Noniterative Phase Equilibrium Calculation in Compositional Reservoir Simulation

In reservoir simulations it is a common practice to use a limited number of components to describe the reservoir fluids, ranging from 2 (black-oil models) to 5-12 (compositional equation of state based models). On the opposite, surface models usually require a greater number of components, typically from 15 to 30. The hydrocarbon components used in the surface model are usually lumped into pseudo-components in the reservoir. This disparity in the description of the mixture may become an issue whenever integrated asset models are developed with the aim of linking surface process and reservoir engineering. Leibovici et al.[10] presented a method to delump pseudo-components and predict the detailed phase compositions according to the results of a flash calculation performed on a lumped mixture. Leibovici et al.[11] proposed to use the delumping in the context of compositional reservoir simulation to predict the detailed composition of the produced fluid. The key-feature of this method is to calculate the phase equilibria with a simplified fluid and transfer the results in the detailed fluid model description applying a linearization in the space of equation of state parameters. They implemented this method by post-processing the results of a compositional simulation and use computed phase velocities to trace the delumped components. The delumping method was then implemented in a commercial compositional reservoir simulator by coupling the algorithm with a tracer advection module. The simulator option is usually referred as compositional tracer option. In this paper we used the Leibovici et al. approach in the context of the integrated asset modeling of complex reservoir fluids. Process models with detailed compositional formulation were linked to compositionally simpler reservoir models, using the compositional tracer option to feed the surface model with the produced reservoir streams. The proposed workflow was applied to simulate two different recovery processes, namely a) a gas injection below the dew-point in a sour gas condensate reservoir; and b) a miscible gas injection in a near critical volatile oil reservoir. Results of this implementation, in which detailed components are traced in the reservoirs, are compared to integrated simulations where all the stages, including reservoirs, are modelled using detailed composition.

Traditional reservoir simulators cannot capture the complicated interactions between fluid production and reservoir rock deformation during hydrocarbon recovery. In particular, in a recovery process when both phase behavior and deformation of reservoir rock play critical roles, a coupled geomechanics and compositional reservoir model can rigorously capture the above physical relations between solid and fluids and thereby perform more precise history matchings and predictions for better well planning and reservoir management decisions. This paper considers a fully coupled geomechanics and compositional modeling process. A novel solution procedure for the associated fully coupled Jacobian system is presented using the Schur complement technique. The proposed approach has the following features: (1) it has similar convergence properties as the fully implicit method, (2) combined with Krylov subspace iterative solvers, it can deal with symmetric and unsymmetric problems, and (3) it can be easily parallelized to solve large-scale reservoir simulation problems. In this work, a parallel, fully implicit, equation of state compositional reservoir simulator was fully coupled with a geomechanics FEM module. For each Newton iteration, an original two by two block Jacobian system is converted to a Schur complement system, and then the associated Schur complement system is solved using a BiCGstab(l) iterative method; subsequently all the unknowns for the reservoir simulator and the geomechanics module are obtained at the same time. The above partition solution procedure is equivalent to a block Gaussian elimination; hence its convergence is similar to a fully implicit coupled procedure. The proposed approach is implemented to solve two- and three-dimensional compositional reservoir simulation case studies considering geomechanics effects. We also compare the results between the fully coupled and the iteratively coupled approaches. It is shown that this approach presents a viable alternative in solution methods for solving general coupled physical problems.

An algorithm for calculating vapour-liquid equilibrium in multi-component systems is presented. The algorithm uses the Newton-Raphson method to solve the non-linear algebraic equations defining equilibrium. Iterative sequences have been organised to trace the progress of a calculation to ensure an efficient and progress of a calculation to ensure an efficient and successful calculation. Initialisation of the calculation procedure is automatic and relies only on parameters procedure is automatic and relies only on parameters characterising the components in a system. Important features discussed in this paper include a generalised flash calculation defined as a calculation of equilibrium conditions specified by a feed and any valid combination of two thermodynamic variables, and the incorporation of any suitable equations of state for evaluating liquid and vapour properties. An example problem is studied to illustrate the efficiency of the algorithm and a comparison is made between the Newton-Raphson method and the successive substitution method which is commonly used in phase-equilibrium calculations. Introduction Phase-equilibrium calculations are often required in the solution of problems involving simultaneous flow of liquid and vapour. Retrograde condensation in gas-condensate reservoirs and gas liberation in volatile oil reservoirs are examples of two-phase flow problems currently attracting interest. In simulating such flows, knowledge of the mutual influences between the flow of the reservoir fluid and the thermodynamics of equilibrium is essential. Some insight into the complex nature of these influences can be gained using compositional reservoir simulators. The basic function of these simulators is to determine numerical solutions of the partial differential equations governing the conservation partial differential equations governing the conservation of mass, momentum etc of each component in a system coupled with the algebraic equations defining vapour-liquid equilibrium throughout the system. During the calculation procedure, a solution of the equations defining equilibrium is required at every time wad space coordinate to evaluate, for example, local flow coefficients etc. The solution of these equations can be either computed internally in a simulator or supplied as multi-dimensional tables, for example, equilibrium ratios as functions of pressure, temperature and composition. In Both cases, a routine is required to determine vapour-liquid equilibrium, hereinafter abbreviated to VLE. The equations defining VLE are non-linear and in general can only be solved by iterative methods. Most commonly used is the method of successive substitution which is based on that proposed by Prausnitz and Chueh. However, this method exhibits slow or no convergence at conditions where the gradients of the thermodynamic variables are large, for example, in the vicinity of dew-, bubble- and critical point. To overcome convergence problems, Fussell and Yanosik developed an iterative sequence based on the Newton-Raphson method. A close examination of their paper revealed that their analysis can be extended. This prompted further investigations, the results of which are described in this paper. The contents of this paper are arranged as follows. The equations defining VLE are first presented. On inspection, it became apparent that a flash calculation can be generalised to a VLE calculation specified by a feed and any valid combination of two thermodynamic variables, An algorithm based on the Newton-Raphson method for generalised flash calculations is then described, followed by a description of an automatic starring procedure which relies only on parameters characterizing the components in a system. A system studied by Yarborough was selected as a basis on which the efficiency of the algorithm was assessed, and a comparison between the Newton-Raphson method and the successive substitution method was made. Phase-equilibrium calculations require equations of state to evaluate relevant properties pertaining to liquid and vapour. With several equations of state, for example ref. 4–7, available to an investigator, an algorithm which incorporates any suitable equations of state for liquid and vapour is desirable. The analysis has therefore been kept general throughout this investigation to allow for the insertion of any suitable equations of state. The experimental results of Yarborough were also used to assess the numerical results computed from the equation of Soave and that of Peng and Robinson.

Recently, use of parallel machines in reservoir simulation has received considerable attention from the petroleum industry. This paper presents parallelization of a 3D compositional, equation-of-state reservoir simulator on the CM-5 and KSR-1. To the best of our knowledge, this is the first time that the parallelization of a compositional reservoir simulator has been performed on both the CM-5 and KSR-1. For new users of the CM-5 machines, the software and hardware of CM-5 architecture is presented, as well as details of the parallelization techniques. For example, domain decomposition, I/O's, phase equilibrium computations, and well model are described. The parallelism techniques on the KSR-1 are presented with the emphasis on the porting of the phase equilibrium calculation. The performance of each machine is evaluated by showing the speedup on different sets of processing nodes. Two test problems were used to explore the capability of the parallelized version of the code; one is a waterflood problem and the other is a CO2 multiple contact miscible flood, both in a West Texas oil field. These field problems were run on 1, 2, 4, 8, 16, and 32 processors to get insight into the locations of communication bottlenecks, generally occurring in the programming with distributed memory machines. The problems of latency and bandwidth which are associated with communication efficiency of the CM-5 are also addressed.

Equations of state are widely used in compositional simulation to predict phase distribution and composition, which in turn facilitates prediction of compositionally dependent properties such as density, viscosity, interfacial tension, and enthalpy. A well-known problem confronted with naturally occuring hydrocarbon systems is characterization of the high molecular weight fraction (usually C7 +). The high molecular weight fraction must be split into an optimum number of pseudo-components each having a prescribed set of critical properties and interaction coefficients. Since procedures for doing this are largely empirical, typical equation of state packages must be tuned to available experimental data. Often available experimental data are limited in scope, or the ultimate simulation involves significant changes in temperature, pressure or composition. In each situation, tuned pseudo-component critical properties and/ or interaction parameters will manifest important temperature, pressure, and composition dependencies. Thus, a versatile phase description can effectively be ensured if tuning is automated internally within the equation of state package. In this paper, a methodology for achieving this task is described. First, the heavy fraction is split into a defined number of pseudo-components and starting values of binary interaction parameters and critical properties are obtained by using readily available correlations in the literature and/or mixing rules. Then, a linear optimization technique (two-phase method) is adopted for non-linear applications by using iterative techniques (iterative variables being pseudo-component binary interaction coefficients with CH4 and pseudo-component critical properties). Tuned sets of binary interaction parameters and critical properties of pseudo-components for a cubic equation of state are generated by using experimental depletion curves. As the computation progresses the tuned set replaces the default set to achieve further improvements. The success of this linear optimization routine is demonstrated for different reservoir oils.

Research on compositional and phase behavior of hydrocarbon mixtures and solvents is of great importance to the improvement of oil recovery. Series of experiments consisting of constant composition expansion, multiple-contact processes and constant volume depletion were performed employing three multicomponent hydrocarbon mixtures and four solvents (carbon dioxide, nitrogen and two flue gases). The results permitted to determine the hydrocarbon recovery efficiency for each solvent as a miscible-immiscible-gas process. In the modeling of IOR projects based on miscible-immiscible-gas processes, complex compositional models are employed. These models are based on equation of state (EOS) calculations. As part of the research work performed, an improved cubic EOS model was developed which can be used for more accurate compositional and phase equilibrium calculations than the previous models. The new model is based on a modification of the function used for calculating the cross-parameters aij and bij of the EOS mixing rules. The new model can be used for various applications, including the improvement of the currently used compositional numerical reservoir simulators. The application of the knowledge produced from the experimental and theoretical work presented permits to answer a series of questions such as "Can a flue gas or a mixture of carbon dioxide be used instead of pure carbon dioxide in a particular miscible project?"

A simple and efficient approach has been developed for improving the predictions of gas and liquid phase viscosities of reservoir fluids. The proposed method was based on the Lohrenz-Bray-Clark (LBC) correlation and applied to the Peng-Robinson equation of state (EOS) as all example although it can be applicable to any EOS. It can significantly improve the predictions of gas and liquid phase viscosities of hydrocarbon mixtures, particularly of high viscosity fluids. The approach is quite suitable for equation of state (EOS) based reservoir simulators and can further improve the prediction of reservoir fluid viscosity with appropriate tuning of the EOS. Introduction Accurate predictions of gas and liquid phase viscosities of reservoir fluids are required in reservoir simulators, particularly in equation of state (EOS) based reservoir simulators. Many researchers have focused their attention on improving the prediction of gas and liquid viscosities. However, most of the methods from mathematically rigorous to completely empirical forms are only applicable to either gas phase or liquid phase within a limited range of composition, temperature and pressure. In reservoir simulations related to oil and gas production, viscosity prediction methods are required to be applicable to a wide range of hydrocarbon mixtures at various conditions. The Lohrenz-Bray-Clark (LBC) correlation is probably the most used one in reservoir simulation models. The correlation was based on the Jossi-Stiel-Thodos correlation and modified by Lohrenz et al. The correlation is related to a fourth-degree polynomial in the reduced density. The disadvantage of the correlation is that the predicted viscosity is very sensitive to the density which is normally determined by a cubic EOS and may be very inaccurate for high viscosity fluids. Alternatively Ely and Hanley presented an extended corresponding states model for predicting the viscosity of non-polar pure fluids and their mixtures. Pedersen and Fredenslund studied this method and proposed a new corresponding states method which is based on the principle of corresponding states with methane as the reference component. Later Aasberg-Petersen et al. further improved the model by introducing n-decane as the second reference component. Little and Kennedy used an analogy with the van der Waals EOS to correlate the viscosity of reservoir fluids. Lawal applied the Lawal-Lake-Silberberg EOS to correlate the viscosity as well. In this study, a simple and efficient approach is proposed to improve the predictions of gas and liquid viscosities of reservoir fluids. The new method is based on the LBC correlation with the introduction of an exponential term and incorporating a cubic EOS. The method has been used to predict the viscosities of pure components, binary mixtures and reservoir fluids over a wide range of pressures and temperatures. The predicted results demonstrate that the new method can significantly improve the accuracy of viscosity predictions when compared with other published methods, particularly for high viscosity fluids. It is shown that the predictions can be further improved by appropriate tuning of the EOS. Viscosity Model As the Lohrenz-Bray-Clark (LBC) correlation is expressed as a function of the reduced density, the calculated viscosity is very sensitive to the density. P. 443

A new equilibrium coefficient or K-value correlation useful in compositional simulation studies was developed to reduce computing times in phase equilibrium calculations while retaining good accuracy in the calculated results. This correlation is based on the convergence pressure concept. Unlike previous works, the mixture critical pressures are correlated from binary critical pressures, the convergence pressures are correlated by treating the mixture as a pseudo-binary mixture, and the functional form of this correlation was derived by considering the asymptotic behavior of K-values. In the low pressure region, it reduces to the Chao-Seader correlation. In the near critical region, it conforms to the critical exponent theory. The correlation is related to the overall fluid composition explicitly, and requires no iteration. This correlation has been tested for three different reservoir fluid systems. The K-values calculated using the new correlation are in good agreement with those using the calibrated Peng-Robinson Equation of State (P-R EOS). The average deviation is less than 3% for the reservoir fluid systems studied. Slim tube displacement simulations, using CO2 and nitrogen as the displacing fluid, were also performed for the three reservoir fluid systems to validate the correlation developed. The predicted oil production rates using the new correlation to calculate phase equilibria are in good agreement with those using P-R EOS. The average deviation is less than 3%; at the same time, the computational cost is reduced by 4 – 14 times.

A fully-coupled flow-reactive-transport formulation based on element conservation, with application to CO2 storage simulations

A comparison of conventional and reduction approaches for phase equilibrium calculations

In this study of the Kaybob area of the Duvernay formation, a process for fracture design optimization utilizing an expanded work flow was developed. Geologic maps of reservoir properties were created using specific wells selected after conducting a field-wide analysis of the geological parameters relative to fracture analysis. The result was an increased understanding of how the reservoir properties relate to various locations throughout the entire reservoir area studied rather than only a well-by-well analysis basis. The following is an overview of the workflow used: Importation of public domain log data for multiple wells in study area.Analysis of raw log data to calculate input data for fracture simulator.Use of calculated log parameters in geological modeling software for creation of representative three-dimensional (3D) reservoir grids of reservoir pressure, relative oil permeability, porosity, and water saturation.Selection of representative wells based on accessing both typical and anomalous geological parameters as observed in the model. In addition, a discussion is provided of the following: 5.Typical fracture design parameter ranges.6.How reservoir maps may be used to populate the grid in a fully compositional numeric reservoir simulator.7.The advantages a compositional reservoir simulator with dual permeability capability provides in characterizing production from a gas-condensate reservoir with complex fracture geometry.8.Fracturing fluid types considerations.

Standard EOS-based phase equilibrium modeling in reservoir simulators involves computationally intensive and time-consuming iterative calculations for stability analysis and flash calculations. Therefore, speeding up stability analysis and flash calculations and improving robustness of the calculations are of utmost importance in compositional reservoir simulation. Prior knowledge of the tie lines traversed by the solution of a gas injection problem translates into valuable information with significant implications for speed and robustness of reservoir simulators. The solution of actual gas injection processes follows a very complex route due to dispersion, pressure variations, and multi-dimensional flow. The multiple mixing-cell (MMC) method originally developed to calculate minimum miscibility pressure of a gas injection process, accounts for various levels of mixing of the injected gas and initial oil. This observation suggests that the MMC tie lines developed upon repeated contacts may represent a significant fraction of the actual simulation tie lines encountered. We investigate this idea and use three tie-line-based K-value simulation methods for application of MMC tie lines in reservoir simulation. In two of the tie-line-based K-value simulation methods, we examine tabulation and interpolation of MMC tie lines in a framework similar to the compositional space adaptive tabulation (CSAT). In the third method, we perform K-value simulations based on inverse distance interpolation of K values from MMC tie lines. We demonstrate that for the displacements examined the MMC tie lines are sufficiently close to the actual simulation tie lines and provide excellent coverage of the simulation compositional route. The MMC-based methods are then compared to the computational time using other methods of phase equilibrium calculations including a modified application of CSAT (an adaptive tie-line-based K-value simulation), a method utilizing only heuristic techniques, and the standard method in an IMPEC-type (implicit pressure, explicit concentration) reservoir simulator. The results show that tabulation and interpolation of MMC tie lines significantly improves phase equilibrium and computational time compared to the standard approach with acceptable accuracy. The results also show that computational performance of the MMC-based methods with only prior tieline tables are very close to that of CSAT, which requires flash calculations during simulation. The K-value simulations using MMC-based tie-line interpolation methods improve the total computational time up to 51% in the cases studied with acceptable accuracy. The results suggest that MMC tie lines represent a significant fraction of the actual tie lines during simulation and can be used to significantly improve speed and robustness of phase equilibrium calculations in reservoir simulators.

Objectives/Scope Recently Abu Dhabi National Oil Company has called Whitson (PERA), a world leading PVT modelling consultancy company, to develop a best practice methodology/tool to quantify the condensate liquid production originating from the gas cap that is produced through oil rim producers' wells. This practice is integrating simulation work with field measured data and provided for the first time a solution to an oil and gas industry challenge, which is causing a conflict of interest between shareholders especially when oil rim and the associated gas cap are belonging to different concessions. Methods, Procedures, Process The work has been done for a giant oil field with large gas cap (rich in condensate) where only the oil is being developed since the 1960s. Initially the production GOR was limited to RS, but in 2010 the development strategy changed, and the field was being produced at GOR higher than RS allowing free gas from Gas Cap (rich with condensate) to be produced with oil. The question then arised of how much condensate is being produced through the oil rim producers. The condensate allocation method makes use of all measured well test data (Qo, GOR and API) and compositional reservoir simulation results. The used EOS (equation of state) model has been tuned to all available laboratory PVT data. This method uses a history-matched, reservoir simulation model run with a "dual-EOS" that is constructed by duplicating the tuned EOS model into two identical EOS models - one for the initial gas cap, and the other one for the initial oil zone. The dual- EOS run gives identical performance to single EOS model run. The generated dual-EOS compositional wellstreams are adjusted (1) to honor exactly the historical well test GOR data for each well, and (2) to honor as best possible the historical well test APIs for each well. The resulting wellstream will honor exactly the simulation model oil rates of each well throughout history, exactly the measured well test GOR, and close-to-exact APIs for each well. The final altered well streams are processed through a 4-stage field separator, yielding the well total stock-tank oil and condensate volumes. Results, Observations, Conclusions Historical gas cap condensate volumes produced from wells completed in the oil rim has been achieved during the field history. This was made possible by using (1) well production test data (GORs and APIs), (2) results from a history-matched compositional model, (3) tracking of components originally found in gas cap and in oil rim, and (4) application of a tuned EOS model. The conclusion is that such an integrated approach will result in a consistent and quantitatively accurate volume of condensate production volumes. Novel/Additive Information An innovative quantitative approach to the accurate estimation of condensate volumes originating in the gas cap - but produced from wells completed in the oil rim zone - has been developed and validated and could be applied for other fields, in addition it is fully flexible for future enhancements if needed. This methodology will definitely save time and unnecessary discussion and will provide more consistent results that will lead to more consensus from different parties.

Reservoir simulation with the cubic plus (cross-) association equation of state for water, CO2, hydrocarbons, and tracers

Compositional reservoir simulation of gas injection processes is commonly performed with a reduced number of components to minimize computation time. However, in simulations of downstream processes, more detailed fluid descriptions are required to design and optimize separation processes. To connect up and down-stream processes, delumping approaches have been proposed to deconvolute the well/production streams from a reservoir simulator into a more detailed fluid description. The accuracy of delumping is closely tied to the lumping approach that is used to generate equation of state (EOS) input for the reservoir simulator. To this end, we have previously introduced a flow-based lumping scheme that includes information from the dynamics of a given displacement process into the grouping of detailed components. In this paper, we investigate the accuracy of the delumped production streams from various gas injection processes and compare results from different lumping schemes. We investigate the benefit of applying flow-based lumping schemes in the selection of pseudo-components for compositional reservoir simulation and find an improvement in terms of uniqueness and overall accuracy of the composition of the delumped well streams. Examples are provided from 3D displacement calculations of enhanced recovery by CO2 and separator gas injection for multicomponent reservoir fluids where a substantial amount of PVT data is available (CME, DDE, swelling test and slimtube MMP). Using a commercial reservoir simulator, the delumped compositions of the produced fluids are compared to detailed displacement calculations for three different lumping schemes and two different gas injection processes. We demonstrate that the proposed lumping/delumping strategy provides additional control of the predictive quality over currently available lumping/delumping approaches. In addition, the proposed workflow provides a unique answer to the question of how to lump a detailed fluid description for displacement calculations. A majority of oil fields currently under production as well as new developments are candidates for EOR processes such as gas floods. The approach presented in this paper are directly applicable to the study and design of such EOR processes an offers an improved workflow for integration of up and down-stream processes. This is, to our best knowledge, the first paper that presents a comparison of delumping results for flow-based lumping schemes in the context of gas injection processes in with emphasis on the accuracy of delumped production stream relative to other available lumping/delumping methods.