2…,N, j hyd and
pure solid).
4) Based on these fugacities (fKj, K = 1,2..,N, j hyd and pure solid), mixture fugacities N)1,2,...,K,f mix
K are
calculated. For the non-water components, a mixture fugacity is calculated as the molar average of the
fugacities of the given component in the present hydrocarbon phases. For water the mixture fugacity is set
equal to the fugacity of water in the water phase.
5) The fugacities of the components present in the hydrate phase are calculated using mix
K
H
K fln fln
where is a correction term identical for all components. is found from
,lnfY1lnνΘfln β
w
NCAV
1i
N
1KKii
mix
w
where w stands for water and refers to the empty hydrate lattice.
6) The hydrate compositions are calculated using the expression
NCAV
1iNHYD
1j
H
jji
H
KKii
w
K ,
fC1
fCν
x
x which enables
calculation of the fugacity coefficients as described below. Non-hydrate formers are assigned large fugacity
coefficients (ln = 50) to prevent them from entering into the hydrate phases.
7) Based on the actual values of the fugacity coefficients for all the components in all the phases (Kj) and the
total overall composition zK an ideal solution (composition independent fugacity coefficients) a multi phase
flash is performed (Michelsen, 1988). The result of this calculation will be compositions and amounts of all
phases (i.e.: xKj and j, K = 1,2,…,N, j = 1,…, number of phases).
8) If not converged repeat from 3).
Calculation of Component Fugacities
Fluid Phases
To use the flash calculation procedure outlined above, expressions must be available for the fugacity of component i
in each phase to be considered. The fugacity of component i in a solution is given by the following expression
Pxf iii
where is the fugacity coefficient, xi the mole fraction and P the pressure.
For the fluid phases, is calculated from the selected equation of state. See Equation of State section for details.
Fugacities calculated with PR will be slightly different from those calculated with SRK, which is why hydrate
parameters specific for the selected equation of state is used.
Hydrate Phases
The fugacities of the various components in the hydrate phases are calculated as described by Michelsen (1991)
PVTsim Method Documentation Modeling of Hydrate Formation 106
Water
2
02
1
0i
β
w
H
wv
θNlnv
v
θ1Nlnvflnfln
Other Hydrate Formers
θ1αθCN
Nf
K20
KH
K
K
In these equations
β
wf = fugacity of water in empty hydrate lattice
vi = number of cavities of type i
N0 = number of empty lattice sites
= ratio of free large lattice sites to total free lattice sites
NK = content of component K per mole of water
CKi = Langmuir constant
K = CK1/CK2
The determination of and N0 follows the procedure described by Michelsen. As the fluid phase fugacities vary
with the equation of state choice, the hydrate model parameters are equation of state specific in order to ensure
comparable model performance for both SRK and PR.
Ice
The fugacity (in atm) of ice is calculated from the following expression
273.15T
P0.0390
T
273.15ln4.710
T
273.1512.064f ice
where P is the pressure in atm and T the temperature in K.
References
Erickson, D.D., ”Development of a Natural Gas Hydrate Prediction Computer Program”, M. Sc. thesis, Colorado
School of Mines, 1983.
Madsen, J., Pedersen, K.S. and Michelsen, M.L., ”Modeling of Structure H Hydrates using a Langmuir Adsorption
Model”, Ind. Eng. Chem. Res., 39, 2000, pp. 1111-1114.
Mehta, P.A. and Sloan, E.D., “Improved Thermodynamic Parameters for Prediction of Structure H Hydrate
Equilibria”, AIChE J. 42, 1996, pp. 2036-2046.
Michelsen, M.L., ”Calculation of Multiphase Equilibrium in Ideal Solutions”, SEP 8802, The Department of
Chemical Engineering, The Technical University of Denmark, 1988.
Michelsen, M.L., ”Calculation of Hydrate fugacities ”, Chem. Eng. Sci. 46, 1991, pp. 1192-1193.
PVTsim Method Documentation Modeling of Hydrate Formation 107
Munck, J., Skjold-Jørgensen S. and Rasmussen, P., ”Computations of the Formation of Gas Hydrates”, Chem. Eng.
Sci. 43, 1988, pp. 2661-2672.
Rasmussen, C.P. and Pedersen, K.S., “Challenges in Modeling of Gas Hydrate Phase Equilibria”, 4th International
Conference on Gas Hydrates Yokohama Japan, May 19 - 23, 2002.
PVTsim Method Documentation Modeling of Wax Formation 108
Modeling of Wax Formation
Modeling of Wax Formation
The wax module of PVTsim may be used to determine the wax appearance temperature (cloud point) at a given
pressure, the wax appearance pressure at a given temperature and to perform PT flash calculations taking into
consideration the possible formation of a wax phase in addition to gas and oil phases. The wax model used is that of
Pedersen (1995) extended as proposed by Rønningsen et al. (1997).
Vapor-Liquid-Wax Phase Equilibria
At thermodynamic equilibrium between a liquid (oil) and a solid (wax) phase, the fugacity, ,f L
i
of component i in the
liquid phase equals the fugacity, ,f L
i of component i in the solid phase
S
i
L
i ff
When a cubic equation of state is used for the liquid phase it is practical to express the liquid phase fugacities in
terms of fugacity coefficients
Pxf L
i
L
i
L
i
In this expression L
ix is the liquid phase mole fraction of component i, L
i the liquid phase fugacity coefficient of
component i and P the pressure. For an ideal solid phase mixture, the solid phase fugacity of component i can be
expressed as
oS
i
S
i
S
i fxf
where S
ix is the solid phase mole fraction of component i, and oS
if the solid standard state fugacity of component i.
The solid standard state fugacity is related to the liquid standard state fugacity as
ref
oL
i
ref
oS
if
iPf
PflnRTΔG
where f
iΔG is the molar change in Gibbs free energy associated with the transition of pure component i from solid
to liquid form at the temperature of the system. To calculate f
iΔG the following general thermodynamic relation is
used
STΔHΔG
PVTsim Method Documentation Modeling of Wax Formation 109
where H stands for change in enthalpy and H for change in entropy. Neglecting any differences between the
liquid and solid phase heat capacities, f
iΔG may be expressed as
f
i
f
i
f
i STΔHΔG
where f
iΔH is the enthalpy and f
iΔS the entropy of fusion of component i at the normal melting point. Again
neglecting any differences between the liquid and solid-state heat capacities, the entropy of fusion may be expressed
as follows in terms of the enthalpy of fusion
f
i
f
if
iT
ΔHΔS
where f
iT is the melting temperature of component i. The following expression may now be derived for the solid
standard state fugacity of component i
RT
PPΔV
T
T1
RT
ΔHexpPff refi
f
i
f
iref
oL
i
oS
i
where V is the difference between the solid and liquid phase molar volumes. Based on experimental observations
of Templin (1956), the difference Vi between the solid and liquid phase molar volumes of component i is assumed
to be 10% of the liquid molar volume, i.e. the solidification process is assumed to be associated with a 10% volume
decrease.
The liquid standard state fugacity of component i may be expressed as follows
Pf oL
i
oL
i
where oL
i is the liquid phase fugacity coefficient of pure i at the system temperature and pressure. This leads to
RT
PPΔV
T
T1
RT
ΔHexpPPf refi
f
i
f
ioL
i
oS
i
The following expression may now be derived for the solid phase fugacity of component i in a mixture
RT
PPΔV
T
T1
RT
ΔHexpPPxf refi
f
i
f
ioL
i
S
i
S
i
oL
i is found using an equation of state on pure i at the temperature of the system and the reference pressure.
Extended C7+ Characterization
To be able to perform wax calculations it is necessary to use an extended C7+ characterization procedure. A
procedure must exist for splitting each C7+ pseudo-component into a potentially wax forming fraction and a fraction,
which cannot enter into a wax phase. In addition correlations are needed for estimating ,ΔH f
i f
iT and f
iV of each
component and pseudo-component.
The wax model is based on the assumption that a wax phase primarily consists of n-paraffins. The user may input the
n-paraffin content contained in each C7+ fraction. Otherwise the following expression is used to estimate the mole
fraction, ,z S
i of the potentially wax forming part of pseudo-component i, having a total mole fraction of ,z tot
i
PVTsim Method Documentation Modeling of Wax Formation 110
C
P
i
P
iii
tot
i
s
iρ
ρρMBA1zz
In this expression Mi is the molecular weight in g/mole and i the density in g/cm3 at standard conditions
(atmospheric pressure and 15 oC) of pseudo-component i. A, B and C are constants of the following values
A = 1.074
B = 6.584 x 10-4
C = 0.1915
P
i is the densities (g/cm3) at standard conditions of a normal paraffin with the same molecular weight as pseudo-
component i. The following expression is used for the paraffinic density.
i
P
i Mln0.06750.3915ρ
For a (hypothetical) pseudo-component for which S
i
P
i z,ρρ will be equal to tot
iz meaning that all the components
contained in that particular pseudo-component are able to enter into a wax phase. In general S
iz will be lower than
tot
iz and the non-wax forming part of the pseudo-component will have a mole fraction of .zz S
i
tot
i
The wax forming and the non-wax forming fractions of the C20+ pseudo-components are assigned different critical
pressures. The critical pressure of the wax-forming fraction of each pseudo-component is found from
3.46
i
P
ici
s
ciρ
ρPP
Pci equals the critical pressure of pseudo-component i determined using the characterization procedure described in
the Characterization section. P
iρ is the density of the wax forming fraction of pseudo-component i and iρ is the
average density of pseudo-component. The critical pressure S-no
ciP of the non-wax forming fraction of pseudo-
component i is found from the equation
S
ci
Sno
ci
S
i
Sno
i
S
ci
2S
i
2
Sno
ci
Sno
i
ci PP
FracFrac2
P
Frac
P
Frac
P
1
where S and no-S are indices used respectively for the wax forming and the non-wax forming fractions (Frac) of
pseudo-component i. By using this relation the contribution to the equation of state a-parameter of pseudo-
component i divided into two will be the same as that of the pseudo-component as a whole.
For the wax forming C7+ components, the following expressions proposed by Won (1986) are used to find the
melting temperature and enthalpy of melting
f
ii
f
i
i
i
f
i
TM0.1426ΔH
M
20172M0.02617374.5T
The division of each C7+-component into a potentially wax forming component and a component, which cannot form
wax, implies that it is necessary to work with twice the number of C7+-components as in other PVTsim modules. The
equation of state parameters of the wax forming and the non-wax forming parts of a pseudo-component are equal,
but the wax model parameters differ. Presence of non-wax forming components in the wax phase is avoided by
assigning these components a fugacity coefficient of exp(50) in the wax phase independent of temperature and
pressure.
PVTsim Method Documentation Modeling of Wax Formation 111
When tuning to an experimentally determined wax content or to an experimental wax appearance. The wax forming
fraction of each pseudo-component is adjusted to match the experimental data.
Viscosity of Oil-Wax Suspensions
Oil containing solid wax particles may exhibit a non-Newtonian flow behavior. This means that the viscosity
depends on the shear rate (dvx/dy). The apparent viscosity of oil with suspended wax particles is in PVTsim
calculated from (Pedersen and Rønningsen (2000) and modified 2006 using proprietary data from Statoil)
dy
dv
F
dy
dv
EDexpηη
x
4
wax
x
waxwaxliq
where liq is the viscosity of the oil not considering solid wax and wax the volume fraction of precipitated wax in the
oil-wax suspension. The parameters D, E and F take the following values (viscosities in mPa s and shear rates in s-1
)
D = 18.12
E = 405.1
F = 7.876106
Correction factor to be multiplied with D, E and F may be determined by regression to experimental viscosity data
for oils with suspended wax. To fully benefit from the model the data material should cover viscosity data for
different shear rates.
Wax Inhibitors
Wax inhibitors are often added to oils being transported in sub-sea pipelines with the purposes of decreasing the
apparent viscosity of the oil. In PVTsim the wax inhibitor effect is modeled as a depression of the melting
temperature of wax components within a given range of molecular weights (Pedersen and Rønningsen, 2003). The
range of affected molecular weights and the depression of the melting temperature may be estimated by entering
viscosity data for the oil with and without wax inhibitor and running a viscosity tuning to this data material.
References
Pedersen, K.S., “Prediction of Cloud Point Temperatures and Amount of Wax Precipitation”, SPE Production &
Facilities, February 1995, pp. 46-49.
Pedersen, K.S. and Rønningsen, H.P., ”Effect of Precipitated Wax on Viscosity – A Model for Predicting Non-
Newtonian Viscosity of Crude Oils”, Energy & Fuels, 14, 2000, pp. 43-51.
Pedersen, K.S. and Rønningsen, H.P., “Influence of Wax Inhibitors on Wax Appearance Temperature, Pour Point,
and Viscosity of Waxy Crude Oils”, Energy & Fuels 17, 2003, pp. 321-328.
Rønningsen, H. P., Sømme, B. and Pedersen, K.S., ”An Improved Thermodynamic Model for Wax Precipitation;
Experimental Foundation and Application, presented at 8th
international conference on Multiphase 97, Cannes,
France, 18-20 June, 1997.
Templin, R.D., “Coefficient of Volume Expansion for Petroleum Waxes and Pure n-Paraffins”, Ind. Eng. Chem., 48,
1956, pp. 154-161.
Won, K.W., ”Thermodynamics for Solid-Liquid-Vapor Equilibria: Wax Phase Formation from Heavy Hydrocarbon
Mixtures”, Fluid Phase Equilibria 30, 1986, pp. 265-279.
PVTsim Method Documentation Asphaltenes 112
Asphaltenes
Asphaltenes
Asphaltene precipitation is in PVTsim modeled using an equation of state is used for all phases including the
asphaltene phase. The equation of state can either be one of the cubic equations of state or it can be PC-SAFT.
By default the aromatic fraction of the C50+ component is considered to be asphaltenes (Rydahl et al. (1997) and
Pedersen and Christensen (2006) Chapter 12). The user may enter an experimental weight content of asphaltenes in
the oil from a flash to standard conditions. If the entered asphaltene content is higher than that initially estimated in
PVTsim, aromatics lighter than C50 are also classified as asphaltenes. The new cut point between non-asphaltenic
and asphaltene aromatics is placed to match the input amount of asphaltenes. If on the other hand the experimental
amount of asphaltenes is lower than initially found in PVTsim, the cut point from which on aromatics are considered
to be asphaltenes is moved upwards from C50. In asphaltene simulations pseudo-components containing asphaltenes
are split into an asphaltene and non-asphaltene component.
In contrast to most other calculation options in PVTsim, the asphaltene module should not be considered a priori
predictive. Being a liquid-liquid equilibrium the oil-asphaltene phase split is extremely sensitive to changes in model
parameters. Consequently the asphaltene module should be considered a correlation tool rather than a predictive
model. It is strongly recommended that an experimental asphaltene onset P,T point is used to tune the model before
further calculations are made.
Cubic Equations of State
The asphaltenes are by default assigned the following properties:
TcA = 1398.5 K/1125.35°C/2057.63°F
PcA = 14.95 bara/14.75 atm/216.83 psia
A = 1.274
The critical temperature Tcino-A
of the non-asphaltene fraction (Fracino-A
) of pseudo-component i is found from the
relation
A
ci
A
i
Ano
ci
Ano
ici TFracTFracT
where Tci is the critical temperature of pseudo-component i before being split. The critical pressure Pcino-A
of the non-
asphaltene forming fraction of pseudo-component i is found from the equation
PVTsim Method Documentation Asphaltenes 113
A
ci
Ano
ci
A
i
Ano
i
A
ci
2A
i
2
Ano
ci
Ano
i
ci PP
FracFrac2
P
Frac
P
Frac
P
1
while the acentric factor of the non-asphaltene forming fraction of pseudo-component i is found from
A
i
A
i
Ano
i
Ano
ii ωFracωFracω
The binary interaction parameters between asphaltene components and C1-C9 hydrocarbons are by default assumed
to be 0.017 where binary interaction parameters of zero are default used for all other hydrocarbon-hydrocarbon
interactions. Tuning the model to an experimental point may either be accomplished by tuning the asphaltene Tc and
Pc or by tuning the asphaltene content in the oil.
PC-SAFT
The default PC-SAFT parameters of asphaltene component i (aromatic C50+ fraction) are found from
i
2
Aspi, M101.449011.60495m
iAspi, Mln93.819694.4396ε
The parameter i,Asp is found to comply with the density of the asphaltene, which density in g/cm3 is assumed to be
iAspi, Mln0.10390.4323ρ
Carbon number fractions containing asphaltenes are split into a non-asphaltene (no-Asp) fraction and an asphaltene
(Asp) fraction. The parameters and m of the non-asphaltenic fraction are found from
2
AspNo
i
Asp
i
Asp
iiiAsp-No
iz
εzεzε
AspNo
i
Asp
i
Asp
iiiAsp-No
iz
mzmzm
and the density from
AspNo
i
Asp
i
Asp
iiiAsp-No
iz
zz
where zi is the total mole fraction of carbon number fraction i, Asp
iz the mole fraction of asphaltenes in carbon
number fraction i and Asp-No
iz the mole fraction of the non-asphaltenic part of carbon fraction i. mi , i and i are the
properties of the pseudo-component i before being split. The parameter Asp-No
i is found to comply with the density
of the non-asphaltenic fraction.
A binary interaction parameters of 0.017 is used for interaction between C1-C9 hydrocarbons and asphaltene
components.
PVTsim Method Documentation Asphaltenes 114
References
Rydahl, A., Pedersen, K.S. and Hjermstad, H.P., ”Modeling of Live Oil Asphaltene Precipitation”, AIChE Spring
National Meeting March 9-13, 1997, Houston, TX, USA.
Pedersen, K.S. and Christensen, P.L., ”Phase Behavior of Petroleum Reservoir Fluids”, CRC Taylor & Francis, Boca
Raton, 2006.
PVTsim Method Documentation H2S Simulations 115
H2S Simulations
H2S Simulations
The H2S module of PVTsim is based on the same PT-flash as is used in many of the other modules. What makes this
module different is the way H2S is treated in the aqueous phase. The dissociation of H2S is considered.
H2S HS- + H
+
The degree of dissociation is determined by the pH
H10
logpH
and pK
SH
HHSlogpK
2
101
pK1 is calculated using considerations based on chemical reaction equilibria. This gives approximately the following
temperature dependence
K10log1pK
0TLn - TLn R
oPC
0T
1
T
1
R
J
0KLn K Ln
where J is calculated as
0
TPCo
H J
Ln K0 is calculated as
0RT
oG
0KLn
T is the temperature in K, ΔH° is the standard enthalpy change of reaction, and ΔG° is the standard Gibbs energy
change of reaction. ΔCPo is the heat capacity change of reaction. ΔH°, ΔG°, ΔCP
o , and R take the following values
ΔH° = 5300 cal/mol
ΔG° = 9540 cal/mol
PVTsim Method Documentation H2S Simulations 116
ΔCPo/R=-29.33
R = 1.986 cal/mol/K
T0=298.15 K
The expression is optimized to experimental data in the temperature range 0-250 °C from Morse et al. (1987)
From the knowledge of the amount of dissolved H2S on molecular form, pH and pK1 it is straightforward to calculate
[HS-].
In principle the following equilibrium should also be considered
HS- S
-- + H
+
Its pK value defined by the following expression
HS
SHlogpK 102
is however of the order 13-14, meaning that the second order dissociation for all practical purposes can be neglected.
It is therefore not considered in the H2S module.
References
Morse, J.W. et al., “The Chemistry of the Hydrogen Sulfide and Iron Sulfide Systems in Natural Waters”, Earth-
Science Reviews, 24, 1-42 , (1987)
PVTsim Method Documentation Water Phase Properties 117
Water Phase Properties
Water Phase Properties
As a rough guideline PVTsim performs full 3-phase flash calculations on mixtures containing aqueous components.
However, the following interface modules treats a possible water phase as pure water, possibly containing salt. This
applies for the interface modules to
Eclipse Black Oil
MORE Black Oil
Prosper/Mbal
Multiphase meter interface (if license does not give access to multiflash module).
The options treating water as pure water calculates the physical properties and transport properties of water using a
separate thermodynamics instead of an EOS. In the OLGA interface the water property routines are used to calculate
the temperature and pressure derivatives of aqueous phases. Use of the water property package (water
thermodynamics) is also an option in the Property Generator. The thermal conductivity of an aqueous phase is
always calculated using the water property package. Independent of composition the thermal conductivity of an
aqueous phase will therefore be output as that of pure water.
Properties of Pure Water
Thermodynamic Properties
The thermodynamic properties of pure water are calculated using an equation for Helmholtz free energy developed
by Keyes et al. (1968)
Tρ,QρρlnRTTΨΨ 0
where
= Helmholtz free energy (J/g)
= Density (g/cm3)
= 1000/T where T is the temperature in K
R = 0.46151 J/(g K)
and
TlnTCCTCTCCTΨ 54
2
3210
PVTsim Method Documentation Water Phase Properties 118
7
2j
8
1i10j9j
Eplo
bij
2j
ac
8
li10,19,1
Eρli
aij
ρAAeρρAττττ
ρAAeρρATρ,Q
where
a = 0.634 g/cm3
b = 1.0 g/cm3
a = 2.5 K-1
c = 1.544912 K-1
E = 4.8 cm3/g
The coefficients C1 – C5 and Aij are given in tables below.
i CI
1 1855.3865
2 3.278642
3 -.00037903
4 46.174
5 -1.02117
Aij-coefficients of the Q-function.
i j
1 2 3 4 5 6 7
1 29.492937 -5.1985860 6.8335354 -01564104 -6.3972405 -3.9661401 -0.69048554
2 -132.13917 7.779182 -26.149751 -0.72546108 26.409282 15.453061 2.7407416
3 274.64632 -33.301902 65.326396 -9.2734289 47.740374 -29.142470 -5.1028070
4 -360.93828 -16.254622 -26.181978 4.3125840 56.323130 29.568796 3.9636085
5 342.18431 -177.31074 0 0 0 0 0
6 -244.50042 127.48742 0 0 0 0 0
7 155.18535 137.46153 0 0 0 0 0
8 5.9728487 155.97836 0 0 0 0 0
9 -410.30848 337.31180 -137.46618 6.7874983 136.87317 79.847970 13.0411253
10 -416.05860 209.88866 733.96848 10.401717 645.81880 399.17570 71.531353
The pressure is given by the following relation
τ
2
τ
2
T
2
ρ
QρQρ1
τ
1000Rρ
ρ
Ψρ
ρ
ΨρP
The pure water density, , is obtained from this equation by iteration. The enthalpy, H, the entropy, S, and the heat
capacity at constant pressure, Cp, are obtained from the following relations
ρ
P
τ
ΨτH
ρ
dT
ΨdTΨ
ρ
Qρ
τ
QτQρ1
τ
R1000 0
0
τ
PVTsim Method Documentation Water Phase Properties 119
dT
Ψd
τ
QτQρρlnR
T
ΨS 0
ρρ
T
ρ
Tρ
p
ρ
P
T
P
ρ
H
T
HC
Viscosity
Four different expressions (Meyer et al. (1967) and Schmidt (1969)) are used to calculate the pure water viscosity.
Which expression to use depends on the actual pressure and temperature. In two of the four expressions an
expression enters for the viscosity, i, at atmospheric pressure (=0.1 MN/m2) valid for 373.15 K/100°C/212°F < T <
973.15 K/700°C/1292°F
6
32
c
11 10bbT
Tbη
Region 1
Psat < P < 80 MN/m2 and 273.15 K/0°C/32°F < T < 573.15 K/300°C/572°F
3c
2
5
c
4
c
sat
c
1
6
aT/T
a10a
T
Ta
P
P
ρ
ρ1a10η
where Tc and Pc are the critical temperature and pressure, respectively and c the density at the critical point.
Region 2
0.1 MN/m2 < P < Psat and 373.15 K/100°C/212°F < T < 573.15 K/300°C/572°F
6
3
c
21
c
6
1 10cT
Tcc
ρ
ρ1010ηη
Region 3
0.1 MN/m2 < P < 80 MN/m
2 and 648.15 K/375°C/707°F < T < 1073.15 K/800°C/1472°F
6
c
1
2
2
3
c
3
6
1 10ρ
ρd
ρ
ρd
ρ
ρd10ηη
Region 4
Otherwise
0.0192
10ηη
Y
1
where
Y = C5kX4 + C4kX
3 + C3kX
2 + C2kX + C1k
PVTsim Method Documentation Water Phase Properties 120
c
10ρ
ρlogX
The parameter k is equal to 1 when /c 4/3.14 and equal to 2 when /c > 4/3.14. The following coefficients are
used in the viscosity equations
a1 241.4
a2 0.3828209486
a3 0.2162830218
a4 0.1498693949
a5 0.4711880117
b1 263.4511
b2 0.4219836243
b3 80.4
c1 586.1198738
c2 1204.753943
c3 0.4219836243
d1 111.3564669
d2 67.32080129
d3 3.205147019
For k = 1
C1k -6.4556581
C2k 1.3949436
C3k 0.30259083
C4k 0.10960682
C5k 0.015230031
For k = 2
C1k -6.4608381
C2k 1.6163321
C3k 0.07097705
C4k -13.938
C5k 30.119832
The vapor pressure, Psat, is calculated from the following correlation
273.15T
D273.15TDD1Plog 2
j7
3jj1sat10
where Psat is in MN/m2 and T in K. The coefficient, Di, are given in the table below.
Coefficients of vapor pressure correlation.
I Di
1 2.9304370
2 -2309.5789
3 .34522497 x 10-1
4 -.13621289 x 10-3
5 .25878044 x 10-6
6 -.24709162 x 10-9
7 .95937646 x 10-13
Thermal conductivity
PVTsim Method Documentation Water Phase Properties 121
Six different expressions (Meyer et al. (1967), Schmidt (1969) and Sengers and Keyes (1971)) are used to calculate
the pure water thermal conductivity (in W/cm/K). Which expression to use depends on the actual pressure and
temperature. The following expression for the thermal conductivity, 1, at atmospheric pressure (=0.1 MN/m2) and
373.15 K/100°C/212°F < T < 973.15 K/700°C/1292°F enters into two of the six expressions
1 = (17.6 + 0.0587 t + 1.04 x 10-4
t2 – 4.51 x 10
-8 t3) x 10
-5
where
t = T – 273.15
Region 1
Psat < P < 55 MN/m2 and 273.15 K/0°C/32°F < T < 623.15K/350°C/662°F
2
3
c
sat
2
c
sat
1 10SP
PPS
P
PPSλ
where
4
0i
i
c
i1T
TaS
3
0i
i
c
i2T
TbS
3
0i
i
c
i3T
TcS
Region 5
When P,T is not in region 1 and P (in MN/m2) and T (in K) are in one of the following ranges
P > 55 and 523.15 K/250°C/482°F < T < 873.15 K/600°C/1112°F
Psat < P < Pc and T <= Tc
16.5 < P 17.5 and T < 653.15 K/380°C/716°F
Y
1 10λλ
where
Y = C5kX4 + C4kX
3 + C3kX
2 + C2kX + C1k
and
c
10ρ
ρlogX
k = 1 for
cρ
ρ
2.5
k = 2 for
cρ
ρ > 2.5
PVTsim Method Documentation Water Phase Properties 122
The constants used in these equations are as follows
for k = 1
C1k -0.5786154
C2k 1.4574646404
C3k 0.17006978
C4k 0.1334805
C5k 0.032783991
for k = 2
C1k -0.70859254
C2k 0.94131399
C3k 0.064264434
C4k 1.85363188
C5k 1.98065901
Region 3
When P,T is not in regions 1 or 5 but in one of the following ranges (P in MN/m2 and T in K)
45 < P and 723.15 K/450°C/842°F < T < 823.15 K/550°C/1022°F
45 < P < Pbound and T < 823.15 K/550°C/1022°F
35 < P and 723.15 K/450°C/842°F< T < 773.15 K/500°C/932°F
27.5 < P < Pbound and T < 723.15 K/450°C/842°F
22.5 < P < Pbound and T < 698.15 K/425°C/797°F
17.5 < P < Pbound and T < 673.15 K/400°C/752°F
where
2
0i
i
c
icboundT
TePP
the thermal conductivity is found from the following expression
1T
Tdexp
P
Pdd
P
Pd1
1T
T9dexp
P
Pd
T
TBd1
T
TA
c
33
c
363512
c
34
c
33
4
c
32
7
c
31
1.445
c
32
c
31 aP
PaA
26.3
32
63.1
31
1
B
c
c
P
Pb
P
Pb
PVTsim Method Documentation Water Phase Properties 123
33
32
1.5
c
31
cB
cP
Pc
C
Region 4
When P,T is not in region 1, 3 or 5 but in one of the following ranges (P in MN/m2 and T in K)
45 < P and Pbound P and T 723.15 K/450°C/842°F
35 < P and Pbound P and T 723.15 K/450°C/842°F
27.5 < P and Pbound P and T < 723.15 K/450°C/842°F
22.5 < P and Pbound P and T < 698.15 K/425°C/797°F
17.5 < P and Pbound P and T < 673.15 K/400°C/752°F
the thermal conductivity is found from the following expression
8
0i
8
0i
i
4i40
c
i
4i
c
kbcP
Pka
T
T
where
k = 100
The solution for is iterative.
Region 6
When P,T is not in region 1, 3, 4 or 5 and in one of the following ranges
15 MN/m2 < P and T > 633.15 K/360°C/680°F
14 MN/m2 < P and T > 618.15 K/345°C/653°F
1
c
vρ
ρ0.20.01λ
where
v1 = 1.76 x 10-2
+ 5.87 x 10-5
t + 1.04 x 10-7
t2 – 4.51 x 10
-11 x t
3
Region 2
Otherwise
52
4.2
1425
1 10xρt
10x2.1482ρt102.771t0.4198103.51λλ
The following coefficients are used in the equations for thermal conductivity
a0 -0.92247
PVTsim Method Documentation Water Phase Properties 124
a1 6.728934102
a2 -10.11230521
a3 6.996953832
a4 -2.31606251
a31 0.01012472978
a32 0.05141900883
a40 1.365350409
a41 -4.802941449
a42 23.60292291
a43 -51.44066584
a44 38.86072609
a45 33.47617334
a46 -101.0369288
a47 101.2258396
a48 -45.69066893
b0 -0.20954276
b1 1.320227345
b2 -2.485904388
b3 1.517081933
b31 6.637426916 x 105
b32 1.388806409
b40 1.514476538
b41 -19.58487269
b42 113.6782784
b43 -327.0035653
b44 397.3645617
b45 96.82365169
b46 -703.0682926
b47 542.9942625
b48 - 85.66878481
c0 0.08104183147
c1 -0.4513858027
c2 0.8057261332
c3 -0.4668315566
c31 3.388557894 x 105
c32 576.8
c33 0.206
c40 1.017179024
d31 2.100200454 x 10-6
d32 23.94
d33 3.458
d34 13.6323539
d35 0.0136
d36 7.8526 x 10-3
e0 50.60225796
e1 -105.6677634
e2 55.96905687
Surface Tension of Water
The surface tension of liquid water (in mN/m) is calculated from the following formula
c
1.256
c T
T10.6251
T
T1235.8τ
where T is the temperature and Tc the critical temperature of water.
PVTsim Method Documentation Water Phase Properties 125
Properties of Aqueous Mixture
Interfacial Tension Between a Water and a Hydrocarbon Phase
The interfacial tension, , between a water phase and a hydrocarbon phase (gas or oil) is calculated from the
following expression (Firoozabadi and Ramey, 1988)
0.3125
r
b1) - (1
11/4
T
ρΔaσ
where:
HCw ρρΔρ
In this equation w is the density of the water phase and HC the density of the hydrocarbon phase. The values of the
constants a1 and b1 are given in the below table as a function of .
Values of the constants a1 and b1 with in dyn/cm (=1 mN/m)
Δρ (g/cm3) a1 b1
< 0.2 2.2062 -0.94716
0.2 - 0.5 2.915 -0.76852
0.5 3.3858 -0.62590
Tr is a pseudo-reduced temperature for the hydrocarbon phase. It equals the temperature divided by a molar average
of the critical temperatures of the individual hydrocarbon phase components.
Salt Water Density
The density of a water phase with dissolved salts is calculated using a correlation suggested by Numbere et al.
(1977)
w
s
ρ
ρ - 1 =CS [7.65 10
-3 – 1.09 10
-7 P + CS (2.16 10
-5 + 1.74 10
-9 P)
-(1.07 10-5
– 3.24 10-10
P)T + (3.76 10-8
–10-12
P)T2]
where s is the salt-water density, w the density of salt free water at the same T and P, Cs is the salt concentration in
weight%, T the temperature in oF and P the pressure in psia.
Salt Water Viscosity
The viscosity of a water phase with dissolved salts is calculated using a correlation suggested by Numbere et al.
(1977)
1.5
s
4
s
320.52.5
s
40.5
s
3
w
s C103.44C102.76T101.35TC102.18C101.871η
η
where s is the salt water viscosity, w the viscosity of pure water at the same T and P, Cs the salt concentration in
weight% and T the temperature in °F.
Viscosity of Water-Inhibitor Mixtures
The viscosities of mixtures of water and inhibitors (alcohols and glycols) are calculated from the viscosities of the
pure fluids using appropriate mixing rules.
PVTsim Method Documentation Water Phase Properties 126
Methanol
The viscosity of saturated liquid methanol can be calculated from the following equation (Alder, 1966)
ln η = A + B/T + CT + DT2
where η is the viscosity in cP, T the temperature in K and
A = -3.94 x 10
B = 4.83 x 103
C = 1.09 x 10-1
D = -1.13 x 10-4
Ethanol
The viscosity of saturated liquid methanol can be calculated from the following equation (Alder, 1966)
ln η = A + B/T + CT + DT2
where η is the viscosity in cP, T the temperature in K and
A = -6.21
B = 1.614 x 103
C = 6.18 x 10-3
D = -1.132 x 10-5
Mono Ethylene Glycol (MEG)
The viscosity of saturated liquid mono ethylene glycol can be calculated from the following equation by Sun and
Teja (2003)
ln η = A1 + A2/(t+A3)
where η is the viscosity in cP, t the temperature in °C and
A1 = -3.61359
A2 = 986.519
A3 = 127.861
Di Ethylene Glycol (DEG)
The viscosity of saturated liquid di-ethylene glycol can be calculated from the following equation by Sun and Teja
(2003)
ln η = A1 + A2/(t+A3)
where η is the viscosity in cP, t the temperature in °C and
A1 = -3.25001
A2 = 901.095
A3 = 110.695
Tri Ethylene Glycol (TEG)
PVTsim Method Documentation Water Phase Properties 127
The viscosity of saturated liquid tri-ethylene glycol can be calculated from the following equation by Sun and Teja
(2003)
ln η = A1 + A2/(t+A3)
where η is the viscosity in cP, t the temperature in °C and
A1 = -3.11771
A2 = 914.766
A3 = 110.068
Saturation Pressures
To be able to determine the pressures corresponding to the above inhibitor viscosities the pure component vapor
pressures are needed. The vapor pressures are determined from the following variations over the Antoine equation.
H2O Ln (P
sat)= A – B/(T + C)
A = 11.6703
B = 3816.44
C = -46.1300
Psat is saturation pressure in bara. T is temperature in K.
Methanol Log10 (P
sat)= A – B/(T + C)
A = 5.20409
B = 1581.341
C = -33.50
Psat is saturation pressure in bara. T is temperature in K.
Ethanol
Log10 (Psat
)= A – B/(T + C)
A = 5.24677
B = 1598.673
C = -46.424
Psat is saturation pressure in bara. T is temperature in K.
MEG
Ln (Psat
) = A – B/(T + C) + D (In (T)) + ETN
A = 84.09
B = 10411
C = 0.0
D = -8.1976
E = 1.6536 x 10-18
N = 6
Psat is saturation pressure in Pa. T is temperature in K.
DEG
Ln (Psat
) = A – B/(T + C) + D (In (T)) + ETN
PVTsim Method Documentation Water Phase Properties 128
A = 116.21594
B = 13273.461
C = 0.0
D = -12.665825
E = 5.9330303 x 10-29
N = 10
Psat is saturation pressure in Pa. T is temperature in K.
TEG
Log10 (Psat
)= A – B/(T + C)
A = 6.75680
B = 3715.222
C = -1.299
Psat is saturation pressure in bara. T is temperature in K.
Effect of Pressure on the Viscosity
The effect of pressure on the pure component liquid viscosity is calculated using the following formula (Lucas,
1981)
r
A
r
SL PωC1
/2.118ΔPD1
η
η
where
η = viscosity of liquid at actual temperature and pressure
η SL = viscosity of saturated liquid at current T
Pr = (P – Psat
)/Pc
= acentric factor
0513.1T 1.0523
104.6749991.0
0.03877-
r
-4
A
208616.0
208616.0T-1.0039
0.32572906.0´2.573
r
D
C = - 0.07921 + 2.1616 Tr – 13.4040 2
rT + 44.1706 3
rT - 84.8291 4
rT + 96.1209 5
rT - 59.8127 6
rT + 15.6719 7
rT
Pc is the critical pressure and Tr the reduced temperature, T/Tc, where Tc is the critical temperature.
Viscosity Mixing Rules
Mixture viscosities are calculated using the following relation (Grunberg and Nissan, 1949)
where wi and wj are the weight fractions of component i and j, respectively and Gij is a binary interaction parameter,
which is a function of the components i and j as well as the temperature. The following temperature dependence is
assumed
where Gij is a fitted parameter to available mixture viscosity data.
PVTsim Method Documentation Water Phase Properties 129
Gij is assumed to be equal to zero for interactions with methanol and glycol. Gij for interactions with water is as
follows
Water – MeOH: Gij = 2.5324
Water – EtOH : Gij = 3.3838
Water – MEG : Gij = -1.3209
Water – DEG : Gij = -0.7988
Water – TEG : Gij = -0.2239
Other glycols
Other glycols are assigned the properties of that of the above glycols that is closest in molecular weight.
Salt Solubility in Pure Water
The solubility in mole salts per mole water is found from the following expressions (with T in K). The remaining
salts in the database are assigned the solubility of CaCl2, if they consist of 3 ions. Otherwise the solubility is assumed
to be equal to that of NaCl.
Sodium Chloride, NaCl
Solubility of NaCl
OHmol
NaClmol
2
T < 268.55 K: OHmol
NaClmol
2
-1.338e-01 + 9.004e-04 T
T < 382.98 K: OHmol
NaClmol
2
7.986e-02 + 1.048e-04 T
T 382.98 K: OHmol
NaClmol
2
1.506e-02 + 2.740e-04
Solubility of NaCl in g NaCl/g H2O = )(
)(
2
2
OHM
NaClMOHmol
NaClmol
Data from CRC Handbook of Chemistry and Physics , Pinho S. P(2005), Pinho S. P(1996) and Farelo F.(2004).
Sodium Bromide, NaBr
Solubility of NaBr
OHmol
NaBrmol
2
T < 327.60 K: OHmol
NaBrmol
2
-1.849e-01 + 1.175e-03 T
T 327.60 K: OHmol
NaBrmol
2
8.362e-02 + 3.553e-04 T
PVTsim Method Documentation Water Phase Properties 130
Solubility of NaBr in g NaBr/g H2O = )(
)(
2
2
OHM
NaBrMOHmol
NaBrmol
Data from CRC Handbook of Chemistry and Physics and Sunler A.A.(1976).
Potassium Bromide, KBr
Solubility of KBr
OHmol
KBrmol
2
OHmol
KBrmol
2
-1.375e-01 + 8.008e-04 T
Solubility of KBr in g KBr/g H2O = )(
)(
2
2
OHM
KBrMOHmol
KBrmol
Data from CRC Handbook of Chemistry and Physics.
Potassium Chloride, KCl
Solubility of KCl
OHmol
KClmol
2
OHmol
KClmol
2
-1.094e-01 + 6.561e-04 T
Solubility of KCl in weight% = )(
)(
2
2
OHM
KClMOHmol
KClmol
Data from CRC Handbook of Chemistry and Physics and Shearman R. W.
Calcium Chloride, CaCl2
Solubility of CaCl2
OHmol
CaClmol
2
2
T < 284.07 K: OHmol
CaClmol
2
2 -6.335e-02 + 5.770e-04 T
T < 322.66 K: OHmol
CaClmol
2
2 -6.724e-01+2.721e-03 T
T 322.66: OHmol
CaClmol
2
2 -2.486e-02 + 7.140e-04 T
PVTsim Method Documentation Water Phase Properties 131
Solubility of CaCl2 in g CaCl2/g H2O = )(
)(
2
2
2
2
OHM
CaClMOHmol
CaClmol
CRC Handbook of Chemistry and Physics
Calcium Bromide, CaBr2
Solubility of CaBr2
OHmol
CaBrmol
2
2
T < 300.81 K: OHmol
CaBrmol
2
2 -8.233e-02 + 7.741e-04 T
T 300.81 K: OHmol
CaBrmol
2
2 -7.516e-01 + 2.999e-03 T
Solubility of CaBr2 in g CaBr2/g H2O = )OH(M
)CaBr(MOHmol
CaBrmol
1002
2
2
2
Data from CRC Handbook of Chemistry and Physics.
Sodium Formate, NaCOOH
Solubility of NaCOOH
OHmol
NaCOOHmol
2
OHmol
NaCOOHmol
2
-6.899e-01 + 3.064e-3 T
Solubility of NaCOOH in g NaCOOH/g H2O = )(
)(
2
2
OHM
NaCOOHMOHmol
NaCOOHmol
Data from Paolo G. C. Et al (1980), Groschuff, E (1903) and Sidgwick (1922).
Potassium Formate, KCOOH
Solubility of KCOOH
OHmol
KCOOHmol
2
OHmol
KCOOHmol
2
= -1.1266 + 6.6623e-03 T
PVTsim Method Documentation Water Phase Properties 132
Solubility of KCOOH in g KCOOH/g H2O = )(
)(
2
2
OHM
KCOOHMOHmol
KCOOHmol
Data from Groschuff, E (1903) and Sidgwick, N.V (1922)
Cesium Formate, CsCOOH
Solubility of CsCOOH
OHmol
CsCOOHmol
2
T < 267.15 K : OHmol
CsCOOHmol
2
-0.1426 + 0.00143 T
267.15 T < 323.15 K : OHmol
CsCOOHmol
2
-1.3669 + 0.006 T
323.15 K T : OHmol
CsCOOHmol
2
0.572
Solubility of CsCOOH in g CsCOOH/g H2O = )(
)(
2
2
OHM
CsCOOHMOHmol
CsCOOHmol
Weight% salt in H2O
Solubility in weight% salt in H2O is calculated as
OHg
saltg
OHg
saltg
2
2
1
100
Salt Solubility Salt-Inhibitor-Water Systems
The maximum solubility of NaCl, KCl, or CaCl2 salt in aqueous inhibitor solutions can be estimated for the
following systems. C is the weight percent of MEG on salt free basis.
NaCl-MEG-Water
Solubility of NaCl in weight% of NaCl + H2O = Weight % NaCl in pure H2O - 0.2824 C
C is weight % MEG of H2O + MEG.
Dara from Masoudi (2004).
PVTsim Method Documentation Water Phase Properties 133
KCl-MEG-Water
Solubility of KCl in weight% of KCl + H2O = Weight% KCl in pure H2O - 0.2589 C
C is weight % MEG of H2O + MEG.
Data fromFilho, O. C. (1993) and Masoudi (2005).
CaCl2-MEG-Water
Solubility of CaCl2 in weight% of CaCl2 + H2O = Weight% CaCl2 in pure H2O - 0.07561 C
C is weight % MEG of H2O + MEG.
NaCl-MeOH-Water
C < 74.04 weight% MeOH:
Solubility of NaCl in weight% of NaCl + H2O = Weight% NaCl in pure H2O - 0.2977 C
C=74.04 weight% MeOH:
Solubility of NaCl in weight% of NaCl + H2O = W74.04_NaCl = Weight% NaCl in pure H2O - 0.2977 * 74.04=
Weight% NaCl in pure H2O - 22.0417
C 74.04 weight/% MeOH:
Solubility of NaCl in weight% of NaCl+ H2O = W74.04_NaCl – 0.1070 (C-74.04)
C is weight % MeOH of H2O + MeOH.
Data from Pinho, S. P (1996).
KCl-MeOH-Water
The same functions as the NaCl-MeOH-Water system.
Data from Pinho, S. P (1996).
Viscosity of water-oil Emulsions
The viscosity of a water-oil emulsion as a function of the water content and temperature, and may exceed the
viscosities of the pure phases by several order of magnitudes.
The maximum viscosity of the emulsion exists at the mixing ratio where the emulsion changes from a water-in-oil to
an oil-in-water emulsion. The following equation (Rønningsen, 1995) is used to predict the viscosity of the water-in-
oil emulsion to the water concentration and the temperature
ln r = -0.06671 – 0.000775 t + 0.03484 + 0.0000500 t·
where
r = relative viscosity (emulsion/oil)
= volume% of water
t = temperature in oC
PVTsim Method Documentation Water Phase Properties 134
Above the inversion point, the viscosity of the oil-in-water emulsion will be calculated as the water phase viscosity,
when the Rønningsen method is applied.
If an experimental point of (,r) is entered, the correlation of Pal and Rhodes (1989) is used.
Invw
2.5
η
h
100η
h
wr,
Invw
2.5
η
w
100η
w
hr,
if,
1.19
1η
if,
1.19
1
100r
r
100r
r
where r,h means the ratio of the water in oil emulsion viscosity and the oil viscosity. r,w is the ratio of the oil in
water emulsion viscosity and the water viscosity. The specified set of and r is used to calculate 100ηr from the
following equation
0.4
r
100ηη11.19r
This value acts as a constant in subsequent calculations, where r is calculated as a function of . is evaluated at
specified temperature and pressure.
References
Alder, B.J., ”Prediction of Transport Properties of Dense Gases and Liquids”, UCRL 14891-T, University of
California, Berkeley, California, May 1966.
Farelo F.; Ana M. C.; Ferra M. I. Solubility Equilibria of Sodium Sulfate at Temperatures of 150 to 350 C. J. Chem.
Eng. Data. 2004, 49 1782-1788.
Filho, O. C., Rasmussen, P.; J. Chem. Eng. Data., 38, no. 3, (1993)
Firoozabadi, A. and Ramey, H.J., “Surface Tension of Water-Hydrocarbon Systems at Reservoir Conditions”,
Journal of Canadian Petroleum Technology 27, 1988, pp. 41-48.
Grunberg, L. and Nissan, A.H., “A Mixture Law for Viscosity”, Nature 164, 1949, pp. 799-800.
Groschuff, E., Ber Dtsch. Chem. Ges., 36, 1783, 4351 (1903).
"Physical Constants of Organic Compounds", in CRC Handbook of Chemistry and Physics, Internet Version 2007,
(87th Edition), David R. Lide, ed., Taylor and Francis, Boca Raton, FL. Table: SOLUBILITY OF COMMON
SALTS AT AMBIENT TEMPERATURES.
"Physical Constants of Organic Compounds", in CRC Handbook of Chemistry and Physics, Internet Version 2007,
(87th Edition), David R. Lide, ed., Taylor and Francis, Boca Raton, FL. Table: AQUEOUS SOLUBILITY OF
INORGANIC COMPOUNDS AT VARIOUS TEMPERATURES.
Keyes, F.G., Keenan, J.H., Hill, P.G. and Moore, J.G., ”A Fundamental Equation for Liquid and Vapor Water”,
presented at the Seventh International Conference on the Properties of Steam, Tokyo, Japan, Sept. 1968.
Lucas, K., “Die Druckabhängikeit der Viskosität vin Flüssigkeiten” (in German), Chem. Ing. Tech. 53, 1981, pp.
959-960.
Masoudi et. al.; Chem. Eng. Sci., 60, 4213-4224, (2005)
PVTsim Method Documentation Modeling of Scale Formation 135
Masoudi et. al.; Fluid Phase Equilibria., 219, 157-163 (2004)
Meyer, C.A., McClintock, R.B., Silverstri, G.J. and Spencer, R.C., Jr., ”Thermodynamic and Transport Properties of
Steam, 1967 ASME Steam Tables”, Second Ed., ASME, 1967.
Numbere, D., Bringham, W.E. and Standing, M.B., ”Correlations for Physical Properties of Petroleum Reservoir
Brines”, Work Carried out under US Contract E (04-3) 1265, Energy Research & Development Administration,
1977.
Pal, R. and Rhodes, E., "Viscosity/Concentration Relationships for Emulsions", J. Rheology, 33, 1989, pp. 1021-
1045.
Paolo G. C. Et al., Solubility of Sodium Formate in Aqueous Hydroxide Solutions.
Chem. Ing. Data. 1980, 25, 170-171.
Pinho S. P.; Macedo E. A. Solubility of NaCl, NaBr, and KCl in Water, Methanol, Ethanol, and Their Mixed
Solvents. J. Chem. Eng. Data. 2005, 50, 29-32.
Pinho, S. P., Macedo, E. A., Fluid Phase Equilibria, 116, 209-216, (1996)
Pinho S. P.; Macedo E. A. Representation of salt solubility in mixed solvents: A comparison of thermodynamic
models. Fluid Phase Equilibria. 1996, 116, 209-216.
Engineering Data. 1976, 21, 3, 335.
Rønningsen, H.P., ”Conditions for Predicting Viscosity of W/O Emulsions based on North Sea Crude Oils”, SPE
Paper 28968, presented at the SPE International Symposium on Oilfield Chemistry, San Antonio, Texas, US,
February 14-17, 1995.
Schmidt, E., ”Properties of Water and Steam in SI-Units”, Springer-Verlag, New York, Inc. 1969.
Sengers, J.V. and Keyes, P.H., ”Scaling of the Thermal Conductivity Near the Gas-Liquid Critical Point”, Tech.
Rep. 71-061, University of Maryland, 1971.
Shearman R. W.; Menzies W. C. The Solubilities of Potassium Chloride in Deuterium Water and Ordinary Water
from 0 to 180 C
Sidgwick, N.V.; Gentle, J.A.H.R., J. Chem. Soc. 121, 1837 (1922)
Sunler A.A.; Baumbach J. The Solubility of Potassium Chloride in Ordinary and Heavy Water. Journal of Chemical
Thomson, G.H. Brabst, K.R. and Hankinson, R.W., AIChE J. 28, 1982, 671.
van Velzen, D., Cordozo, R.L. and Langekamp, H., Ind. Eng. Chem. Fundam. 11, 1972, 20.
Modeling of Scale Formation
Modeling of Scale Formation
PVTsim Method Documentation Modeling of Scale Formation 136
The scale module considers precipitation of the minerals BaSO4, SrSO4, CaSO4, CaCO3, FeCO3 and FeS. The input
to the scale module is:
A water analysis, including the concentrations (mg/l) of the inorganic ions Na+, K
+, Ca
++, Mg
++, Ba
++, Sr
++, Fe
++,
Cl-, SO4
-, of organic acid and the alkalinity.
Contents CO2 and H2S
Pressure and temperature.
Since the major part of the organic acid pool is acetic acid and since the remaining part behaves similar to acetic
acid, the organic acid pool is taken to be acetic acid.
The alkalinity is defined in terms of the charge balance. If the charge balance is rearranged with all pH-dependent
contributions on one side of the equality sign and all pH-independent species on the other, the alkalinity appears, i.e.
the alkalinity is the sum of contributions to the charge balance from the pH-independent species. Therefore the
alkalinity has the advantage of remaining constant during pH changes.
The calculation of the scale precipitation is based on solubility products and equilibrium constants. In the
calculation, the non-ideal nature of the water phase is taken into account.
Thermodynamic equilibria
The thermodynamic equilibria considered are
Acid-equilibria
H2O(l) H+ + OH
-
H2O(l) + CO2(aq) H+ + HCO3
-
HCO3- H
+ + CO3
--
HA(aq) H+ + A
-
H2S(aq) = H+ + HS
-
Sulfate mineral precipitation reactions
Ca++
+ SO4-- CaSO4(s)
Ba++
+ SO4-- BaSO4(s)
Sr++
+ SO4-- SrSO4(s)
Ferrous iron mineral precipitation reactions
Fe++
+ CO3-- FeCO3 (s)
Fe++
+ HS- H
+ + FeS(s)
Calcium carbonate precipitation reaction
Ca++
+ CO3-- CaCO3(s)
The thermodynamic equilibrium constants for these reactions are
O(l)H
OHH
OHHOH
2
2 a
γγmmK
O(l)H(aq)CO
HCOH
CO
HCOH
1CO
22
3
2
3
2 aγ
γγ
m
mmK
PVTsim Method Documentation Modeling of Scale Formation 137
3
3
3
3
2
HCO
COH
HCO
COH
,2COγ
γγ
m
mmK
HA(aq)
AH
HA(aq)
AH
HAγ
γγ
m
mmK
S(aq)H
HSH
S(aq)H
HSH
SH
22
2 γ
γγ
m
mmK
444 SOCaSOCaCaSO γγmmK
444 SOBaSOBaBaSO γγmmK
444 SOSrSOSrSrSO γγmmK
333 COFeCOFeFeCO γγmmK
H
HSFe
H
HSFe
FeSγ
γγ
m
mmK
333 COCaCOCaCaCO γγmmK
The temperature dependence of the thermodynamic equilibrium constants is fitted to a mathematical expression of
the type
2T
EDTlnTC
T
BATKln
A, B, C, D and E for each reaction are listed in the table below.
A B C 1000D E Ref.:
2,1COK -820.4327 50275.5 126.8339 -140.2727 -3879660.2 Haarberg
(1989)
2,2COK -248.419 11862.4 38.92561 -74.8996 -1297999 Haarberg
(1989)
HAK -10.937 0 0 0 0
SH2K -16.1121 0 0 0 0 Kaasa and
Østvold
(1998)
4CaSOK 11.6592 -2234.4 0 -48.2309 0 Haarberg
(1989)
O2HCaSO 24K 815.978 -26309.0 -138.361 167.863 18.6143 Haarberg
(1989)
4BaSOK 208.839 -13084.5 -32.4716 -9.58318 2.58594 Haarberg
(1989)
4SrSOK 89.6687 -4033.9 -16.0305 -1.34671 31402.1 Haarberg
(1989)
3FeCOK 21.804 56.448 -16.8397 0.02298 0 Kaasa and
Østvold
(1998)
FeSK -8.3102 0 0 0 0 Kaasa and
Østvold
(1998)
PVTsim Method Documentation Modeling of Scale Formation 138
3CaCOK -395.448 6461.5 71.558 -180.28 24847 Haarberg
(1989)
Coefficients in expression for T-dependence of equilibrium constants. T is in Kelvin.
The temperature dependence of the self-ionization of water is described by Olofsson and Hepler (1982)
2
10OH10 T0.0129638T9.7384Tlog4229.195T
142613.6TKlog
2
8908.483T104.602T101.15068 4935
The pressure dependence is given by
RT
ΔVΔZP
P
lnK i
Where Z is the partial molar compressibility change of the reaction, V is the partial molar volume change of the
reaction and R is the universal gas constant. Z for the sulfate precipitation reactions is expressed by a third degree
polynomial
10-3
Z = a + bt + ct2 + dt
3
Where t is the temperature in oC. The coefficients a, b, c and d for each of the sulfate precipitation reactions are listed
in the below table
Coefficients in compressibility change expression for sulfate mineral precipitation reactions. Units: t in oC and Z in
cm3 /mole/bar.
a 100b 1000c 106d
BaSO4 17.54 -1.159 -17.77 17.06
SrSO4 17.83 -1.159 -17.77 17.06
CaSO4 16.13 -0.944 -16.52 16.71
CaSO4-2H2O 17.83 -1.543 -16.01 16.84
Reference: Atkinson and Mecik (1997)
The compressibility changes associated with both of the CO2 acid equilibria are (Haarberg, 1989)
2
CO
3
CO
3 T0.000371T0.23339.3ΔK10ΔZ102,22,1
For the calcium carbonate and ferrous carbonate precipitation reactions the compressibility changes are –0.015
cm3/mole and are considered as independent of temperature (Haarberg et al., 1990).
The partial molar volume changes of the sulfate precipitation reactions are described by the expression
V = A + BT + CT2 + DI + EI
2
where I is the ionic strength. The constants A through E for the sulfate mineral precipitation reactions are listed in
the below table
Coefficient in volume change expression for sulfate mineral precipitation reactions. Units, T in Kelvin, I in moles/kg
solvent and V in cm3/mole.
A B 1000C D E
BaSO4 -343.6 1.746 -2.567 11.9 -4
SrSO4 -306.9 1.574 -2.394 20 -8.2
CaSO4 -282.3 1.438 -2.222 21.7 -9.8
PVTsim Method Documentation Modeling of Scale Formation 139
CaSO4-2H2O -263.8 1.358 -2.077 21.7 -9.8
Reference: Haarberg (1989).
For the calcium carbonate and ferrous carbonate precipitation reactions, the partial molar volume change are
described by (Haarberg, 1989)
2
FeCOCaCO T0.002794T1.738328.7ΔVΔV33
The partial molar volume changes of both of the acid equilibria of CO2 are (Haarberg, 1989)
2
,2CO,1CO T0.0019T0.735141.4ΔVΔV22
For all other reactions than those explicitly mentioned above, the pressure effects on the equilibrium constants are
not considered.
Amounts of CO2 and H2S in water
The potential scale forming aqueous phase will in principle always be accompanied by a hydrocarbon fluid phase.
The hydrocarbon fluid phase is the source of CO2 and H2S. The calculation of the amounts of CO2 and H2S dissolved
in the water phase is determined by PT flash calculations. The aqueous phase and the hydrocarbon fluid are mixed in
the ratio 1:1 on molar basis. An amount of CO2 and H2S is added to the mixture, and a flash calculation is performed.
When the content of CO2 and H2S in the resulting hydrocarbon phase (oil and gas) equals that of the initially
specified hydrocarbon fluid, the water phase CO2 and H2S concentrations will equal the amounts of CO2 and H2S
dissolved in the water phase.
The amounts of CO2 and H2S consumed by scale formation is assumed to be negligible compared to the amounts of
CO2 and H2S in the system. The concentration of CO2 and H2S in the aqueous phase are therefore assumed to be
constant.
Activity coefficients of the ions
The activity coefficients used in the scale module come from the Pitzer model (Pitzer, 1973, 1975, 1979, 1986, 1995
and Pitzer et al., 1984). According to the Pitzer model the activity coefficients of the ionic species in a water solution
are
a c aMcaaMccMaMaa
2
MM Ψm2φmZC2BmFzγln
a' a'a c a
caacMMaa'a'a CmmzΨmm
for the cat ions, and
c a ccXacXaacXcXc
2
XX Ψm2φmZC2BmFzγln
c' c'c c a
caacXX'cc'c'c CmmzΨmm
for the anions. c denotes a cat ion species, whereas a denotes an anion species. m is the molality (moles/kg solvent)
and I is the ionic strength (moles/kg solvent)
i
2
ii zm2
1I
PVTsim Method Documentation Modeling of Scale Formation 140
z is the charge of the ion considered in the unit of elementary units. ijk is a model parameter that is assigned to each
cat ion-cat ion-anion triplet and to each cat ion-anion-anion triplet. The remaining quantities in the activity
coefficient equations are
c a
caac
1/2
1/2
1/2
B'mmbI1lnb
2
bI1
IAF
c' c'c a' a'a
aa'a'acc'c'c φmmφmm
where b is a constant with the value 1.2 kg 1/2
/mole1/2
and
3/2
0
21/2
w0φDkT4ππ
ed2ππ
3
1A
N0 is the Avogadro number, dw is the water density, e is the elementary charge, D is the dielectric constant of water
and k is the Boltzman constant.
1/2
2
(2)
MX
1/2
1
(1)
MX
(0)
MXMX IαgβIαgββB
where
2x
xexpx112xg
(2)
ij
(1)
ij
(0)
ij βandβ,β are model parameters. One of each parameter is assigned to each cat ion-anion pair. 1 and
are constants, with 1 = 2 kg1/2
mole-1/2
and 2 =12 kg1/2
mole-1/2
. However, for pairs of ions with charge +2 and –
2, respectively, the value for 1 is 1.2 kg1/2
mole-1/2
.
Further
i
ii zmZ
1/2
XM
φ
MX
MX
zz2
CC
III ij
E
ij
E
ij
s
ij
Iθθφ ij
E
ij
sφ
ij
φ
ijC is yet another model parameter assigned to each cat ion-anion pair.
ij
Sθ is a model parameter assigned to each cat ion-cat ion pair and to each anion-anion pair and
ij
Eθ is an electrostatic term
jjiiij
ji
ij
E xJ2
1xJ
2
1xJ
4I
zzθ
where
1/2
φjiij IAz6zx
10.5280.7231 0.0120xexpx4.5814xxJ
PVTsim Method Documentation Modeling of Scale Formation 141
Also the Pitzer model describes the activity of the water in terms of the osmotic coefficient
i c a
ca
φ
caac1/2
3/2
φ
i ZCBmmbI1
I2Am1φ
c' c'c a' a'a caca'c
φ
aa'a'aa
acc'a
φ
cc'c'c ΨmφmmΨmφmm
where
1/2
2
(2)
MX
1/2
1
(1)
MX
(0)
MX
φ
MX IαexpβIαexpββB
and the relation between the osmotic coefficient and the activity of the water is
i
iOHOH mφMaln22
Model parameters at 25°C are listed below.
(0)
parameters at 25°C
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
Fe++
OH-
0.00000 0.08640 0.12980 0.00000 -0.17470 0.00000 0.17175 0.00000
Cl-
0.17750 0.07650 0.04810 0.35090 0.30530 0.28370 0.26280 0.44790
SO4--
0.02980 0.01810 0.00000 0.21500 0.20000 0.20000 0.20000 -4.70500
HCO3-
0.00000 0.02800 -0.01070 0.32900 -1.49800 0.00000 0.00000 0.00000
CO3--
0.00000 0.03620 0.12880 0.00000 -0.40000 0.00000 0.00000 1.91900
HS-
(1)
parameters at 25°C
(2)
parameters at 25°C
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
Fe++
OH-
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Cl-
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
SO4-
0.00000 0.00000 0.00000 -32.74000 -54.24000 -54.24000 -54.24000 0.00000
HCO3-
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
CO3--
0.00000 0.00000 0.00000 0.00000 879.20000 0.00000 0.00000 0.00000
HS-
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
C parameters at 25°C
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
Fe++
OH-
0.00000 0.00410 0.00410 0.00000 0.00000 0.00000 0.00000 0.00000
Cl-
0.00080 0.00127 -0.00079 0.00651 0.00215 -0.00089 -0.01938 0.00000
SO4-
0.04380 0.00571 0.01880 0.02797 0.00000 0.00000 0.00000 0.00000
HCO3-
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
CO3--
0.00000 0.00520 0.00050 0.00000 0.00000 0.00000 0.00000 0.00000
HS-
S parameters at 25°C
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
H-
0.00000
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
Fe++
OH-
0.00000 0.25300 0.32000 0.00000 -0.23030 0.00000 1.20000 0.00000
Cl-
0.29450 0.26640 0.21870 1.65100 1.70800 1.62600 1.49630 2.04300
SO4--
0.00000 1.05590 1.10230 3.36360 3.19730 3.19730 3.19730 17.00000
HCO3-
0.00000 0.04400 0.04780 0.60720 7.89900 0.00000 0.00000 14.76000
CO3--
0.00000 1.51000 1.43300 0.00000 -5.30000 0.00000 0.00000 -5.13400
HS-
PVTsim Method Documentation Modeling of Scale Formation 142
Na+
0.03600 0.00000
K+
0.00500 -0.01200 0.00000
Mg++
0.10000 0.07000 0.00000 0.00000
Ca++
0.06120 0.07000 0.03200 0.00700 0.00000
Sr++
0.06500 0.05100 0.00000 0.00000 0.00000 0.00000
Ba++
0.00000 0.06700 0.00000 0.00000 0.00000 0.00000 0.00000
OH-
Cl-
SO4--
HCO3-
CO3--
OH-
0.00000
Cl-
-0.05000 0.00000
SO4-
-0.01300 0.02000 0.00000
HCO3-
0.00000 0.03590 0.01000 0.00000
CO3--
0.10000 -0.05300 0.02000 0.08900 0.00000
parameters at 25°C
Anion 1 fixed as Cl-
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
H-
0.00000
Na+
-0.00400 0.00000
K+
-0.01100 -0.00180 0.00000
Mg++
-0.01100 -0.01200 -0.02200 0.00000
Ca++
-0.01500 -0.00700 -0.02500 0.01200 0.00000
Sr++
0.00300 -0.00210 0.00000 0.00000 0.00000 0.00000
Ba++
0.01370 -0.01200 0.00000 0.00000 0.00000 0.00000 0.00000
Anion 1 fixed as SO4--:
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
H-
0.00000
Na+
0.00000 0.00000
K+
0.19700 -0.01000 0.00000
Mg++
0.00000 -0.01500 -0.04800 0.00000
Ca++
0.00000 -0.05500 0.00000 0.02400 0.00000
Sr++
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Ba++
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Anion 1 fixed as HCO3-
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
H-
0.00000
Na+
0.00000 0.00000
K+
0.00000 -0.00300 0.00000
Mg++
0.00000 0.00000 0.00000 0.00000
Ca++
0.00000 0.00000 0.00000 0.00000 0.00000
Sr++
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Ba++
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Anion 1 fixed as CO3--
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
H-
0.00000
Na+
0.00000 0.00000
K+
0.00000 -0.00300 0.00000
Mg++
0.00000 0.00000 0.00000 0.00000
Ca++
0.00000 0.00000 0.00000 0.00000 0.00000
Sr++
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Ba++
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Cat ion 1 fixed as Na+
OH-
Cl-
SO4-
HCO3-
CO3--
PVTsim Method Documentation Modeling of Scale Formation 143
OH-
0.00000
Cl-
-0.00600 0.00000
SO4--
-0.00900 0.00140 0.00000
HCO3-
0.00000 -0.01430 -0.00500 0.00000
CO3--
0.01700 0.00000 -0.00500 0.00000 0.00000
Cat ion 1 fixed as K+
OH-
Cl-
SO4-
HCO3-
CO3--
OH-
0.00000
Cl-
-0.00800 0.00000
SO4--
-0.05000 0.00000 0.00000
HCO3-
0.00000 0.00000 0.00000 0.00000
CO3--
-0.01000 0.02400 -0.00900 -0.03600 0.00000
Cat ion 1 fixed as Mg++
OH-
Cl-
SO4--
HCO3-
CO3--
OH-
0.00000
Cl-
0.00000 0.00000
SO4-
0.00000 -0.00400 0.00000
HCO3-
0.00000 -0.09600 -0.16100 0.00000
CO3--
0.00000 0.00000 0.00000 0.00000 0.00000
Cat ion 1 fixed as Ca++
OH-
Cl-
SO4--
HCO3-
CO3--
OH-
0.00000
Cl-
-0.02500 0.00000
SO4-
0.00000 -0.01800 0.00000
HCO3-
0.00000 0.00000 0.00000 0.00000
CO3--
0.00000 0.00000 0.00000 0.00000 0.00000
All parameters not listed here are equal to zero.
The Pitzer parameters ijk and Sij are temperature independent parameters, whereas ,
(0)
ij (1)
ij and (2)
ij and ijC are
temperature dependent parameters (=X). Their temperature dependence is described by (Haarberg, 1989) for
temperatures in K
2
2
2
298.15TT
X
2
1298.15T
T
X298.15XTX
Due to the appearance of Na and Cl in many systems, Pitzer et al. (1984) have developed a more sophisticated
description of the temperature dependence of the parameters for these species. Also a pressure dependence is
included in the description. The functional form is for temperatures in K
TPQQTlnQPQQT
QTX 65432
1
T680
PQQ
227T
PQQTPQQ 12111092
87
The temperature coefficients T
X
and
2
2
T
X
and the coefficient Q1, Q2…..,Q12 are listed below.
First order temperature derivative of (0)
100.
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
Fe++
OH-
0.00000 -0.01879 0.00000 0.000000 0.00000 0.00000 0.00000 0.00000
Cl-
-0.18133 0.007159 0.03579 -0.05311 0.02124 0.02493 0.06410 0.00000
PVTsim Method Documentation Modeling of Scale Formation 144
SO4--
0.00000 0.16313 0.09475 0.00730 0.00000 0.00000 0.00000 0.00000
HCO3-
0.00000 0.10000 0.10000 0.00000 0.00000 0.00000 0.00000 0.00000
CO3--
0.00000 0.17900 0.11000 0.00000 0.00000 0.00000 0.00000 0.00000
HS-
Second order temperature derivative of (0)
100.
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
Fe++
OH-
0.00000 0.00003 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Cl-
0.00376 -0.00150 -0.00025 0.00038 -0.00057 -0.00621 0.00000 0.00000
SO4--
0.00000 -0.00115 0.00008 0.00094 0.00000 0.00000 0.00000 0.00000
HCO3-
0.00000 -0.00192 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
CO3--
0.00000 -0.00263 0.00102 0.00000 0.00000 0.00000 0.00000 0.00000
HS-
First order temperature derivative of (1)
100.
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
Fe++
OH-
0.00000 0.27642 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Cl-
0.01307 0.07000 0.11557 0.43440 0.36820 0.20490 0.32000 0.00000
SO4--
0.00000 -0.07881 0.46140 0.64130 5.46000 5.46000 5.46000 0.00000
HCO3-
0.00000 0.11000 0.11000 0.00000 0.00000 0.00000 0.00000 0.00000
CO3--
0.00000 0.20500 0.43600 0.00000 0.00000 0.00000 0.00000 0.00000
HS-
Second order temperature derivative of (1)
100.
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
Fe++
OH-
0.00000 -0.00124 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Cl-
-0.00005 0.00021 -0.00004 0.00074 0.00232 0.05000 0.00000 0.00000
SO4--
0.00000 0.00908 -0.00011 0.00901 0.00000 0.00000 0.00000 0.00000
HCO3-
0.00000 0.00263 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
CO3--
0.00000 -0.04170 0.00414 0.00000 0.00000 0.00000 0.00000 0.00000
HS-
First order temperature derivative of (2)
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
Fe++
OH-
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Cl-
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
SO4--
0.00000 0.00000 0.00000 -0.06100 -0.51600 -0.51600 -0.51600 0.00000
HCO3-
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
CO3--
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
HS-
Second order temperature derivative of (2)
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
Fe++
OH-
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Cl-
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
SO4--
0.00000 0.00000 0.00000 -0.01300 0.00000 0.00000 0.00000 0.00000
HCO3-
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
CO3--
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
HS-
First order temperature derivative of C 100.
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
Fe++
OH-
0.00000 -0.00790 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Cl-
0.00590 -0.01050 -0.00400 -0.01990 -0.01300 0.00000 -0.01540 0.00000
SO4--
0.00000 -0.36300 -0.00625 -0.02950 0.00000 0.00000 0.00000 0.00000
HCO3-
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
CO3--
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
PVTsim Method Documentation Modeling of Scale Formation 145
HS-
Second order temperature derivative of C 100..
H+
Na+
K+
Mg++
Ca++
Sr++
Ba++
Fe++
OH-
0.00000 0.00007 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Cl-
-0.00002 0.00015 0.00003 0.00018 0.00005 0.00000 0.00000 0.00000
SO4--
0.00000 0.00027 -0.00023 -0.00010 0.00000 0.00000 0.00000 0.00000
HCO3-
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
CO3--
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
HS-
Temperature coefficients in expression for temperature dependence of the Pitzer parameters for NaCl
NaCl(0)
NaCl(1) φ
NaClC
Q1 -6.5684518102
1.1931966102 -6.1084589
Q2 2.486912950101
-4.830932710-1
4.021779310-1
Q3 5.38127526710-5 0 2.290283710
-5
Q4 -4.4640952 0 -7.535464910-4
Q5 1.11099138310-2
1.406809510-3
153176729510-4
Q6 -2.65733990610-7 0 -9.055090110
-8
Q7 -530901288910-6 0 -1.5386008210
-8
Q8 8.63402332510-10 0 8.6926610
-11
Q9 -1.579365943 -4.2345814 3.5310413610-1
Q10 0.002202282079010-3 0 -4.331425210
-4
Q11 9.706578079 0 -9.18714552910-2
Q12 -2.68603962210-2 0 5.19047710
-4
The coefficients correspond to units of pressure and temperature in bars and Kelvin, respectively.
Reference: Pitzer (1984)
Calculation procedure
The amount of minerals that precipitates from a specified aqueous solution is evaluated by calculating the amount of
ions that stays in solution when equilibrium has established. This amount is given as the solution to the system of
thermodynamic equilibrium constant equations. Only the solubility products of the salts precipitating, need be
fulfilled. Solving the system of equations is an iterative process
The thermodynamic equilibrium constants are calculated for the specified solution at the specified set of conditions,
pressure and temperature.
The activity coefficients of all components are set equal to one.
The stoichiometric equilibrium constants are calculated from the thermodynamic ones and from the activity
coefficients.
The ratio of CO2(aq) to H2S(aq) is calculated. This determines if any of the ferrous iron minerals FeCO3 and FeS will
precipitate. Only one can precipitate, since both H2S and CO2 are fixed in concentration, and then the Fe++
concentration cannot fulfill both solubility products at the same time.
The equilibrium in the acid/base reactions is determined without considering the precipitation reactions. The
convergence criterion is that the charge balance must be fulfilled.
The amount of sulfate precipitation (independent of the acid/base reactions) is calculated, with none of the other
precipitation reactions taken into account.
The ion product of the iron mineral identified at a previous step is checked against the solubility product. If the
solubility product is exceeded, the amount of precipitate of the iron mineral is determined. The convergence
criterion in this iteration is the charge balance. Precipitation of calcium carbonate is not included in the
PVTsim Method Documentation Modeling of Scale Formation 146
calculation.
The ion product of calcium carbonate is checked against its solubility product. If the solubility product is exceeded,
simultaneous precipitation of calcium carbonate and the iron mineral is calculated. A double loop iteration is
applied. The inner loop: With a given amount of ferrous iron mineral precipitation (which comes from the outer
loop), the amount of calcium carbonate precipitate is determined. During the calcium carbonate precipitation,
the sulfate precipitate is influenced since some Ca++
is removed from the solution. The state in the sulfate system
is therefore corrected in each of these inner loop iterations. In the inner loop, the charge balance is used to check
for convergence. The outer loop: The iteration variable is the amount of ferrous iron mineral precipitate.
Convergence is achieved when the ion product of the ferrous mineral matches the thermodynamic solubility
product.
The resulting amount of each precipitate is compared to that of the previous iteration. If the weighted sum of relative
changes in the amounts of precipitates exceeds 10-6
, then all activity coefficients are recalculated from Pitzers
activity coefficient model for electrolytes. The procedure is then repeated from the 3rd
step.
References
Atkinson, A. and Mecik, M., “The Chemistry of Scale Prediction”, Journal of Petroleum Science and Engineering
17, 1997, pp. 113-121.
Haarberg, T. “Mineral Deposition During Oil Recovery”, Ph.D. Thesis, Department of Inorganic Chemistry,
Trondheim, Norway (1989).
Haarberg, T., Jakobsen, J.E., and Østvold, T., “The effect of Ferrous Iron on Mineral Scaling During Oil Recovery”,
Acta Chemica Scandinavia 44, 1990, pp. 907-915.
Kaasa, B. and Østvold, T., “Prediction of pH and Mineral Scaling in Waters with Varying Ionic Strength Containing
CO2 and H2S for 0<T(oC)<200 and 1<p(bar)<500” Presented at the conference “Advances in Solving Oilfield
Scaling” held January 28 and 29, 1998 in Aberdeen, Scotland.
Pitzer, K.S., “Thermodynamics of Electrolytes I. Theoretical basis and general equations”, Journal of Physical
Chemistry 77, 1973, pp. 268-277.
Pitzer, K.S., “Thermodynamics of Electrolytes V. Effects of Higher-Order Electrostatic Terms”, Journal of Solution
Chemistry 4, 1975, pp. 249-265.
Pitzer, K.S., “Theory: Ion Interaction Approach. Activity Coefficients in Electrolyte Solutions”, Book by Pytkowicz,
R.M., pp. 157-208, CRC Press, Boca Raton, Florida (1979).
Pitzer, K.S., Peiper, J.C. and Busey, R.H., “Thermodynamic Properties of Aqueous Sodium Chloride Solutions”.
Journal of Physical Chemistry 13, 1984, pp. 1-102.
Pitzer, K.S., “Theoretical Considerations of Solubility with Emphasis on Mixed Aqueous Electrolytes”, Pure and
Applied Chemistry 58, 1986, pp. 1599-1610.
Pitzer, K.S., “Thermodynamics” 3. edition, McGraw-Hill, Inc. (1995)
PVTsim Method Documentation Wax Deposition Module 147
Wax Deposition Module
Modeling of wax deposition
The wax deposition module, DepoWax, is fundamentally a steady state compositional pipeline simulator, in which
wax deposition on the pipe wall is overlaid on the steady state results. The steady state approach is chosen because
wax deposition is a very slow process relative to typical residence times. In the following, the methods of Lindeloff
and Krejbjerg (2001 and 2002) used for numerical discretization, heat transfer, energy balances, thermodynamic
equilibrium, and wax deposition will be described.
Discretization of the Pipeline into Sections
The simulator is based on an approach where the pipeline is divided into a number of cells. In the following, these
will be referred to as segments and sections. Segments are larger entities, which are user specified in terms of inlet
and outlet position in the x-y coordinate space, where x is the horizontal coordinate and y the vertical.
Each segment consists of a number of sections, the locations of which are generated automatically by the program.
The user may affect the selection of the sections by altering the maximum section length and maximum temperature
drop over a section, which by default are set to 500 m/1640 ft and 5°C/9°F, respectively. The max temperature drop
is calculated in an approximate manner assuming that the bulk fluid temperature will exhibit an exponential decline
as the fluid passes through the pipeline. Assuming single phase flow and steady state in the simulation, a temperature
profile may be estimated analytically from the following expression
x
mC
πDUexpTTT
p
totambinambx
T
The equation states that under the above assumptions, the temperature Tx at a given position x can be calculated
based on the mass flow rate,
,m the heat capacities Cp, the pipeline diameter D, and the overall heat transfer
coefficient Utot. Tamb is the ambient temperature, while Tin is the fluid temperature at the pipeline inlet. This
expression may be exploited to optimize the discretization of the pipeline by assigning section lengths in such a way
that the temperature only declines a predefined amount in each section. This results in short section lengths near the
inlet, while sections are longer further down the pipeline where the temperature changes less.
Energy balance
The energy balance calculations are sketched in the figure below. The mass flowrate, temperature, pressure and
composition at inlet are known. Also insulation and temperature of the surroundings are known. This allows the
program to calculate heat loss from the pipeline, enthalpy of the exiting fluid, and pipe wall temperatures.
A steady state flow model calculates pressure drop, flow regime, and liquid hold-up, based on information about the
amounts and properties of the phases.
PVTsim Method Documentation Wax Deposition Module 148
Knowing pressure, enthalpy, and feed composition at the outlet of the section, an integrated wax-PH flash is used to
calculate the temperature and phase compositions. These values are then used as inlet conditions for the next section.
This proceeds until the calculation has been completed for the entire pipeline in the current time step. Subsequent
time steps are calculated similarly, the only change from one time step to the next being that the pipeline diameter
and insulation have changed due to a layer of deposited wax on the pipe wall.
Q=UAT
Tamb
Ti, Pi
mi
Hi Ho
To, Po
The structure of the algorithm, as described above, can be summarized by the following four points that are further
illustrated in the above figure
Heat balance, HO = Hi – (Q + W)
Pressure drop and flow regime, OLGAS 2000 PO
Wax flash at wall and deposition
PH-wax flash, (PO, HO) TO
The enthalpy (H) of the fluid exiting the section depends on the amount of heat transferred through the pipeline walls
(Q) and the work done (W) due to changes in elevation. The work term, which becomes significant for instance in a
riser, is calculated from
hgρW bulk
In this equation bulkρ is the average bulk fluid density, g is the gravitational acceleration and h the elevation change.
The heat loss is calculated as
ambbulktot TTAUQ
where A is the pipe wall area, Tamb is the ambient temperature, and bulkT is the mean bulk temperature in the section.
Utot is the overall heat transfer coefficient.
Overall heat transfer coefficient
The overall heat transfer coefficient is calculated from the equation
1
NLAY
1ioutout
i1,i
1i
i
inin
1
inhr
1
k
r
rln
hr
1rU in
PVTsim Method Documentation Wax Deposition Module 149
In this equation, the heat transfer coefficient is referred to the inner radius of the pipeline rin. ki-1,i is the thermal
conductivity of the layer between the radii ri-1 and ri. Deposited wax is included as an additional layer at radius rwax =
rin – xwax, where xwax is the deposit layer thickness. hin and hout are the inside film heat transfer coefficient and outside
film heat transfer coefficient, respectively. For a more detailed description of this, please refer to example 9.6-1 of
the textbook by Bird et al. (1960).
Inside film heat transfer coefficient
The inside film heat transfer coefficient hin is calculated from the Nusselt number
D
kNh Nu
in
where k is the thermal conductivity of the fluid and D is the inside diameter of the pipeline. The following
dimensionless numbers enter into the correlations for the Nusselt number
Reynolds number
DuN Re
Prandtl number
pPr
CN
Grashof number
2
3
Gr
L T gN
where
Cp Heat capacity at constant pressure
D Inner diameter
g Gravitational acceleration
T Absolute temperature
u Linear velocity
β Thermal expansion coefficient (1/V dV/dT)
ΔT Absolute temperature difference between warmer and cooler side
η Viscosity
λ Thermal conductivity
ρ Density
DepoWax supports four sets of correlations for the Nusselt number. Each set has different correlations for different
flow regimes. DepoWax has been developed for turbulent flow. Reliable wax deposition results cannot be expected
for laminar flow (NRe < 2300).
Sieder-Tate (default selection) 0.25
w
b1/3
Pr
0.8
ReNu
4
Reη
ηNN0.027N10
N
0.25
w
b1/3
PrGrNu
0.25
w
b
1.8
Re
51/3
Pr
0.8
ReNu
4
Re
η
η3.66,NN0.184maxN
η
η
N
1061NN0.027N
ofvaluehigherThe10N3002
0.25
w
b1/3
PrGrNuReη
η3.66,NN0.184maxN2300N
Dittus-Bölter
PVTsim Method Documentation Wax Deposition Module 150
0.25
w
b0.3
Pr
0.8
ReNu
4
Reη
ηNN0.023N10N
25.0
w
b1/3
PrGrNu
0.25
w
b
1.8
Re
50.3
Pr
0.8
ReNu
4
Re
η
η3.66,NN0.184maxN
η
η
N
1061NN0.023N
ofvaluehigherThe10N2300
0.25
w
b1/3
PrGrNuReη
η3.66,NN0.184maxN2300N
Petukhov-Gnielinski
0.25
w
b1/3
PrGrNu
2
Re10
0.25
w
b
2/3
Pr
PrRe
Nu
Re
η
η3.66,NN0.184maxN
1.64)N(1.82log
1ξwhere
η
η
1N8
ξ12.71
N1000N8
ξ
N
ofvaluehigherThe2300N
0.25
w
b1/3
PrGrNuReη
η3.66,NN0.184maxN2300N
Petukov/ESDU
2
Re10
0.25
w
b
2/3
Pr
1/2
PrRe
Nu
3
Re1.64)N4(1.82log
1fwhere
η
η
1N2
f12.71.07
NN2
f
(tur)N104N
0.25
w
b1/3
PrGrNuReη
η3.66,NN0.184max(lam)N2300N
6000
N1.33εwhere(tur)Nε)(1(lam)NεN104N2300 Re
NuNuNu
3
Re
Outside Film Heat Transfer Coefficient
The outside film heat transfer coefficient is specified as a constant value for each segment along with the insulation
properties of that segment. The value may be entered by the user or may be selected from the default values given
for free and forced convection in air and water. The actual outside film heat transfer coefficient will of course vary
with the environment outside the pipeline, but the default values will at least have the right order of magnitude. In
the case where the pipeline is covered by soil, the soil is added as a layer of insulation of thickness reflecting the
depth of burial. A film heat transfer coefficient for air or water is then specified reflecting whether the pipeline is
located offshore or onshore.
Pressure drop models
The total pressure drop Total
L
P
over a given pipeline section in DepoWax is calculated as the sum of three
contributions
PVTsim Method Documentation Wax Deposition Module 151
onalAcceleratiFrictionalcHydrostatiTotalL
P
L
P
L
P
L
P
where
cHydrostatiL
P
is the contribution to the pressure drop from elevation changes in the pipeline.
FrictionalL
P
represents the irreversible losses due to shear of fluids at the pipe wall and internally in the fluid.
onalAcceleratiL
P
is the pressure drop contribution from acceleration.
The three contributions to the total pressure drop are given as
θρgsin
L
P
cHydrostati
2d
ρgvReε,f
L
P 2
Frictional
L
Pgvu
L
P
onalAccelerati
where is the density, g is the gravitational acceleration, is the inclination of the pipeline, f(Re) is the friction
factor, which is a function of the section’s wall roughness () and the Reynold’s number (Re), v is the linear velocity,
u is the superficial velocity, and P is the pressure.
Single-phase flow When a single fluid phase is present in a pipeline section, the properties of the phase are assumed to be constant
throughout the entire section. The pressure drop from acceleration in single-phase flow is negligible, and the total
pressure drop over a pipeline section with single-phase flow is
2d
vRefθsinρg
L
P 2
Total
In the above equation, the Reynold’s number is given by
dvRe
and the friction factor (Bendiksen et al., 1991) by
PVTsim Method Documentation Wax Deposition Module 152
3
1
643
Re
10
d
ε1021105.5f(Re)
where d is the section diameter and is the viscosity.
Two-phase flow The pressure drop in two-phase flow in DepoWax may either be determined using the OLGAS two-phase steady-
state pressure drop model (Bendiksen et al., 1991) or using the two-phase steady-state pressure drop model proposed
by Mukherjee and Brill (1985).
Mukherjee and Brill pressure drop model When two fluid phases are present in a pipeline section, the properties of the total fluid are dependent on the flow
regime in that particular section.
Based on an evaluation of appropriate dimensionless variables, Mukherjee and Brill (1985) set up the flow sheet
below, which determines whether the two-phase flow is stratified smooth, annular, slug, or bubble.
NGV > NGVSMYesAnnular
θ > 0 Bubble
No
|θ| > 6
π
Yes NGV > NGVBS
No
Yes
NLV > NLVBS
Yes
No Slug
No
Yes
NLV > NLVST
Yes
NoStratified
NoNLV > NLVST
Yes
NGV > NGVBS
Bubble
No
YesSlug
No
NGV > NGVSMNGV > NGVSMYesYesAnnularAnnular
θ > 0θ > 0 BubbleBubble
NoNo
|θ| >|θ| > 6
π
YesYes NGV > NGVBSNGV > NGVBS
NoNo
YesYes
NLV > NLVBSNLV > NLVBS
Yes
NoNo SlugSlug
NoNo
YesYes
NLV > NLVSTNLV > NLVST
Yes
NoNoStratifiedStratified
NoNoNLV > NLVSTNLV > NLVST
YesYes
NGV > NGVBSNGV > NGVBS
BubbleBubble
NoNo
YesYesSlugSlug
NoNo
Determination of flow regime (Mukherjee and Brill, 1985).
The dimensionless numbers NGV, NGVSM, NGVBS, NLV, NLVBS, and NLVST are determined as follows
4
1
LiquidGas
Liquid
GasGVg
uN
4
1
LiquidGas
Liquid
LiquidLVg
uN
PVTsim Method Documentation Wax Deposition Module 153
10GVBSN
10GVSMN
10LVBSN
10LVSTN
where σGas-Liquid is the interfacial tension between the gas and liquid phases.
The parameters α, β, γ, and δ are given as
θ1.132SinNLogθ0.429Sin-θSinN1.138Log-3.003N-0.431α LV
2
10LV10L
0.329
LVL 0.521N+2.694N-1.401
L
2
GV10 N*3.695+θ0.855Sin-θ0.074SinNLog0.94γ
θ3.925Sin-N0.033Log-2.972N-θ4.267Sin-0.017N-0.321δ 2
GV
2
10LGV
where NL is a dimensionless number given as
4
1
3
LiquidGasLiquid
LiquidLσρ
gμN
Mukherjee and Brill (1985) have proposed the following correlation for determining the liquid hold-up
6
54321Liquid
P
PPPPPexpH
where P1 to P6 are
11 CP
θSinCP 22
θSinCP 2
33
2
L44 NCP
55
C
GVNP
PVTsim Method Documentation Wax Deposition Module 154
66
C
LVNP
Constants C1 to C6 are flow regime dependent and given in the table below.
Flow
Uphill & horizontal Stratified smooth and wavy flow
– downhill
Other flow regimes – downhill
C1 -0.380113 -1.330282 -0.516644
C2 0.129875 4.808139 0.789805
C3 -0.119788 4.171584 0.551627
C4 2.343227 56.262268 15.519214
C5 0.475686 0.079951 0.371771
C6 0.288657 0.504887 0.393952
Constants C1-C6 dependent on flow regime.
Handling of an aqueous phase in the model
An aqueous phase is assumed to be completely immiscible with gas and oil. Average properties of oil and an
aqueous phase are calculated and these are assumed to be representative for the liquid phase as a whole. Only the
wax deposition model distinguishes between hydrocarbon phases and an aqueous phase. The wax deposition can
only take place from the hydrocarbon-wetted part of the inner pipe wall.
Wax deposition
Wax deposition from the oil phase is always considered. Furthermore it is optional whether or not wax deposition
from the gas phase should be considered. The wax deposition mechanisms considered for the gas and oil phases are
molecular diffusion and shear dispersion.
The volume rate of wax deposited by molecular diffusion for a given wax-forming component i is calculated from
the relation
NWAX
1i i
iwet
w
i
b
iidiff
waxδρ
MWSccDVol
where b
ic is the molar concentration of wax component i in the bulk phase and w
ic is the molar concentration of
wax component i in the phase at the wall. Swet is the fraction of the perimeter wetted by the current phase. NWAX is
the number of wax components, Mi the molecular weight and i the density of wax component i. L is the length of
the pipeline section and r the current inner pipeline radius considering wax deposition.
The thickness of the laminar film layer inside the pipeline is calculated from the expression
(Bendiksen et al., 1991)
f
D 1
Re26.11αδ
where is a user defined thickness correction factor. The allowed values of are between 0 and 100. The
introduction of provides the user with the possibility of tuning a predicted thickness of a wax layer to experimental
data, since a very narrow film layer will result in an increase in wax deposition and vice versa.
The diffusion coefficient, Di of the wax-forming component is calculated from a correlation by Hayduk and Minhas
(1982).
PVTsim Method Documentation Wax Deposition Module 155
71.0
iwax,
iwax,w,
791.0
ρ
M
10.2
1.4712
iρ
MηT1013.3βD
iwax,
iwax,w,
where is a user defined diffusion coefficient factor. The allowed values of are between 0 and 100. The
introduction of provides the user with yet another possibility of tuning a predicted wax layer thickness to
experimental data, since a large diffusion coefficient for a given wax component will result in an increased
deposition of that particular component and vice versa.
For systems with a large oil fraction, it is generally expected that deposition is dominated by oil phase deposition to
an extent where contributions from the gas phase are negligible. For rich gases and lean condensate systems, it may
however be of interest to include contributions from the gas as well. The model considers wax deposition from the
gas phase as results of both molecular diffusion and shear dispersion. The same assumptions are used as for the oil
phase. Whether wax deposition from the gas phase should be considered or not is selected on the ’Simulation
Options’ menu.
Shear dispersion accounts for deposition of wax already precipitated in the bulk phase. The volume rate of wax
deposited from shear dispersion is estimated from the following correlation of Burger et al. (1981)
wax
wall
*
shear
waxρ
γAckVol
where k* is a shear deposition rate constant, Cwall is the volume fraction of deposited wax in the oil in the bulk, is
the shear rate at the wall, A is the surface area available for deposition and wax is the average density of the wax
precipitated in the bulk phase. The shear dispersion mechanism is often assumed to be negligible as compared with
molecular diffusion (Brown et al. (1993) and Hamouda (1995)). Therefore the allowed values of k* is set to
[0;0.0001 g/cm2] or [0;0.025 lb/ft
2] or [0;0.001 kg/m
2].
Boost pressure
It is possible to specify a pressure increase or boost pressure at the entrance of each user specified segment. The
boost pressure may originate from a pump or a compressor, which is located between two sections. In plots the boost
pressure will show up at the end of the subsequent section.
Porosity
The porosity of the deposited wax is understood as the space between the wax crystals occupied by captured oil. This
porosity is reported to be quite significant in many cases (70%) and to depend on the shear rate. The program has
the possibility of treating the porosity as a constant or to depend linearly on shear rate. The expression used is:
BσA
In this expression, is the porosity and the shear rate. The constants A and B are determined from two input data
points of shear rate and corresponding porosity. If a constant porosity is to be used, A = 0 and B is the constant
porosity value.
Boundary conditions
By boundary conditions is understood the fluid inlet specifications to the pipeline. This includes pressure,
temperature, flow rate and fluid composition. One or more boundary conditions may be changed during the
simulation at specified time steps. In case the inlet composition is to be changed.
PVTsim Method Documentation Wax Deposition Module 156
Mass Sources
A mass source in this context is understood as a side stream to the pipeline. Mass sources may be defined to enter in
a specified segment inlet in a given time step. Mass sources cannot be specified to enter into the first segment. A
change of boundary conditions may be specified instead. Temperature and flow rate of the source are specified. The
pressure in the source is assumed to be equal to that of the fluid at the current position in the pipeline. The fluid
composition for the source is specified by referring to a fluid in the current fluid database. It is possible to change
conditions for the source in a later time step, or to change the composition of that source. The source composition is
mixed into the main pipeline stream, and a PH-flash determines the phase distribution and temperature of the mixed
stream. This is done by first determining the enthalpy of the source through a PT-flash and then determine the
mixture enthalpy based on the molar flow rates. Fluids entered as sources must be characterized to the same pseudo-
components as the original fluid in the simulation.
References
Bendiksen, K.H., Maines, D., Moe, R., Nuland, S., SPE 19451, “The Dynamic Two-Fluid Model OLGA: Theory
and Application”, SPE Production Engineering, May 1991, pp. 171-180.
Bird, R.B., Steward, W.E., Lightfoot, E.N., Transport Phenomena, Wiley, NY. 1960, pp. 286-28.
Brown, T.S., Niesen, V.G. and Erickson, D.P., ”Measurement and Prediction of Kinetics of Paraffin Deposition”,
SPE 26548, 68th
Annual Technical Conference and Exhibition of SPE Houston, Tx, 3-6 October, 1993.
Burger, E.D., Perkins, T.K. and Striegler, I.H., ”Studies of Wax Deposition in the Trans Alaska Pipeline”, Journal of
Petroleum Technology, June 1981, 1075-1086.
ESDU 93018 and 92003: ”Forced convection heat transfer in straight tubes”, ESDU 1993.
Hamouda, A., ”An Approach for Simulation of Paraffin Deposition in Pipelines as a Function of Flow
Characteristics with a Reference to Teeside Oil Pipeline”, SPE 28966, presented at SPE Int. Symposium on Oilfield
Chemistry, San Antonio, 14-17 February, 1995.
Hayduk, W. and Minhas, B.S., ”Correlations for Predictions of Molecular Diffusivities in Liquids”, The Canadian
Journal of Chemical Engineering 60, 1982, pp. 295-299.
Lindeloff, N. and Krejbjerg, K., “Compositional Simulation of Wax Deposition in Pipelines: Examples of
Application”, Presented at Multiphase ’01, Cannes, France, June 13-15, 2001.
Lindeloff, N. and Krejbjerg, K., “A Compositional Model Simulating Wax Deposition in Pipeline Systems”, Energy
& Fuels, 16, pp. 887-891, 2002.
Szilas, A.P.: ”Production and Transport of Oil and Gas, part B”, 2. Ed. Developments in Petroleum Science, 18B,
Elsevier, Amsterdam, 1986.
PVTsim Method Documentation Clean for Mud 157
Clean for Mud
Clean for Mud
Reservoir samples are often contaminated by base oil from drilling mud. The Mud module of PVTsim (Pedersen and
Christensen (2006) Chapter 2) has been implemented for the purpose of estimating the composition of a reservoir
fluid from the composition of the fluid with a certain content of base oil contaminate.
It is possible to make regression to experimental PVT data for a contaminated fluid and afterwards make use of the
regressed component parameters for the non-contaminated fluid.
Cleaning Procedure
In order to use the Mud module, the following compositional data are needed:
Composition of contaminated reservoir fluid. It is customary to analyze to either C7+, C10+, C20+, or C36+.
Composition of base oil contaminate. It will usually consist of components in the carbon number range C11 – C30
(defined components not accepted)
Weight% contaminate in stock tank oil (optional for extended compositions)
The cleaning procedure will differ depending on the extent of the compositional analysis
Reservoir fluids to C7+ or C10+
With a composition to C7+ or C10+ all base oil contaminate will be contained in the plus fraction of the contaminated
reservoir fluid. The base oil affects molar amount, density and molecular weights of the plus fraction. The weight%
contaminate in the oil from a flash of the contaminated reservoir fluid to standard conditions is required input.
1) Characterization of contaminated reservoir fluid as for a usual plus composition.
2) PT-flash to standard conditions
3) Weight% contaminate of total reservoir fluid initially estimated as weight% contaminate of the STO oil
(input) multiplied by the weight fraction of oil from flash.
4) Contaminated reservoir fluid cleaned.
5) Usual characterization of cleaned fluid.
6) Weaving of cleaned fluid with mud contaminate.
7) PT flash to standard conditions. Check whether calculated amount of contaminate in STO oil agrees with
input. Otherwise make new estimate of weight% contaminate in reservoir fluid and return to 4.
Reservoir fluids to C20+
Most base oil contaminates will contain components lighter than C20 as well as components heavier than C20. Some
contaminate is therefore contained in the plus fraction and some in the lighter fractions. It is practical to have all the
contaminate contained in the plus fraction before performing the cleaning calculation. The carbon number fractions
with contaminate are therefore combined into a plus fraction ending at the carbon number of the lightest base oil
component. Say the base oil composition starts at C15, the C15 – C20+ fractions of the contaminated reservoir fluid are
combined into a C15+ fraction.
PVTsim Method Documentation Clean for Mud 158
After the contraction of the contaminated reservoir fluid composition the cleaning procedure is the same as for a C7+
or a C10+ composition.
Reservoir fluids to C36+
With a composition to C36+ the carbon number fraction C7-C10 will usually be free of contamination and the same
will be the case for the fractions C30-C36. This allows the percent contamination to be estimated.
For a clean reservoir fluid PVTsim assumes the following relation between the mole fraction (z) of C7+ fractions and
carbon number i.
iCNBAlnzi
A and B are estimated by a fit to mole%’s for C7+ mole fractions against carbon number.
The above relation will not apply for fractions contaminated by base oil, but it will still be true for uncontaminated
C7+ fractions. A and B may be determined by a linear fit to zi versus CNi, where i stands for uncontaminated C7+
fractions. Using A and B, the mole fractions of the remaining C7+ fractions in the uncontaminated fluid may be
estimated. The remaining molar amount of each carbon number fraction is assumed to originate from the base oil,
which enables the composition of the contaminate to be estimated. The estimated base oil composition will not
necessarily be identical to the input composition.
Cleaning with Regression to PVT Data
Any PVT data will be for the contaminated sample. It is obviously of more interest to know the PVT properties of
the uncontaminated fluid. It is therefore desirable to have the option to carry out a regression for the contaminated
composition and afterwards be able to apply the regressed component parameters for the uncontaminated fluid.
The contaminated composition is initially cleaned as above. A regression is performed as for a usual plus fraction
composition, where the cleaned reservoir fluid composition in each iterative step is weaved with the base oil
contaminate in the pertinent weight ratio. Weaving is a mixing where each component of the individual fluids is
retained. The base oil contaminate is lumped into pseudo-components (default is 4 pseudo-components). Only the
components originating from the cleaned reservoir fluid are regressed on, i.e. the base oil components are left out of
the regression. The weaving procedure is selected because it enables regression to be performed directly on the
component properties of the reservoir fluid.
Regression on the characterized contaminated fluid is also an option, in which case the same regression parameters
are used as with ordinary regression for characterized fluids. To allow the program identify the mud components in
the contaminated fluids, the characterized mud must be saved in the database prior to the regression and selected as
mud contaminate in the Clean for Mud menu. The result of the regression is a cleaned, tuned and characterized
reservoir fluid composition.
References
Pedersen, K.S. and Christensen, P.L., ”Phase Behavior of Petroleum Reservoir Fluids”, CRC Taylor & Francis, Boca
Raton, 2006.
PVTsim Method Documentation Black Oil Correlations 159
Black Oil Correlations
Black Oil Correlations
“Black Oil” type correlations may be used in PVTsim to generate PVT tables for the Eclipse Black Oil reservoir
simulator. Only a minimum set of information is needed, i.e. reservoir temperature, API gravity of the fluid, gas
gravity and pressure stages.
The following “black oil” type correlations are available in PVTsim (references in Whitson and Brule, 2000)
Bubble-point Pressure
Standing
Required input: Rs (scf/STB), T (oF), g, API
Output Units: psia
Expression:
1.4A18.2Pb
API0.0125γ0.00091T
0.83
g
s 10γ
RA
Lasater
Required input: Rs (scf/STB), T (oR), g, API, o
Output Units: psia
Expression:
1.958.26yγ
Tp:0.6y
0.3232.786y0.679expγ
Tp:0.6y
3.56
g
g
bg
g
g
bg
oos
s
g/M350γ/379.3R
/379.3Ry
The stock-tank oil molecular weight Mo can be calculated from
1.562
APIo
APIo
γ73,110M:40API
10γ630M:40API
PVTsim Method Documentation Black Oil Correlations 160
The oil gravity o can be calculated from
API
oγ131.5
141.5γ
Glasø
Required input: Rs (scf/STB), T (oF), g, API
Output Units: psia
Expression:
2
b logA0.302181.7447logA1.7669Plog
0.989
API
0.1720.816
g
s
γ
T
γ
RA
Beggs-Vazquez
Required input: Rs (scf/STB), T (oF), g, API
Output Units: psia
Expression:
0.8425
460T
10.393γ
g
s
b
0.9143
460T
11.172γ
g
s
b
API
API
10γ
R56.06P:30API
10γ
R27.64P:30API
Dindoruk-Christman
Required input: Rs (scf/STB), T (oF), g, API
Output Units: psia
Expression:
11
A
a
g
a
s
8b a10γ
RaP
10
9
2
a
g
a
s5
a
API3
a
1
7
6
42
γ
2Ra
γaTaA
Coefficient Value
a1 1.42828E-10
a2 2.844591797
a3 -6.74896E-04
a4 1.225226436
a5 0.033383304
a6 -0.272945957
a7 -0.084226069
a8 1.869979257
a9 1.221486524
a10 1.370508349
PVTsim Method Documentation Black Oil Correlations 161
a11 0.011688308
Saturated Gas/Oil Ratio
Standing
Required input: P (psia), T (oF), g, API
Output Units: scf/STB
Expression:
1.205
0.00091T
0.0125γ
gs10
101.40.055pγR
API
Lasater
Required input: P (psia), T (oR), g, API, o
Output Units: scf/STB
Expression:
go
go
sy1M
y132755γR
0.281
g
g
g
g
g
g
0.236T
0.121pγy:3.29
T
pγ
0.476T
1.473pγ0.359lny:3.29
T
pγ
where
1.562
APIo
APIo
γ73,110M:40API
10γ630M:40API
Vazquez-Beggs
Required input: P (psia), T (oF), g, API
Output Units: scf/STB
Expression:
460T
γCexppγCR API3C
g1s2
°API 30 °API > 30
C1 0.0362 0.0178
C2 1.0937 1.1870
C3 25.724 23.931
Glasø:
Required input: P (psia), T (oF), g, API
Output Units: scf/STB
PVTsim Method Documentation Black Oil Correlations 162
Expression:
1/0.816
0.989
API
0.172
0.60436
logP1.76691.208723.0441.7447
gs/γT
10γ
R
Dindoruk-Christman
Required input: P (psia), T (oF), g, API
Output Units: scf/STB
Expression:
11
10
a
Aa
g9
8
s 10γaa
PR
2
a
a
API5
a
3
a
API1
7
6
42
P
2γa
TaγaA
Coefficient Value
a1 4.86996E-06
a2 5.730982539
a3 9.92510E-03
a4 1.776179364
a5 44.25002680
a6 2.702889206
a7 0.744335673
a8 3.359754970
a9 28.10133245
a10 1.579050160
a11 0.928131344
Oil Formation Volume Factor
Standing
Required input: Rs (scf/STB), T (oF), g, API
Output Units: bbl/STB
Expression:
Bubblepoint Bo
1.25
ob A10120.9759B
1.25T/γγRA0.5
ogs
The oil gravity o can be calculated from:
PVTsim Method Documentation Black Oil Correlations 163
API
oγ131.5
141.5γ
Saturated Bo
Same expression as the one used at the bubble point but in this case the Rs is a function of the pressure p.
The Rs values are determined as in the saturated GOR (Rs) section.
Glasø
Required input: Rs (scf/STB), T (oF), g, API
Output Units: bbl/STB
Expression:
Bubblepoint Bo
2
ob logA0.27682.9133logA6.5851Blog
0.968T/γγRA0.526
ogs
The oil gravity o can be calculated from:
API
oγ131.5
141.5γ
Saturated Bo
Same expression as the one used at the bubble point but in this case the Rs is a function of the pressure p.
The Rs values are determined as in the saturated GOR (Rs) section.
Al-Marhoun
Required input: Rs (scf/STB), T (oF), g, API
Output Units: bbl/STB
Expression:
Bubblepoint Bo
60T100.528707
γ160TR104.292580/γγR100.220163R100.1773421.0B
3
os
6
ogs
3
s
3
ob
The oil gravity o can be calculated from:
API
oγ131.5
141.5γ
Saturated Bo
Same expression as the one used at the bubble point but in this case the Rs is a function of the pressure p.
The Rs values are determined as in the saturated GOR (Rs) section.
PVTsim Method Documentation Black Oil Correlations 164
Vazquez-Beggs
Required input: Rs (scf/STB), T (oF), g, API
Output Units: bbl/STB
Expression:
Bubblepoint Bo
gcAPIs3gcAPI2s1ob /γγ60TRC/γγ60TCRC1B
°API 30 °API > 30
C1 4.677 x 10-4
4.670 x 10-4
C2 1.751 x 10-5
1.100 x 10-5
C3 -1.811 x 10-8
-1.337 x 10-9
Saturated Bo
Same expression as the one used at the bubble point but in this case the Rs is a function of the pressure p.
The Rs values are determined as in the saturated GOR (Rs) section.
Dindoruk-Christman
Required input: Rs (scf/STB), T (oF), g, API
Output Units: bbl/STB
Expression:
Bubblepoint Bo
g
API
14
2
131211obγ
γ60TaAaAaaB
2
a
g
a
S8
a
s6
a
4a
o
a
g
a
s
60)(Tγ
2Ra
Ra60)(Taγ
γR
A
10
9
7
5
3
21
The oil gravity o can be calculated from
API
oγ131.5
141.5γ
Coefficient Value
a1 2.510755
a2 -4.852538
a3 1.183500E+01
a4 1.365428E+05
a5 2.252880
a6 1.007190E+01
a7 4.450849E-01
a8 5.352624
a9 -6.309052E-01
PVTsim Method Documentation Black Oil Correlations 165
a10 9.000749E-01
a11 9.871766E-01
a12 7.865146E-04
a13 2.689173E-06
a14 1.100001E-05
Saturated Bo
Same expression as the one used at the bubble point but in this case the Rs is a function of the pressure p.
The Rs values are determined as in the saturated GOR (Rs) section.
Dead-Oil Viscosity
Beal-Standing
Required input: T (oF), API
Output Units: cP
Expression:
A
4.53
API
7
oD200T
360
γ
101.80.32η
APIA/33.843.0
10
Beggs-Robinson
Required input: T (oF), API
Output Units: cP
Expression:
F 70Tfor ,101η o0.04658γ6.9824expT
oDAPI
1.163
For T < 70 °F it should be substituted (according to Bergman) by:
310TlnAA1ηln 10oD
2
APIAPI0 0.00033γ0.194γ22.33A
API1 0.0185γ3.20A
Glasø
Required input: T (oF), API
Output Units: cP
Expression:
36.44710.313logT
API
3.44410
oD logγT103.141η
Al-Khafaji
Required input: T (oF), API
PVTsim Method Documentation Black Oil Correlations 166
Output Units: cP
Expression:
2.709
API
0.00488T4.9563
oD14.29T/30γ
10η
Dindoruk-Christman
Required input: T (oF), API, Pb (psia), Rsb (scf/STB)
Output Units: cP
Expression:
86
4
a
sb7
a
b5
A
API
a
3
oDRaPa
)(logγTaη
21 alogTaA
Coefficient Value
A1 14.505357625
A2 -44.868655416
A3 9.36579E+09
A4 -4.194017808
A5 -3.1461171E-9
A6 1.517652716
A7 0.010433654
A8 -0.000776880
Saturated Oil Viscosity
Standing
Required input: Rs (scf/STB), oD (cp)
Output Units: cP
Expression:
2A
oD1o ηAη
2s
7s
4 R102.2R107.4
1 10A
s3
s3
s5 R103.74R101.1R108.622
10
0.062
10
0.25
10
0.68A
The Rs values are determined as in the saturated GOR (Rs) section.
Beggs-Robinson
Required input: Rs (scf/STB), oD (cp)
Output Units: cP
Expression:
2A
oD1o ηAη
0.515
s1 100R10.715A
PVTsim Method Documentation Black Oil Correlations 167
0.338
s2 150R5.44A
The Rs values are determined as in the saturated GOR (Rs) section.
Bergman
Required input: Rs (scf/STB), oD (cp)
Output Units: cP
Expression:
2A
oD1o ηAη
300R0.8359ln4.768lnA s1
300R
133.50.555A
s
2
The Rs values are determined as in the saturated GOR (Rs) section.
Aziz et al.
Required input: Rs (scf/STB), oD (cp)
Output Units: cP
Expression:
2A
oD1o ηAη
s0.00081R
1 100.800.20A
s0.00072R
2 100.570.43A
The Rs values are determined as in the saturated GOR (Rs) section.
Al-Khafaji
Required input: Rs (scf/STB), oD (cp)
Output Units: cP
Expression:
2A
oD1o ηAη
4
0
3
0
2
001 0.0631A0.4065A0.5657A0.2824A0.247A
4
0
3
0
2
002 0.01008A0.0736A0.07667A0.0546A0.894A
)log(RA S0
The Rs values are determined as in the saturated GOR (Rs) section.
Dindoruk-Christman
Required input: Rs (scf/STB), oD (cp)
Output Units: cP
Expression:
2A
oD1o ηAη
PVTsim Method Documentation Black Oil Correlations 168
)Rexp(a
Ra
)Rexp(a
aA
s5
a
s3
s2
1
1
4
)Rexp(a
Ra
)Rexp(a
aA
s10
a
s8
s7
6
2
9
Coefficient Value
a1 1
a2 4.740729E-04
a3 -1.023451E-02
a4 6.600358E-01
a5 1.075080E-03
a6 1
a7 -2.191172E-05
a8 -1.660981E-02
a9 4.233179E-01
a10 -2.273945E-04
Gas Formation Volume Factor
Calculation of Bg
Required input: T (oR), P (psia), Tsc (
oR), Psc (psia), Z
Output Units: ft3/scf
Expression:
ZP
T
T
PB
sc
sc
g
Calculation of Z
Required input: T (oR), Tpc (
oR), P (psia), Ppc (psia)
Output Units: Dimensionless
Expression:
y
t11.2texp0.06125pZ
2
pr
where
/TT1/Tt pcpr
pcpr P/Pp
y (the “reduced” density) is obtained by solving
0y42.4t242.2t90.7t
y4.58t9.76t14.76ty1
yyyyt11.2texp0.06125p
2.82t2.1832
232
3
4322
pr
Through a Newton-Raphson scheme
PVTsim Method Documentation Black Oil Correlations 169
dy
df/fyy
oldoldoldnew
Where f is the function above and df/dy is
2.82t1.1832
32
4
432
y42.4t242.2t90.7t2.82t2.18
y9.16t19.52t29.52ty1
y4y4y4y1
dy
df
Use as an initial estimate y = 0.001 and as a convergence criteria 81x10f(y)
Calculation of Tpc and Ppc
Sutton
Required input: g
Output Units: Tpc (oR) and Ppc (psia)
Expression:
2
ggpc
2
ggpc
3.6γ131γ756.8P
74.0γ349.5γ169.2T
Gas Viscosity
Dempsey
Required input: P (psia), T (oF), g
Output Units: cP
Expression:
prprpr P,TfTln
pr
g
g eT
T,γgη
3
pr15
2
pr14pr1312
3
pr
3
pr11
2
pr10pr98
2
pr
3
pr7
2
pr6pr54pr
3
pr3
2
pr2pr10prprpr
PaPaPaaTPaPaPaaT
PaPaPaaTPaPaPaaP,TfTln
2
g
2
8
2
g7
2
g6g
2
5g4g3
2
210g MTbTMbMbMTbTMbMbTbTbbT,γg
The molecular weight and reduced properties can be obtained from
g
pr
g
pr
gg
47.44γ700.55
PP
307.97γ175.59
460TT
28.97γM
Coefficient Value
PVTsim Method Documentation Black Oil Correlations 170
a0 -2.46211820
a1 2.97054714
a2 -2.86264054e-1
a3 8.05420533e-3
a4 2.80860949
a5 -3.49803305
a6 3.60373020e-1
a7 -1.04432413e-2
a8 -7.93385684e-1
a9 1.39643306
a10 -1.49144925e-1
a11 4.41015512e-3
a12 8.39387178e-2
a13 -1.86408848e-1
a14 2.03367881e-2
a15 -6.09579263e-4
Coefficient Value
b0 1.11231913e-2
b1 1.67726604e-5
b2 2.11360496e-9
b3 -1.09485050e-4
b4 -6.40316395e-8
b5 -8.99374533e-11
b6 4.57735189e-7
b7 2.12903390e-10
b8 3.97732249e-13
Note: the correlation is valid only in the range 1.2 Tpr 3 and 1 Ppr 20
Lee-Gonzalez
Required input: g (g/cm3), T (
oR), Mg
Output Units: cP
Expression:
3A
g2
4
1g ρAexp10Aη
where
T19.26M209.2
T0.01607M9.379A
g
1.5
g
1
g2 0.01009M986.6/T3.448A
23 0.2224A2.447A
The gas molecular weight Mg can be calculated from
gg 28.97γM
Additionally, the gas density gρ can be calculated from
ZRT
PMρ
g
g
PVTsim Method Documentation Black Oil Correlations 171
Where P is in psia, T is in oR, R = 10.732 (psia * ft
3 / lb mole *
oR) and g is in lbm/ft
3
Lucas
Required input: T (oR), Tpc (
oR), P (psia), Ppc (psia), Mg
Output Units: cP
Expression:
1A
pr3
A
pr2
1.3088
pr1
gsc
g
45 pA1pA
pA1
η
η
pr
0.3286
pr
3
1T
5.1726Texp101.245A
1.27231.6553TAA pr12
pr
37.7332
pr
3T
3.0578T0.4489expA
pr
7.6351
pr
4T
2.2310T1.7368expA
0.4489
pr5 0.1853T0.9425expA
0.0184.058T0.340exp0.449T0.357exp0.807Tξη prpr
0.618
prgsc
1/6
4
pc
3
g
pc
pM
T9490ξ
The gas molecular weight Mg can be calculated from
gg 28.97γM
The Tpc and Ppc can be calculated from the Sutton correlations for pseudo-critical properties.
Note: the correlation is valid only in the range 1 Tpr 40 and 0 Ppr 100
Nomenclature
gB Gas formation volume factor
oB Oil formation volume factor
obB Oil formation volume factor at the bubble point
oM Stock-tank oil molecular weight
gM Gas molecular weight
sR Solution gas oil ratio
T Temperature
PVTsim Method Documentation Black Oil Correlations 172
pcT Pseudo critical temperature
prT Pseudo reduced temperature
P Pressure
bP Bubblepoint pressure
pcP Pseudo critical pressure
prp Pseudo reduced pressure
Z Z factor
APIγ Oil API gravity
gγ Gas specific gravity
oγ Oil gravity
gρ Gas density
gη Gas viscosity
oη Oil Viscosity
oDη Dead oil viscosity
References
Dempsey, J.R.: “Computer Routine Treats Gas Viscosity as a Variable,” Oil & Gas Journbal, August 1965, pp. 141-
143.
Dindoruk, B. and Christman, P.G.: “PVT Properties and Viscosity Correlations for Gulf of Mexico Oils,” SPE paper
71633 presented at the SPE ATCE, New Orleans, September 30 – October 3, 2001.
Society of Petroleum Engineers: “Petroleum Engineering Handbook,” Richardson, Texas.
Whitson, C.H. and Brule, M.: “Phase Behavior,” SPE Monograph Volume 20, Richardson, Texas, 2000.
PVTsim Method Documentation STARS 173
STARS
VISCTABLE This section outlines how experimental viscosity data can be used when generating VISCTABLE values for a
STARS table.
Introduction
In STARS, the viscosity of a fluid at temperature T is calculated as
(1)
is the natural logarithm to the viscosity contribution of component at the temperature T, and is the mole
fraction of component . The -values appear in the STARS interface table under the VISCTABLE keyword.
In the procedure, two compositions will be referred to, dead oil and live oil. The live oil composition is the
composition of the selected fluid, i.e. . The dead oil composition is the composition of the liquid from a
flash of the live oil to standard conditions, i.e. .
In the following, both -values and -values are referred to as ’viscosities’. The notation for corresponding values
of temperature and viscosity data is where k can be dead oil or live oil.
The purpose of the procedure is to find physically reasonable values of which fulfill that they
1. Represent the variation of the fluid viscosity with temperature for both dead oil and live oil, using Eq. (1),
over the temperature range of interest.
2. Follow certain rules wrt. the variation of with temperature and molecular weight of component i.
These rules are outlined in a later section.
Two temperature ranges are referred to
1. The tabulation temperature range ( ) which is the temperature range of interest. It covers the range given
by and which must be input. The range is split into 40 equidistant temperature tabulation points.
2. The data range which covers to , where and
, i.e. the common temperature range in which dead oil viscosity
data and live oil viscosity data are input.
To make use of the procedure dead oil viscosities must be input. It is optional to input live oil viscosities. It is
required that for all temperatures, since the dead oil will contain relatively more of the heavy
components making the dead oil more viscous than the live oil at a given temperature. The live oil must also release
a gas phase at standard conditions, i.e. is not allowed.
PVTsim Method Documentation STARS 174
Outline of Procedure
The input to the procedure is a live oil composition, viscosity data points for dead oil and optionally viscosity data
points for live oil. The procedure consists of the following steps
1. In case no live oil viscosity data has been input, generating artificial live oil viscosity data ( ,
based on the dead oil viscosity data ( .
2. Generating 40 equidistant tabulation viscosity data points ( ) covering the temperature range of
interest, one set for dead oil and one set for live oil. The tabulation is done using a cubic spline with the
input viscosity data as fix points, combined with extrapolation.
3. Calculating the component viscosity contribution for each component at each tabulation temperature.
4. Checking if required monotonicity is found in the obtained -values. Correction of the -values is
performed in case the required monotonicity is not found. Checking if required monotonicity is found for
fluid viscosity calculated from Eq. (1).
Generating Artificial Live Oil Viscosity Data from Dead Oil Viscosity Data
In case no live oil viscosity data, ( has been input, such data is created artificially as follows
1. Tuning of the 3rd
CSP coefficient to match the dead oil viscosity at the given temperature.
2. Calculate the live oil viscosity at the same temperature using the tuned 3rd
CSP coefficient. The calculated
live oil viscosity is then used as an artificial data point, i.e. .
3. 1 and 2 are repeated for each dead oil data point temperature ( ).
Generating Tabulation Viscosity Data Points
A cubic spline procedure combined with extrapolation is used to obtain sets of tabulation data points ( ),
one set for dead oil and one set for live oil.
First the viscosity data are matched using the cubic spline. For tabulation temperatures within the data range,
i.e., for , the viscosities at the tabulation temperatures ( ) are calculated using the cubic
spline data match.
For tabulation temperatures outside the data range, extrapolated viscosities are generated. Two temperature points
are used, where the first point is at the data range temperature end point, and the second point is
1 K inside the data range from the data range temperature end point. Corresponding values of are calculated
using the cubic spline data match. From the two points a slope of a straight ( )-line is calculated. This slope is
used for extrapolation starting at the data range temperature end point and the corresponding viscosity. This is done
on both sides of the data range.
The dead and live oil viscosities obtained are checked as follows
1. A check if the ( , )-curves cross inside the data range thereby violating the
requirement. If that is the case, calculation is stopped and an error returned.
2. A check if the -curve and the -curve diverge from each other for temperatures above ,
i.e., a check if the curves ’open’ for temperatures outside the data range on the high temperature side. If that
is the case, all live and dead oil viscosities for temperatures above are replaced by linearly
extrapolated values. The extrapolation is done starting at and the corresponding viscosity using a
slope equal to the average of the slopes of the dead oil and the live oil extrapolation lines.
3. A check if the -curve and the -curve cross for temperatures above i.e., a check if the
criterion is violated for temperatures outside the data region on the high temperature
side. If this is the case, the values are replaced by the values.
PVTsim Method Documentation STARS 175
Calculating Component Viscosity Contributions
The basic assumption is that the contribution of a component to the viscosity of the fluid increases with increasing
molecular weight (Mw). This is to reflect that the fluid viscosity should decrease if a light or intermediate
component (solvent) is injected into a heavy highly viscous fluid.
The relation assumed is
(2)
where is an exponent, and is the molecular weight of component . For n > 0 this relation will cause to
increase with increasing molecular weight.
The relation in Eq. (2) cannot represent a temperature dependent fluid viscosity as the relation has no built in
temperature dependency. To compensate for this a temperature dependent scaling factor is introduced so that
the relation becomes
(3)
For a given exponent, , the task is thus to calculate so that the dead oil and the live oil viscosity data points
are matched, i.e.
(4)
where k can be dead oil or live oil. Using this approach the difference, , between the dead oil and the live oil
viscosity calculated from Eq. (1) becomes
(5)
Given that the dead oil with composition should always be richer in high molecular weight components than the
live oil composition , will be positive if is positive. This is in line with the requirement of .
However, is independent of the temperature as cancels out and is assumed to be constant in temperature.
The difference in viscosity between live oil and dead oil cannot be expected to be constant over a temperature range.
More flexibility is required, so different values of at different temperatures are required.
If is allowed to vary with Eq. (5) can be rewritten as
(6)
can be calculated directly from Eq. (6) given the viscosity data points and the live oil and dead oil
compositions. can be calculated directly from Eq. (4) using the same information. Because Eq. 3 is common
for dead oil and live oil, can be calculated using either the dead oil or the live oil composition with the same
result. If dead oil is chosen the expression becomes
(7)
Finally is calculated from Eq. (3).
PVTsim Method Documentation STARS 176
represents the difference between dead oil and live oil viscosities at varying temperature. represents the
general level of the viscosities at varying temperature.
Checking for Monotonicity and Performing Corrections
The -values must meet the following criteria
1. For each component , must decrease with increasing temperature.
2. For each temperature , must increase with increasing molecular weight of component .
3. The viscosity , calculated from Eq. (1), must decrease with increasing temperature for both dead oil and
live oil.
and calculated from Eqs. (6) and (7) only ensure a match of the dead oil and the live oil viscosity data. It
is not ensured that the -values calculated from Eq. (3) meet the above criteria. The -values are
checked as follows
1. For each component, a search is performed from high to low temperatures for the first high temperature
point that breaks the required monotonicity in . If such point is identified, a search from low to high
temperatures is performed to identify the last low temperature point with an -value larger than the
-value at the previously identified high temperature point. A straight line is then used to connect
these two points. If such low temperature point is not found, because the high temperature point is a global
maximum, an -value 1% larger than the high temperature -value is assigned at the
minimum tabulation temperature, and a straight line is used to connect the minimum tabulation temperature
and the high temperature point.
2. must increasing with increasing molecular weight of component i. To ensure this, If
, is replaced by . This is done for each tabulation temperature, one
temperature at a time.
The above checks and corrections and repeated until both criteria are met. If this does not succeed in 100 such
repeats the procedure is stopped and an error is returned.
Finally it is checked if fluid viscosities calculated from Eq. (1) decrease with increasing temperature. This is done
over the tabulation temperature range for both dead oil and live oil. This is mainly a final precaution as the checking
and correction of the -values should ensure decreasing fluid viscosity with increasing temperature.
PVTsim Method Documentation Allocation 177
Allocation
Allocation The PVTsim Allocation Module allocates the gas, oil and water production from a process plant to the different
producers feeding the plant. The module in other words determines the volumetric contributions from each feed to
the product streams.
The required input is
Molar composition of each feed stream.
Total volumetric flow rate (gas+oil+water) of each feed stream at given P&T (often flow meter conditions).
Process plant (separator) configuration.
The below figure shows schematically how the process plant is simulated. The number of separator stages may vary
from 1 (single stage flash) to 6.
Feed
WaterWater Production
Oil Production
Gas Production
Reference Condtions
Oil Oil
PVTsim Method Documentation Allocation 178
The allocation principle (Pedersen, 2005) is shown below for two hydrocarbon feed streams with no water.
Component i is followed from feed to product streams.
kiz
feediz
vapiy
liqix
kix
kiy
1
kiz
feediz
vapiy
liqix
kix
kiy
1
Yellow color is used for component i in the upper (and largest) feed stream, in which component i is present in low
concentration. In the two remaining streams component i is the most abundant component and its concentration
illustrated using red and green colors. Component i could be methane and the upper feed stream could be a stabilized
oil and the two lower feed streams could be a volatile oil and a gas condensate.
The feed compositions are first characterized to the same pseudo-components using the same principles as in the
same pseudo-components option. Each fluid composition influences the pseudo-component properties with a weight
proportional to its mass flow rate.
In the Allocation Module the volumetric flow rates are converted to molar flow rates, and the molar feed
composition, ,z Feed
i to the process plant determined through
N1,2,...,i,n
zn
zFeed
k
i
M
1kk
Feed
i
where kn is the molar flow rate of the k’th feed stream and
M
1kk
Feed nn the total molar flow rate fed to the plant. N
is the number of components, and k
iz is the mole fraction of component i in the k’th feed stream.
The Allocation module assumes complete mixing in the process plant, meaning that
Aqueous
i
k
i
Oil
i
k
i
Gas
i
k
i
Feed
i
k
i
w
w
x
x
y
y
z
z
where Gas
iy , Oil
ix andAqueous
iw are the mole fractions of component i in the export gas, oil and water streams. k
iy , k
ix
and k
iw are the mole fractions of component i originating from the k’th feed stream in the export gas, oil and water
streams.
If the terms βg, βo, and βw are used for the gas, oil and water mole fractions of the total product, the total molar gas
production of component i originating from stream k may be determined to
gk
k
i
kGas,
i βnyn
PVTsim Method Documentation Allocation 179
and the total volumetric gas production from feed stream k becomes
Gas
Gas
iN
1i
k
igk
Gas,
kV
V~
yβnV
where Gas
iV~
is the partial molar volume of component i in the product gas phase and Vgas
the molar volume of the
produced gas. Similarly the volumetric oil and water production originating from feed stream k become
Oil
Oil
iN
1i
k
iok
Oil
kV
V~
xβnV
Aqueous
Aqueous
iN
1i
k
iwk
Aqueous
kV
V~
wβnV
where Oil
iV~
and Aqueous
iV~
are the partial molar volume of component i in the produced oil and water phases and Voil
and VAqueous
the molar volume of the produced oil and water.
References
Pedersen, K.S., “PVT Software Applied With Multiphase Meters for Oil & Gas Allocation”, Presented at the Flow
2005: Modelling, Metering and Allocation conference, Aberdeen, March 14–15, 2005.