Issue 
Sci. Tech. Energ. Transition
Volume 78, 2023
Characterization and Modeling of the Subsurface in the Context of Ecological Transition



Article Number  20  
Number of page(s)  19  
DOI  https://doi.org/10.2516/stet/2023016  
Published online  14 August 2023 
Regular Article
Calculation of JouleThomson and isentropic expansion coefficients for twophase mixtures
CNRS UMR 5150, Laboratoire des Fluides Complexes et leurs Réservoirs, Université de Pau et des Pays de l’Adour, B.P. 1155, 64013 Pau Cedex, France
^{*} Corresponding author: dbroseta@univpau.fr
Received:
3
February
2023
Accepted:
10
July
2023
Joule–Thomson (JT) and isentropic expansion coefficients describe the temperature change induced by a pressure variation under isenthalpic and isentropic conditions, respectively. They are commonly used to model a variety of processes in which either fluid compression or expansion is involved. While a lot of work has been devoted to inferring the JT coefficient from an equation of state when the fluid is a single phase, little attention has been paid to multiphase fluids, where phase equilibrium has to be taken into account; previous work has only addressed the construction on the JT inversion curve. In the present paper, we describe and implement an approach to calculate these two coefficients for multicomponent fluid systems, including when they form two different phases, liquid, and vapor, in thermodynamic equilibrium. The only ingredients are an equation of state and expressions for the ideal part of the specific heats of the fluid components. We make use of cubic equations of state, but any thermodynamic model can be used in the proposed framework. Calculations conducted with typical geofluids, some of them containing CO_{2}, show that these coefficients are discontinuous at phase boundaries (where enthalpy and entropy variations exhibit angular points), as expected with any thermodynamic quantity built from firstorder derivatives of state functions, and cannot be simply inferred from the coefficients of the liquid and vapor phases.
Key words: Joule–Thomson coefficient / Isentropic expansion coefficient / Carbon dioxide / Inversion curve
© The Author(s), published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Depending on whether the enthalpy H or entropy S of a given fluid system is kept constant, the variation of temperature induced by a pressure variation is described by, respectively, the Joule–Thomson (JT) coefficient$${\mu}_{\mathrm{JT}}={\left(\frac{\partial T}{\partial p}\right)}_{H},$$(1)and the coefficient of isentropic expansion$${\mu}_{S}={\left(\frac{\partial T}{\partial p}\right)}_{S}.$$(2)These coefficients are the slopes of the isenthalps (H = constant) and isentrops (S = constant) in the p–T (pressure–temperature) plane. A positive (resp. negative) value means the fluid experiences cooling (resp. heating) when submitted to depletion. As a rule, the coefficient of isentropic expansion is positive, except in rare instances, e.g., liquid water at temperatures near 0 °C or liquid He^{3} near 0 K [1], whereas JT coefficients commonly exhibit positive (usually for gases) or negative (for liquids and supercritical fluids) values, depending on the p and T conditions. The Joule–Thomson Inversion Curve (JTIC) in the p–T plane separates the cooling and the heating regions; it connects points where the JT coefficients are equal to zero (or where they change sign when a discontinuity arises). This curve delineates the T and p conditions where cooling occurs upon depletion (${\mu}_{\mathrm{JT}}>0$) from conditions where heating occurs (${\mu}_{\mathrm{JT}}<0$).
These coefficients are key for describing a variety of processes in which a variation of fluid pressure, whether a compression or an expansion, generates a change in temperature, which is either exploited or avoided, depending on the application. Refrigeration and cryocooling processes exploit the Joule–Thomson effect, by passing a gas through a valve or a throttling device [2]. The efficiency of compressors relies on the isentropic expansion coefficient μ_{S} of the refrigerant used, which should indeed not be too high [3]. Hydrocarbon mixtures and other fluids such as CO_{2} rapidly flowing to or from an underground porous reservoir, and passing through the wellbore and pipelines and the numerous chokes and valves, experience rapid pressure variations that generate temperature changes. These changes may be so large that a second fluid phase, or even a solid phase (e.g., hydrates [4] or paraffin [5, 6]) appears, leading to flow impairment [7]. Temperature well testing is another area where the knowledge of the JT coefficients helps interpret the data in terms of the composition of the produced fluids–gas, oil, or water [8]. Finally, it is worth mentioning here the recent surge in interest for the JT coefficients of CO_{2}rich fluids in the context of CO_{2} permanent storage in underground reservoirs. On the one hand, these coefficients play a role if CO_{2} is injected in a depleted reservoir, because Joule–Thomson cooling may cause ice or hydrate formation and therefore severe injectivity problems [9]. On the other hand, the same effect provides a safety factor in case of CO_{2} leakage from the reservoir, provided the temperature is low enough – below 10 °C, the maximum temperature of CO_{2} hydrate stability – as is commonly encountered in offshore conditions [8, 10].
These coefficients are difficult to measure experimentally, and most often are estimated from an Equation of State (EoS) or from molecular simulations [11]. Kortekaas et al. [12] calculated the JT coefficients of North Sea gas condensates in the singlephase region using conventional cubic EoS. These calculations were however limited to conditions where the fluid forms a singlephase in the p–T domain investigated.
To date and the best of our knowledge, JT and isentropic expansion coefficients have been calculated for singlephase fluids only, except in the case of onecomponent fluids where simple closedform expressions exist [12]. In the case of multicomponent fluids, the calculations have been limited to the determination of the JTIC in the p–T plane, see e.g. Nichita and Leibovici [13] and Refs. [5, 6].
The purpose of this paper is to describe and implement on a few multicomponent fluid examples simple calculation schemes based on an EoS for estimating the JT and isentropic expansion coefficients of fluids, both in the single and twophase domains. The multicomponent fluids chosen are representative of geofluids – namely, hydrocarbonrich fluids of underground reservoirs, some of them containing CO_{2}. One important expected feature is the discontinuity of these coefficients at the crossing of twophase boundaries, which stems from the fact that these coefficients are related to firstorder derivatives of state functions. This feature has been analyzed for onecomponent fluids by Sychev [14]. In this work, discontinuities for twophase, multicomponent systems are calculated.
The paper is structured as follows. The next section is a brief reminder of how these two coefficients are related to each other and to other thermodynamic parameters, which are readily calculated from an EoS. The calculation procedure is outlined for a fluid system containing two phases in thermodynamic equilibrium. Examples are given for a variety of oils and gas condensates, i.e., hydrocarbon mixtures exhibiting bubble or dew points, as well as a more complex fluid system. Conclusions are presented at the end of the paper. Three appendices give the expressions of required thermodynamic functions calculated with an EoS, ideal gas heat capacities, and mixture compositions and component properties used in test examples.
2 The Joule–Thomson and isentropic expansion coefficients
The Joule–Thomson and isentropic expansion coefficients are related to the partial derivatives with respect to pressure p and temperature T of, respectively, the enthalpy H and entropy S of the fluid system by$${\mu}_{\mathrm{JT}}={\left(\frac{\partial T}{\partial p}\right)}_{H}={\left(\frac{\partial H}{\partial p}\right)}_{T}{\left(\frac{\partial H}{\partial T}\right)}_{p}^{1}$$(3)and$${\mu}_{S}={\left(\frac{\partial T}{\partial p}\right)}_{S}={\left(\frac{\partial S}{\partial p}\right)}_{T}{\left(\frac{\partial S}{\partial T}\right)}_{p}^{1}.$$(4)H and S also depend on the number of moles n_{i} of the various constituents $i=1,\dots ,\mathrm{nc}$, which are kept constant and for simplicity omitted in the above expressions. In the case of a multicomponent fluid system, with molar content $\mathit{n}=({n}_{1},{n}_{2},\dots ,{n}_{\mathrm{nc}})$, the enthalpy and entropy are split into an ideal gas term, H ^{ig } and S ^{ig }, and a departure term directly related to the EoS, H ^{dep} and S ^{dep} $$H={H}^{\mathrm{ig}}\left(\mathit{n}\right)+{H}^{\mathrm{dep}}\left(\mathit{n}\right)$$(5)and$$S={S}^{\mathrm{ig}}\left(\mathit{n}\right)+{S}^{\mathrm{dep}}\left(\mathit{n}\right).$$(6)The above expressions (Eqs. (3)–(6) hold when substituting H and S and their ideal and departure parts with their molar counterparts h = H/n, s = S/n, ${h}^{\mathrm{id}}={H}^{\mathrm{id}}/n$, etc., where $n={n}_{1}+{n}_{2}+\cdots +{n}_{\mathrm{nc}}$ is the total number of moles. For cubic EoS, simple closedform expressions exist for the departure parts ${h}^{\mathrm{dep}}={H}^{\mathrm{dep}}/n$ and ${s}^{\mathrm{dep}}={S}^{\mathrm{dep}}/n$ as a function of compositions or mole fractions, z_{i} = n_{i}/n. These expressions are given in Appendix A, for a general form of twoparameter cubic EoS, containing the Soave [15] and PengRobinson [16, 17] EoS. Note that any thermodynamic model can be used; only the expressions of enthalpy and entropy departures are required. On the other hand, the ideal parts ${h}^{\mathrm{id}}={H}^{\mathrm{id}}/n$ and ${s}^{\mathrm{id}}={S}^{\mathrm{id}}/n$ have partial derivatives with respect to temperature expressed as$${\left(\frac{{\partial h}^{\mathrm{ig}}}{\partial T}\right)}_{n}=\sum _{i=1}^{\mathrm{nc}}{z}_{i}{C}_{\mathrm{pi}}\left(T\right)$$(7)and$${\left(\frac{{\partial s}^{\mathrm{ig}}}{\partial T}\right)}_{n}=\sum _{i=1}^{\mathrm{nc}}{z}_{i}\frac{{C}_{\mathrm{pi}}\left(T\right)}{T}$$(8)where ${C}_{\mathrm{pi}}\left(T\right)$ is the ideal gas isobaric heat capacity per mole of component i (see Appendix B for the expressions used in this work). The ideal gas enthalpy and entropy do not depend on pressure.
When two distinct phases (liquid and vapor, hereafter identified with subscripts L and V) are present, the enthalpy and entropy of the twophase fluid system to consider in equations (3)–(6) are the sum of the enthalpies and entropies of the liquid and vapor phases, i.e., H_{t} = H_{V} + H_{L} and S_{t} = S_{V} + S_{L} or, expressed on a molar basis:$$h={h}^{\mathrm{ig}}\left(\mathit{z}\right)+\theta {h}_{V}^{\mathrm{dep}}\left(\mathit{y}\right)+\left(1\theta \right){h}_{L}^{\mathrm{dep}}\left(\mathit{x}\right)$$(9)and$$s={s}^{\mathrm{ig}}\left(\mathit{z}\right)+\theta {s}_{V}^{\mathrm{dep}}\left(\mathit{y}\right)+\left(1\theta \right){s}_{L}^{\mathrm{dep}}\left(\mathit{x}\right),$$(10)where θ is the molar fraction of the vapor phase, and x and y are the vectors of liquid and vapor phase compositions, respectively (subscript t stands for total enthalpy and entropy). We assume in this work that thermodynamic equilibrium is ensured, which is legitimate when there are strong enough interdispersion and small domains of the two phases: these conditions hold for instance for a fluid in a porous medium, at least under incipient phase separation, i.e., near bubble or dew point conditions [1]. The quantities θ, x, and y are then readily obtained from an EoS by a flash calculation for any given T and p. The partial derivatives of interest are obtained by numerical derivation of the enthalpy and entropy (whose expressions are given below for the ideal parts and in Appendix A for the departure part) along constantpressure or constanttemperature paths around T and p.
The isobaric ideal gas heat capacity of one mole of the twophase fluid is$${C}_{p}^{\mathrm{ig}}=\sum _{i=1}^{\mathrm{nc}}{z}_{i}{C}_{\mathrm{pi}}^{\mathrm{ig}}\left(T\right).$$(11)The ideal gas part of the molar enthalpy is$${H}^{\mathrm{ig}}={H}_{0}^{\mathrm{ig}}+\sum _{i=1}^{\mathrm{nc}}{z}_{i}\underset{{T}_{0}}{\overset{T}{\int}}{C}_{p}^{\mathrm{ig}}\left(T\right)\mathrm{d}T$$(12)and the ideal part of entropy is$${S}^{\mathrm{ig}}={S}_{0}^{\mathrm{ig}}+\sum _{i=1}^{\mathrm{nc}}{z}_{i}\underset{{T}_{0}}{\overset{T}{\int}}\frac{{C}_{p}^{\mathrm{ig}}}{T}\left(T\right)\mathrm{d}T,$$(13)where ${H}_{0}^{\mathrm{ig}}$ and ${S}_{0}^{\mathrm{ig}}$ are the ideal gas enthalpy and entropy at the reference conditions, respectively. These ideal parts depend only on temperature (not on pressure), hence the partial derivatives of the total enthalpy H_{t} with respect to pressure and temperature (see Eq. (3)) can be expressed as follows:$${\left(\frac{\partial {H}_{t}}{\partial p}\right)}_{T,\mathit{n}}={\left(\frac{\partial {H}_{L}^{\mathrm{dep}}}{\partial p}\right)}_{T,\mathit{n}}+{\left(\frac{\partial {H}_{V}^{\mathrm{dep}}}{\partial p}\right)}_{T,\mathit{n}}$$(14)and$${{\left(\frac{\partial {H}_{t}}{\partial T}\right)}_{p,\mathit{n}}=\left(\frac{\partial {H}_{L}^{\mathrm{dep}}}{\partial T}\right)}_{p,\mathit{n}}+{\left(\frac{\partial {H}_{V}^{\mathrm{dep}}}{\partial T}\right)}_{p,\mathit{n}}+\sum _{i=1}^{\mathrm{nc}}{z}_{i}{C}_{\mathrm{pi}}.$$(15)Similarly, the partial derivatives of the total entropy S_{t} with respect to pressure and temperature (see Eq. (4)) are:$${\left(\frac{\partial {S}_{t}}{\partial p}\right)}_{T,\mathit{n}}={\left(\frac{\partial {S}_{L}^{\mathrm{dep}}}{\partial p}\right)}_{T,\mathit{n}}+{\left(\frac{\partial {S}_{V}^{\mathrm{dep}}}{\partial p}\right)}_{T,\mathit{n}}$$(16)and$${{\left(\frac{\partial {S}_{t}}{\partial T}\right)}_{p,\mathit{n}}=\left(\frac{\partial {S}_{L}^{\mathrm{dep}}}{\partial T}\right)}_{p,\mathit{n}}+{\left(\frac{\partial {S}_{V}^{\mathrm{dep}}}{\partial T}\right)}_{p,\mathit{n}}+\sum _{i=1}^{\mathrm{nc}}{z}_{i}\frac{{C}_{\mathrm{pi}}}{T}.$$(17)Note that the partial derivatives with respect to pressure and temperature in equations (14)–(17) are at constant n (mixture composition). In this work, these derivatives are calculated numerically. An analytical treatment would require a framework similar to that presented in Ref. [13], relating the partial derivatives at constant n to those at constant n _{k } (phase compositions) obtained from the EoS, by differentiation of equations (A10) and (A11).
Another route to calculate the JT and isentropic expansion coefficients makes use of the differential expression of the enthalpy H $$\mathrm{d}H=\mathrm{d}S+V\mathrm{d}p$$(18)and the Maxwell relation$${\left(\frac{\partial S}{\partial p}\right)}_{T}={\left(\frac{\partial V}{\partial T}\right)}_{p},$$(19)leading to$${\left(\frac{\partial H}{\partial p}\right)}_{T}=T{\left(\frac{\partial S}{\partial T}\right)}_{p}+V$$(20)and therefore, inserting equations (20) into (3) and equations (19) into (4),$${\mu}_{\mathrm{JT}}=\frac{1}{{C}_{p}}\left[T{\left(\frac{\partial V}{\partial T}\right)}_{p}V\right]=\frac{V\left(\mathrm{T\alpha}1\right)}{{C}_{p}},$$(21) $${\mu}_{S}=\frac{1}{{C}_{p}}\left[T{\left(\frac{\partial V}{\partial T}\right)}_{p}\right]=\frac{\mathrm{VT\alpha}}{{C}_{p}}$$(22)where V is the volume and α is the isobaric thermal expansivity$$\alpha =\frac{1}{V}{\left(\frac{\partial V}{\partial T}\right)}_{p}$$(23)and$${C}_{p}={\left(\frac{\partial H}{\partial T}\right)}_{p}=T{\left(\frac{\partial S}{\partial T}\right)}_{p}$$(24)the isobaric heat capacity. The two coefficients are therefore related as follows:$${\mu}_{\mathrm{JT}}={\mu}_{S}\frac{V}{{C}_{p}}.$$(25)Equations (21) and (22) are equivalent to equations (3) and (4), and offer an alternative route for estimating these coefficients. These equations hold at twophase pressure and temperature conditions if the extensive thermodynamic functions involved are the sum of the contributions of the two equilibrium phases. When the fluid system contains two phases (liquid and vapor), the volume is the total volume$${V}_{t}={V}_{L}+{V}_{V}=\frac{\mathrm{RT}}{p}\left({n}_{L}{Z}_{L}+{n}_{V}{Z}_{V}\right)$$(26)with Z_{L} and Z_{V} the compressibility factors of the liquid and vapor phases (obtained from an EoS, see Appendix A, and a flash calculation), respectively. The twophase expansivity and isobaric heat capacity are obtained by carrying out flash calculations at various temperatures near the temperature of interest along a constant pressure path.
The singlephase JTIC is calculated analytically, as detailed in Refs. [19, 20]. In the twophase region, the location of the JTIC is calculated by tracking sign changes of the numerators (the denominators are always positive) in equations (3) or (21) on a given isotherm or isobar. Partial derivatives are calculated numerically by finite differences. It is incorrect to calculate the JTIC at feed composition in the twophase domain.
3 Results and discussion
The singlephase and twophase Joule–Thomson and isentropic expansion coefficients of several geofluids reported in the literature are calculated here with the Peng–Robinson EoS [16, 17] using the approaches described above. Some of the fluid examples and the pressure and temperature conditions chosen here are close to those considered in Ref. [13] where the focus was on the JTIC. The JT coefficients are therefore expected to be small, whether the fluid considered is an oil or a gas condensate at reservoir temperature (a few K/bar), much smaller than the typical values of gases at low pressure. All input data, that is, feed compositions z_{i}, component critical properties (temperature and pressure) acentric factors${\omega}_{i}$, molecular weights ${M}_{\mathrm{Wi}},$as well as Binary Interaction Parameters (BIPs) k_{ij} are taken from the literature. Riazi [21] and Kesler and Lee correlations [22] are used for the ideal gas isobaric heat capacity, for the light and heavy components, respectively. In the twophase domain, the partial derivatives of enthalpy and entropy with respect to pressure and temperature are calculated numerically using a secondorder central finite difference method (the pressure and temperature derivation steps used were Δp = 0.0001 bar and ΔT = 0.00001 K, respectively; the sensitivity on the results to perturbations used in the numerical differentiation was carefully checked).
3.1 Reservoir fluid
The first example is the reservoir fluid (described by 20 components) studied by Stenby et al. [23]. The feed composition and component critical properties are taken from Nichita et al. [12] and are listed in Table C1. The nonzero BIPs in the EoS, taken also from Ref. [12] are:${k}_{{\mathrm{N}}_{2}{\mathrm{C}}_{1}}=0.02;{k}_{{\mathrm{N}}_{2}{\mathrm{C}}_{2}}=0.06;{k}_{{\mathrm{N}}_{2}j\ge 5}=0.08;$ ${k}_{{\mathrm{CO}}_{2}{\mathrm{C}}_{1}}=0.12;$ ${k}_{{\mathrm{CO}}_{2}j\ge 4}=0.15;$ ${k}_{{\mathrm{C}}_{1}{\mathrm{C}}_{6}}=0.0298;$ ${k}_{{\mathrm{C}}_{1}{\mathrm{C}}_{7}}=0.035;$ ${k}_{{\mathrm{C}}_{1}{\mathrm{C}}_{11}}=0.0442$; ${k}_{{\mathrm{C}}_{1}{\mathrm{C}}_{14}}=0.04488$; ${k}_{{\mathrm{C}}_{1}{\mathrm{C}}_{16}}=0.0512$; ${k}_{{\mathrm{C}}_{1}{\mathrm{C}}_{20}}=0.0544$; ${\mathrm{k}}_{{\mathrm{C}}_{1}{\mathrm{C}}_{23}}=0.0565$; ${\mathrm{k}}_{{\mathrm{C}}_{1}{\mathrm{C}}_{27}}=0.0586$; ${k}_{{\mathrm{C}}_{1}{\mathrm{C}}_{32}}=0.0609$; ${k}_{{\mathrm{C}}_{1}{\mathrm{C}}_{38}}=0.0627$; ${k}_{{\mathrm{C}}_{1}{\mathrm{C}}_{48+}}=0.08$.
The phase envelope and the JTIC are drawn in Figure 1. The JTIC exhibits three distinct branches: a branch in the singlephase region and an “Sshaped” branch in the twophase region, which are connected by a segment of the bubble point curve. This allows up to three Joule–Thomson inversions on isotherms below the intersection of the twophase branch with the phase boundary. The minimum inversion temperature located in the twophase region is around 185 K and the maximum inversion temperature located in the singlephase region is about 1625 K. The variation of the Joule–Thomson coefficient in the P–T plane for temperatures from 225 K to 675 K and for pressures between 100 and 500 bar is depicted in Figure 2 (an arrow indicates the location of the phase boundary).
Figure 1 Joule–Thomson inversion curve and phase envelope of reservoir fluid (the dotted line indicates the extrapolation of the singlephase JTIC in the twophase domain). 
Figure 2 Joule–Thomson coefficient of the reservoir fluid in the intervals 225–675 K and 100–500 bar. 
Figure 3 plots the reduced departure enthalpy H ^{dep}/RT (note that the ideal part of the enthalpy does not depend on pressure) of this reservoir oil at four isotherms: T = 400 K, T = 540 K, T = 615 K, and T = 650 K, in the pressure interval from 100 to 400 bar. The Bubble Point (BP) pressures calculated with the PR EoS [16] for these isotherms are 305.4 bar, 275.65 bar, 240.28 bar, and 220.23 bar, respectively. The figure shows that the departure enthalpy exhibits angular points at the bubble point, i.e. a discontinuity of its derivative with respect to temperature. The single and twophase Joule–Thomson coefficients, calculated on the above isotherms in the pressure interval from 100 to 400 bar are plotted in Figure 4. The figure shows that the Joule–Thomson coefficient is negative at 400 K, it changes sign once at 540 and 650 K and three times at 615 K. The results agree with those from Ref. [12]. Except for the first isotherm where it varies in a monotonous manner, the Joule–Thomson coefficient exhibits discontinuities (corresponding to the angular points of the enthalpy in Fig. 3).
Figure 3 Reduced departure enthalpy of the reservoir fluid at: (a) T = 400 K, (b) T = 540 K, (c) T = 615 K and (d) T = 650 K. 
Figure 4 Joule–Thomson coefficient of reservoir fluid at: (a) T = 400 K, (b) T = 540 K, (c) T = 615 K and (d) T = 650 K. 
The isentropic expansion coefficient of this reservoir fluid is represented in Figure 5 vs. pressure on four isotherms: T = 400 K, T = 540 K, T = 615 K, and T = 650 K, in the pressure interval from 100 to 400 bar. As expected, the isentropic expansion coefficient is positive at all four isotherms, therefore it presents no inversion. In a similar manner to the Joule–Thomson coefficient, the isentropic expansion coefficient varies in a monotonous manner, except at phase boundaries where it exhibits discontinuities (this corresponds to the angular points of the entropy variation with pressure).
Figure 5 Isentropic expansion coefficient of reservoir fluid at: (a) T = 400 K, (b) T = 540 K, (c) T = 615 K and (d) T = 650 K. 
3.2 North Sea gas condensate (Lille Frigg reservoir)
The second example is a 27component North Sea (Lille Frigg reservoir) gas condensate in the temperature range from 300 to 500 K (the fluid system has no T_{c}), with feed composition, component properties taken from Nichita and Leibovici [13], see Table C2. The nonzero BIPs in the EoS are: ${k}_{{\mathrm{N}}_{2}{\mathrm{CO}}_{2}}=0.15,\hspace{0.5em}{k}_{{\mathrm{N}}_{2}\mathrm{j}}=0.12,{k}_{{\mathrm{CO}}_{2}j}=0.15$, ${k}_{{\mathrm{C}}_{1}\mathrm{CN}1}=0.03$, ${k}_{{\mathrm{C}}_{1}\mathrm{CN}2}=0.05$ and ${k}_{{\mathrm{C}}_{2}\mathrm{CN}2}=0.03$.
The phase envelope and the JTIC are depicted in Figure 6. This mixture exhibits an openshape phase envelope with no critical point. A dotted line indicated the extrapolation of the singlephase JTIC in the twophase domain; it has no physical significance since the stable state of the mixture is a twophase one and it is incorrect to calculate the JT inversion this way (as done for instance in Ref. [13]). There are three distinct branches of the JTIC: a singlephase branch, a portion of the dew point curve, and a twophase branch. In a small temperature interval, there are three inversion pressures on a given isotherm. The variation of the Joule–Thomson coefficient in the P–T plane for a temperature range from 390 K to 470 K and a pressure range from 350 to 500 bar is drawn in Figure 7 (the location of the dew point curve is indicated by an arrow).
Figure 6 Joule–Thomson inversion curve and phase envelope of NorthSea gas condensate (the dotted line indicates the extrapolation of the singlephase JTIC in the twophase domain). 
Figure 7 Joule–Thomson coefficient of the North Sea gas condensate in the intervals 390–470 K and 350–500 bar. 
The reduced enthalpy departures of this mixture along three isotherms (T = 400 K, T = 435 K, and T = 470 K) are plotted in Figure 8 against pressure in the interval 400–600 bar. The dewpoint pressures calculated with the PR EoS for the three isotherms are 545 bar, 495 bar, and 439 bar, respectively. The departure enthalpies exhibit angular points at dewpoint pressures, corresponding to a discontinuity of their derivatives with respect to pressure. The results match those reported in Ref. [13].
Figure 8 Reduced departure enthalpy of North Sea gas condensate at: (a) T = 400 K, (b) T = 430 K and (c) T = 470 K. 
The Joule–Thomson coefficients are plotted against pressure in Figure 9 for the above three isotherms in the pressure interval from 400 to 600 bar, with discontinuities at the phase boundary. At T = 430 K, Figure 9b shows three inversion pressures on this isotherm. The isentropic expansion coefficient is drawn vs. pressure for the same three isotherms in Figure 10, exhibiting discontinuities at dew points; it is always positive with no inversion.
Figure 9 Joule–Thomson coefficient of the North Sea gas condensate at: (a) T = 400 K, (b) T = 430 K and (c) T = 470 K. 
Figure 10 Isentropic expansion coefficient of the North Sea gas condensate at: (a) T = 400 K, (b) T = 430 K and (c) T = 470 K. 
3.3 Bakken fluid
The third example, referred to as the Bakken fluid, is described by 8 components (four lowmolecularweight alkanes, two intermediate, and two heavy pseudocomponents). The composition, component properties (given in Tab. C3) are taken from Nojabaei et al. [24]. The nonzero binary interaction parameters in the EoS are:${k}_{12}=0.005$, ${k}_{13}={k}_{14}=0.0035$, ${k}_{15}=0.0035$, ${k}_{1j\ge 6}=0.0033$, ${k}_{23}={k}_{24}={k}_{25}=0.0031$, ${k}_{2j\ge 6}=0.0026$.
The phase envelope and the JTIC calculated with the PR EoS are drawn in Figure 11 (the critical point coordinates are T_{c} = 584.45 K and p_{c} = 259.25 bar). The JTIC follows the bubblepoint curve, from the intersection of the singlephase JTIC with the phase boundary towards low temperatures. This behavior is a reminiscence from pure fluids, for which the Joule–Thomson inversion takes place on the vapor pressure curve at low temperatures, as described earlier by Nichita and Leibovici [13] (see also Sychev [14]).
Figure 11 Joule–Thomson inversion curve and phase envelope of Bakken fluid (the dotted line indicates the extrapolation of the singlephase JTIC in the twophase domain). 
Figure 12 shows the variation of the Joule–Thomson coefficient in the P–T plane for temperatures from 200 K to 500 K and for pressures between 40 and 250 bar.
Figure 12 Joule–Thomson coefficient for the Bakken fluid in the intervals 200–500 K and 40–250 bar. 
The calculations are carried out along the isotherm T = 389.3 K in the pressure interval from 100 to 400 bar, where the bubble point pressure is 197.69 bar. The results for the JT and isentropic expansion coefficients are depicted in Figures 13 and 14, respectively. When the pressure decreases along this isotherm and bubble point conditions are crossed, both coefficients experience a discontinuous increase: in the case of the JT coefficient, the jump is from a negative to a positive value.
Figure 13 Joule–Thomson coefficient of Bakken fluid along the isotherm T = 389.3 K. 
Figure 14 Isentropic expansion coefficient of Bakken fluid along the isotherm T = 389.3 K. 
3.4 SJ15 fluid
The last example is a 15component mixture, denoted here SJ15, with composition and component properties taken from Sherafati and Jessen [25] (listed in Tab. C4). The nonzero binary interaction parameters in the EoS are:${k}_{{\mathrm{N}}_{2}{\mathrm{CO}}_{2}}=0.017,{k}_{{\mathrm{N}}_{2}{\mathrm{C}}_{1}}=0.0311$, ${k}_{{\mathrm{N}}_{2}{\mathrm{C}}_{2}}=0.0515$, ${k}_{{\mathrm{N}}_{2}{\mathrm{C}}_{3}}=0.0852$, ${k}_{{\mathrm{N}}_{2}{i\mathrm{C}}_{4}}=0.1033$, ${k}_{{\mathrm{N}}_{2}n{\mathrm{C}}_{4}}=0.0800$, ${k}_{{\mathrm{N}}_{2}{i\mathrm{C}}_{5}}=0.0922$, ${k}_{{\mathrm{N}}_{2}n{\mathrm{C}}_{5}}=0.10$, ${k}_{{\mathrm{N}}_{2}j\ge 10}=0.08$, ${k}_{{\mathrm{CO}}_{2}j\mathrm{\u03f5}[29]}=0.12$, ${k}_{{\mathrm{CO}}_{2}j9}=0.1$, ${k}_{{\mathrm{C}}_{1}\mathrm{PS}1}=0.028349$, ${k}_{{\mathrm{C}}_{1}\mathrm{PS}2}=0.044813$, ${k}_{{\mathrm{C}}_{1}\mathrm{PS}3}=0.062256$, ${k}_{{\mathrm{C}}_{1}\mathrm{PS}4}=0.077679$, ${k}_{{\mathrm{C}}_{1}\mathrm{PS}5}=0.0952$.
The phase envelope and the JTIC are plotted in Figure 15 (the critical point temperature is 721.34 K). The Joule–Thomson and isentropic expansion coefficients of the SJ15 fluid at T = 383.15 K in the pressure interval 50–200 bar are shown in Figures 16 and 17, respectively. These coefficients increase with decreasing pressure in a continuous manner, except at the bubble point pressure of 101.96 bar, where the JT coefficient jumps from a negative value in the singlephase (undersaturated liquid) region to a positive value in the twophase liquid–vapor region. The isentropic expansion also experiences a similar jump but remains positive in the whole pressure interval.
Figure 15 Joule–Thomson inversion curve and phase envelope of SJ15 fluid (the dotted line indicates the extrapolation of the singlephase JTIC in the twophase domain). 
Figure 16 Joule–Thomson coefficient of SJ15 fluid along the isotherm T = 383.15 K. 
Figure 17 Isentropic expansion coefficient of SJ15 fluid along the isotherm T = 383.15 K. 
4 Conclusions
We have defined and implemented two different but equivalent approaches to extend the calculation of both Joule–Thomson and isentropic expansion coefficients of fluid mixtures from the singlephase to the twophase region. If one of these coefficients is known, the other can easily be obtained (by adding or subtracting 1/C_{p}, where C_{p} is the total isobaric heat capacity at twophase conditions). For both coefficients, a discontinuity occurs at liquid/vapor phase boundaries, which correspond to angular points of the enthalpy and entropy variations, respectively.
The JTIC may exhibit several branches: a branch in the singlephase region (down to the intersection with the phase boundary), a branch in the twophase region, and a portion of the phase boundary itself. Thus, several inversion temperatures/pressures may occur at given pressure/temperature conditions. A rule seems to emerge from JTIC calculations for the test mixtures in this work and confirmed for many other mixtures (not reported here), that is, a twophase branch exists for openshaped (or Sshaped) phase envelopes). For closedphase envelopes, the JT inversion takes place at the phase boundary at low temperatures, as a reminiscence from the behavior of pure fluids, where the JT inversion takes place on the vapor pressure curve at low enough temperatures.
A cubic equation of state is used here, but any thermodynamic model can be used; beyond a twophase flash calculation routine, only the expressions of the ideal part and of the residual (or departure) part (which is specific to a given EoS) of the enthalpy and entropy are required. In the twophase region, the required partial derivatives can be calculated numerically, by a finitedifference scheme with the inputs obtained from flash calculations along a constant temperature or constant pressure path.
This work is a first step of a larger project, aimed to study the JT effect and other secondorder derivative properties for CO_{2} injection in depleted oil and gas reservoirs, as well as the difference between using conventional (at constant pressure and temperature) and isobarisenthalpic phase equilibrium calculations in compositional reservoir simulation. From practical reasons, in hydrocarbonsbrineCO_{2} systems, it is sufficient to use a robust multiphase flash routine and an accurate evaluation of enthalpies in the multiphase region; an accurate equation of state must be used for the aqueous phase and an adapted thermodynamic model is required to account for salinity.
Acknowledgments
AOU thanks UPPA for financial support.
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Appendix A
A general form of twoparameter cubic EoS is used in this work. Its explicit in pressure form is$$P=\frac{\mathrm{nRT}}{V\mathrm{bn}}\frac{{\mathrm{an}}^{2}}{\left(V+{\delta}_{1}\mathrm{bn}\right)\left(V+{\delta}_{2}\mathrm{bn}\right)},$$(A1)where ${\delta}_{\mathrm{1,2}}=1\pm \sqrt{2}$ for the Peng–Robinson (PR) EoS [16, 17] and ${\delta}_{1}=1,{\delta}_{2}=0$ for the Soave–Redlich–Kwong (SRK) EoS [15].
The van der Waals mixing rules are used for the energy, a for the volume and b parameters of the EoS$$a=\sum _{i=1}^{\mathrm{nc}}\sum _{j=1}^{\mathrm{nc}}{x}_{i}{x}_{j}\sqrt{{a}_{i}}\sqrt{{a}_{j}}\left(1{k}_{\mathrm{ij}}\right),$$(A2) $$b=\sum _{j=1}^{\mathrm{nc}}{x}_{j}{b}_{j},$$(A3)where$${a}_{i}={\mathrm{\Omega}}_{a}\frac{{R}^{2}{T}_{\mathrm{ci}}^{2}}{{p}_{\mathrm{ci}}}{\left[1+m\left({\omega}_{i}\right)\left(1\sqrt{\frac{T}{{T}_{\mathrm{ci}}}}\right)\right]}^{2},$$(A4) $${b}_{i}={\mathrm{\Omega}}_{b}\frac{{\mathrm{RT}}_{\mathrm{ci}}}{{p}_{\mathrm{ci}}},$$(A5)in which k_{ij} is the Binary Interaction Parameter (BIP) between components i and j.
The values of Ω_{a} and Ω_{b} and the expression of the function m(ω_{i}) are specific to a given EoS. For the PR EoS Ω_{a} = 0.45724, Ω_{a} = 0.0778 and m(ω_{i}) is [15, 16]$$m\left(\omega \right)=0.37464+1.54226\omega 0.26992{\omega}^{2};\omega 0.49,$$(A6) $$m\left(\omega \right)=0.379642+1.48503\omega 0.164423{\omega}^{2}+0.016667{\omega}^{3};\omega \ge 0.49.$$
For the SRK EoS, Ω_{a} = 0.42748, Ω_{b} = 0.08664 [14] and$$m\left(\omega \right)=0.48508+1.55171\omega 0.15613{\omega}^{2}.$$(A7)The implicit form (in compressibility factor Z) of the EoS is obtained by substituting A = ap/R ^{2}T ^{2}, B = bp/RT, and Z = pv/RT into equation (A1) $${Z}^{3}+\left[\left({\delta}_{1}+{\delta}_{2}1\right)B1\right]{Z}^{2}+\left[A+{\delta}_{1}{\delta}_{2}{B}^{2}\left({\delta}_{1}+{\delta}_{2}\right)B\left(B+1\right)\right]Z\left[\mathrm{AB}+{\delta}_{1}{\delta}_{2}{B}^{2}\left(B+1\right)\right]=0.$$(A8)The fugacity coefficients are, for i = 1, nc and k = V, L $$\mathrm{ln}{\phi}_{\mathrm{ik}}=\left({Z}_{k}1\right)\frac{{B}_{i}}{{B}_{k}}\mathrm{ln}\left({Z}_{k}{B}_{k}\right)\frac{1}{\left({\delta}_{1}{\delta}_{2}\right){B}_{k}}\left(2\sum _{j=1}^{\mathrm{nc}}{x}_{j}{A}_{\mathrm{ij}}{A}_{k}\frac{{B}_{i}}{{B}_{k}}\right)\mathrm{ln}\left(\frac{{Z}_{k}+{\delta}_{1}{B}_{k}}{{Z}_{k}+{\delta}_{2}{B}_{k}}\right).$$(A9)From the general form of the cubic equations of state, ${h}_{k}^{\mathrm{dep}}$ is expressed as follows:$${h}_{k}^{\mathrm{dep}}=\mathrm{RT}\left\{{Z}_{k}1\frac{1}{({\delta}_{1}{\delta}_{2}){b}_{k}\mathrm{RT}}\left[{a}_{k}T\frac{{\mathrm{d}a}_{k}}{\mathrm{d}T}\right]\mathrm{ln}\left(\frac{{Z}_{k}+{\delta}_{1}{B}_{k}}{{Z}_{k}+{\delta}_{2}b{B}_{k}}\right)\right\}$$(A10)and ${s}_{k}^{\mathrm{dep}}$ is$${s}_{k}^{\mathrm{dep}}=R\left[\mathrm{ln}\left(\frac{p}{{p}_{0}}\right)\mathrm{ln}\left({Z}_{k}{B}_{k}\right)\frac{\frac{\mathrm{d}{a}_{k}}{\mathrm{d}T}}{\left({\delta}_{1}{\delta}_{2}\right){b}_{k}R}\mathrm{ln}\left(\frac{{Z}_{k}+{\delta}_{1}{B}_{k}}{{Z}_{k}+{\delta}_{2}{B}_{k}}\right)+\sum _{i=1}^{\mathrm{nc}}{x}_{\mathrm{ik}}\mathrm{ln}\left({x}_{\mathrm{ik}}\right)\right],$$(A11)where p_{0} is the reference pressure (taken equal to 1 bar).
Appendix B
The ideal gas isobaric heat capacity ${C}_{p}\left(T\right)$ for one mole of light hydrocarbon components (specific carbon number less than 7) and nonhydrocarbon components is calculated with the correlation of Riazi [21], which consists in a four degree polynomial function in temperature$${C}_{p}\left(T\right)={C}_{p,0}+{C}_{p,1}T+{C}_{p,2}{T}^{2}+{C}_{p,3}{T}^{3}+{C}_{p,4}{T}^{4}.$$(B1)The constants ${C}_{p,i},\hspace{0.5em}i=0,4$ are tabulated in Ref. [20].
Kesler and Lee [21] give the constants ${C}_{p,i},\hspace{0.5em}i=0,2$for a single heavy hydrocarbon fraction (specific carbon number greater than 7), as follows:$${C}_{p,0}=0.33886+0.02827{K}_{w}0.06105\mathrm{CF}+0.59332\omega \mathrm{CF},$$ $${C}_{p,1}=\left[0.9291+1.1543{K}_{w}0.0368{K}_{w}^{2}+\mathrm{CF}\left(4.569.48\omega \right)\right]{10}^{4},$$ $${C}_{p,2}=1.6659*{10}^{7}+\mathrm{CF}\left(0.5360.6828\omega \right){10}^{7},$$ $${C}_{p,3}={C}_{p,4}=0,$$where$$\mathrm{CF}={\left[\left(12.8{K}_{w}\right)\left(10{K}_{w}\right)/\left(10\omega \right)\right]}^{2}$$ $\omega $ and ${K}_{w}$ are respectively, the acentric factor and Watson factor:$${K}_{w}=4.5579{{M}_{W}}^{0.15178}{\mathrm{SG}}^{0.84573},$$(B2)where M_{W} (in g/mol) and SG are the molar weight and the specific gravity, respectively. In this correlation, the temperature is in Rankin (°R) and ${C}_{p}$ in Btu·Ib^{−1}°F^{−1} unit (1 Btu·Ib^{−1}°F^{−1} = 4.187 KJ/(Kg·K)).
The specific gravity is obtained from the Riazi and AlSahhaf correlation [25] (valid from C_{6} to C_{50})$$\mathrm{SG}=1.07\mathrm{exp}\left(3.560732.93886{{M}_{W}}^{0.1}\right).$$(B3)
Appendix C
Feed composition, component critical properties, and molar weight of reservoir fluid.
Feed composition, component critical properties, and molar weight of North Sea gas condensate.
Feed composition, component critical properties, and molar weight of Bakken fluid.
Feed composition, component critical properties, and molar weight of SJ15 fluid.
All Tables
Feed composition, component critical properties, and molar weight of reservoir fluid.
Feed composition, component critical properties, and molar weight of North Sea gas condensate.
Feed composition, component critical properties, and molar weight of Bakken fluid.
Feed composition, component critical properties, and molar weight of SJ15 fluid.
All Figures
Figure 1 Joule–Thomson inversion curve and phase envelope of reservoir fluid (the dotted line indicates the extrapolation of the singlephase JTIC in the twophase domain). 

In the text 
Figure 2 Joule–Thomson coefficient of the reservoir fluid in the intervals 225–675 K and 100–500 bar. 

In the text 
Figure 3 Reduced departure enthalpy of the reservoir fluid at: (a) T = 400 K, (b) T = 540 K, (c) T = 615 K and (d) T = 650 K. 

In the text 
Figure 4 Joule–Thomson coefficient of reservoir fluid at: (a) T = 400 K, (b) T = 540 K, (c) T = 615 K and (d) T = 650 K. 

In the text 
Figure 5 Isentropic expansion coefficient of reservoir fluid at: (a) T = 400 K, (b) T = 540 K, (c) T = 615 K and (d) T = 650 K. 

In the text 
Figure 6 Joule–Thomson inversion curve and phase envelope of NorthSea gas condensate (the dotted line indicates the extrapolation of the singlephase JTIC in the twophase domain). 

In the text 
Figure 7 Joule–Thomson coefficient of the North Sea gas condensate in the intervals 390–470 K and 350–500 bar. 

In the text 
Figure 8 Reduced departure enthalpy of North Sea gas condensate at: (a) T = 400 K, (b) T = 430 K and (c) T = 470 K. 

In the text 
Figure 9 Joule–Thomson coefficient of the North Sea gas condensate at: (a) T = 400 K, (b) T = 430 K and (c) T = 470 K. 

In the text 
Figure 10 Isentropic expansion coefficient of the North Sea gas condensate at: (a) T = 400 K, (b) T = 430 K and (c) T = 470 K. 

In the text 
Figure 11 Joule–Thomson inversion curve and phase envelope of Bakken fluid (the dotted line indicates the extrapolation of the singlephase JTIC in the twophase domain). 

In the text 
Figure 12 Joule–Thomson coefficient for the Bakken fluid in the intervals 200–500 K and 40–250 bar. 

In the text 
Figure 13 Joule–Thomson coefficient of Bakken fluid along the isotherm T = 389.3 K. 

In the text 
Figure 14 Isentropic expansion coefficient of Bakken fluid along the isotherm T = 389.3 K. 

In the text 
Figure 15 Joule–Thomson inversion curve and phase envelope of SJ15 fluid (the dotted line indicates the extrapolation of the singlephase JTIC in the twophase domain). 

In the text 
Figure 16 Joule–Thomson coefficient of SJ15 fluid along the isotherm T = 383.15 K. 

In the text 
Figure 17 Isentropic expansion coefficient of SJ15 fluid along the isotherm T = 383.15 K. 

In the text 
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