q913 rfp w2 lec 8

Reservoir Fluid Properties Course (1st Ed.)

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http://www.about.me/AlamiNia

mailto:[email protected]

http://www.about.me/AlamiNia

1. Differential Liberation Experiment: Data set

2. Separator Experiment

3. Swelling Experiment

4. Other Experiments

2013 H. AlamiNia Reservoir Fluid Properties Course: Equations of State and Compressibility Factor 2

1. General Notes about EoS

2. Ideal Gas EoS

3. Compressibility Factor

4. Van Der Waals EoS


Correlation of Properties

An equation of state (EoS) is an algebraic relation between P, V, and T.

The physical properties of every substance depend directly on the nature of the molecules of the substance.

In the nineteenth century, the observations of Charles and Gay-Lussac were combined with Avogadro’s hypothesis to form the gas ‘‘law,’’ PV = NRT, which was perhaps the first important correlation of properties.


Deviations from the Ideal Gas Law

Deviations from the ideal gas law, though often small, were finally tied to the fundamental nature of the molecules.

The equation of van der Waals, the virial equation, and other equations of state express these quantitatively.

Such extensions of the ideal-gas law have not only facilitated progress in the development of a molecular theory but, more important for our purposes here, have provided a framework for correlating physical properties of fluids.


Semi theoretical and Empirical EoS

Semi theoretical EoS are cubic or quartic in volume, and therefore whose volumes can be found analytically from specified P and T. These equations can represent both liquid and vapor

behavior over limited ranges of temperature and pressure for many but not all substances.

In empirical EoS volume cannot be found analytically. Nonanalytic equations are applicable over much broader

ranges of P and T than are the analytic equations, but they usually require many parameters that require fitting to large amounts of data of several properties.


Analytical Equations of State

An EoS used to describe both gases and liquids must be at least cubic in V.

The term ‘‘analytical equation of state’’ implies that the function ƒV (T, V) has powers of V no higher than quartic.

Then, when T and P are specified, V can be found analytically rather than only numerically.

We focus here on cubic EoS because of their widespread use and simple form.


Introduction to Cubic EoS

The majority of PVT calculations carried out for oil and gas mixtures are based on a cubic equation of state, the most popular type of thermodynamic model.

This type of equations dates back more than 100 years to the famous van der Waals equation (van der Waals, 1873).

The cubic equations of state most commonly used in the petroleum industry today are very similar to the van der Waals equation, but it took almost a century for the petroleum industry to accept this type of equation as a valuable engineering tool.


History of Cubic EoS

The first cubic equation of state to obtain widespread use was the one presented by Redlich and Kwong in 1949.

Soave (1972) and Peng and Robinson (1976 and 1978) further developed this equation in the 1970s.

In 1982 Peneloux et al. presented a volume-shift concept with the purpose of improving liquid density predictions of the two former equations.


Status Quo of Cubic EoS

The increased use of cubic equations of state seen over the past 30 years is greatly due to the availability of affordable computer power that has made it possible, within seconds, to perform millions of multicomponent phase equilibrium and physical property calculations using an equation of state as the thermodynamic basis.

When deriving the first cubic equation of state, van der Waals used the phase behavior of a pure component as the starting point.


Challenges to EoS Models: the Critical RegionFluid properties in states near a pure component’s

vapor-liquid critical point are the most difficult to obtain from both experiments and from models such as EoS.

The principal experimental difficulty is that the density of a near-critical fluid is so extremely sensitive to variations in P, T, that maintaining homogeneous and stable conditions takes extreme care. Even gravity influences the measurements.

The principal model difficulty is that near-critical property variations do not follow the same mathematics as at conditions well-removed from the critical.


Challenges to EoS Models: High Pressure RegionThe other region where EoS are often inaccurate is

at very high pressures both above and below the critical temperature.

The form of the PV isotherms of EoS functions often do not correspond to those which best correlate data, unless careful modifications are made.



PV Curve for Pure Component at Various Temperatures

At temperatures far above the critical (T 1), the PV curves exhibit a

hyperbolic shape suggesting that the pressure is inversely

proportional to the molar volume.

Ideal Gas Law

This behavior is known from the ideal gas law:

Where R equals the gas constant and T the absolute temperature.

The molar volume of a component behaving like an ideal gas also at high pressures would asymptotically approach zero for the pressure going towards infinity.


𝑷 =𝑹𝑻

𝑽

Deviation from Ideal Gas Behavior

At standard conditions a gas will behave approximately like an ideal gas, for which the ideal gas law applies: PV/RT=1.

Any deviation from ideal gas behavior may be expressed through the compressibility factor, Z:

For an ideal gas, Z equals 1. For real gases, Z is somewhat less than one except at high reduced temperatures and pressures. (Z depends on P, T and composition)


𝒁 =𝑷𝑽

𝑹𝑻

Experimentally Determination of Z

Z is defined as the ratio of the actual volume of n-moles of gas at T and p to the ideal volume of the same number of moles at the same T and p. (Z=Va/Vi)

The value of Z at any given pressure and temperature can be determined experimentally by measuring the actual volume of some quantity of gas at the specified p and T and solving (pV=ZnRT).



A Typical Z-P Diagram

A typical curve of the Z-factor for a natural gas, where the Z-factor is

plotted as a function of pressure for a given constant temperature

Compressibility Factor Scaling

Since the compressibility factor is dimensionless, it is often represented by a function of dimensionless (reduced) temperature, Tr = T/T*, and dimensionless (reduced) pressure, Pr = P/P*, where T*, and P* are characteristic properties for the substance, such as the component’s vapor-liquid criticals, Tc, and Pc.

This scaling allows many substances to be represented graphically in generalized form.



Generalized Compressibility Chart For All Pr.

For example, ƒ (Tr, Pr) was obtained by Nelson and Obert (1954) for

several Pr substances from experimental PVT data and they

constructed this graph. (V = V/ (RTc /Pc))

Sweet and Sour Gases

Natural gases frequently contain materials other than hydrocarbon components, such as nitrogen, carbon dioxide, and hydrogen sulfide. Hydrocarbon gases are classified as sweet or sour

depending on the hydrogen sulfide content.

Both sweet and sour gases may contain nitrogen, carbon dioxide, or both.


Effect of Non-Hydrocarbon Components on the Z-FactorA hydrocarbon gas is termed a sour gas if it

contains one grain of H2S per 100 cubic feet.

Concentrations of up to 5% of nitrogen and carbon dioxide components will not seriously affect accuracy.

Errors in compressibility factor calculations as large as 10% may occur in higher concentrations of non-hydrocarbon components in gas mixtures.


Relation between the Graph & EoS

Equations of state (EoS) are mathematical representations of graphical information such as shown in previous slide.

Modern computers obviate the need to manually obtain volumetric behavior such as from graphs and also allow more accurate results by using equations with component-specific parameters.

There have been an enormous number of EoS functions generated, especially in the last few years.


Virial Equation

The virial equation is a polynomial in P or 1/V (or density) which, when truncated at the Second or Third Order term (The lower the molar volume (the higher the pressure), the more terms are needed.), can represent modest deviations from ideal gas behavior, but not liquid properties.

For lower molar volumes (higher pressures), Z deviates from 1, which can be expressed through a Taylor Series expansion

This Z factor expression is called the virial equation, and A, B, and C are called virial coefficients (which are functions only of T for a pure fluid)


𝒁 = 1 + 𝑩𝑷

𝑹𝑻+ 𝑪 − 𝑩2

𝑷

𝑹𝑻

2

+⋯ = 1 +𝑨

𝑽+𝑩

𝑽2+𝑪

𝑽3+⋯

Virial Truncation

Because 1) The virial expansion is not rigorous at higher

pressures,

2) Higher order molecular force relations are intractable, and

3) Alternative EoS forms are more accurate for dense fluids and liquids,

The virial equation is usually truncated at the second or third term and applied only to single-phase gas systems.


The Van Der Waals Equation

As is seen from Figure, this is not the case.

With increasing pressure, the molar volume approaches a limiting value, which van der Waals named b. Rearrangement of Equation to

Suggests that the b-parameter should enter the equation as follows:

Which would give the following expression for P:


𝑽 =𝑹𝑻

𝑷

𝑽 =𝑹𝑻

𝑷+ 𝒃

𝑷 =𝑹𝑻

𝑽 − 𝒃

PV Curve Explanation for Temperatures below the CriticalAt temperatures below the critical (T 3), a vapor-to-

liquid phase transition may take place. Consider a component at temperature T 3 initially at a low pressure and in vapor form. By decreasing the volume while maintaining a constant

temperature, T 3, the pressure will increase and at some stage a liquid phase may start to form showing that the dew point pressure has been reached.

A further lowering of the volume will take place at a constant pressure until all the vapor has been transformed into liquid.

As a liquid is almost incompressible, a further reduction of the volume will be associated with a steep increase in pressure.


Intermolecular Attractive Forces Consideration The fact that the substance may undergo a

transition from a gaseous form with the molecules far apart to a liquid form with the molecules much closer together shows the existence of attractive forces acting between the molecules.These attractive forces are not accounted for in (P=RT/

(V-b)), which is therefore not capable of describing a vapor-to-liquid phase transition.



Interaction between Two Volume Elements

Interaction between Two Volume Elements in a Container Filled With

Gas

Intermolecular Attractive Forces Consideration (Cont.)The figure shows a container filled with gas.

The two small volume elements, v 1 and v 2, initially contain one molecule each. Suppose the force between the two volume elements is f. If just another molecule is added to v 2, the force acting between the two elements will be 2f.

So the force of attraction between the two volume elements is therefore proportional to c (1 or 2), the concentration of molecules in v (1 or 2).

Thus, the force acting between the two volume elements is proportional to c 1 × c 2.

In reality, the concentration in the gas is everywhere the same, i.e., c = c 1 = c 2, where c is the molecular concentration in the container.

The concentration c is inversely proportional to the molar volume, V, implying that the attractive force is proportional to the 1/V^2.


Final Form of the Van Der Waals EquationBased on that type of considerations, van der

Waals found that the attractive term should be a constant a times 1/V 2 leading to:

Which is the final form of the van der Waals equation.

The constants a and b are equation of state parameters, the values of which are found by evaluating the PV curve for the critical temperature. This curve is also called the critical isotherm.


𝑷 =𝑹𝑻

𝑽 − 𝒃−𝒂

𝑽2

The Van Der Waals Equation (Cubic in V)Van der Waals two-parameter cubic equation of

stateBy rearranging the equation to

It is seen that the van der Waals equation is cubic in V, which explains why the van der Waals and related equations are called cubic equations of state.


𝑽3 − 𝒃 +𝑹𝑻

𝑷𝑽2 +

𝒂

𝑷𝑽 −

𝒂𝒃

𝑷= 0


The Volumetric Behavior of a Pure Component

The Volumetric Behavior of a Pure Component As Predicted By the Van

Der Waal S EoS

Solution of Van Der Waals Equation

Assume that the substance is kept at a constant temperature T below its critical temperature.

At this temperature, (V^3-(b+RT/P) V^2+a/P V-ab/P=0) has three real roots (volumes) for each specified pressure p.


Van Der Waals Equation Roots

The largest root (volume), as indicated by point D, corresponds to the volume of the saturated vapor. While the smallest positive volume, as indicated by point B, corresponds to the volume of the saturated liquid. The third root, point E, has no physical meaning.

It should be noted that these values become identical as the temperature approaches the critical temperature Tc of the substance.


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

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.

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


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


q913 rfp w2 lec 8

Technology

equation of state eos

cubic equation of state

analytical equations

nonanalytic equations

waals equation

equation of van

history of cubic eos

type of equations dates