Multiphase Equilibrium Calculations Research Articles

Rapid Hydrate Formation Conditions Prediction in Acid Gas Streams

Sour gas in hydrocarbon reservoirs contains significant amounts of H2S and smaller amounts of CO2. To minimize operational costs, meet air emission standards and increase oil recovery, operators revert to acid gas (re-)injection into the reservoir rather than treating H2S in Claus units. This process requires the pressurization of the acid gas, which, when combined with low-temperature conditions prevailing in subsurface pipelines, often leads to the formation of hydrates that can potentially block the fluid flow. Therefore, hydrates formation must be checked at each pipeline segment and for each timestep during a flow simulation, for any varying composition, pressure and temperature, leading to millions of calculations that become more intense when transience is considered. Such calculations are time-consuming as they incorporate the van der Walls–Platteeuw and Langmuir adsorption theory, combined with complex EoS models to account for the polarity of the fluid phases (water, inhibitors). The formation pressure is obtained by solving an iterative multiphase equilibrium problem, which takes a considerable amount of CPU time only to provide a binary answer (hydrates/no hydrates). To accelerate such calculations, a set of classifiers is developed to answer whether the prevailing conditions lie to the left (hydrates) or the right-hand (no hydrates) side of the P-T phase envelope. Results are provided in a fast, direct, non-iterative way, for any possible conditions. A set of hydrate formation “yes/no” points, generated offline using conventional approaches, are utilized for the classifier’s training. The model is applicable to any acid gas flow problem and for any prevailing conditions to eliminate the CPU time of multiphase equilibrium calculations.

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.

Fate of trace elements in Oxygen Carrier Aided Combustion (OCAC) of municipal solid waste

Oxygen Carrier Aided Combustion is a novel fluidized bed concept for burning waste. This study analyzed solid samples from an industrial OCAC application using municipal solid waste and the oxygen carrier ilmenite. The presence of oxygen carriers impacts the ash chemistry, which can influence corrosion and ash characteristics. By investigating samples obtained from industrial applications, unique and highly relevant information on the solid-state chemistry and the fate of important elements can be obtained. In total, 20 bottom ashes and 17 fly ashes were sampled over a period of 38 days. In a preceding study, the surface interaction between ilmenite and Zn, Cu and Pb was investigated. In this paper, the distribution of these elements throughout the particle cross-section and the influence of residence time has been studied using XRD, SEM-EDX and XPS. The results show that Zn is incorporated in the Fe-rich ash layer over time in the form of Zn ferrites, while Cu accumulates inside the ilmenite particles with time, and Cr is enriched in the magnetically separated bottom ash. Low concentrations of Pb were detected in the bottom ashes, suggesting that a significant part is released in the gas phase. The influence of temperature, bed material and reduction potential were evaluated using multicomponent, multiphase equilibrium calculations. It is shown that an ilmenite bed is less prone to form melts in comparison to a bed of silica sand and that the addition of sulfur could decrease the volatilization of Pb.

Simple and Robust Algorithm for Multiphase Equilibrium Computations at Temperature and Volume Specifications

Summary Two-phase and three-phase equilibria are frequently encountered in a variety of industrial processes, such as carbon dioxide (CO2) injection for enhanced oil recovery in oil reservoirs, multiphase separation in surface separators, and multiphase flow in wellbores and pipelines. Simulation and engineering design of these processes using isothermal/isochoric (VT) multiphase equilibrium algorithms are sometimes more convenient than that using the conventional isothermal/isobaric (PT) algorithms. This work develops a robust algorithm for VT multiphase equilibrium calculations using a nested approach. The proposed algorithm is simple because a robust PT multiphase equilibrium algorithm is used in the inner loop without any further modifications, while an effective equation-solving method (i.e., Brent’s method; Brent 1971) is applied in the outer loop to solve the pressure corresponding to a given volume/temperature specification. The robustness of the VT algorithm is safeguarded by using a highly efficient trust-region-method-based PT algorithm. We demonstrate the good performance of the newly developed algorithm by applying it to calculate the isochores of fluid mixtures that exhibit both two-phase and three-phaseequilibria.

A 6M digital twin for modeling and simulation in subsurface reservoirs

A 6M digital twin for modeling and simulation in subsurface reservoirs

Full-EoS based thermal multiphase compositional simulation of CO2 and steam injection processes

Several compositional reservoir simulators based on equations of state (EoS) have been designed. Yet, few of them can deal with thermal processes such as steam injection and no fully compositional thermal simulation of the steam injection process has been proposed with extra-heavy oils yet. Those simulations appear essential and will offer better tools to decide whether to carry out the exploitation of a heavy oil field or not. In those processes, the accurate modelling of the water/steam phase plays an important role and accurate multiphase equilibrium calculations are necessary. Thermodynamics generate highly nonlinear problems. Besides, in reservoir simulation a huge amount of phase equilibrium calculations is required, and a single failure may cause significant error propagations leading to false solutions. This study presents several improvements leading to a more robust and efficient phase equilibrium calculation (stability and flash) program. A general multiphase flash implementing all the developed algorithms is presented and tested against experimental and literature data. It can handle any number of phases; a numerical example consists in a four-phase simulation of CO2 injection. Simulations of steam flooding are performed with highly heterogeneous reservoirs. Besides, a fully compositional simulation of the SAGD process for an extra heavy bitumen is performed, which appears to be the one of the first simulation of the kind in the literature.

Reply to “Comment on ‘Multiphase Equilibrium Calculation Framework for Compositional Simulation of CO2 Injection in Low-Temperature Reservoirs’”

Reply to “Comment on ‘Multiphase Equilibrium Calculation Framework for Compositional Simulation of CO₂ Injection in Low-Temperature Reservoirs’”

Thermodynamic Modelling of Iron Ore Sintering Reactions

Silico-ferrite of calcium and aluminum (SFCA) is one of the most commonly-produced phases in fluxed iron-ore sintering, and has long been regarded as an important bonding phase in industrial sinters. It is thus considered to have a significant effect on sinter quality. In this study, a solid solution model and database has been developed for the SFCA phase, and has been incorporated into the thermodynamic software, Multi-Phase Equilibrium (MPE). MPE calculations were compared with the in situ X-ray powder diffraction (XRD) observations of the formation of SFCA phase during sintering. The effects of the raw material composition, temperature and the oxygen partial pressure on the formation of mineral phases in the sinter, as well as the viscosity of the melt formed during sintering under equilibrium conditions, were modelled using MPE. The results show that the formation of SFCA phase can be promoted by increasing oxygen partial pressure and basicity of the raw material. Increases of Al2O3 and MgO content have no significant effect on the SFCA formation under equilibrium condition. The increase of oxygen partial pressure (10−3 atm or above) and basicity also leads to a decrease in melt viscosity, which enhances the fluidity of the melt, and hence, the assimilation of the sinter. However, increases of Al2O3 and MgO result in the increase of melt viscosity.

A new efficient algorithm to determine three-phase equilibrium conditions in the presence of aqueous phase: Phase stability and computational cost

Three-phase flash calculation is a crucial part of the modeling and simulation of the transport phenomenon in various chemical and petroleum processes. To obtain the compositions of phases in equilibrium, utilization of Equations of State (EOSs) and thermodynamic phase equilibria rules/laws (e.g., Henry's law, Lewis/Randall rule, and modified Raoult's law) seem inevitable to model phase behaviors of liquid, vapor, and aqueous phases. This study plans to develop two thermodynamic modeling approaches (on the basis of Henry's law and EOS) to conduct three-phase flash calculations in the presence of water. It is found that both developed algorithms are capable of determining the equilibrium conditions in the three-phase region such that a desired precision is obtained. In this research study, a new stability algorithm based on a negative flash procedure is introduced to forecast the formation and/or condition of the stable phases. The modeling results reveal that the aqueous phase equilibrium calculations by using EOS lead to an inaccurate prediction of stable phases in a vicinity of phase change regions. Improving the accuracy and reliability of the phase equilibria calculations, a new analytical derivative of fugacity coefficient with respect to mole fraction is obtained so that a better convergence and less runtime (compared to numerical derivatives) are achieved. An appropriate method to guess the initial equilibrium ratio is presented in this research investigation. The best roots of the cubic EOS are then selected for all three phases by minimizing Gibbs free energy. The proposed procedure can be implemented in different multiphase systems (and simulation softwares) to increase the accuracy of multiphase equilibrium calculations with an acceptable computational cost.

An Experimental and Modeling Study of Carbon Dioxide/Bitumen and C4/Bitumen Phase Behavior at Elevated Temperatures Using Cold Lake Bitumen

Summary The coinjection of carbon dioxide (CO2) or light hydrocarbons with steam in the steam-assisted-gravity-drainage (SAGD) process might enhance bitumen mobility and reduce the steam/oil ratio (SOR). Understanding and modeling the phase behavior of solvent/bitumen systems are essential for the development of in-situ processes for bitumen recovery. In this paper, an experimental and modeling study is undertaken to characterize the phase behavior of bitumen/CO2 and bitumen/C4 systems. Produced and dewatered oil from the Cenovus Osprey Pilot is used for the experiments. The Osprey Pilot produces oil from the Clearwater Formation. Constant-composition-expansion (CCE) experiments are conducted for characterizing Clearwater bitumen, CO2/bitumen mixture, and C4/bitumen mixture. The Peng and Robinson (1978) equation of state (EOS) (PR-EOS) is calibrated using the measured data and is used for pressure/volume/temperature (PVT) modeling. Multiphase equilibrium calculations are performed to predict the solubility of CO2 and C4 in the temperature range of 393.2 to 453.2 K. The potential of asphaltene precipitation for CO2/bitumen and C4/bitumen mixtures is also investigated using three screening criteria. According to the CCE tests and multiphase equilibrium calculations, C4 has much higher solubility in bitumen than does CO2 at operating pressure of 3997.9 kPa and temperature between 393.2 and 453.2 K (393.2 K < T < 453.2 K). During the CCE tests, coexistence of three equilibrium phases is observed for the C4/bitumen system with high C4 concentration. The three phases consist of a heavy oleic phase (L1), gaseous phase (V), and a light (solvent-rich) oleic phase (L2). Compositional analysis of the samples from L1 and L2 phases shows that C4 can extract light hydrocarbon components from bitumen into the L2 phase and preserve the heavy components in the L1 phase. Also, the L2 phase becomes darker by increasing the pressure, suggesting the extraction of heavier hydrocarbon components at higher pressures. Similar tests on the CO2/bitumen system show only two effective phases over a similar temperature range. The two phases consist of a heavy oleic phase (L1) and a gaseous phase (V). Phase-equilibrium regions are predicted using the regressed EOS model in the compositional space for the solvent/bitumen system. EOS predictions indicate two types of two-phase regions in the composition space for the C4/bitumen system (i.e., L1/L2 when 393.2 K < T < 421.2 K and L1/V when 421.2 K < T < 453.2 K). However, only one type of two-phase region (i.e., L1/V) exists in a similar temperature range for a CO2/bitumen system. The EOS predictions show that 1.8 wt% CO2 can reduce bitumen viscosity by up to 1.4 times, and 16.3 wt% C4 can reduce bitumen viscosity by up to 20 times when 393.2 K < T < 453.2 K. Viscosity calculations indicate that oil dilution by CO2 and C4 dissolution is more effective at lower temperatures, especially for C4. This shows the potential of injecting hot hydrocarbon solvents for bitumen recovery. The results show that asphaltene might precipitate in a system of C4/bitumen with high C4 concentration.

A RQPSO Algorithm for Multiphase Equilibrium Calculation in the KIVCET Process

The prediction of phase configurations and the optimization of operating parameters in the metallurgical process are normally achieved by the multiphase equilibrium calculation (MEC), which is formulated as a constrained optimization problem based on the principle of Gibbs free energy minimization. A revised quantum-behaved particle swarm optimization (RQPSO) algorithm has been proposed to solve the optimization problem using three-part improved strategies. Based on the KIVCET smelting characteristics, a MEC model for the KIVCET process is established and solved using the RQPSO algorithm. The calculated and industrial data of the lead grade are 96.20% and 96.33%, respectively, those of the matte grade are 19.39% and 19.68%, and the mass fractions of Pb in the predicted and industrial matte are 3.96% and 3.34%, respectively. The calculated results of the phase configuration are consistent with the actual production data, which indicates that the MEC model and RQPSO algorithm are accurate and reliable.

Modelling of trace material flow distribution in a melter gasifier based on thermodynamic multiphase equilibrium calculations

In combination with a complex global market environment, the iron and steel industry is dealing with increasing demands for reduction of environmental and greenhouse emissions. Alternative iron making technologies such as Corex® and Finex® can offer solutions for these challenges. The melter gasifier is the central unit operation of these well-established alternative iron making processes. Both offer considerable advantages with respect to economic characteristics and environmental emissions. Similar to the conventional iron making process of the blast furnace route, a variety of different input materials is required for these processes including coal, coke, lump or fine ore, iron ore pellets as well as limestone and dolomite. Most of these raw materials also carry minor amounts of unwanted compounds such as alkali or zinc-based species. The presence of these materials inside a melter gasifier can lead to instability, problematic process conditions and refractory damages. A better understanding of the distribution of these problematic species is therefore of great importance to plant manufacturers and operators. A variety of different multiphase reactions are responsible for the distribution of trace materials within the melter gasifier. In order to get a better understanding of the complex nature of these reactions, a thermodynamic investigation was carried out using calculation routines of HSC Chemistry and FactSage. Based on the findings of this investigation, a multi-zone model was developed using the simulation platform of the gPROMS ModelBuilder®. The model is separated into distinct reaction zones across the height of the melter gasifier to depict the main iron ore reduction process as well as the governing reactions of trace material components. Validation of the model was carried out using reference data of existing plants. The model described in this work is used to analyse the behaviour of trace materials within a melter gasifier. Due to the highly integrated material flows of the Corex® and Finex® iron making processes, the impact on their operational characteristics can be investigated as well. Through this it is possible to optimise the raw material input of these processes, potentially leading to reduced fuel demand and environmental emissions.

Hybrid Newton-Successive Substitution Method for Multiphase Rachford-Rice Equations.

In multiphase (≥3) equilibrium calculations, when the Newton method is used to solve the material balance (Rachford-Rice) equations, poorly conditioned Jacobian can lead to false convergence. We present a robust successive substitution method that solves the multiphase Rachford-Rice equations sequentially using the method of bi-section while considering the monotonicity of the equations and the locations of singular hyperplanes. Although this method is slower than Newton solution, as it does not rely on Jacobians that can become poorly conditioned, it can be inserted into Newton iterations upon the detection of a poorly conditioned Jacobian. Testing shows that embedded successive substitution steps effectively improved the robustness. The benefit of the Newton method in the speed of convergence is maintained.

Phase stability testing and phase equilibrium calculation at specified internal energy, volume, and moles

We have developed a fast and robust algorithm for the general multi-phase equilibrium calculation at constant internal energy, volume, and moles (specification UVN). The algorithm is based on the direct maximization of the total entropy of the system subject to the internal energy-, volume-, and mole-balance constraints. The algorithm uses the Newton-Raphson method with line-search and modified Cholesky decomposition of the Hessian matrix to produce a sequence of states with increasing values of the total entropy of the system. Unlike the previously published formulations, our method uses results of the UVN-phase stability testing for initialization of the UVN-flash calculation. As the number of phases is not known a-priori, the proposed strategy is based on repeated UVN-stability testing and UVN phase-split calculation until a stable phase split is found. The performance of the algorithm is demonstrated on many examples of different complexity.

Computational methods for reactive transport modeling: An extended law of mass-action, xLMA, method for multiphase equilibrium calculations

We present an extended law of mass-action (xLMA) method for multiphase equilibrium calculations and apply it in the context of reactive transport modeling. This extended LMA formulation differs from its conventional counterpart in that (i) it is directly derived from the Gibbs energy minimization (GEM) problem (i.e., the fundamental problem that describes the state of equilibrium of a chemical system under constant temperature and pressure); and (ii) it extends the conventional mass-action equations with Lagrange multipliers from the Gibbs energy minimization problem, which can be interpreted as stability indices of the chemical species. Accounting for these multipliers enables the method to determine all stable phases without presuming their types (e.g., aqueous, gaseous) or their presence in the equilibrium state. Therefore, the here proposed xLMA method inherits traits of Gibbs energy minimization algorithms that allow it to naturally detect the phases present in equilibrium, which can be single-component phases (e.g., pure solids or liquids) or non-ideal multi-component phases (e.g., aqueous, melts, gaseous, solid solutions, adsorption, or ion exchange). Moreover, our xLMA method requires no technique that tentatively adds or removes reactions based on phase stability indices (e.g., saturation indices for minerals), since the extended mass-action equations are valid even when their corresponding reactions involve unstable species. We successfully apply the proposed method to a reactive transport modeling problem in which we use PHREEQC and GEMS as alternative backends for the calculation of thermodynamic properties such as equilibrium constants of reactions, standard chemical potentials of species, and activity coefficients. Our tests show that our algorithm is efficient and robust for demanding applications, such as reactive transport modeling, where it converges within 1–3 iterations in most cases. The proposed xLMA method is implemented in Reaktoro, a unified open-source framework for modeling chemically reactive systems.

Teaching the Concept of Gibbs Energy Minimization through Its Application to Phase-Equilibrium Calculation

Robust and fast methods for chemical or multiphase equilibrium calculation are routinely needed by chemical-process engineers working on sizing or simulation aspects. Yet, while industrial applications essentially require calculation tools capable of discriminating between stable and nonstable states and converging to nontrivial solutions, students are often introduced to the simplest and less efficient equilibrium-calculation methods. In this article, the Gibbs energy minimization technique is presented as a robust alternative way to perform phase-equilibrium calculations and is applied to the determination of 2-fluid phase equilibria in low-pressure binary systems. It is carefully explained how the combination of this technique with an efficient quasi-global minimization method makes it possible to derive a robust PT-flash algorithm. In the frame of project-based learning devoted to develop a tool for building complex phase diagrams, this approach was successfully tested with a group of 100 students.

Enabling Gibbs energy minimization algorithms to use equilibrium constants of reactions in multiphase equilibrium calculations

The geochemical literature provides numerous thermodynamic databases compiled from equilibrium constants of reactions. These databases are typically used in speciation calculations based on the law of mass action (LMA) approach. Unfortunately, such LMA databases cannot be directly used in equilibrium speciation methods based on the Gibbs energy minimization (GEM) approach because of their lack of standard chemical potentials of species. Therefore, we present in this work a simple conversion approach that calculates apparent standard chemical potentials of species from equilibrium constants of reactions. We assess the consistency and accuracy of the use of apparent standard chemical potentials in GEM algorithms by benchmarking equilibrium speciation calculations using GEM and LMA methods with the same LMA database. In all cases, we use PHREEQC to perform the LMA calculations, and we use its LMA databases to calculate the equilibrium constants of reactions. GEM calculations are performed using a Gibbs energy minimization method of Reaktoro — a unified open-source framework for numerical modeling of chemically reactive systems. By comparing the GEM and LMA results, we show that the use of apparent standard chemical potentials in GEM methods produces consistent and accurate equilibrium speciation results, thus validating our new, practical conversion technique that enables GEM algorithms to take advantage of many existing LMA databases, consequently extending and diversifying their range of applicability.

Robustness of three-phase equilibrium calculations

Phase equilibrium calculations converge less often as the number of phases and components increase, and for overall compositions closer to phase split boundaries and critical points. Computational speed and robustness of flash calculations are very important aspects for pipeline transmission and for reservoir simulation where billions of flash calculations may be done. Reduced methods have the potential to improve the robustness of phase equilibrium calculations because if chosen properly they can linearize the equations, use fewer independent variables and can be unbounded. Improved robustness could further improve the speed and accuracy of compositional simulation and avoid false two-phase solutions, where three or more phases may be present. In this paper, we use the reduced variables of Gorucu and Johns (2014) to test robustness in performing two- and three-phase stability analysis and corresponding flash calculations. These multi-phase equilibrium calculations are compared with the conventional phase equilibrium calculations based on minimization of Gibbs energy and the reduced method proposed by Okuno et al. (2010a). Using thousands of equally-spaced and unbiased multi-phase equilibrium calculations, the proposed multi-phase equilibrium calculations are shown to be more robust than the other two tested algorithms.

Phase diagrams for hydrates beyond incipient condition — Complex behavior in methane/propane and carbon dioxide/iso-butane hydrates

We have recently developed an algorithm for multiphase equilibrium calculations and stability analysis involving hydrates. This algorithm allows phase equilibrium calculations in scenarios presenting different number of phases that may also be in distinct physical states.Here, we present P×T phase diagrams calculated using our algorithm that involve mixed hydrates of natural gas beyond incipient condition and show complex behavior. Our phase diagrams show a global overview of the behavior of phase equilibrium involving hydrates in a mixture of water, methane and propane and in a mixture of water, carbon dioxide and iso-butane. We accomplish that using a global phase diagram in a wide range of pressure and temperature to show diverse phase equilibrium regions.We present an approach to tune the Peng–Robinson equation of state to improve the calculations of the chemical potential of water in liquid state at high pressure and therefore, improve the prediction of the phase equilibrium between this phase and ice or hydrates, up to pressure conditions as high as 100 MPa.Our calculations predict 3-phase equilibrium regions and 4-phase equilibrium univariant lines in binary hydrates of methane and propane and in binary hydrates of carbon dioxide and iso-butane. We use local phase diagrams highlighting details in regions with complex behavior of phase equilibrium to show how the phase boundary isopleths and univariant lines relate in a phase diagram for water, methane and propane.Our calculations for a mixture of water, carbon dioxide and iso-butane predicts structural transitions occurring below the dew point line of the guest components, in a mixture with excess of guest components, and 5- phase equilibrium invariant point. We show equilibrium lines that present retrograde behavior in the phase diagram for this mixture.

Computational methods for reactive transport modeling: A Gibbs energy minimization approach for multiphase equilibrium calculations

We present a numerical method for multiphase chemical equilibrium calculations based on a Gibbs energy minimization approach. The method can accurately and efficiently determine the stable phase assemblage at equilibrium independently of the type of phases and species that constitute the chemical system. We have successfully applied our chemical equilibrium algorithm in reactive transport simulations to demonstrate its effective use in computationally intensive applications. We used FEniCS to solve the governing partial differential equations of mass transport in porous media using finite element methods in unstructured meshes. Our equilibrium calculations were benchmarked with GEMS3K, the numerical kernel of the geochemical package GEMS. This allowed us to compare our results with a well-established Gibbs energy minimization algorithm, as well as their performance on every mesh node, at every time step of the transport simulation. The benchmark shows that our novel chemical equilibrium algorithm is accurate, robust, and efficient for reactive transport applications, and it is an improvement over the Gibbs energy minimization algorithm used in GEMS3K. The proposed chemical equilibrium method has been implemented in Reaktoro, a unified framework for modeling chemically reactive systems, which is now used as an alternative numerical kernel of GEMS.