Flow Analysis for Hydrocarbon Pipeline Engineering

By Alessandro Terenzi

Flow Analysis for Hydrocarbon Pipeline Engineering gives engineers a tool to help them determine fluid dynamics. The book describes hydrocarbon fluid transport in pipelines by presenting useful applied thermodynamic derivations specialized for pipelines. All transport phenomena is covered, such as heat, momentum and mass transport. Moving past the fundamentals, the reference addresses the complexity of these fluids and dedicates a chapter on multiphase mixtures, including slugging, hydrates, wax and sand. Rounding out with practical case studies, this book delivers a critical reference for engineers and flow assurance experts that will help them correlate basic fluid principles with applied engineering practices.

• Includes discussions on sustainable operations such as CO2 transport in pipelines utilized in carbon capture and hydrocarbon recovery operations
• Delivers multiple case studies for practical applications and lessons learned
• Describes hydrocarbon fluid transport in pipelines by presenting useful applied thermodynamic derivations specialized for pipelines

• Language: English
• Publisher: Elsevier Science
• Release date: May 11, 2022
• ISBN: 9780128224830

Alessandro Terenzi

Alessandro Terenzi is currently a Flow Assurance Lead for a major Italian oil and gas pipeline engineering company, managing flow assurance, pre-commissioning and commissioning specifications on projects, supporting operation activities around the world, including the US, Uganda, Oman, Lybia, Iran, Italy, and Saudi Arabia. Previously, he worked on many flow assurance studies requiring technical coordination of design activities, interface with clients, and support existing pipeline operations and worked as a Senior Process Specialist working in gas dynamics, multiphase flow, heat transfer, and economical optimization. He earned a degree in physics from Bologna University. Alessandro is an active member of Italian Association for Multiphase Flow In Industrial Plants. He is also a past member of Past Member of European Consortium for Mathematics in Industry. He is a lecturer for University of Bologna and has authored many journal contributions and conference proceedings.

Book preview

CHAPTER ONE: Basics of hydrocarbons thermodynamics

Abstract

The behavior of hydrocarbon mixtures depends on their compositions as well as the pressure and temperature conditions encountered. The thermodynamics of mixtures is determined by the contributions of single components; this Chapter is then developed by treating the thermodynamic behavior of single components at first, and then extending to binary and multicomponent mixtures. Different equations of state are presented, starting with the simplest approach based on ideal gas model, followed by the description of the most popular models in the oil and gas field, as Peng–Robinson and Soave–Redlich–Kwong, belonging to the cubic equations family. Thermodynamic equilibrium conditions are discussed, by considering the analysis of vapor–liquid phase transformations, referred to both single components and binary mixtures. The final part of this Chapter is dedicated to the multicomponent mixtures, as typical of hydrocarbon fluids. Reference is made to the interaction of different components, the phase behavior including a vapor–liquid coexistence in a finite region of the pressure–temperature diagram, retrograde condensation phenomena, the different behaviors of reservoir fluid as depending on their dominant components, and the interaction with the water phase.

Keywords

Binary and multicomponent mixtures; Equation of state; Equilibrium and stability; Phase diagrams; Phase transformation; Thermodynamics

1.1 Introduction

1.2 Equation of state

1.3 Other real fluid properties

1.4 The principle of corresponding states

1.5 Equilibrium and stability of one-component fluids

1.6 Vapor–liquid equilibrium for one-component fluids

1.7 Vapor–liquid equilibrium for multicomponent fluids

1.8 Equation of state for multicomponent mixtures

1.9 Phase diagrams of reservoir fluids

References

Further reading

1.1. Introduction

Hydrocarbons transported by pipeline systems are usually composed by a large number of components; at ambient temperatures, some of them are present in the gas phase, while others are liquids. Some heavier components may also appear as solid in some critical operating conditions (hydrates, waxes). Water is often present in the transported fluid mixture, and it is usually treated as a separate phase, even if a hydrocarbon liquid is present. The behavior of hydrocarbon mixtures depends on their compositions as well as the pressure and temperature conditions encountered. The thermodynamics of mixtures is determined by the contributions of single components; for this reason, the properties of single components are covered first. The treatment begins with the description of the equations of state, defining the relationship between the main fluid properties; starting with the simplest approach based on ideal gas model, more complex and realistic models are introduced. Thermodynamic equilibrium conditions are discussed by considering the relevant trends of entropy and free energies, for specific systems. Stable equilibrium occurs when it is independent on time and its previous history, and it is able to resist to small thermodynamic parameter fluctuations. These concepts are applied to the analysis of vapor–liquid phase transformations, referred to both single components and binary mixtures. The discussion is further extended to multicomponent mixtures, as typical of hydrocarbon fluids. The description of their properties is developed by treating the interaction of different components, the phase behavior including a vapor–liquid coexistence condition occupying a finite region of the pressure–temperature diagram, retrograde condensation phenomena, the different behaviors of reservoir fluid as depending on their dominant components, and the interaction with the water phase.

1.2. Equation of state

The thermodynamic properties of a fluid can be defined by using a volumetric equation of state, providing information about the relationship between pressure, temperature, and volume (or density).

The simpler EoS (Equation of Sate) is that relevant to the ideal gas model, which assumes no interaction among fluid particles and neglects their volume. It is expressed by the following:

(1.1)

where:

=fluid pressure

=fluid specific volume

=universal gas constant

=fluid absolute temperature

In term of compressibility factor Z, defined in this way,

(1.2)

the ideal gas model is represented by the condition .

The compressibility factor is a parameter giving the indication of the deviation of a real gas thermodynamic behavior from the ideal gas approximation; actually, the real substances can be described by Eq. (1.1) only in a limited range of conditions, i.e., at low pressures and moderate temperatures. Hence, other equations of state have been developed on the basis of experimental data of real fluids; the most important from a historical point of view is the classical Van der Waals equation, given by the following:

(1.3)

In this expression, two parameters have been introduced to represent real fluid effects, namely:

=it is a parameter relevant to the Van der Waals intermolecular forces, which are included as an equivalent pressure term proportional to the inverse square of the volume

=it is a parameter representing the finite volume of molecules

Eq. (1.3) is a so-called cubic EoS, since a cubic algebraic equation must be solved in order to obtain the volume.

The Van der Waals equation is not very accurate, but it was the first equation capable of predicting the transition between vapor and liquid; furthermore, it has also been the prototype for modern, more accurate equations of state as the Peng–Robinson one (Peng and Robinson, 1976), given by:

(1.4)

or the Soave–Redlich–Kwong (Soave, 1972):

(1.5)

The parameters of the above equations can be obtained or by fitting the equations to PVT data for the fluid of interest or by using general relations between them and critical point properties (see Section 1.4).

All the above expressions can be written in the following form:

(1.6)

This is again a cubic algebraic equation that must be solved to get the compressibility factor Z, as must be for the fluid volume.

All cubic equations of state are approximate; in general, they provide a reasonable description of thermodynamic properties of vapor and liquid phases of hydrocarbons, and of vapor region only for many other pure fluids. The Peng–Robinson and Soave–Redlich–Kwong are, at present, among the more popular cubic equations of state in oil and gas applications.

A different type of equation of state is the virial equation (Sandler, 1999):

(1.7)

where and are the temperature-dependent second and third virial coefficients. This expression is of theoretical interest since it can be derived from statistical mechanics with explicit formulations of the virial coefficients in terms of particle interactions. It is a power series expansion in specific volume about the ideal gas result. With a sufficient number of coefficients, the virial equation can predict the vapor phase properties with good precision, but it is not applicable to the liquid phase. However, even in the case of the gases, it is not recommended for pressure above 10bar.

1.3. Other real fluid properties

One of the quantities appearing in the fundamental thermodynamic relationships is the heat capacity. It is possible to show that, given a volumetric EoS, and heat capacity data as a function of temperature at a single value of pressure P 1 or volume V 1, the value of the heat capacity in any other state (represented by pressure P 2 or volume V 2) can be computed by the following (Sandler, 1999):

(1.8)

(1.9)

where C P and C V are the constant pressure and constant volume heat capacities, respectively.

In practice, heat capacity data are usually collected for low pressure conditions or large specific volumes where all fluids are ideal gases. Hence, if P 1 or V 1 are taken as 0 and ∞, respectively in Eqs. (1.8) and (1.9) above, it is possible to write:

(1.10)

(1.11)

where the asterisk denotes the ideal gas heat capacity. Usually, the temperature dependence of the ideal gas heat capacity is given in polynomial form as follows:

(1.12)

Other thermodynamic variables that allow to calculate the deviations of the real fluid behavior from the ideal gas status are the Departure Functions.

In particular, it is possible to show that, for what concerns the enthalpy H and the entropy S, their changes for a real fluid are equal to that of an ideal gas undergoing the same transformation plus the departure of the fluid from the ideal gas behavior at the end state less the departure from the ideal gas behavior at the initial state. These Departure Functions can be computed from the equation of state, based on the following:

(1.13)

(1.14)

1.4. The principle of corresponding states

The analysis presented in Sections 1.2 and 1.3 has shown that it is possible to calculate the thermodynamic characteristics of a real substance given only the ideal gas heat capacity and the volumetric EoS. However, the necessary information on the EoS is not always available for all fluids. Hence, it is possible to resort to the principle of corresponding states, which allows to predict thermodynamic properties of fluids from generalized property correlations based on experimental data for similar fluids.

We concentrate our attention on a volumetric EoS, which is determined by the intermolecular interactions. From the study of molecular behavior, it has been found that molecules can be grouped into classes, such as spherical molecules, nonspherical molecules, molecules having a permanent dipole, and so on, and that within any one class molecular interactions are similar.

It has been also found that all the members of a class obey the same volumetric EoS, i.e., the volumetric data of each member are fitted by simply changing the parameters of the EoS. The fact that several different molecular species may be described by a volumetric EoS of the same form suggests that it might be possible to construct generalized correlations for both the EoS and density-dependent contribution to enthalpy, entropy, and other thermodynamic variables. The first historical generalized correlation arose from the study of the Van der Waals EoS. Fig. 1.1 shows the isotherms of this equation in a P versus V diagram, for five values of temperature including the critical temperature T C , which is defined as the maximum temperature at which a liquid phase can exist. To be noted that this definition holds for a single component fluid only.

It can be observed in this figure that the isotherms with temperature below the critical one exhibit a non-monotone behavior, with a local minimum followed by a local maximum over part of the specific volume—pressure range. Such trend is associated with a liquid–vapor transition. While this behavior is absent for T > T C , in such a way that the liquid phase cannot exist in this range, in the critical temperature curve, the two extremes coincide, and this point (called critical point) is an inflection point, where the first and second derivative of P with respect to V vanishes simultaneously. This is expressed analytically by the following requirements:

Figure 1.1  Isotherms of the Van der Waals EOS in a plane (P,V).

(1.15)

By replacing Eq. (1.3) in the relations (1.15), the following expressions of the Van der Waals parameters and are obtained:

(1.16)

Using (1.16) in the Van der Waals Eq. (1.3), it is possible to get the pressure P C and the compressibility factor Z C at the critical point:

(1.17)

By using the above expressions and by defining the dimensionless variables called reduced temperature T r , reduced pressure P r and reduced volume V r

(1.18)

the following form of the Van der Waals equation of state is obtained:

(1.19)

From (1.19) it is concluded that all the fluids obeying the Van der Waals equation of state have the same numerical value of reduced volume for given values of P r and T r . Two fluids having the same values of reduced pressure and temperature are said to be in corresponding states.

The principle of the corresponding states has been historically represented by drawing the compressibility factor Z as function of reduced pressure and temperature, as shown in Fig. 1.2, where the compressibility data for several fluids have been reported.

Fig. 1.2 demonstrates that the idea of the corresponding states is reasonable, since the general trend of the data confirms the similarity represented by (1.19). However, it can be observed that compressibility factors for inorganic fluids are almost always below those for hydrocarbons. Furthermore, if (1.19) were universally valid, all fluids would have the same value of the critical compressibility Z C =0.375, as shown by (1.17), while for most fluids, the critical compressibility is in the range 0.23–0.31.

Figure 1.2  Compressibility factor for several fluids as a function of reduced pressure and temperature. From Su and Chang (1946).

These deviations have led to the development of more complicated corresponding states principles. The simplest generalization is based on the adoption of a family of different relations Z = Z(P r ,T r ), with different values of Z C , i.e., the critical compressibility factor is used as an additional parameter of the compressibility law:

(1.20)

In fact, other quantities have been considered as additional corresponding states parameters, since the critical compressibility factor Z C is not known with enough accuracy for many substances.

Pitzer (1995) has proposed the so-called acentric factor as third correlative parameter, which is defined in this way:

(1.21)

Here is the vapor pressure of the fluid at .

Even these extensions of the corresponding state concept have been found not suitable to represent the thermodynamic behavior of some kinds of molecules, in particular that having dipoles or quadrupoles. Hence, this application remains limited to some classes of molecules only, as in case of most hydrocarbons.

The modern version of the corresponding states idea is the use of generalized EoS.

The Van der Waals equation is the first example of application of this approach. In fact, the expressions (1.16), rewritten in terms of the critical pressure and temperature in this way,

(1.22)

demonstrate that the thermodynamic properties of a class of fluids with parameters and that have not been derived from a set of experimental data can be described by knowing the fluid critical properties only. As it was said before, the Van der Waals equation is not accurate enough, and other more sophisticated equations are used.

In particular, we consider here the Peng–Robinson equation already presented in Section 1.1, by showing that its generalized form is given by the following:

(1.23)

with:

(1.24)

(1.25)

(1.26)

(1.27)

The above relationships were obtained by Peng and Robinson in order to improve the predictions of the boiling point pressure versus temperature, and the function was chosen by fitting the vapor pressure values for many fluids. With respect to the Van der Waals scheme, the generalized Peng–Robinson approach uses one parameter more other than P C and T C , namely the acentric factor .

This equation can be used to calculate not only the compressibility factor, but also the departure functions for the other thermodynamic properties.

Another very popular cubic equation of state used in the oil and gas industry is the SRK (Soave–Redlich–Kwong):

(1.28)

with:

(1.29)

(1.30)

(1.31)

(1.32)

The current industrial practice is based on the application of the corresponding states concept, by assuming that different fluids are described by the same generalized equations of state, and that their thermodynamic behavior in an extended pressure and temperature range is described by knowing the generalized parameters only. Such kind of equations can give a very accurate thermodynamic picture of many substances, in particular hydrocarbons.

1.5. Equilibrium and stability of one-component fluids

The equilibrium state of a closed thermodynamic system can be derived by using the energy and entropy balances derived from the first and second principles of thermodynamics:

(1.33)

(1.34)

where:

=specific internal energy

=heat transfer rate

=rate of internal generation of entropy by irreversible processes ≥0

In (1.33) only work due to deformation of the system boundaries is considered.

For an adiabatic ( =0) and constant volume system, the system of energy and entropy balances equations becomes

(1.35)

(1.36)

Since the entropy function can only increase during the approach to equilibrium due to (1.36), the entropy must be a maximum at equilibrium.

Hence, the equilibrium criterion for a closed and isolated system is as follows:

(1.37)

This principle can be illustrated by referring to the simple system shown in Fig. 1.3.

The system is composed by two subsystems containing different amounts of the same molecular species of particles of a single-component fluid (N represents the number of moles). They are connected by a channel of infinitesimal volume equipped by an ideal valve opening instantaneously. The system is assumed adiabatic and constant volume, exchanging neither mass neither energy with the external environment. However, the two subsystems can exchange heat and mass across the communication channel when the valve is open. For the global system, the total number of moles N, the total internal energy U, the total volume, and the total entropy are given by the addition of the two subsystems contributions:

Figure 1.3  Isolated nonequilibrium system.

(1.38)

(1.39)

(1.40)

(1.41)

By using the fundamental thermodynamic relationships, the entropy change of each i-th subsystem after communication and mixing of particles can be expressed as a function of changes in the number of moles, volumes, and internal energy by the following:

(1.42)

where G i is the i-th subsystem molar Gibbs energy.

The entropy change of the overall system can be easily obtained by summing the two subsystems contributions, by considering that the total quantities given by (1.38), (1.39), and (1.40) are constant:

(1.43)

Since the entropy must be a maximum at equilibrium, the (1.25) expression must vanish for system variations, thus obtaining the following:

(1.44)

(1.45)

(1.46)

Hence, the equilibrium condition for an adiabatic isolated system states that all its subsystems must have the same temperature, the same pressure, and the same molar Gibbs energy.

The above discussion is based on the condition , which is necessary but not sufficient for S to reach a maximum value. The further condition assures that a maximum entropy value, giving a true equilibrium state, has been identified. On the contrary, a minimum of the entropy corresponds to an unstable state. Therefore, the sign of determines the stability of the thermodynamic state found from .

It is possible to determine the equilibrium and stability for other kinds of systems also.

For instance, for a closed system kept at constant temperature and volume, the energy and entropy balance equations become

(1.47)

(1.48)

By combining the above equations, the following relationship, written in term of the molar Helmholtz free energy , is obtained:

(1.49)

Since the molar Helmholtz free energy can only decrease during the approach to equilibrium due to (1.49), it must be a minimum at equilibrium.

Hence, the equilibrium criterion for a closed system kept at constant temperature and volume is the following:

(1.50)

For a closed system kept at constant temperature and pressure, the energy and entropy balance equations become the following:

(1.51)

(1.52)

By combining the above equations, the following relationship, written in term of the molar Gibbs free energy , is obtained:

(1.53)

which leads to the following equilibrium criterion:

(1.54)

Table 1.1

Actually for all the systems considered here, the application of the equilibrium criteria determines that they are at uniform temperature, pressure, and molar Gibbs free energy.

The equilibrium and stability conditions for the system under consideration, written for a single-component fluid, are summarized in Table 1.1. It is possible to show that they are also valid for multicomponent fluids.

From the stability criteria listed in Table 1.1, three fundamental thermodynamic inequalities can be derived for systems considered in a stable equilibrium state:

(1.55)

1.6. Vapor–liquid equilibrium for one-component fluids

Now we are going to analyze the equilibrium of two phases (vapor and liquid) on the basis of the discussion exposed in Section 1.5. In Fig. 1.4, some isothermal curves in a plane (P,V) are shown, as given by a typical cubic EoS. These curves are represented for increasing temperature (T 1 < T 2 < T 3 < T 4 < T 5). We can observe that the isotherms labeled by T 1 and T 2 do not have a monotone trend, but they present a minimum followed by a maximum in a certain region of the (P,V) diagram. Therefore, while for the high-temperature isotherms the stability condition is fulfilled everywhere, this is not true for the low-temperature isotherms. Actually, the regions of the diagram where the stability violation occurs are not physically realizable and cannot be observed in practice. Thus, in this particular thermodynamic region, the cubic equations of state fail to predict the real behavior of the fluid, and their trend must be replaced by another physical description. The isotherm at temperature T 3 has a single point for which the first and second derivatives of pressure which respect to volume vanish simultaneously: this is called critical point and T 3 is the critical temperature (usually identified as T C ).

Figure 1.4  Isotherms on a plane (P,V) predicted by a cubic equation of state.

A low-temperature isotherm is represented alone in Fig. 1.5; it is shown that the intersection of a horizontal line at constant pressure with the curve gives three possible values for the volume of the fluid, , , and . One of these, , is on the part of the isotherm that is unphysical for the above mentioned stability reasons. The other two values have a physical meaning instead, and they represent the volumes of the liquid and vapor phase, respectively, at the specified pressure and temperature.

The true thermodynamic behavior in the region between and is shown in Fig. 1.6, where a phase change occurs along the considered isotherm isobar represented by the horizontal section which boundary volumes are the single-phase liquid and vapor volumes labeled as and . This is the vapor–liquid coexistence region. The two-phase mixture specific volume varies in this region since, although the specific volumes of the single phases are fixed, the vapor fraction in the mixture varies from 0 to 1, thus giving the following:

(1.56)

Figure 1.5  Low-temperature isotherm in a plane (P,V) predicted by a cubic EoS.

Figure 1.6  Real low-temperature isotherm in a plane (P,V) including the vapor–liquid phase transition.

Thus, the isotherm section predicted by a cubic EoS in the two-phase region of a single-component fluid must be replaced by the isobaric horizontal line shown in Fig. 1.6.

The question that remains is the value of pressure corresponding to the vapor–liquid coexistence region. The criterion to be applied in order to find this pressure is the equality of the two phases Gibbs free energies, as derived from Section 1.5. From fundamental thermodynamic relationships, the free Gibbs energy variation along an isotherm is given by the following integral:

(1.57)

Thus, by using the equation of state which provides the relationship among and , the value of the phase change pressure is obtained by the