The phase equilibrium is characterized. Phase equilibria
The description of phase equilibrium is one of the most important tasks in the calculation of separation processes. Knowing the equilibrium conditions allows not only to fundamentally solve the problem of the possibility of separating a multicomponent mixture by methods of rectification, absorption, extraction, but also to choose a separation scheme. The most general method for calculating equilibrium is based on the application of some equation (equation of state) to all phases of the vapor-liquid system. However, the use of the equations of state is possible only in the case of simple systems that are formed by substances with similar properties, for example, non-polar substances that make up natural gas.
In the presence of polar molecules, these equations are not applicable. For such systems, which for the most part constitute the processes of basic chemistry, the equations of state are used only for the vapor phase, and the liquid phase is considered taking into account deviations from the ideal behavior for real mixing conditions and in accordance with the theory of solutions.
When considering multiphase systems, the achievement of phase equilibrium is interpreted as a state in which the rates of mutual transition of component molecules from one phase to another are equal. Thermodynamically, this condition is expressed as the equality of the chemical potentials of the phases for each component, i.e.
where c/ is the chemical potential equal to the partial derivative of the isobaric-isothermal potential (Gibbs free energy) with respect to composition at constant pressure and temperature of the system.
For a chemical reaction occurring in an ideal gas phase, we have
where d„ bj- stoichiometric coefficients of initial substances and final reaction products; A it Bj- name of initial components and reaction products.
If in (1.18) we substitute the expression for the chemical potential C/ = po + R Tin/>, then after appropriate transformations for the chemical equilibrium constant we can write
where Y^. 9 SW(~ concentrations of the initial components of the gas mixture and reaction products, respectively; R- system pressure; To is the equilibrium constant of a chemical reaction.
From (1.19) it follows that with a change in pressure, the equilibrium composition will change depending on the ratio of the stoichiometric coefficients a, and b,. If the reaction proceeds without
changes in the total number of moles in the system (i.e. ? i / \u003d Z b t), then
pressure does not affect the equilibrium composition. The temperature dependence of the equilibrium composition is determined by the nature of the temperature dependence of the equilibrium constant (increasing or decreasing function of temperature). Note that equation (1.19) is called the “law of mass action”.
If some of the components of reaction (1.17) are in a condensed state, then with insignificant changes in the external pressure, their concentrations are not indicated on the left side of equation (1.19), however, when calculating the equilibrium constant through the Gibbs free energy, the condensed state of these components is taken into account.
Since phase equilibrium in closed systems is achieved in the presence of chemical equilibrium in the system, it is obvious that the calculation of equilibrium in systems in which several simultaneous reactions occur presents certain difficulties.
The disadvantage of describing the conditions of phase equilibrium in terms of chemical potentials is that these quantities cannot be measured directly. Therefore, to obtain the calculated ratios, it is necessary to express them in terms of the system state parameters. This approach is thermodynamically justified when considering ideal systems, when there is an equation of state. However, for real systems, there is still no generally accepted equation of state for a real gas, and therefore the derivation of relations for thermodynamic functions and the description of the conditions of phase equilibrium is extremely difficult. Using the fugacity function, the equilibrium condition for real systems is expressed as
where - fugacity, determined by the fact that, firstly, thermodynamic relationships are valid for ideal and real systems; secondly, it coincides with the pressure if the latter is so small that the gas becomes ideal.
If we consider an ideal solution as a mixture of components, the molecules of which behave in the same way as in a pure liquid, and the non-ideal behavior is reflected by introducing a measure of non-ideality in the form of coefficients, then the fugacity of individual components should be proportional to their concentrations, i.e., for a two-phase system, we have :
where y;, f, - coefficients characterizing the imperfection of the liquid and vapor phases, t.s. activity and volatility coefficients, respectively: /,° ,//^ - fugacity of components under standard conditions (L- liquid; V- steam); x, -, yj are the molar concentrations of the liquid and vapor phases, respectively.
If the vapor phase obeys the laws of ideal gases, then instead you can use R(system pressure). Similarly, at moderate pressures, instead of f^L can be used R j° (vapor pressure of the pure component). Thus, the equilibrium condition for any component of a three-phase multicomponent mixture can be written as
This equation is the main one in calculating the phase equilibrium of multicomponent three-phase systems. The activity and volatility coefficients, as a measure of phase deviation from ideality, are equal to unity for an ideal mixture.
As follows from equation (1.21), the parameters characterizing the equilibrium conditions (composition, temperature, pressure) can only be calculated if there are fugacity values in the standard state. In principle, the choice of the standard state is arbitrary and is dictated only by the convenience of calculation. It is only necessary that the standard fugacity be calculated at the same temperature as the temperature of the mixture. Pressure and composition can be chosen arbitrarily.
The change in potential when mixing two solutions under isobaric-isothermal conditions is
The first term of this expression corresponds to a change in potential when mixing ideal components, and the second term corresponds to a change in potential due to the imperfection of the mixture. Thus, the excess Gibbs mixing free energy is:
where xi- average mole fraction of the i-th component,
Since the excess energy is a measure of the non-ideality of the mixture, expression (1.23) is widely used to estimate the degree of this non-ideality.
In accordance with Wilson's model, in which he introduced the concept of “local” volume fractions of components, equation (1.23) takes the form for the free energy of mixing:
where is the local volume fraction of the /-th component relative to the central molecule of the same type; G E- Gibbs free energy of mixing.
The main idea behind this concept is that, when viewed microscopically, a liquid mixture is not homogeneous. The composition at one point in the mixture may be different from the composition at another. Although in engineering applications only the average concentration of components in mixtures is used, in order to build an adequate model of a liquid mixture, it is necessary to operate with local compositions. According to the concept of local compositions introduced by Wilson, the distribution of molecules relative to the central molecule has the form:
where Xji- local composition, defined as the number of type molecules ] - / in the vicinity of the central molecule, divided by the total number of molecules; (Hd- Xd) - parameter characterizing the energy of interaction between pairs i - / and
Taking into account the introduced local concentrations, the local volume fractions of the i-th component in equation (1.24) can be determined as
where V „ Vj- molar volumes of pure / and j components; gy = Hu- X#; gji = - Xd.
By substituting equation (1.25) into (1.24), we obtain the dependence of the Gibbs excess mixing energy on the composition and temperature of the mixture. Differentiation of the obtained expression by the composition of the mixture gives the well-known Wilson equation, which relates the activity coefficient of the component in the mixture y, with the composition and temperature of the mixture:
To describe the equilibrium state of multicomponent mixtures (MCM), the Wilson model requires only the assignment of experimental data on binary pairs included in a given MCM and the physicochemical properties of pure components.
The equilibrium concentration of the /-th component in the n-component mixture is expressed as:
where R® is the vapor pressure of the pure component; R- pressure in the system, MPa.
The activity coefficient of any component y/ can be determined from equation (1.26), and the activity coefficient in a binary pair can be calculated using the following equations:
The vapor pressure of the /-th component R, 0 is calculated using the Antoine equation:
where cj,.,Cj are empirical constants determined by the least squares method.
Algorithm for determining the equilibrium concentration y*= / (x,) next: parameters are defined k/j - c and - kjj from equations (1.27) for all pairs of components that make up the MCS, according to the experimental data of binary equilibrium. The solution is carried out by the method of successive approximations, and the degree of approximation is controlled by the sums of the squared deviations of the equilibrium compositions of the vapor phase at the control points, calculated and taken experimentally. The direction of the search for unknown parameters is indicated by the minimized function R:
where to = 1, 2, ... , m- number of control points; at composition of the vapor phase at the control point.
The search algorithm must be designed in such a way that it is possible to determine the values of the unknown parameters X).
For any point in a single-phase area, for example for a point Z(liquid), according to the formula (3.36) FROM= 3 – 1 = 2. The system is bivariant , i.e., within certain limits, independently of each other, you can change 2 external conditions (T and R
For any point corresponding to two equilibrium phases, for example, for a point X(ice/liquid), FROM= 3 – 2 = 1. The system is monovariant , i.e., only one external condition can be changed (either T or R) without changing the number and type of phases.
Finally, for the triple point O(ice/liquid/steam) FROM= 3 – 3 = 0. The system is invariant (non-variant ), i.e., without changing the number and type of phases, it is impossible to change any of the external conditions.
Shown in Fig. 3.5. the state diagram of water is valid for low pressures. At high pressures, ice can exist in several crystalline modifications. The diagram looks much more complicated [Kireev].
Rice. 3.5. Diagram of a water state diagram
[Ravdel], [Chemist's Handbook 1], [Eisenberg]
Federal Agency for Education
Astrakhan State University
Department of Analytical and
physical chemistry
coursework
Phase equilibria
Performed
DXM Student 311
Shamaeva Amina
Checked
Ph.D. Associate Professor Dzhigola L.A.
Astrakhan 2008
Introduction
2. Gibbs phase rule
3. Equilibrium states during phase transitions
3.1 Gas-liquid equilibrium in two-component systems
3.1.1 Dependence of the solubility of gases in liquids on the nature of the gas and solvent
3.1.2 Dependence of the solubility of gases in liquids on pressure
3.1.3 Dependence of the solubility of gases in liquids on temperature
3.2 Liquid-liquid equilibrium in two-component systems
3.3 Vapor-liquid equilibrium in two-component systems
3.3.1 Equilibrium vapor-liquid solution in systems with unlimited mutual solubility of liquids
3.3.2 Equilibrium vapor - liquid solution in systems with limited mutual solubility of liquids
3.3.3 Vapor-liquid equilibrium in systems with mutually insoluble liquids
4. Clapeyron equation
5. Clausius-Clapeyron equation
6. Practical part
6.1 Questions
6.2 Tasks
Conclusion
Bibliographic list
Introduction
Heterogeneous system - system consisting of several phases.
A phase is a set of homogeneous (homogeneous) parts of a system that are identical in composition, chemical and physical properties and separated from other parts of the system by a surface. The phase interfaces are formed by a certain number of molecules located on the boundary of the region filled with a given phase. The molecules that form the surface layer are in special conditions, as a result of which the surface layer has special properties that are not inherent in the substance located in the depth of the phase. Formations composed of a small number of molecules cannot be divided into the surface layer and the internal mass of the substance; therefore, the concept of a phase is inapplicable to formations with a very small volume. Liquid and solid phases are called condensed.
The constituent substance of the system - it is every substance of the system that can be isolated from the system and exist outside of it. If a chemical reaction does not occur in the system, then the amount of each of the substances does not depend on the amounts of other substances. If a chemical reaction occurs in the system, then the composition of the phases of the equilibrium system can be determined knowing the concentration of only a part of the substances. The constituent substances, the concentrations of which determine the composition of the phases of a given equilibrium system, are called independent constituent substances or components of the system. The properties of a system are determined not only by the nature of the components, but also by their number. The number of components is the same as the number of constituent substances in the absence of a chemical reaction or less in the presence of a chemical reaction.
2. Gibbs phase rule
In an equilibrium system, the relationship between the numbers of phases, components, and thermodynamic degrees of freedom is expressed by the Gibbs phase rule or the law of phase equilibrium. Let us consider an equilibrium thermodynamic system consisting of F phases, each of which contains K components. Let us take as the parameters that determine the state of the system, pressure, temperature and concentrations of components, expressed in mass or mole fractions or percentages. Let us estimate the total number of state parameters and the number of equations relating them. The number of parameters that are the same in all phases of the system is two (P and T). Since the concentrations of the components are expressed in mass or mole fractions or percentages, it is sufficient to set (K - 1) concentrations to characterize the composition of one phase. The concentration of one of the components of this phase will be determined if the concentrations of the remaining components are known. To characterize the composition of all phases of an equilibrium system, it is necessary to know F(K - 1) concentrations. Then the total number of parameters that determine the state of the equilibrium system will be equal to F(K - 1) + 2.
Since at R = const T=const the chemical potentials of the components are functions of their concentrations, for example, in an ideal solution
, then the number of equations relating the concentrations of the components of the equilibrium system can be determined from the equality of the chemical potentials of each component in all phases:
(1)
The number of equations relating the concentrations of one component is (Ф - 1), and all K components - K(Ф - 1). The number of independent system state parameters, i.e. the number of thermodynamic degrees of freedom is equal to the difference between the total number of parameters that determine the state of the equilibrium system and the number of equations relating these parameters:
C \u003d F (K - 1) + 2 - K (F - 1),
S=K-F + 2. (2)
Equation (2) is called Gibbs phase rule or phase balance law : in an equilibrium thermodynamic system, which of the external factors is influenced only by pressure and temperature. The number of thermodynamic degrees of freedom is equal to the number of components minus the number of phases plus two. The number of degrees of freedom increases with an increase in the number of components and decreases with an increase in the number of phases. Since the number of degrees of freedom cannot be negative, the number of phases in an equilibrium system cannot exceed K + 2.
The phase rule was developed by the American physicist J. Gibbs in 1876. The doctrine of phases was later used in the works of J. Van't Hoff, B. Rosebom, N.S. Kurnakova and others and was the basis for the study of equilibria in heterogeneous systems. If only pressure (P=const) or temperature (T=const) influences the system from external factors, the number of degrees of freedom decreases by one and the equation of the phase rule (2) takes the form
Su SL \u003d K-F + 1 (3)
The variance of the system calculated according to this equation is called conditional (Cusl), and the system is called conditionally invariant, conditionally monovariant, etc. At constant pressure and temperature
C cond=K-F (4)
If the compositions of two equilibrium phases, for example, liquid and vapor, are the same, then when calculating the number of degrees of freedom, one more equation should be taken into account that relates the concentrations of the components X, (l) = X, (n). In this case, instead of equation (2), one should use the equation
Su SL = K-F+1 (5)
and instead of equation (3) - by the equation
C cond=K-F (6)
If the state of the system is also determined by such external factors as an electric or magnetic field, a gravitational field, etc., then they should also be taken into account when calculating the number of degrees of freedom in an equilibrium system. In general, when the system is subject to P various factors, then
S=K-F+ n (7)
3. Equilibrium states during phase transitions
3.1 Equilibrium gas-liquid solution in two-component systems
When gases are dissolved in liquids, an equilibrium is established between the liquid phase containing the solvent and the dissolved gas, and the gas phase containing the given gas and solvent vapors. If the solvent is relatively little volatile, then an equilibrium will be established between the solution and the practically pure dissolved gas. The solubility of a substance is determined by its concentration in a saturated solution. The solubility of gases in liquids depends on the nature of the dissolved gas and solvent, gas pressure, temperature, and the presence of various substances in the solution, especially electrolytes. The numerical value of the solubility of a gas in a liquid depends on the way it is expressed. The solubility of gases is expressed by the number of grams of gas in 100 g of pure solvent or in 100 g of solution, the number of moles of gas in 1000 g of solvent or in 1 liter of solution, mole fraction. In addition, the solubility of gases in liquids is characterized by the solubility coefficient o or the absorption coefficient
The method of determining the equilibrium constants according to the NGAA atlas issued by the American Gasoline Production Association has received the greatest distribution both in our country and abroad. The atlas contains equilibrium constants from methane to decane inclusive, as well as nitrogen and carbon dioxide. The equilibrium constant at a given convergence pressure is determined from an atlas based on pressure and temperature data. When calculating on a computer, it is difficult to use graphs of equilibrium constants, because one has to repeatedly remove the values of the constants from the graphs and enter them into the computer memory. In this regard, the NGAA atlases are translated into tables:
Recommendations for automating the choice of equilibrium constants of hydrocarbon systems on a computer. Tables of equilibrium constants. All-Russian Research Institute of Gas, Moscow, 1972. A partial atlas of equilibrium constants is presented in the reference book: Katz, "Gas Production and Transportation".
Phase equilibrium equations.
For calculations, the following initial data are needed:
Initial reservoir pressure
Initial formation temperature
Composition of the hydrocarbon system
Equilibrium constants of individual constituent components, K=y/x, where
y is the molar fraction of the component in the gas phase,
x-molar fraction of a component in the liquid phase in equilibrium with the gas phase.
Each component of the mixture at a given temperature and pressure has its own equilibrium constants. They are determined experimentally. Dependences of the equilibrium constant are expressed in terms of the given parameters:
, , 
.
In the case of a multicomponent mixture, the critical pressure is called the convergence pressure. Since there is no difference between liquid and vapor (gas) in the critical and supercritical regions, then for a mixture:
The convergence pressure corresponds to a certain composition of the mixture and is a numerical characteristic of the composition.
The atlas of constants was compiled for a wide range of changes in P cx.
For a multicomponent mixture, an equation of phase concentrations is compiled. N is the mass of all components in a certain volume V. N G is the mass of components in a gas, N W is the mass of components in a liquid N = N G +N L.
If we divide by the sum of the molecular weights of all components contained in the volume V, we get n M = n MG + n MF is the number of moles of the components in the gas and liquid phases.
Mole fraction of components in gas "y i" and in liquid "x i":
N G i - mass of the i-th component in the gas phase
N w i is the mass of the i-th component in the liquid phase
M i - molecular weight of the i-th component
The molar fraction of the i-th component in the volume as a whole is expressed:
N i is the mass of the i-th component in the volume V.
From the above expressions it follows: 
(*)
Denote:
n MG / n M = Y is the mole fraction of all components in the gas phase.
n MF / n M \u003d X is the mole fraction of all components in the liquid phase
Considering y i = K i x i ? 1=X+Y, substitute into equality (*)
, 
Equations of phase concentrations
When determining the phase state, various problems can be solved.
For example, given: ν i (composition), P,T,Y → then determine x i and y i from ur-th concentrations. More often there is a problem of finding Y,X according to the known composition ν i , P,T. Then the equality is used 
. The equation is solved by an iterative method. The original equation is solved in the form: 
.
It is obvious that for Y=0 f(Y)=0.
The maximum value of the proportion of the gas phase Y=1. hence the solution is sought in the interval 0
Taking as the initial approximation Y 0 =0.5 and applying sequentially the iterative formulas of the Newton method - the method of chords, a solution is found with a small number of iterations:
Two-phase filtration.
In connection with the design and analysis of the development of oil and gas fields, it is necessary to study the joint flow in a porous medium of several liquids, most often water, oil and gas, which are separate phases that do not mix with each other.
The formation of deposits occurs as a result of the displacement of the water that was originally there. Therefore, along with oil and gas in the reservoirs there is a certain amount (10-30%) of buried water. In addition, many deposits are filled with oil and gas only in the upper dome part, while the underlying zones are filled with water with its initial content that was not pushed aside during the formation process. The uppermost parts of the reservoir contain gas that may be present or collected during development. A two- or three-phase flow occurs during the development of oil deposits, oil and gas, gas condensate, and simply gas deposits in the presence of water underlying the gas cap, i.e. almost always, except for dry gas traps.
When filtering two liquids (oil-water), or liquid and gas (oil-gas, water-gas), Darcy's law has a different form than with a single-phase function:
, 
Here K 1 (S)? K 2 (S) - relatively phase permeability, depending on S - saturation of the 2nd phase, usually water, 1st phase - oil and gas.
In hydrodynamic calculations, it is often convenient to use empirical dependences of relative phase permeability on saturation obtained from experimental data. Let's consider the empirical formulas obtained by Chen-Zhong-Xiang, which can be used in estimating calculations.
1. For water and oil (s-water saturation):
2. for gas and water (s-gas saturation):
The behavior of relative phase permeabilities is described by graphs of the form:
Dependencies have two characteristic points S st, S *
At the point S \u003d Sv relative water permeability \u003d 0 \u003d K 2 (S)
At point S=S * relative permeability of oil (gas) = 0
At these points, the phase with zero permeability is dispersed and occupies isolated dead ends in a porous medium, and therefore is not mobile. Simultaneous filtration of 3 phases has been studied less than two-phase. Use this approach. S n + S in \u003d S well, considering 2 phases - liquid and gas S G + S well \u003d 1 two-phase system.
two-phase system, K n (S), K in (S)
K G (S f), K f (S f)
All relative permeabilities are determined from two-phase diagrams, (S G, S W) and 
Then the relative permeability for oil is K f (S f) K n (S)
for water - K w (S w) K in (S)
for gas - K G (S well)
For thick seams, or sloping seams, where gravity must be taken into account, if the Z axis, then the vertical component of the two-three-phase filtration rate instead contains:
P are the same pressures in the phases.
P * = P + ρgZ reduced pressures.
We have considered expressions for the filtration rate for two, three-phase filtration flow. If two or three immiscible phases (oil, gas, water) are moving, then the same type of equations will be written for each separately:
I=1, 2, 3 where 1 is gas
2 - oil
3 - water or:
The difference from the equation of continuity of a single-phase liquid - the equation includes - the saturation of the phases. On the left side, S i affect the phase permeabilities. On the right side, when compiling the mass balance for an element, we must take into account for a separate phase not the entire pore volume, but its share occupied by the i-th phase.
If we substitute expressions for velocities as functions of pressure and saturation into the system of equations for the continuity of phases, as well as expressions for phase densities as functions of pressure, then for a system of 3 equations we have unknowns 4-P, S 1, S 2 , S 3.
The system is closed by the relation: S 1 + S 2 + S 3 = 1.
In this case, it was assumed that the pressures in the phases are the same Р.
capillary forces. Interfacial (capillary) pressure forces arise in pore channels, for example, in a two-phase flow.
R g - R f = R k (S f)
Since the functions P k (S w) have been studied experimentally, the input cap. forces in the equation does not add the number of unknowns.
7. Multi-phase multi-component filtration. Three-phase - two-three-dimensional filtration.
We consider a system from " n l" phases, for example:
1st phase - oil wetting by gas, non-wetting by water;
2nd phase - water, wetting;
3rd phase - gas, non-wetting.
In general, the system consists of nc"chemical components. When moving, changing the pressure, temperature of the mixture, individual hydrocarbon components can transfer from the oil phase to the gas phase and vice versa. The transition of water into a gaseous state is not excluded during thermal action on the formation. Mass transfer between the phases occurs by various components. In this case, the material the balance when deriving the continuity equation is written for each component separately, and as a result we have " nc"continuity equations.
Fraction of the pore space of the mesh element occupied by " 1 "th phase - S 1 ;
C e j - concentration j- th component in 1 is the phase in the grid volume under consideration.
Then the change in mass j- th component in the grid element must be considered as the sum of its changes in each phase, taking into account ( S 1 C e j) – fraction of the pore volume of the element occupied j- component in 1 – oh phase;
S l C lj ρ l- mass fraction j- 1 – oh phase;
Total mass fraction j- th component in the pore volume of the grid element 1 – oh phase;
- mass fraction j- th component in the pore volume of the grid element;
Ω el is the volume of the grid element.
Mass change j- th component in a short period of time Δt, in the case of a Cartesian coordinate system, we write the form:
Let be the density of the source (sink) 1 – oh phase, - concentration j- th component in the source 1 - oh phase.
Then - the total density of the source according to j- oh component.
The flow terms in the continuity equation, in contrast to the single-phase motion, contain
Bulk speed j- th component in the stream 1 - oh phase.
In the material balance equation j- th component, the flows of the considered component over all phases are summed up.
As a result, the continuity equation for j- th component looks like:
The number of system equations is determined by the number of moving components j=1,2,…,n c .
Three-phase model of an oil reservoir.
In the design of oil reservoirs, the non-volatile oil model (beta model) has been widely used. The hydrocarbon system is approximated by two components: non-volatile (oil) and volatile (gas) soluble in oil. It is assumed that three separate phases coexist in a porous medium: oil, gas, and water.
Water and oil do not mix, do not exchange masses and do not change phases.
The gas is soluble in oil, insoluble in water.
Fluids are assumed to be in thermodynamic equilibrium at constant temperature.
Consider the movement of a three-phase fluid: oil, gas, water (g, o, c):
the gas phase consists of one component - free gas;
water phase - from one water component;
oil phase - 2-component, oil and gas dissolved in it.
Let us determine the concentrations of the components in the phase; 1,2 - oil, gas.
C H1 is the concentration of oil in the oil phase;
C H2 - gas concentration in the oil phase;
C B1 \u003d C B2 \u003d 0, i.e. the water phase does not contain a component of gas and oil;
C G2 = 1, i.e. the gas phase contains only gas;
FROM lj– mass concentration j- th component in 1 - oh phase.
Consider the left side of the equation for the continuity of a multiphase fluid.
(sums for all phases of flows j- th component).
Let us introduce the concept of volumetric phase coefficients: the ratio of the phase volume in reservoir conditions to the volume in standard conditions.
For the gas phase:
For the aqueous phase:
For the oil phase:
here - the volume of oil in reservoir conditions, taking into account the gas dissolved in it;
.
Main literature
additional literature
1. Lysenko VD Innovative development of oil fields. - M.: Nedra-Business Center, 2000. - 516s. - Bibliography: pp.513-514
2. Zakirov, S. N. Development of gas, gas condensate and oil and gas condensate fields / S.N. Zakirov. - M. : Struna, 1998. - 626 p. - Bibliography: p. 597-620. - ISBN 5-85926-011-3
3. Zheltov, Yu. P. Development of oil fields: a textbook for universities / Yu. P. Zheltov. - 2nd ed., revised. and additional - M. : Nedra, 1998. - 365 p. : ill. - Bibliography. With. 359. - ISBN 5-247-03806-1
4. Kanevskaya R.D. Mathematical modeling of hydrodynamic processes in the development of hydrocarbon deposits. - M. - Izhevsk: Institute of Computer Research, 2002. - 140 p.
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1 Method for constructing phase equilibrium constants of multicomponent solutions EV Koldoba New analytical formulas for phase equilibrium constants are proposed in this work, taking into account the influence of fluid composition and more accurately conveying the phase behavior of multicomponent solutions. The approach makes it possible to build a thermodynamically consistent model of multicomponent filtration convenient for numerical simulation: the required computing resources are reduced, and the reliability of calculations is increased. Key words: phase equilibrium constants, EOS, phase transition. Introduction To predict the development of oil and gas fields, numerical modeling methods using three-dimensional hydrodynamic models are widely used. Calculations can take from several minutes to several months, depending on the complexity and accuracy of the model and computer performance. Oil and gas condensate contain hundreds of components, and even small concentrations of one of them can change the phase state of the mixture, so one of the urgent problems is to take into account as many components as possible in order to more accurately describe the complex phase 269
2 E. V. Koldoba of system behavior. Phase transitions in solutions are accompanied not only by the formation of new phases, but also by a continuous change in the component composition of the phases. When distributing the components between the gas and liquid phases, phase equilibrium constants (distribution coefficients) are used. In the Russian-language literature, the terms "constants" or "coefficients" are traditionally used to designate these quantities, although for the class of problems under consideration, these are complex functions of pressure, temperature, and composition of solutions. To simulate phase transitions in multicomponent solutions, modern hydraulic simulators use compositional and thermal models. thermal model, multidimensional phase diagrams (if there are N components in the solution, then the phase diagram is N-dimensional) are not calculated, the distribution of components over phases is carried out using phase equilibrium constants, so the correct setting of these functions can significantly increase the accuracy of calculations. takes the calculation of phase equilibria of multicomponent solutions (flash). Iterative methods calculate "the exact values of the phase equilibrium constants, but the stability of these complex and time-consuming calculations depends largely on the accuracy of their initial approximations, which for given by formulas. It is possible to increase the stability of numerical calculations of phase equilibrium in the compositional model and significantly reduce its time if the formulas that specify the phase equilibrium constants are more accurate. There are many explicit and implicit (iterative) methods for calculating phase equilibrium constants. In iterative methods, equations of state for solutions (EOS) are used to calculate phase equilibrium. The accuracy of which can be improved by adjusting the pair interaction constants . Explicit methods for calculating phase equilibrium constants use reference data on the properties of each component and some characteristics of the mixture known from laboratory tests. 270
3 Method for constructing phase equilibrium constants for multicomponent solutions For an N-component solution with a total molar concentration z, the phase equilibrium constants are functions of pressure, temperature, and composition K = K (p, T, z 1, z 2,..., z N). K are defined as the ratio of the concentrations of the -th component in the gas and liquid phases: K = y /x (1) where is the component number. For weak solutions, K is determined, according to Raoult's laws, as the ratio of the partial pressure of the saturated vapor of the th component p s to the total pressure in the system: K = p s (T) / p (2) The most well-known methods of calculating by formulas are listed below. The formula proposed by Wilson: K = p c p exp (5.31(1 + ω) (1 T c T)) where p is the pressure in the solution, T is the temperature. The following reference data are used to calculate the -th component: p c - critical pressure, T c - critical temperature, ω - acentric factor. Thor and Witson's modification: K = (pc p) A 1 (ps p A = 1) (exp 5.31(1 + ω)(1 T) c T) () 0.6 p 14.7 p k 14.7 in (4) there are several approaches. Praza, for example, proposed the following relation: p k = (MW γ) C7+ (3) (4) 271
4 E. V. Koldoba + 3 [ a (MW γ)c7+)/(T 460) ] ln p + a 5 p 2 + a 6 p where a 1, a 2, a 3, a 4, a 5, a 6 are constants known for each component. Modified by Reid et al., used in modern thermal simulators, has the following form: () a1 K = p + a 2 + a 3 p exp [ a 4 /(T a 5)] (6) where a 1, a 2, a 3, a 4, a 5, a 6 are constants known for each component. The listed models (2-5) do not take into account the composition of real solutions and the features of the dissolution of the components, which sometimes leads to significant errors. For example, in a two-component C 4 H 10 C 10 H 22 solution, the C 4 H 10 component is lighter and its concentration in the gas phase is higher than in the liquid phase, i.e. by definition (1): K C4 H 10 >> 1. However, according to formulas (2-5) it turns out K C4 H 10< 1. В работах - был предложен способ построения констант фазового равновесия с настраиваемыми параметрами, учитывающими поведение конкретного реального раствора в заданном диапазоне давлений и температур: (5) K = A (p + p) α p β (7) где A, p, α, β - настраиваемые параметры, учитывающие свойства реального флюида. В работе была продемонстрирована эффективность и быстродействие такого подхода при моделировании многокомпонентной фильтрации с фазовыми переходами. 272
5 Method for constructing phase equilibrium constants for multicomponent solutions In this paper, we propose a further development of this approach. The assessment of the accuracy of the proposed model and its adjustment was carried out on a more complex and resource-intensive compositional model using the Peng-Robinson equation of state (PR) and the corresponding procedure for calculating phase equilibrium. The accuracy is compared with other methods for calculating K. Problem Statement Let an N-component solution with a total molar concentration z be in a two-phase state, separating into a gas with a concentration y and a liquid with a concentration x. Indices G and L here and below will denote quantities related to the gas and liquid phases, respectively. For the mixture and each phase separately, the following normalization conditions must be met: N z = 1, N x = 1, N y = 1 The gas phase is a non-ideal gas that can be in a supercritical state. To describe the pvt-properties, the paper proposes to use the hyperbolic equation of state (EOS): V G = βrt p + b (8) where V G is the molar volume of the gas phase, T is temperature, p is pressure, β, b are adjustable parameters. The hyperbola (8) has two asymptotes: 1) at V p = 0 2) p V G = b (curve 2 in Fig.1). The liquid phase can be either incompressible or compressible, and near the critical point, the EOS of the liquid must change into the EOS for the gas, so the EOS of the liquid phase is written 273
6 E. V. Koldoba Fig. 1. Hyperbolic equations of state: 1 - for liquid, 2 for gas, ABCD - isotherm of the Peng-Robinson equation. also in hyperbolic form: VL = αrt +b p + p (9) where VL is the molar volume of the liquid phase, α, p, b are adjustable parameters. The hyperbola (9) has two asymptotes: 1) for V p = p 2) p VG = b. Let us construct the Gibbs molar potential for the gas phase: X X gg = βrt ln p + b p + RT y ln B y + y χ (10) The first and second terms of equation (10) are obtained by integrating the EOS of the gas. The remaining terms of the expression are added according to the mixing rules of the physics of solutions and describe the processes of dissolution. Correction coefficients B are introduced in the third term, taking into account the imperfection of the dissolution of the components. χ are functions that depend only on temperature and characterize the pure th component; these functions are the same both in the gas and in the liquid phase. 274
7 Method for constructing phase equilibrium constants of multicomponent solutions After transformations, the Gibbs molar potential for the gas phase has the form: g G = RT ln (p β exp(b p/rt)) + RT y ln B y + y χ (11) It is known that the potential Gibbs is a first order homogeneous function of the number of moles of the components, so the Gibbs molar potential is a first order homogeneous function of the component concentrations. For this property to hold, it is necessary to multiply the first term of equation (11) by the sum of concentrations N y = 1, after which the chemical potentials of the components in the gas phase are calculated: or () gg µ,g = y p = RT ln (p β exp(b p/rt ) + RT y ln B y + χ) µ,g = RT ln (B y p β exp(b p/rt) + χ) (12) The chemical potentials of the components for the liquid phase are calculated similarly: µ,l = RT ln (A x ( p + p) α exp(bp/rt)) + χ (13) where A are some correction factors characterizing the imperfection of the processes of component dissolution in the liquid. From the equality of the chemical potentials of the components µ,l = µ,g in the phases (one of the conditions for phase equilibrium) we have: From the equality we obtain an expression for y /x, i.e. an expression for the phase equilibrium constants K: exp(bp/rt) B p β exp(b p/rt) = C (p + p) α p β exp((b b)p/rt) (14) 275
8 EV Koldoba where C = A /B are integral correction factors that simultaneously characterize the imperfection of dissolution processes in both gas and liquid. In this model, the equilibrium constant of the th component does not explicitly depend on the concentrations and characteristics of other components. However, by adjusting the parameters C, β, b, α, p, b, we thereby take into account the properties of a real solution in the considered range of pressures and temperatures. The parameters C are determined from the calculated or measured values of the concentrations y 0 and x 0 at pressure p 0, we calculate the values K 0 = y 0 /x 0 and find C. Finally, we get: K = K 0 (p + p) α p β p β 0 (p 0 + p) α exp((b b)(p p 0)/RT) (15) Model parameter calculations Parameters K 0, β, b, α, p, b are calculated at some reference pressure p 0 from experimental data or data , obtained from more complex and costly models, we call them "exact" models: α, p, b - are determined from the approximation of the "exact" urs of the fluid by the model; β, b - are determined from the approximation of the "exact" gas EOS by the model one, K 0 - from the values of the concentrations y 0 and x 0 at the equilibrium node, corresponding to the total fluid concentration z. Compositional model
9 Method for constructing the phase equilibrium constants of multicomponent solutions del, using the iterative method and the Peng-Robinson equation of state, which is given as follows: where p = b = RT V b a V (V + b) + b(v + b) p b =1 a = N =1 b = RT c, P c, N j a a j (1 k j) j=1 a = R2 T 2 c, P c, [ T 1 + m (1) T c, ] 2 (16) f ω m = ω ω 2 f ω > m = ω ω ω 3 , then c = x, if gas, then c = y) The parameter b in the Peng-Robinson EOS has the physical meaning of the volume of molecules, therefore V > b is naturally always satisfied, and, moreover, V = b is the pole of the function. The same parameter b is used in the model hyperbolic EOS of a fluid (9) and in it V = b is the asymptote. Thus, the parameter b in the EOS equation (9) is calculated in the same way as in the Peng-Robinson model. The ABCD curve in Fig. 1 is the Peng-Robinson EOS isotherm, branch AB describes the liquid state of the fluid, CD is the gas state. Hyperbola 1 given by equation (9) approximates the liquid branch of the EOS, hyperbola 2 given by equation (8) approximates the gas branch. 277
10 EV Koldoba In the compositional model, the Peng-Robinson EOS is used to calculate the chemical potentials of components in gas and liquid. The system of equations for chemical potentials is solved by iterative methods: µ,g = µ,l, = 1, 2,...N 0. To solve system (17), a computer program was used that simulates the calculation of phase equilibrium in a compositional model (flash), a description of such calculations can be found in . Substituting the concentrations in the gas and the pressure into equation (16), we find the gas root (the largest of the roots, moreover, V > b). Substituting the concentrations in the liquid into equation (16), we find the liquid root (the smallest of the roots, moreover, V > b). We find the derivatives p for the Peng-Robinson EOS: V p V = RT (V b) + 2a(V + b) 2 (V 2 + 2bV b 2) 2 We calculate the derivatives at pressure p 0. On the other hand, we calculate the derivatives of the equations ( 8-9): () p = αrt 2 () p = βrt 2 V L V b V G V b Equating the values of the EOS and their derivatives for the gas and liquid phases, we find the values of the parameters β, b, α, p. For a C 1 H 4 C 10 H 22 solution, the results obtained by formulas (3-6) were compared using the proposed method and the "exact" compositional model. The comparison results are shown in Fig. 2. In the pressure range under consideration (atm), a good agreement between the new model and the "exact" solution was obtained. 278
11 Method for constructing phase equilibrium constants for multicomponent solutions Fig. 2. Phase equilibrium constants for methane in solution C1 H4 C10 H22: K - "exact" solution, W - according to the Wilson formula, R - according to the Reid formula, Kol - according to the new formula References Habbalah W.A., Startzman R.a., Barrafet M.A. use of neural networks for prediction of vapour/lqud equlbrum K-values for lght hydrocarbon mxture, SPE Reservor Engneerng, May Wlson G.M. A modfed Redlch-Kwong EOS. Applcaton to General Physcal Data Calculatons. Paper 15c presented at the 1969 AlChE Natl.Meetng, Cleveland, Oho. Whtson C.H. and Torp S.B. Evaluatng Constant Volume Depleton Data. JPT (March 1983), Trans., AIME,
12 E. V. Koldoba Red R. C., Prausntz J. M. and Sherwood T.K. The properties of Gases and Lquds, 3rd edton, McGraw-Hll, New York, Koldoba A.V., Koldoba E.V. Model Equation of State and Gibbs Potential for Numerical Calculation of Multicomponent Filtration Problems with Phase Transitions. - Geochemistry, 2004, N 5, c Koldoba A.V., Koldoba E.V. Thermodynamically consistent model of a multicomponent mixture with phase transitions. Mathematical modeling, 2010, v.22, N 4, with Koldoba A.V., Koldoba E.V., Myasnikov A.V. Efficient thermodynamically consistent approach for numerical modeling of oil displacement processes, - Mathematical Modeling, 2009, N 10, with Sivukhin D.V. Thermodynamics and molecular physics. 5th ed., rev. - M.: FIZMATLIT, 2005, 544 p. Brusilovsky A.I. Phase transformations in the development of oil and gas. Publishing house "Grail Moscow 2002, 575 p. 280
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