Phase-split Calculations Research Articles

Robust Multiphase-Split Calculations Based on Improved Successive Substitution Schemes

Summary Successful large-scale compositional reservoir simulations require robust and efficient phase-split calculations. In recent years, there has been progress in three-phase-split calculations. However, there may be convergence issues when the number of equilibrium phases increases to four. Part of the problem is from the poor initial guesses. In phase-split computations, the results from stability provide good initial guesses. Successive substitution (SS) is a key step in phase-split calculations. The method, if efficient, can provide good initial guesses for the final step, the Newton method that has a rapid rate of convergence. In this contribution, we present a robust algorithm with high efficiency and robustness in phase-split calculations in two, three, and four phases. We find that a key step is the SS. The convergence may even be very slow away from the critical point and phase boundaries. A modified SS is used which may reduce the number of iterations many times. In the course of this investigation, we observe some regions often inside the phase envelopes (far from the phase boundary or critical points) with a very high number of SS iterations. The adoption of the improved SS iterations leads to a significant speedup of the multiphase-split computations. In some mixtures, the average reduction is more than 70%.

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.

A generalized flash algorithm to bridge stability analysis and phase split calculations

Phase split and phase stability algorithms traditionally have been the backbone of all phase behavior calculations in the presence of an Equation of State model. Phase stability examines whether a feed composition at given specifications is stable, or if it will split in two or more phases. In the latter case, phase split will further provide the composition and molar fraction of all equilibrating compositions.Despite the extension and improvement of the original algorithms developed by Michelsen as far as both speed and robustness are concerned, those two calculations are still treated as independent. In this work, we explore the relationship between them and develop a new algorithm called “the generalized flash” which acts as a bridge between stability and phase split. By utilizing suitable values of its parameters, namely, the stability distance θ and the molar fraction β, it is shown that the generalized algorithm can reduce to any of the two basic ones. Additionally, it can offer intermediate results between stability and phase split which also guarantee minimum Gibbs energy. The algebraic, geometric and computational aspects of the proposed algorithm are investigated. Finally, applications of the new algorithm with emphasis to negative flash calculations are discussed.

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.

Thermodynamics-Informed Neural Network (TINN) for Phase Equilibrium Calculations Considering Capillary Pressure

The thermodynamic properties of fluid mixtures play a crucial role in designing physically meaningful models and robust algorithms for simulating multi-component multi-phase flow in subsurface, which is needed for many subsurface applications. In this context, the equation-of-state-based flash calculation used to predict the equilibrium properties of each phase for a given fluid mixture going through phase splitting is a crucial component, and often a bottleneck, of multi-phase flow simulations. In this paper, a capillarity-wise Thermodynamics-Informed Neural Network is developed for the first time to propose a fast, accurate and robust approach calculating phase equilibrium properties for unconventional reservoirs. The trained model performs well in both phase stability tests and phase splitting calculations in a large range of reservoir conditions, which enables further multi-component multi-phase flow simulations with a strong thermodynamic basis.

FAST AND ACCURATE PHASE EQUILIBRIUM CALCULATIONS FOR CONDENSATE SHALE GAS RESERVOIRS

Phase equilibrium calculations of complex fluids in shale gas reservoirs require the establishment of advanced numerical models that consider capillary effects, and the design of fast and reliable algorithms to handle the various components in the reservoir fluids in practical working conditions. In this study, we develop a thermodynamically consistent VT-type pore-scale flash calculation scheme based on realistic equations of state suitable for oil/gas reservoirs, e.g. the Peng-Robinson equation of state. The effect of capillarity has been incorporated in the scheme for a more accurate description of the thermodynamic properties of shale gas, and the diffuse interface model is applied to establish a dynamic evolution scheme in the phase equilibrium process, and a convex splitting method is used to model the evolution of compositional moles and volume. In order to accelerate the iterative flash calculations for realistic reservoir fluids containing a large number of components, a self-adaptive deep learning algorithm is developed in this paper with a novel structure to achieve wider applicability to various components in different fluids. The input and output features of the neural network are selected as the key thermodynamic features on the basis of thermodynamic analysis, and the network hyper-parameters have been carefully tuned to achieve a better performance on both accuracy and efficiency. Advanced deep learning technics resolving overfitting problems have been applied in our algorithm. The trained model significantly accelerates the conventional flash calculation based on iterative methods, while a good prediction accuracy has been preserved. Phase stability test and phase splitting calculations are automatically incorporated in our prediction, and we can significantly capture the effect of capillarity on phase equilibrium behaviors. Such a fast, accurate and reliable shale gas phase equilibrium calculation scheme using deep learning algorithms can provide an initial phase distribution field with physical meanings for subsequent multiphase flow simulations, while the number of phases can be also determined. The thermodynamic information and analysis can also be used as a thermodynamic basis for a multiphase numerical model with built-in physical conservation.

A soft computing method for rapid phase behavior calculations in fluid flow simulations

To describe the flow of hydrocarbons in the reservoir and in pipelines upstream petroleum engineers utilize compositional simulation models which allow for the accurate description of complex phase behavior phenomena by means of EoS models and detailed compositional characterization. To speed up the conventional time-consuming phase stability and phase split calculations, soft computing has been recently utilized by means of methods such as proxy models and on-the-fly tabulation of phase behavior results.

Reduced method for rapid multiphase isenthalpic flash in thermal simulation

Thermal enhanced oil recovery (EOR) involves the complex interplay of mass and energy transport processes with phase behavior. Hydrocarbon and water components partition across multiple fluid phases as a function of composition, pressure and temperature. Thermal compositional simulation thus requires phase-equilibrium calculations for at least three fluid phases. Isenthalpic flash (phase-split) is needed for thermal compositional simulation when the molar variables are used in the solution of the global mass and energy balances. Isenthalpic flash solves for phase compositions and phase amounts, in addition to temperature.Isothermal flash algorithms have been consistently refined over the past four decades, whereas isenthalpic flash remains an outstanding challenge in the study of petroleum reservoir thermodynamics. Isenthalpic flash is more challenging than isothermal flash for a number of reasons. First, temperature is unknown a priori, resulting in uncertainty regarding the phase state. Second, narrow boiling point behavior results in extremely nonlinear behavior of enthalpy with respect to temperature. Finally, thermal EOR involves the injection of steam which results in hydrocarbon–water interactions at elevated temperatures. Water is the thermodynamically dominant component in this context, and cannot be excluded from phase equilibrium calculations in the simulation of thermal recovery processes.In this research we address the isenthalpic flash problem using a reduced variables formulation. For the first time, we extend the use of reduced variables beyond the isothermal-isobaric domain for phase-split calculations. We introduce a fully coupled Newton method, which exhibits rapid convergence even for narrow boiling point problems. Excellent performance of our novel formulation is demonstrated through comprehensive testing of characterized fluids from the literature. We expect the developments in this research to fortify the coupling of phase-equilibrium calculations with the mass and energy conservation laws in thermal compositional simulation.

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We present continuum-scale modeling of multiphase compressible flow in porous media with applications to hydrocarbon reservoir engineering. A new black-oil model is developed and compared with a fully compositional simulator to model the thermodynamic phase behavior. In the context of black-oil modeling, where components are lumped into a gas and liquid pseudocomponent with only the gas transferring between liquid and gas phases, we allow for a variable bubble point pressure (e.g., when gas enters an undersaturated zone). Traditionally, a primary variable switching strategy has been used, which is known to be prone to convergence and phase identification issues. Instead, we adopt an overall molar composition-based framework that can robustly model phase appearance or disappearance. Phase properties across a broad range of pressures for different black-oil compositions are constructed from compositional phase split calculations to correctly model the phase transition. Mass transport is updated explicitly by a locally mass conserving multilinear discontinuous Galerkin method. Globally continuous pressure and velocity fields are obtained through an implicit mixed hybrid FE scheme. The robustness and accuracy of our FE simulator are demonstrated in several problems, where we have attained considerable speed-up and maintained the accuracy with the new black-oil model.

Practical application of machine learning on fast phase equilibrium calculations in compositional reservoir simulations

To accurately describe the fluid phase behaviour in reservoir simulation, Equation-of-State-based compositional models are usually used. However, phase equilibrium calculations, including stability tests and phase splitting calculations, may require huge computational costs. An improved artificial neural network model is developed based on our previous work to achieve the prediction of phase stability with high accuracy and reduce the computational costs in order of magnitude. This model does not directly tell if a hydrocarbon mixture at given compositions, pressure and temperature is stable or unstable, and it is able to predict the saturation pressures. By comparing the given pressure with the predicted saturation pressure, it is natural to tell the stability of a hydrocarbon mixture. For the phase splitting calculations, another artificial neural network model is developed to provide more reliable initial guesses for commonly used equilibrium methods to reduce their nonlinear iterations. Compared with our previous work, multi-layer neural networks are employed in the models. The improved models have more efficient training processes and more accurate prediction results. Additionally, an adaptive data generation process is introduced to optimize the quality and size of training data; the data transformation and normalization are discussed to reduce the skewness of the data and rescale the data; and a model training and selection strategy is also discussed to efficiently train a high-quality model. The efficiency of the ANN models is first validated by standalone phase equilibrium calculations with a three-component hydrocarbon mixture. Then the ANN models are successfully applied in compositional simulation examples, which demonstrates the practical value of these ANN models on speeding up the phase equilibrium calculations during compositional simulations.

Artificial neural network assisted two-phase flash calculations in isothermal and thermal compositional simulations

Flash calculations are widely employed in compositional simulations in the determination of a number of phases, phase amount and phase composition at each grid block. The efficiency and robustness of a flash algorithm is critical to saving simulation run time and producing reliable simulation results. Traditional flash calculation methods that consist of a stability test and phase splitting calculations often require tremendous computational time to iteratively solve nonlinear equations. In general, the stability test is the most time-consuming part as it aims at calculating the compositions of an unstable trial phase from various empirical initial estimates. For the phase splitting calculations, their convergence behavior highly depends on the quality of initial guesses of K-values. Good initial guesses can significantly reduce the number of iterations, avoid converging to an incorrect solution and, therefore, enhance the efficiency and robustness of a flash algorithm. Conventionally, initial guesses for the phase splitting calculations come from the values at the previous time step, the neighboring grid blocks or the stability test, but sometimes they do not work well.In this work, we use artificial neural networks (ANNs) to assist the traditional flash calculations in achieving their fast and robust convergence. ANN models are built standalone prior to a simulation and can be considered as a preparation stage. Once the ANN models are successfully built, they can benefit a large number of simulation runs during history matching and production optimization. There are two ANN models developed in this work: an ANN-STAB model and an ANN-SPLIT model. The ANN-STAB model is built to fit a saturation pressure diagram and the ANN-SPLIT model is generated to regress mole fractions and K-values. At given pressure, temperature and feed composition, the ANN-STAB model is first used to label the stability of a system; if it shows instability, the ANN-SPLIT model is then used to predict the mole fractions and K-values, which are to be used as initial guesses in the following phase splitting calculations. Compared with the traditional stability test algorithms, the ANN-STAB model gives almost the same accuracy but consumes much less computational time. The ANN-SPLIT model provides thoroughly reliable initial guesses for phase splitting calculations. As a result, it requires far fewer Newton iterations.

Higher-order black-oil and compositional modeling of multiphase compressible flow in porous media

We present continuum-scale modeling of multiphase compressible flow in porous media with applications to hydrocarbon reservoir engineering. A new black-oil model is developed and compared with a fully compositional simulator to model the thermodynamic phase behavior. In the context of black-oil modeling, where components are lumped into a gas and liquid pseudocomponent with only the gas transferring between liquid and gas phases, we allow for a variable bubble point pressure (e.g., when gas enters an undersaturated zone). Traditionally, a primary variable switching strategy has been used, which is known to be prone to convergence and phase identification issues. Instead, we adopt an overall molar composition-based framework that can robustly model phase appearance or disappearance. Phase properties across a broad range of pressures for different black-oil compositions are constructed from compositional phase split calculations to correctly model the phase transition. Mass transport is updated explicitly by a locally mass conserving multilinear discontinuous Galerkin method. Globally continuous pressure and velocity fields are obtained through an implicit mixed hybrid FE scheme. The robustness and accuracy of our FE simulator are demonstrated in several problems, where we have attained considerable speed-up and maintained the accuracy with the new black-oil model.

A Unified Algorithm for Phase-Stability/Split Calculation for Multiphase Isobaric-Isothermal Flash

Summary The conventional method for multiphase flash is the sequential usage of phase-stability and phase-split calculations. Multiphase flash requires the conventional method to obtain multiple false solutions in phase-split calculations and correct them in phase-stability analysis. Improvement of the robustness and efficiency of multiphase flash is important for compositional flow simulation with complex phase behavior. This paper presents a new algorithm that solves for stationary points of the tangent-plane-distance (TPD) function defined at an equilibrium-phase composition for isobaric-isothermal (PT) flash. A solution from the new algorithm consists of two groups of stationary points: tangent and nontangent stationary points of the TPD function. Hence, equilibrium phases, at which the Gibbs free energy is tangent to the TPD function, are found as a subset of the solution. Unlike the conventional method, the new algorithm does not require finding false solutions for robust multiphase flash. The advantage of the new algorithm in terms of robustness is more pronounced for more-complex phase behavior, for which multiple local minima of the Gibbs free energy are present. Case studies show that the new algorithm converges to a lower Gibbs free energy compared with the conventional method for the complex fluids tested. It is straightforward to implement the algorithm because of the simple formulation, which also allows for an arbitrary number of iterative compositions. It can be robustly initialized even when no K value correlation is available for the fluid of interest. Although the main focus of this paper is on robust solution of multiphase flash, the new algorithm can be used to initialize a second-order convergent method in the vicinity of a solution.

Unconstrained Gibbs Free Energy Minimization for Phase Equilibrium Calculations in Nonreactive Systems, Using an Improved Cuckoo Search Algorithm

This study introduces a strategy to improve the effectiveness of Cuckoo Search (CS) algorithm for the unconstrained Gibbs free energy minimization in phase equilibrium calculations of nonreactive systems. Specifically, the gradient information of the unconstrained Gibbs free energy function, which is readily available, is used to enhance the balance between diversification and intensification stages of the CS algorithm for phase-split calculations in multicomponent systems. The results showed that it is feasible to improve the numerical performance of the CS algorithm using the gradient information of the Gibbs free energy function; this improved method provides better results for phase equilibrium calculations in nonreactive systems with insignificant additional computational effort. This gradient-based Cuckoo Search (GBCS) algorithm outperformed the conventional CS algorithm, in terms of its reliability and efficiency in solving phase equilibrium problems, especially for multicomponent systems.

Transient, thermal wellbore flow of multispecies carbon dioxide mixtures with phase transition during geological storage

A non-isothermal and unsteady wellbore flow model for multispecies mixtures is presented. The model uses a rigorous thermodynamic approach to describe the isothermal flash process for phase transitions, and contains both stability analyses and phase split calculations. For two-phase state flow, the thermodynamic restraint for the pertinent gas and liquid chemical potential of each species is strictly met. This process is fully coupled with the relevant flow equations. Two examples of applications of the model are presented: a practical field trial case study and a theoretical problem. In the first example, the simulations agree reasonably well with measurements from the field case study, demonstrating that the modelling approach can characterise the unsteady dynamic process observed. In the second example, we demonstrate that the model can handle multispecies carbon dioxide flow with complicated phase transition processes.

Development of a thermodynamically consistent, robust and efficient phase equilibrium solver and its validations

An applied phase-related equilibrium (APPLE) solver using only the Peng–Robinson equation of state is developed based on rigorous classical thermodynamics. The solver is theoretically and thermodynamically consistent with the stringent equilibrium criterion. It is mainly composed of phase stability and phase splitting calculations, which will be called routinely in the course of searching for the globally stable equilibrium state. It also makes use of various robust and efficient numerical methods. To demonstrate its performance, the solver is tested against various mixtures, such as oil and gas mixtures, hydrocarbon mixtures and hydrocarbon–nitrogen mixtures. Phase diagrams of these mixtures are constructed and verified with available experimental data or other researchers’ calculations. Results show that the APPLE solver is reliable and fast to solve phase equilibrium problems, including three-phase equilibrium. Finally, its potential applications to droplet evaporation and computational fluid dynamics (CFD) calculations are discussed.

Modified Bounded Homotopies in the Solving of Phase Stability Problems for Liquid–Liquid Phase-Splitting Calculations

In this study, the modified bounded homotopies presented by Malinen and Tanskanen (Malinen, I.; Tanskanen, J. Modified bounded homotopies to enable a narrow bounding zone. Chem. Eng. Sci.2008, 63, 3419) are investigated in order to solve phase stability analysis problems in liquid–liquid equilibrium cases for phase-splitting calculations. The tangent-plane distance criterion is used to analyze the phase stability. The emphasis is on approaching the first root on the homotopy path. According to the observations, the bounding of the homotopy path with respect to the problem variables aids in the solving of a phase stability analysis problem. The main shortcomings of the modified bounded Newton, affine, and fixed-point homotopies are the starting point isolas, the existence of unfeasible solutions, and the convergence to only certain roots, respectively. The attraction domain of the global minimum was observed to be the largest with the fixed-point homotopy. Sequential usage of the fixed-point homotopy and s...

Improved Two-Sided Tangent Plane Initialization and Two-Phase-Split Calculations

The initialization procedure proposed by Michelsen for locating stationary points in the tangent plane distance function was applied to several cases considered as challenging in the literature concerning phase equilibrium calculations. For most of the systems studied in the literature, such a simple procedure is enough to detect the phase instability; thus it is unnecessary to resort to more robust, more elaborated, and more time-consuming initialization schemes. For the conditions and systems where problems were found, a simple procedure was proposed to extend Michelsen's original scheme with the aim of increasing the probability of obtaining a negative value of the tangent plane distance function. Two algorithms for phase-split calculations were used. Both algorithms use a relaxation parameter to increase their convergence regions. No difficulties were found to calculate the equilibrium distributions in contrast with several convergence problems reported, in which other approaches were applied to the s...

Equilibrium and rate-based approaches to liquid–liquid phase splitting calculations

Several approaches are presented that can be used to compute liquid–liquid phase splitting. These approaches are based on homotopy continuation methods or on non-equilibrium thermodynamics. It is shown that they exhibit quite different computational demands with respect to computation times and a suggestion is made as to which algorithm to use. For chemical systems with only a few components, the computation times encountered on a normal PC in most cases are only a few milliseconds, in some cases even less—suggesting the general use of such algorithms to assure that phase splitting is always detected when present.

Multiphase equilibria calculation by direct minimization of Gibbs free energy with a global optimization method

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