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The major specifications for phase equilibrium are: pressure and temperature (PT), volume, temperature and moles (VTN), pressure, enthalpy and moles (PHN), pressure, entropy and moles (PSN), internal energy, volume and moles (UVN) and entropy, volume and moles (SVN). For some of these specifications, stability testing is not well documented in the literature. It appears that derivations of stability criteria for PHN, PSN and SVN stability are presented here for the first time, as well as the possibility to solve the stability problem in their hyperspaces. Two general stability criteria are derived, which include all state functions and specifications. The general tangent plane distance (TPD) function degenerates to the TPD function corresponding to any desired specifications. The formalism is not model dependent. It is shown that stability testing for any specifications naturally reduces to either PT stability (when pressure is a specification, as in PHN and PSN stability) or VTN stability (when volume is a specification, as in UVN and SVN stability), by solving the appropriate nonlinear equation in temperature. In other words, there is no need (moreover, it is shown that there it is not even recommended) to minimize a specific TPD function in its hyperspace (defined by natural variables of the state function). This paper presents the first unified treatment of phase stability testing including all major specifications; beyond its theoretical importance, an important practical consequence is a unified calculation framework for phase stability testing, allowing a versatile numerical treatment and using existing highly robust and efficient procedures for PT, volume-based PT or VTN stability, whatever the specifications. [ABSTRACT FROM AUTHOR]

A recently proposed density-based phase envelope construction method (Nichita, Fluid Phase Equilib. 478, 100–113, 2018) is adapted to account for capillary effects. The set of saturation point equations is selected such as both the zero tangent plane distance (TPD) function (in terms of component molar densities and temperature) and the Young-Laplace equation are honored. The set of variables and potential specifications includes mixture molar density, temperature and the modified equilibrium constants (defined as the ratios of reference to incipient phase component molar density). The number of equations and the variables are the same as in the bulk fluid case. The density-based method including capillary pressure is not dependent on the thermodynamic model; any pressure-explicit equation of state and volume-explicit interfacial tension model can be used. The equation of state (EoS) must not be solved for volume and the elements of the Jacobian matrix have simpler expressions than those in conventional (pressure-based) methods. A code for phase envelope construction of bulk fluids can be easily modified by adding the capillary terms to the residual functions and Jacobian matrix. The additional partial derivatives of capillary terms have very simple expressions due to the explicit in volume form of the interfacial tension model. Unlike in conventional formulations, negative pressures in the reference phase can be handled. The proposed method is tested for several hydrocarbon mixtures, ranging from natural gases to heavy oils. As compared to a bulk fluid, under capillary pressure influence the bubble point pressures are suppressed and the dew point locus is expanded, with a shift of cricondentherm points towards higher temperatures. For the mixtures investigated, the computational results are practically identical to those reported in the literature. The computational procedure is robust, there are no problems neither in crossing the critical region (where interfacial tensions are very low) nor at important negative pressures and for very large capillary pressures (of the order of hundreds bar in some test examples). [ABSTRACT FROM AUTHOR]

The effect of capillary pressure on phase equilibrium is very important in tight formations, such as shale oil and gas reservoirs. In confined fluids, the capillary pressure induced by highly curved interfaces causes bubble points suppression and an inflated dew point locus, with a shift of cricondentherm points towards higher temperatures, as compared to the bulk fluid. In the conventional phase stability testing including capillary pressure, the problem is solved using the classical tangent plane distance (TPD) function (related to the Gibbs free energy surface) and mole numbers as primary variables. In this work, a volume-based approach (in which the equation of state needs not to be solved for volume) is used to solve this problem, as a bound and linear inequality constrained minimization of a TPD function with respect to the component molar densities. A modified Cholesky factorization (to ensure a descent direction) and a two-stage line search procedure (to ensure that iterates remain in the feasible domain and the objective function is decreased) are used in Newton iterations; a proper change of variables strengthens the robustness. The Weinaug-Katz equation (widely used in chemical and petroleum industry), which is a function of molar densities only, is used for interfacial tensions; the additional partial derivatives in the gradient vector and Hessian matrix have very simple expressions, unlike in conventional formulations. The proposed method is tested for two hydrocarbon mixtures (an oil and a gas condensate) in a wide range of pressures, temperatures and curvature radii, with a special attention paid to the convergence behavior near the singularities, that is, the stability test limit locus (STLL) and the spinodal. The method proves to be highly robust and exhibits fast convergence, showing a similar convergence behavior and almost the same computational effort as in the case of a bulk fluid. The results are only slightly different from conventional methods (except at high curvatures near the cricondentherm and at low pressures); differences are due to the fact that the dependence of capillary pressure on composition is taken into account in deriving the stationarity conditions, unlike in the conventional approach. The proposed method is not dependent on the thermodynamic model and can be used with any capillary pressure representation in which the interfacial tension model is explicit in molar densities. [ABSTRACT FROM AUTHOR]

Abstract A new density-based phase envelope construction procedure is proposed. The equations for two-phase saturation molar density and temperature computations are established by equating to zero the tangent plane distance (TPD) function in terms of component molar densities and temperature to find a non-trivial set of component molar densities at various specifications: mixture molar density, temperature and the modified equilibrium constants. The latter are defined as the ratios of feed to incipient phase component molar density. The nonlinear system of equations is solved by the full Newton method. The sequence of calculations is started at some conditions where convergence is easily obtained and an extrapolation procedure provides high quality initial guesses for subsequent calculations. The phase envelope is traced in the molar density-temperature plane, where a unique saturation temperature exists at specified mixture molar density; the representation in the pressure-temperature plane is obtained by calculating explicitly the pressure from the equation of state at given temperature and component molar densities on the saturation curve. The equation of state (EoS) must not be solved for volume and the elements of the Jacobian matrix have simpler expressions than their conventional (pressure-based) counterparts. The proposed method is successfully tested for a variety of mixtures, ranging from binary and ternary mixtures to reservoir fluids, exhibiting usual (closed) and unusual phase envelopes, such as open-shaped envelopes characterized by a bubble point branch extending to infinity, branches with negative pressures constructed in a single run (unlike in the conventional case), several branches, double retrograde behavior and swallowtail pattern corresponding to liquid demixing at low temperatures. The proposed method is not dependent of the thermodynamic model and any pressure-explicit equation of state can be used, provided the required partial derivatives of fugacity and pressure with respect to mole numbers, temperature and volume are available. [ABSTRACT FROM AUTHOR]

Conventional phase equilibrium calculations at temperature and pressure specifications require the resolution of the equation of state (EoS). In volume-based calculations, the EoS must not be solved for volume, which is a primary variable. The tangent plane distance (TPD) function can be expressed in terms of mole numbers and volume or of component molar densities. The stationary points of the TPD function, as well as the location of the stability test limit locus (STLL) are different for the two formulations. A modified TPD function in terms of mole numbers and volume is proposed here. Using the block structure of the Hessian matrix, it is shown that Newton iterations in volume-based stability are inherently slower than those in the conventional PT stability, since the Hessian matrix is evaluated at a lower implicitness level. The minimization of several TPD functions (in volume-based and conventional stability testing) is analyzed using various sets of independent variables and scaling procedures. A modified Cholesky factorization and a two-stage line search procedure ensure a sequence of decreasing TPD functions in all cases. The proposed methods are tested for several mixtures, with emphasis on the vicinities of singularities (STLL and spinodal). With proper scaling, the modified Newton iterations are robust and converge very fast for most conditions and reasonably fast for very difficult conditions. The proposed algorithm is not model-dependent; any pressure explicit EoS can be used, provided the required partial derivatives are available. [ABSTRACT FROM AUTHOR]

In this paper, a robust and efficient phase stability testing procedure at isotherm-isochoric conditions is proposed. The tangent plane distance function is expressed in terms of component molar densities. Convergence behavior using various sets of independent variables is analyzed and it is shown that the choice of independent variables is very important for both robustness and computational speed. A particular attention is paid for conditions in which the Hessian matrix is notoriously ill-conditioned, that is, in the vicinity of the stability test limit locus (STLL) and of the spinodal, where the convergence is slow and problematic. It is found that poor conditioning occurs also at certain conditions in the two-phase region at low temperatures and towards high densities. A modified Cholesky factorization (to ensure a descent direction) and a line search procedure are used in Newton iterations. The equation of state (EoS) must not be solved for volume and any pressure explicit EoS can be used; only expressions of pressure, fugacity and their partial derivatives with respect to mole numbers are required. The proposed method is tested on several examples from the literature, using a refined grid in the temperature-molar density plane. The variables obtained by scaling the ideal part of the Hessian matrix lead by far to the most robust formulation, with no failure observed for the investigated mixtures, and much faster than previous methods, with a low average number of iterations required for convergence for all mixtures. [ABSTRACT FROM AUTHOR]

Beyond the well-documented conventional phase equilibrium calculations at pressure and temperature (PT) specifications, other specifications, such as volume, temperature and moles (VTN) and internal energy, volume and moles (UVN) have received an increasingly interest in the last years. Whatever the selected set of independent variables, phase stability at UVN conditions can be asserted by solving simpler VTN or PT stability problems (a non-linear equation is solved only once at UVN specifications for temperature). In VTN stability, modified Newton iterations were previously used to achieve convergence, since successive substitution iterations (SSI) may exhibit problematic convergence. A mathematical analysis of convergence properties of the successive substitution (SSI) method for VTN stability reveals its potential instability, unlike for PT stability, in which SSI is very robust. This problem can be theoretically overcome by using a damping factor, but severe damping may lead to unacceptable slow convergence rates. It is shown that the SSI method should never be used alone, combined undamped SSI-standard Newton method is neither robust nor efficient as claimed previously and must be avoided; moreover, damped SSI can be used with caution only in early iteration stages. The convergence behavior of several methods for VTN stability is analyzed for a variety of mixtures and the numbers of iterations are compared with those reported in the literature. Several domains in the molar density-temperature plane were identified, where SSI is either systematically not detecting a phase split (converging to the trivial solution from all initial guesses) or systematically diverges. Our calculation procedures are highly robust and systematically faster than previous methods (for some test points up to one or even two orders of magnitude) and are able to find the global minimum of the TPD function in all test cases, unlike previously proposed SSI/Newton methods, which either are strongly attracted by local minima or they diverge. This paper gives for the first time a detailed mathematical and numerical analysis of the convergence behavior of SSI and combined SSI-Newton methods in VTN stability testing. [ABSTRACT FROM AUTHOR]

The phase stability analysis problem is highly important in phase equilibrium calculations. The stability test limit locus (STLL) is an important underlying property of multi-component system phase diagrams, because in its vicinity the number of iterations for phase stability testing dramatically increases and divergence may occur. The cause of convergence problems in phase stability calculations, as well as of the existence of a discontinuity of the TPD function in the single-phase state, is the topology of the TPD surface. At the STLL, the stationary point of the TPD function is a saddle point (the Hessian is indefinite), and for pressures just above the STLL the Hessian matrix is ill-conditioned in a domain of the hyperspace that must be “crossed” by iterates starting from one of the initial guesses. This makes stability testing in the vicinity of the STLL really challenging, and any algorithm will experience difficulties in this (fortunately tiny in most cases) region. A change of variables would not eliminate this problem, since the TPD function in the new hyperspace inherits certain properties from the original one. In this work we propose a modified objective function which exhibits multiple global minima corresponding to the stationary points of the original (TPD) function. The highly desirable feature of the modified objective functions is that the Hessian matrix is positive definite in the vicinity of the STLL (the nature of the singularity is changed). One additional derivative level is required for minimizing the new objective function, but a normal termination of the iterative sequence in a reasonable number of iterations is worth this effort. The minimization is performed by a quasi-Newton BFGS method with line search, using a suitable change of variables which avoids improper scaling, as well as by Newton iterations. Criteria to switch from the original TPD formulation to the new one are required, since the use of the more complex formulation is justified only near the STLL. Results show how application of the proposed stability testing method for a number of typical extremely difficult situations ensures convergence within tens of iterations, while all other iterative methods for minimizing the TPD function are extremely slow, unstable or even divergent. [ABSTRACT FROM AUTHOR]

Abstract: Phase stability testing is an important subproblem in phase equilibrium calculations. Phase stability analysis consists in finding either all stationary points or only the global minimum of the tangent plane distance (TPD) function. The TPD surface is non-convex and may be highly non-linear, and many phase stability calculations are rather difficult. In this work we are solving the phase stability problem using the tunneling global optimization method and a modified objective function; all stationary points of the TPD function are global minima of this objective function. For finding all its global minima at the same level (known, with objective function equal to zero), we exploit a unique feature of the tunneling method, which is able to find efficiently and reliably multiple minima at the same level. Numerical experiments for various difficult phase equilibrium problems show that the tunneling method is a powerful and reliable tool for global phase stability testing. [Copyright &y& Elsevier]

Abstract: Phase stability calculation is a very important topic in phase equilibrium modeling. Usually the phase stability problem is solved by minimization of the tangent plane distance (TPD) function, the sign of the objective function at its global minimum indicating the state of the mixture at given conditions. The TPD function is non-convex and may be highly non-linear, many phase stability problems being really challenging. The tunneling global optimization method had been successfully used for solving a variety of phase equilibrium problems, including stability, with cubic equations of state (EoS). In this work, we test the ability of the tunneling method to solve the phase stability problem for more complex EoS like PC-SAFT. Calculations are performed for several benchmark problems, for mixtures of non-associating molecules, from binaries to multicomponent. In one example, the mixture contains hydrogen sulphide, for which the three parameters required by the PC-SAFT EoS were unavailable in the literature. These parameters, as well as the binary interaction parameter (BIP) between hydrogen sulphide and methane, were calculated based on experimental data. [Copyright &y& Elsevier]

Abstract: The convergence locus (CL) and stability test limit locus (STLL) are important underlying properties of multicomponent system phase diagrams. In the pressure–temperature plane, the CL separates the region where the negative flash has non-trivial solutions. The mathematical domain of flash calculations is significantly wider than the physical domain; if a negative flash is performed, the equilibrium constants are continuously derivable when a phase boundary is crossed. Criticality criteria are met for the phase compositions resulting from the negative flash and the minimum eigenvalue of a quadratic form evaluated with these compositions (which are intrinsically stable) is used to locate the CL. An efficient negative flash procedure is also proposed. The STLL is important because in its vicinity the number of iterations for phase stability testing increases dramatically and divergence may occur. Outside the STLL the tangent plane distance function has only a trivial solution; between STLL and the phase boundary, there is a non-trivial positive solution. The spinodal criterion is met at the STLL for trial phase compositions. The CL and STLL are located at given pressure or temperature based on rigorous thermodynamic criteria in only few Newton iterations. The proposed method avoids repeated expensive negative flash calculations in the vicinity of the CL and phase stability calculations in the vicinity of the STLL. Results are presented for representative hydrocarbon mixtures with different amounts of classical contaminants and different shapes of phase envelopes. [Copyright &y& Elsevier]

This paper presents a new method for multiphase equilibria calculation by direct minimization of the Gibbs free energy of multicomponent systems. The methods for multiphase equilibria calculation based on the equality of chemical potentials cannot guarantee the convergence to the correct solution since the problem is non-convex (with several local minima), and they can find only one for a given initial guess. The global optimization methods currently available are generally very expensive. A global optimization method called Tunneling, able to escape from local minima and saddle points is used here, and has shown to be able to find efficiently the global solution for all the hypothetical and real problems tested. The Tunneling method has two phases. In phase one, a local bounded optimization method is used to minimize the objective function. In phase two (tunnelization), either global optimality is ascertained, or a feasible initial estimate for a new minimization is generated. For the minimization step, a limited-memory quasi-Newton method is used. The calculation of multiphase equilibria is organized in a stepwise manner, combining phase stability analysis by minimization of the tangent plane distance function with phase splitting calculations. The problems addressed here are the vapor–liquid and liquid–liquid two-phase equilibria, three-phase vapor–liquid–liquid equilibria, and three-phase vapor–liquid–solid equilibria, for a variety of representative systems. The examples show the robustness of the proposed method even in the most difficult situations. The Tunneling method is found to be more efficient than other global optimization methods. The results showed the efficiency and reliability of the novel method for solving the multiphase equilibria and the global stability problems. Although we have used here a cubic equation of state model for Gibbs free energy, any other approach can be used, as the method is model independent. [Copyright &y& Elsevier]

In this paper, we propose a new method for phase stability analysis with cubic equations of state by minimization of the tangent plane distance (TPD) function.A global optimization method called tunneling, able to escape from local minima and saddle points is used here. The tunneling method has two phases. In phase one, a local bounded optimization method is used to minimize the TPD function. In phase two (tunneling), either global optimality is ascertained, or a feasible initial estimate for a new minimization is generated. For the minimization step, a limited-memory quasi-Newton method is used.The tunneling method is used for the conventional approach, and for the reduced variables approach. In the latter case, the number of independent variables does not depend on the number of components in the mixture. The solution of the minimization of TPD function should be found in a space with a significantly reduced number of dimensions.The new method is used for testing phase stability for a variety of representative systems. The examples show the robustness of the method even in the most difficult situations. The proposed method is more efficient than other global optimization methods. Furthermore, in many cases, the reduced approach is faster than the conventional approach. The results showed the efficiency and reliability of the novel method for solving the global stability problem. [ABSTRACT FROM AUTHOR]