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Collaborating Authors
Accurate Two-Phase Flow Simulation in Faulted Reservoirs by Combining Two-Point Flux Approximation and Mimetic Finite Difference Methods
Dong, Rencheng (The University of Texas at Austin (Corresponding author)) | O. Alpak, Faruk (Shell International Exploration and Production Inc.) | F. Wheeler, Mary (The University of Texas at Austin)
Summary Faulted reservoirs are commonly modeled by corner-point grids (CPGs). Because the two-point flux approximation (TPFA) method is not consistent on non-K-orthogonal grids, multi-phase flow simulation using TPFA on CPGs may have significant discretization errors if grids are not K-orthogonal. We developed a novel method to improve the simulation accuracy where the faults are modeled by polyhedral cells, and mimetic finite difference (MFD) methods are used to solve flow equations. We use a cut-cell approach to build the mesh for faulted reservoirs. A regular K-orthogonal grid is first constructed, and then cells are divided where fault planes are present. Most cells remain K-orthogonal while irregular non-K-orthogonal polyhedral cells can be formed with multiple cell divisions. We investigated three spatial discretization methods for solving the pressure equation on general polyhedral grids, including the TPFA, MFD, and TPFA-MFD hybrid methods. In the TPFA-MFD hybrid method, the MFD method is only applied to the part of the domain with severe grid non-K-orthogonality, while the TPFA method is applied to the rest of the domain. We compare flux accuracy between TPFA and MFD methods by solving a single-phase flow problem. The reference solution is obtained on a rectangular grid, while the same problem is solved by TPFA and MFD methods on a grid with non-K-orthogonal cells near a fault. Fluxes computed using TPFA exhibit larger errors in the vicinity of the fault, while fluxes computed using MFD are still as accurate as the reference solution. We also compare saturation accuracy for two-phase (oil and water) flow in faulted reservoirs when the pressure equation is solved by different discretization methods. Compared with the reference saturation solution, saturation exhibits non-physical errors near the fault when the pressure equation is solved by the TPFA method. Because the MFD method yields accurate fluxes over general polyhedral grids, the resulting saturation solutions agree with reference saturation solutions with an enhanced accuracy when the pressure equation is solved by the MFD method. Based on the results of our simulation studies, we observe that the accuracy of the TPFA-MFD hybrid method is very close to the accuracy of the MFD method, while the TPFA-MFD hybrid method is computationally cheaper than the MFD method.
accuracy, Cartesian grid, displacement, equation, flow in porous media, Fluid Dynamics, fracture characterization, grid, Magnitude, matrix, method, mimetic finite difference method, Modeling & Simulation, multiphase flow, permeability, polyhedral grid, Reservoir Characterization, reservoir simulation, seismic surveying, setup, SPE Journal, TPFA, Upstream Oil & Gas
Country:
- Europe (0.93)
- North America > United States > Texas (0.46)
- Asia > Middle East > UAE (0.28)
SPE Disciplines:
Abstract Faulted reservoirs are commonly modeled by corner-point grids. Since the two-point flux approximation (TPFA) method is not consistent on non-orthogonal grids, multi-phase flow simulation using TPFA on corner-point grids may have significant discretization errors if grids are not K-orthogonal. To improve the simulation accuracy, we developed a novel method where the faults are modeled by polyhedral cells, and mimetic finite difference (MFD) methods are used to solve flow equations. We use a cut-cell approach to build the mesh for faulted reservoirs. A regular orthogonal grid is first constructed,and then fault planes are added by dividing cells at fault planes. Most cells remain orthogonal while irregular non-orthogonal polyhedral cells can be formed with multiple cell divisions. We investigated three spatial discretization methods for solving the pressure equation on general polyhedral grids, including the TPFA, MFD and TPFA-MFD hybrid methods. In the TPFA-MFD hybrid method, the MFD method is only applied to part of the domain while the TPFA method is applied to rest of the domain. We compared flux accuracy between TPFA and MFD methods by solving a single-phase flow problem. The reference solution is obtained on a rectangular grid while the same problem is solved by TPFA and MFD methods on a grid with distorted cells near a fault. Fluxes computed using TPFA exhibit larger errors in the vicinity of the fault while fluxes computed using MFD are still as accurate as the reference solution. We also compared saturation accuracy of two-phase (oil and water) flow in faulted reservoirs when the pressure equation is solved by different discretization methods. Compared with the reference saturation solution, saturation exhibits non-physical errors near the fault when pressure equation is solved by the TPFA method. Since the MFD method yields accurate fluxes over general polyhedral grids, the resulting saturation solutions match the reference saturation solutions with an enhanced accuracy when the pressure equation is solved by the MFD method. Based on the results of our simulation studies, the accuracy of the TPFA-MFD hybrid method is very close to the accuracy of the MFD method while the TPFA-MFD hybrid method is computationally cheaper than the MFD method.
accuracy, Cartesian grid, flow in porous media, flow simulation, Fluid Dynamics, fracture characterization, grid, MFD method, Modeling & Simulation, multiphase flow, polyhedral cell, polyhedral grid, pressure equation, Reservoir Characterization, reservoir simulation, saturation solution, tpfa method, tpfa-mfd hybrid method, two-phase flow problem, Upstream Oil & Gas, vector, velocity magnitude
Country:
- Europe (0.93)
- North America > United States > Texas (0.46)
SPE Disciplines:
- Reservoir Description and Dynamics > Reservoir Simulation (1.00)
- Reservoir Description and Dynamics > Reservoir Fluid Dynamics > Flow in porous media (1.00)
- Reservoir Description and Dynamics > Reservoir Characterization > Faults and fracture characterization (1.00)
- Reservoir Description and Dynamics > Reservoir Fluid Dynamics > Multiphase flow (0.73)
Abstract Discretization methods have been developed to accompany a novel cut-cell gridding technique for reservoir simulation that preserves the orthogonality characteristic in the lateral direction. A major drawback of the cut-cell gridding method is that polyhedral cells emerge near faults that have relatively small volumes. Pragmatic but non-rigorous approximation methods have been developed in the past to merge these cells with their neighbors so that the grid representation fits the two-point flux approximation (TPFA) framework. In this work, we take a different approach and investigate the global and local applications of select consistent discretization methods in the vicinity of fault representations on cut-cell grids. We develop and test consistent discretization methods that are of low computational cost and do not require major intrusive changes to the solver structure of commercial reservoir simulators. Cell-centered methods such as multi-point flux approximation (MPFA), average multi-point flux approximation (AvgMPFA), and nonlinear two-point flux approximation (NTPFA) methods fit naturally into the framework of existing industrial-grade simulators. Therefore, we develop and test variants of the AvgMPFA and NTPFA methods that are specifically designed to operate on cut-cell grids. An implementation of the well-established but computationally expensive MPFA method is also made for cut-cell grids to serve as a reference to computations with AvgMPFA and NTPFA. All investigated methods are implemented within the framework of a full-physics 3D research simulator with a general compositional formulation, which encompasses black-oil models. We use a set of synthetic cut-cell grid models of varying complexity including conceptual models and a field-scale model. We compare the novel cut-cell adapted AvgMPFA and NTPFA simulation results in terms of accuracy and computational performance against the ones computed with reference MPFA and TPFA methods. We observe that AvgMPFA consistently yields more accurate and computationally efficient simulations than NTPFA on cut-cell grids. Moreover, AvgMPFA hybrids run faster than NTPFA hybrids when compared on the same problem for the same hybridization strategy. On the other hand, the computational performance of AvgMPFA degrades more rapidly compared to NTPFA with increasing "rings" of orthogonal blocks around cut-cells owing to its relatively wider stencil. Auspiciously, only one or two "rings" of orthogonal blocks around cut cells are sufficient with AvgMPFA to deliver high accuracy.
accuracy, Artificial Intelligence, avgmpfa, cross-section, cut-cell grid, enhanced recovery, equation of state, europe government, Figure, flow in porous media, Fluid Dynamics, flux approximation, fracture characterization, grid, Implementation, Journal, Modeling & Simulation, neighborhood, ntpfa, optimization problem, Reservoir Characterization, reservoir simulation, scaling method, seismic surveying, simulator, stencil, united states government, Upstream Oil & Gas, water saturation cross-section, waterflooding
Country:
- Europe (1.00)
- North America > United States > Texas (0.46)
SPE Disciplines:
- Reservoir Description and Dynamics > Reservoir Fluid Dynamics > Flow in porous media (1.00)
- Reservoir Description and Dynamics > Reservoir Characterization > Faults and fracture characterization (1.00)
- Reservoir Description and Dynamics > Improved and Enhanced Recovery > Waterflooding (0.93)
- (2 more...)
Technology:
Summary The representation of faults and fractures using cut-cell meshes often results in irregular non-orthogonal grids. Simple finite volume approaches fail to handle complex meshes because they are highly prone to grid orientation effects and only converges for K-orthogonal grids. Wide stencil approaches and higher order methods are computationally expensive and impractical to adopt in commercial reservoir simulators. In this work, we implement an Enriched Galerkin (EG) discretization for the flow and transport problems on non-orthogonal grids. The EG approximation space combines continuous and discontinuous Galerkin methods. The resulting solution lies in a richer space than the the two-point flux approximation (TPFA) method and allows a better flux approximation. It also resolves the inconsistencies that are usually associated with TPFA scheme. The method is tested for various non-orthogonal mesh configurations arising from different fault alignments. The performance of the scheme is also tested for reservoirs with strong anisotropy as well as reservoirs with heterogeneous material properties.
Computational physics, configuration, enriched galerkin discretization scheme, equation, Figure, flow in porous media, Fluid Dynamics, flux approximation, geometry, grid, Journal, mesh configuration, method, Modeling & Simulation, Reservoir Characterization, reservoir simulation, saturation equation, SPE Journal, united states government, Upstream Oil & Gas, Wheeler
SPE Disciplines:
Toward Accurate Reservoir Simulations on Unstructured Grids: Design of Simple Error Estimators and Critical Benchmarking of Consistent Discretization Methods for Practical Implementation
Raynaud, X. (SINTEF) | Pizzolato, A. (Eni (Corresponding author) | Johansson, A. (email: alberto.pizzolato@eni.com)) | Caresani, F. (SINTEF) | Ferrari, A. (Eni) | Møyner, O. (Eni) | Nilsen, H. M. (SINTEF) | Cominelli, A. (SINTEF) | Lie, K.-A. -A. (Eni)
Summary In this paper, we aim to identify discretization errors caused by non-K-orthogonal grids upfront through simple preprocessing tools and perform a comparative study of a set of representative, state-of-the-art, consistent discretizations [multipoint flux approximation (MPFA-O), mimetic finite difference (MFD), nonlinear two-pointflux approximation (NTPFA, TPFA), and average multipoint flux approximation (AvgMPFA)] to select the method most suited for inclusion in a commercial reservoir simulator. To predict the potential impact of discretization errors, we propose two types of error indicators. Static indicators measure the degree of nonconsistency of the two-point method at a cell level, and dynamic indicators measure how local discretization errors affect flow paths. The latter are computed using a series of idealized tracer simulations. By changing monitoring and injection points, one can mimic the reservoir-development strategy and thus focus on the errors introduced on quantities of real interest. To assess the practical usability of various consistent methods and validate our new error indicators, we use a set of representative grid models generated by contemporary commercial tools, for which we discuss static error indicators and compare tracer responses for the various discretization methods. We also compare degrees of freedom, sparsity, and the condition number of the alternative methods and discuss challenges related to their practical implementation. Our results indicate that tracer simulations constitute an efficient tool to identify and classify discretization errors and quantify their potential impact. We observe distinctively different behavior with the inconsistent two-point method and the consistent methods, which agree closely in terms of accuracy of the response. We also note a deficiency in the commercial realization of so-called Depogrids, which can result in unnecessarily complicated polytopal cells with hundreds of faces. Our overall conclusion is that NTPFA and AvgMPFA are the most viable solutions for integration into a commercial simulator, with the linear AvgMPFA method being the least invasive.
approximation, Artificial Intelligence, chemical tracer, condition number, consistent method, cut-cell grid, directional bias, discretization error, discretization method, equation, error indicator, flow in porous media, Fluid Dynamics, flux, grid, Implementation, Indicator, mismatch, Modeling & Simulation, Multiphase Simulation, pressure field, Reservoir Characterization, reservoir simulation, Simulation, TPFA, tracer test analysis, Upstream Oil & Gas
Country:
- Europe > Norway (0.46)
- Asia > Middle East (0.46)
- North America > United States > Texas (0.28)
Geophysics:
- Geophysics > Seismic Surveying (0.46)
- Geophysics > Time-Lapse Surveying > Time-Lapse Seismic Surveying (0.46)
Oilfield Places:
- Europe > Norway > Norwegian Sea > Halten Terrace > PL 128 > Block 6608/10 > Norne Field > Tofte Formation (0.99)
- Europe > Norway > Norwegian Sea > Halten Terrace > PL 128 > Block 6608/10 > Norne Field > Not Formation (0.99)
- Europe > Norway > Norwegian Sea > Halten Terrace > PL 128 > Block 6608/10 > Norne Field > Ile Formation (0.99)
- (3 more...)
SPE Disciplines:
- Reservoir Description and Dynamics > Reservoir Simulation (1.00)
- Reservoir Description and Dynamics > Reservoir Fluid Dynamics > Flow in porous media (1.00)
- Reservoir Description and Dynamics > Reservoir Characterization > Geologic modeling (1.00)
- Reservoir Description and Dynamics > Formation Evaluation & Management > Tracer test analysis (1.00)
Technology:
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