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ECMOR XV - 15th European Conference on the Mathematics of Oil Recovery
- Conference date: 29 Aug 2016 - 01 Sep 2016
- Location: Amsterdam, Netherlands
- ISBN: 978-94-6282-193-4
- Published: 29 August 2016
141 - 160 of 163 results
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Fully Implicit Simulation of Polymer Flooding with MRST
More LessThe present work describes a fully implicit simulator for polymer injection implemented in the free, open-source MATLAB Reservoir Simulation Toolbox (MRST). Polymer injection is one of the widely used enhanced oil recovery (EOR) techniques and complicated physical process is involved, which makes accurate simulation very challenging. The proposed work is intended for providing a powerful and flexible tool to investigate the polymer injection process in realistic reservoir scenarios. Within the model, the polymer component is assumed to be only transported in the water phase and adsorbed in the rock. The hydrocarbon phases are not influenced by the polymer and they are described with the standard, three-phase, black-oil equations. The effects of the polymer are simulated based on the Todd--Longstaff mixing model, accounting for adsorption, inaccessible pore space, and permeability reduction effects. Shear-thinning/thickening effects based on shear rate are also included by the means of a separate inner-Newton iteration process within the global nonlinear iteration. The implementation is based on the automatic differentiation framework in MRST (MRST-AD), and an iterative linear solver with a constrained pressure residual (CPR) preconditioner is used to solve the resulting linear systems efficiently. We discuss certain implementation details to show how convenient it is to use the existing functionality in MRST to develop an accurate and efficient polymer flooding simulator for real fields. With its modular design, vectorized implementation, support for stratigraphic and general unstructured grids, and automatic differentiation framework, MRST is a very powerful prototyping and experimentation platform for development of new reservoir simulators. To verify the simulator, we first compare it with a commercial simulator and good agreement is achieved. Then, we apply the new simulator to a few realistic reservoir models to investigate the benefit of adding polymer injection. Computational efficiency is demonstrated, and we argue that the presented model can be used as an efficient prototyping tool to evaluate new models for polymer-water-flooding processes in real reservoir fields.
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Modelling of Reactive Flow and Transport in the Presence of a Complex Phase Transition Phenomena
Authors D.V. Voskov, A. Lucia and H. HenleyWe present a novel simulation approach for modeling of reactive flow and transport in multiphase multi-component mixtures that include light gases, hydrocarbon components, and different ions present in an aqueous electrolyte phase. The phase behavior in these systems involves both thermodynamically-driven phase transitions (e.g. between supercritical vapor and liquid phases) and chemically driven precipitation and dissolution of solid (mineral) phases. All phases are modeled using the multi-scale Gibbs-Helmholtz Constrained Equation of State (GHC EoS), which up-scales molecular length scale information from a priori Monte Carlo simulations to help build accurate estimates of the energy parameter. Our proposed approach is implemented in the combined software system included the Automatic Differentiation General Purpose Research Simulator (ADGPRS) developed at Stanford University and the GFLASH library developed at University of Rhode Island. The extended variable substitution schema for a natural fully implicit formulation is designed to support the potential coexistence of an arbitrary set of phases in the flow. The classical reduction in the number of conservation equations based on element balances is combined with specific local constraints describing simultaneous thermodynamic and chemical equilibrium. Rigorous flash solutions for detecting phase changes in each grid block are computed using phase splitting and phase/chemical equilibrium to ensure equality of component chemical potentials and by monitoring the Gibbs free energy of the system to guarantee a global minimum is found. We present examples that cover a wide range of physical processes related to CO2 sequestration in saline aquifers.
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IORSim - A Simulator for Fast and Accurate Simulation of Multi-phase Geochemical Interactions at the Field Scale
Authors A. Hiorth, J. Sagen, A. Lohne, J. Nossen, J.L. Vinningland, E. Jettestuen and T. SiraOptimal injection strategies for a single core in the lab can usually be found, but there is a significant challenge to translate the lab results to field scale. In this paper we present a simulator, IORSim, that can be calibrated to lab experiments and used together with industry standard reservoir simulator to predict chemical alteration on field scale. Based on the flow velocities predicted by the reservoir simulator, IORSim advects the chemical species in the reservoir, taking into account chemical interactions that may affect the flow of the fluids. As a finer grid than the reservoir simulator and a block sorting technique is used in IORSim, the species transport and rock-fluid interactions can be performed at considerably higher speed and accuracy compared to a built-in geochemical model. The sorting algorithm allows us to solve the whole transport problem implicitly without solving for all blocks simultaneously, and thus greatly improve the stability of the numerical problem. We present two applications (1) simulation of produced water for a sector of the Ekofisk field, which is compared with data, (2) sodium silicate injection in a high permeable zone. IORSim modifies the reservoir permeability due to the gelling and the macroscopic sweep is improved.
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A Fully Implicit Reactive Flow Formulation for Low Salinity Waterflooding Process
Authors G. Singh, A. Venkatraman, G. Pencheva and M. WheelerThe non-stochiometric formulation is widely used to obtain compositions resulting from reactive flow systems. The current reactive flow simulators either assume ideality or activity coefficient models such as the Debye-Huckel model for aqueous components. Such models are not applicable for higher ionic strength solutions where Pitzer activity coefficient model best represent the polar forces of concentrated solutions. In this research, we model the low salinity waterflooding process using a non-stochiometric formulation for equilibrium reactions between aqueous phase components and the solid rock mineral. In particular, a set of ion-exchange as well as precipitation/dissolution reactions between aqueous phase components and the rock mineral are formulated as an equivalent Gibbs free energy minimization problem. The concentrations are obtained by solving the resulting non-linear optimization problem satisfying the elemental balance constraint and the non-negativity constratint on the species mole number. The appearance/disappearance of solid rock minerals are handled as complementarity condition using the perturbed KKT approach. Model results are compared with low salinity water flooding laboratory experimental data. The presented model can be extended to find compositions in higher ionic strength solutions for other processes such as CO2 sequestration and aquifer remediation.
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Geologic Modelling Using Parametric NURBS Surfaces
Authors C. Jacquemyn, Y. Melnikova, M.D. Jackson, G.J. Hampson and C.M. JohnMost reservoir modelling/simulation workflows represent geological heterogeneity on a pillar-grid defined early in the modelling process. However, it is challenging to represent many common geological features using pillar grids: examples include intersecting faults, recumbent folds, slumps, and non-monotonic injection structures such as salt diapirs. It is also challenging to represent multi-scale features, because the same number of pillars must be present in all layers so there is little flexibility to adjust the areal grid resolution. We present a surface-based geological modelling (SBGM) workflow that uses NURBS (Non-Uniform Rational B-Splines) surfaces to represent geological heterogeneities without reference to a pre-defined grid. The NURBS surfaces represent a broad range of heterogeneity types, including faults, fractures, stratigraphic surfaces across a range of length-scales, and boundaries between different facies or lithologies. The geological model is constructed using the NURBS surfaces and a mesh created only when required for flow simulation or other calculations. The mesh preserves the geometry of the modelled surfaces. NURBS surfaces are an efficient and flexible tool to model complex geometries and are common in many modelling and engineering disciplines; however, they are rarely used in reservoir modelling. Complex surfaces can be created using a small number of control points; modelling with NURBS surfaces is therefore computationally efficient. We report here a variety of new stochastic approaches to create geological NURBS surfaces, including (1) extrusion of spatially variable cross-sections, (2) parametric 3D geometry templates, and (3) perturbation of control points to yield similar results to some pixel-based geostatistical methods. Surface interactions, such as erosion, stacking or conforming, are enforced to ensure geological relationships are preserved and the boundary representation is watertight. We illustrate our NURBS SBGM approach via a number of examples, including channelized sandbodies, clinoforms, sedimentary cycles, fractures, crosscutting faults, recumbent folds and combinations thereof.
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Novel Basin Modelling Concept for Simulating Deformation from Mechanical Compaction Using Level Sets
Authors S. McGovern, S. Kollet, C. Bürger, R. Schwede and O. PodlahaAs sedimentation progresses in the formation and evolution of a depositional geologic basin, the rock strata are subject to various stresses. With increasing lithostatic pressure, compressional forces act to compact the porous rock matrix, leading to overpressure buildup, changes in the fluid pore pressure and fluid flow. In the context of petroleum systems modeling, the present study concerns the geometry changes that a compacting basin experiences subject to deposition. The purpose is to track the positions of the rock facies interfaces as compaction occurs. To handle the challenge of potentially large geometry deformations, a new modeling concept is proposed that couples the pore pressure equation with a level set method to determine the movement of lithostratigraphic interfaces. The level set method propagates an interface according to a prescribed speed. The coupling term for the pore pressure and level-set equations consists of this speed function, which was derived from the compaction law. A first evaluation of the model concept is presented based on an implementation for one spatial dimension accounting for vertical effective stress. Thus, the propagation speed is determined by the sedimentation rate and the change in porosity. Both, the case of a linear compaction law and that of Athy’s exponential compaction law were studied for multiple rock facies, consisting of varying lithologies defined by initial porosities, permeabilities and grain densities. Isothermal conditions with a constant fluid density and viscosity were assumed. The accuracy of the implemented numerical solution for the case of a single stratigraphic unit with a linear compaction law was compared to the available analytical solution (Wangen 2010). The multi-facies setup and the nonlinear case were tested for plausibility. Reference: Wangen, A. (2010), Physical Principles of Sedimentary Basin Analysis, Cambridge University Press, 527 p.
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Geostatistical Simulations on Irregular Reservoir Models
Authors V.N. Zaytsev, P. Biver, H. Wackernagel and D. AllardThe new generation of reservoir computer models based on unstructured meshes requires appropriate methods for filling the cells with petrophysical data. Classical methods are not applicable to unstructured reservoir models since they can neither treat complicated geometry nor reproduce correctly the statistical properties of the model (as marginal distributions and covariance between the blocks). In order to overcome these difficulties, we develop new statistical methods for model filling which work directly on unstructured grids. The most important problem in the model filling is the transfer of the small scale data available from well cores and laboratory analysis to multiple different scales which are defined by various grid blocks. This problem is known in as the volume support effect and can be addressed by the methods of non-linear geostatistics. In the proposed solution, we use non-linear geostatistics to address the problem of simulating heterogeneities on unstructured grids (Zaytsev et al., 2015). We present a workflow for approximating the block distributions by the means of the discrete Gaussian model and for performing geostatistical simulations directly on block support. Theoretical model and case studies are presented.
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Reservoir Modelling Using Parametric Surfaces and Dynamically Adaptive Fully Unstructured Grids
Authors Y. Melnikova, C. Jacquemyn, H. Osman, P. Salinas, G. Gorman, G.J. Hampson and M.D. JacksonGeologic heterogeneities play a key role in reservoir performance. Surface based geologic modeling (SBGM) offers an alternative approach to conventional grid-based methods and allows multi-scale geologic features to be captured throughout the modeling process. In SBGM, all geologic features that impact the distribution of material properties, such as porosity and permeability, are modeled as a set of volumes bounded by surfaces. Within these volumes, the material properties are constant. The surfaces have parametric, grid-free representation, which, in principle, allows for unlimited complexity, since no resolution is implied at the stage of modeling and features of any scale can be included. Surface based models are discretized only when required for numerical analysis. We report here a new automated and integrated workflow for creating and meshing stochastic, surface-based models. Surfaces are represented through non-uniform rational B-splines (NURBS). Multiple relations between surfaces are captured through geologic rules that are translated into Boolean operations (intersection, union, subtraction). Finally, models are discretized using fully unstructured tetrahedral meshes coupled with a geometry-adaptive sizing function that efficiently approximate complex geometries. We demonstrate the new workflow via examples of multiple erosional channelized geobodies, fault models and a fracture network. We also show finite element flow simulations of the resulting geologic models, using the Imperial College Finite Element Reservoir Simulator (IC-FERST) that features dynamic adaptive mesh optimization. Mesh adaptivity allows us to focus computational effort on the areas of interest, such as the location of water saturation front. The new approach has broad application in modeling subsurface flow.
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Hybrid Multiscale Formulation for Coupled Flow and Geomechanics
Authors N. Castelletto, H. Hajibeygi and H.A. TchelepiWe devise a hybrid MultiScale Finite Element-Finite Volume (h-MSFE-FV) framework to simulate single-phase flow through elastic deformable porous media. The coupled problem is solved based on a two-field fine-scale mixed finite element-finite volume formulation of the governing equations, namely conservation laws of linear momentum and mass, in which the primary unknowns are the displacement vector and pressure. For the MSFE displacement stage, we develop sets of local basis functions for the displacement vector over coarse cells, subject to reduced boundary condition. This MSFE stage is then coupled with the MSFV method for flow, where coarse and dual-coarse grids are imposed to obtain approximate but conservative multiscale solutions. Numerical experiments are presented to demonstrate accuracy and robustness of the proposed h-MSFE-FV method---both as an approximate, non-iterative solver, and a preconditioner.
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A Hybrid Approach to Parallel Multiscale Reservoir Simulator
Authors A. Kozlova, D. Walsh, S. Chittireddy, Z. Li, J. Natvig, S. Watanabe and K. BratvedtIn the wake of advancements in multi-core/multi-processor NUMA systems and associated hardware architectures, there is a need and an opportunity for software to be enhanced in lock-step with these hardware developments, in order to reap performance gains. However, Giga cell reservoir models still may require days of simulation on available hardware. Recently in reservoir simulation various multiscale methods have been developed in order to create faster computation algorithm. The key idea of these methods is to solve the problem on a coarse level and then prolong it to fine level using a set of basis functions. These prolongation operators map between fine grid geological properties of reservoir model and the coarse grid that is used for simulation. One of these methods has been implemented in a commercially available simulator. The next natural step to further speed-up simulation is to use high performance technologies for parallelization. We illustrate how this was accomplished by reusing and extending the existing parallel pattern for distributed, MPI, and by adding shared, OpenMP, memory techniques. Finally results are presented for large highly heterogeneous real-field models showing performance and scalability of the new industrial strength multiscale reservoir simulator running on a multi-node cluster.
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Multiscale projection-based Embedded Discrete Fracture Modeling approach (F-AMS-pEDFM)
Authors M. Tene, M.S. Al Kobaisi and H. HajibeygiThis work presents the formulation of a novel Projection-based Embedded Discrete Fracture Model (pEDFM), and its integration into an algebraic multiscale procedure. Similar to EDFM, pEDFM constructs independent grids for the matrix and fracture domains. However, as a significant step forward, it is able to accurately model the effect of fractures with general conductivity contrasts relative to the matrix, including impermeable flow barriers. This is achieved by automatically adjusting the matrix transmissibility field, in accordance to the conductivity of neighboring fracture networks. Then, in order to extend the pEDFM to real-field applications, F-AMS-pEDFM is introduced, which is an extension of the recently developed algebraic multiscale solver, F-AMS [Ţene et al., 2016], to include pEDFM. The performance (efficiency and scalability) of F-AMS-pEDFM is investigated extensively for challenging two- and three-dimensional scenarios with complex fracture geometries and a wide range of conductivity contrasts. Moreover, F-AMS-pEDFM is benchmarked against the commercial SAMG solver, where CPU time is monitored during both the setup and solution phases. The results support the conclusions that (1) pEDFM significantly outperforms the original EDFM model, and (2) the F-AMS-pEDFM approach proposed in this work is an accurate and efficient method for field-scale simulation of flow in fractured reservoirs.
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Multiscale Gradient Computation for Subsurface Flow Models
Authors R. Moraes, J.R.P. Rodrigues, H. Hajibeygi and J.D. JansenWe present an efficient multiscale (MS) gradient computation that is suitable for reservoir management studies involving optimization techniques for, e.g., computer-assisted history matching or life-cycle production optimization. The general, algebraic framework allows for the calculation of gradients using both the Direct and Adjoint derivative methods. The framework also allows for the utilization of any MS formulation in the forward reservoir simulation that can be algebraically expressed in terms of a restriction and a prolongation operator. In the implementation, extra partial derivative information required by the gradient methods is computed via automatic differentiation. Numerical experiments demonstrate the accuracy of the method compared against those based on fine-scale simulation (industry standard).
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A Multiscale Finite Volume Method with Oversampling for Geophysical Electromagnetic Simulations
Authors L.A. Caudillo Mata, E. Haber and C. Schwarzbachon, we develop a Multiscale finite volume (MSFV) method with oversampling for the quasistatic Maxwell’s equations in the frequency domain. Our method begins by assuming a coarse mesh nested into a fine mesh, which accurately discretizes the setting. For each coarse cell, we solve independently a local version of the original Maxwell’s system subject to linear boundary conditions on an extended domain, which includes the coarse cell and a neighborhood of fine cells around it. To solve the local Maxwell’s system, we use the fine mesh contained in the extended domain and the Mimetic Finite Volume method. Afterwards, these local solutions, called basis functions, together with a weak continuity condition are used to construct a coarse-scale version of the global problem that is much cheaper to solve. The basis functions can be used to obtain the fine-scale details from the solution to the coarse-scale problem. Our approach leads to a significant reduction in the size of the final system of equations to be solved and in the amount of computational time of the simulation, while accurately approximating the behavior of the fine-mesh solutions. We demonstrate the performance of our method using a heterogeneous 3D mineral deposit model.
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Successful Application of Multiscale Methods in a Real Reservoir Simulator Environment
Authors K.A. Lie, O. Møyner, J.R. Natvig, A. Kozlova, K. Bratvedt, S. Watanabe and Z. LiFor the past 10 years or so, a number of so-called multiscale methods have been developed as an alternative approach to upscaling and to accelerate reservoir simulation. The key idea of all these methods is to construct a set of prolongation operators that map between unknowns associated with cells in a fine grid holding the petrophysical properties of the geological reservoir model and unknowns on a coarser grid used for dynamic simulation. The prolongation operators are computed numerically by solving localized flow problems, much in the same way as for flow-based upscaling methods, and can be used to construct a reduced coarse-scale system of flow equations that describe the macro-scale displacement driven by global forces. Unlike effective parameters, the multiscale basis functions have subscale resolution, which ensures that fine-scale heterogeneity is correctly accounted for in a systematic manner. Among all multiscale formulations discussed in the literature, the multiscale restriction-smoothed basis (MsRSB) method has proved to be particularly promising. This method has been implemented in a commercially available simulator and has three main advantages. First, the input grid and its coarse partition can have general polyhedral geometry and unstructured topology. Secondly, MsRSB is accurate and robust when used as an approximate solver and converges relatively fast when used as an iterative fine-scale solver. Finally, the method is formulated on top of a cell-centered, conservative, finite-volume method and is applicable to any flow model for which one can isolate a pressure equation. We discuss numerical challenges posed by contemporary geomodels and report a number of validation cases showing that the MsRSB method is an efficient, robust, and versatile method for simulating complex models of real reservoirs.
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Enriched Algebraic Multiscale Linear Solver
Authors A. Manea, H. Hajibeygi, P. Vassilevski and H.A. TchelepiWe describe an Enriched Algebraic Multiscale Solver (EAMS) that overcomes the deficiency of existing multiscale methods for flow in heterogeneous media with large coherent correlation structures and high contrasts. For a given multiscale method, EAMS enriches the coarse space with local basis-functions specifically aimed at the largest error components in the solution space. For this purpose, the discrete error equation is used to identify the solution modes that are missing from the multiscale operator. The identified error modes, which are complex combinations of a spectrum of wave numbers, are then localized (truncated) and added to the prolongation operator. The enrichment process is repeated iteratively until the desired convergence rate is reached. The identification and enrichment processes are algebraic, and they are performed adaptively during the iterative solution process. Using challenging test cases from the literature, we show that EAMS leads to great improvements in the robustness and efficiency of existing state-of-the-art multiscale linear solvers. In most settings, the convergence rate of AMS is improved significantly by supplementing the standard basis functions with a few basis functions guided by the error equation. Since the enrichment is adaptive and algebraic, it can be integrated into any existing multiscale linear-solver implementation. EAMS is expected to be most useful in modeling evolution multiphase problems in heterogeneous reservoirs, whereby the changes in the character of the linear system - across Newton iterations and time steps - are relatively mild.
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New Sequential Scheme for Mixed-implicit Compositional Flow Simulation
Authors A. Moncorge, H. Tchelepi and P. JennyThe Fully Implicit (FI) and the Adaptive Implicit (AI) methods are widely used for general-purpose reservoir simulation. There has been growing interest, however, in sequential-implicit schemes. The common sequential solution strategies are: IMplicit Pressure Explicit Saturations (IMPES) and Sequential Fully Implicit (SFI) methods. In highly heterogeneous domains with tight coupling between the multiphase flow and the multi-component transport, IMPES suffers from severe restrictions on the size of the stable timestep, and SFI suffers from slow convergence of the sequential updating between the flow and transport problems. Here, we describe a modified SFI (m-SFI) scheme that improves the convergence behavior substantially. The modification entails additional coupling terms to the pressure equation that are limited in both space and time. Specifically, the pressure equation is complemented with a local approximation of the pressure-saturation/composition coupling terms that are brought about by the appearance of the gas-phase during iterations. This modification is also applied to the modified Sequential Adaptive Implicit (m-SAI) scheme. We consider several very challenging compositional processes whereby the SFI method suffers from severe nonlinear convergence behaviors, and we demonstrate using numerical experiments and analysis that the modified algorithms have convergence properties that are quite close to those of the FI and AI methods.
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Monotone Non-Galerkin Algebraic Multigrid Method Applied to Reservoir Simulations
Authors T.B. Jonsthovel, A.A. Lukyanov, E.D. Wobbes and C. VuikCommercial reservoir simulators must be very robust and fast. Moreover, current hardware requires the simulators to scale over multiple number of computing nodes and for a fixed (‘strong scalability’) as well as an increasing problem size per computing node (‘weak scalability’). In most current commercial reservoir simulators, due to the different geological structures and properties of hydrocarbon reservoirs and the use of enhanced oil recovery (EOR) techniques, the governing equations are strongly nonlinear and hard to solve. The Jacobian system is solved by FGMRES preconditioned by the two-level constrained pressure residual (CPR) preconditioner. The driving force of the CPR preconditioner is the solution of the pressure equation. The industry standard for solving the pressure equation is the algebraic multigrid (AMG) solver. AMG is well known for its ‘weak scalability’. However, in these applications, AMG has unfavorable ‘strong’ scalability properties. This degradation in scalability is due to the increased level of inter-processor communication in the algorithm. In this paper, a monotone non-Galerkin AMG (MNG-AMG) method is presented. The aim of the method is to reduce the overall communication in MNG-AMG by enforcing a predefined nonzero pattern and monotonicity property (i.e., M-matrices) on each multigrid level. This paper describes the application of the MNG-AMG method in the context of reservoir simulations. We will compare the parallel scalability of the default solver with the MNG-AMG solver and discuss the optimal values for the MNG-AMG solver for a variety of test cases based on full field reservoir simulations.
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Convergence and Error Analysis of Fully Discrete Iterative Coupling Schemes for Coupling Flow with Geomechanics
Authors T. Almani, K. Kumar, G. Singh and M. WheelerThe modeling of fluid flow coupled with mechanical deformations is needed for reliable production forecasts, especially in stress sensitive, and structurally weak reservoirs. It also plays a critical role in accurately predicting surface subsidence, well stability, CO2 sequestration, and sand production. Traditionally, changes in mechanical deformations are visible to fluid flow through a pore compressibility factor, which is insufficient for purposes described above. In fact, it is only through the coupling between flow and mechanics that reliable reservoir models can be obtained. In this work, we consider the multirate fixed-stress split iterative coupling scheme for the Biot system modeling coupled flow with geomechanics in a poro-elastic medium. Due to its physical nature, the geomechanics problem can cope with a coarser time step compared to the flow problem. This makes the multirate coupling scheme, in which several flow finer time steps are solved within one coarser mechanics time step, a natural candidate in this setting. For the multirate iterative scheme proposed here, our contributions are: 1. We establish the contracting behavior of the two successive iterates leading to geometric speed of convergence, 2. We derive error estimates for quantifying the error between any iterate and the true solution. Error estimates of the fully discrete fixed-stress split iterative coupling are derived for the first time in this work. 3. We show the sharpness of our derived theoretical estimates by implementing the proposed scheme numerically for field scale problems. Our approach is based on studying the equations satisfied by the difference of iterates and utilizing a Banach contraction argument to show that the corresponding scheme is a fixed point iteration. Obtained contraction results are then used to derive theoretical convergence error estimates for the iterative coupling scheme. Moreover, by comparing theoretical contraction estimates against numerical computations, we conclude that theoretical estimates can predict the contracting behavior, and subsequently, the rate of convergence of the iterative scheme with high accuracy.
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Localized Computation of Newton Updates for General Fully-implicit Reservoir Simulation
Authors S.M. Sheth and R.M. YounisReported observations suggest that Newton updates that are computed during the course of a fully-implicit time step are often sparse. The level of sparsity can vary dramatically from nonlinear iteration to the next, and across time steps. Reported observations suggest that the level of sparsity can be as large as 95%. This work develops an algorithm that accurately predetermines the nonzero elements of the Newton update, and subsequently, can compute it by only solving a truncated linear system. Several alternative ad hoc sparsity prediction strategies have been proposed. Due to their inability to consistently and accurately predetermine the sparsity set, the resulting Newton updates that are computed are inaccurate, leading to a severe degradation of the nonlinear convergence rate. An exact strategy based on an analysis of the sparsity graph of the Jacobian matrix was also proposed for two phase incompressible flow without gravity. Although exact, the proposed strategy cannot be generalized to more complex physics or numerical approximations. Recently, a theoretically sharp and conservative estimate for the sparsity set was derived specifically for the pressure and saturation variables in two-phase sequential-implicit simulation. In this strategy, the discrete Newton update was related to analytical solutions of linear Partial Differential Equations for flow and transport independently. The analytical solutions were evaluated and projected onto the computational domain, thereby providing an estimate of the sparsity set. The theoretically reliable algorithm was demonstrated to reduce the sequential-implicit simulation time for general two phase flow in the full SPE 10 comparative geological model by 5 fold. In this work, the approach is extended to general fully-implicit simulation of coupled flow and multicomponent transport. This is accomplished by considering a canonical functional form of the equations for flow and a system of transported quantities. The analytical estimate is derived by solving the system of linear differential equations using the Schur complement decomposition in functional space. When applied to various simulations of three-phase flow recovery processes in the full SPE 10 model, the observed reduction in computational effort ranged between four and tenfold depending on the level of total compressibility in the system and on the time step size. To investigate the scalability of the algorithm, we applied it to refined models of the SPE 10 case and to multicomponent problems. The improvement in computational speed scales strongly with the number of transport components, and to a lesser degree with problem size.
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Nonlinear Solver for Three-phase Transport Problems Based on Approximate Trust Regions
By O. MøynerImplicit transport solvers used in reservoir simulation can take longer time steps than explicit solvers, but for long time steps the commonly used Newton-Raphson's method will often fail to converge. The convergence issues will manifest themselves as oscillating residuals even though the implicit discretization itself is stable. This behavior occurs because the fractional flow-type flux functions often change between convex and concave during long time steps, resulting in multiple contraction regions for the Newton-Raphson solver. The common strategy to overcome this is to set limits on the saturation changes during the nonlinear iteration, but such a limit has to be determined on a case by case basis, excess iterations may be required and practical convergence is not guaranteed for a given problem. Previous work on this problem by multiple authors has resulted in solvers based on trust regions, where unconditional convergence can be obtained for incompressible two-phase flow provided a priori analytical knowledge of the flux function exists. The goal of our work is to extend this methodology to a solver where inflection points demarking the different contraction regions does not need to be explicitly known. Instead, these values are estimated during the solution process, giving improved convergence by a local computation for each interface in the simulation model. By following the Newton path, it is possible to greatly reduce the computational expense, making the same formulation suitable for an arbitrary number of components. We present a series of numerical results, including arbitrary time-step lengths for two and three-phase gravity segregation, as well as three-dimensional gas and water injection problems with wells and a mixture of both viscous and gravity-dominated flow regimes. The test cases are a systematic validation on a wide variety of both analytical and tabulated relative permeability curves.
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