q913 rfp w3 lec 9

Reservoir Fluid Properties Course (1st Ed.)

mailto:[email protected]

http://www.about.me/AlamiNia

mailto:[email protected]

http://www.about.me/AlamiNia

1. General Notes about EoS

2. Ideal Gas EoS

3. Compressibility Factor

4. Van Der Waals EoS

2013 H. AlamiNia Reservoir Fluid Properties Course: Advanced EoS and C7+ Characterization 2

1. Cubic EoS:A. SRK EoS

B. PR EoS

C. Other Cubic EoS

2. Non Cubic EoS

3. EoS for Mixtures

4. Hydrocarbons A. Components

B. Mixtures

C. Heavy Oil


Developments of Cubic Equations of StateThe van der Waals equation is seen to qualitatively

describe the pure-component phase behavior at temperatures above, equal to, and below the critical temperature.

Later developments of cubic equations of state have primarily served to improve the quantitative predictions of either vapor pressure or phase properties.

In addition, much effort has been used to extend the application area of cubic equations of state from pure components to mixtures.


Redlich and Kwong Equation

The equation of Redlich and Kwong (1949) is, by many, considered the first modern equation of state and takes the form

By comparing this equation with the van der Waals equation, it is seen that the attractive term has a more complicated temperature dependence. This temperature modification serves to improve the

vapor pressure predictions.

The parameters a and b are found by imposing the critical point criteria.


𝑷 =𝑹𝑻

𝑽 − 𝒃−

𝒂

𝑻𝑽 𝑽 + 𝒃

The Soave–Redlich–Kwong (SRK) EquationSoave (1972) found the pure-component vapor

pressures calculated from the Redlich–Kwong (RK) equation to be somewhat inaccurate.

He suggested replacing the term in the RK equation by a more general temperature dependent term, a (T), giving an equation of state of the form:

This equation is usually referred to as the Soave-Redlich-Kwong or just SRK equation.


𝑷 =𝑹𝑻

𝑽 − 𝒃−𝒂 𝑻

𝑽 𝑽 + 𝒃

Terms in SRK Equation

ω is the acentric factor


𝒃 =0.08664𝑹𝑻𝒄𝑷𝒄

𝒂 𝑻 = 𝒂𝒄𝜶 𝑻 , 𝒂𝒄 =0.42747𝑹2𝑻𝒄

2

𝑷𝒄

𝜶 𝑻 = 1 +𝒎 1 −𝑻

𝑻𝒄

2

,𝒎 = 0.48 + 1.574𝝎 − 0.176𝝎2

SRK Equation in Terms of Z

With the classical Soave temperature dependence, α (T) = 1 at the critical temperature, where a (T) therefore becomes equal to ac.

Recalling that the compressibility factor Z is defined as Z= (PV)/ (RT), SRK Equation (P=RT/ (V-b)-a (T)/ (V (V+b))) may be rewritten in terms of Z:

With the SRK equation, the compressibility factor of a pure component at its critical point will always be equal to 0.333.


𝒁3 − 𝒁2 + (𝑨 − 𝑩 + 𝑩2)𝒁 − 𝑨𝑩 = 0

𝑨 =𝒂 𝑻 𝑷

𝑹2𝑻2, 𝑩 =𝒃𝑷

𝑹𝑻

Peng–Robinson (Pr) Equation

The liquid-phase densities predicted using the SRK equation are in general too low.

Peng and Robinson (1976) traced this deficiency to the fact that the SRK equation predicts the pure component critical compressibility factor to be 0.333. The critical compressibility factors are generally of the order 0.25 to 0.29, i.e., somewhat lower than simulated using the SRK equation.


PR Formulation

Peng and Robinson suggested an equation of the form:


𝑷 =𝑹𝑻

𝑽 − 𝒃−

𝒂 𝑻

)𝑽 𝑽 + 𝒃 + 𝒃(𝑽 − 𝒃

𝒃 =0.07780𝑹𝑻𝒄𝑷𝒄

𝒂 𝑻 = 𝒂𝒄𝜶 𝑻 , 𝒂𝒄 = 0.45724𝑹2𝑻𝒄2

𝑷𝒄

𝜶 𝑻 = 1 +𝒎 1 −𝑻

𝑻𝒄

2

,𝒎 = 0.37464 + 1.54226𝝎 − 0.26992𝝎2

Other Cubic Equations of State

The increasing popularity of cubic equations of state in the 1970s and 1980s inspired thermodynamics research groups to propose alternatives to the SRK and PR equations. Many of these equations have the general form:

The Equation offers the opportunity to include three different volumetric correction parameters, δ1, δ2, and δ3.


𝑷 =𝑹𝑻

𝑽 + 𝜹1−

𝒂 𝑻

𝑽 + 𝜹2 𝑽 + 𝜹3

Which EoS to Use?

In the petroleum industry, it is important with some kind of industrial standards to enable different companies working on the same project to produce consistent calculation results. PR seems to be the preferred choice in North America.

Europe generally prefers SRK, while

The rest of the world is more divided between the two equations of state.


Other Equations of State

Much exploration activity is currently directed towards deep reservoirs at high temperature and high pressure.

The ability of the classical cubic equations of state to represent the molecular interactions at such conditions has often been questioned.

More sophisticated equations of state have been proposed, some of which include terms to account for the strong repulsive forces acting at high pressures.


Other Equations of State (Cont.)

There is little evidence that any of these equations should be more suited for representing the PVT properties of petroleum reservoir fluids at elevated pressures and temperatures than a conventional cubic equation of state.

When it comes to simulating hydrocarbon liquid–liquid split as, for example, oil–asphaltene equilibria, more advanced equations of state, for example, the PC-SAFT equation may be needed.


Differences between EoS for a Pure Component and MixturesTypically, a model for a pure component physical

property contains parameters that are constant or temperature-dependent and found either by fitting to data or by CSP.

Thus, the EoS models for pure gases and liquids express the relationship among the variables P, V, and T.


Differences between EoS for a Pure Component and Mixtures (Cont.)To describe mixture properties, it is necessary to

include composition dependence which adds considerable richness to the behavior, and thus complicates modeling.

Therefore, a mixture equation of state (EoS) is an algebraic relation between P, V, T, and {y}, where {y} is the set of n-1 independent mole fractions of the mixture’s n components.


Challenges to EoS Models for Mixtures

Composition Dependence of Liquid Partial Properties,

Multiphase Equilibria,

The Critical Region and High Pressures


Composition Dependence of Liquid Partial PropertiesThe composition dependence of the properties of

liquid mixtures is fundamentally different from that of a vapor or gas.

The strongest effect on gaseous fluids is caused by changes in system density from changes in pressure; composition effects are usually of secondary importance, especially when mixing is at constant volume.

Except at high pressures, vapors are not dissimilar to ideal gases and deviations from ideal mixing are small.


Applications of the Equation of State

The Determination of the Equilibrium RatiosThe system temperature T, the system pressure p, and

the overall composition of the mixture zi must be known.

Determination of the Dew-Point PressureA saturated vapor exists for a given temperature at the

pressure at which an infinitesimal amount of liquid first appears (Pd).

Determination of the Bubble-Point PressureThe pressure at which the first bubble of gas is formed

Determination of the Mixture Critical Properties


C7+ Characterization

Naturally occurring oil or gas condensate mixtures may contain thousands of different components. Such high numbers are impractical in flash calculations.

Some components must be lumped together and represented as pseudocomponents. C 7 + characterization consists of representing the

hydrocarbons with seven and more carbon atoms (the heptane plus or C 7 + fraction) as a convenient number of pseudocomponents and to find the needed equation of state parameters (T c, P c, and ω) for each of these pseudocomponents.


The Characterization (or Lumping) Problem


Classes of Components

The components contained in oil and gas condensate mixtures can be divided into three classes: Defined components:

The defined components contained in petroleum reservoir fluids are N 2 , CO 2 , H 2 S, C 1 , C 2 , C 3 , iC 4 , nC 4 , iC 5 , nC 5 , and C 6

C 7 + fractions:Each C 7 + fraction contains hydrocarbons with boiling points

within a given temperature interval.

Plus fraction: The plus fraction consists of the components that are too

heavy to be separated into individual C 7 + fractions.



Sample of Components

Molar Composition of North Sea Gas Condensate

Mixing of Multiple Fluids

There is often a need to mix a number of reservoir fluid compositions into one. This is, for example, the case when multiple fluids are let

to the same process plant.

When representing the mixed stream, one may either work With a weaved composition where the

pseudocomponents of each stream are retained or

With a truly mixed composition.



Weaved Composition

The two compositions have initially been characterized individually. For

both fluids, the C7+ fraction is represented using three

pseudocomponents. As is seen, the pseudocomponent properties differ

between the two fluids.

The Molar Amounts of the Weaved CompositionIn the weaved composition, the molar amounts of

the defined components have been obtained as a simple average of the molar concentrations of these compounds in each individual composition.

The weaved composition contains all the pseudocomponents found in each of the two compositions.


Mixing

It is recommended to carry out the mixing before lumping into pseudocomponents. Say NFLUID different fluids are to be mixed, the properties of carbon number fraction i of the mixed fluid are found from

In these equations, zij is the molar fraction of carbon number fraction i in the j-th composition to be mixed. Frac (j) is the mole fraction of the j-th composition of the total mixture.


𝑻𝒄𝒊𝒎𝒊𝒙 =

𝒋=1𝑵𝑭𝑳𝑼𝑰𝑫𝑭𝒓𝒂𝒄 𝒋 𝒛𝒊

𝒋𝑻𝒄𝒊𝒋


𝒋 , 𝑷𝒄𝒊𝒎𝒊𝒙 =


𝒋𝑷𝒄𝒊𝒋


𝒋 ,

𝝎𝒄𝒊𝒎𝒊𝒙 =


𝒋𝝎𝒄𝒊𝒋


𝒋 ,

𝒛𝒊𝒎𝒊𝒙 =

𝒋=1

𝑵𝑭𝑳𝑼𝑰𝑫

𝑭𝒓𝒂𝒄 𝒋 𝒛𝒊𝒋, 𝑴𝒊𝒎𝒊𝒙 =


𝒋𝑴𝒊𝒋


𝒋 ,

Heavy Oil Composition Simulation

PVT simulations on heavy oil mixtures have traditionally been carried out using black oil correlations expressing the fluid properties in terms of easily measurable quantities such as API oil gravity, gas gravity, and gas/oil ratio.

With the application of secondary recovery techniques such as gas injection and thermal stimulation it has become more interesting also for heavy reservoir oils to make compositional equation-of-state-based simulations.


Heavy Oil Compositions

A heavy oil is one of a high density at standard conditions. Crude oils are essentially mixtures of paraffinic (P), naphthenic (N), and aromatic (A) compounds.

The densities of aromatics are higher than those of naphthenes and paraffins of the same molecular weight. This is consistent with chemical analyses showing that

heavy oil mixtures are rich in aromatic compounds.

The term heavy oil may be used for oil mixtures of an API gravity below 30.


Wax Precipitation

The majority of the C10+ aromatics present in crude oil mixture will be components containing one or more aromatic ring structures with paraffinic side branches.

The melting temperature of that type of compounds is low as compared to that of normal and slightly branched paraffins of approximately the same molecular weight. For this reason, wax precipitation is unlikely to take place

from a heavy oil mixture.


Viscosity of Heavy Oil Mixtures

Since high-molecular-weight compounds may be kept in solution in the oil at low temperatures, the viscosity of heavy oil mixtures can be very high indeed at production conditions and even at reservoir conditions.

Gas injection is often applied to heavy oil reservoirs. If the gas is dissolved in the oil, it will lower the oil viscosity and facilitate production and possibly also enhance the recovery rate.


1. Pedersen, K.S., Christensen, P.L., and Azeem, S.J. (2006). Phase behavior of petroleum reservoir fluids (CRC Press). Ch4 & Ch5.

2. Poling, B.E., Prausnitz, J.M., John Paul, O., and Reid, R.C. (2001). The properties of gases and liquids (McGraw-Hill New York). Ch1 & Ch4 & Ch5 & Ch8.

3. Tarek, A. (1989). Hydrocarbon Phase Behavior (Gulf Publishing Company, Houston). Ch3.


1. Phase Equilibrium Calculations

2. Tc, Pc, and ω Calculation

3. K-Factor & Delumping


q913 rfp w3 lec 9

Technology

advanced eos

srk eos

pr eos

waals equation

kwong equation

equation of redlich

srk equation p

modern equation of state