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ECMOR XVI  16th European Conference on the Mathematics of Oil Recovery
 Conference date: September 36, 2018
 Location: Barcelona, Spain
 Published: 03 September 2018
1  20 of 172 results


Reduced Variables Method for FourPhase Equilibrium Calculations of HydrocarbonH2OCO2 Mixtures at a Low Temperature
Authors M. Imai, H. Pan, M. Connolly, H. Tchelepi and M. KuriharaSummaryCarbon dioxide (CO2) flooding has been widely applied to enhance oil recovery. In low temperature CO2 injection cases, three hydrocarbon phases may be formed at equilibrium. Given the fact that connate water always exists in formations, and in many cases water injection precedes CO2 injection, fourphase equilibrium may arise where one aqueous phase plus three hydrocarbon phases coexist. In cases where CO2 dissolution into water cannot be ignored, a robust and efficient fourphase equilibrium calculation framework 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 fourphase equilibrium calculation framework is proposed for a compositional reservoir simulator. A reduced variables method is adopted to solve fourphase 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 ). Also RV solves numerical problem in Newton iterations with trace components in 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 Kvalues. Sets of different initial Kvalues are appropriately tested in each stability analysis.
We perform comprehensive testing using characterized fluids found in publications, in order to validate robustness of the proposed procedure. The fourphase regions in pressurecomposition (PX) space can be accurately identified using our procedure. On the other hand, some points are mistakenly evaluated as the threephase state if the existing approaches such as Li and Firoozabadi (2012) are used, as in the case of previously published articles on fourphase equilibrium calculations. Our framework proposed in this paper is a promising fourphase equilibrium calculation framework for a compositional reservoir simulator. The procedure achieves excellent robustness and efficiency with minimal modification to the conventional two or threephase equilibrium calculation framework.



Production Optimization Of Thermodynamically Rigorous Isothermal And Compositional Models
Authors T.K.S. Ritschel and J.B. JørgensenSummaryIn this work, we consider algorithms for solving production optimization problems that involve isothermal (constant temperature) and compositional oil production processes. The purpose of production optimization is to compute a longterm production strategy that is economically optimal. We present a thermodynamically rigorous model of isothermal oil production processes. We derive the model from first principles by applying a number of assumptions including the assumption of constant temperature. The model is based on two key principles, namely phase equilibrium and conservation of mass and energy. The conservation equations are expressed as partial differential equations, and we model the phase equilibrium as a VT flash process. It is common to formulate the phase equilibrium conditions in oil reservoir flow models as the fugacities being equal. We describe how to derive that condition from the phase equilibrium conditions from the VT flash problem. The VT flash is an adaption of the second law of thermodynamics, i.e. the entropy of a closed system in equilibrium is maximal, to isothermal systems. The VT flash can therefore be formulated as an inner optimization problem that needs to be solved for each grid cell in the discretized reservoir in the forward simulation of the oil production process. We demonstrate that it is natural to model such isothermal production processes with differentialalgebraic equations in a semiexplicit index1 form. We describe a singleshooting algorithm for solving the production optimization problem efficiently. It is key to the efficiency of such algorithms to compute gradients. For that purpose, we use an adjoint algorithm. We implement the singleshooting algorithm in C/C++ using the opensource software DUNE, the opensource thermodynamic software ThermoLib, and the numerical optimization software KNITRO. Finally, we present a numerical example that involves optimal waterflooding.



Parameterization Of Element Balance Formulation In Reactive Compositional Flow And Transport
More LessSummaryParameterization of element balance formulation in reactive compositional flow and transport
K. Kala1, D. Voskov1,2
1 Department of Geoscience and Engineering, TU Delft
2 Department of Energy Resources Engineering, Stanford University
We present a novel nonlinear formulation for modeling reactivecompositional transport in the presence of complex phase behavior related to dissolution and precipitation in multiphase systems. This formulation is based on the consistent element balance reduction of the molar (overall composition) formulation. To predict a complex phase behavior in such systems, we include the chemical equilibrium constraints to the multiphase multicomponent negative flash calculations and solve the thermodynamic and chemical phase equilibrium simultaneously. In this solution, the phase equilibrium is represented by the partition coefficients whereas the chemical equilibrium reaction is represented by the activity coefficients model. This provides a generic treatment of chemical and thermodynamic equilibrium within an EOS SSI loop by modification of the multiphase flash to accommodate chemical equilibrium. Using the Equilibrium Rate Annihilation matrix allows us to reduce the governing unknowns to the primary set only while the coupling between chemical and thermodynamic equilibrium is captured by a simultaneous solution of modified multiphase flash equations. An input in this thermodynamic computation is an element composition of the mixture when an output contains fractions of components in each phase, including solids. This element balance molar formulation along with the modified formulation for multiphase flash has been tested in a simple transport model with dissolution and precipitation reactions. The same approach will be later used to model a system involving kinetic reactions. The simulation of more general practical models is performed using the recently developed OperatorBased Linearization (OBL) technique. In the modified version of the OBL, the nonlinear element based governing equations are formulated in terms of space and statedependent parameters constrained by the solution of the extended multiphase flash based on molar element compositions. This approach helps us to add equilibrium reaction capabilities to the computationally efficient OBL technique.



Compositional Reservoir Simulation Using (T,V) VariablesBased Flash Calculation
Authors D. Paterson, M.L. Michelsen, E.H. Stenby and W. YanSummaryPhase equilibrium calculation is at the heart of compositional reservoir simulation. The conventional example is the isothermal isobaric flash (T,P) which must be solved in every gridblock at each iteration during simulation. This is conventionally solved through the repeated use of stability analysis and phasesplit calculation. To accurately represent the fluid properties it is useful to use an equation of state.
Most compositional reservoir simulations use the cubic equations of state (SRK or PR). More advanced equations of state which, for example, take account of association are attractive alternatives (e.g. CPA or PCSAFT) for some simulations. Each of these models are functions for the Helmholtz energy with the natural variables (T,V,n), using the conventional flash framework it is necessary to solve the equation for volume (at a given pressure) at each iteration. Though simple for the cubic equations of state this is a more significant issue when using advanced equations of state where the association equations must also be solved iteratively at each volume iteration. This increase in computational cost is one reason that the more advanced equations of state are not yet in common use.
An alternative framework to solve the isothermal isobaric flash problem is possible. Instead of solving the equation of state for volume at each iteration the pressure of the phases is matched only at the final equilibrium point. This allows for the natural variables, (T,V,n), of the equation of state to be used to cosolve the equation of state with the equilibrium equations. Using this framework means that the equation of state does not need to be solved for volume at each iteration. This means that the more complex equations of state are only marginally more computationally expensive than the simple cubics.
In this work we will present a method to solve the (T,P) flash problem using the natural variables of the equation of state, (T,V,n). The resulting framework will be used in a multiphase, compositional, 3D reservoir simulator and demonstrated using a number of examples. The computational cost of the proposed method will be compared with the conventional method for solving the (T,P) flash problem when solving the same simulation problem.



Theoretical Investigation Of TwoEndsOpen Free Spontaneous Imbibition
More LessSummaryTwoendsopen free spontaneous imbibition refers to a laboratory core experiment, with one end face exposed to the wetting phase and the other end exposed to the nonwetting phase. Spontaneous imbibition leads to the production of nonwetting phase both cocurrently and countercurrently. This paper extends previous work on systems of infinite length and presents the exact onedimensional semianalytic solution for such a system, and validates the solution with numerical simulation.
The methodology solves the partial differential equation of unsteady state immiscible, incompressible flow with arbitrary relative permeability and capillary pressure functions using a fractional flow concept as a function of saturation and time. The solution strategy uses backward finite differences on both the temporal variable and water saturation to solve for the instantaneous and average normalized water fluxes. The approach avoids the evaluation of implicit integral solutions and applies iterations on the flux ratio to satisfy both the flow and pressure boundary conditions. The solutions are obtained through two shooting processes for both normalized water flux and the pressure profile at all temporal steps.
The wetting phase is continuously imbibed into the core with initial flux being close to infinity. As the imbibition front propagates, the ratio of the cocurrent nonwetting phase flux to the inlet water flux increases from zero to a finite value below one which is dependent on the intrinsic properties of the system. This indicates that the production of nonwetting phase at the inlet will not cease before the front reaches the outlet, irrespective of the length of the system. The time for the front to reach the outlet, the ending ratio of cocurrent oil flux to inlet water flux and the variation of total produced volume from both ends, along with their sensitivities with respect to the absolute permeability, system length and wettability are analyzed in this study as well. The results also indicate the solution is independent of system length and permeability in its dimensionless form.
Unlike previous literature, we have not assumed selfsimilar solutions or treated the flow as purely cocurrent or countercurrent. The boundary conditions for the system analyzed here are easily achievable in the lab and have been discussed in the literature. The results from this study could be used to serve as a benchmark for numerical simulations, in future applications such as relative permeability and capillary pressure estimation, or improved interpretation of lab to field relationships through scaling group analysis.



Field Scale Modeling Of BioReactions During Underground Hydrogen Storage
Authors B. Hagemann, L. Ganzer and M. PanfilovSummaryThe energy transition from fossil and nuclear energy towards an energy supply system from renewable sources will require an enormous extension of the existing energy storage capacity. For this intention depleted oil and gas reservoirs could play a key role when they are used as storage reservoirs for hydrogen or other energy carriers in a seasonal or more frequent cycle.
In previous studies it was shown that chemical reactions catalyzed by anaerobic microorganisms and mixing phenomena between gases with different composition have important influences in underground storage of hydrogen. In particular hydrogenotrophic microorganisms could produce methane by metabolizing hydrogen and carbon dioxide. To describe these effects a model was developed which couples the compositional twophase transport of gas and water to microbial population dynamics and biochemical reactions.
In this work the numerical model was applied to a field scale storage scenario using a real geological model and several storage operation wells. The complex multiphysical model applied on around 200.000 grid cells results in approximately 2 million degrees of freedom. In addition the strong coupling between the microbial population dynamics and transport of chemical components is numerically difficult to handle and consequently small times steps not larger than one or two days have to be calculated. To overcome the computational effort the simulation study was executed on a high performance computing cluster.
The interpretation of the simulation results shows that a significant amount of the stored hydrogen was transformed into methane due to the biochemical reactions. In addition it was demonstrated that the produced gas contains H2S in the range of some parts per thousand when sulfate reducing bacteria were present.



Well Modelling By Means Of Coupled 1D3D Flow Models
Authors I.G. Gjerde, K. Kumar and J.M. NordbottenSummaryIn this work we present a new numerical method for solving coupled 1D3D flow models, which can be used, for example, to model the flow in a well and the surrounding reservoir. Assuming 1D Poiseuille flow in the well, we use Stirling’s law of filtration to couple it to the 3D flow model used for the reservoir. The presence of a line source in the governing equations of the reservoir is known to cause a logarithmic type singularity in the solution around the well. For this reason, the solution is difficult to approximate numerically. We therefore introduce a decomposition technique where the solution is split into an explicitly known logarithmic term capturing the singularity and a wellbehaved correction term. After a decoupling, the coupled 1D3D model can then be reformulated as a fixed point iteration scheme iterating over the 1D pressure in the well and the 3D correction term for the reservoir. The iteration scheme can be implemented using both Galerkin and mixed finite element methods, the former of which leads to mass conservative solutions. The advantage of the decoupling and reformulation is twofold: Firstly, it recasts the model into a system for which the discretization schemes and solution methods are readily available. Secondly, it recovers optimal convergence rates without needing to perform a meshrefinement around the well.



Vanishing Artificial Capillary Pressure For Improved RealSize Reservoir Simulations
Authors P. SalinasCortes, C.C. Pain, H. Osman, C. Heaney, D. Pavlidis and M.D. JacksonSummaryA common approach to stabilise the system arising from the discretisation of the advection equation is to introduce artificial diffusion. However, introducing artificial diffusion affects the result, therefore a balance has to be found so that the introduced artificial diffusion does not affects the final result.
Recently, a vanishing artificial diffusion was presented. In that method, the diffusion was controlled by the convergence of the nonlinear solver by multiplying the artificial diffusion term by the difference between the most recent saturation estimation and the one obtained in the previous nonlinear iteration. This approach showed that it is capable to help to reduce the computational effort required by the nonlinear solver, as classical artificial diffusions do. However, this approach could lead to an introduction of an artificial source/sink in the system, therefore not conserving mass.
A conservative vanishing artificial diffusion is presented here. It improves the convergence and convergence rate of the nonlinear solver by reducing the nonlinearity of the equations. Moreover, it is tailored to specially help to deal with the capillary pressure. The vanishing artificial diffusion is introduced using the same model employed to introduce the capillary pressure, obtaining a vanishing artificial capillary pressure diffusion term. By solving this term implicitly in the saturation equation, a very efficient method to model multiphase porous media flow with physical capillary pressure is obtained. This is tested in realsize reservoir simulations with realistically high capillary numbers to prove its efficiency.
The presented method provides accurate results and significantly reduces the effort required by the nonlinear solver to achieve convergence. It enables to carry out very demanding numerical simulations, e.g. when the physical capillary pressure effects are dominant, with Courant numbers that are at least two orders of magnitude bigger than without it.



Application of a quasiimplicit scheme to the simulation of nonNewtonian flows through porous media
Authors R. de Loubens, L. Léon and L. PatacchiniSummaryWhen modeling nonNewtonian flows through porous media, numerical difficulties arise due to the velocitydependence of phase mobilities. While a fully implicit treatment is unconditionally stable in the von Neumann sense, it leads to at least a 19point stencil on 3D hexahedral grids; its implementation is therefore complex and its computational cost is high. Simpler semiimplicit schemes are often encountered, whereby the pressure gradient driving the flow is treated implicitly while the velocityargument of mobilities is evaluated explicitly. These are however only conditionally stable, and in some practical situations of interest, clearly nonmonotone. In this context, a quasiimplicit discretization has recently been proposed, where the velocityargument of the mobilities is evaluated at cell edges with an implicit normal component, and an explicit transverse component. It has better stability and monotonicity properties than conventional semiimplicit schemes, while requiring only a 7point stencil on 3D hexahedral grids.
This quasiimplicit scheme was implemented in our parallel inhouse research reservoir simulator to model nonNewtonian polymer flows. A detailed description of its implementation is provided, accommodating different rheological models. Extensive numerical tests in various geometries are then performed to validate the implementation and illustrate the advantages of the new scheme.



Robust And Accurate Formulation For Modeling Of Acid Stimulation
More LessSummaryAccurate representation of processes associated with energy extraction from subsurface formations often requires models which account for chemical interactions between different species in the presence of multiphase flow. In this study, we focus on modeling of acid stimulation in the nearwell region. For the chemical processes which include a dissolution of rock material, an issue arises with the predictive representation of flow. Taking into account the spatial scale of discretization, some of simulation control volumes can have values of porosity close to 1, which makes an application of Darcy’s law inconsistent and requires employing a true momentum equation such as the DarcyBrinkmanStokes (DBS) equation. The DBS equation automatically switches the description between Darcy equation in control volumes with low porosity and Stokes equation in grid blocks with high porosity. For chemical reactions, we propose a local nonlinear solution technique that allows solving the balance of solid species separately yet retaining the full coupling with rest of the equations. Finally, we study the impact of multiphase flow. The DBS approach is not well established for multiphase flow description. Therefore we employ a hybrid approach, where we assume that the singlephase DBS flow and the multiphase Darcy flow occur in separate regions. We test the accuracy and performance of both approaches on realistic models of practical interest.



A Novel AMG Approach Based On Adaptive Smoothing And Prolongation For Reservoir Simulations
Authors V.A. Paludetto Magri, M. Ferronato, A. Franceschini and C. JannaSummaryReservoir models can easily incorporate millions or even billions of unknowns. Algebraic multigrid (AMG) methods are often the standard choice as iterative solvers or preconditioners for the solution of the resulting linear systems. These comprise a family of techniques built on a hierarchy of levels associated with decreasingsize problems. In this way, optimality and efficiency are achieved by combining two complementary processes, i.e. relaxation and coarsegrid correction. One of the key factors defining a fast AMG method consists of capturing accurately the nearnull space of the system matrix for the construction of suitable prolongation operators.
In this work, we propose a novel AMG package, aSPAMG, where aSP means “adaptive Smoothing and Prolongation” and the “adaptive” attribute implies that we follow the perspective of adaptive and bootstrap AMG. We construct a space of smooth vectors of limited size (test space) using the Lanczos method and introduce the factorized sparse approximate inverse (FSAI) as a smoother. This improves the smoothing capabilities of the aSPAMG as FSAI is more effective than Jacobi and much sparser than GaussSeidel. Moreover, FSAI has been shown to be strongly scalable. The coarsening phase is carried out as in classical AMG, but the strength of connection is computed by means of the affinity based on the test space. Finally, three new techniques are developed for building the prolongation operator: i) ABF, running few iterations of the aFSAI algorithm; ii) LSABF, updating the ABF coefficients with a least squares minimization; iii) DPLS, considering a leastsquares process only.
The aSPAMG performance is assessed through the solution of reservoir engineering problems including both fluid flow and geomechanical test cases. Comparisons are made with the FSAI and BoomerAMG preconditioners, showing that the new method is generally superior to both approaches.



On The Acceleration Of IllConditioned Linear Systems: A PodBased Deflation Method For The Simulation Of TwoPhase Flow
Authors G.B. Diaz Cortes, J.D. Jansen and C. VuikSummaryWe explore and develop PODbased deflation methods to accelerate the solution of largescale linear systems resulting from twophase flow simulation.
The techniques here presented collect information from the system in a POD basis, which is later used in a deflation scheme.
The snapshots required to obtain the POD basis are captured in two ways: a moving window approach, where the most recently computed solutions are used, and a training phase approach, where a full presimulation is run. We test this methodology in highly heterogeneous porous media: a full SPE 10 model containing O(10^6) cells, and in an academic layered problem presenting a contrast in permeability layers up to 10^6. Among the experiments, we study cases including gravity and capillary pressure terms.
With the PODbased deflated procedure, we accelerate the convergence of a Preconditioned Conjugate Gradient (PCG) method, reducing the required number of iterations to around 10–30 %, i.e., we achieve speedups of factors three to ten.



Localized Linear Systems For FullyImplicit Simulation Of Multiphase Multicomponent Flow In Porous Media
Authors S. Sheth, A. Moncorgé and R.M. YounisSummaryDuring the solution of fullyimplicit reservoir simulation timesteps, it is often observed that the computed Newton updates may be very sparse, considering computer precision. This sparsity can be as high as 95% and can vary largely from one iteration to the next.
In recent work, a mathematically sound framework was developed to predict the sparsity pattern before the full linear system is solved. The theory is restricted to general, scalar nonlinear advectiondiffusionreaction problems in multidimensional and heterogeneous settings. This theory had been applied to reduce the size of the linear systems that were computed during sequential implicit timesteps for twophase flow. The results confirmed that the linearization computations and the linear solution processes may be localized by as much as 95% while retaining the exact Newton convergence behavior and final solution. Inspired by the great success of that methodology, this work develops algorithmic extensions in order to devise localization algorithms for fullyimplicit coupled multicomponent problems.
We propose, apply, and test a novel algorithm to resolve a system of hyperbolic equations obtained from an Equation of State (EOS) based compositional simulator. When applied to various fullyimplicit flow and multicomponent transport simulations, involving six thermodynamic species, on the full SPE 10 geological model, the observed reduction in computational effort ranges between six to fortynine fold depending on the level of locality present in the system. We apply this algorithm to several injection and depletion scenarios with and without gravity and capillarity in order to investigate the adaptivity and robustness of the proposed method to the underlying heterogeneity and complexity. We demonstrate that the algorithm enables efficient and robust fullresolution fully implicit simulation without the errors introduced by adaptive discretization methods or the stability concerns of semiimplicit approaches.



Transition Of Algebraic Multiscale To Algebraic Multigrid
Authors S. Ehrmann, S. Gries and M.A. SchweitzerSummaryAlgebraic Multiscale (AMS) is a recent development for the construction of efficient linear solvers in certain reservoir simulations. It employs analytical upscaling ideas to coarsen the respective linear system and provides a high amount of inherent parallelism.
However, it has the drawback that it can currently only be applied to problems for a single scalar physical unknown, e.g., a pressure subproblem. Moreover, typical AMS exploits a structured grid and results in a twolevel scheme only. Generalizing the AMS approach to overcome these limitations requires substantial efforts and is not straightforward.
To exploit the benefits of AMS, however, we integrate its core ideas in an Algebraic Multigrid (AMG) method. Thus, all results and techniques from the wellestablished AMG are directly available in conjunction with (core ingredients of) AMS. This holds regarding multilevel usage and applicability for unstructured problems. But it also holds for the SystemAMG approach that allows to consider additional thermal and mechanical unknowns. In this paper, we identify the algorithmic similarities between both approaches, AMS and AMG. In fact, the basic coarsening idea of AMS corresponds to the socalled aggregative AMG approach. However, plain aggregative AMG suffers from the drawback of simplified transfer operation, or interpolation, within the hierarchy. By the integration of the AMS transfer we overcome this limitation and significantly improve the robustness of the aggregative AMG; especially in cases with inhomogeneous material coefficients. Yet, the method is not as robust as classical AMG, though. However, the setup phase is significantly simplified.
Moreover, we adjust the AMSlike interpolation to work purely algebraically, independent of any grid structure. This involves certain compromises to the original AMS idea. However, it can now be used as an interpolation in any aggregative AMG approach.
An additional advantage of our approach is the improved controllability of the hierarchy’s operator complexity, i.e., its memory consumption. This is especially important with increasing density of the matrix stencils, e.g., in (geo)mechanics or data science.



On the Development of a Relative Permeability Equation of State
Authors P. Purswani, M.S. Tawfik, Z.T. Karpyn and R.T. JohnsSummaryStandard compositional simulators use compositiondependent cubic equationsofstate (EoS), but saturationdependent relative permeability (kr). This discrepancy causes discontinuities, increasing computational time and reduced accuracy. To rectify this problem, kr has been recently defined as a state function, so that it becomes compositional dependent. Such a form of the kr EoS can significantly improve the convergence in compositional simulation, in that time step sizes are near the IMPEC stability limit and flash calculation convergence is improved.
This paper revisits the development of kr EoS by defining relevant state variables and deriving functional forms of the state function via a methodical approach. The state variables include phase saturation, phase connectivity, wettability, capillary number, and pore topology. The developed EoS is constrained to physical boundary conditions. The model coefficients are estimated through linear regression on data collected from a porescale simulation study that estimates kr based on microCT image analysis. The results show that a simple quadratic expression gives an excellent match with simulation measurements from the literature. The goodness of fit (R2) value is 0.97 for kr at variable phase saturation and phase connectivity, and constant wettability, pore structure, and capillary number (∼104). The quadratic response for kr also shows excellent predictive capabilities.



Efficient Modeling Of Near Wellbore Phenomena For Large Scale GasCondensate Systems In Massively Parallel Reservoir Sim
Authors S. Manzoor, U. Middya, T.J. Byer and P.I. CrumptonSummaryGas condensate reservoirs exhibit complex behaviour when they are produced below dew point pressure under isothermal conditions; this is due to the appearance of a twophase gascondensate in the near wellbore region. In addition, at high flow rates in the near wellbore region, inertial forces counteract with velocity dependence of relative permeability. This behaviour can be resolved using local grid refinements; however, the computation burden becomes excessive, especially in a full field simulation. Alternatively pseudopressure approach can be used which iteratively solves a nonlinear equation at each integration point, and is also computationally expensive. Furthermore, the conventional pseudopressure method lacks efficient coupling of the complex interaction of fluid composition, liquid dropout rate, gasoil relative permeability, gasoil interfacial tensions, and nonDarcy flow effects. Development of a computationally efficient and accurate approach to model near wellbore phenomena without increasing grid resolution is presented. An adaptive piecewise representation of pseudopressure is used, replacing nonlinear equation solved at each integration point, thus drastically reducing computational cost without undue loss of accuracy. NonDarcy flow effects and interaction of rock and fluid properties are captured in the pseudopressure integrand in a unified manner. The results are validated against a commercial simulator, and fine grid results, which demonstrate the accuracy and consistency of the approach. Finally the efficiency of the approach is demonstrated by simulating a giant gascondensate model with thousands of wells and millions of cells, all solved on a massively parallel computer.



Phase behavior computations using Gibbs free energy minimization on GPUs for speeding up compositional simulations
Authors S. Shiozawa, A. Venkatraman and B. DindorukSummaryCO2 flooding in a preferred method of enhancing oil recovery as it has the dual benefit of sequestration and increasing oil recovery. In order to evaluate and design these processes, compositional simulations are used to track component changes during the course of flow. However, the phase behavior, as well as the stability computations associated with compositional simulations are timeconsuming. In a compositional reservoir simulator, EOS, typically a cubic EOS (i.e., PengRobinson) for hydrocarbons, must be solved in all the grid blocks at every time step in order to accurately predict the phase behavior. In typical fieldscale simulations, millions of grid blocks are used, and hence solving EOS for multicomponent hydrocarbon systems really slows down the simulation. There is a need to increase the speed and improve the efficiency of phase equilibrium computation for fieldscale simulations. In this research, we develop a Gibbs free energy framework optimized for graphic processing unit (GPU) implementation. The Gibbs free energy minimization is preferred as it is a unifying function to combine components described using different thermodynamic models  Equation of State (EOS) as well as activity coefficient models. We use a combination of CPUs and GPUs to solve a constrained minimization problem and the solution is the equilibrium composition at a fixed temperature and pressure (PT flash). The NVIDIA Tesla GPUs help parallelize multiple functions evaluations simultaneously, which results in a significant speedup in computation times. A comparison of computation times of our approach with other approaches to compute and also algorithms to obtain equilibrium compositions is presented 1. CPU versus GPU. 2. Kvalue versus Gibbs free energy approach. The proposed model can easily be incorporated into existing reservoir simulators to decrease computational times.



Nonlinear Domain Decomposition Scheme For Sequential Fully Implicit Formulation Of Compositional Multiphase Flow
Authors O. Møyner and A. MoncorgéSummaryNew Sequential Fully Implicit (SFI) methods for compositional flow simulation have been recently investigated. These SFI schemes decouple the fully coupled problem into separate pressure and transport problems and have convergence properties comparable to the Fully Implicit (FI) method. The pressure system is a parabolic problem with fixed overallcompositions and the transport system is a hyperbolic problem with fixed pressure and totalvelocity. We discuss some aspects of how to design optimal SFI schemes for compositional flow with general equationofstates by localizing the computations. The different systems are solved sequentially and the Fully Implicit solution is recovered by controlling the aposteriori splitting errors due to the choice of decoupling.
When the parabolic and the hyperbolic operators are separated, it is possible to design nonlinear domain decomposition schemes taking the advantage of the specific properties of each operator. Usually, for reservoir simulation models, most of the reservoir is converged with SFI methods in one outeriteration. However, in some localized regions with strong coupling between the pressure and the compositions, the SFI algorithms may need several outeriterations. Here we propose a domain decomposition method based on a predictorcorrector strategy. As a first step, the nonlinear parabolic pressure equation is solved on the whole domain with the Multiscale RestrictionSmooth Basis (MsRSB) method used as a linear domain decomposition solver. In a second step the compositions system is solved. At the end of this first outeriteration, most of the reservoir is converged. Based on aposteriori splittingerrors of the SFI scheme in volume and velocity, we define local regions where additional global outeriterations would be required in the conventional SFI scheme. We then fix Dirichlet boundary conditions for the pressure and the compositions and solve local problems in these nonconverged regions. After convergence of these smaller nonlinear problems, if the boundary conditions are changed by the updated regions, the global pressure problem is revisited. An additional postprocessing of local transport iterations makes sure mass is conserved everywhere. The resulting algorithm converges to the same solution as the FI solver, with all simultaneous updates to composition and pressure in localized regions.
We demonstrate the robustness of this nonlinear domain decomposition algorithm across a wide parameter range. Realistic compositional models with gas and water injection are presented and discussed.



Consistent Upwinding for Sequential Fully Implicit Compositional Simulation
Authors A. Moncorge, H.A. Tchelepi and P. JennySummaryThere is strong interest to design Sequential Fully Implicit (SFI) methods for compositional flow simulations with convergence properties that are comparable to Fully Implicit (FI) methods. SFI methods decompose the fully coupled system into a pressure equation and a transport system of the components. During the pressure update, the compositions are frozen, and during the transport calculations, both the pressure and totalvelocity are kept constant. The two systems are solved sequentially, and the solution, which is a fully implicit one, is obtained by controlling the splitting errors due to the decoupling. Having an SFI scheme that enjoys a convergence rate similar to FI makes it possible to design specialized numerical methods optimized for the different parabolic and the hyperbolic operators, as well as the use of highorder spatial and temporal discretization schemes. Here, we show that phasepotential upwinding is incompatible with the totalvelocity formulation of the fluxes, which is common in SFI schemes. We observe that in cases with strong gravity or capillary pressure, it is possible to have flow reversals. These reversals can strongly affect the convergence rate of SFI methods. In this work, we employ implicit hybrid upwinding (IHU) with a SFI method. IHU determines the upwinding direction differently for the viscous, buoyancy, and capillary pressure terms in the phase velocity expressions. The use of IHU leads to a consistent SFI scheme in terms of both pressure and compositions, and it improves the SFI convergence significantly in settings with strong buoyancy or capillarity. We demonstrate the robustness of the IHUbased SFI algorithm across a wide parameter range. Realistic compositional models with gas and water injection are presented and discussed.



Nonlinear GaussSeidel Solvers With Higher Order For BlackOil Models
Authors Ø.S. Klemetsdal, A.F. Rasmussen, O. Møyner and K.A LieSummaryThe fully implicit method is the most commonly used approach to solve blackoil problems in reservoir simulation. The method requires repeated linearization of large nonlinear systems and produces illconditioned linear systems. We present a strategy to reduce computational time that relies on two key ideas: i) a sequential formulation that decouples flow and transport into separate subproblems, and ii) a highly efficient GaussSeidel solver for the transport problems. This solver uses intercell fluxes to find all cells that only depend on their upstream neighbors and groups all remaining cells into local clusters of cells that are mutually dependent because of countercurrent flow. The single cells and local clusters can then be sorted and solved in sequence, starting from the inflow and moving gradually downstream, since each new cell or local cluster will only depend on upstream neighbors that have already been computed. Altogether, this gives optimal localization and control of the nonlinear solution process.
This method has been successfully applied to realfield problems using the standard firstorder finite volume discretization. Here, we extend the idea to firstorder dG methods on fully unstructured grids. We also demonstrate proof of concept for the reordering idea by applying it to the full simulation model of the Norne oil field, using a prototype variant of open source OPM Flow simulator.
