Rachford Rice Research Articles

Unified flash calculations with isenthalpic and isochoric constraints

In the unified flash procedure, a persistent set of unknowns and equations are solved in equilibrium calculations, allowing for simultaneous phase stability and split calculations. For fluids in a subcritical thermodynamic state characterized by pressure and temperature, modeling both liquid and gas phases with inequality conditions for phase fractions has been shown to incorporate the tangent-plane criterion and results in a consistent formulation of compositions for both present and absent phases. However, applications such as high-enthalpy systems in subsurface flow require a state definition using other state variables than pressure and temperature, as well as the capability to represent supercritical phases. Furthermore, the robustness of the flash across a wide range of state values is required if equilibrium dynamics are to be included in a flow and transport problem.This work introduces constraints in terms of enthalpy and volume to allow pressure and temperature to vary in the unified setting. The constraints are shown to arise from equilibrium conditions for the relevant state functions to be minimized. To increase the range of applicability, a modified extension procedure for the compressibility factor was devised, as well as procedures for the flash initialization.The unified formulation is extendable to allow isenthalpic and isochoric flash calculations. The initialization was devised using methodologies from the negative flash and Rachford–Rice equations. Extensive numerical tests with multiple equilibrium definitions in terms of state variables were performed. Gas–liquid, binary and a multicomponent mixture using Peng–Robinson EoS showed consistency of results which are verified using third-party code.

Rachford–Rice equation – A look from a different perspective

Rachford–Rice equation – A look from a different perspective

Robust and efficient methods for solving the Rachford–Rice equation in flash equilibrium calculation

Robust and efficient methods for solving the Rachford–Rice equation in flash equilibrium calculation

A generalization of associated polynomial for calculating flash equilibrium

Standard techniques for isothermal flash calculation of multi-component mixtures present convergence issues, especially in extreme conditions like root near poles or almost flat-slope near the root. A generalized expression for associated polynomials is presented here. It covers alternative flash equation criteria obtained for the sums of restrictions on liquid and vapor equilibrium phases, which, sometimes, represent an advantage over traditionally used Rachford–Rice equation.

Avoiding round-off error in the Rachford–Rice equation

The Rachford–Rice equation defines the inner loop of the 2-phase vapor–liquid equilibrium flash calculation. It is an objective function which determines the molar vapor or liquid phase split (V or L) and phase compositions (yi, xi) for an overall mixture composition (zi) with constant equilibrium K-values (Ki). The Rachford–Rice equation can suffer from severe round-off error due to catastrophic cancellation. This paper identifies where round-off errors can occur, how to identify it, and how to avoid the round-off error. Furthermore, this paper identifies two methods for mitigating catastrophic cancellation of the second kind by coordinate transforming the Rachford–Rice equation. These two methods are compared with regards to round-off error relative to a naive solution.

A Machine Learning Approach for Phase-Split Calculations in n-Octane/Water and PASN/Water Systems

Flash calculations, including phase split and phase classification for both n-octane/water blends and paraffinic aromatic synthetic naphtha (PASN)/water blends present significant computational challenges. Calculations to establish the two-phase and three-phase regions, as well as the transitions between regions, were addressed by a phase classification method proposed in a recent contribution involving machine learning (ML). This work focusses on the phase-split calculations, considering (a) the lack of numerical convergence of the traditional calculations and their related numerical issues for water/n-octane and PASN/water systems based on the Rachford–Rice derived surfaces and (b) the successful implementation of an ML approach based on a K-nearest-neighbor (KNN) algorithm, which uses the abundant experimental data obtained in a CREC-VL cell.

Solution of the Rachford Rice equation using perturbation analysis

Numerical simulation of two-phase fluid flow requires the Rachford Rice equation to be solved repeatedly at each grid block and at each time step to estimate the distribution of mass between the gas and the liquid phase. Despite its simplicity and monotonicity, the non-linearity of the Rachford Rice function imposes the need to utilize iterative methods which may take a considerable amount of time depending on the shape of the specific function. Therefore, due to the extensive number of solutions needed during a simulation run, any attempt to increase the speed or the robustness of the solution method is welcome.In this work, a new solution method is proposed that is based on perturbation theory. The idea lies in the fact that the shape and the root location of the Rachford Rice equation is usually dominated by only few of the mixture components and a closed form solution is directly available for their subset. The effect of the ignored components on the exact solution can be taken into account by considering their contribution as a perturbation of the subset solution. This way very accurate approximations of the solution are obtained at a very low computation cost. Unlike most available methods based on the Newton-Raphson technique, the method in this work does not need an initial estimate of the solution. The value of the proposed method is demonstrated through its application to three example fluids.

A New Experimental and Modeling Investigation of Wax Precipitation in Pipelines Based on Asphaltene Content

In this paper, a new model is proposed for wax precipitation. Fugacity of the solid phase is calculated indirectly by considering the fugacity of the pure liquid phase. In this approach, crude oil is divided into five pseudo components using the SARA test results. Solubility parameters were calculated by SRK (Soave Modification of Redlich Kwong) equation of state, and the composition of components is calculated by employing Rachford Rice equation. Also, geneticalgorithm was used to improve the accuracy of results in WAT and wax weight. The average deviation of WAT (Wax Appearance Temperature) was 0.11 and 0.15% in the multi solid and solid solution models respectively, also the average deviation of the wax weight was 10.32 and 5.8% in the multi solid and solid solution models, respectively. The results showed that the solid solution model was more accurate in the prediction of wax precipitation in pipelines, as compared to the multi solid model. The optimized parameters in this solid solution model were more effective and had a dynamic influence. The multi solid and solid solution models had a remarkable accuracy with the experimental data. There was no logical explanation for asphaltene on wax precipitation and WAT. As a result, asphaltene was omitted in five crude oils in order to investigate the effect on the wax weight percentage and WAT. Results reveal that after omitting asphaltene, WAT and wax weight percentage were decreased.

Solution of multiphase Rachford-Rice equations by trust region method in compositional and thermal simulations

Robust and efficient multiphase flash calculations are crucial in compositional and thermal simulations for complex fluid systems in which three or four phases may co-exist. Solution of the Rachford-Rice (RR) equations is an important operation in the multiphase flash. The Newton method generally does not converge during solution of the RR equations unless very good initial values are provided. In this paper, the solution of the RR equations is formulated as a minimization of a convex function problem. For the first time, we use a trust-region (TR) method to solve the RR equations through minimization of the convex function. The Hessian matrix of the convex function is always positive-definite, and the TR-based solver guarantees convergence. The key to successful implementation is to determine the relaxation parameter in the Newton update. We select this relaxation parameter to meet the boundary of the objective function and to ensure an adequate step length. We tested the RR solver for three and four phase RR problems in the construction of phase diagrams. The test cases are representative of complex fluid systems encountered in enhanced oil recovery, including injection of CO2 into low temperature reservoirs and steam injection into heavy oil reservoirs at elevated temperatures. The test results reveal our RR solver to be robust, efficient, insensitive to initial values, and capable of handling negative phase amounts. We also evaluated the effect of the initial values on convergence and recommend methods to estimate the initial values in our RR solver. In summary, our RR solver greatly improves the multiphase flash calculations and strengthens the coupling of phase equilibrium calculations to the governing equations in multiphase compositional and thermal simulation.

Calculation of the Thermodynamic Equilibrium of a Multicomponent Two-Phase System Based on Minimization of the Gibbs Potential

The authors have proposed a method to model the thermodynamic equilibrium of the two-phase multicomponent system “solution–vapor” by solving a system of ordinary differential equations following from the principles of nonequilibrium thermodynamics and statistical physics. Calculation of the thermodynamics of liquid and vapor phases was based on the equation of state. During the solution of the system of ordinary differential equations, the authors determined the molar volumes of the solution and the vapor, the molar composition of the phases, and the pressures of the onset of boiling of the solution and condensation of the vapor. The proposed method is fundamentally different from the popular empirical Rachford–Rice method. Results of calculating with this method have been presented.

An augmented free-water three-phase Rachford-Rice algorithm for CO2/hydrocarbons/water mixtures

Multiphase equilibria for CO2/water/hydrocarbons mixtures are frequently encountered in using CO2 for flooding or storing CO2 in depleted reservoirs. The presence of an aqueous phase in multiphase equilibrium poses difficulties associated with how to ensure the convergence of stability test. The free-water assumption, where the aqueous phase is considered as pure water, provides a better solution to this problem compared with the conventional full three-phase flash method. As for CO2/hydrocarbons/water mixtures, the free-water assumption might not be valid since the solubility of CO2 in the aqueous phase may not be negligible. This research presents a robust and efficient algorithm to perform three-phase flash calculation for CO2/hydrocarbons/water mixtures based on the assumption that only the presence of CO2 and water is considered in the aqueous phase, i.e., the so-called augmented free-water concept. The new algorithm is comprised of two loops. In the outer loop, we first develop a pragmatic method for initializing the equilibrium ratios of CO2 and water in the aqueous phase with respect to the reference phase (i.e., the hydrocarbon-rich liquid phase). In the inner loop, we solve the Rachford-Rice (RR) equation that has been simplified based on the augmented free-water assumption. In addition, we incorporate this new augmented free-water three-phase flash algorithm to a flash package which can deal with the single-phase, two-phase, and three-phase equilibria. Both positive and negative flashes have been applied in the flash package. Also, a new criterion is developed to properly switch between two-phase flash and three-phase flash; such criterion has been used as the supplementary judgment for the negative flash, which is proven to be reliable and efficient after extensive testing. The example calculations using the new algorithm show that the augmented free-water method is more accurate than the traditional free-water method since the solubility of CO2 is considered in the augmented one. The flash package that couples the augmented free-water flash requires less computational time and a fewer number of iterations than the conventional full three-phase flash package.

On the improvement of computational performance of a vapor-liquid equilibria solver for mixtures

This work deals the computational performance improvement of vapor–liquid equilibria solver for fluids mixtures. The code here developed is based on the chemical potential equality (expressed in terms of fugacity) and implements Soave–Redlich–Kwong and Peng–Robinson equations of state with classical van der Waals mixing rules. To reduce the bulk of the computational effort required by the solver we propose the following approaches: (i) exploit high-order methods for the solution of Rachford–Rice equation; (ii) develop an efficient programming methodology for the sub-routines devoted to the fugacity coefficients computation in order to reduce their overall impact on the CPU-time exploiting the parallelism at CPU level, i.e. CPU pipelining, and cache blocking. In this paper we have carefully evaluated the effectiveness of the aforementioned approaches performing a suite of computations of the equilibrium properties of several literature mixtures. The pros and cons of the strategies here suggested are outlined and discussed.

On a choice of independent variables in Newton iterations for multiphase flash calculations

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%.

A new parametric algorithm for isothermal flash calculations by the Adomian decomposition of Michaelis–Menten type nonlinearities

We develop a robust algorithm for isothermal flash calculations of multi-component mixtures. The algorithm provides an explicit, parametric solution for a recent variation of the Rachford–Rice equation by employing a powerful analytical technique, namely the Adomian decomposition method. The computational speed of the algorithm is shown to be further enhanced by applying the iterated Shanks transformation. Unlike previous methods, our algorithm obviates the troublesome need for an initial guess, guarantees an excellent rate of convergence and is demonstrated to be more computationally economic than prior art. Several reliable examples from the literature, including the flash calculation for a mixture with nineteen components, as well as an extensive set of hypothetical, randomly-generated problems are presented to illustrate the exceptional efficacy of our new approach.

Negative Flash for Calculating the Intersecting Key Tielines in Multicomponent Gas Injection

Gas injection is a widely used enhanced oil recovery method, and its application is expected to increase in the foreseeable future. In order to build a method of characteristics solution to a two-phase gas injection system, we must construct the composition route from the injection gas to the initial oil where all the intersecting key tielines must be identified. Calculation of these intersecting tielines requires a series of special negative flashes, which allow not only phase fractions outside the physical interval [0,1] but also negative feed compositions. The phase compositions from one negative flash are used to recombine the feed for the next negative flash. Despite the apparent complexity due to multicomponent phase equilibrium and transport, for pure component gas injection, negative flash and elimination of components can be performed in an alternating manner. In particular, if K-values are constant, there exists a simple feature that the vapor fraction roots (β-roots) for the Rachford–Rice equat...

A rapid and robust method for solving the Rachford–Rice equation using convex transformations

The Rachford–Rice equation must be solved in an inner loop of flash calculations. It consists in a sum of hyperbolas, and the desired root is located between two asymptotes which define the negative flash window. First, a simple procedure to remove the adjacent poles by changing the variable is proposed. Then, a class of convex transformations is identified, and several elementary functions that respect the general convexity condition are recognized. A new extremely simple solution window is proposed (depending only on compositions of the lightest and heaviest components), as well as an initialization procedure. Based on several convex formulations, two solution algorithms are proposed, based on the selection of the appropriate convex function (such that the monotonic convexity condition is fulfilled), and several Newton methods (including higher order) can be used without any control (it is guaranteed that an overshoot of the solution cannot occur). The transformations to make the functions convex within the search region, enable the use of Newton-type updates without the need for bounds checking and interval reduction lead to significant increase in speed of solution. The proposed algorithms are simple, very easy to implement, robust and rapid; as shown by various test examples (for selected very difficult cases and for a very large number of randomly generated flashes), only few iterations are required for convergence and the proposed method compares favorably to previous solution methods.

An efficient method to calculate multisolid-(wax) precipitation

Based on the multisolid-wax precipitation assumption for petroleum mixtures, a monotonic equation is proposed for the calculation of liquid and multisolid equilibria. Owing to the fact that the number of precipitating species can be determined instantly at fixed temperature and pressure through a stability analysis, the computation is fast and reliable. Although the developed iteration procedure is focus to liquid mixtures it is shown that can be easily incorporate into existing vapor–liquid iteration procedures based on the Rachford–Rice equation to get a vapor–liquid–multisolid-(wax) flash calculations. Numerical experimentations are shown and even more the proposed iteration method is used in the construction of phase boundaries without any problem.

Initialization of phase fractions in Rachford–Rice equations for robust and efficient three-phase split calculation

A key element of efficient and robust three-phase split calculation is the initialization of phase mole fractions. While the initial guess of equilibrium ratios is made from two-phase split calculation and two-phase stability testing, initializing phase fractions through solving three-phase Rachford–Rice equations can be a challenge especially close to the bicritical points and phase boundaries. In this research, we examine three different methods: the improved two-dimensional bisection method, the minimization method, and the direct Newton method. We present a large number of three-phase examples of various degrees of complexity to demonstrate both robustness and efficiency of all these methods. The direct Newton method combined with the starting guess of phase fractions from two-phase split calculation and two-phase stability testing is the most efficient approach.

An Iterated Eigenvalue Algorithm for Vapor–Liquid Split in Flash Calculations

Flash calculations are used extensively in computer-aided process simulation programs. Inherent to flash calculations is the determination of the vapor–liquid split. Currently, to calculate the vapor–liquid split, the Rachford–Rice equation is the most accepted method. In this work, it is shown that the Rachford–Rice equation can be posed as a generalized eigenvalue problem, where each eigenvalue of the resulting system is exactly a root of the Rachford–Rice equation. In this way, an ad hoc strategy based on linear algebra is proposed for finding the vapor–liquid split. The main advantage of the proposed strategy is that the resulting linear system has an almost-diagonal structure and generates a discrepancy function with a more linear shape than that of the Rachford–Rice equation. In this way, the solution of the system is simple and computationally inexpensive. Numerical experiments show the reliability of the proposed solution strategy for calculating the vapor–liquid split.

A general framework of model functions for rapid and robust solution of Rachford–Rice type of equations

For most phase equilibrium calculations, a non-linear function in one unknown, known as the Rachford–Rice equation for vapor–liquid equilibrium problems, needs to be solved for determining the molar fractions of the co-existing phases required for achieving the desired feed separation. Two issues appear when solving this type of equations. Firstly, the existence of as many asymptotes as the number of components present in the mixture splits these equations to a set of n−1 branches, each one containing a different solution. Therefore, care should be taken to ensure that the obtained solution lies within the branch of interest. Secondly, conventional solution methods often exhibit slow convergence rates which is an issue when repeated solutions are required as in the case of compositional reservoir simulation and pipeline flow.In this work, a general framework for developing new, rapid and robust solution methods is presented utilizing function classes, the members of which exhibit behavior similar to that of the equation to be solved. Fitting such functions, instead of a simple straight line at each iteration of the solution process, as is the case of the Newton–Raphson approach, leads to better subsequent estimates and thus to faster convergence. Moreover, a new initial value estimation method is presented for cases in which the Rachford–Rice function exhibits a very abrupt shape and it is shown that, for such cases, the number of iterations required by the conventional methods can be reduced to only one or two.