L9-High Pressure VLE - EOS

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Termodinamica de Termodinamica de HydrocarburosHydrocarburos

High Pressure Phase EquilibriaHigh Pressure Phase Equilibria

EOSEOS

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Petroleum Engineering Applications of Petroleum Engineering Applications of Phase EquilibriaPhase Equilibria

High Pressure Phase Equilibria High Pressure Phase Equilibria Applications (Reservoir).Applications (Reservoir).

Equations of State Models (EOS). Cubic Equations of State Models (EOS). Cubic EOS. Root Selection.EOS. Root Selection.

Evaluation of Fugacity Coefficients from Evaluation of Fugacity Coefficients from Equations of State.Equations of State.

Evaluation of Phase Boundaries (Dew and Evaluation of Phase Boundaries (Dew and Bubble Points) and Flash Equilibrium with Bubble Points) and Flash Equilibrium with EOS. EOS.

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Petroleum Engineering Applications of Petroleum Engineering Applications of Phase EquilibriaPhase Equilibria

Tuning of Equations of State (EOS). Tuning of Equations of State (EOS). Miscible Gas Injection. Swelling TestsMiscible Gas Injection. Swelling Tests

Constant Volume Depletion Studies (Gas Constant Volume Depletion Studies (Gas Condensates and Volatile Oils).Condensates and Volatile Oils).

Determination of Oil and Gas in Place by Determination of Oil and Gas in Place by Recombination. Recombination.

Additional Reading: Selected SPE papersAdditional Reading: Selected SPE papers

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Instructional ObjectivesInstructional Objectives

After seeing this module the student After seeing this module the student should be able to: should be able to:

Evaluate volume roots from a cubic equation Evaluate volume roots from a cubic equation of state when two-phases coexist.of state when two-phases coexist.

Derive and evaluate fugacity coefficients Derive and evaluate fugacity coefficients from cubic EOS.from cubic EOS.

Evaluate phase boundaries (dew and bubble Evaluate phase boundaries (dew and bubble points) using EOS.points) using EOS.

Evaluate flash separations using an EOS.Evaluate flash separations using an EOS.

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Instructional Objectives (cont.)Instructional Objectives (cont.)

Understand the production mechanism for Understand the production mechanism for a gas condensate or a volatile oil a gas condensate or a volatile oil (Constant Volume Depletion, CVD).(Constant Volume Depletion, CVD).

Determine oil and gas in place and Determine oil and gas in place and recoveries using CVD compositional data.recoveries using CVD compositional data.

Determine oil and gas in place from Determine oil and gas in place from recombination.recombination.

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Equations of State (EOS)Equations of State (EOS)

Single Component SystemsSingle Component Systems

Equations of State (EOS) are Equations of State (EOS) are mathematical relations between mathematical relations between pressure (P) temperature (T), and molar pressure (P) temperature (T), and molar volume (V). volume (V).

For a pure component in a single phase For a pure component in a single phase (i.e. gas or liquid) given any (i.e. gas or liquid) given any pairpair of of these variables the third can be these variables the third can be evaluated. evaluated.

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If the component exhibits phase If the component exhibits phase equilibrium an additional constraining equilibrium an additional constraining equation is placed and only one equation is placed and only one variable, either P or T are needed to variable, either P or T are needed to specify the STATE of the system. specify the STATE of the system. (Recall phase rule)(Recall phase rule)

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For VLE, the constraining equation is For VLE, the constraining equation is the equality of the Gibb’s energy of the the equality of the Gibb’s energy of the GAS and LIQUID phases. GAS and LIQUID phases.

A CUBIC EOS is one of the many A CUBIC EOS is one of the many different models available to evaluate different models available to evaluate this ENERGY this ENERGY

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The ideal gas EOS was the first mathematical The ideal gas EOS was the first mathematical expression used to describe the PVT behavior expression used to describe the PVT behavior of gases of gases

Its use is limited to low pressures (near Its use is limited to low pressures (near atmospheric) atmospheric)

Since only provides a single volume for a Since only provides a single volume for a given pressure and temperature, it cannot given pressure and temperature, it cannot describe phase transitionsdescribe phase transitions

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Equations of State (EOS)Equations of State (EOS)Multicomponent SystemsMulticomponent Systems

For multicomponent mixtures in addition to For multicomponent mixtures in addition to variables (P, T & V) , the overall molar variables (P, T & V) , the overall molar composition and a set of mixing rules are composition and a set of mixing rules are needed.needed.

Modern EOS’s are versatile tools for reservoir Modern EOS’s are versatile tools for reservoir engineering applications. They can be used for engineering applications. They can be used for all states (gas, liquid, and solid), and all states (gas, liquid, and solid), and they can they can describe phase transition conditions and describe phase transition conditions and properties of the coexisting phasesproperties of the coexisting phases. .

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Multicomponent SystemsMulticomponent Systems

Some of the EOS uses include: Some of the EOS uses include:

evaluation of gas injection processes evaluation of gas injection processes (miscible and immiscible), (miscible and immiscible),

evaluation of properties of a reservoir evaluation of properties of a reservoir oil (liquid) coexisting with a gas cap oil (liquid) coexisting with a gas cap (gas), (gas),

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Multicomponent SystemsMulticomponent Systems

Some of the EOS uses include: Some of the EOS uses include:

simulation of volatile and gas condensate simulation of volatile and gas condensate production through constant volume production through constant volume depletion evaluations, depletion evaluations,

recombination tests using separator oil and recombination tests using separator oil and gas streams, gas streams,

evaluation of paraffin deposition in the evaluation of paraffin deposition in the wellbore, etc.wellbore, etc.

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Types of EOSTypes of EOS

There are many families of EOS’, There are many families of EOS’, suitable for different purposes and suitable for different purposes and substances i.e. hydrocarbons, substances i.e. hydrocarbons, electrolytes, solids, gas-liquid-solid electrolytes, solids, gas-liquid-solid equilibria, etc. equilibria, etc.

In petroleum engineering the most In petroleum engineering the most commonly used EOS are cubic commonly used EOS are cubic polynomials in volume. polynomials in volume.

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Cubic equations are EXPLICIT in Cubic equations are EXPLICIT in pressure and can be written as the sum pressure and can be written as the sum of a term indicating repulsion forces of a term indicating repulsion forces and a term indicating attraction forces and a term indicating attraction forces

attrrep PPP

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The Father of Cubic The Father of Cubic Equations of State Equations of State

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One of the most used EOS’ in One of the most used EOS’ in petroleum engineering is the Peng-petroleum engineering is the Peng-Robinson EOS (1975), which is a Robinson EOS (1975), which is a three-parameter corresponding three-parameter corresponding states model.states model.

)()( bVbbVV

a

bV

RTP

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The critical point conditions are used The critical point conditions are used to determine the EOS parametersto determine the EOS parameters

0

0

2

2

c

c

T

T

V

P

V

P

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Solving these two equations Solving these two equations simultaneously for the Peng-simultaneously for the Peng-Robinson EOS providesRobinson EOS provides

c

ca P

TRa

22

c

cb P

RTb andand

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Where

07780.0

45724.0

b

a

211 rTm

22699.054226.137464.0 m

and

with

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PR equation can be expressed as a PR equation can be expressed as a cubic polynomial in (V) or (Z). cubic polynomial in (V) or (Z).

RT

bPB

RT

PaA

23 2

2

2 3

( 1)

( 3 2 )

( ) 0

Z B Z

A B B Z

AB B B

withwith

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Cubic EOS’s are the simplest Cubic EOS’s are the simplest polynomials that can provide an polynomials that can provide an adequate description of both: liquid adequate description of both: liquid and gas propertiesand gas properties

EOS’s can describe the state of pure EOS’s can describe the state of pure fluids and mixtures (single or fluids and mixtures (single or multiphase) and their properties.multiphase) and their properties.

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When working with mixtures the When working with mixtures the same expressions apply except that same expressions apply except that (a(a) and (b) are evaluated for a ) and (b) are evaluated for a mixture using a set of mixing rules. mixture using a set of mixing rules. The most commonly used mixing The most commonly used mixing rules (MR) are:rules (MR) are:

Quadratic MR for a Quadratic MR for a

Linear MR for bLinear MR for b

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Quadratic MR for aQuadratic MR for a

0.5

1 1

1Nc Nc

i j i j i j i jmi j

a x x a a k

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Linear MR for bLinear MR for b

where the kij’s are called interaction where the kij’s are called interaction parameters and by definitionparameters and by definition

1

Nc

m i ii

b x b

0

ij ji

ii

k k

k

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ExampleExample

For a three-component mixture (Nc = For a three-component mixture (Nc = 3) the attraction (a) and the repulsion 3) the attraction (a) and the repulsion constant (b) are given byconstant (b) are given by

1

0.5 0.5

1 2 1 2 1 2 12 2 3 2 3 2 3 23

0.5 2 21 3 1 3 1 3 13 1 1 2 2 2

23 3 3

2 (1 ) 2 (1 )

2 (1 )

ma x x a a k x x a a k

x x a a k x a x a

x a

1 1 2 2 3 3

mb x b x b x b

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A system may not exist as a SINGLE A system may not exist as a SINGLE PHASE at the specified P, T, and overall PHASE at the specified P, T, and overall composition zi. Energy requirements may composition zi. Energy requirements may cause the system to split into two (or cause the system to split into two (or more) phases such that the more) phases such that the Gibb’s Gibb’s ENERGY of the system (i.e VAPOR + ENERGY of the system (i.e VAPOR + LIQUID)LIQUID) is at a is at a MINIMUMMINIMUM. These two . These two phases will have different compositions phases will have different compositions from the original system (from the original system (yyii & & xxii).).

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The overall mixture composition will be The overall mixture composition will be used to evaluate a unique set of liquid used to evaluate a unique set of liquid and gas compositions such that and gas compositions such that material balance is satisfied. material balance is satisfied.

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The overall composition zi is normally The overall composition zi is normally provided to you, this composition may provided to you, this composition may coincide with the composition of a coincide with the composition of a phase boundary , or you may need to phase boundary , or you may need to evaluate gas and liquid compositions evaluate gas and liquid compositions ((yyii, , xxii, , ii = = 1, 2…Nc1, 2…Nc) which will normally ) which will normally be evaluated from a flash computation.be evaluated from a flash computation.

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Therefore the constants a and b can Therefore the constants a and b can be evaluated usingbe evaluated using

Overall compositions Overall compositions zzii with with ii = = 1, 2…Nc1, 2…Nc

Liquid compositions Liquid compositions xxii with with ii = = 1, 2…Nc1, 2…Nc

Vapor compositions Vapor compositions yyii with with ii = = 1, 1, 2…Nc2…Nc

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The cubic expression for a mixture is then The cubic expression for a mixture is then evaluated usingevaluated using

2 mm

m m

a P b PA B

RTRT

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Analytical Solution of Cubic Analytical Solution of Cubic EquationsEquations

The cubic EOS can be arranged into The cubic EOS can be arranged into a polynomial and be solved a polynomial and be solved analytically as follows. analytically as follows.

3 2

2

2 3

( 1)

( 3 2 )

( ) 0

Z B Z

A B B Z

AB B B

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Let’s write the polynomial in the Let’s write the polynomial in the following wayfollowing way

3 2 31 2 0x a x a x a

Note:Note: “x” could be either the molar “x” could be either the molar volume, or the density, or the z-volume, or the density, or the z-factorfactor

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When the equation is expressed in When the equation is expressed in terms of the z factor, the coefficients terms of the z factor, the coefficients aa11 to to aa33 are: are:

1

22

2 33

( 1)

( 3 2 )

( )

a B

a A B B

a AB B B

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Procedure to Evaluate the Roots Procedure to Evaluate the Roots of a Cubic Equation Analyticallyof a Cubic Equation Analytically

22 1

31 2 3 1

3 23

3 23

3

9

9 27 2

54

a aQ

a a a aR

S R Q R

T R Q R

LetLet

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1 1

2 1

3 1

1

31 1 1

32 3 21 1 1

32 3 2

x S T a

x S T a i S T

x S T a i S T

The solutions are,The solutions are,

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If If aa11, , aa22 and and aa33 are real and if are real and if D = QD = Q33 + + RR22 is the discriminant, then is the discriminant, then

One root is real and two complex One root is real and two complex conjugate if conjugate if D > 0D > 0;;

All roots are real and at least two are All roots are real and at least two are equal if equal if D = 0D = 0; ;

All roots are real and unequal if All roots are real and unequal if D < 0D < 0..

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wherewhere

1 1

2 1

3 1

1 12 cos

3 3

1 1If 0 2 cos 120

3 3

1 12 cos 240

3 3

x Q a

D x Q a

x Q a

3cos

R

Q

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where where xx11, , xx22 and and xx33 are the three roots. are the three roots.

1 2 3 1

1 2 2 3 3 1 2

1 2 3 3

x x x a

x x x x x x a

x x x a

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The range of solutions that are used The range of solutions that are used for the engineer are those for for the engineer are those for positive volumes and pressures, we positive volumes and pressures, we are not concerned about imaginary are not concerned about imaginary numbers.numbers.

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Solutions of a Cubic PolynomialSolutions of a Cubic Polynomial

Basically, from the general shape of

the polynomial we are interested in the

first quadrant.

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Solutions of a Cubic PolynomialSolutions of a Cubic Polynomial

http://www.uni-koeln.de/math-nat-http://www.uni-koeln.de/math-nat-fak/phchem/deiters/quartic/fak/phchem/deiters/quartic/quartic.htmlquartic.html contains Fortran codes contains Fortran codes to solve the roots of polynomials up to solve the roots of polynomials up to fifth degree.to fifth degree.

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Web site to download Fortran source codes Web site to download Fortran source codes to solve polynomials up to fifth degreeto solve polynomials up to fifth degree

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Phase equilibrium for a single Phase equilibrium for a single component at a given temperature component at a given temperature can be graphically determined by can be graphically determined by selecting the saturation pressure selecting the saturation pressure such that the areas above and below such that the areas above and below the loop are equal, these are known the loop are equal, these are known as the van der Waals loops.as the van der Waals loops.

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van der Waals loops showing the van der Waals loops showing the Maxwell Equal Area rule (Maxwell Equal Area rule (AA11 = = AA22).).

-10

0

0

100

200

300

400

500

600

700

2 4 6 8 10

12

14

A1

A2

Pre

ssu

re

Mo la r V olum e

Tc

T2

T1P1

v

L

2 - P has es

CP

V

L

V

1

2

3

4

7 6

5

0

TV~P

-10

0

0

100

200

300

400

500

600

700

2 4 6 8 10

12

14

A1

A2

Pre

ssu

re

-10

0

0

100

200

300

400

500

600

700

2 4 6 8 10

12

14

A1

A2

Pre

ssu

re

Mo la r V olum e

Tc

T2

T1P1

v

L

2 - P has es

CP

V

L

V

1

2

3

4

7 6

5

0

TV~P

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Although the EOS does not provide Although the EOS does not provide viscosities (a transport property), it viscosities (a transport property), it gives phase compositions that are gives phase compositions that are used in the evaluation of viscosities.used in the evaluation of viscosities.

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Along the production path we will Along the production path we will need to evaluate:need to evaluate:

Bubble point pressure (Bubble point pressure (PPbb) at given ) at given

reservoir reservoir TT and overall and overall zzii..

Properties of gas & liquid, below (Properties of gas & liquid, below (PPbb), ),

through equilibrium computations in through equilibrium computations in the two-phase region (flash the two-phase region (flash computations).computations).

04/19/2304/19/23 4747


For a retrograde fluid, along the For a retrograde fluid, along the production path we will need to production path we will need to evaluate:evaluate:

Dew point pressure (Dew point pressure (PPdd) at given ) at given

reservoir T and overall reservoir T and overall zzii..

Properties of gas & liquid, below (Properties of gas & liquid, below (PPdd), ),

through equilibrium computations in the through equilibrium computations in the two-phase region (flash computations).two-phase region (flash computations).

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The number of equilibrium equations The number of equilibrium equations to be solved is equal to the number to be solved is equal to the number of components that make up the of components that make up the mixture times the number of phases mixture times the number of phases minus one. minus one. (i.e. 10 components and (i.e. 10 components and three-phases (gas / liquid 1/liquid 2 three-phases (gas / liquid 1/liquid 2 20 equations). 20 equations).

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Two-phase VLETwo-phase VLE

For two-phase vapor-liquid-For two-phase vapor-liquid-equilibrium VLE these equations are equilibrium VLE these equations are expressed as the equal fugacity expressed as the equal fugacity constraintsconstraints

((i = 1, 2, 3, …Nci = 1, 2, 3, …Nc))ˆ ˆv li if f

04/19/2304/19/23 5050

Physically, the difference of the Physically, the difference of the fugacities of one component in one fugacities of one component in one phase with respect to another phase phase with respect to another phase gives a measure of the potential for gives a measure of the potential for transfer of that component between transfer of that component between these phases. Equal fugacities of a these phases. Equal fugacities of a

component in the two (or more component in the two (or more phases) results in zero net mass phases) results in zero net mass

transfer across the phases, or transfer across the phases, or equilibrium.equilibrium.

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When an EOS is used to evaluate the When an EOS is used to evaluate the fugacities an alternative expression fugacities an alternative expression is using the fugacity coefficients . is using the fugacity coefficients .

ˆ ˆ,v li i ˆ ˆ

ˆ ˆ

l li i i

v li i i i

f x P

y x

wherewhere

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Dependent and independent variables used Dependent and independent variables used in typical phase equilibrium problemsin typical phase equilibrium problems

Given Variables(independent)

Unknown Variables(dependent)

Problem Type Example Application

P, zi = xi T, yi Bubble Point Distillation

T, zi = xi P,yi Bubble PointGas injection,Production

P, zi = yi T,xi Dew Point SeparationsT, zi = yi P,yi Dew Point Gas Condensates,

ProductionP, T, zi xi, yi, fv Flash Production

Separation

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We will present the algorithms for We will present the algorithms for the: DEW – BUBBLE & FLASH the: DEW – BUBBLE & FLASH equilibrium calculations which are equilibrium calculations which are general regardless of the EOS and general regardless of the EOS and mixing rule used. mixing rule used.

OUR GOAL: Solve phase equilibrium for OUR GOAL: Solve phase equilibrium for binary mixtures and compare our binary mixtures and compare our results from those evaluated with a results from those evaluated with a commercial package.commercial package.

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The phase equilibria equations are The phase equilibria equations are expressed in terms of the expressed in terms of the equilibrium ratios, or more equilibrium ratios, or more commonly called the “K-values”. commonly called the “K-values”. The K-value of component “i” is The K-value of component “i” is defined as:defined as:

ˆ

ˆ

li i

i vi i

yK

x

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Dew Point CalculationsDew Point Calculations

Equilibrium is always stated as:Equilibrium is always stated as:

ˆ ˆl vi i i ix P y P

1 1 1

1, 1, 1Nc Nc Nc

i i ii i i

x y z

(i = 1, 2, 3 ,…Nc) (i = 1, 2, 3 ,…Nc)

with the following material balance with the following material balance constrainsconstrains

04/19/2304/19/23 5656


At the dew-point zAt the dew-point zii = y = yii . Therefore . Therefore

making use of the K-value definition:making use of the K-value definition:

ˆ ˆl vi i i i

i i i

x z

x K z

(i = 1, 2, 3 ,…Nc)(i = 1, 2, 3 ,…Nc)

04/19/2304/19/23 5757


For a Dew-Point equilibrium For a Dew-Point equilibrium calculation the objective is to find a calculation the objective is to find a uniqueunique set of K-values that satisfies, set of K-values that satisfies,

1

1 0Nc

i

i i

z

K

04/19/2304/19/23 5858

Bubble Point Equilibrium Bubble Point Equilibrium CalculationsCalculations

For a Bubble-point equilibrium For a Bubble-point equilibrium calculation, the objective function is calculation, the objective function is derived following the same derived following the same reasoning as:reasoning as:

1

1 0Nc

i ii

z K

04/19/2304/19/23 5959

Flash Equilibrium CalculationsFlash Equilibrium Calculations

The objective is to find the The objective is to find the ffvv in a VL in a VL

mixture at a specified T and P such mixture at a specified T and P such thatthat

1

( 1)0

1 ( 1)

cNi i

i v i

z K

f K

04/19/2304/19/23 6060

Flash Equilibrium CalculationsFlash Equilibrium Calculations

This is also known as the Rachford-This is also known as the Rachford-Rice function and it is derived from:Rice function and it is derived from:

1 1

0Nc Nc

i ii i

y x

04/19/2304/19/23 6161

Numerical Behavior of Flash FunctionsNumerical Behavior of Flash Functions

Numerical Behavior of Flash Functions

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

0.00 0.20 0.40 0.60 0.80 1.00

Molar Fraction of vapor (fv)

F(fv

)

Sum XiSum YiRachford Rice

Notice that the three have the same solution but only the Rachford Rice function is monotonic and therefore it is well behaved from a numerical stand point.

04/19/2304/19/23 6262


The two-phase equilibria criteria is The two-phase equilibria criteria is that both functions (dew and bubble) that both functions (dew and bubble) must be greater than one.must be greater than one.

1 1

1, 1Nc Nc

ii i

i i i

zz K

K

04/19/2304/19/23 6363


The equilibrium GAS compositions The equilibrium GAS compositions are evaluated as:are evaluated as:

1 ( 1)i i

iv i

z Ky

f K

ii

i

yx

K

with the liquid compositions evaluated with the liquid compositions evaluated as,as,

04/19/2304/19/23 6464

Two-phase VLE - Two-phase VLE - ExampleExample

zzii may indicate the composition of the may indicate the composition of the

reservoir oil fluid above the Bubble-reservoir oil fluid above the Bubble-Point. Point.

Later in production, the pressure in Later in production, the pressure in the reservoir drops and this initial the reservoir drops and this initial single-phase fluid separates into two single-phase fluid separates into two phases. phases.

04/19/2304/19/23 6565

Compositional Changes that May Compositional Changes that May Occur due to Production or InjectionOccur due to Production or Injection

Temperature

t1

Composition Changes Due to Production

and Gas InjectionP

ress

ure

t3

t2

GasInjection

Production

Temperature

t1

Composition Changes Due to Production

and Gas InjectionP

ress

ure

t3

t2

GasInjection

Production

04/19/2304/19/23 6666

TD

T B

Ta

z1 y1x1 10

11

Nc

i i

i

K

z

11

Nc

i i

i

K

z

11

Nc

i i

i

K

z P = Pa

Tem

pera

ture

TX diagram for the TX diagram for the behavior of the Dew behavior of the Dew Point function at the Point function at the boundaries and in boundaries and in the 2-phase regionthe 2-phase region

04/19/2304/19/23 6767

Two-phase VLE - Two-phase VLE - Dew Point TDew Point T

When the objective is to find the DEW When the objective is to find the DEW POINT temperature at a fixed pressure POINT temperature at a fixed pressure ((PPaa) and overall composition:) and overall composition:

1

1Nc

i

i i

z

K

1

1Nc

i

i i

z

K

Superheated fluid Superheated fluid lower T lower T

Subcooled fluid Subcooled fluid increase T increase T

IfIf

IfIf

04/19/2304/19/23 6868


One way of updating the temperature isOne way of updating the temperature is

1

1

Nck k i

i i

zT T

K

1

1Nc

i

i i

z

K

At the dew pointAt the dew point

04/19/2304/19/23 6969


The schemes indicated are secure The schemes indicated are secure but slow, there are other well but slow, there are other well documented numerical accelerating documented numerical accelerating schemes, but numerical methods is schemes, but numerical methods is not the main objective of this course.not the main objective of this course.

04/19/2304/19/23 7070

Behavior of the Dew point Behavior of the Dew point function in a PX diagramfunction in a PX diagram

11

Nc

i i

i

K

z

11

Nc

i i

i

K

z

11

Nc

i i

i

K

zPr e

ssur

e

PD

PB

Pa

z1 y1x1 10

T= Ta

04/19/2304/19/23 7171

Two-phase VLE - Two-phase VLE - Dew Point PDew Point P

Under pressurized fluid Under pressurized fluid increase Pincrease P

1

1Nc

i

i i

z

K

1

1Nc

i

i i

z

K

When the objective is to find the DEW POINT When the objective is to find the DEW POINT PRESSURE at a fixed temperature (PRESSURE at a fixed temperature (TTaa) and ) and overall composition,overall composition,

Over pressurized fluid Over pressurized fluid lower P lower P

If If

If If

04/19/2304/19/23 7272

Two-phase VLE - Two-phase VLE - Dew Point PDew Point P

One way of updating the pressure isOne way of updating the pressure is

1

1

Nck k i

i i

zP P

K

1

1Nc

i

i i

z

K

At the dew pointAt the dew point

04/19/2304/19/23 7373

Two-phase VLE - Two-phase VLE - Dew PointDew Point

For the upper DEW POINT of GAS For the upper DEW POINT of GAS CONDENSATES we must use other CONDENSATES we must use other numerical approaches. When the numerical approaches. When the initial guess is in the two-phase initial guess is in the two-phase region the approach just explained region the approach just explained will converge to the lower dew point.will converge to the lower dew point.

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Two-phase VLE - BubbleTwo-phase VLE - Bubble Point P Point P

When the objective is to find the BUBBLE When the objective is to find the BUBBLE POINT PRESSURE at a fixed temperature (Ta) POINT PRESSURE at a fixed temperature (Ta) and overall composition, thenand overall composition, then

1

1Nc

i ii

z K

1

1Nc

i ii

z K

Over pressurized fluid Over pressurized fluid lower P lower P

Under pressurized fluid Under pressurized fluid increase P increase P

If If

If If

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Two-phase VLE - BubbleTwo-phase VLE - Bubble Point P Point P

One way of updating the pressure isOne way of updating the pressure is

1

1

k

Nck

i ii

P P z K

1

1Nc

i ii

z K

At the bubble pointAt the bubble point

04/19/2304/19/23 7676


To ensure the existence of two-To ensure the existence of two-phases both functions (DEW & phases both functions (DEW & BUBBLE) must be:BUBBLE) must be:

1 1

1, 1Nc Nc

ii i

i ii

zz K

K

04/19/2304/19/23 7777

Behavior of the Bubble Point Behavior of the Bubble Point function in a PX diagramfunction in a PX diagram

11

Nc

iiiKz

11

Nc

iiiKz

11

Nc

iiiKz

Pr e

ssur

e

PD

PB

Pa

z1 y1x1 10

T= Ta

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Behavior Behavior of the of the

Bubble Bubble function function on a TX on a TX

and a PX and a PX diagram.diagram.

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Dew and Bubble functions Dew and Bubble functions behavior on a PT diagrambehavior on a PT diagram

.

Pb

Pd

Pre

ssu

re

Temperature

CP

Bubble-Curve

Dew

-Cur

ve

11

Nc

i i

i

Kz

11

Nc

i i

i

Kz

11

Nc

i i

i

Kz

11

Nc

iiiKz

11

Nc

iiiKz

11

Nc

iiiKz

2-phases

Tr

A

B

L9-High Pressure VLE - EOS

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