Auxiliary variables for 3D multiscale simulations in heterogeneous porous media

Sandvin, Andreas; Keilegavlen, Eirik; Nordbotten, Jan Martin

doi:10.1016/j.jcp.2012.12.016

Cited by 8 publications

(5 citation statements)

References 30 publications

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“…The multiscale method itself can be generalized to N-spatial dimensions [83]. Note however that the quality of the coarse discretization may degenerate as the number of dimensions increase [96]. In three dimensions the fractures, now as planes, still form natural boundary conditions for the local problems and represent the global connections in the domain.…”

Section: Further Directionsmentioning

confidence: 99%

An efficient multi-point flux approximation method for Discrete Fracture–Matrix simulations

Sandve

Berre

Nordbotten

2012

Journal of Computational Physics

266

158

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Section: Further Directionsmentioning

confidence: 99%

An efficient multi-point flux approximation method for Discrete Fracture–Matrix simulations

Sandve

Berre

Nordbotten

2012

Journal of Computational Physics

266

158

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“…The model contains two formations: The Tarbert formation at the top of the model represents a prograding near-shore formation, which even though it exhibits orders of magnitude variations in the permeabilities, is relatively smooth and tends to be well resolved by most multiscale methods. The Upper Ness formation in the lower part of the model is fluvial and has proved to be more challenging, in particular for finite-volume type multiscale methods that rely on some form of pressure extrapolation for localization, see e.g., [38,48,56] and references therein.…”

Section: Spe10: Horizontal Layersmentioning

confidence: 99%

A Multiscale Restriction-Smoothed Basis Method for Compressible Black-Oil Models

Møyner

Lie

2016

SPE Journal

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Simulation problems encountered in reservoir management are often computationally expensive because of the complex fluid physics for multiphase flow and the large number of grid cells required to honor geological heterogeneity. Multiscale methods have been proposed as a computationally inexpensive alternative to traditional fine-scale solvers for computing conservative approximations of the pressure and velocity fields on high-resolution geo-cellular models. Although a wide variety of such multiscale methods have been discussed in the literature, these methods have not yet seen widespread use in industry. One reason may be that no method has been presented so far that handles the combination of realistic flow physics and industrystandard grid formats in their full complexity. Herein, we present a multiscale method that handles both the most widespread type of flow physics (black-oil type models) and standard grid formats like corner-point, stair-stepped, PEBI, as well as general unstructured, polyhedral grids. Our approach is based on a finite-volume formulation in which the basis functions are constructed using restricted smoothing to effectively capture the local features of the permeability. The method can also easily be formulated for other types of flow models, provided one has a reliable (iterative) solution strategy that computes flow and transport in separate steps. The proposed method is implemented as open-source software and validated on a number of two and three-phase test cases with significant compressibility and gas dissolution. The test cases include both synthetic models and models of real fields with complex wells, faults, and inactive and degenerate cells. Through a prescribed tolerance, the solver can be set to either converge to a sequential or the fully implicit solution, in both cases with a significant speedup compared to a fine-scale multigrid solver. Altogether, this ensures that one can easily and systematically trade accuracy for efficiency, or vice versa.

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“…Whereas the solution quality is generally very good for both solvers for Tarbert, MsRSB clearly outperforms MsFV on Upper Ness. Several authors have independently shown that the MsFV method has issues with coarse scale stability in the presence of channelized, high-contrast formations and will suffer from strong unphysical oscillations that may prevent iterative versions of the method from converging properly, see e.g., (Sandvin et al 2013;Wang et al 2014;Møyner and Lie 2014a) and references therein. MsRSB is much more robust and does not suffer from such problems and therefore gives solutions of similar accuracy for the smooth and channelized layers.…”

Section: Accuracy For Single-phase Flowmentioning

confidence: 99%

A Multiscale Method Based on Restriction-Smoothed Basis Functions Suitable for General Grids in High Contrast Media

Møyner

Lie

2015

SPE Reservoir Simulation Symposium

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A wide variety of multiscale methods have been proposed in the literature to improve simulation runtimes and provide better scaling to large models. With a few notable exceptions, the methods proposed so far are mostly limited to structured grids. We present a new multiscale restricted-smoothed basis (MsRSB) method that is designed to be applicable to stratigraphic and fully unstructured grids. Like many other multiscale methods, it is based on a coarse partition of an underlying fine grid with a set of prolongation operators (also called multiscale basis functions) that map from unknowns associated with the fine grid cells to unknowns associated with the coarse grid blocks. These mappings are constructed by restricted smoothing: starting from a constant, a localized iterative scheme is applied directly to the fine-scale discretization to give prolongation operators that are consistent with the local properties of the differential operators. The resulting method has three main advantages: First, there are almost no requirements on the geometry and topology of the fine and the coarse grids. Coarse partitions and good prolongation operators can therefore easily be constructed on complex models involving high media contrasts and unstructured cell connections introduced by faults, pinch-outs, erosion, local grid refinement, etc. Moreover, the coarse grid can easily be adapted to features in the geo-cellular model or any precomputed flow field to improve accuracy. Secondly, the method is accurate and robust when compared to existing multiscale methods. In particular, the method does not need to recompute local basis functions to account for transient behavior: dynamic mobility changes are incorporated by continuing previous iterations with a few extra steps. This way, the cost of updating the prolongation operators will be proportional to the amount of change in fluid mobility and one avoids tolerance-based updates. Finally, since the MsRSB method is formulated on top of a cell-centered, conservative, finite-volume method, it is applicable to any flow model in which one can isolate a pressure equation; in the paper, we discuss incompressible two-phase flow and compressible, three-phase, black-oil type models. For our fine-scale discretization, we use the standard two-point flux-approximation scheme, but the method could equally well have been formulated using a multipoint discretization.Several numerical examples are presented to highlight features of the method. First, we compare the MsRSB method with the multiscale finite-volume (MsFV) method for single-phase flow problems with petrophysical parameters from the SPE 10 benchmark. Then we perform several validation studies using two-phase flow geometry and petrophysical properties from simulation models of two real fields (Gullfaks and Norne), as well as a compressible gas-injection case described by the black-oil equations.

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Auxiliary variables for 3D multiscale simulations in heterogeneous porous media

Cited by 8 publications

References 30 publications

An efficient multi-point flux approximation method for Discrete Fracture–Matrix simulations

An efficient multi-point flux approximation method for Discrete Fracture–Matrix simulations

A Multiscale Restriction-Smoothed Basis Method for Compressible Black-Oil Models

A Multiscale Method Based on Restriction-Smoothed Basis Functions Suitable for General Grids in High Contrast Media

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