Day 2 Thu, April 11, 2019最新文献

{"title":"Distributed Parallel Hybrid CPU-GPGPU Implementation of the Phase-Field Method for Accelerated High-Accuracy Simulations of Pore-Scale Two-Phase Flow","authors":"C. Thiele, M. Araya-Polo, F. Alpak, B. Rivière","doi":"10.2118/193922-MS","DOIUrl":"https://doi.org/10.2118/193922-MS","url":null,"abstract":"\u0000 Direct numerical simulation of multi-phase pore-scale flow is a computationally demanding task with strong requirements on time-to-solution for the prediction of relative permeabilities. In this paper, we describe the hybrid-parallel implementation of a two-phase two-component incompressible flow simulator using MPI, OpenMP, and general-purpose graphics processing units (GPUs), and we analyze its computational performance. In particular, we evaluate the parallel performance of GPU-based iterative linear solvers for this application, and we compare them to CPU-based implementations of the same solver algorithms. Simulations on real-life Berea sandstone micro-CT images are used to assess the strong scalability and computational performance of the different solver implementations and their effect on time-to-solution. Additionally, we use a Poisson problem to further characterize achievable strong and weak scalability of the GPU-based solvers in reproducible experiments. Our experiments show that GPU-based iterative solvers can greatly reduce time-to-solution in complex pore-scale simulations. On the other hand, strong scalability is currently limited by the unbalanced computing capacities of the host and the GPUs. The experiments with the Poisson problem indicate that GPU-based iterative solvers are efficient when weak scalability is desired. Our findings show that proper utilization of GPUs can help to make our two-phase pore-scale flow simulation computationally feasible in existing workflows.","PeriodicalId":246878,"journal":{"name":"Day 2 Thu, April 11, 2019","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125265901","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Comprehensive Adaptive Forecasting Framework for Optimum Field Development Planning","authors":"A. Salehi, Gill Hetz, Feyisayo Olalotiti, N. Sorek, H. Darabi, D. Castineira","doi":"10.2118/193914-MS","DOIUrl":"https://doi.org/10.2118/193914-MS","url":null,"abstract":"\u0000 An integral aspect of smart reservoir management of oil and gas fields is the process of identifying and performance forecasting of the remaining, feasible, and actionable field development opportunities (FDOs). In the present work, we introduce an adaptive full-physics simulation-based forecasting framework that applies a series of cutting-edge technologies to provide short- and long-term forecasts for both field- and well-level performance. Our workflow can be applied to a comprehensive opportunities inventory including behind-pipe recompletion, infill drilling, and sidetrack opportunities. In our approach, we begin with a model order reduction technique, which involves a parsimonious elimination of redundancies existing in a given geologic model. This involves an adaptive model upscaling strategy that retains fine details in the vicinity of critical geological features by locally varying the resulting model grid resolution. Reduced models, which are validated using streamline-based flow metrics, are passed into an automated sensitivity study and model calibration engine for efficient reconciliation of observed production trends in the field. Here, we apply a recently proposed Ensemble Smoother robust Levenberg- Marquardt (ES-rLM) method to generate plausible model realizations that replicate the reservoir energy. Representative models are further improved in a sensitivity-based local inversion step to match multiphase production data at the well level. An approach alternative to streamlines, which is compliant with a general unstructured grid format, is utilized to directly compute production data sensitivities on the underlying grid in the local inversion module. Finally, calibrated models are directly passed to the optimization and forecasting engine to assess and optimize field opportunities and development scenarios. This framework has been successfully applied to several giant mature assets in the Middle East, North America, and South America. A case study for one of the giant reservoirs in Latin America is presented where hundreds of field development opportunities are initially identified. We then apply our forecasting framework to the various scenarios including all opportunities to deliver the optimum field development plan. We propose a systematic workflow for field-scale modeling and optimization using an adaptive framework. Our approach facilitates a flexible framework to rapidly generate reliable forecasts and quantify associated uncertainties in a robust manner. This advantage in flexibility and robustness is tied to our fast and automated two-stage model calibration module that leads to substantial savings in computational time. This makes it an efficient method for quantifying the uncertainty as demonstrated through improved estimation of the faults’ connectivity, permeability distribution, fluid saturation evolution, and swept volume.","PeriodicalId":246878,"journal":{"name":"Day 2 Thu, April 11, 2019","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127655667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Identifiability of Model Discrepancy Parameters in History Matching","authors":"M. H. Rammay, A. Elsheikh, Yan Chen","doi":"10.2118/193838-MS","DOIUrl":"https://doi.org/10.2118/193838-MS","url":null,"abstract":"\u0000 In this work, we investigate different approaches for history matching of imperfect reservoir models while accounting for model error. The first approach (base case scenario) relies on direct Bayesian inversion using iterative ensemble smoothing with annealing schedules without accounting for model error. In the second approach the residual, obtained after calibration, is used to iteratively update the covariance matrix of the total error, that is a combination of model error and data error. In the third approach, PCA-based error model is used to represent the model discrepancy during history matching. However, the prior for the PCA weights is quite subjective and is generally hard to define. Here the prior statistics of model error parameters are estimated using pairs of accurate and inaccurate models. The fourth approach, inspired from Köpke et al. (2017), relies on building an orthogonal basis for the error model misfit component, which is obtained from difference between PCA-based error model and corresponding actual realizations of prior error. The fifth approach is similar to third approach, however the additional covariance matrix of error model misfit is also computed from the prior model error statistics and added into the covariance matrix of the measurement error. The sixth approach, inspired from Oliver and Alfonzo (2018), is the combination of second and third approach, i.e. PCA-based error model is used along with the iterative update of the covariance matrix of the total error during history matching. Based on the results, we conclude that a good parameterization of the error model is needed in order to obtain good estimate of physical model parameters and to provide better predictions. In this study, the last three approaches (i.e. 4, 5, 6) outperform the others in terms of the quality of the estimated parameters and the prediction accuracy (reliability of the calibrated models).","PeriodicalId":246878,"journal":{"name":"Day 2 Thu, April 11, 2019","volume":"144 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134466849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Modeling of Cosolvents in a Fully-Implicit Surfactant Flood Simulator Using the Three-Level Framework","authors":"Choongyong Han, Xundan Shi, Yih-Bor Chang, Christian Wolfsteiner, B. Guyaguler","doi":"10.2118/193913-MS","DOIUrl":"https://doi.org/10.2118/193913-MS","url":null,"abstract":"\u0000 Cosolvents are commonly injected along with surfactants for successful enhanced oil recovery as they help control aqueous stability, salinity gradient, and microemulsion phase viscosity. Therefore, modeling capability for numerical simulation of cosolvent injection is essential in helping design optimal surfactant floods. Also, the numerical implementation in the simulator should be fully implicit, fully coupled, and highly-scalable to enable full-field models and the higher resolutions often required by chemical flood simulations.\u0000 We propose a novel numerical approach to model cosolvents in a fully implicit, fully coupled, parallel, four-phase surfactant flood simulator using the three-level (phase/pseudocomponent/pure component) framework. Three pseudoalcohol components are introduced to the framework for efficient modeling of surfactant phase behavior with alcohols that are partitioned to pseudooil, pseudowater, and pseudosurfactant, respectively. They consist of pure alcohol components which are partitioned to the same pseudocomponent and are distributed to phases as required by the phase behavior equations. New nonlinear solution variables of concentrations are proposed to model transport of pure alcohols, their partitioning into pseudcomponents, and distribution of the pseudoalcohols to phases, along with corresponding equations. The physical properties critical for surfactant flood simulation such as interfacial tension, phase relative permeability, viscosity, and mass density are extended to consider the effect of alcohols.\u0000 It is shown that the new numerical approach significantly simplifies implementation of the cosolvent simulation functionality. This is because time consuming and error prone conversion between variables and derivatives, and local iterative solve for the concentrations, are not needed. This simplification enables us to significantly reduce implementation efforts, even within the fully implicit, fully coupled framework. The implementation is validated with various test cases against a widely referenced chemical flood simulator. A large-scale surfactant/polymer flood case with cosolvent injection is successfully simulated with all the important physical processes modeled, with the simulator exhibiting good performance.\u0000 Large field scale, four-phase chemical flood simulations with surfactant phase behavior with cosolvents are now practically achievable with the novel numerical approach using the three-level framework without compromising comprehensive physics.","PeriodicalId":246878,"journal":{"name":"Day 2 Thu, April 11, 2019","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133064177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Simulation of Radio Frequency Heating of Heavy Oil Reservoir Using Multi-Physics Coupling of Reservoir Simulation with Electromagnetic Solver","authors":"Gary Li, X. Guan, Hanming Wang, S. Du, Dagang Wu, Ji Chen","doi":"10.2118/193836-MS","DOIUrl":"https://doi.org/10.2118/193836-MS","url":null,"abstract":"\u0000 Steam-Assisted Gravity Drainage (SAGD) is one of the popular methods for heavy oil production. The process is efficient and economical. However, it requires the use of large quantity of water and disposal of waste water can be costly. In addition, burning of natural gas for steam generation contributes to additional carbon dioxide generation, a known greenhouse gas, which is also undesirable. A method to heat up the in-situ oil without the use of injected water is highly desirable. Radio frequency (RF) heating of heavy oil reservoir is a potential method for oil recovery without steam injection. The evaluation of the potential of such method requires the coupling of a reservoir simulator with an electromagnetic (EM) simulator.\u0000 This paper describes the development and implementation of a flexible interface in a reservoir simulator that allows the runtime loading of third party software libraries with additional physics. Data is exchanged between the reservoir simulator and externally loaded software libraries through memory, therefore there is minimal communication overhead. The implementation allows for iterative coupling, explicit coupling and periodic coupling. This paper describes the mathematical coupling of the mass and energy conservation equations in the reservoir simulator with the Maxwell equations in an external library. The electromagnetic properties in the reservoir are highly dependent on temperature and water saturation, this dependence is accounted for in the coupled code using table look-up properties.\u0000 Canadian heavy oil and reservoir properties were used in our simulation investigation. We found that RF heating alone can be effective in heating up the in-situ water and reducing heavy oil viscosity by several orders of magnitude. As the in-situ water near wellbore was vaporized by RF heating, electrical conductivities were reduced to zero and thus allowed the EM wave to propagate further into the formation and heat up the water further away from the wellbore. With properly designed RF heating field pilots and tuning of EM and reservoir parameters, the coupled reservoir/EM simulator can be a powerful tool for the evaluation and optimization of RF heating operations.\u0000 The interface is sufficiently flexible to allow different types of multi-physics coupling. In addition to RF heating, it has also been used for reaction kinetics and geomechanics coupling with a reservoir simulator. It has been used for large scale coupled full field simulation with over 30 million cells.","PeriodicalId":246878,"journal":{"name":"Day 2 Thu, April 11, 2019","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132895503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"System-AMG for Fully Coupled Reservoir Simulation with Geomechanics","authors":"S. Gries, B. Metsch, K. Terekhov, P. Tomin","doi":"10.2118/193887-MS","DOIUrl":"https://doi.org/10.2118/193887-MS","url":null,"abstract":"\u0000 The consideration of geomechanical effects is becoming more and more important in reservoir simulations. Ensuring stable simulation processes often enough requires handling the entire process with all types of physical unknowns fully implicitly. However, the resulting fully coupled linear systems pose challenges for linear solvers. The number of approaches that can efficiently handle a fully coupled system is extremely limited.\u0000 System-AMG has demonstrated its efficiency for isothermal and thermal reservoir simulations. At the same time, AMG is known to be a robust and highly efficient linear solver for mere linear elasticity problems. This paper will discuss the combination of the advantages that AMG approaches have for both types of physics. This results in a robust and efficient solution scheme for the fully coupled linear system. The Automatic Differentiation General Purpose Research Simulator (AD-GPRS) is used to produce the Jacobians that are guaranteed to be exact.\u0000 In a single-phase case, the overall Jacobian matrix takes the form of a constrained linear elasticity system where the flow unknowns serve as a Lagrangian multiplier. In other words, a saddle point system needs to be solved, where the flow and the mechanics problem might come at very different scales. A natural relaxation method for this kind of systems is given by Uzawa smoothing schemes which provide a way to overcome the difficulties that other smoothers may encounter.\u0000 This approach appears intuitive for single-phase problems, where Gauss-Seidel can be applied in an inexact Uzawa scheme. However, in the multiphase case, incomplete factorization smoothers are required for the flow and transport part. We will discuss the incorporation in an inexact Uzawa scheme, where different realizations are possible, with different advantages and disadvantages. Finally, we propose an adaptive mechanism along with the outer Krylov solver to detect the best-suited realization for a given linear system. In the multiphase case, also the matrix preprocessing, for instance, by Dynamic Row Summing, needs to be considered. However, the process now also needs to reflect the requirements of the Uzawa scheme to be applicable.\u0000 We demonstrate the performance for widely used test cases as well as for real-world problems of practical interest.","PeriodicalId":246878,"journal":{"name":"Day 2 Thu, April 11, 2019","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127021815","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Natural Variable Well Model for Advanced Thermal Simulation","authors":"Yifan Zhou, Gary Li, V. J. Zapata","doi":"10.2118/193835-MS","DOIUrl":"https://doi.org/10.2118/193835-MS","url":null,"abstract":"\u0000 For numerical reservoir simulation, the well model has always been a critical component that can have significant impact on the results and performance of the simulation. A new well model has been developed in a commercially available simulator to provide additional capabilities and improved robustness for advanced thermal simulation.\u0000 A Natural Variable (NV) formulation, similar to that used in the reservoir solution, has been adopted for the new well model. The NV formulation enables the well model to reuse many of the reservoir solution computations hence allows for rapidly adding support of new features as they get implemented in the reservoir solution. In addition, to model the steam injection process more accurately, we adopted full-upstream weighted mobility for injection connections, for which the amount of steam injected depends on the wellbore instead of reservoir cell condition. The NV well model also supports advanced features such as thermal multi-segment wells with loops for modeling annular flow, thermal drift flux model for counter-current flow, and dynamic coefficients for conductive heat transfer.\u0000 We present numerical results using real field data to demonstrate the new capabilities. Comparisons between the NV well model and the original Mass Variable (MV) well model, as well as between the full-upstream weighted and traditional cell voidage injection mobility are included. For cases using cell-voidage mobility and no advanced features, both well models produce similar results. On the other hand, for the cases tested using recommended full-upstream weighted mobility and advanced features such as thermal multi-segment wells with drift flux, and dynamic heat transfer, NV well model produces more stable results with superior convergence behavior. We also observed that, compared with cell-voidage mobility, full-upstream weighted mobility yields more realistic (higher) injectivity, which is critical in modeling steam injection processes.\u0000 With NV well model and its advanced features, we can obtain more efficient, accurate, and robust performance predictions for thermal recovery processes for better reservoir management of heavy oil fields. In addition, algorithmic reuse of reservoir calculations within the reservoir simulator enable easier extension of the well-model to support additional complex physics.","PeriodicalId":246878,"journal":{"name":"Day 2 Thu, April 11, 2019","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132469011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ranking Fractured Reservoir Models Using Flow Diagnostics","authors":"V. Spooner, S. Geiger, D. Arnold","doi":"10.2118/193861-MS","DOIUrl":"https://doi.org/10.2118/193861-MS","url":null,"abstract":"\u0000 This paper describes the application and testing of innovative dual porosity flow diagnostics to quantitatively rank large ensembles of fractured reservoir models. Flow diagnostics can approximate the dynamic response of multi-million cell models in seconds on standard hardware. The need for new faster screening methods stems from the challenge of making robust forecasts for naturally fractured carbonate reservoirs. First order uncertainties including the distribution and properties of natural fractures, matrix heterogeneity and wettability can all negatively impact on recovery. A robust multi-realisation approach to production forecasting is often rendered impractical due to the time cost for simulating many models.\u0000 We have extended existing flow diagnostics techniques to dual porosity systems by accounting for the matrix-fracture exchange. New metrics combine the transfer rate with the advective time of flight in the fractures identifying risk factors for early water breakthrough and providing quantitative measures of dynamic heterogeneity.\u0000 We have compared ranking a large ensemble of synthetic fractured reservoir models using dual porosity flow diagnostics and using full-physics simulation. The synthetic ensemble explores a number of different geological concepts around the fracture distributions, wettability and matrix heterogeneity which can. Not only does the flow diagnostic ranking agree well with the cumulative oil ranking the run time for the flow diagnostics is <0.25% of the total simulation time. This significant reduction in the time to compare models allows more time to spend running full physics simulation on the important and geologically diverse cases that offer the most insight.","PeriodicalId":246878,"journal":{"name":"Day 2 Thu, April 11, 2019","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129162613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"INSIM-FT-3D: A Three-Dimensional Data-Driven Model for History Matching and Waterflooding Optimization","authors":"Zhenyu Guo, A. Reynolds","doi":"10.2118/193841-MS","DOIUrl":"https://doi.org/10.2118/193841-MS","url":null,"abstract":"\u0000 We previously published a two-dimensional data-driven model (INSIM-FT) for history matching waterflooding production data and to identify flow barriers and regions of high connectivity between injector-producer pairs. This two-dimensional INSIM model assumed vertical wells. The history-matched models can be used for prediction of waterflooding performance and life-cycle waterflooding optimization. The INSIM-FT-3D model presented here extends INSIM-FT to three dimensions, considers gravity and enables the use of arbitrary well trajectories. INSIM-FT-3D places nodes at each well perforation and then adds nodes throughout the reservoir. Flow occurs through \"streamtubes\" between each pair of connected nodes. Mitchell's best candidate algorithm is used to place nodes and a three-dimensional (3D) connection map is generated with Delaunay triangulation. Pressures and saturations at nodes, respectively, are obtained from IMPES-like pressure equations and a Riemann solver that include gravity effects. With history-matched model(s) as the forward model(s), we estimate the optimal well controls (pressure or rates at control steps) that maximize the life-cycle net-present-value (NPV) of production under waterflooding using a gradient-based method that employs a stochastic gradient. Two 3D reservoirs are considered to establish the viability of using INSIM-FT-3D history-matched models for waterflooding optimization, a channelized reservoir and the Brugge reservoir. Unlike history-matching and waterflooding optimization based on reservoir simulation models, INSIM-FT-3D is not a detailed geological model. Moreover, the time required to run INSIM-FT-3D is more than one order of magnitude less the cost of running a comparable reservoir simulation model.","PeriodicalId":246878,"journal":{"name":"Day 2 Thu, April 11, 2019","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115956092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Three-Dimensional Symmetric Positive Definite Control-Volume Distributed Multi-Point Flux Approximation for Flow Computation on Tetrahedral Grids","authors":"Raheel Ahmed, M. Edwards","doi":"10.2118/193919-MS","DOIUrl":"https://doi.org/10.2118/193919-MS","url":null,"abstract":"\u0000 A three-dimensional symmetric positive definite (SPD) cell-centred control-volume distributed multi-point flux approximation (CVD-MPFA) is presented for porous media flow simulation on tetrahedral grids. The scheme depends on a single degree of freedom per control-volume and is derived in physical space where the continuous fluxes are resolved directly along the face normals of the tetrahedra. We believe this is the first and possibly only general three-dimensional full-tensor finite-volume scheme that is flux- continuous and SPD in physical space, while depending on a single degree of freedom per control-volume. The equivalent general two-dimensional CVD-MPFA scheme that is SPD in physical space is presented in (Friis et al., 2008).","PeriodicalId":246878,"journal":{"name":"Day 2 Thu, April 11, 2019","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116792223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}