Method Documentation

PVTsim 13

CALSEP

Contents Introduction 5

Introduction ...............................................................................................................................5

Pure Component Database 6 Pure Component Database.........................................................................................................6

Component Classes .....................................................................................................6 Component Properties .................................................................................................9 User Defined Components ........................................................................................10 Missing Properties .....................................................................................................10

Composition Handling 13 Composition Handling.............................................................................................................13

Types of fluid analyses..............................................................................................13 Handling of pure components heavier than C6 ..........................................................14 Fluid handling operations ..........................................................................................15 Mixing .......................................................................................................................15 Weaving ....................................................................................................................15 Recombination...........................................................................................................15 Characterization to the same pseudo-components.....................................................15

Flash Algorithms 17 Flash Algorithms .....................................................................................................................17

PT Flash.....................................................................................................................17 Flash Algorithms .......................................................................................................17 Other Flash Specifications.........................................................................................22 Phase Identification ...................................................................................................22 Components Handled by Flash Algorithms...............................................................23 References .................................................................................................................23

Phase Envelope and Saturation Point Calculation 25 Phase Envelope and Saturation Point Calculation ...................................................................25

No aqueous components............................................................................................25 Mixtures with Aqueous Components ........................................................................26 Components handled by Phase Envelope Algorithm ................................................26 References .................................................................................................................27

Equations of State 28 Equations of State ....................................................................................................................28

SRK Equation............................................................................................................28 SRK with Volume Correction ...................................................................................30 PR/PR78 Equation.....................................................................................................31 PR/PR78 with Volume Correction ............................................................................31 Classical Mixing Rules..............................................................................................32 The Huron and Vidal Mixing Rule............................................................................33 Phase Equilibrium Relations .....................................................................................34 References .................................................................................................................35

Characterization of Heavy Hydrocarbons 37 Characterization of Heavy Hydrocarbons................................................................................37

Classes of Components..............................................................................................37 Properties of C7+-Fractions ........................................................................................38 Extrapolation of the Plus Fraction .............................................................................39 Estimation of PNA Distribution ................................................................................39 Grouping (Lumping) of Pseudo-components ............................................................40 Delumping.................................................................................................................42 Characterization of Multiple Compositions to the Same Pseudo-Components .........43 References .................................................................................................................44

Thermal and Volumetric Properties 45 Thermal and Volumetric Properties.........................................................................................45

Density ......................................................................................................................45 Enthalpy ....................................................................................................................45 Internal Energy ..........................................................................................................46 Entropy ......................................................................................................................47 Heat Capacity ............................................................................................................47 Joule-Thomson Coefficient .......................................................................................47 Velocity of sound ......................................................................................................48 References .................................................................................................................48

Transport Properties 49 Transport Properties.................................................................................................................49

Viscosity....................................................................................................................49 Thermal Conductivity................................................................................................55 Gas/oil Interfacial Tension ........................................................................................57 References .................................................................................................................58

PVT Experiments 60 PVT Experiments.....................................................................................................................60

Constant Mass Expansion..........................................................................................60 Differential Depletion................................................................................................61 Constant Volume Depletion ......................................................................................61 Separator Experiments...............................................................................................62 Viscosity Experiment ................................................................................................62 Swelling Experiment .................................................................................................62 References .................................................................................................................63

Compositional Variation due to Gravity 63 Compositional Variation due to Gravity..................................................................................63 Isothermal case ........................................................................................................................64 Systems with a Temperature Gradient .....................................................................................65

Prediction of Gas/Oil Contacts ..................................................................................66 References .................................................................................................................67

Regression to Experimental Data 68 Regression to Experimental Data.............................................................................................68

Experimental data......................................................................................................68 Object Functions and Weight Factors........................................................................69 Regression for Plus Compositions.............................................................................70 Regression for already characterized compositions...................................................71 Regression on fluids characterized to the same pseudo-components ........................72 Regression Algorithm................................................................................................72 References .................................................................................................................72

Minimum Miscibility Pressure Calculations 73 Minimum Miscibility Pressure Calculations............................................................................73

Minimum Miscibility Pressure Calculations .............................................................73 Combined drive mechanism ......................................................................................75 References .................................................................................................................76

Unit Operations 77 Unit Operations........................................................................................................................77

Compressor................................................................................................................77 Expander....................................................................................................................79 Cooler ........................................................................................................................80 Heater ........................................................................................................................80 Pump..........................................................................................................................80 Valve .........................................................................................................................80 Separator....................................................................................................................80 References .................................................................................................................80

Modeling of Hydrate Formation 81 Hydrate Formation...................................................................................................................81

Types of Hydrates .....................................................................................................81 Hydrate Model...........................................................................................................82 Hydrate P/T Flash Calculations.................................................................................85

Calculation of Fugacities .........................................................................................................86 Fluid Phases...............................................................................................................86 Hydrate Phases ..........................................................................................................86 Ice ..............................................................................................................................87 Salts ...........................................................................................................................87 References .................................................................................................................88

Modeling of Wax Formation 90 Modeling of Wax Formation ...................................................................................................90

Vapor-Liquid-Wax Phase Equilibria .........................................................................90 Extended C7+ Characterization ..................................................................................92 Viscosity of Oil-Wax Suspensions ............................................................................93 Wax Inhibitors...........................................................................................................94 References .................................................................................................................94

Asphaltenes 96 Asphaltenes..............................................................................................................................96

Asphaltene Component Properties ............................................................................96 References .................................................................................................................97

H2S Simulations 98 H2S Simulations.......................................................................................................................98

Water Phase Properties 99 Water Phase Properties ............................................................................................................99

Properties of Pure Water ...........................................................................................99 Properties of Aqueous Mixture................................................................................ 108 Viscosity of water-oil Emulsions ............................................................................ 111 References ............................................................................................................... 112

Modeling of Scale Formation 114 Modeling of Scale Formation ................................................................................................ 114

Thermodynamic equilibria ...................................................................................... 114 Amounts of CO2 and H2S in water .......................................................................... 118 Activity coefficients of the ions............................................................................... 118 Calculation procedure.............................................................................................. 125 References ............................................................................................................... 126

Wax Deposition Module 128 Modeling of wax deposition .................................................................................................. 128

Discretization of the Pipeline into Sections............................................................. 128 Energy balance ........................................................................................................ 129 Overall heat transfer coefficient .............................................................................. 130 Inside film heat transfer coefficient......................................................................... 130 Outside Film Heat Transfer Coefficient .................................................................. 132 Pressure drop models............................................................................................... 132 Handling of an aqueous phase in the model ............................................................ 132 Wax deposition........................................................................................................133 Boost pressure ......................................................................................................... 134 Porosity.................................................................................................................... 134 Boundary conditions................................................................................................ 134 Mass Sources........................................................................................................... 135 References ............................................................................................................... 135

Clean for Mud 137 Clean for Mud........................................................................................................................ 137

Cleaning Procedure ................................................................................................. 137 Cleaning with Regression to PVT Data................................................................... 138

Introduction

Introduction

This document describes the calculation procedures used in PVTsim. When installing PVTsim the Method Documentation is copied to the installation directory as a PDF document (pvtdoc.pdf). It may further be accessed from the <Help> menu in PVTsim. The <Help> menu also gives access to a Users Manual. This is during installation copied to the PVTsim installation directory as the PDF document pvthelp.pdf.

Pure Component Database

Pure Component Database The Pure Component Database contains approximately 100 different pure components and pseudo-components. The different component classes are described in the following.

Component Classes PVTsim distinguishes between the following 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 2,3-dim-C4 2,3-Dimethyl-butane C6H14 2-m-C5 2-Methyl-pentane C6H14 3-m-C5 3-Methyl-pentane C6H14 nC6 Normal-hexane C6H14 C6 Hexane -------- m-c-C5 Methyl-cyclo-pentane C6H12 Benzene Benzene C6H6 Napht Naphthalene C10H8 c-C6 Cyclo-hexane C6H12 223-tm-C4 2,2,3-Trimethyl-butane C7H16 3,3-dim-C5 3,3-Dimethyl-butane C7H16 2-m-C6 2-Methyl-hexane C7H16 c13-dm-cC5 Cis-1,3-Dimethyl-cyclo-pentane C7H14 t13-dm-cC5 Trans-1,3-Dimethyl-cyclo-pentane C7H14 3-m-C6 3-Methyl-hexane C7H16 t12-dm-cC5 Trans-1,2-Dimethyl-cyclo-pentane C7H14 nC7 Normal-heptane C7H16 m-c-C6 Methyl-cyclo-hexane C7H14 et-c-C5 Ethyl-cyclo-pentane C7H14 113-tr-cC5 1,1,3-Trimethyl-cyclo-pentane C8H16

Toluene Toluene C7H8 2-m-C7 2-Methyl-heptane C8H18 c-C7 Cyclo-heptane C7H14 3-m-C7 3-Methyl-heptane C8H18 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 the carbon number fractions from a C21 fraction to a C100 fraction. 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. The only properties given in the database are the molecular weight, and

, and the molecular weight will usually also have to be modified by the user. Other component properties must be entered manually.

Component Properties For each component the database holds the following 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 either the SRK or PR equations

• Melting point depression ( ) • 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)

• Enthalpy of melting ( ) • 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)

• and 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, , , , and molecular weight are required input for all components to perform simulations. Whether the remaining component properties are needed or not depends on the simulation to be performed.

The below table shows what component properties are needed to calculate a given property for gas and oil phases.

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 these as possible. The following properties must be entered • Component type • Name • Critical temperature (Tc) • Critical pressure (Pc) • Acentric factor ( )

• and • 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, , , , 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 Section where described

needed for estimation 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 pseudo-components

SRK with Volume Correction or PR with Volume Correction

Melting point depression

( )

Only for pseudo-components. Viscosity data for an uninhibited/inhibited fluid.

Ideal gas absolute enthalpy at 273.15 K/0°C/32°F (Href)

Molecular weight Compositional variation due to gravity

Ideal gas Cp coefficients Not estimated for defined components. Liquid density for pseudo-components

Enthalpy

Melting temperature (Tf) Irrelevant for defined components. None for pseudo-components

Extended C7+ Characterization

Enthalpy of melting ( ) Irrelevant for defined components. None for pseudo-components

Extended C7+ Characterization

PNA distribution Irrelevant for defined components. Liquid density for pseudo-components

Estimation of PNA Distribution

Wax fraction Irrelevant for defined components. None for pseudo-components.

Extended C7+ Characterization

Asphaltene fraction Irrelevant for defined components. Liquid density for pseudo-components

Asphaltenes

Parachor Not estimated for defined components. Liquid density for pseudo-components

Gas/Oil interfacial tension.

Hydrate former or not Not estimated - Hydrate Langmuir constants Not estimated - Number of ions in aqueous solution (only for salts)

Not estimated -

Number of crystal water molecules per salt molecule (only for salts)

Not estimated -

Pc of wax forming fraction Irrelevant for defined components. None for pseudo-components

Extended C7+ Characterization

Composition Handling

Composition Handling PVTsim distinguishes between the following 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. For this type of compositions the required input is mol%’s of all components and molecular weights and densities of all C7+ components (carbon number fractions). 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 and no extrapolation is performed. Gas mixtures with only a marginal content of C7+ components are usually classified as compositions with No plus fraction. In the simulations characterized compositions are used. 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 When considering fluid composition input a distinction is made between the light components up to C6 which are always identified by gas chromatographic analysis, and the components heavier than C6 which may be analyzed in different ways. Generally two types of fluid analyses are used for the C7+ components, both of which must deal with the fact that the number of isomeric components for the larger molecules makes a detailed analysis of all chemical species impossible. These are true boiling point analyses (TBP) and a gas chromatographic (GC) analyses.

GC analysis The GC analysis in various modifications is often used as it is relatively cheap, very fast, and because only a very small sample volume is required. Furthermore the GC analysis is much more detailed than a TBP analysis. A GC analysis on the other hand suffers from the problem that

heavy ends may be lost in the analysis, especially heavy aromatics such as 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 C6. These are instead estimated from correlations. This in particular is a problem for the plus fraction residue properties, which are essential for a proper representation of the heaviest constituents of the fluid. To remedy this problem a GC composition may be entered into PVTsim as follows. Often a set of residue properties is available say for the C7+ fraction, while the measured GC composition often extends to e.g. C30. In this case one may enter the mol%'s to C30 together with the M and density of the total C7+ fraction leaving the M and density fields blank for the higher C8 - C30 fractions. With this input, the program will be extrapolating from the C7+ fraction properties, while honoring the reported composition for the fractions up to C30 under the mass balance constraints. If no information is available on the residue properties, one may as an alternative lump back the composition to C7+ and estimate properties from there, which will often provide equally accurate simulation results as with the detailed GC composition.

TBP Distillation The TBP distillation requires a larger sample volume, typically 50 – 200 cc and is more time consuming. 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, which distil off between the boiling point of nC6 + 0.5°C/0.9°F, and the boiling point of nC7 + 0.5°C/0.9°F, regardless of how many carbon atoms these components contain. Each of the fractions distilled off is weighed and the molecular weights and densities are determined experimentally. The density and molecular weight in combination provide valuable information to the characterization procedure on the PNA distribution. Aromatic components for instance have a higher density and a lower Mw than paraffinic components. The residue from the distillation is also analyzed for amount, M and density. These properties are important in the characterization procedure. 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. While the default values in PVTsim are generally considered to be reasonably accurate, they can never be expected to match the characteristics on any given crude exactly, and thus experimental values are much to be preferred.

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 fraction because it has a boiling point higher than nC6 + 0.5°C/0.9°F. 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 are split from the pseudo-component it ended up in, and the properties adjusted

accordingly. This procedure ensures that discrepancies between different component classes are avoided in the characterization.

Fluid handling operations Quite often it becomes practical to mix two or more fluids and continue simulations with the mixed composition. In PVTsim there are a number of facilities available for this purpose. These are ‘Mixing’, ‘Weaving’, ‘Recombination’ and ‘Characterization to the same Pseudo-components’.

Mixing PVTsim may be used to mix or weave from 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 on the level where the fluid has been characterized but not yet lumped. Each set of discrete fractions is mixed and the properties of the mixed fraction averaged on a mass basis. Afterwards the mixed fluid is lumped to the specified number of components. If the total number of C7+ components in the fluids to be mixed exceeds the defaults number of pseudo-components (12), pseudo-components of approximately the same weight are lumped to get down to the desired number of pseudo-components in the mixed fluid.

Weaving Weaving will maintain the pseudo-components of the individual compositions and can only be performed for characterized compositions. When weaving two fluids, 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 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 mols corresponding to the input volumes and simply mixes the two fluids based on this. When the saturation pressure is specified, the recombination is iterative (i.e. how much of the gas should be added to yield this saturation pressure).

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 a very powerful tool, and can be applied 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. In this case the data sets will also affect the characterization of the fluids for which no PVT data exist. Characterization to same pseudo-components is described in more detail in the section of Characterization of Heavy Hydrocarbons.

Flash Algorithms

Flash Algorithms The flash algorithms of PVTsim are the backbone of all equilibrium calculations performed in the various simulation options. The terminology behind the different flash options are described in the following.

PT Flash The input to a PT flash calculation consists of • Molar composition of feed (z) • Pressure (P) and temperature (T) A flash results consists of • Number of phases • Amounts and molar compositions of each phase • Compressibility factor (Z) or density of each phase

Flash Algorithms PVTsim makes use of the following flash algorithms • PT non aqueous (Gas and oil)

• PT aqueous (Gas, oil, and aqueous)

• PT multi phase (Gas, max. two oils, and aqueous)

• PH (Gas, oil, and aqueous)

• PS (Gas, oil, and aqueous)

• VT (Gas, oil, and aqueous)

• UV (Gas, oil, and aqueous)

• HS (Gas, oil, and aqueous) Specific PT flash options considering the appropriate solid phases are used in the hydrate, wax, and asphaltene options. A flash calculation assumes thermodynamic equilibrium. The thermodynamic models available in PVTsim are the Soave-Redlich-Kwong (SRK) equation of state, the Peng-Robinson (PR) equation of state, and the Peng-Robinson 78 (PR78) equation of state. These equations are presented in Equation of State section. To apply an equation of state, a number of properties are needed for each component contained in the actual mixture. These are established through a C7+-characterization as outlined in the section on Characterization of Heavy Hydrocarbons. 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 mols present of component i and 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

IIi

Ii µµ =

The final number of phases and the phase compositions are determined as those with the lowest total Gibbs energy. The calculation of 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 mols 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 - , n2 - , n3 - …., nN - ) and ( , , ……, ) where is small. The Gibbs energy of phase I may be approximated by a Taylor series expansion truncated after the first order term

∑=

∂∂

−=N

1i ni

ii01 n

GεGG

The Gibbs energy of the second phase is found to be

GII = G ( , , ,……, ) The change in Gibbs energy due to the phase split is hence

( ) ∑∑==

−=−=−+=N

1i0iIIii0iIIi

N

1ii0III ))(µ)((µyε))(µ(µεGGG∆G

where , 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 is greater

than zero for all possible trial compositions of phase II. The chemical potential, , may be expressed in terms of the fugacity, fi, as follows

)P1nlnzRT(1nµf1nRTµµ ii0ii

0ii +++=+= ϕ

where is a standard state chemical potential, a fugacity coefficient, z a mol 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 may then be rewritten to

∑=

−−+=N

1i0iiIIiii ))1n(zln)1n(y(1ny

εRT∆G ϕϕ

where zi is the mol fraction of component i in the total mixture. The stability criterion can now be expressed in terms of mol fractions and fugacity coefficients. Only one phase exists if

∑=

>−−+N

1i0iiIIiii 0))ln(zln)ln(y(lny ϕϕ

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 =−−+ ϕϕ

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( )0 – 1n( )II

PVTsim uses the following initial estimate for the ratio Ki between the mol fraction of component i in the vapor phase and in the liquid phase

−= )

TT(15.42exp

PPK 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, ( )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

Materiel balance equations

( ) ( N1,2,3,...,i,zxβ1βy iii )==−+ Equilibrium equations

( )N1,2,3,...,i,xy Lii

Vii == ϕϕ

Summation of mol fractions

( )N1,2,3,...,i,0)x(yN

1iii ==−∑

=

In these equations xi, yi and zi are mol fractions in the liquid phase, the vapor phase and the total

mixture, respectively. is the molar fraction of the vapor phase. and 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,K Vi

Li

i ==ϕϕ

The above (2N+1) equations may then be reduced to the following (N+1) equations

( )N1,2,3,...,i,lnlnKln V

i

Li

i ==ϕϕ

∑ ∑=

=−+−=−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

mi

mi −+= ∑

−

=

mβ is the molar fraction of phase m. equals the ratio of mol fractions of component i in phase m and phase J. The phase compositions may subsequently be found from

( )

( )N1,2,3,...,i,Hzy

J1,2,3,...,mN;1,2,3,...,i,HKzy

i

iJi

i

miim

i

==

===

where and are the mol fractions of component i in phase m and phase J, respectively.

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 an estimate has to be provided for the temperature. PVTsim assumes 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. The iteration procedure is described in Michelsen (1986). 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.

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 following phase identification criteria are used in PVTsim Liquid if

1. The pressure is lower than the critical pressure and the temperature lower than the bubble point temperature.

2. The pressure is above the critical pressure and the temperature lower than the critical temperature.

Gas if 1. The pressure is lower than the critical pressure and the temperature higher than the dew

point temperature.

2. 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 mol% total of the components water, hydrate inhibitors and salts is identified as an aqueous phase.

Components Handled by Flash Algorithms The non-aqueous PT-flash algorithm handles the following component classes • Other inorganic • Organic defined • Pseudo-components The PT aqueous and multiflash algorithms handle • Water • Hydrate inhibitors • Other inorganic • Organic defined • Pseudo-components • Salts The PH, PS, VT, UV, and HS flash algorithms handle • Water • Hydrate inhibitors • Other inorganic • Organic defined • Pseudo-components

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., “Multiphase Isenthalpic and Isentropic Flash Algorithms”, SEP Report 8616, Institut for Kemiteknik, The Technical University of Denmark, 1986

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 mol 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 Bar/4.93 atm/72.52 psi. An initial estimate of the equilibrium factors (Ki = yi/xi) is obtained from the following equation

−= )

TT5.42(1exp

PPK cici

i

This equation 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 mol fraction. The correct value of T is subsequently calculated by solving this equation in conjunction with

Vi

Li

i lnlnlnK

ϕϕ

=

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 Heideman (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 (2002).

Components handled by Phase Envelope Algorithm The algorithm handles the component 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 77385, SPE ATCE in San Antonio, Tx, September 29 – October 2, 2002. Michelsen, M.L., “Calculation of Phase Envelopes and Critical Points for Multicomponent Mixtures”, Fluid Phase Equilibria, 1980, 4, 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.

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).

SRK Equation The SRK equation takes the form

b)V(Va(T)

bVRTP

+−

−=

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()VP(( =

∂∂

=∂∂

The following relation is then obtained for parameter a of component i at the critical point

ci

2ci

2

aci PTRΩa =

and for parameter b

ci

cibi P

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). 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) = aci i(T) The parameter is by default obtained from the following expression

20.5

ci T

T1m1(T)α

−+=

where

2iii 0.176ω1.574ω0.480m −+=

It is seen that i(T) equals 1 at critical temperature at which ai(T) therefore becomes equal to aci.

is the acentric factor that is defined as follows (Pitzer, 1955)

1Plogω0.7T

Vapri10i

r−−=

=

where 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) r23

r3

2

r2r1 <−+−+−+=

1T,))T(1C(1(T)α r2

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

( )( )2cbVcVa

bVRTP

+++−

−=

The SRK molar volume, , and the Peneloux molar volume, V, are related as follows

cVV~

−=

The b parameter in the Peneloux equation is similarly related to the SRK b-parameter as follows

cb~b −= The parameter c can be regarded as a volume translation parameter, and it 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

( )RAc

c Z0.29441P

RT0.40768c' −=

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

( ) ( )bVbbVVa(T)

bVRTP

−++−

−=

where a(T) = ac (T)

c

2c

2

ac PTRΩa =

20.5

cTT1m1α(T)

−+=

c

cb P

TRb Ω=

where

aΩ = 0.45724

bΩ = 0.07780 The parameter m is for the PR equation found from m = 0.37464 + 1.54226 - 0.26992 2 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 + 0.01666 2) 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

( )( ) ( )( )bVcbb2cVcVa(T)

bVRTP

−+++++−

−=

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' RAc

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”.

Classical Mixing Rules The classical mixing rules for a, b and c are

∑∑= =

=N

1i

N

1jijji azza

∑=i

iibzb

∑=i

iiczc

where zi and zj are mol fractions, i and j component indices, and

( )ijjiij k1aaa −= The parameter kij is a binary interaction coefficient, which by default is zero for hydrocarbon-hydrocarbon interactions and different from zero for interactions between a hydrocarbon and a non-hydrocarbon and between unlike pairs of non-hydrocarbons. The greater part of the interaction coefficients in the PVTsim database has been found in Knapp et al. (1982). The interaction coefficients between hydrogen and other components have been found using the Tc-correlation of Tsonopoulos and Heidman (1986). It is given by

3

3

ij X1BXAk+

+=

1000T50;T1000

50TX cj

cj

cj ≤≤−

−=

where Tcj is the critical temperature of the component interacting with hydrogen. The values of A and B are given in Tsonopoulos and Heidman (1986) for the SRK and PR/PR78 equations of state

A B SRK 0.0067 0.63376

PR/PR78 0.0736 0.58984

Furthermore the option exists to calculate interaction parameters from critical volumes using the following equation (Chueh and Prausnitz, 1967)

n

31

cj31

ci

31

cj31

ciij

VV

VV21k

+

×−=

In PVTsim the exponent n is user specified with a default value of 1.

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, alcohols, glycols, ethers, and salts. The H&V a-parameter mixing rule takes the form

−

= ∑

=

∞N

1i

E

i

ii λ

Gbazba

where is specific for the selected equation of state. For SRK and PR the values for are

−+

=

=

1212ln

221λ:PR

ln2λ:SRK

EG∞ is the excess Gibbs energy at infinite pressure. is found using a modified NRTL mixing

rule

EG∞

( )

( )∑

∑

∑=

=

=∞

−

−=

N

liN

1kkikikk

N

1jjijijjji

i

E

ταexpzb

ταexpzbτz

RTG

where is a non-randomness parameter, i.e. a parameter for taking into account that the mol fraction of molecules of type i around a molecule of type j may deviate from the overall mol

fraction of molecules of type i in the mixture. When is zero, the mixture is completely random. The parameter is defined by the following expression

RTgg

τ iijiji

−=

where gji 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 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. should be chosen equal to zero. By further selecting the following expressions for the interaction energy parameters

λbag

i

iii −=

( ) ( ij0.5

jjiiji

jiji k1gg

bbbb

2g −+

−= )

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. The H&V mixing rule can for PVTsim version 13 and onwards be used both with the SRK and PR equations of state.

Phase Equilibrium Relations In case of two phases, each component will have equal fugacities, fi, in both phases

Li

Vi ff =

The following general thermodynamic relation exists for determination of the fugacity coefficient

( )( )∫∞−−∂∂−=

V

nV,T,ii lnZdVRT/VnP/1/RTlnj

ϕ

where ni is the number of mols of type i. Subsequently when the SRK equation is used, the following relation can be derived for the fugacity coefficient

( )( ) ( )[ ] ( )( ) ( )[ ]V/bVln/aaak1z2/bbRTa/bbVV/lnlnZ1Z/bblnN

1j

0.5jiijiiii +

−−+−+−−= ∑

=

ϕ

With two phases present, the phase compositions are related to the total composition as follows

( )1Kβ1zx

i

ii −+=

( )1Kβ1zKy

i

iii −+

=

where zi is the mol 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

Chueh, P.L., and Prausnitz, J.M., “Vapor-Liquid Equilibrium at High Pressures: Calculation of Partial Molar Volumes in Non-Polar Liquid Mixtures”, AIChE J 6, 13, 1967, pp. 1099. Dahl, S., “Phase Equilibria for Mixtures Containing Gases and Electrolytes”, Ph.D. thesis, Department of Chemical Engineering, Technical University of Denmark, 1991. 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., Milter, J. and Sørensen, H., “Cubic Equations of State Applied to HT/HP and Highly Aromatic Fluids”, SPE 77385, SPE ATCE in San Antonio, Tx, September 29-October 2, 2002. 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, 1197. Sørensen, H., Pedersen, K.S. and Christensen, P.L., "Modeling of Gas Solubility in Brine", Organic Geochemistry 33, 2002, pp. 35-642. Tsonopoulos, C., and Heidman, J.L., “High-Pressure Vapor-Liquid Equilibria with Cubic Equations of State”, Fluid Phase Equilibria 29, 1986, pp. 391-414.

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 of components exceeds what is practical in a usual phase equilibrium calculation. Some of the components must be grouped 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 + 0.5°C/0.9°F and nCn + 0.5°C/0.9°F. The C7 fraction for example consists of the components with a boiling point between those of nC6 + 0.5°C/0.9°F and nC7 + 0.5°C/0.9°F . For the C7+-fractions the density at standard conditions (1 atm/14.969 psi 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 separated in 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 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 + d2

d5 + d3/M + d4/M2 m = e1 + e2 M + e3 + e4 M2 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 103 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 (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 1/3 e 3.7377 x 101 5.4927 x 10-3 1.1793 x 10-2 -4.9305 x 10-6 -

Heavy oil characterization – SRK (Pedersen et al., 2002)

Sub-index/ Coefficient

1 2 3 4 5

c 3.04143 × 102 4.84052 ×10 7.10774 × 0-1 3.80073 × 103 - d 3.05081 -9.03352×10-1 2.33768×102 -1.27154× 104 0.25 e 4.96902×10-1 5.58442×10-3 1.01564×10-2 -5.24300×10-6

Heavy oil characterization - PR (Pedersen et al., 2002)

Sub-index/ Coefficient

1 2 3 4 5

c 3.26725×102 5.23447×10-1 5.77248×10-1 1.77498×103 - d 2.68058 -5.32274×10 2.04507×102 -9.45434×103 0.25 e 1.89723×10-1 7.42901×10-3 3.28795×10-2 -7.36151×10-6

M is in g/mol, is in g/cm3, Tc is in K and Pc in atm. The correlations are the same with and without volume correction.

Extrapolation of the Plus Fraction Characterization of the plus fraction consists in • Estimation of the molar distribution, i.e. mol fraction versus carbon number. • Estimation of the density distribution, i.e. the density versus carbon number. • Estimation of the boiling point distribution, i.e. boiling point 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 TBP-residue 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 lnzN A1 and B1 are determined from the measured mol 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. 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)

I12I1n

−+

=

I is a characterization factor, which is found from the following correlation

I = 0.3773 0.9182 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 mol%) 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 default number of C7+ components in PVTsim is 12. The component reduction is accomplished through a grouping or lumping. 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

ckMz

TMzT

∑

∑=

=

=L

Miii

L

Miciii

ckMz

PMzP

∑

∑=

=

=L

Miii

L

Miiii

ckMz

ωMzω

where zi is the mol 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. ab Grouping This represents a lumping scheme, which minimizes the variation in the equation of state parameters a and b within a pseudo-component (Lomeland and Harstad, 1994). The terms used in the following are further explained in the Equation of State section. The a-parameter may for a pure component i be written

−+=

ciiicii T

Tmm1aa

or

( )Ta1aa 2i1ii −= where

( )ici1i m1aa +=

( ) cii

i2i Tm1

ma+

=

The expression for the parameter a of an N component mixture may similarly be rewritten to

( ) ( )∑∑= =

−

+

+−=

N

1i

N

1jij2j2i1j1i

2j2i1j1i1j1iji k1aaaa

Taaaa

Taa

zzTa

For pseudo-component k comprising the carbon number fractions from Ln to Un the average a1 and a2 parameters are calculated by

( )2

nU

nLii

nU

nLi

nU

nLjij1j1iji

2lk

z

k1aazza

∑

∑ ∑ −=

=

= =

( ) ( )

2nU

nLii

nU

nLi

nU

nLjij1j2j2i1iji

21k2k

z

k1aaaazzaa2

∑

∑ ∑ −+=

=

= =

Similarly the average parameter b is found from

∑

∑=

=

=

nU

nLii

nU

nLiii

k

z

bzb

The sub-components of pseudo-component n is found by minimizing the function

∑ ∑

−+

−+

−=

= =

pcN

sLn

nU

nLi

2

i

ki

2

2i

2k2i

2

1i

lk1i

bbb

aaa

aaaS

by varying Ln and Un. Ls is lowest carbon number considered for grouping and Npc is the final number of pseudo-components. The parameters n, Tcn and Pcn are found by back-calculation using the following formulas

k

2k1k

a

bk

baa

ΩΩm =

( )2

2kk

kck am1

mT

+=

k

ckbck b

TRΩP =

where m is a second order polynomial in as defined in the Equation of State section. In case non-zero binary interaction coefficients are used for the hydrocarbon-hydrocarbon interactions, the binary interaction coefficient between pseudo-component n and m is determined from the formula

∑ ∑

∑ ∑

= =

= ==n

n

m

m

n

n

m

m

U

Li

U

Ljjimn

U

Li

U

Ljijjiji

nm

zzMM

kMMzzk

where the pseudo-component m comprises the carbon number fractions for Lm to Um and

and are the average molecular weights of pseudo-components m and n, respectively. For interactions with methane the following correction term is to be added to the binary interaction parameters calculated from the above formula

( )mn MM

mn MMC −

where

Npc is the number of pseudo-components.

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 component lumping. Not only C7+ components but also some of the defined components will usually have to be lumped. In subsequent process simulations it may be desirable to reestablish all the defined components and possibly also to increase the number of C7+ pseudo-components. This 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

ji

NFL

1j

jci

ji

uniqueci

zjWgt

TzjWgtT

( )

( )∑

∑

=

== NFL

1j

ji

NFL

1j

jci

ji

uniqueci

zjWgt

PzjWgtP

( )

( )∑

∑

=

== NFL

1j

ji

NFL

1j

jji

mixi

zjWgt

ωzjWgtω

NFL is the number of compositions to be characterized to the same pseudo-components, is the mol fraction of component i in composition number j, and Wgt(j) is the weight 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 mol fraction of

( )

( )∑

∑

=

== NFL

1j

NFL

1j

ji

uniquei

jWgt

zjWgtz

and a molecular weight of

( )

( )∑

∑=

=

=

NFl

1j

ji

ji

NFL

1j

ji

uniquei

zjWgt

MzjWgtM

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 mol fraction of this pseudo-component in the j’th composition will be

∑=

=L

Mi

ji

jk zz

References Katz, D.L. and Firoozabadi, A., ”Predicting Phase Behavior of Condensate/Crude-Oil Systems Using Methane Interaction Coefficients”, J. Pet. Technol. 20, 1998, pp. 1649-1655. 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 77385, SPE ATCE in San Antonio, Tx, September 29-October 2, 2002. 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.

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

residii HHzH

where N is the number of components, zi is the mol fraction of component i in the phase

considered and is the molar ideal gas enthalpy of component i.

∫=T

Tres

idpi

idi dTCH

Tref is a reference temperature (273.15 K (= 0°C/32°F) in PVTsim). is the molar ideal gas enthalpy of component i, which is calculated from a third degree polynomial in temperature

3i4,

2i3,i2,i1,

idpi 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 heavy hydrocarbons coefficients 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 1/3B=

TB is the normal boiling point in °R and SG the specific gravity, which is approximately equal to the liquid density in g/cm3. 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

TlnRTH 2res

∂∂

−=ϕ

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

residii SSzS

The ideal gas term at the temperature T is calculated from

∫ −−=T

Ti

ref

idpiid

i

ref

zlnRPPTlndT

TC

S

Pref is a reference pressure (1 atm/14.696 psi in PVTsim). is the molar ideal gas enthalpy of component i, which is calculated as outlined in the Enthalpy section. The residual term is calculated from

ϕlnRT

HSref

ref −=

Heat Capacity The heat capacity at constant pressure is calculated from

PP T

HC

∂∂

=

and the heat capacity at constant volume from

VPPV T

PTVTCC

∂∂

∂∂

−=

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

TpHjT P

HC1

PTµ

∂∂

−=

∂∂

=

Velocity of sound The velocity of sound is derived as

PVV

P

Ssonic V

TTP

CC

MWV

VP

MWVu

∂∂

∂∂

=

∂∂

−=

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, 153. Reid, R.C., Prausnitz, J. M. and Sherwood, J.K., ”The Properties of Gases and Liquids”. McGraw-Hill, New-York 1977.

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). The idea behind the corresponding states principle is that the relation between the reduced viscosity

( )1/22/3c

-1/6cr MPT/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

coo

1/2

o

x

2/3

co

cx

1/6

co

cxrrx T

TT,P

PPηMM

PP

TTT,Pη

where o refers to the reference component. In PVTsim methane is used as reference component. 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))

( ) ( ) ( ) ( ) ( )oooomix1/2

omix2/3

comixc,1/6

comixc,mix T,Pη)/α(α/MM/PP/TTTP,η −=

where

mixmixc,

ocoo αT

αTTP =

The critical temperature and the critical molar volume for unlike pairs of molecules (i and j) are found using the below formulas

mixmixc,

ocoo αT

αTTT =

cjcicij TTT =

( )31/3cj

1/3cicij VV

81V +=

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

81Vcij = constant

The critical temperature of a mixture is found from

∑∑

∑∑

= =

= == N

1i

N

1jcijji

N

1i

N

1jcijcijji

mixc,

Vzz

VTzzT

where zi and zj are mol 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

1/2cjci

31/3

cj

cj

1/3

ci

ciji

mixc,

PT

PTzz

TTPT

PTzz

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

1/31/3

cj

cj1/3

ci

ciji

N

1i

1/2cjci

3N

1j

1/3

cj

cj1/3

ci

ciji

mixc,

PT

PTzz

TTPT

PTzz8

P

+

+

=

∑∑

∑∑

= =

= =

The applied mixing rules are those recommended by Mo and Gubbins (1976). The mixture molecular weight is found as follows

( ) n

2.303n

2.303w

4mix MMM101.304M +−×= −

where and are the weight average and number average molecular weights, respectively.

∑ ∑== =

N

1i

N

1jii

2iiw Mz/MzM

∑==

N

1iiin MzM

The constants 1.304×10-4 and 2.303 in the above equation are the main tuning parameters applied when performing regression on data with the CSP model. The viscosity correction factors referred to in the regression output are multiplication factors for these coefficients, i.e. with a default value of 1. The first viscosity correction factor is multiplied onto the coefficient 1.304×10-4 while the second viscosity correction factor is multiplied onto the exponent 2.303. This is further described in the section about Regression to Experimental PVT Data. The parameter of the mixture is found from the expression

0.5173mix

1.847r

3mix Mρ107.3781.000α −×+=

The reduced density is defined as

co

mixc,

co

mixc,

coo

r ρPPP,

TTTρ

ρ

=

The reference viscosity correlation is based on the methane viscosity model of Hanley et al. (1975)

( ) ( ) ( ) ( )Tρ,∆η'ρTηTηTρ,η' 1o ++=

where are functions defined in the above reference. The methane density is found using the B R-equation in the form suggested by McCarty (1974). In the dense liquid region

ainly governed by the term

W

this expression is m

( ) ( )

−

+++

++= 1.0

Tj

Tjjθρ

TjjρexpT/jjexpTρ,∆η' 2

765

0.52/33

20.1

41

In the work of Hanley (1975) the coefficients j1 – j7 have the following values (viscosities in µP) 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

( ) cc /ρρρθ −= The presented viscosity calculation method presents some problems when methane is in a solid form at its reference state. This is the case when the reduced temperature is below approximately

0.4. This problem is overcome by replacing by the following term (Pedersen and Fredenslund (1987))

( ) ( )

−

+++

++= 1.0

Tk

Tkkθρ

Tkkρexp/TkkexpTρ,'∆η' 2

765

0.52/33

20.1

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,'∆η'FTρ,∆η'FρTηTηTρ,η 2110 ++++=

21HTANF1

+=

2HTAN1F2

−=

( ) ( )( ) ( )∆Texp∆Texp

∆Texp∆TexpHTAN−+−−

=

with

FTT∆T −= where TF is the freezing point of methane. 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, ( )[ ] 4

r53r4

2r3r21

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 mol fraction of component i.

The critical density, , is calculated from the critical volume

( ) ( )1N

1icii

1cc VzVρ

−

=

−

== ∑

For C7+ fractions the critical volume in ft3/lb mol 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)

[ ] 1c

c

cc 7.0)0.26(α3.72

PRTV −−+=

−+=

c

b

cc

b

c

TT1

lnPTT

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 ω−=

The dilute gas mixture viscosity is determined from (Herning and Zippener, 1936)

∑

∑

=

== N

1i

1/2ii

N

1i

1/2i

*ii

*

MWz

MWηzη

The following expressions (Stiel and Thodos, 1961) are used for the dilute gas viscosity of the

individual components,

1.5T,Tξ11034η ri

0.94ri

i

5*i <×= −

( ) 1.5T,1.67T4.58ξ11017.78η ri

5/8ri

5*i >−×= −

where is given by

2/3ci

1/2i

1/6ci

i PMTξ =

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.

Thermal Conductivity The thermal conductivity is defined as the proportionality constant, , in the following relation (Fourier’s law)

−=dXdTλq

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=λ 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

( ) ( ) [ ]1/22/3c

1/6cr MPT/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

( ) ( ) ( ) ( ) ( )ooo1/2

ox2/3

cocx1/6

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

1/2omix

2/3comixc,

1/6comixc,mix

+−

×= −−

where

=

=

oco

mixmixc,o

coco

mixmixc,o αP

αPP/Pand

αTαT

T/T

The mixture molecular weight Mmix is found from Chapman-Enskog theory as described by Mo and Gubbins (1976)

( ) ( )[ ] ( ) ( ) ][ ] 4/3mixc,

1/3mixc,

221/3cjcj

1/3cici

i j

1/4cjci

1/2jijimix PT/PT/PT/T/T1/M1/Mzz

161M −

−

+∑∑ +=

where z are mol 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

r2rrr

ridpiint

ρ0.029725ρ0.030182ρ0.0534321ρf

/Mρf2.5RC1.18653ηλ

−−+=

−=

is the gas viscosity at the actual temperature and a pressure of 1 atm, 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))

1.086i

2.043rii Mρ0.00060041α +=

where

( ) cocicocicoori /ρ/PPP,T/TTρρ = αmix is found using the mixing rule

( )∑∑= =

=N

1i

N

1j

0.5jijimix ααzzα

which ensures that components having small -values, i.e. small molecules, are attributed more importance than those having larger -values. The smaller molecules are more mobile than the larger ones. They thereby contribute relatively more to the transfer of energy than 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,∆λTρ,∆λ'ρTλTλTρ,λ c1o +++=

In the dense liquid region the major contribution to this expression comes from , which

has the same functional form as the expression for 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,∆λTρ,'∆λ'FTρ,∆λ'FρTλTρ,λ c210 +++=

F1 and F2 are defined in the viscosity section. The following expression is used for

( ) ( )

−

+++

++= 1.0

Tl

Tllθρ

Tllρexp/TllexpTρ,'∆λ'

2

765

0.52/33

20.1

41

where l1= -8.55109 l2= 12.5539 l3= -1020.85 l4= 238.394 l5= 1.31563 l6= -72.5759 l7= 1411.60

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 are the molar densities in mol/cm3 (the density divided by the molecular weight) of the oil and gas phases, respectively and xi and yi are the mol 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, p. 871. Hanley, H.J.M., McCarty, R.D. and Haynes, W.M., ”Equation for the Viscosity and Thermal Conductivity of the Individual Gases”, Cryogenics 15, 1975, 413. Hanley, H.J.M., ”Prediction of the Viscosity and Thermal Conductivity Coefficients of Mixtures”, Cryogenics 16, 1976, p. 643. 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. 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, 276. Mo, K.C. and Gubbins, K.E., ”Conformal Solution Theory for Viscosity and Thermal Conductivity of Mixtures”, Mol. Phys. 31, 1976, 825. Pedersen, K.S., Fredenslund, Aa., Christensen, P.L. and Thomassen, P., ”Viscosity of Crude Oils”, Chem. Eng. Sci. 39, 1984, 1011. 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, 182. Reid, R. C. and Sherwood, T. K., "The Properties of Gases and Liquids", 2nd ed. Chap 2, McGraw-Hill, New York, 1966. Reidel L., Chem. Ing. Tech., 26, 1954, 83

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.

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). PVT experiments are carried out with reference to standard conditions that may be specified in PVTsim. Default values are default 1 atm/14.696 psi 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 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 - V is the actual volume and Vb is bubble point or saturation point volume. Compressibility (only for pressures above the saturation point)

To P

VV1c

∂∂

−=

Y factor (only for pressures above the saturation point)

−

−=

1VVP

PPY

sat

t

sat

Vt is the total gas and liquid volume.

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 Depletion 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 mol% 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. Separator Test with Recirculation PVTsim has an option for simulating a special two stage separator set-up. The gas stream from the first separator is separated in a second separator. The liquid stream from the second separator is mixed into the feed to the first separator. The product streams are the liquid stream from the first separator and the gas stream from the second separator.

Viscosity Experiment A viscosity experiment is performed at the reservoir temperature. The pressure is reduced in steps as in a differential depletion experiment. At each step the phase 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 also 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

Mol% Cumulative mol% 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).

References 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, KS., Fredenslund Aa. and Thomassen, P., ”Properties of Oils and Natural Gases”, Gulf Publishing Company, Houston, 1989.

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 mol fractions of the lighter components decrease, whereas the mol 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 options of PVTsim consider - Isothermal reservoirs - Reservoirs with a vertical temperature gradient. In the first case temperature is assumed constant over the entire fluid coloumn and the compositional variations with depth are assumed only to originate from the effect of gravitational forces. In the second case temperature varies over the fluid coloumn and compositional variations with depth are affected both by the temperature gradient and by the gravitational forces.

Isothermal case For an isothermal system the chemical potentials, , of component i located in height h and in height h0 are related as follows

( ) ( ) ( )0w,i

0 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

0w,ih

ih

i

o −=−

The fugacity of component i is related to the fugacity coefficient of component i as

Pzf iii ϕ= which gives the following relation between the fugacity coefficients of component i in height h and in height h0

( ) ( ) ( )RT

hhgMPzlnPzln

0iwhh

ihi

hhi

hi

000 −=− ϕϕ

This equation is valid for any component i. For a system with N components there are N such equations. The mol fractions of the components must sum to 1.0 giving one additional equation

∑=

=N

1ii 1z

If the pressure and the composition are known in the reference height h0,

there are N + 1 variables for a given height h, namely 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

RTPclnln i

PENi,SRKi, =− ϕϕ

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.

Systems 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 0.02°C/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, 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. The model that has been choosen in PVTsim (Pedersen and Lindeloff, 2003) for describing the non-isothermal case was first proposed by Haase (1971). A number of models have been proposed to describe this problem, but the Haase model is attractive because it can be derived from first principles. The approach can be summarized as follows

N1,i;T∆T

MH~

MH~M)hg(hM)PzRTln()PzRTln(

i

ii

0i

h0h0i

h0i

hhi

hi =

−−−=− ϕϕ

Relative to the isothermal expression by Schulte, an additional term including the effect of the temperature gradient has been added. The term furthermore contains average molecular weight M, component molecular weight M

∆Ti and partial molar enthalpies H and Hi.

It follows that 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 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 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

ig273.15K

ig273.15K

igresig273.15K

PVTsimabs H)H(HHHHH +−+=+= A simultaneous parameter fit has been carried out to thermal diffusion data for the mixtures C1-C3 and C1-nC4. The enthalpy of C1 as ideal gas at 273.15 K/0°C/32°F was assumed to be zero. The data were fitted using

iRM00.1

H :N

ig273.15

2 =

iRM1.70

H :CO

ig273.15

2 =

iRM.933

H :C

ig273.15

2 =

iRM15.8

H :C

ig273.15

3 =

iRM7.07

H :C

ig273.15

4 =

iRM.337

H :C

ig273.15

5 =

iRM.448

H :C

ig273.15

6 =

For all other components the reference ideal gas enthalpy is assumed to be

iM50Hig273.15 =

When tuning to experimental data is performed, two multiplication factors are used, one for the ideal gas reference of the defined components and a second one for the ideal gas reference enthalpy of the C7+ components.

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, to be presented at the SPE ATCE in Denver, 5-8 October, 2003. Rutherford, W.M. and Roof, J.G., "Thermal diffusion in methane n-butane mixtures in the critical region", J. Phys. Chem. 63, 1959, pp. 1506-1511.

Regression to Experimental Data

Regression to Experimental Data PVTsim is basically a predictive program. 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 required or it is desirable with a heavy lumping, 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

Saturation Point

*) x x x x x x 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 critical point.

Gas condensate mixtures

Sat. points CME CVD Separator Viscosity Saturation Point

*) x x x x 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 Mol% 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

wr

OBJ

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

exp

calcexpj OBS

OBSOBSr

−=

where OBS stands for the observed value and the sub-indices exp and calc stand for experimental and calculated, respectively. For liquid dropout curves from a constant mass expansion and constant volume depletion experiment, a constant is added to all OBS-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

2jw

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 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) 1. Coefficient c2 in Tc correlation. 2. Coefficient d2 in Pc correlation. 3. Peneloux volume shift parameter. 4. Coefficient c3 in Tc correlation. 5. Coefficient d3 in Pc correlation. 6. Coefficient e2 in m correlation. 7. Coefficient e3 in m correlation. 8. Coefficient c4 in Tc correlation. 9. Coefficient d4 in Pc correlation. 10. Coefficient e4 in m correlation. In each iteration the parameters c1, d1 and e1 are recalculated to give the same Tc, Pc and m of a component with a molecular weight of 94 and a density of 0.745 g/cm3 as is obtained with the standard coefficients. This is done to ensure that Tc, Pc and m of the lower C7+ fractions are assigned properties, which are physically meaningful. The user therefore has no control of the parameters c1, d1 and e1 in the regression input menu. The user may modify the default selection of regression parameters, but the number of regression parameters must not exceed the number of experimental data points. Regression to Viscosity Data The regression parameters depend on which viscosity correlation is used. With the corresponding states model the assumed mixture molecular weight is found from the following equation

( ) n,WVISC2

n,WVISC2

w,Wmixw, MMMVISC1M +−= With no regression VISC1 = 1.304 x 10-4 and VISC2 = 2.303. During a regression to viscosity data, VISC1 and VISC2 are multiplied by two viscosity correction factors to give the best possible agreement with the experimental viscosity data. The optimum viscosity correction factors may be viewed in the Char Options menu accessed from the composition input menu. 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 - ω- VPEN (volume shift parameter) - Vc

- aΩ

- bΩ- kij (binary interaction parameter) The mentioned properties are all defined in the Equation of State section. A total of up to 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 has been characterized to the same set of pseudo-components. This feature can be very powerful, for instance when data are only available for some fluid samples. 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.

Minimum Miscibility Pressure Calculations

Minimum Miscibility Pressure Calculations Injection of gas into oil fields is commonly used to enhance the recovery from the field. 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 gas 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. With a vaporizing drive, miscibility is achieved at the oil/gas front. At the injection well, both gas and liquid are present at equilibrium. Compared to the injection gas, the gas phase that has been in contact with oil contains more intermediate molecular weight components, extracted from the oil phase. The gas phase has a higher mobility than the oil phase. Hence, gas which has taken up intermediate molecular weight components will move forwards and contact original reservoir oil. With the increasing content of intermediate molecular weight components in the gas phase, miscibility may be achieved in some distance from the injection well. In case of a condensing drive, the original reservoir fluid and the injection gas are not miscible, but miscibility may be achieved later on at the injection well. The reservoir fluid takes up intermediate molecular weight components from the injection gas. Since the liquid phase has a lower mobility than the gas phase, liquid which has taken up intermediate molecular weight components will be contacted by injection gas. Further transfer of intermediate molecular weight components from the gas to the liquid phase may then occur. After some time, miscibility may be obtained between the injection gas and the liquid phase at the injection well.

Minimum Miscibility Pressure Calculations The degree of miscibility between a reservoir oil and an injection gas is often expressed in terms of the minimum miscibility pressure (MMP). The first contact minimum miscibility pressure

(FCMMP) is the lowest pressure at which the reservoir oil and the injection gas are miscible in all ratios. The multiple contact minimum miscibility pressure (MCMMP or just MMP) is the lowest pressure at which the oil and the gas phases resulting from a multi-contact process (vaporizing or condensing) between a reservoir oil and an injection gas are miscible in all ratios. The MMP module of PVTsim permits calculation of FCMMP as well as MCMMP. The FCMMP may be calculated by tracking the saturation pressure as a function of oil/gas mixing ratio. The highest saturation point pressure located during the tracking procedure equals FCMMP. Two different procedures are used in PVTsim for calculating vaporizing and condensing MCMMP's. One is based on an extension of the procedures behind a ternary diagram (e.g. Stalkup, 1984) to multi-component mixtures. With the second procedure the miscibility process for a vaporizing gas drive is simulated as a continuous addition of reservoir oil at constant pressure and temperature to a cell initially containing injection gas. Any oil phase formed in the cell is continuously removed. The initial situation, where the cell contains pure injection gas corresponds to a situation just before the gas enters into the well. Starting the continuous addition of oil corresponds to the situation where the gas moves into the reservoir where it becomes saturated with oil. Further addition of oil to the now saturated gas causes an oil phase to be formed in the cell. The removal of this oil from the cell corresponds to the saturated gas moving in the reservoir leaving the oil behind. It is seen that the experiment corresponds to a cell moving with the gas front in the reservoir. Simulating a condensing gas drive, the cell initially contains reservoir oil. Injection gas is added and any formed gas is removed. A set of differential mass balances and algebraic equilibrium relations of the following form describe the vaporizing drive

( )

∑ ∑= =

==

==

=−=

N

1i

N

1iii

Vi

iLi

i

iii

vV;oO

N1,2,...,i;Vv

Oo

N1,2,...,i;dtofdv

ϕϕ

fi and oi are respectively the rates of addition to and removal of component i from the cell in mols/time. vi is the number of mols of component i in the cell. Since the feed rate and feed composition are constant, integration over an arbitrary time step gives

( ) ( ) N1,2,....,i;dto∆tftv∆ttv i∆tt

tiii =∫−+=+ + Using the trapezoidal rule for evaluation of the integral one gets

( ) ( ) ( ) ( ) N1,2,....,i;∆t2

∆ttoto∆tftv∆ttv iiii =

++−+=+

The compositions of the first equilibrium phases (t = 0) are known from the calculation of FCMMP. Adding mols of reservoir oil to a gas at equilibrium causes mols of oil to be formed, and the initial relative flow rate of oil, / , can be determined manipulating the equilibrium relations. Successive solution with a chosen time step simulates the miscibility process. If an additional time step at some point does not result in a two phase solution, miscibility has been established. If on the other hand the end result is constant compositions, miscibility cannot be obtained at the specified conditions. Simulations at different pressures determine MCMMP as the lowest pressure at which miscibility is obtained.

Combined drive mechanism While the above mentioned method for determination of the MCMMP describes the situation where miscibility develops at the flood front or at the injection point, a more sophisticated approach is required to properly account for the situation where miscibility develops between the flood front and the injection point. In this case a key tie line approach is applied. Tie-lines are lines that connect points in composition space, for instance between an oil composition and the composition of the gas that contacts it. When a tie-line becomes a point, the two phases are miscible (they have the same composition). Ignoring dispersion, it can be shown that there exists a series of key tie-lines which control the development of miscibility. These represent the path which the composition changes in the system theoretically will follow. In order to locate the MMP the algorithm tracks this path at increasing pressures until one of the key tie lines reduce to a point. Key tie lines are either connected by continuous variations or by shocks. The shocks represent the situations where oil is contacted by gas with which it is not in equilibrium, causing an abrupt change in composition. In a fully self-sharpening system all key tie lines are connected by shocks (In a fully self sharpening system the gas moves faster than the contacted oil anywhere in the displacement process). Even when the system is not fully self sharpening, the present method is considered to give a very good estimate of the true solution. It has been shown (Wang and Orr, 1998) that neighboring key tie lines are coplanar and hence have a point of intersection. This information is used to locate the next key tie line in the series. In order to determine co-planarity, the method by Jessen et al. (1998) is applied. In this approach, the coplanarity criterion

( ) ( ) (1)(2)(2)(1)* xβ1yβxα1yαz −+=−+= constrains the values of and to lie in the interval of [0;1]. This makes the algorithm quite robust. The succession of N-1 intersecting key tie lines can then be written as

( ) ( )2N1,...,j,1N1,...,i

0βyβ1xαyα1x j1j

ijjij

jij

1ji

−=−=

=−−−+− ++

where i are the component number and j the tie line number. The first and last tie lines in the sequence are specified by the tie-lines through the original oil and the injection gas respectively. Following the above nomenclature, these two compositions are specified as follows:

( )( )1N1,...,i

βyβ1xz

βyβ1xz

inj1Nj

iinj1ncj

iinji

oil1j

ioil1j

ioili

−=

+−=

+−=−=−=

==

The above mentioned equations are solved subject to the usual phase equilibrium and mass balance constraints

0yx

1N1,...,j,N1,...,i0yxN

1i

ji

ji

Vi

ji

Li

ji

=−

−===−

∑=

ϕϕ

All of this may be rearranged to a set of non-linear equations to be solved for the co-planarity parameters ( , ) and the phase compositions. A more thorough description may be found in Jessen et al. (1998)

References Jensen, F. and Michelsen, M.L., ”Calculation of First Contact and Multiple Contact Minimum Miscibility Pressure” In Situ 14, 1990, pp. 1-17. Jessen, K.; Michelsen, M.L. and Stenby, E.H.: ”Effective Algorithm for Calculation of Minimum Miscibility Pressure”, SPE Paper 50632, 1998. Stalkup, F.I., ”Miscible Displacement”, Monograph Volume 8, H.L. Doherty Series, Society of Petroleum Engineers, 1984. Wang, Y., and Orr, F.M., ”Calculation of Minimum Miscibility Pressure”, SPE paper 39683, 1998.

Unit Operations

Unit Operations

Compressor PVTsim supports two compressor options • Compressor with classical isentropic efficiency. • Compression following constant efficiency path (polytropic compression). The two options differ in the way the compression path is corrected for isentropic efficiency. The isentropic efficiency, , is defined as

dHdPVη =

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 =0 and that

VdPdH

s

=

meaning that the definition of the efficiency can be rewritten to

dH(dH)

dPdHdPdH

η Ss =

=

Neglecting variations in efficiency along the compression path, one arrives at the classical definition of the efficiency

∆HH)(η s∆

=

where ( )S is the enthalpy change of a compression following an isentropic path (=reversible adiabatic compression) and 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. 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 ( )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.

P

Pout

in

S

HP

P

P

P1

2

..

..

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 as illustrated by the dotted line in the figure. Each segment is simulated as an isentropic compression with the pressure increase . The corresponding enthalpy change ( )S is derived. The actual enthalpy change, =( )S/ , and 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 =(Pout-Pin)/n. 2. Perform a PT-flash for Tin, Pin. Flash determines Sin and Hin. 3. Perform a PS-flash for P2=Pin + , Sin. Flash determines isentropic outlet temperature (T2)S

and (H2)S from segment.

4. Determine 5. Determine T2 and S2 by PH-flash for P2,H2. 6. Perform a PS-flash for P3=P2+ , S2. Flash determines isentropic temperature (T3)S and

(H3)S

7. Determine 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

PV

VPY

1TV

VTX

∂∂

=

−

∂∂

=

Also given in the output is the HEAD defined as

fmgWORKHEAD =

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 compresor.

As can be seen from the above equation, the unit of HEAD is m or ft depending on selected unit. HEAD therefore expreses 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 which 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

( )s∆H∆Hη=

where is the enthalpy change by an isentropic expansion and 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 which is 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 which is 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

∆HP)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.

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 called guest molecules occupy either a fraction or all 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 Cavities

sI - Large cavities

sII - Small cavities

sII - Large 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). Neglecting these components (Danesh et al. (1993), Tohidi et al. (1996), Tohidi et al., (1997)) show that may lead to mal-predictions of the hydrate formation temperatures of heavy reservoir oil mixtures by more than 2°C/3.6°F. 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 molecules 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 1. pure water ( ) → empty hydrate lattice ( )

2. 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. Which state is energetically favorable depends on which state has 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 can be regarded as the stabilizing effect on the hydrate lattice caused by the adsorption of the 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

ljjjiKKiKi 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. The structure I and II hydrate parameters have been obtained from e.g. Munck et al. (1988) and Rasmussen and Pedersen (2002), and the parameters for structure H are from Madsen et al. (2000).

The term 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∆VdT

RT∆H

RT∆µd 2 +−=

where R is the gas constant and and 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

∆VdTRT∆H

RTP,T∆µ

RTPT,∆µ

RTµµ

where T0, P0 indicates a reference state at which is known. In this equation it has been assumed that is independent of pressure. The temperature dependence of the second term has been approximated by the average temperature

2TTT 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

∆VdTRT∆H

RTP,T∆µ

RTPT,∆µ

RTµµ

∆H is calculated from the difference, , in the molar heat capacities of the - and the -states

( ) ∫=T

T0 pdT∆CT∆H

The constants needed in the calculation of for the transition at a given temperature and pressure are taken from the work of Erickson (1983) and shown below.

Property Unit Structure I Structure II Structure H 0∆µ (liq) J/mol 1264 883 1187.33

0∆H (liq) J/mol -4858 -5201 -5162.43

0∆H (ice) J/mol 1151 808 846.57

0∆V (liq) cm3/mol 4.6 5.0 5.45

0∆V (ice) cm3/mol 3.0 3.4 3.85

p∆C (liq) J/mol/K -39.16 -39.16 -39.16

Using the procedure outlined above, the difference in chemical potentials 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

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

where is a correction term identical for all components. is found

from where w stands for water and refers to the empty hydrate lattice.

6. The hydrate compositions are calculated using the expression

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 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 ϕ= where , is the fugacity coefficient, xi the mol 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.

Hydrate Phases The fugacities of the various components in the hydrate phases are calculated as described by Michelsen (1991) Water:

( )

+

−=−

2

02

1

0i

βw

Hw v

θNlnvv

θ1Nlnvflnfln

Other Hydrate Formers:

( )( )θ1αθCNNf

k2k0

kHk −+

=

In these equations

βwf = fugacity of water in empty hydrate lattice

vi = number of cavities of type i N0 = number of empty lattice sites θ = ratio of free large lattice sites to total free lattice sites NK = content of component k per mol of water Cki = Langmuir constant α 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

273.15TP0.0390

T273.15ln4.710

T273.1512.064f ice

++

−

−−=

where P is the pressure in atm and T the temperature in K.

Salts The fugacities of a salt in pure solid form is assumed to be equal to the fugacity of the mentioned salt in saturated liquid solution in water. The solubilities in mol salts per mol water are found from the following expressions (with T in °C) Sodium Chloride, NaCl

T0.0001250.108waterMolsaltMol

+=

Calcium Chloride, CaCl2

T < 3.91°C : Solubility in wgt% =

3.91°C ≤ T < 30.35°C : Solubility in wgt% = 30.35°C ≤ T : Solubility in wgt% = 3.85 Potassium Chloride, KCl

waterMolsaltMol = 0.0674 + 0.000544 T

Sodium Formate, HCOONa

T < 50°C : waterMolsaltMol = 0.145 + 0.00355 T

T ≥ 50°C : waterMolsaltMol = 0.313

Potassium Formate, HCOOK

T < 20°C : waterMolsaltMol = 0.712 + 0.00705 T

T ≥ -20°C : waterMolsaltMol = 0.964 + 0.0174 T

Cesium Formate, HCOOCs

T < -6°C : waterMolsaltMol = 0.0248 + 0.00143 T

-6°C ≤ T < 50°C : waterMolsaltMol = 0.272 + 0.006 T

50oC ≤ T : waterMolsaltMol = 0.572

The remaining salts in the database are assigned the solubility of CaCl2, if they consist of 3 ions and the solubility of NaCl, if they consist of a number of ions different from 3.

References Danesh, A., Tohidi, B., Burgass, R.W., and Todd, A.C., "Benzene Can Form Gas Hydrates", Trans. IChemE, Vol. 71 (Part A), pp. 457-459, July, 1993.

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. 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, 1192-1193. Munck, J., Skjold-Jørgensen S. and Rasmussen, P., ”Computations of the Formation of Gas Hydrates”, Chem. Eng. Sci. 43, 1988, 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. Tohidi, B., Danesh, A., Burgass, R.W., and Todd, A.C., “Equilibrium Data and Thermodynamic Modelling of Cyclohexane Gas Hydrates”, Chem. Eng. Sci., Vol. 51, No. 1, pp. 159-163, 1996. Tohidi, B., Danesh, A., Todd, A.C., Burgass, R.W., Østergaard, K.K., "Equilibrium Data and Thermodynamic Modelling of Cyclopentane and Neopentane Hydrates", Fluid Phase Equilibria 138, pp. 241-250, 1997.

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, ,

of component i in the liquid phase equals the fugacity, , of component i in the solid phase

Si

Li 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 Li

Li

Li ϕ=

In this expression is the liquid phase mol fraction of component 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

oSi

Si

Si fxf =

where is the solid phase mol fraction of component i, and Six the solid standard state fugacity

of component i. The solid standard state fugacity is related to the liquid standard state fugacity as

( )( )

=

refoLi

refoSif

i PfPflnRT∆G

where 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 the following general thermodynamic relation is used

ST∆H∆G ∆−= where stands for change in enthalpy and for change in entropy. Neglecting any

differences between the liquid and solid phase heat capacities, may be expressed as

fi

fi

fi ST∆H∆G ∆+=

where is the enthalpy and 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

fi

fif

i T∆H∆S =

where is the melting temperature of component i. The following expression may now be derived for the solid standard state fugacity of component i

( ) ( )

−+

−

−=

RTPP∆V

TT1

RT∆HexpPff refi

fi

fi

refoLi

oSi

where is the difference between the solid and liquid phase molar volumes. Based on experimental observations of Templin (1956), the difference i 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 oLi

oLi ϕ=

where is the liquid phase fugacity coefficient of pure i at the system temperature and pressure. This leads to

( ) ( )

−+

−

−=

RTPP∆V

TT1

RT∆HexpPPf refi

fi

fi

refoLi

oSi ϕ

The following expression may now be derived for the solid phase fugacity of component i in a mixture

( ) ( )

−∆+

−

∆−=

RTPPV

TT

RTHPPxf refi

fi

fi

refoLi

Sii 1expϕ

is found using an equation of state on pure i at the temperature of the system and the reference pressure.

Extended C7+ Characterization To be able to perform wax calculations it is necessary to use an extended C7+ characterization procedure. A procedure must exist for splitting each C7+ pseudo-component into a potentially wax forming fraction and a fraction, which cannot enter into a wax phase. In addition

correlations are needed for estimating , and 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 mol fraction, , of the potentially wax forming part of pseudo-

component i, having a total mol fraction of

( )

−×+−=

C

Pi

Pii

itoti

si ρ

ρρMBA1zz

In this expression Mi is the molecular weight in g/mol 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 = 1.0744 B = 6.584 x 10-4 C = 0.1915

Piρ 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.

iPi Mln0.06750.3915ρ +=

For a (hypothetical) pseudo-component for which will be equal to meaning that all the components contained in that particular pseudo-component are able to enter into a wax

phase. In general will be lower than and the non-wax forming part of the pseudo-

component will have a mol fraction of . 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

Pi

cisci ρ

ρPP

=

Pci equals the critical pressure of pseudo-component i determined using the characterization

procedure described in the Characterization section. is the density of the wax forming fraction

of pseudo-component i and is the average density of pseudo-component. The critical pressure

of the non-wax forming fraction of pseudo-component i is found from the equation

( ) ( )Sci

Snoci

Snoi

Sci

2Si

2

Snoci

Snoi

ci PP2Frac

PFrac

PFrac

P1

−

−

−

−

++=

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

fii

fi

ii

fi

TM0.1426∆HM

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)

( )

++=

dydv

F

dydv

E1Dexpηηx

4wax

x

waxwaxliq

ϕϕϕ

where is the viscosity of the oil not considering solid wax and 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 = 37.82 E = 83.96 F = 8.559×106 Having performed regression of the viscosity model to experimental data, the modified model can be applied in the wax module by entering the multiplication factors for the coefficients D, E and F. For the original model these multiplication factors should all have values of 1. Viscosity values at different T and P can then be calculated by specifying the P,T grid of interest.

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 molecules 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”, 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.

Asphaltenes

Asphaltenes Asphaltene precipitation is in PVTsim considered as a that can be described by equilibrium thermodynamics. An equation of state is used for all phases including the asphaltene phase.

By default the aromatic fraction of the C50+ component is considered to be asphaltenes (Rydahl et al, 1997). The user may enter an experimental weight content of asphaltenes in the oil from a flash to standard conditions. This will change the cut point between asphaltenic and non-asphaltenic aromatics from C50 to a carbon number that will make the total asphaltene content agree with that measured. In asphaltene simulations pseudo-components containing asphaltenes are split into an asphaltene and non-asphaltene component. Having completed an asphaltene simulation, selecting ’Split Pseudos’ will maintain the split fluid.

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. Having tuned the model to a single data point, the model will in general correlate the remaining part of the asphaltene precipitation envelope quite well.

Asphaltene Component Properties The asphaltenes are by default assigned the following properties: Tc

A = 1398.5 K/1125.35°C/2057.63°F Pc

A = 14.95 Bar/14.75 atm/216.83 psi ωA = 1.274

The critical temperature Tcino-A of the non-asphaltene fraction (Fraci

no-A) of pseudo-component i is found from the relation

2A

ciAi

Aci

Anoci

Ai

Anoi

2Anoci

Anoici )T(FracTTFracFrac2)T(FracT +×+= −−−−

where Tci is the critical temperature of pseudo-component i before being split. The critical

pressure of the non-asphaltene forming fraction of pseudo-component i is found from the equation

( ) ( )Aci

Anoci

Anoi

Aci

2Ai

2

Anoci

Anoi

ci PP2Frac

PFrac

PFrac

P1

−

−

−

−

++=

while the acentric factor of the non-asphaltene forming fraction of pseudo-component i is found from

Ai

Ai

Anoi

Anoii 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.

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.

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

[ ]+−= HlogpH 10 and pK

[ ][ ]SHHHSlogpK

2101

+−

=

pK1 is calculated using considerations based on chemical reaction equilibria. This gives approximately the following temperature dependence pK1 = 7.2617 – 0.01086(T – 273.15) where T is the temperature in K. 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.

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 - Dynalog - Prosper/Mbal - Multiphase meter interface if license does not give access to multiflash options. The options treating water as pure water calculates the physical properties and transport properties using a separate thermodynamics instead of an EOS. In the OLGA2000 interface the water property routines are used in cases where no hydrate inhibitors are present. This is also an option in the Property Generator.

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)

( ) ( )[ ]TQRTT ,ln0 ρρρ ++Ψ=Ψ where Ψ = Helmholtz free energy (J/g) ρ = Density (g/cm3) τ = 1000/T where T is the temperature in K R = 0.46151 J/(g K) and

( ) ( ) TTCCTCTCCT ln542

3210 ++++=Ψ

( ) ( ) ( )

( ) ( ) ( ) ( )

++−−−+

++−=

∑ ∑

∑

= =

−−−

=

−−

7

2

8

1109

2

8

1,101,9,

j ijj

Eplobij

jac

li

Eliaij

AAeA

AAeATQ

ρρρττττ

ρρρρ ρ

where

aρ = 0.634 g/cm3

bρ = 1.0 g/cm3

aτ = 2.5 K-1

cτ = 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

∂∂

++=

∂Ψ∂

=

∂Ψ∂

=ττ ρ

ρρτ

ρρ

ρρ

ρ QQRPT

222 11000

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

( )ρτ

τ

ρ

PH +

∂Ψ∂

=

dTdTQQQR 0

011000 Ψ−Ψ+

∂∂

+

∂∂

++=τρ

ρτ

τρτ

dTdQQR

TS 0ln Ψ

−

∂∂

−+−=

∂Ψ∂

−=ρρ τ

τρρ

∂∂

∂∂

∂∂

−

∂∂

=

T

Tp P

TP

HTHC

ρρ

ρ

ρ

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, , at atmospheric pressure (=0.1 MN/m2) valid for 373.15 K/100°C/212°F < T < 973.15 K/700°C/1292°F

63211 10−×

+

−= bb

TTb

c

η

Region 1: Psat < P < 80 MN/m2 and 273.15 K/0°C/32°F < T < 573.15 K/300°C/572°F

( )

−

×

−×

−+= −

3

2541

6

/10110

aTTaa

TTa

PPa

ccc

sat

cρρη

where Tc and Pc are the critical temperature and pressure, respectively and 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

6321

61 101010 −×

−−−×= c

TTcc

ccρρηη

Region 3:

0.1 MN/m2 < P < 80 MN/m2 and 648.15 K/375°C/707°F < T < 1073.15 K/800°C/1472°F

61

2

2

3

36

1 1010 −×

+

+×=

cc

dddρρ

ρρ

ρρηη

Region 4: Otherwise

0192.010

1

Y

+=ηη

where Y = C5kX4 + C4kX3 + C3kX2 + C2kX + C1k

=

c

Xρρ

10log

The parameter k is equal to 1 when / <= 4/3.14 and equal to 2 when / > 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

( ) ( )15.273

15.2731log 27

3110 −

+−++= ∑= T

DTDDPj

jjsat

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

2321 10−×

−+

−+= S

PPPS

PPPS

c

sat

c

satλ

where

∑=

=

4

01

i

i

ci T

TaS

∑=

=

3

02

i

i

ci T

TbS

∑=

=

3

03

i

i

ci T

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

Y101 +=λλ where Y = C5kX4 + C4kX3 + C3kX2 + C2kX + C1k and

=

c

Xρρ

10log

k = 1 for <= 2.5

k = 2 for > 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

cicbound T

TePP

the thermal conductivity is found from the following expression

−−

−×

+

−−

+

−

= −− 1exp

1

19exp

133363512

34

33

4

32

7

31

445.1

cc

c

cc

c

c

TTd

PPdd

PPd

TTd

PPd

TTBd

TTA

λ

3231 aPPaA

c

+

=

26.3

32

63.1

31

1

+

=

c

c

PPb

PPb

B

33

32

5.1

31

cB

cPPc

C 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 P ≤ P and T ≤ 723.15 K/450°C/842°F bound

- 27.5 < P and P ≤ P and T < 723.15 K/450°C/842°F bound

- 22.5 < P and Pbound ≤ P and T < 698.15 K/425°C/797°F - 17.5 < P and Pbound ≤ P and T < 673.15 K/400°C/752°F the thermal conductivity is found from the following expression

where k = 100 The solution for is iterative.

When P,T is not in region 1, 3, 4 or 5 and in one of the following ranges

- 15 MN/m < P and T > 633.15 K/360°C/680°F 2

- 14 MN/m < P and T > 618.15 K/345°C/653°F 2

v = 1.76 x 10 + 5.87 x 10 t + 1.04 x 10 t – 4.51 x 10 x t -2 -5 -7

∑ ∑= =

−+=

8

0

8

04404

i i

ii

c

ii

c

kbcPPka

TT

Region 6:

+= 12.001.0 v

cρρλ

where

12 -11 3

Region 2: Otherwise

( ) 522.4

1425

1 10101482.210771.24198.051.103 −−

+×−++= x

txtt ρρλλ

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

−−

−=

cc TT

TT 1625.0118.235

256.1

τ

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)

3125.0

114/1

r

b

Ta ρσ ∆

=

where:

HCw ρρρ −=∆ 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)

∆ρ (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

ρρ - 1 =CS [7.65 x 10-3 – 1.09 · 10-7 P + CS (2.16 x 10-5 + 1.74 x 10-9 P)

-(1.07 x 10-5 – 3.24 x 10-10 P)T + (3.76 x 10-8 – 1.0 x 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)

( )( )5.14325.05.245.03 1044.31076.21035.11018.21087.11 ssssw

s CCTxTCC −−−−− ×−×−+×+×−=−ηη

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, methanol and/or mono ethylene glycol (MEG) 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 = A + B/T + CT + DT2 where is the viscosity in cP, T the temperature in K and A = -2.687 x 10 B = 1.150 x 103 C = 1.875 x 10-1 D = -5.211 x 10-4 Mono Ethylene Glycol (MEG) The viscosity of saturated liquid mono ethylene glycol can be calculated from the following equation (Alder, 1966) ln = A + B/T where is the viscosity in cP, T the temperature in K and A = -7.811 B = 3.143 x 103 Di-Ethylene Glycol (DEG) The viscosity of saturated liquid di-ethylene glycol is calculated from the following equation (van Velzen et al., 1972)

−=

010

11logTT

Bη

where is in cP, B = 1385.09, T0 = 495.54 and the temperature T is in K. Tri-Ethylene Glycol (TEG)

The viscosity of saturated liquid tri-ethylene glycol is calculated from the following equation (van Velzen et al., 1972)

−=

010

11logTT

Bη

where is in cP, B = 1453.34, T0 = 523.83 and the temperature T is in K. 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 Antoine equation

CTBAPsat

+−=ln

where the vapor pressure, Psat, is in atm, the temperature, T, in Kelvin and A, B and C are constants for which values are given in the below table. Antoine constants for methanol (MEOH), mono ethylene glycol (MEG), di-ethylene glycol (DEG), tri-ethylene glycol (TEG) and water. T is in K and P is in atm.

Component A B C MeOH 11.9542 3626.55 -34.290 MEG 13.6168 6022.18 -28.250 DEG 10.3993 4122.52 -122.50 TEG 0.0784 8699.44 2.2040 Water 11.6703 3816.44 -46.1300

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

Ar

SL PCPD

ωηη

+∆+1

=1

118.2/

where η = viscosity of liquid at actual temperature and pressure η SL = viscosity of saturated liquid at current T

rP∆ = (P – Psat)/Pc ω = acentric factor

A = 0.9991 – (4.674 x 10-4 / (1.0523 - 1.0513))

D = (0.3257 / ) - 0.208616

C = - 0.07921 + 2.1616 Tr – 13.4040 + 44.1706 - 84.8291 + 96.1209 - 59.8127 +

15.6719

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)

∑ ∑∑>

+=i j

ijjii

iimix Gzzz ηη lnln

where zi and zj are the mol 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

( )275

573))298(1(1 TGTG ijij−

−−=

where Gij (298) is the value of Gij at T = 298.15 K/25°C/77°F. Gij (298) is assumed to be equal to zero for interactions with methanol and glycol. Gij (298) for interactions with water is as follows Water – MeOH: Gij (298) = 3.02 Water – MEG : Gij (298) = 3.24 Water – DEG : Gij (298) = 3.43 Water – TEG : Gij (298) = 3.62 Other glycols Other glycols are assigned the properties of that of the above glycols that is cloest in molecular weight.

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 = 0.04120 – 0.002605 t + 0.03841 + 0.0002497 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

h

h

hw

Invww

w

hr

if

if

r

r

r

r

φφ

ϕφ

φφ

η

φφ

ϕφ

φφ

η

η

η

η

η

>

−+=

<

−+=

=

=

=

=

,19.1

1

,19.1

1

5.2

100,

5.2

100,

100

100

The specified set of and r is used to calculate from the following equation

( )4.0100 119.1 −= −=

rr η

φφη

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. Firoozabadi, A. and Ramey, H.J., Journal of Canadian Petroleum Technology 27, 1988, pp. 41-48. Grunberg, L. and Nissan, A.H., Nature 164, 1949, 799. 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., Chem. Ing. Tech. 53, 1981, 959.

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(7), 1989, 1021. 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. 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 In the scale module, precipitation is calculated 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

)(2

2lOH

OHHOHHOH a

mmK−+

−+=γγ

)()(1

22

3

2

3

2lOHaqCO

HCOH

CO

HCOHCO am

mmK

γ

γγ −+−+

=

−

−−+

−

−−+

=3

3

3

3

2 2,HCO

COH

HCO

COHCO m

mmK

γ

γγ

)()( aqHA

AH

aqHA

AHHA m

mmK

γγγ −+−+

=

)()( 22

2aqSH

HSH

aqSH

HSHSH m

mmK

γγγ −+−+

=

−−++−−++=

444 SOCaSOCaCaSO mmK γγ

−−++−−++=

444 SOBaSOBaBaSO mmK γγ

−−++−−++=

444 SOSrSOSrSrSO mmK γγ

−−++−−++=

333 COFeCOFeFeCO mmK γγ

+

−++

+

−++

=H

HSFe

H

HSFeFeS m

mmK

γγγ

−−++−−++=

333 COCaCOCaCaCO mmK γγ

The temperature dependence of the thermodynamic equilibrium constants is fitted to a mathematical expression of the type

( ) 2lnlnTEDTTC

TBATK ++++=

A, B, C, D and E for each reaction are listed in the table below.

A B C 1000D E Ref.:

1,2COK -820.433 50275.5 126.8339 -140.273 -3879660 Haarberg (1989)

2,2COK -248.419 11862.4 38.92561 -74.8996 -1297999 Haarberg (1989)

HAK -10.937 0 0 0 0

SHK2

-16.112 0 0 0 0 Østvold (1998)

4CaSOK 11.6592 -2234.4 0 -48.2309 0 Haarberg (1989)

OHCaSOK24 2−

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 -4033.3 -16.0305 -1.34671 31402.1 Haarberg (1989)

3FeCOK 21.804 56.448 16.8397 0.02298 0 Østvold (1998)

FeSK -8.3102 0 0 0 0 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)

( )( ) 21010 0129638.07384.9log195.42296.142613log

2TTT

TTK OH +−+=−

483.890810602.41015068.1 4935 −×+×− −− TT The pressure dependence is given by

RTVZP

PKi ∆−∆

=∂

∂ ln

Where is the partial molar compressibility change of the reaction, is the partial molar volume change of the reaction and R is the universal gas constant. for the sulfate precipitation reactions is expressed by a third degree polynomial

- 10-3 = a + bt + ct2 + dt3 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 Coefficient in compressibility change expression for sulfate mineral precipitation reactions. Units: t in oC and in cm3 /mol/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)

233 000371.0233.03.3910102,21,2

TTKZ COCO −+−=∆=∆ For the calcium carbonate and ferrous carbonate precipitation reactions the compressibility changes are –0.015 cm3/mol 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

= A + BT + CT2 + DI + EI2

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 mols/kg solvent and in cm3/mol.

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)

2002794.0738.17.32833

TTVV FeCOCaCO −+−=∆=∆

The partial molar volume changes of both of the acid equilibria of CO2 are (Haarberg, 1989)

22,1, 0019.0735.04.141

22TTVV COCO −+=∆=∆

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

( ) +

Ψ++++= ∑ ∑ ∑

a c aMcaaMccMaMaaMM mmZCBmFz φγ 22ln 2

∑∑ ∑∑

>

+Ψ' '

''a aa c a

caacMMaaaa Cmmzmm

for the cations, and

( ) +

Ψ++++= ∑ ∑ ∑

c a ccXacXaacXcXcXX mmZCBmFz φγ 22ln 2

∑∑ ∑∑

>

+Ψ' '

'''c cc c a

caacXXcccc Cmmzmm

for the anions. c denotes a cation species, whereas a denotes an anion species. m is the molality (mols/kg solvent) and I is the ionic strength (mols/kg solvent)

∑=i

ii zmI 2

21

z is the charge of the ion considered in the unit of elementary units. ijk is a model parameter that is assigned to each cation-cation-anion triplet and to each cation-anion-anion triplet. The remaining quantities in the activity coefficient equations are

( ) ∑∑ ++

+++

−=c a

caac BmmbIbbI

IAF '1ln21

2/12/1

2/1

ϕ

∑∑ ∑∑

> >

+' ' ' '

''''c cc a aa

aaaacccc mmmm φφ

where b is a constant with the value 1.2 kg 1/2/mol1/2 and

( )2/3

0

22/1

0 42

31

=

DkTedNA w πε

πφ

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.

( ) ( )2/12

)2(2/11

)1()0( IgIgB MXMXMXMX αβαββ ++= where

( ) ( ) ( ) 2exp112

xxxxg −+−

=

(2)

ij(1)

ij(0)

ij βandβ,β are model parameters. One of each parameter is assigned to each cation-anion pair. 1 and 2 are constants, with 1 = 2 kg1/2 mol-1/2 and 2 =12 kg1/2mol-1/2. However, for pairs of ions with charge +2 and –2, respectively, the value for 1 is 1.2 kg1/2mol-1/2. Further

∑=i

ii zmZ

2/12 XM

MXMX

zzCC

φ

=

( ) ( )III ijE

ijE

ijs

ij θθθφφ ++=

( )IijE

ijs

ij θθφφ +=

φijC is yet another model parameter assigned to each cation-anion pair.

ijSθ is a model parameter assigned to each cation-cation pair and to each anion-anion pair and

ijEθ is an electrostatic term

( ) ( ) ( )

−−= jjiiij

jiij

E xJxJxJIzz

21

21

4θ

where

2/16 IAzzx jiij φ=

( ) ( ) ( ) 1528.07231.0 0120.0exp581.44 −− −+= xxxxJ Also the Pitzer model describes the activity of the water in terms of the osmotic coefficient

( ) ( )∑ ∑∑ ++++

−=−i c a

cacaaci ZCBmmbIIA

m φφφ 2/1

2/3

12

1

∑∑ ∑∑ ∑∑> >

Ψ++

Ψ+

' ' ' '''''''

c cc a aa cacacaaaa

aaccacccc mmmmmm φφ φφ

where

( ) ( )2/12

)2(2/11

)1()0( expexp IIB MXMXMXMX αβαββφ −−+=

and the relation between the osmotic coefficient and the activity of the water is

∑=i

iOHOH mMa22

ln φ

Model parameters at 25°C are listed below.

( )0β parameters at 25°C H Sr+ Na+ K+ Mg++ Ca++ ++ Ba++ Fe++

OH- 0.00000 0.08640 0.12980 0.00000 -0.17470 0.00000 0.17175 0.00000 - 0.17750 0.07650 0.04810 0.35090 0.30530 0.28370 0.26280 0.44790

SO4-- 0.02980 0.01810 0.21500 0.20000 0.20000 0.20000 0.00000 -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-

Cl

(1β ) parameters at 25°C

H+ Na+ K+ Mg++ Ca++ Sr++ Ba++ Fe++

OH- 0.00000 -0.230300.25300 0.32000 0.00000 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-

( )2β parameters at 25°C

H Mg Ca+ Na+ K+ ++ ++ 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 0.00000 0.00000 879.20000 0.00000 - 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 HS

φC 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 0.00000 0.00127 -0.00079 0.00651 0.00215 -0.00089 -0.01938 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-

θS parameters at 25°C

H+ Na+ K+ Mg++ Ca++ Sr++ Ba++

H- 0.00000 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

++ 0.06700 0.00000 0.00000 0.00000 0.00000 0.00000

Ba 0.00000

OH- Cl- SO4-- HCO3

- CO3--

OH- 0.00000 Cl- -0.05000 0.00000

- -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

SO4

Ψ parameters at 25°C

HAnion 1 fixed as Cl-

+ 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 -0.04800 ++ -0.01500 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

HAnion 1 fixed as HCO3

- + 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 CO

H Ba3

— + Na+ K+ Mg++ Ca++ Sr++ ++

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

Cation 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

Cation 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

Cation 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

Cation 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 ij are temperature independent parameters, whereas

, and are temperature dependent parameters (=X). Their temperature dependence is described by (Haarberg, 1989) for temperatures in K

( ) ( ) ( ) ( )22

2

15.2982115.29815.298 −

∂∂

+−∂∂

+= TTXT

TXXTX

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

( ) ( ) ( )TPQQTQPQQTQTX 65432

1 ln +++++=

( )T

PQQT

PQQTPQQ−

++

−+

+++680227

1211109287

The temperature coefficients and and the coefficient Q1, Q2…..,Q12 are listed below.

First order temperature derivative of x 100.

H Ca+ Na+ K+ Mg++ ++ 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 x 100.

H+ Na+ K+ Mg++ Ca++ Sr++ Ba++ Fe++

OH- 0.00000 0.00003 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Cl- 0.00376 -0.00150 -0.00025 0.00038 -0.00057 -0.00621 0.00000 0.00000 SO4

-- 0.00000 -0.00115 0.00008 0.00094 0.00000 0.00000 0.00000 0.00000 HCO3

- 0.00000 -0.00192 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 CO3

-- 0.00000 -0.00263 0.00102 0.00000 0.00000 0.00000 0.00000 0.00000 HS-

First order temperature derivative of x 100. H Na Mg+ + K+ ++ Ca++ Sr++ Ba++ Fe++

- 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-

OH

Second order temperature derivative of x 100. H Na Mg Fe+ + K+ ++ Ca++ Sr++ Ba++ ++

OH- 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -0.00124 Cl- -0.00005 0.00021 -0.00004 0.00074 0.00232 0.05000 0.00000 0.00000 SO4

-- 0.00000 0.00908 -0.00011 0.00901 0.00000 0.00000 0.00000 0.00000 HCO3

- 0.00000 0.00263 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 CO3

-- 0.00000 -0.04170 0.00414 0.00000 0.00000 0.00000 0.00000 0.00000 HS-

First order temperature derivative of H K Mg Ca Sr Fe+ Na+ + ++ ++ ++ Ba++ ++

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 3

-- 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 HS- CO

Second order temperature derivative of 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 x 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 x 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

βNaCl(0) βNaCl

(1) φNaClC

Q1 -6.5684518×102 1.1931966×102 -6.1084589 Q2 2.486912950×101 -4.8309327×10-1 4.0217793×10-1

Q3 5.381275267×10-5 0 2.2902837×10-5

Q4 -4.4640952 0 -7.5354649×10-4

Q5 1.110991383×10-2 1.4068095×10-3 1531767295×10-4

Q6 -2.657339906×10-7 0 -9.0550901×10-8

Q7 -5309012889×10-6 0 -1.53860082×10-8

Q8 8.634023325×10-10 0 8.69266×10-11

Q9 -1.579365943 -4.2345814 3.53104136×10-1

Q10 0.0022022820790×10-3 0 -4.3314252×10-4

Q11 9.706578079 0 -9.187145529×10-2

Q12 -2.686039622×10-2 0 5.190477×10-4

The coefficients correspond to units of pressure and temperature in bars and Kelvin, respectively. Reference: Pitzer (1984)

Calculation procedure The amount of minerals that precipitate from a specified aqueous solution is evaluated by calculating the amount of ions that stay 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 ratio of CO his determines if any of the ferrous iron

minerals FeCO ill precipitate. Only one can precipitate, since both Hare fixed in concentration, and then the Feproducts at the same time.

•

•

The stoichiometric equilibrium constants are calculated from the thermodynamic ones and from the activity coefficients.

2(aq) to H2S(aq) is calculated. T3 and FeS w 2S and CO2

++ concentration cannot fulfil both solubility

• 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).

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 pipewall 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 only 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. Since the temperature of the fluid as it enters into the pipeline is generally higher than that of the surroundings, the bulk fluid temperature will generally 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

mCDUTTTT

p

totambinambx &

πexp

The equation states that under the above assumptions, the temperature Tx at a given position x can be calculated based on the mass flowrate , 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 principles of the algorithm are illustrated in the below figure. The inlet conditions to a section, such as mass flowrate, temperature, pressure and composition are known. Also, the pipeline specifications such as 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 pipewall temperatures. A steady state flow model, OLGAS 2000, calculates pressure drop, flow regime, and liquid holdup, based on information about phase equilibria and viscosity passed from the thermodynamic models in PVTsim. Subsequently the wax model in PVTsim is used to calculate wax concentrations, which are needed to determine deposition on the walls of the pipeline. 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 timestep. Subsequent timesteps are calculated similarly, the only change from one timestep to the next being that the pipeline diameter and insulation have changed due to a layer of deposited wax on the pipewall.

Q=UA∆T

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 energy balance determines the total fluid enthalpy at the outlet of the section based on the enthalpy of the fluid entering into the section. This enthalpy is known from the last section. For the first section, the inlet enthalpy is obtained from a PT-flash. The enthalpy 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 the following equation.

hgW bulkρ=

In this equation 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 pipewall area, Tamb is the ambient temperature, and is the mean bulk temperature in the section extrapolated from the expectation of an exponential decline of the temperature. Utot is the overall heat transfer coefficient.

Overall heat transfer coefficient The calculation of the overall heat transfer coefficient is calculated from the below equation.

1

1,111 1

ln1

−

=−

−−

++= ∑NLAY

i outoutii

i

i

ininin hrk

rr

hrrU 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 text by Bird et al. (1960).

Inside film heat transfer coefficient

The inside film heat transfer coefficient hin is estimated from the flow regime, based on the definition of the Nusselt number

kDhN in

Nu =

where k is the thermal conductivity of the fluid and D is the inside diameter of the pipeline. The Nusselt number has been related to the Reynolds and Prandtl numbers through different correlations depending on flow regime. Four sets of correlations are available, of which the Sieder-Tate and Dittus-Bölter are probably the most popular and the Petukov/ESDU (recommended set) the most reliable and well documented. Sieder-Tate

25.03/1

Pr8.0

Re4

Re 027.010

=>

w

bNu NNNN

µµ

25.0

8.1Re

53/1

Pr8.0

Re4

Re1061027.0:102300

×−=<<

w

bNu N

NNNNµµ

( )( )66.3,184.0max:2300 3/1PrRe NNNN GrNu =>

Dittus-Bölter

25.03.0

Pr8.0

Re4

Re 023.010

=>

w

bNu NNNN

µµ

25.0

8.1Re

53.0

Pr8.0

Re4

Re1061023.0:102300

×−=<<

w

bNu N

NNNNµµ

( )( )66.3,184.0max:2300 3/1PrRe NNNN GrNu =<

Petukov-Gnielinski

( )

( )

25.03/2

3/2Pr

PrRe

Re 11

87.121

10008:2300

+

−+

−=<

w

bNu L

D

N

NNNN

µµ

ξ

ξ

where ( )2

Re 64.1log82.11

−N=ξ

25.0

3.0

RePr

3/4

Pr

Re

1.01

0677.0657.3:2300

+

+=>w

bre

Nu

LDNN

LDNN

NNµµ

Petukov/ESDU (recommended set)

( )( ) ( )

25.0

3/2Pr

2/1PrRe

Re 12/7.1207.12/:4000

−+=>

w

bNu Nf

NNfNNµµ

( )2

Re 64.1log82.141

−=

Nf

( )[ ] 25.03/133/133

Re 7.077.17.066.3:2300

−++=<

w

bGzNu NNN

µµ

( ) turlam NuNuNuN <−+=<< εε 140002300 Re

6000

33.1 ReN−=ε

Depending on the insulation of the pipeline, the film heat transfer coefficient may strongly affect the wax deposition calculation through the temperature gradient over the laminar film layer inside the pipeline. For that reason, the choice of correlation is important.

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 pressure drop in a given section will depend on a number of factors. Different methods are applied depending on whether single phase or multiphase flow is considered. For single phase flow, pressure drop is calculated as a combination of frictional and elevational pressure drop. Friction factors are calculated from:

Re

64:N

flam =

+×+=

3/1

Re

64 1010210055.0:

NDfturb ε

The method is described by Bendiksen et al. (1991). For multiphase flow the steady state flow model OLGAS 2000 is applied. Input to the OLGAS 2000 model includes viscosities and superficial fluid velocities of the different phases. The aqueous and oil phases are combined as one liquid phase based on the volumetric phase fractions. Further information about the OLGAS 2000 model can be obtained from Bendiksen et al. (1991). The liquid holdup is returned from OLGAS 2000. Based on this and geometrical considerations the wetted perimeter of each phase is calculated. The following equations relate the holdup to the wetted perimeter

rSwet ϕ2=

πϕϕ

2sin−

=HOL

Here HOL is the hold-up, is the angle corresponding to the part of the pipeline circumference swept by liquid and Swet is the wetted perimeter corresponding to that angle. The wetted perimeter is corrected for presence of an aqueous phase based on the phase volume fractions.

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 pipewall.

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

i i

iwetwi

biidiff

waxMWSccDVol

1 δρ

where is the molar concentration of wax component i in the bulk phase and 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 Blasius (1913) expression

87

Re58 NDαδ = 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

,

,,

791.0

,,,

2.10

47.112103.13−

−

××=

iwax

iwaxwiwax

iwaxwM

i

MTD

ρηβ ρ

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

wallshearwax

AckVolρ

γ*

=

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/ft2] or [0;0.001 kg/m2].

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.

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:

BA += σφ 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 timesteps. 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 timestep. Mass sources cannot be specified to enter in the first segment. In this case 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 timestep, 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, (1991), SPE Production Engineering, May, pp. 171-180. Bird, R.B., Steward, W.E., Lightfoot, E.N., Transport Phenomena, (1960), Wiley, NY. pp. 286-28. Blasius, H., ”Das Ähnlichkeitsgesetz bei Reibungsvorgängen in Flüssigkeiten”, Forch. Ver. Deut. Ing. 131, 1913. 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, 1975-1086. 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 (1995), 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, (1986), Elsevier, Amsterdam.

Clean for Mud

Clean for Mud Reservoir samples are often contaminated by base oil from drilling mud. The Mud module of PVTsim 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.

• Weight% contaminate in stock tank oil (optional for compositions to C36+) The cleaning procedure will differ depending on the extent of the compositional analysis

Reservoir fluids to C7+ or C10+The base oil contaminate will seldom contain components lighter than C11. 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. Otherwize make new estimate of eeight% contaminate in reservoir fluid and return to 4.

Chapter 1 Clean for Mud • 137

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.

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 degree of contamination to be estimated.

For a clean reservoir fluid PVTsim assumes the following relation between the mol fraction (z) of C7+ fractions and carbon number i.

ii CNBAz ×+=ln

A and B are estimated by a fit to mol%’s for C7+ mol 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 mol 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.

Chapter 1 Clean for Mud • 138

Chapter 1 Clean for Mud • 139

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 to 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 comtaminate in the Clean for Mud menu. The result of the regression is a cleaned, tuned and characterized reservoir fluid composition.