PVTSim Method Doc

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


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-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


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


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)


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)


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


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


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


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.


Asphaltene fraction Irrelevant for defined components.


components

Asphaltenes

Parachor Not estimated for defined


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.


components


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)


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.


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.


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)


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).


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


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


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


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

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


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).


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


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)


provided of the plus component amount (C20+ if the compositional analysis ends at C20+). The plus component

amount is calculated from


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


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


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


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)


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.


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 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


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.


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.


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


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.


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.


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


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).


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


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


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


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


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


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


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


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


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


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.


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.


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.


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


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)


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


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


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


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


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


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)


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


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


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


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.


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


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.


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.


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 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


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


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


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


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.


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


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


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.


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).


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)


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.


- 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


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


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.


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.


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


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 e2 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


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.


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


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.


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.


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.


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


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)


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


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.


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)


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.


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


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.


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


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.


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


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.


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.


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


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


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


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


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


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


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



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


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.


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.


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)


A1 = -3.25001

A2 = 901.095

A3 = 110.695

Tri Ethylene Glycol (TEG)


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)


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


Ethanol

Log10 (Psat

)= A – B/(T + C)

A = 5.24677

B = 1598.673

C = -46.424


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


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


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.


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


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


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


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).


KCl-MEG-Water

Solubility of KCl in weight% of KCl + H2O = Weight% KCl in pure H2O - 0.2589 C


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


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


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


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


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)


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


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


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


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-


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--


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


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-


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-


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-


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-


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


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


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.


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


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


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


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


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


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


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).


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.


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


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


The oil gravity o can be calculated from

API

oγ131.5

141.5γ

Glasø


Output Units: psia

Expression:

2

b logA0.302181.7447logA1.7669Plog

0.989

API

0.1720.816

g

s

γ

T

γ

RA

Beggs-Vazquez


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


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


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


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



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:




Expression:

1/0.816

0.989

API

0.172

0.60436

logP1.76691.208723.0441.7447

gs/γT

10γ

R

Dindoruk-Christman



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


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:


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



Expression:

Bubblepoint Bo

2

ob logA0.27682.9133logA6.5851Blog

0.968T/γγRA0.526

ogs


API

oγ131.5

141.5γ

Saturated Bo



Al-Marhoun



Expression:

Bubblepoint Bo

60T100.528707

γ160TR104.292580/γγR100.220163R100.1773421.0B

3

os

6

ogs

3

s

3

ob


API

oγ131.5

141.5γ

Saturated Bo




Vazquez-Beggs



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



Dindoruk-Christman



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


a10 9.000749E-01

a11 9.871766E-01

a12 7.865146E-04

a13 2.689173E-06

a14 1.100001E-05

Saturated Bo



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


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


Output Units: cP

Expression:

36.44710.313logT

API

3.44410

oD logγT103.141η

Al-Khafaji



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


Beggs-Robinson


Output Units: cP

Expression:

2A

oD1o ηAη

0.515

s1 100R10.715A


0.338

s2 150R5.44A


Bergman


Output Units: cP

Expression:

2A

oD1o ηAη

300R0.8359ln4.768lnA s1

300R

133.50.555A

s

2


Aziz et al.


Output Units: cP

Expression:

2A

oD1o ηAη

s0.00081R

1 100.800.20A

s0.00072R

2 100.570.43A


Al-Khafaji


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


Dindoruk-Christman


Output Units: cP

Expression:

2A

oD1o ηAη


)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


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


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


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


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.


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.


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).


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


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


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.

Documents

PVTSim Method Doc