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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]

In this paper, a new method for phase equilibrium calculations at constant temperature, volume and moles (VTN flash) is presented. The problem is formulated as an unconstrained minimization of the Helmholtz free energy with respect to mole numbers. Volume is a dependent variable and its dependence on mole numbers is taken into account by solving a nonlinear equation (the volume balance equation) for pressure at each iteration level. By analyzing the block structure of the Hessian matrix in the constrained minimization of the Helmholtz free energy with respect to mole numbers and volume (used in previous approaches), it is shown that the Hessian in the proposed method has a higher implicitness level and Newton iterations converge faster (in terms of number of iterations). The calculation framework is similar to that in flash calculations at pressure and temperature specifications (PT flash). A combined successive substitution - Newton method is used, with mole numbers or natural logarithms of equilibrium constants as independent variables. The iterations in the VTN flash are “PT-like” (for a sequence of pressures converging to the pressure at which the volume specification is met) and the number of iterations required for convergence is comparable to that of the PT flash performed at the final pressure. The proposed method is tested for several mixtures of various complexities and proved to be rapid and robust. For vapor-liquid equilibrium, the convergence is obtained starting from the ideal equilibrium constants, even very close to phase boundaries and at near-critical conditions. Existing codes for PT flash calculations can be easily modified to incorporate the new method for VTN flash calculations. [ABSTRACT FROM AUTHOR]

In a recent study (D.V. Nichita, C.F. Leibovici, A Rapid and Robust Method for Solving the Rachford-Rice Equation Using Convex Transformations, Fluid Phase Equilib. 353, 2013, 38–49) we have shown that using appropriate changes of variables the original function can be transformed in convex functions for which the sufficient condition of monotonic convergence of the Newton method is fulfilled, thus Newton iterations can be used without interval control. For the proposed algorithms, the quality of the initial guess, thus the convergence speed, are directly related to the existence of narrow solution windows. In this work, new, extremely simple solution windows are proposed, which depends only on feed compositions. It is proved that novel solution windows are always smaller than previously proposed ones. Numerical experiments are carried out for a very large number of randomly generated flashes, showing that the use of new solution windows systematically decreases the number of iterations required for convergence. In the range of number of components usually encountered in compositional reservoir simulations, the reduction in number of iterations is about 15% as compared to the fastest previous formulation. [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]

Multiphase split calculations consist in the minimization of the Gibbs free energy. If the second-order Newton method is used, either component mole numbers or the logarithms of equilibrium constants (lnK) are selected as independent variables. Several recent papers advocated the use of the phase mole fractions together with lnK as independent variables in Newton iterations (lnK-θ). Unlike in the lnK approach, there is no explicit resolution of the Rachford-Rice (RR) equations, which are solved implicitly during the Newton update. The resulting linear system is non-symmetric and is solved by Gaussian elimination. In this study, this latter choice of variables is analyzed in detail, and an efficient formulation is proposed in which algebraic transformations are used to recast the lnK-θ method in a way that enables using the symmetry efficiently, both for constructing the Jacobian matrix and for solving the linear system of equations using a Cholesky factorization. Taking advantage of the block structure of the Jacobian matrix, it is revealed that the matrix of the linear system is formally identical to the Jacobian of lnK (evaluated at a lower implicitness level) and phase mole fraction can be updated separately using the partial solution of this system. Numerical experiments carried out on several multiphase mixtures exhibiting complex phase envelopes show the robustness and the efficiency of the proposed approach as compared to the original lnK-θ methodology, with the observed computational time savings up to 25%. [ABSTRACT FROM AUTHOR]

In this work, a new method is proposed for solving Underwood's equations. Newton methods cannot be used without interval control, and may require many iterations or experience severe convergence problems if the roots are near poles and the initial guess is poor. It is shown that removing only one adjacent asymptote leads to convex functions, while removing both asymptotes leads to quasi convex functions which are close to linearity on wide intervals. Using a change of variable, a pair of convex functions is defined; at each point within the search interval one of the two functions is guaranteed to satisfy the monotonic convergence condition for Newton methods. The search interval is restricted to narrow solution windows (simple and costless) and a simple high quality initial guess can be obtained using their bounds. Two solution algorithms are presented: in the first one, Newton (including higher-order) methods can be safely applied without any interval control using the appropriate convex function; in the second one, Newton iterations are applied to a quasi-convex function, and convex functions are used only if an iterate goes out of its bounds. The algorithms are tested on several numerical examples, some of them recognized as very difficult in the literature. The proposed solution methods are simple, robust, very rapid (quadratic or super-quadratic convergence) and easy to implement. In most cases, convergence is obtained in 2–3 Newton iterations, even for roots extremely close to a pole. [ABSTRACT FROM AUTHOR]

Abstract: Phase equilibrium calculations require the most important computational effort in many process simulators and in reservoir compositional simulations. In this work, a new reduction method for phase equilibrium calculations is proposed. A new set of independent variables (and the related set of error equations) is introduced, based on the observation that, under certain conditions, the equilibrium ratios can be related only to some component properties (elements of the reduction matrix) and to equation of state parameters. The new formulation leads to simpler expressions of the elements of the Jacobian matrix. Some important features are presented, and an interesting and useful link with classical flash calculation methods is revealed. The reliability and efficiency of the proposed method are tested on several synthetic and reservoir hydrocarbon mixtures. The proposed method proves to be robust and it performs in all cases better than previous reduction methods. Finally, it is discussed how the new set of independent variables can be used for a variety of phase equilibrium calculations. [Copyright &y& Elsevier]

Highlights: [•] An analytical solution of the Frost–Kalkwarf vapor pressure equation is established. [•] A simple approximation is provided for generating a good, simple and inexpensive starting value usable by any root finder. [•] As a result, the numerical solution is obtained in one or two Newton iterations. [Copyright &y& Elsevier]

A new simple density-based method is proposed for approximate phase envelope construction of constant composition multicomponent mixtures. The phase envelope is traced in the molar density-temperature plane (with a unique saturation temperature at given mixture molar density) and the pressure is calculated explicitly from the equation of state (EoS). The computational procedure is very easy to implement and a reduced system of only three equations must be solved for three variables (molar densities of feed and incipient phases and temperature), irrespective of the number of components in the mixture. The EoS must not be solved for volume and the elements of the Jacobian matrix have simple forms; the partial derivatives of fugacity coefficients with respect to compositions are not required. A simple and computationally inexpensive correction procedure highly improves results by updating the reference conditions at each point on the phase envelope. Simple equations for calculation of maximum temperature and pressure points are presented. The proposed method was tested for a variety of mixtures, ranging from natural gases to heavy oils, using a general form of two-parameter cubic EoS. For usual (closed) phase envelopes, the entire phase boundary is remarkably well reproduced. Certain unusual (open-shaped, with a bubble point branch extending to infinity) phase envelopes are also entirely traced up to very high pressures, while in certain cases the method fails at low temperatures (where the approximation of equilibrium constants becomes inadequate), but provides an excellent approximation for a wide temperature range. The proposed method is not dependent of the thermodynamic model (any pressure-explicit EoS can be used), it is recommended for mixtures with many components and it may be particularly attractive for complex EoS. [ABSTRACT FROM AUTHOR]