Multiscale Mimetic Solvers for Efficient Streamline Simulation of Fractured Reservoirs

Natvig, Jostein R.; Skaflestad, Bård; Bratvedt, Frode; Bratvedt, Kyrre; Lie, Knut–Andreas; Laptev, Vsevolod; Khataniar, Sanjoy Kumar

doi:10.2118/119132-pa

Cited by 52 publications

(13 citation statements)

References 20 publications

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“…When free-flow regions are confined to a single coarse block, the multiscale method delivers qualitatively correct solutions with good accuracy. If the free-flow regions extend beyond a single coarse block, the multiscale method is able to reproduce major parts of the flow patterns, but to get all small details correct, one may have to introduce an adaptive coarsening in which free-flow regions are represented as extra coarse blocks (as discussed in by Aarnes et al (2006) for long-correlation shale objects and by Natvig et al (2009) for fracture corridors). This is a topic of ongoing research.…”

Section: Discussionmentioning

confidence: 99%

“…8 this amounts to splitting block i into three blocks that all are coupled with block j and together take care of the flow in the fracture and in the porous region above and below the horizontal fracture. The efficiency of this approach has been demonstrated by Natvig et al (2009), but has so far not been implemented in our simple Darcy/Stokes-Brinkman multiscale solver.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Multiscale methods have proved capable of handling industry-standard complexity both with respect to grid representation ) and flow physics (Lee et al 2008;Zhou and Tchelepi 2008;Hajibeygi et al 2008;Krogstad et al 2009). Natvig et al (2009) recently demonstrated how multiscale methods can be used to simulate geological models with fracture corridors modeled as volumetric objects with high permeability. To date, however, multiscale methods have not been applied to simulate flow in naturally-fractured and vuggy media using a multiphysics approach with different flow models on the fine and the coarse scale.…”

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A Multiscale Mixed Finite Element Method For Vuggy and Naturally Fractured Reservoirs

2009

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Vugs, caves, and fractures can significantly alter the effective permeability of carbonate reservoirs and should be accurately accounted for in a geomodel. Accurate modeling of the interaction between free-flow and porous regions is essential for flow simulations and detailed production engineering calculations. However, flow simulation of such reservoirs is very challenging because of the co-existence of porous and free-flow regions on multiple scales that need to be coupled.Multiscale methods are conceptually well-suited for this type of modeling as they allow varying resolution and provide a systematic procedure for coarsening and refinement. However, to date there are hardly no multiscale methods developed for problems with both free-flow and porous regions. Our work is a first step to make a uniform multiscale framework, where we develop a multiscale mixed finite-element (MsMFE) method for detailed modeling of vuggy and naturally-fractured reservoirs. The MsMFE method uses a standard Darcy model to approximate pressure and fluxes on a coarse grid, but captures fine-scale effects through basis functions determined from numerical solutions of local Stokes-Brinkman flow problems on the underlying fine-scale geocellular grid. The Stokes-Brinkman equations give a unified approach to simulating free-flow and porous regions using a single system of equations, avoid explicit interface modeling, and reduce to Darcy or Stokes flow by appropriate choices of parameters.In the paper, the MsMFE solutions are compared with fine-scale Stokes-Brinkman solutions for test cases including both short-and long-range fractures. The results demonstrate how fine-scale flow in fracture networks can be represented within a coarse-scale Darcy flow model by using multiscale elements computed solving the Stokes-Brinkman equations. The results indicate that the MsMFE method is a promising path toward direct simulation of highly detailed geocellular models of vuggy and naturally-fractured reservoirs. IntroductionNaturally fractured and carbonate reservoirs are composed of porous material, but will typically also contain relatively large void spaces in the form of fractures, small cavities, and caves, which are called vugs in the geological literature. Flow simulation of such formations is very challenging because of the co-existence of porous and free-flow domains on multiple scales that require coupling (Wu et al. 2006).The Darcy-Stokes equations have been used to model industrial infiltration processes and coupled surface and subsurface flow, for which the porous and the free-flow domains are well separated. The Darcy-Stokes model consists of Darcy's law combined with mass conservation in the porous subdomain and the Stokes equations in the free-flow subdomain. To close the model, one must specify conditions on the interface between the Darcy and Stokes subdomains. All these conditions require continuity of mass and momentum over the interface, but differ in the way they allow the tangential component to jump across the interface.In a ca...

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Section: Discussionmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

mentioning

confidence: 99%

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A Multiscale Mixed Finite Element Method For Vuggy and Naturally Fractured Reservoirs

2009

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“…The coarse grid (thick lines) is formed based on a partition of the fine grid (thin lines) so that each coarse block B j consists of a connected set of cells from the fine grid. Each block can, in principle, have arbitrary shape, but the best numerical accuracy is obtained if the blocks are somewhat regular, follow the layered structures of stratigraphic grids [10], and/or adapt to high-contrast features [30,31].…”

Section: Multiscale Approximationmentioning

confidence: 99%

Validation of the multiscale mixed finite‐element method

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Lie

et al. 2014

Numerical Methods in Fluids

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SUMMARYSubsurface reservoirs generally have a complex description in terms of both geometry and geology. This poses a continuing challenge in modeling and simulation of petroleum reservoirs owing to variations of static and dynamic properties at different length scales. Multiscale methods constitute a promising approach that enables efficient simulation of geological models while retaining a level of detail in heterogeneity that would not be possible via conventional upscaling methods. Multiscale methods developed to solve coupled flow equations for reservoir simulation are based on a hierarchical strategy in which the pressure equation is solved on a coarsened grid and the transport equation is solved on the fine grid, and the two equations are treated as a decoupled system. In particular, the multiscale mixed finite-element (MsMFE) method attempts to capture subgrid geological heterogeneity directly into the coarse-scale equations via a set of numerically computed basis functions. These basis functions are able to capture the predominant multiscale information and are coupled through a global formulation to provide good approximation of the subsurface flow solution. In the literature, the general formulation of the MsMFE method for incompressible two-phase and compressible three-phase flow has mainly addressed problems with idealized flow physics. In this paper, we first outline a recent formulation that accounts for compressibility, gravity, and spatially dependent rockfluid parameters. Then, we validate the method by evaluating its computational efficiency and accuracy on a series of representative benchmark tests that have a high degree of realism with respect to flow physics, heterogeneity in the petrophysical models, and geometry/topology of the corner-point grids. In particular, the MsMFE method is validated and compared against an industry-standard fine-scale solver. The fine-scale flux, pressure, and saturation fields computed by the multiscale simulation show a noteworthy improvement in resolution and accuracy compared with coarse-scale models.

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“…To solve the transport without upscaling-using the finescale, approximate fluxes-the best choice is probably a streamline method, e.g. as described in [2,20], or one can use similar operator-splitting techniques to develop highly efficient finite-volume solvers [18,19] that use flow information to obtain an optimal ordering of the nonlinear discrete transport equations so that these can be solved in a cell-by-cell or block-by-block fashion. This gives local control over the (computationally expensive) nonlinear iterations and can significantly reduce the computational cost compared with standard (implicit) finite-volume methods.…”

Section: Introductionmentioning

confidence: 99%

Flow-based coarsening for multiscale simulation of transport in porous media

2011

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Geological models are becoming increasingly large and detailed to account for heterogeneous structures on different spatial scales. To obtain simulation models that are computationally tractable, it is common to remove spatial detail from the geological description by upscaling. Pressure and transport equations are different in nature and generally require different strategies for optimal upgridding. To optimize the accuracy of a transport calculation, the coarsened grid should generally be constructed based on a posteriori error estimates and adapt to the flow patterns predicted by the pressure equation. However, sharp and rigorous estimates are generally hard to obtain, and herein we therefore consider various ad hoc methods for generating flow-adapted grids. Common for all is that they start by solving a single-phase flow problem once and then continue to form a coarsened grid by amalgamating cells from an underlying fine-scale grid. We present several variations of the original method. First, we discuss how to include a priori information in the coarsening process, e.g. to adapt to special geological features or to obtain less irregular grids in regions where flowadaption is not crucial. Second, we discuss the use of bi-directional versus net fluxes over the coarse blocks and show how the latter gives systems that better represent the causality in the flow equations, which can be exploited to develop very efficient nonlinear solvers. Finally, we demonstrate how to improve simulation accuracy by dynamically adding local resolution near strong saturation fronts.

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Multiscale Mimetic Solvers for Efficient Streamline Simulation of Fractured Reservoirs

Cited by 52 publications

References 20 publications

A Multiscale Mixed Finite Element Method For Vuggy and Naturally Fractured Reservoirs

A Multiscale Mixed Finite Element Method For Vuggy and Naturally Fractured Reservoirs

Validation of the multiscale mixed finite‐element method

Flow-based coarsening for multiscale simulation of transport in porous media

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