Multiphase Flash Calculations Research Articles

Improved phase-state identification bypass approach of the hydrocarbons-CO2-H2O system for compositional reservoir simulation

CO2 injection is a highly effective technique to enhance oil recovery, achieved through continuous or alternative injection. However, the intricate interactions between different phases within porous media present significant challenges when predicting the performance of CO2 injection. To address this, it is crucial to employ compositional simulation, which accounts for the multiphase multicomponent transport. Nonetheless, conventional multiphase flash calculations can be computationally inefficient for large-scale reservoir simulations. Therefore, it is necessary to accelerate the Equation-of-State (EoS)-based compositional simulation, given the widespread use of CO2 enhanced oil recovery (CO2-EOR) in recent years. The phase-state identification bypass method has proven to be superior to other methods in terms of efficiency. However, this approach struggles with regions near phase boundaries, resulting in reduced computational efficiency in those areas.In this study, an enhanced phase-state identification bypass approach is developed to address this limitation. The first step involves discretising the pressure-temperature space using rectangular grids. Additionally, the tie-simplexes, which represent regions defined by the maximum number of phases formed by the fluid under consideration, are discretized in the phase-fraction space at the pressure and temperature of each discretization node. Subsequently, the discretization grid associated with the given point (the overall composition, pressure, and temperature) is located, and the phase states of the grid nodes are determined using the conventional multiphase flash method. If all nodes exhibit the same phase state, that phase state is assigned to the given point. However, if multiple phase states are obtained, a novel process is proposed to determine the phase state of the given point. To validate this improvement to the phase-state identification bypass method, phase diagram calculations and simulation cases are conducted, and the results demonstrate the robustness of the proposed method and its superior computational efficiency compared to the previous method.

A Line-Search-Based Algorithm for Multiphase Flash Calculations with CO2-Hydrocarbon System.

Multiphase flash calculations are pivotal in compositional simulation, necessitating a robust and efficient computational algorithm. In this work, we have developed a line-search-based algorithm framework for stability analysis and multiphase flash calculations. This algorithm is rooted in the modified Newton step and line-search method. The modified Newton step, derived from modified Cholesky factorization, ensures a descent direction, while the line search determines the degree of decrease. This combination facilitates convergence even in challenging regions for phase stability analysis and multiphase flash calculations, exhibiting superlinear convergence speed. Unlike traditional approaches that rely on successive substitution iteration and may resort to Newton iteration only if the Hessian matrix is positive definite, our algorithm incorporates modification via modified Cholesky factorization upon encountering a nonpositive definite Hessian matrix. We tested our algorithm with several classical fluids, demonstrating its efficiency and robustness. Furthermore, we assessed the algorithm's performance by computing the pressure-composition diagram for the CO2-hydrocarbon system, where all calculations achieved rapid convergence without failure. This newly developed algorithm for phase stability analysis and multiphase flash calculations represents a significant advancement for compositional or chemical process simulations.

A novel method to accelerate the three-phase flash calculation of the hydrocarbon-CO2 system

To achieve high computational efficiency and simulation accuracy in the equation-of-state-based compositional simulation, the fast multiphase flash calculation approach is required for the hydrocarbon-CO2 system, allowing for the simultaneous existence of vapour, CO2-lean, and CO2-rich phases. However, the conventional step-by-step method for multiphase flash calculation is associated with a high cost in identifying the three-phase equilibrium, thereby significantly impeding the computation efficiency of compositional simulation when the three-phase equilibrium appears in most grids.To address this challenge, a novel approach to multiphase flash calculation is proposed. In this study, all trial phases exhibiting negative tangent plane distance are employed to identify the three-phase equilibrium, eliminating the need for additional phase stability analysis and two-phase split calculations. Furthermore, this new method incurs minimal computational cost and requires only minor modifications to the existing stepwise procedure. Phase diagram calculations conducted on four cases demonstrate that the newly proposed method dramatically accelerates the identification of the three-phase equilibrium.

Hybrid Multiphase Flash Calculation Algorithm Based on Simultaneous Solution of Equilibrium Ratios and Phase Fractions

A robust flash calculation algorithm can increase the possibility of correctly performing multiphase equilibrium calculations. This work develops a new hybrid multiphase flash calculation algorithm adopting both equilibrium ratios and phase fractions as iteration variables. Different from the standalone implementation of the Newton–Raphson method documented in the literature, the hybrid algorithm properly hybridizes Newton–Raphson and trust-region methods to achieve a higher computational efficiency and better robustness. We show the fast convergence behavior of the hybrid algorithm in some scenarios where the conventional Newton–Raphson method encounters convergence difficulties. Specifically, the hybrid algorithm is capable of reducing the number of iterations consumed in the flash calculations from several hundreds to only tens. We also demonstrate the good performance of the hybrid algorithm by constructing phase diagrams for several reservoir fluid mixtures.

An efficient, robust and reliable initialization of phase fractions in Rachford-Rice system of equations for three-phase split calculation

Multiphase flash calculations are crucial in compositional and thermal simulations for complex fluid systems in which three phases may co-exist. Solution of the Richford-Rice system of equations is an important operation in the multiphase flash. Within the literature, solution methods have been proposed, based on optimization methods and on nonlinear equation solution methods. Many methods in these two approaches may fail if the initial estimate is too inaccurate, and may require a substantial amount of computation only to arrive at a trivial solution in cases where the assumed number of phases is too high. The present work deals with an initialization method for the solution of the Rachford-Rice system of equations based on approximating each of the functions by means of tangent lines. It is proposed to solve the system of equations by applying a two-step Newton-type method. Through extensive numerical experimentation, the efficiency, robustness and reliability of the proposed initialization method for the solution of the Rachford-Rice system of equations by means of a Newton-type method is shown.

Simultaneous Multiphase Flash and Stability Analysis Calculations Including Solid CO2 for CO2–CH4, CO2–CH4–N2, and CO2–CH4–N2–O2 Mixtures

This work deals with solid–vapor equilibria and solid–liquid–vapor equilibria of carbon dioxide in mixtures of interest for natural gas purification and biogas upgrading. Experimental data available in the literature are reviewed and an algorithm for solving an isobaric–isothermal flash coupled to a phase stability analysis is presented, which does not require a-priori knowledge of the number and type of phases existing at equilibrium. The good agreement between calculation results, also performed with a tool that makes use of Gibbs free energy minimization, and experimental data suggests that the proposed approach can be used for determining suitable operating conditions for processes aimed at separating CO2 out of a gas by freezing it. This work points out that more experimental studies should be performed on phase equilibria in the presence of solid CO2 for multicomponent mixtures containing species other than methane (e.g., nitrogen and oxygen), which are representative of gaseous streams from which CO2 needs to be removed, such as natural gas, biogas, and flue gas from power plants. Such data are important for a proper calibration of thermodynamic models that must be selected for reliable process simulations.

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.

Experimental study and thermodynamic modeling of clathrate hydrate stability conditions in carbon dioxide + cyclopentane + water system: Retrograde region

In this study, hydrate dissociation conditions for the carbon dioxide + cyclopentane + water system were measured. The pressure-search method at constant volume was employed to measure equilibrium conditions in the temperature range from 287.4–293.1 K and pressure range from 0.99–4.78 MPa. The results exhibit a retrograde behavior at pressures above 4 MPa for this system. Also, the experiments show that cyclopentane would significantly promote carbon dioxide hydrate stability conditions. In addition, based on the minimization of Gibbs free energy, a multiphase flash calculation was performed using the van der Waals and Platteeuw (vdW-P) solid solution theory along with the cubic plus association (CPA) equation of state to predict gas hydrate stability conditions. Modeling results are found in good agreement with the experimental data even in the retrograde region.

Reduced variables method for four-phase equilibrium calculations of hydrocarbon-water-CO2 mixtures at a low temperature

Carbon dioxide (CO2) flooding has been widely applied to enhance oil recovery. In low temperature reservoirs, CO2 injection may result in the formation of three hydrocarbon phases at thermodynamic equilibrium. In addition, an aqueous phase is always present in the reservoir. The aqueous phase can coexist with three hydrocarbon phases to form a four-phase equilibrium system. To evaluate CO2 utilization and storage in these low temperature petroleum reservoirs, a robust and efficient four-phase equilibrium calculation framework involving water is necessary for a compositional reservoir simulator. This is challenging not only because the number of variables increases but also because stability analysis becomes much complicated as the number of phases increases.In this research a novel four-phase equilibrium calculation framework is proposed for compositional reservoir simulation. A reduced variables method is used to solve four-phase flash problems efficiently and robustly. Multiphase flash calculations using reduced variables (RV) can converge to the equilibrium solution faster than formulations using conventional variables (Petitfrere and Nichita, 2015). Secondly, RV does not suffer from numerical problems in the Newton iterations when trace components are present in the aqueous phase. In addition to the implementation of the RV formulation, a systematic procedure consisting of stability analysis and flash calculations is proposed without any prior knowledge of initial K-values. Sets of different initial K-values are appropriately tested in each stability analysis.We performed comprehensive testing using characterized fluids found in the literature in order to validate the robustness of the proposed procedure. The four-phase regions in pressure-composition (Px) space could be accurately identified using our procedure. On the other hand, in some instances mixtures were incorrectly evaluated as three-phase when using existing approaches such as Li and Firoozabadi (2012). Our approach proposed in this paper is a promising four-phase equilibrium calculation framework for compositional reservoir simulation. The procedure achieved excellent robustness and efficiency with minimal modification to the conventional two or three-phase equilibrium calculation framework.

Multiphase flash calculation for system containing TBAB semiclathrate: Application to semiclathrate-based post-combustion CO2 capture

A algorithm coupled with modified Patel-Teja equation of state and Chen-Guo hydrate model is developed to conduct vapor-liquid-semiclathrate multiphase flash calculation for TBAB containing systems. To quantitatively describe the gas consumption ability of semiclathrate slurry, a new concept, i.e., the apparent liquid phase, is introduced. The Patel-Teja equation of state (PT EOS) combined with a Debye-Hückel electrostatic term is chosen to describe the fugacity of TBAB in aqueous solution and calculated gas solubility in aqueous phase. The Chen-Guo hydrate model is extended to calculate the fugacity of TBAB in semiclathrate phase. Several parameters in Chen-Guo model for TBAB are retuned by correlating phase equilibrium data. A mathematical approach based on the secant iteration is successfully applied to solve flash calculation. Results from flash calculations are in satisfactory agreement with experimental data reported in other literature. This new algorithm provides an extended range of calculations for semiclathrate phases, such as amounts of semiclathrate formed, and then, the contributions of absorption and hydration upon separation performance can be quantitatively described. In addition, multistage separation are simulated using this flash algorithm. The effects of TBAB concentration of solution, initial gas liquid volume ratio, temperature, and pressure on separation efficiency are systematically discussed from the perspective of thermodynamics and mass balance.

Multiphase flash calculations for gas hydrates systems

In this study, the van der Waals and Platteeuw model was coupled with the Cubic Plus Association (CPA) equation of state (EoS) for equilibrium calculations in systems with gas hydrates. It has been applied to simple and complex multicomponent systems involving methane, ethane, propane, isobutane, carbon dioxide, nitrogen and hydrogen sulfide. Methanol, ethanol, monoethylene glycol, calcium chloride, sodium chloride and potassium chloride were contemplated as thermodynamic hydrate inhibitors. The calculations were performed in the presence of single and mixed inhibitors. The mole fraction of components in all phases were determined using flash algorithm procedures to improve the calculations accuracy. To evaluate the ability of the methodology, the prediction of hydrate phase behavior in the presence and absence of inhibitors was compared with the experimental data. Additionally, the binary interaction coefficients were considered as a linear function of temperature. Furthermore, the Langmuir constants were optimized for each gas hydrate former. The results obtained with the proposed approach agreed well with the experimental data, predominantly for hydrate systems comprising mixture of different inhibitors as salts and alcohols.

New two‐phase and three‐phase Rachford‐Rice algorithms based on free‐water assumption

Based on the free‐water assumption that the water‐rich liquid phase contains only pure water, we develop two simple free‐water Rachford‐Rice methods: 1) a two‐phase free‐water Rachford‐Rice method where the phase fractions can be analytically solved; and 2) a three‐phase free‐water Rachford‐Rice method where there is only one unknown in the objective function (i.e. the vapour‐phase fraction) that is used for solving the phase fractions. Combining these two Rachford‐Rice methods, a new free‐water flash algorithm is developed to perform multiphase flash calculations where single‐phase equilibria, two‐phase equilibria, and three‐phase vapour‐liquid‐aqueous equilibria can be considered. In this free‐water flash algorithm, we first test if the mixture is stable; if the mixture is found to be unstable, we directly initiate the three‐phase free‐water flash. A set of criteria is developed for one to properly switch from a three‐phase free‐water flash to either a two‐phase free‐water flash or a conventional two‐phase flash, depending on whether a water‐rich phase is present in the two‐phase equilibrium. We also develop efficient and robust methods for initializing the equilibrium ratios for the two‐phase flashes. The negative flash is allowed in the flash calculation algorithms. A number of example calculations are carried out to demonstrate the robustness of the newly developed algorithm. A good agreement can be achieved between the flash results obtained by the new flash algorithm and those obtained by the conventional full three‐phase flash algorithm.

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%.

Asphaltene Deposition Modeling during Natural Depletion and Developing a New Method for Multiphase Flash Calculation

The specific objective of this paper is to develop a fully implicit compositional simulator for modeling asphaltene deposition during natural depletion. In this study, a mathematical model for asphaltene deposition modeling is presented followed by the solution approach using the fully implicit scheme. A thermodynamic model for asphaltene precipitation and the numerical methods for performing flash calculation with a solid phase are described. The pure solid model is used to model asphaltene precipitation. The transformation of precipitated solid into flocculated solid is modeled by using a first order chemical reaction. Adsorption, pore throat plugging, and re-entrainment were considered in the deposition model. The simulator has the capability of predicting formation damage including porosity and permeability reduction in each block. A new set of independent unknowns in a fully implicit scheme is presented for asphaltene deposition modeling. In order to find the solution of these variables, the same number of equations is also presented. The description of how to solve the nonlinear system of equations is also described.

Simultaneous multiphase flash and stability analysis calculations including hydrates

We proposed an algorithm for phase equilibrium calculations involving hydrates. Our algorithm applies stability analysis simultaneously to flash calculations in order to handle scenarios differing in pressure, temperature and composition, through ranges presenting different number of phases that may also be in distinct physical states. Our work is based on the algorithms of Gupta et al. [Fluid Phase Equilibria, 65–89 (1991) 63] and Ballard and Sloan [Fluid Phase Equilibria 15–31 (2004b) 218], with modifications that contribute to both speed and robustness of these calculations. Our algorithm depends on fugacity coefficients that have to be calculated by suitable thermodynamic models using input variables from a given iteration. We treat pressure, temperature and guest components fugacities as input variables and present expressions for the calculations of fugacity coefficients for all components included in hydrate phases from these variables. We obtain these expressions through a combination of the hydrate phase thermodynamic model described by van der Waals and Platteuw with suitable thermodynamic modeling for the reference condition of the empty lattice of water. We applied the algorithm in P×T phase equilibrium diagram calculations involving liquid and vapor fluid phases, and ice and hydrates with sI and sII crystalline structures solid phases.

Multiphase equilibrium calculations using a reduction method

Phase equilibrium problems (phase stability testing and multiphase flash) have to be solved repeatedly, sometimes a very large number of times, in process simulators and in compositional reservoir simulation. The computational effort can be significant, particularly when a detailed description (with a large number of components nc) of the mixture is required. The reduction method represent an attractive technique in the attempt to reduce the computational time, by significantly reducing the dimensionality of the problem, from nc×(np−1) in the conventional methods to a maximum of (2c+3)×(np−1), where np is the number of equilibrium phases and c the number of components with non-zero binary interaction parameters (BIP). The number of independent variables does not depend on nc, but only on c, and the computation time increases linearly with nc. Traditionally, the reduction approaches use the reduction parameters and the phase mole fractions as independent variables. In this work, a recent improvement of two-phase reduced flash calculations (Nichita and Graciaa, Fluid Phase Equilib. 302, 2011, 226–233) is extended to multiphase equilibrium calculation with any number of phases. Two approaches are proposed: (i) a direct extension of the reduction method for two-phase flashes and (ii) a constrained minimization Gibbs free energy with respect to a specific set of variables and constraints (taking advantage of symmetry). The proposed algorithms are tested for several multiphase systems (with up to four phases and exhibiting complex phase envelopes) containing hydrocarbon components, carbon dioxide and hydrogen sulfide. Numerical experiments show that the reduction methods for multiphase flash calculations are robust and they become faster than the conventional methods when (i) the number of components increases, (ii) the number of equilibrium phases increases and (iii) the number of BIP families decreases. For mixtures with many components and few BIP families, the reduction method may be at least one order of magnitude faster than conventional methods.

Fully implicit compositional simulator for modeling of asphaltene deposition during natural depletion

This paper describes development of a fully implicit compositional simulator for modeling of asphaltene deposition during natural depletion. In this paper, a new approach for multiphase flash calculation has been developed. This approach provides a more detailed description of the kinetic part of asphaltene deposition which needs to be explained more clearly. Due to a large number of unknowns, there are many ways to solve such a system by choosing different sets of independent variables. A new set of independent variables in a fully implicit model is considered for asphaltene deposition modeling. By incorporating an asphaltene precipitation model into a compositional simulator where the phase equilibrium equations, the volumetric constraint equation, the component transport equations, the multiphase flash equations and the deposition equation are solved simultaneously; a simulator for asphaltene deposition was developed with respect to natural depletion. The pure solid model is used to model asphaltene precipitation. The solid particles are considered to be separated into three parts: precipitated, flocculated and deposited solid. A first-order chemical reaction is used which models forward and reverse rates for the conversion of precipitated asphaltene to flocculated asphaltene. Also, a deposition model including adsorption, pore throat plugging, and re-entrainment was used. The simulator can also predict formation damage including porosity and permeability reduction in each block.

Multiphase equilibrium calculations using the semi-continuous thermodynamics of hydrocarbon mixtures

Natural hydrocarbon mixtures are composed of very many components. Usually, several components are lumped into pseudo-components to reduce the dimensionality of the equilibrium flash calculations. An interesting alternative to lumping is the use of semi-continuous thermodynamics, which enables a better characterization of mixture compositions. Traditionally, in the semi-continuous approach, the feed composition is approximated by a standard distribution function. Then, a Gaussian quadrature is used to find discrete carbon numbers representing the “pseudo-components”, for which a discrete flash can be performed. However, the feed composition can be random and/or far from classical distributions. In the quadrature method of moments (QMoM), developed by Lage (Lage, Comput. Chem. Eng. 31 (2007) 782–799), an optimal quadrature rule is used, considering the feed composition distribution as the weight function. In this work, a hidden feature of QMoM is revealed, by identifying a distribution function used to approximate a newly considered dependence, which involves the feed composition and the component liquid mole numbers. This dependence is extremely smooth, and, the most important, the shape of the function remains identical for any composition, pressure and temperature. The smoothness of the distribution enables a really good accuracy in solving the quadrature. The calculation of the quadrature is based on a procedure which avoids problems due to the ill-conditioned nature of the problem, even for a larger number of pseudo-components (unlike the original QMoM formulation). An analytical delumping procedure can be used (with an extremely small computational effort) to recover detailed composition in each equilibrium phase. Comparisons between a full discrete flash and the semi-continuous description proved the QMoM to be highly accurate. Various test problems, including two-phase and multiphase flash calculations, showed that Lage's methodology is generally applicable, even in cases when no distribution function can model the feed composition, or several distribution functions are needed to model different portions of the mixture.

Robust and efficient Trust-Region based stability analysis and multiphase flash calculations

Multiphase flash calculations and phase stability analysis are central in compositional reservoir and chemical process simulators. For instance, in some simulations, a huge amount of phase equilibrium calculations is required (the most important part of the computational effort). Moreover, a single failure may cause significant error propagations leading to false solutions. Thus, it is imperative that calculation algorithms are efficient and highly robust. The most difficult regions in mixture phase envelopes are in the vicinity of singularities: critical points for flash calculations and the stability test limit locus for stability analysis. For these conditions, all algorithms have difficulties to converge. Traditionally, a number of successive substitution iterations (SSI) are performed before switching to the second-order Newton method (many SSI iteration may be required before switch very close to singularities). The Trust-Region method has the advantage of performing a Newton step whenever the Hessian is definite positive; otherwise, the Trust-Region corrects the Hessian matrix by adding a diagonal element to make it positive definite, thus a descent direction is guaranteed. The Trust-Region limits the solution within a trust-radius, which is updated automatically at each iteration level, depending on the quality of the quadratic approximation. If the function is convex, the trust-radius enables larger changes in iteration variables, otherwise restricted steps are used to ensure a progress towards the solution. The proposed Trust-Region algorithm, as well as a hybrid methodology that combines SSI, Newton and Trust-Region steps, are tested for multiphase flash calculations and stability analysis on a variety of mixtures involving hydrocarbon components, carbon dioxide and hydrogen sulfide, exhibiting complicated phase envelopes. The proposed method compares favorably to the widely used SSI-Newton methods with various independent variables. The more difficult a test point is, the more spectacular the algorithm acts from both efficiency and reliability perspectives.

Prediction of vapor-liquid-liquid-hydrate phase equilibrium for multicomponent systems containing tetrahydrofuran

Numbers of hydrate-based new techniques require the algorithm to be able to perform multiphase flash calculation where one phase is a gas hydrate phase. Tetrahydrofuran (THF) is frequently used as a thermodynamic promoter in the development of hydrate-based technique, which can reduce the hydrate formation pressure, especially when methane or hydrogen exists. However, it is a hard work to describe accurately the phase behavior of THF-water system due to their polarities. It has been proved that the water-in-oil (W/O) emulsion can raise the hydrate formation rate and improve the single stage separation efficiency, and furthermore prevent the hydrate from agglomeration to plug the facilities. The goal of this work is to present an extension of our previous work to the prediction of the vapor-liquid-liquid-hydrate equilibrium of such complex systems. These include ternary and quaternary mixtures with W/O emulsion containing THF. The proposed algorithm of four phase equilibrium calculation is very simple due to avoiding the complexity of simultaneous solution of the sophisticated equation group. The calculation results were found to be in satisfactory agreement with the experimental data.