PVTsim Tutorial Calsep
Short Description
Detailed tutorial...
Description
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 pseudocomponents..................................................... 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 Pseudocomponents ............................................................ 40 Delumping ................................................................................................................. 42 Characterization of Multiple Compositions to the Same PseudoComponents ......... 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 JouleThomson 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 pseudocomponents ........................ 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 VaporLiquidWax Phase Equilibria ......................................................................... 90 Extended C7+ Characterization .................................................................................. 92 Viscosity of OilWax 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 wateroil 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 menu in PVTsim. The 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 pseudocomponents. 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 Pseudocomponents
The program is delivered with a pure component database consisting of the following components Short Name Water H2O Hydrate inhibitors MeOH EtOH PG DPGME MEG PGME DPG DEG TEG Glycerol Salts NaCl
Systematic Name
Formula Name
Water
H2O
Methanol Ethanol Propyleneglycol Dipropyleneglycolmethylether Monoethyleneglycol Propyleneglycolmethylether Dipropyleneglycol Diethyleneglycol Triethyleneglycol Glycerol
CH4O C2H6O C6H8O2 C7H16O3 C2H6O2 C7H10O2 C6H14O3 C4H10O3 C6H14O4 C3H8O3
Sodium chloride
NaCl
KCl NaBr CaCl2 HCOONa HCOOK KBr HCOOCs CaBr2 ZnBr2 Other inorganic He H2 N2 Ar O2 CO2 H2S Organic defined C1 C2 C3 cC3 iC4 nC4 2,2dimC3 cC4 iC5 nC5 cC5 2,2dimC4 2,3dimC4 2mC5 3mC5 nC6 C6 mcC5 Benzene Napht cC6 223tmC4 3,3dimC5 2mC6 c13dmcC5 t13dmcC5 3mC6 t12dmcC5 nC7 mcC6 etcC5 113trcC5
Potassium chloride Sodium bromide Calcium chloride (anhydrous) Sodium formate (anhydrous) Potassium formate (anhydrous) Potassium bromide Caesium formate (anhydrous) Calcium bromide (anhydrous) Zinc bromide
KCl NaBr CaCl2 HCOONa HCOOK KBr HCOOCs CaBr2 ZnBr2
Helium4 Hydrogen Nitrogen Argon Oxygen Carbon dioxide Hydrogen sulfide
He(4) H2 N2 Ar O2 CO2 H2S
Methane Ethane Propane Cyclopropane Isobutane Normalbutane 2,2Dimethylpropane Cyclopropane 2methylbutane Normalpentane Cyclopentane 2,2Dimethylbutane 2,3Dimethylbutane 2Methylpentane 3Methylpentane Normalhexane Hexane Methylcyclopentane Benzene Naphthalene Cyclohexane 2,2,3Trimethylbutane 3,3Dimethylbutane 2Methylhexane Cis1,3Dimethylcyclopentane Trans1,3Dimethylcyclopentane 3Methylhexane Trans1,2Dimethylcyclopentane Normalheptane Methylcyclohexane Ethylcyclopentane 1,1,3Trimethylcyclopentane
CH4 C2H6 C3H8 C3H6 C4H10 C4H10 C5H12 C4H8 C5H12 C5H12 C5H8 C6H14 C6H14 C6H14 C6H14 C6H14 C6H12 C6H6 C10H8 C6H12 C7H16 C7H16 C7H16 C7H14 C7H14 C7H16 C7H14 C7H16 C7H14 C7H14 C8H16
Toluene 2mC7 cC7 3mC7 11dmcC6 c13dmcC6 t12dmcC6 nC8 c12dmcC6 EtcC6 etBenzene pXylene mXylene 2mC8 oXylene 1m3ecC6 1m4ecC6 cC8 4mC8 nC9 Mesitylene PsCumene nC10 Hemellitol nC11 nC12 nC13 1mNapht nC14 nC15 nC16 nC17 nC18 nC19 nC20 nC21 … nCn … nC40
Toluene 2Methylheptane Cycloheptane 3Methylheptane 1,1Dimethylcyclohexane Cis1,3Dimethylcyclohexane Trans1,2Dimethylcyclohexane Normaloctane Cis1,2Dimethylcyclohexane Ethylcyclohexane EthylBenzene Paraxylene Metaxylene 2Methyloctane Orthoxylene 1Methyl3Ethylcyclohexane 1Methyl4Ethylcyclohexane Cyclooctane 4Methyloctane Normalnonane 1,3,5TrimethylBenzene 1,2,4TrimethylBenzene Normaldecane 1,2,3TrimethylBenzene Normalundecane Normaldodecane Normaltridecane 1methylNaphthalene Normaltetradecane Normalpentadecane Normalhexadecane Normalheptadecane Normaloctadecane Normalnonadecane Normaleicosane NormalC21 … NormalCn … NormalC40
C7H8 C8H18 C7H14 C8H18 C8H16 C8H16 C8H16 C8H18 C8H16 C8H16 C8H10 C8H10 C8H10 C9H20 C8H10 C9H18 C9H18 C8H16 C9H20 C9H20 C9H12 C9H12 C10H22 C9H12 C11H24 C12H26 C13H28 C11H10 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42 C21H44 … CnH2n+2 … 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 nCn1 + 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 pseudocomponents. 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 pseudocomponents) Critical volume (Vc) Vapor pressure model (classical or MathiasCopeman) MathiasCopeman 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 pseudocomponents) Wax fraction (only for nparaffins and pseudocomponents) Asphaltene fraction (only for pseudocomponents) 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 nparaffins and pseudocomponents)
•
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 Volume Density Z factor Enthalpy (H) Entropy (S) Heat capacity (CP) Heat capacity (CV) Kappa (CP/ CV) JouleThomson coefficient Velocity of sound Viscosity Thermal conductivity Surface tension
Component properties needed Peneloux parameter*1) Peneloux parameter*1) Peneloux parameter*1) Ideal gas CP coefficients, Peneloux parameter*1) Ideal gas CP coefficients, Peneloux parameter*1) Ideal gas CP coefficients Ideal gas CP coefficients, Peneloux parameter*1) Ideal gas CP coefficients, Peneloux parameter*1) Ideal gas CP coefficients, Peneloux parameter*1) Peneloux parameter*1) Weight average molecular weight*2), Vc*3) Parachor, Peneloux parameter*1)
*1)
Only if an equation of state with Peneloux volume correction is used. Only if corresponding states viscosity model selected. *3) Only if LBC viscosity model selected. *2)
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 pseudocomponents it is highly recommended also to enter the liquid density.
Missing Properties PVTsim has a 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 T independent term of Peneloux parameter None Assumed equal to number average molecular weight None
Liquid density Normal boiling point Weight average molecular weight Critical volume Vapor pressure model MathiasCopeman coefficients Tindependent term of SRK or PR Peneloux parameter Tdependent term of SRK or PR Peneloux parameter Melting point depression (
)
Ideal gas absolute enthalpy at 273.15 K/0°C/32°F (Href) Ideal gas Cp coefficients Melting temperature (Tf)
Enthalpy of melting (
)
PNA distribution Wax fraction Asphaltene fraction Parachor Hydrate former or not Hydrate Langmuir constants Number of ions in aqueous solution (only for salts) Number of crystal water molecules per salt molecule (only for salts)
Not estimated Not estimated for defined components. Liquid density for pseudocomponents Not estimated for defined components. Liquid density for pseudocomponents Only for pseudocomponents. Viscosity data for an uninhibited/inhibited fluid. Molecular weight Not estimated for defined components. Liquid density for pseudocomponents Irrelevant for defined components. None for pseudocomponents Irrelevant for defined components. None for pseudocomponents Irrelevant for defined components. Liquid density for pseudocomponents Irrelevant for defined components. None for pseudocomponents. Irrelevant for defined components. Liquid density for pseudocomponents Not estimated for defined components. Liquid density for pseudocomponents Not estimated Not estimated Not estimated Not estimated
SRK with Volume Correction. PR with Volume Correction. Extrapolation of Plus Fraction. LohrenzBrayClark (LBC) part of Viscosity section. SRK with Volume Correction or PR with Volume Correction SRK with Volume Correction or PR with Volume Correction
Compositional variation due to gravity Enthalpy Extended C7+ Characterization Extended C7+ Characterization Estimation of PNA Distribution Extended C7+ Characterization Asphaltenes Gas/Oil interfacial tension. 
Pc of wax forming fraction
Irrelevant for defined components. None for pseudocomponents
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 pseudocomponents 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 TBPterminology 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 pseudocomponent 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 Pseudocomponents’.
Mixing PVTsim may be used to mix or weave from 2 to 50 fluid compositions. A mixing will not necessarily retain the pseudocomponents of the individual compositions. Averaging the properties of the pseudocomponents in the individual compositions generates new pseudocomponents. 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 pseudocomponents (12), pseudocomponents of approximately the same weight are lumped to get down to the desired number of pseudocomponents in the mixed fluid.
Weaving Weaving will maintain the pseudocomponents of the individual compositions and can only be performed for characterized compositions. When weaving two fluids, all pseudocomponents 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 pseudocomponents The goal of characterizing fluids to the same pseudocomponents 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 pseudocomponent contained in each individual fluid. The characterization to the same pseudocomponents 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 pseudocomponents. 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 pseudocomponents 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 SoaveRedlichKwong (SRK) equation of state, the PengRobinson (PR) equation of state, and the PengRobinson 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
G =∑ n iµ i i =1
is the chemical potential is at a minimum. ni is the number of mols present of component i and 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 µ iI = µ iII 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 ∂G G1 = G 0 − ∑ ε i i i =1 ∂n i n
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
N
i =1
i =1
∆G = G I + G II − G 0 = ∑ ε i ((µ i )II − (µ i )0 ) = ε ∑ yi ((µ i ) II − (µ i ) 0 )
, and yi is the mol fraction of component i in phase II. The subindices 0 and II where 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, expressed in terms of the fugacity, fi, as follows
, may be
µ i = µ i0 + RT1n f i = µ i0 + RT(1n z i + lnϕ i + 1n P ) is a standard state chemical potential, a fugacity coefficient, z a mol fraction, P the where pressure, and the subindex 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 ∆G N = ∑ y i (1n y i +1n(ϕ i ) II − ln z i − 1n(ϕ i )0 ) εRT i =1
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
∑ y (ln y + ln(ϕ ) i =1
i
i
i II
− ln z i − ln(ϕ i ) 0 ) > 0
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 ln yi + ln(ϕ i ) II − lnzi − ln(ϕ i )0 = k 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 Ki =
Pci T exp 5.42 (1 − ci ) P T
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 Yivalue are determined, and so on. For a singlephase mixture this direct substitution calculation will either converge to the trivial solution (i.e. to two identical phases) or to Yivalues fulfilling the criterion N
∑Y ≤1 i =1
i
which corresponds to a nonnegative value of the constant k. A negative value of k would be an indication of the presence of two or more phases. In the twophase 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
βyi + (1 − β )x i = zi ,
(i = 1,2,3,..., N )
Equilibrium equations yiϕ iV = x iϕ iL ,
(i = 1,2,3,..., N )
Summation of mol fractions N
∑ (y − x ) = 0, (i = 1,2,3,..., N ) i =1
i
i
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 Kfactor, Ki = yi/xi. The following expressions may then be derived for xi and yi
xi =
zi , 1 + β(K i − 1)
(i = 1,2,3,..., N )
(i = 1,2,3,..., N )
yi = K i x i ,
and for Ki
ϕ iL Ki = V , ϕi
(i = 1,2,3,..., N )
The above (2N+1) equations may then be reduced to the following (N+1) equations ln K i =
ln ϕ iL , ln ϕ iV
(i = 1,2,3,..., N ) N
∑ (y − x ) = ∑ z (K i
i
i =1
i
i
i
− 1)/(1 + β(K i − 1)) = 0
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
zi (K im − 1) =0 Hi i =1 N
∑
where j −1
H i = 1 + ∑ β m (K im − 1) m =1
m
β 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
y im =
z i K im , Hi
y iJ =
zi , Hi
where
(i = 1,2,3,..., N; m = 1,2,3,..., J )
(i = 1,2,3,..., N ) 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 twophase PH flash N
g1 = ∑ z i (K i − 1)ζ i i =1
g 2 = H − H spec
where
ς i = 1 + β(K i − 1) 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 singlephase 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 nonaqueous PTflash algorithm handles the following component classes • • •
Other inorganic Organic defined Pseudocomponents
The PT aqueous and multiflash algorithms handle • • • • • •
Water Hydrate inhibitors Other inorganic Organic defined Pseudocomponents Salts
The PH, PS, VT, UV, and HS flash algorithms handle • • • • •
Water Hydrate inhibitors Other inorganic Organic defined Pseudocomponents
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: PhaseSplit 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 It is seen that the proposed temperature dependence reduces to the default (classical) one for C1 = m and C2 = C3 = 0. In general the MathiasCopeman (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 P=
RT a − V − b (V + c )(V + b + 2c )
The SRK molar volume,
, and the Peneloux molar volume, V, are related as follows
~
V =V −c The b parameter in the Peneloux equation follows
is similarly related to the SRK bparameter as
~ b= b −c
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+ pseudocomponents. In general the temperature independent Peneloux volume correction for defined organics and “other organics” is found from the following expression c' = 0.40768
RTc (0.29441 − ZRA ) Pc
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 125080 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 P=
RT a(T) − V − b V(V + b ) + b(V − b )
where a(T) = ac a c = Ωa
(T)
R 2Tc2 Pc
T 0.5 α(T) = 1 + m1 − Tc R Tc b = Ωb Pc
2
where Ω a = 0.45724 Ω b = 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 correlation is used m = 0.379642 +
(1.48503 − 0.164423
+ 0.01666
2
0 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 nonwater 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.
from where w stands for water and empty hydrate lattice. 6. The hydrate compositions are calculated using the expression
is found refers to the
which enables calculation of the fugacity coefficients as described below. Nonhydrate 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 1,…, number of phases). 8. If not converged repeat from 3.
j,
K = 1,2,…,N, j =
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
fi = ϕi x i P 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:
N (1 − θ ) N 0θ ln f wH − ln f wβ = v i ln 0 + v 2 ln v1 v2 Other Hydrate Formers:
f kH =
Nk N 0 C k 2 (θ + α k (1 − θ ))
In these equations f wβ = 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 273.15 273.15 0.0390 P f ice = − 2.064 1− − 4.710 ln + T T T + 273.15
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 Mol salt = 0.108 + 0.000125 T Mol water
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 Mol salt = 0.0674 + 0.000544 T Mol water
Sodium Formate, HCOONa
T < 50°C :
Mol salt = 0.145 + 0.00355 T Mol water
T ≥ 50°C :
Mol salt = 0.313 Mol water
Potassium Formate, HCOOK
T < 20°C :
Mol salt = 0.712 + 0.00705 T Mol water
T ≥ 20°C :
Mol salt = 0.964 + 0.0174 T Mol water
Cesium Formate, HCOOCs
T < 6°C :
Mol salt = 0.0248 + 0.00143 T Mol water
6°C ≤ T < 50°C :
50oC ≤ T :
Mol salt = 0.272 + 0.006 T Mol water
Mol salt = 0.572 Mol water
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. 457459, 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. 11111114. 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, 11921193. Munck, J., SkjoldJørgensen S. and Rasmussen, P., ”Computations of the Formation of Gas Hydrates”, Chem. Eng. Sci. 43, 1988, 26612672. 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. 159163, 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. 241250, 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).
VaporLiquidWax 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
f iL = f iS 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 f iL =x iL ϕ iL P is the liquid phase mol fraction of component i, the liquid phase In this expression 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 f iS = x Si f ioS the solid standard state fugacity where x Si is the solid phase mol fraction of component i, and of component i. The solid standard state fugacity is related to the liquid standard state fugacity as f ioS (Pref ) ∆G = RT ln oL ( ) f P ref i f i
is the molar change in Gibbs free energy associated with the transition of pure
where
component i from solid to liquid form at the temperature of the system. To calculate following general thermodynamic relation is used
the
∆G = ∆H − T∆S
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
∆G if = ∆H if + T∆Sif is the enthalpy and the entropy of fusion of component i at the normal melting where 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 ∆Sif =
∆H if Tif
where is the melting temperature of component i. The following expression may now be derived for the solid standard state fugacity of component i − ∆H if f ioS = f ioL (Pref )exp RT
T 1 − f Ti
∆Vi (P − Pref ) + RT
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 f ioL = ϕ ioL P is the liquid phase fugacity coefficient of pure i at the system temperature and where pressure. This leads to − ∆H if f ioS = ϕ ioL (Pref ) P exp RT
T 1 − f Ti
∆Vi (P − Pref ) + RT
The following expression may now be derived for the solid phase fugacity of component i in a mixture
− ∆H i f f i =xiS ϕ ioL (Pref ) P exp RT
T 1− f Ti
∆Vi (P − Pref ) + RT
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+ pseudocomponent into a potentially wax forming fraction and a fraction, which cannot enter into a wax phase. In addition correlations are needed for estimating component.
,
and
of each component and pseudo
The wax model is based on the assumption that a wax phase primarily consists of nparaffins. The user may input the nparaffin 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 P − ρ ρ i i z = z 1 − (A + B× M i ) P ρ i s i
tot i
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 pseudocomponent i. A, B and C are constants of the following values A = 1.0744 B = 6.584 x 104 C = 0.1915
ρ iP is the densities (g/cm3) at standard conditions of a normal paraffin with the same molecular weight as pseudocomponent i. The following expression is used for the paraffinic density.
ρ iP = 0.3915 + 0.0675 ln M i For a (hypothetical) pseudocomponent for which will be equal to meaning that all the components contained in that particular pseudocomponent are able to enter into a wax phase. In general
will be lower than
component will have a mol fraction of
and the nonwax forming part of the pseudo.
The wax forming and the nonwax forming fractions of the C20+ pseudocomponents are assigned different critical pressures. The critical pressure of the wax forming fraction of each pseudocomponent is found from
ρP P = Pci i ρi
3.46
s ci
Pci equals the critical pressure of pseudocomponent i determined using the characterization
procedure described in the Characterization section. of pseudocomponent i and
is the density of the wax forming fraction
is the average density of pseudocomponent. The critical pressure
of the nonwax forming fraction of pseudocomponent i is found from the equation
(
1 Fracino−S = Pci Pcino−S
) + (Frac ) 2
S 2 i S ci
P
+
2Fracino−S Pcino−S PciS
where S and noS are indices used respectively for the wax forming and the nonwax forming fractions (Frac) of pseudocomponent i. By using this relation the contribution to the equation of state aparameter of pseudocomponent i divided into two will be the same as that of the pseudocomponent 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 Tif = 374.5 + 0.02617 M i −
20172 Mi
∆H if = 0.1426 M i Tif
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 nonwax forming parts of a pseudocomponent are equal, but the wax model parameters differ. Presence of nonwax 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 pseudocomponent is adjusted to match the experimental data.
Viscosity of OilWax Suspensions Oil containing solid wax particles may exhibit a nonNewtonian 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)
4 Eϕ wax Fϕ wax η = η liq exp(Dϕ wax )1 + + dv x dv x dy dy
is the viscosity of the oil not considering solid wax and the volume fraction of where precipitated wax in the oilwax suspension. The parameters D, E and F take the following values (viscosities in mPa s and shear rates in s1) 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 subsea 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. 4649. Pedersen, K.S. and Rønningsen, H.P., ”Effect of Precipitated Wax on Viscosity – A Model for Predicting NonNewtonian Viscosity of Crude Oils”, Energy & Fuels, 14, 2000, pp. 4351. 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. 321328. 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, 1820 June, 1997.
Templin, R.D., “Coefficient of Volume Expansion for Petroleum Waxes and Pure nParaffins”, Ind. Eng. Chem., 48, 1956, pp. 154161. Won, K.W., ”Thermodynamics for SolidLiquidVapor Equilibria: Wax Phase Formation from Heavy Hydrocarbon Mixtures”, Fluid Phase Equilibria 30, 1986, pp. 265279.
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 nonasphaltenic aromatics from C50 to a carbon number that will make the total asphaltene content agree with that measured. In asphaltene simulations pseudocomponents containing asphaltenes are split into an asphaltene and nonasphaltene 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 liquidliquid equilibrium the oilasphaltene 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: TcA = 1398.5 K/1125.35°C/2057.63°F PcA = 14.95 Bar/14.75 atm/216.83 psi ω A = 1.274 The critical temperature TcinoA of the nonasphaltene fraction (FracinoA) of pseudocomponent i is found from the relation
Tci = (Fracino−A Tcino−A ) 2 + 2 × Fracino−A FraciA Tcino−A TciA + (FraciA TciA ) 2
where Tci is the critical temperature of pseudocomponent i before being split. The critical pressure equation
(
1 Fracino−A = Pci Pcino−A
of the nonasphaltene forming fraction of pseudocomponent i is found from the
) + (Frac ) 2
A 2 i A ci
P
+
2Fracino−A Pcino−A PciA
while the acentric factor of the nonasphaltene forming fraction of pseudocomponent i is found from ω i = Fracino−A ω ino−A + FraciA ω iA The binary interaction parameters between asphaltene components and C1C9 hydrocarbons are by default assumed to be 0.017 where binary interaction parameters of zero are default used for all other hydrocarbonhydrocarbon 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 913, 1997, Houston, TX, USA.
H2S Simulations
H2S Simulations The H2S module of PVTsim is based on the same PTflash 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
[ ]
pH = − log10 H +
and pK pK 1 = log10
[HS ][H ] −
+
H 2S
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
pK 2 = − log10
[H ][S ] [HS ] +
−−
−
is however of the order 1314, 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) Ψ = Ψ0 (T ) + RT [ln ρ + ρ Q( ρ , T )] where Ψ = ρ = τ = R =
Helmholtz free energy (J/g) Density (g/cm3) 1000/T where T is the temperature in K 0.46151 J/(g K)
and Ψ0 (T ) = C1 + C2T + C3T 2 + (C4 + C5T ) ln T
8
i −l Q( ρ , T ) = ∑ Aij ( ρ − ρ a ) + e − Eρ ( A9,1 + A10,1 ρ ) i =l
8 7 j −2 o −l + (τ −τ c )∑ (τ −τ a ) ∑ Aij (ρ − ρb ) + e − Ep (A9 j + A10 j ρ ) i =1 j=2
where
ρ a = 0.634 g/cm3 ρb = 1.0 g/cm3 τ a = 2.5 K1 τ c = 1.544912 K1
E = 4.8 cm3/g
The coefficients C1 – C5 and Aij are given in tables below. i 1 2 3 4 5
CI 1855.3865 3.278642 .00037903 46.174 1.02117
Aijcoefficients of the Qfunction. i 1 2 3 4 5 6 7 8 9 10
1 29.492937 132.13917 274.64632 360.93828 342.18431 244.50042 155.18535 5.9728487 410.30848 416.05860
2 5.1985860 7.779182 33.301902 16.254622 177.31074 127.48742 137.46153 155.97836 337.31180 209.88866
3 6.8335354 26.149751 65.326396 26.181978 0 0 0 0 137.46618 733.96848
j 4 01564104 0.72546108 9.2734289 4.3125840 0 0 0 0 6.7874983 10.401717
5 6.3972405 26.409282 47.740374 56.323130 0 0 0 0 136.87317 645.81880
6 3.9661401 15.453061 29.142470 29.568796 0 0 0 0 79.847970 399.17570
7 0.69048554 2.7407416 5.1028070 3.9636085 0 0 0 0 13.0411253 71.531353
The pressure is given by the following relation ∂Ψ ∂Ψ 1000 2 ∂Q = ρ 2 = ρ R P = ρ 2 1 + ρ Q + ρ τ ∂ρ T ∂ρ τ ∂ρ τ
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
P ∂ (Ψτ ) H = + ∂τ ρ ρ =
∂Q d Ψ0 1000 R ∂Q + Ψ0 − T + ρ 1+ ρ Q +τ τ dT ∂τ ∂ρ τ
∂Ψ S =− =− R ∂T ρ
∂Q d Ψ0 − ln ρ + ρ Q −τ dT ∂τ ρ
∂H ∂H Cp = − ∂T ρ ∂ρ T
∂P ∂T ρ ∂P ∂ρ T
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 T − b2 + b3 ×10 −6 Tc
η1 = b1
Region 1: Psat < P < 80 MN/m2 and 273.15 K/0°C/32°F < T < 573.15 K/300°C/572°F
ρ
ρc
η =10 −6 a1 1+
−
Psat Pc
T a2 × a4 − a5 ×10 Tc (T / Tc ) − a3
where Tc and Pc are the critical temperature and pressure, respectively and critical point.
the density at the
Region 2: 0.1 MN/m2 < P < Psat and 373.15 K/100°C/212°F < T < 573.15 K/300°C/572°F
η = η1 ×10 6 −10 Region 3:
T ρ −6 c1 − c2 − c3 ×10 ρc Tc
0.1 MN/m2 < P < 80 MN/m2 and 648.15 K/375°C/707°F < T < 1073.15 K/800°C/1472°F 3 ρ 6 η = η1 ×10 + d 3 + d 2 ρc
ρ ρ
2 ρ ×10 −6 d1 ρ c
Region 4: Otherwise
η = η1 +
10Y 0.0192
where Y = C5kX4 + C4kX3 + C3kX2 + C2kX + C1k ρ X = log10 ρc The parameter k is equal to 1 when / 4/3.14. The
C1k C2k C3k C4k C5k
6.4608381 1.6163321 0.07097705 13.938 30.119832
The vapor pressure, Psat, is calculated from the following correlation j
7
log10 Psat = (1 + D1 ) + ∑ D j (T − 273.15) + j =3
D2 T − 273.15
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 1 2 3 4 5 6 7
Di 2.9304370 2309.5789 .34522497 x 101 .13621289 x 103 .25878044 x 106 .24709162 x 109 .95937646 x 1013
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 104 t2 – 4.51 x 108 t3) x 105
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
P − Psat Pc
λ = S1 + where
P − Psat S 2 + Pc
S 3 ×10 −2
T S1 = ∑ ai i =0 Tc 4
i
T S 2 = ∑ bi i =0 Tc 3
T S3 = ∑ ci i =0 Tc 3
i
i
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 φ Inv
r
is used to calculate
from the following equation
)
This value acts as a constant in subsequent calculations, where . is evaluated at specified temperature and pressure.
r
is calculated as a function of
References Alder, B.J., ”Prediction of Transport Properties of Dense Gases and Liquids”, UCRL 14891T, University of California, Berkeley, California, May 1966. Firoozabadi, A. and Ramey, H.J., Journal of Canadian Petroleum Technology 27, 1988, pp. 4148. 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 (043) 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 1417, 1995. Schmidt, E., ”Properties of Water and Steam in SIUnits”, SpringerVerlag, New York, Inc. 1969. Sengers, J.V. and Keyes, P.H., ”Scaling of the Thermal Conductivity Near the GasLiquid Critical Point”, Tech. Rep. 71061, 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 pHdependent contributions on one side of the equality sign and all pHindependent species on the other, the alkalinity appears, i.e. the alkalinity is the sum of contributions to the charge balance from the pHindependent 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 nonideal nature of the water phase is taken into account.
Thermodynamic equilibria The thermodynamic equilibria considered are •
Acidequilibria H2O(l) ↔ H+ + OHH2O(l) + CO2(aq) ↔ H+ + HCO3HCO3 ↔ H+ + CO3HA(aq) ↔ H+ + AH2S(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
γ H γ OH +
K H 2O = mH + mOH −
K CO2 1 =
a H 2O ( l )
mH + m HCO −
γ H γ HCO
mCO2
γ CO ( aq ) a H O (l )
3
K CO2 , 2 =
+
3
2
2
mH + mCO −− γ H + γ CO −− 3
γ HCO
mHCO − 3
K HA =
−
3
3
−
mH + mA − γ H + γ A −
KH2S =
mHA( aq ) γ HA( aq )
mH + mHS − γ H + γ HS − mH 2 S ( aq ) γ H 2 S ( aq )
K CaSO4 = mCa ++ mSO −− γ Ca + + γ SO −− 4
4
K BaSO4 = mBa + + mSO −− γ Ba ++ γ SO −− 4
4
K SrSO4 = mSr ++ mSO −− γ Sr ++ γ SO −− 4
4
K FeCO3 = mFe ++ mCO −− γ Fe + + γ CO −− 3
K FeS =
3
mFe + + mHS − γ Fe ++ γ HS −
γH
mH +
+
K CaCO3 = mCa + + mCO − − γ Ca + + γ CO − − 3
3
−
The temperature dependence of the thermodynamic equilibrium constants is fitted to a mathematical expression of the type ln K (T ) = A +
B E + C ln T + DT + 2 T T
A, B, C, D and E for each reaction are listed in the table below.
K CO2 ,1
A 820.433
B 50275.5
C 126.8339
1000D 140.273
E 3879660
KCO2 , 2
248.419
11862.4
38.92561
74.8996
1297999
K HA KH 2S
10.937 16.112
0 0
0 0
0 0
0 0
K CaSO4
11.6592
2234.4
0
48.2309
0
K CaSO 4 − 2 H 2 O 815.978
26309.0
138.361
167.863
18.6143
K BaSO4
208.839
13084.5
32.4716
9.58318
2.58594
K SrSO4
89.6687
4033.3
16.0305
1.34671
31402.1
K FeCO3
21.804
56.448
16.8397
0.02298
0
K FeS
8.3102
0
0
0
0
K CaCO3
395.448
6461.5
71.558
180.28
24847
Ref.: Haarberg (1989) Haarberg (1989) Østvold (1998) Haarberg (1989) Haarberg (1989) Haarberg (1989) Haarberg (1989) Østvold (1998) Østvold (1998) Haarberg (1989)
Coefficients in expression for Tdependence of equilibrium constants. T is in Kelvin. The temperature dependence of the selfionization of water is described by Olofsson and Hepler (1982)
(
)
142613.6 + 4229.195 log10 T − 9.7384 T + 0.0129638 T 2 T − 1.15068×10 −5 T 3 + 4.602 × 10 −9 T 4 − 8908.483
− log10 K H 2O (T ) =
The pressure dependence is given by ∂ ln K i ∆ZP − ∆V = ∂P RT
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
 103
= 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. in cm3 /mol/bar. Units: t in oC and a 17.54 17.83 16.13 17.83
BaSO4 SrSO4 CaSO4 CaSO42H2O
100b 1.159 1.159 0.944 1.543
1000c 17.77 17.77 16.52 16.01
106d 17.06 17.06 16.71 16.84
Reference: Atkinson and Mecik (1997) The compressibility changes associated with both of the CO2 acid equilibria are (Haarberg, 1989) 103 ∆Z CO2 ,1 =103 ∆K CO2 , 2 = − 39.3 + 0.233T − 0.000371T 2 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. BaSO4 SrSO4 CaSO4 CaSO42H2O
A 343.6 306.9 282.3 263.8
B 1.746 1.574 1.438 1.358
1000C 2.567 2.394 2.222 2.077
D 11.9 20 21.7 21.7
E 4 8.2 9.8 9.8
Reference: Haarberg (1989). For the calcium carbonate and ferrous carbonate precipitation reactions, the partial molar volume change are described by (Haarberg, 1989)
∆VCaCO3 = ∆VFeCO3 = − 328.7 +1.738T − 0.002794T 2
The partial molar volume changes of both of the acid equilibria of CO2 are (Haarberg, 1989)
∆VCO2 ,1 = ∆VCO2 , 2 =141.4 + 0.735T − 0.0019T 2 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 ln γ M = z M2 F + ∑ ma (2 BMa + ZC Ma ) + ∑ mc 2φ Mc + ∑ ma ΨMca + a c a
∑∑ m m Ψ a ' a >a '
a
a'
Maa '
+ zM
∑∑ m m C c
c
a
ca
a
for the cations, and ln γ X = z X2 F + ∑ mc (2 BcX + ZCcX ) + ∑ ma 2φ Xa + ∑ mc ΨcXa + c a c
∑∑ m m Ψ c ' c >c '
c
c'
cc ' X '
+ zX
∑∑ m m C c
c
a
ca
a
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=
1 2 mi z i ∑ 2 i
z is the charge of the ion considered in the unit of elementary units. ijk is a model parameter that is assigned to each cationcationanion triplet and to each cationanionanion triplet. The remaining quantities in the activity coefficient equations are
I 1/ 2 2 + ln 1 + bI 1 / 2 F = − Aϕ 1/ 2 b 1 + bI
(
∑∑ m m φ c ' c >c '
c
c ' cc '
) + ∑∑ m m B'
c
c
a
ca
+
a
+ ∑∑ ma ma 'φaa ' a ' a >a '
where b is a constant with the value 1.2 kg 1/2/mol1/2 and 1 e2 1/ 2 Aφ = (2πN 0 d w ) 3 4πε 0 DkT
3/ 2
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. (0) (1) ( 2) BMX = β MX + β MX g (α1I 1 / 2 )+ β MX g (α 2 I 1 / 2 )
where g (x ) = (0)
2{1 − (1 + x ) exp (− x )} x2 (1)
β ij , β ij and β ij
(2)
are model parameters. One of each parameter is assigned to each cationanion
pair. 1 and 2 are constants, with 1 = 2 kg1/2 mol1/2 and 2 =12 kg1/2mol1/2. However, for pairs of ions with charge +2 and –2, respectively, the value for 1 is 1.2 kg1/2mol1/2. Further Z = ∑ mi z i i
CMX =
φ CMX
2 zM z X
1/ 2
φijφ = sθ ij + Eθ ij (I ) + I Eθ ij (I )
φijφ = sθ ij + Eθ ij (I )
Cijφ is yet another model parameter assigned to each cationanion pair. S
θ ij is a model parameter assigned to each cationcation pair and to each anionanion pair and
E
θ ij is an electrostatic term
E
θ ij =
zi z j 1 1 J (xij )− J (xii ) − J (x jj ) 4I 2 2
where
xij = 6 zi z j Aφ I 1/ 2
J ( x ) = x{4 + 4.581(x − 0.7231 )exp(− 0.0120 x 0.528 )}
−1
Also the Pitzer model describes the activity of the water in terms of the osmotic coefficient
(φ − 1)∑ mi = − i
∑∑ m m c ' c >c '
c
c'
2 Aφ I 3 / 2 1 + bI
1/ 2
(
)
φ + ∑∑ mc ma Bca + ZCca + c
a
φ φ φcc ' + ∑ ma Ψcc 'a + ∑∑ ma ma ' φaa ' + ∑ mc Ψca 'a a c a ' a>a '
where φ (0) (1) ( 2) BMX = β MX + β MX exp(− α1I 1 / 2 ) β MX exp(− α 2 I 1 / 2 )
and the relation between the osmotic coefficient and the activity of the water is ln aH 2 O = φM H 2 O ∑ mi i
Model parameters at 25°C are listed below. β (0 ) parameters at 25°C H+ Na+ OH 0.00000 0.08640 Cl0.17750 0.07650 SO4 0.02980 0.01810 HCO3 0.00000 0.02800 CO30.00000 0.03620 HSβ (1) parameters at 25°C H+ Na+ OH 0.00000 0.25300 Cl0.29450 0.26640 SO4 0.00000 1.05590 HCO3 0.00000 0.04400 CO30.00000 1.51000 HS
K+ 0.12980 0.04810 0.00000 0.01070 0.12880
Mg++ 0.00000 0.35090 0.21500 0.32900 0.00000
Ca++ 0.17470 0.30530 0.20000 1.49800 0.40000
K+ 0.32000 0.21870 1.10230 0.04780 1.43300
Mg++ 0.00000 1.65100 3.36360 0.60720 0.00000
Ca++ 0.23030 1.70800 3.19730 7.89900 5.30000
Sr++ 0.00000 0.28370 0.20000 0.00000 0.00000
Sr++ 0.00000 1.62600 3.19730 0.00000 0.00000
Ba++ 0.17175 0.26280 0.20000 0.00000 0.00000
Ba++ 1.20000 1.49630 3.19730 0.00000 0.00000
Fe++ 0.00000 0.44790 4.70500 0.00000 1.91900
Fe++ 0.00000 2.04300 17.00000 14.76000 5.13400
β (2 ) parameters at 25°C H+ Na+ OH 0.00000 0.00000 Cl0.00000 0.00000 SO4 0.00000 0.00000 HCO3 0.00000 0.00000 CO30.00000 0.00000 HS 0.00000 0.00000
K+ 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
C φ parameters at 25°C H+ Na+ OH0.00000 0.00410 Cl 0.00080 0.00127 SO4 0.04380 0.00571 HCO3 0.00000 0.00000 CO3 0.00000 0.00520 HSS
θ parameters at 25°C H+ H0.00000 Na+ 0.03600 + K 0.00500 ++ Mg 0.10000 Ca++ 0.06120 ++ Sr 0.06500 Ba++ 0.00000
OHClSO4HCO3CO3
Mg++ 0.00000 0.00000 32.74000 0.00000 0.00000 0.00000
K+ 0.00410 0.00079 0.01880 0.00000 0.00050
Mg++ 0.00000 0.00651 0.02797 0.00000 0.00000
Ca++ 0.00000 0.00000 54.24000 0.00000 879.20000 0.00000 Ca++ 0.00000 0.00215 0.00000 0.00000 0.00000
Sr++ 0.00000 0.00000 54.24000 0.00000 0.00000 0.00000 Sr++ 0.00000 0.00089 0.00000 0.00000 0.00000
Ba++ 0.00000 0.00000 54.24000 0.00000 0.00000 0.00000 Ba++ 0.00000 0.01938 0.00000 0.00000 0.00000
Fe++ 0.00000 0.00000 0.00000 0.00000 0.00000
Na+
K+
Mg++
Ca++
Sr++
Ba++
0.00000 0.01200 0.07000 0.07000 0.05100 0.06700
0.00000 0.00000 0.03200 0.00000 0.00000
0.00000 0.00700 0.00000 0.00000
0.00000 0.00000 0.00000
0.00000 0.00000
0.00000
Cl
SO4
HCO3
CO3
0.00000 0.08900
0.00000
OH0.00000 0.05000 0.01300 0.00000 0.10000
0.00000 0.02000 0.00000 0.03590 0.01000 0.05300 0.02000
Fe++ 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
Ψ parameters at 25°C
Anion 1 fixed as ClH+ H0.00000 + Na 0.00400 + K 0.01100 Mg++ 0.01100 ++ Ca 0.01500 Sr++ 0.00300 ++ Ba 0.01370
Na+
K+
Mg++
Ca++
Sr++
Ba++
0.00000 0.00180 0.01200 0.00700 0.00210 0.01200
0.00000 0.02200 0.02500 0.00000 0.00000
0.00000 0.01200 0.00000 0.00000
0.00000 0.00000 0.00000
0.00000 0.00000
0.00000
Anion 1 fixed as SO4: H+ H0.00000 + Na 0.00000 + K 0.19700 Mg++ 0.00000 ++ Ca 0.00000 Sr++ 0.00000 Ba++ 0.00000 Anion 1 fixed as HCO3H+ H0.00000 + Na 0.00000 K+ 0.00000 ++ Mg 0.00000 Ca++ 0.00000 ++ Sr 0.00000 Ba++ 0.00000 Anion 1 fixed as CO3— H+ H 0.00000 Na+ 0.00000 + K 0.00000 Mg++ 0.00000 ++ Ca 0.00000 ++ Sr 0.00000 Ba++ 0.00000 Cation 1 fixed as Na+ OHClSO4HCO3CO3Cation 1 fixed as K+ 
OH ClSO4HCO3CO3Cation 1 fixed as Mg++
Na+
K+
Mg++
Ca++
Sr++
Ba++
0.00000 0.01000 0.01500 0.05500 0.00000 0.00000
0.00000 0.04800 0.00000 0.00000 0.00000
0.00000 0.02400 0.00000 0.00000
0.00000 0.00000 0.00000
0.00000 0.00000
0.00000
Na+
K+
Mg++
Ca++
Sr++
Ba++
0.00000 0.00300 0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000
0.00000 0.00000
0.00000
Na+
K+
Mg++
Ca++
Sr++
Ba++
0.00000 0.00300 0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000
0.00000 0.00000
0.00000
Cl
SO4
HCO3
CO3
0.00000 0.00140 0.01430 0.00000
0.00000 0.00500 0.00500
0.00000 0.00000
0.00000
OH0.00000 0.00600 0.00900 0.00000 0.01700 OH0.00000 0.00800 0.05000 0.00000 0.01000 OH
Cl
SO4
HCO3
CO3
0.00000 0.00000 0.00000 0.02400
0.00000 0.00000 0.00900
0.00000 0.03600
0.00000
SO4
HCO3
Cl
CO3
OHClSO4HCO3CO3Cation 1 fixed as Ca++ 
OH ClSO4HCO3CO3
0.00000 0.00000 0.00000 0.00000 0.00000
0.00000 0.00400 0.09600 0.00000
OH0.00000 0.02500 0.00000 0.00000 0.00000
0.00000 0.16100 0.00000
0.00000 0.00000
0.00000
Cl
SO4
HCO3
CO3
0.00000 0.01800 0.00000 0.00000
0.00000 0.00000 0.00000
0.00000 0.00000
0.00000
All parameters not listed here are equal to zero. The Pitzer parameters , and
ijk
and ij are temperature independent parameters, whereas are temperature dependent parameters (=X). Their temperature
dependence is described by (Haarberg, 1989) for temperatures in K
X (T ) = X (298.15) +
2 ∂X (T − 298.15)+ 1 ∂ X2 (T − 298.15)2 ∂T 2 ∂T
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 Q1 + Q2 + Q3 P + Q4 ln (T ) + (Q5 + Q6 P )T T Q + Q10 P Q11 + Q12 P + (Q7 + Q8 P )T 2 + 9 + T − 227 680 − T X (T ) =
The temperature coefficients
and
First order temperature derivative of OHClSO4HCO3CO3HS
H+ 0.00000 0.18133 0.00000 0.00000 0.00000
Na+ 0.01879 0.007159 0.16313 0.10000 0.17900
and the coefficient Q1, Q2…..,Q12 are listed below. x 100.
K+ 0.00000 0.03579 0.09475 0.10000 0.11000
Second order temperature derivative of
Mg++ 0.000000 0.05311 0.00730 0.00000 0.00000
x 100.
Ca++ 0.00000 0.02124 0.00000 0.00000 0.00000
Sr++ 0.00000 0.02493 0.00000 0.00000 0.00000
Ba++ 0.00000 0.06410 0.00000 0.00000 0.00000
Fe++ 0.00000 0.00000 0.00000 0.00000 0.00000
Ca++ 0.00000 0.00057 0.00000 0.00000 0.00000
Sr++ 0.00000 0.00621 0.00000 0.00000 0.00000
Ba++ 0.00000 0.00000 0.00000 0.00000 0.00000
First order temperature derivative of x 100. Na+ K+ Mg++ H+ OH 0.00000 0.27642 0.00000 0.00000 Cl0.01307 0.07000 0.11557 0.43440 SO4 0.00000 0.07881 0.46140 0.64130 HCO3 0.00000 0.11000 0.11000 0.00000 CO3 0.00000 0.20500 0.43600 0.00000 HS
Ca++ 0.00000 0.36820 5.46000 0.00000 0.00000
Sr++ 0.00000 0.20490 5.46000 0.00000 0.00000
Ba++ 0.00000 0.32000 5.46000 0.00000 0.00000
Fe++ 0.00000 0.00000 0.00000 0.00000 0.00000
Second order temperature derivative of Na+ K+ H+ OH0.00000 0.00124 0.00000 Cl 0.00005 0.00021 0.00004 SO4 0.00000 0.00908 0.00011 HCO3 0.00000 0.00263 0.00000 CO3 0.00000 0.04170 0.00414 HS
x 100. Mg++ 0.00000 0.00074 0.00901 0.00000 0.00000
Ca++ 0.00000 0.00232 0.00000 0.00000 0.00000
Sr++ 0.00000 0.05000 0.00000 0.00000 0.00000
Ba++ 0.00000 0.00000 0.00000 0.00000 0.00000
Fe++ 0.00000 0.00000 0.00000 0.00000 0.00000
First order temperature derivative of Na+ K+ H+ OH0.00000 0.00000 0.00000 Cl 0.00000 0.00000 0.00000 SO40.00000 0.00000 0.00000 HCO3 0.00000 0.00000 0.00000 CO30.00000 0.00000 0.00000 HS
Mg++ 0.00000 0.00000 0.06100 0.00000 0.00000
Ca++ 0.00000 0.00000 0.51600 0.00000 0.00000
Sr++ 0.00000 0.00000 0.51600 0.00000 0.00000
Ba++ 0.00000 0.00000 0.51600 0.00000 0.00000
Fe++ 0.00000 0.00000 0.00000 0.00000 0.00000
Second order temperature derivative of Na+ K+ H+ OH0.00000 0.00000 0.00000 Cl 0.00000 0.00000 0.00000 SO40.00000 0.00000 0.00000 HCO3 0.00000 0.00000 0.00000 CO30.00000 0.00000 0.00000 HS
Mg++ 0.00000 0.00000 0.01300 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
Ba++ 0.00000 0.00000 0.00000 0.00000 0.00000
Fe++ 0.00000 0.00000 0.00000 0.00000 0.00000
Ca++
Sr++
Ba++
Fe++
OHClSO4HCO3CO3HS
H+ 0.00000 0.00376 0.00000 0.00000 0.00000
Na+ 0.00003 0.00150 0.00115 0.00192 0.00263
K+ 0.00000 0.00025 0.00008 0.00000 0.00102
First order temperature derivative of Na+ K+ H+
Mg++ 0.00000 0.00038 0.00094 0.00000 0.00000
x 100. Mg++
Fe++ 0.00000 0.00000 0.00000 0.00000 0.00000
OHClSO4HCO3CO3HS
0.00000 0.00590 0.00000 0.00000 0.00000
0.00790 0.01050 0.36300 0.00000 0.00000
0.00000 0.00400 0.00625 0.00000 0.00000
0.00000 0.01990 0.02950 0.00000 0.00000
0.00000 0.01300 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000 0.00000
0.00000 0.01540 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000 0.00000
Second order temperature derivative of Na+ K+ H+ OH 0.00000 0.00007 0.00000 Cl0.00002 0.00015 0.00003 SO40.00000 0.00027 0.00023 HCO3 0.00000 0.00000 0.00000 CO30.00000 0.00000 0.00000 HS
x 100. Mg++ 0.00000 0.00018 0.00010 0.00000 0.00000
Ca++ 0.00000 0.00005 0.00000 0.00000 0.00000
Sr++ 0.00000 0.00000 0.00000 0.00000 0.00000
Ba++ 0.00000 0.00000 0.00000 0.00000 0.00000
Fe++ 0.00000 0.00000 0.00000 0.00000 0.00000
Temperature coefficients in expression for temperature dependence of the Pitzer parameters for NaCl φ βNaCl(1) βNaCl(0) C NaCl Q1 6.1084589 6.5684518×102 1.1931966×102 1 1 Q2 2.486912950×10 4.8309327×10 4.0217793×101 Q3 0 5.381275267×105 2.2902837×105 Q4 4.4640952 0 7.5354649×104 Q5 1.4068095×103 1531767295×104 1.110991383×102 7 Q6 0 2.657339906×10 9.0550901×108 Q7 0 5309012889×106 1.53860082×108 Q8 0 8.634023325×1010 8.69266×1011 Q9 1.579365943 4.2345814 3.53104136×101 Q10 0 0.0022022820790×103 4.3314252×104 Q11 9.706578079 0 9.187145529×102 Q12 0 2.686039622×102 5.190477×104 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 stoichiometric equilibrium constants are calculated from the thermodynamic ones and from the activity coefficients.
•
The ratio of CO2(aq) to H2S(aq) is calculated. This determines if any of the ferrous iron minerals FeCO3 and FeS will precipitate. Only one can precipitate, since both H2S and CO2 are fixed in concentration, and then the Fe++ concentration cannot fulfil both solubility products at the same time.
•
The equilibrium in the acid/base reactions is determined without considering the precipitation reactions. The convergence criterion is that the charge balance must be fulfilled.
•
The amount of sulfate precipitation (independent of the acid/base reactions) is calculated, with none of the other precipitation reactions taken into account.
•
The ion product of the iron mineral identified at a previous step is checked against the solubility product. If the solubility product is exceeded, the amount of precipitate of the iron mineral is determined. The convergence criterion in this iteration is the charge balance. Precipitation of calcium carbonate is not included in the calculation.
•
The ion product of calcium carbonate is checked against its solubility product. If the solubility product is exceeded, simultaneous precipitation of calcium carbonate and the iron mineral is calculated. A double loop iteration is applied. The inner loop: With a given amount of ferrous iron mineral precipitation (which comes from the outer loop), the amount of calcium carbonate precipitate is determined. During the calcium carbonate precipitation, the sulfate precipitate is influenced since some Ca++ is removed from the solution. The state in the sulfate system is therefore corrected in each of these inner loop iterations. In the inner loop, the charge balance is used to check for convergence. The outer loop: The iteration variable is the amount of ferrous iron mineral precipitate. Convergence is achieved when the ion product of the ferrous mineral matches the thermodynamic solubility product.
•
The resulting amount of each precipitate is compared to that of the previous iteration. If the weighted sum of relative changes in the amounts of precipitates exceeds 106, 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. 113121. 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. 907915.
Kaasa, B. and Østvold, T., “Prediction of pH and Mineral Scaling in Waters with Varying Ionic Strength Containing CO2 and H2S for 0
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