Multiscale Finite Volume Method Research Articles

An oversampling technique for the multiscale finite volume method to simulate electromagnetic responses in the frequency domain

In order to reduce the computational cost of the simulation of electromagnetic responses in geophysical settings that involve highly heterogeneous media, we develop a multiscale finite volume method with oversampling for the quasi-static Maxwell’s equations in the frequency domain. We assume a coarse mesh nested within a fine mesh that accurately discretizes the problem. For each coarse cell, we independently solve 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. The local Maxwell’s system is solved using the fine mesh contained in the extended domain and the mimetic finite volume method. Next, these local solutions (basis functions) together with a weak-continuity condition are used to construct a coarse-mesh version of the global problem. The basis functions can be used to obtain the fine-mesh details from the solution of the coarse-mesh problem. Our approach leads to a significant reduction in the size of the final system of equations and the computational time, while accurately approximating the behavior of the fine-mesh solutions. We demonstrate the performance of our method using two 3D synthetic models: one with a mineral deposit in a geologically complex medium and one with random isotropic heterogeneous media. Both models are discretized using an adaptive mesh refinement technique.

FV-MHMM method for reservoir modeling

The present paper proposes a new family of multiscale finite volume methods. These methods usually deal with a dual mesh resolution, where the pressure field is solved on a coarse mesh, while the saturation fields, which may have discontinuities, are solved on a finer reservoir grid, on which petrophysical heterogeneities are defined. Unfortunately, the efficiency of dual mesh methods is strongly related to the definition of up-gridding and down-gridding steps, allowing defining accurately pressure and saturation fields on both fine and coarse meshes and the ability of the approach to be parallelized. In the new dual mesh formulation we developed, the pressure is solved on a coarse grid using a new hybrid formulation of the parabolic problem. This type of multiscale method for pressure equation called multiscale hybrid-mixed method (MHMM) has been recently proposed for finite elements and mixed-finite element approach (Harder et al. 2013). We extend here the MH-mixed method to a finite volume discretization, in order to deal with large multiphase reservoir models. The pressure solution is obtained by solving a hybrid form of the pressure problem on the coarse mesh, for which unknowns are fluxes defined on the coarse mesh faces. Basis flux functions are defined through the resolution of a local finite volume problem, which accounts for local heterogeneity, whereas pressure continuity between cells is weakly imposed through flux basis functions, regarded as Lagrange multipliers. Such an approach is conservative both on the coarse and local scales and can be easily parallelized, which is an advantage compared to other existing finite volume multiscale approaches. It has also a high flexibility to refine the coarse discretization just by refinement of the lagrange multiplier space defined on the coarse faces without changing nor the coarse nor the fine meshes. This refinement can also be done adaptively w.r.t. a posteriori error estimators. The method is applied to single phase (well-testing) and multiphase flow in heterogeneous porous media

Zonal Multiscale Finite-Volume framework

In this work, the zonal Multiscale Finite Volume (zMSFV) framework is presented. The framework generalizes previous extensions of the Multiscale Finite Volume (MSFV) method and allows to integrate additional models in a unified way. The new approach is based on splitting the domain of interest into a classical MSFV domain and additional zones. Within each zone, the solution space is spanned by so-called zonal functions. Coupling of the zonal solutions to the surrounding domain is achieved via extended zonal functions. The construction of basis, zonal and extended zonal functions can be performed such that the functions form a partition of unity. The zMSFV framework allows to include the influence of the zonal functions directly in the coarse-scale problem. Similar to previous MSFV method formulations, the zMSFV framework can ensure conservative coarse-scale solutions and allows for the construction of conservative fine-scale solutions. The main features of the zMSFV framework are used to derive robust preconditioners for iterative solutions of high-contrast problems as encountered in subsurface flow and transport simulations. It is shown that the obtained convergence rates are independent of the contrast.

Hybrid Multiscale Finite Volume Method for Advection-Diffusion Equations Subject to Heterogeneous Reactive Boundary Conditions

We present a hybrid scheme for the coupling of macro- and microscale continuum models for reactive contaminant transport in fractured and porous media. The transport model considered is the advection-diffusion equation, subject to linear heterogeneous reactive boundary conditions. The multiscale finite volume method (MsFV) is employed to define an approximation to the microscale concentration field defined in terms of macroscopic or global degrees of freedom, together with local interpolator and corrector functions capturing microscopic spatial variability. The macroscopic mass balance relations for the MsFV global degrees of freedom are coupled with the macroscopic model, resulting in a global problem for the simultaneous time-stepping of all macroscopic degrees of freedom throughout the domain. In order to perform the hybrid coupling, the micro- and macroscale models are applied over overlapping subdomains of the simulation domain, with the overlap denoted as the handshake subdomain $\Omega^{hs}$, over which continuity of concentration and transport fluxes between models is enforced. Continuity of concentration is enforced by posing a restriction relation between models over $\Omega^{hs}$. Continuity of fluxes is enforced by prolongating to microscopic resolution the macroscopic model fluxes across the boundary of $\Omega^{hs}$. The microscopic interpolator and corrector functions are solutions to local microscopic advection-diffusion problems decoupled from the global degrees of freedom and from each other by virtue of the MsFV decoupling ansatz. The error introduced by the decoupling ansatz is reduced iteratively by the preconditioned GMRES algorithm, with the hybrid MsFV operator serving as the preconditioner.

Multiscale Finite-Volume CVD-MPFA Formulations on Structured and Unstructured Grids

This paper presents the development of finite-volume multiscale methods for quadrilateral and triangular unstructured grids. Families of Darcy-flux approximations have been developed for consistent approximation of the general tensor pressure equation arising from Darcy's law together with mass conservation. The schemes are control-volume distributed (CVD) with flow variables and rock properties sharing the same control-volume location and are comprised of a multipoint flux family formulation (CVD-MPFA). The schemes are used to develop a CVD-MPFA based multiscale finite-volume (MSFV) formulation applicable to both structured and unstructured grids in two dimensions. The basis functions are a key component of the MSFV method, and are a set of local solutions, usually defined subject to Dirichlet boundary conditions. A generalization of the Cartesian grid Dirichlet basis functions described in [P. Jenny, S. H. Lee, and H. A. Tchelepi, J. Comput. Phys., 187 (2003), pp. 47--67] is presented here for unstructured grids. Whilst the transition from a Cartesian grid to an unstructured grid is largely successful, use of Dirichlet basis functions can still lead to pressure fields that exhibit spurious oscillations in areas of strong heterogeneity. New basis functions are proposed in an attempt to improve the pressure field solutions where Neumann boundary conditions are imposed almost everywhere, except corners which remain specified by Dirichlet values.

Investigating Darcy-scale assumptions by means of a multiphysics algorithm

Multiphysics (or hybrid) algorithms, which couple Darcy and pore-scale descriptions of flow through porous media in a single numerical framework, are usually employed to decrease the computational cost of full pore-scale simulations or to increase the accuracy of pure Darcy-scale simulations when a simple macroscopic description breaks down. Despite the massive increase in available computational power, the application of these techniques remains limited to core-size problems and upscaling remains crucial for practical large-scale applications. In this context, the Hybrid Multiscale Finite Volume (HMsFV) method, which constructs the macroscopic (Darcy-scale) problem directly by numerical averaging of pore-scale flow, offers not only a flexible framework to efficiently deal with multiphysics problems, but also a tool to investigate the assumptions used to derive macroscopic models and to better understand the relationship between pore-scale quantities and the corresponding macroscale variables. Indeed, by direct comparison of the multiphysics solution with a reference pore-scale simulation, we can assess the validity of the closure assumptions inherent to the multiphysics algorithm and infer the consequences for macroscopic models at the Darcy scale. We show that the definition of the scale ratio based on the geometric properties of the porous medium is well justified only for single-phase flow, whereas in case of unstable multiphase flow the nonlinear interplay between different forces creates complex fluid patterns characterized by new spatial scales, which emerge dynamically and weaken the scale-separation assumption. In general, the multiphysics solution proves very robust even when the characteristic size of the fluid-distribution patterns is comparable with the observation length, provided that all relevant physical processes affecting the fluid distribution are considered. This suggests that macroscopic constitutive relationships (e.g., the relative permeability) should account for the fact that they depend not only on the saturation but also on the actual characteristics of the fluid distribution.

Spatiotemporal adaptive multiphysics simulations of drainage-imbibition cycles

We present a spatiotemporal adaptive multiscale algorithm, which is based on the Multiscale Finite Volume method. The algorithm offers a very efficient framework to deal with multiphysics problems and to couple regions with different spatial and temporal resolution. We employ the method to simulate two-phase flow through porous media. At the fine scale, we consider a pore-scale description of the flow based on the Volume Of Fluid method. In order to construct a global problem that describes the coarse-scale behavior, the equations are averaged numerically with respect to auxiliary control volumes, and a Darcy-like coarse-scale model is obtained. The spatial adaptivity is based on the idea that the fine-scale description is only required in the front region, whereas the resolution can be coarsened elsewhere. Temporal adaptivity relies on the fact that the fine-scale and the coarse-scale problems can be solved with different temporal resolution (longer time steps can be used at the coarse scale). By simulating drainage-imbibition cycles for different flow regimes, we show that the method is able to capture the coarse-scale behavior outside the front region and to reproduce complex fluid patterns in the front region.

Monotone multiscale finite volume method

The MultiScale Finite Volume (MSFV) method is known to produce non-monotone solutions. The causes of the non-monotone solutions are identified and connected to the local flux across the boundaries of primal coarse cells induced by the basis functions. We propose a monotone MSFV (m-MSFV) method based on a local stencil-fix that guarantees monotonicity of the coarse-scale operator, and thus, the resulting approximate fine-scale solution. Detection of non-physical transmissibility coefficients that lead to non-monotone solutions is achieved using local information only and is performed algebraically. For these ‘critical’ primal coarse-grid interfaces, a monotone local flux approximation, specifically, a Two-Point Flux Approximation (TPFA), is employed. Alternatively, a local linear boundary condition can be used for the dual basis functions to reduce the degree of non-monotonicity. The local nature of the two strategies allows for ensuring monotonicity in local sub-regions, where the non-physical transmissibility occurs. For practical applications, an adaptive approach based on normalized positive off-diagonal coarse-scale transmissibility coefficients is developed. Based on the histogram of these normalized coefficients, one can remove the large peaks by applying the proposed modifications only for a small fraction of the primal coarse grids. Though the m-MSFV approach can guarantee monotonicity of the solutions to any desired level, numerical results illustrate that employing the m-MSFV modifications only for a small fraction of the domain can significantly reduce the non-monotonicity of the conservative MSFV solutions.

Adaptive algebraic multiscale solver for compressible flow in heterogeneous porous media

This paper presents the development of an Adaptive Algebraic Multiscale Solver for Compressible flow (C-AMS) in heterogeneous porous media. Similar to the recently developed AMS for incompressible (linear) flows (Wang et al., 2014) [19], C-AMS operates by defining primal and dual-coarse blocks on top of the fine-scale grid. These coarse grids facilitate the construction of a conservative (finite volume) coarse-scale system and the computation of local basis functions, respectively. However, unlike the incompressible (elliptic) case, the choice of equations to solve for basis functions in compressible problems is not trivial. Therefore, several basis function formulations (incompressible and compressible, with and without accumulation) are considered in order to construct an efficient multiscale prolongation operator. As for the restriction operator, C-AMS allows for both multiscale finite volume (MSFV) and finite element (MSFE) methods. Finally, in order to resolve high-frequency errors, fine-scale (pre- and post-) smoother stages are employed. In order to reduce computational expense, the C-AMS operators (prolongation, restriction, and smoothers) are updated adaptively. In addition to this, the linear system in the Newton–Raphson loop is infrequently updated. Systematic numerical experiments are performed to determine the effect of the various options, outlined above, on the C-AMS convergence behaviour. An efficient C-AMS strategy for heterogeneous 3D compressible problems is developed based on overall CPU times. Finally, C-AMS is compared against an industrial-grade Algebraic MultiGrid (AMG) solver. Results of this comparison illustrate that the C-AMS is quite efficient as a nonlinear solver, even when iterated to machine accuracy.

Constrained pressure residual multiscale (CPR-MS) method for fully implicit simulation of multiphase flow in porous media

We develop the first multiscale method for fully implicit (FIM) simulations of multiphase flow in porous media, namely CPR-MS method. Built on the FIM Jacobian matrix, the pressure system is obtained by employing a Constrained Pressure Residual (CPR) operator. Multiscale Finite Element (MSFE) and Finite Volume (MSFV) methods are then formulated algebraically to obtain efficient and accurate solutions of this pressure equation. The multiscale prediction stage (first-stage) is coupled with a corrector stage (second-stage) employed on the full system residual. The converged solution is enhanced through outer GMRES iterations preconditioned by these first and second stage operators. While the second-stage FIM stage is solved using a classical iterative solver, the multiscale stage is investigated in full detail. Several choices for fine-scale pre- and post-smoothing along with different choices of coarse-scale solvers are considered for a range of heterogeneous three-dimensional cases with capillarity and three-phase systems. The CPR-MS method is the first of its kind, and extends the applicability of the so-far developed multiscale methods (both MSFE and MSFV) to displacements with strong coupling terms.

Local–global splitting for spatiotemporal-adaptive multiscale methods

We present a novel spatiotemporal-adaptive Multiscale Finite Volume (MsFV) method, which is based on the natural idea that the global coarse-scale problem has longer characteristic time than the local fine-scale problems. As a consequence, the global problem can be solved with larger time steps than the local problems. In contrast to the pressure-transport splitting usually employed in the standard MsFV approach, we propose to start directly with a local–global splitting that allows to locally retain the original degree of coupling. This is crucial for highly non-linear systems or in the presence of physical instabilities. To obtain an accurate and efficient algorithm, we devise new adaptive criteria for global update that are based on changes of coarse-scale quantities rather than on fine-scale quantities, as it is routinely done before in the adaptive MsFV method. By means of a complexity analysis we show that the adaptive approach gives a noticeable speed-up with respect to the standard MsFV algorithm. In particular, it is efficient in case of large upscaling factors, which is important for multiphysics problems. Based on the observation that local time stepping acts as a smoother, we devise a self-correcting algorithm which incorporates the information from previous times to improve the quality of the multiscale approximation. We present results of multiphase flow simulations both for Darcy-scale and multiphysics (hybrid) problems, in which a local pore-scale description is combined with a global Darcy-like description. The novel spatiotemporal-adaptive multiscale method based on the local–global splitting is not limited to porous media flow problems, but it can be extended to any system described by a set of conservation equations.

A multiscale finite volume method for Maxwell's equations at low frequencies

A multiscale finite volume method for Maxwell's equations at low frequencies

Iterative Galerkin-enriched multiscale finite-volume method

The multiscale finite volume (MSFV) method was designed to efficiently compute approximate numerical solutions of elliptic and parabolic problems with highly heterogeneous coefficients. For a wide range of test cases, it has been demonstrated that MSFV solutions are in excellent agreement with reference data obtained with a standard fine-scale finite-volume method. However, for some problems involving large coherent structures with strong contrasts (e.g., channels and shale layers), the MSFV method produces unsatisfactory results due to the dwindling validity of the localization assumption. Recently, the iterative MSFV (i-MSFV) method was introduced to iteratively improve the localization assumption, and it was shown that the method converges to the fine-scale reference solution. In this work, it is explained how the convergence rate of the i-MSFV method can be enhanced by consistent enrichment of the initial coarse space spanned by nodal basis functions. A new set of enrichment basis functions associated with additional coarse-scale degrees of freedom (DoF) is introduced. By construction, the sum of all nodal and enrichment basis functions is one in the entire computational domain. Further, a hybrid finite-volume/Galerkin formulation for the coarse-scale problem and a generic placement strategy for the additional DoF are devised. The resulting iterative Galerkin-enriched MSFV (i-Ge-MSFV) method has all features of the i-MSFV method, but is more robust and has improved convergence properties.

A multiscale two-point flux-approximation method

A large number of multiscale finite-volume methods have been developed over the past decade to compute conservative approximations to multiphase flow problems in heterogeneous porous media. In particular, several iterative and algebraic multiscale frameworks that seek to reduce the fine-scale residual towards machine precision have been presented. Common for all such methods is that they rely on a compatible primal–dual coarse partition, which makes it challenging to extend them to stratigraphic and unstructured grids. Herein, we propose a general idea for how one can formulate multiscale finite-volume methods using only a primal coarse partition. To this end, we use two key ingredients that are computed numerically: (i) elementary functions that correspond to flow solutions used in transmissibility upscaling, and (ii) partition-of-unity functions used to combine elementary functions into basis functions. We exemplify the idea by deriving a multiscale two-point flux-approximation (MsTPFA) method, which is robust with regards to strong heterogeneities in the permeability field and can easily handle general grids with unstructured fine- and coarse-scale connections. The method can easily be adapted to arbitrary levels of coarsening, and can be used both as a standalone solver and as a preconditioner. Several numerical experiments are presented to demonstrate that the MsTPFA method can be used to solve elliptic pressure problems on a wide variety of geological models in a robust and efficient manner.

Local modeling of instability onset for global finger evolution

Simulation of density-driven instabilities requires flexible methods to deal with the different spatial and temporal scales involved. Downscaling approaches based on standard adaptive grid refinement aim at resolving the fine-scale details only in the region of interest, but they may become computationally expensive in presence of very corrugated unstable fronts because the problem to be solved approaches the size of the fully refined system. The Downscaling Multiscale Finite-Volume (DMsFV) method overcomes this issue by splitting the problems into a set of localized subproblems that interact only through a global problem. However, in presence of convective instabilities (e.g. density-driven fingers) the diffusion scale has to be resolved only at early times to capture the evolution of infinitesimal random perturbations, whereas at later times fingers have developed and merged, allowing the use of a coarser numerical description. Based on this observation, we present an adaptive algorithm which splits the simulation into three stages: an onset stage in which a set of localized problems is solved independently to capture the initial growth of the instabilities; a transition stage in which the DMsFV method is used to couple local and global scales; and a global stage in which only a fully coarsened description of the problem is employed. The dissolution–diffusion–convection problem (which is typically studied in the context of CO2 sequestration) is chosen as an example to evaluate the accuracy of the adaptive algorithm. For this problem, the use of a coarse grid that does not resolve the fine-scale details at earlier times leads to a dramatic underestimation of mass influx and penetration depth. On the contrary, the solutions obtained with the adaptive algorithm are in good agreement with the reference solution (obtained with a fully refined discretization) and are able to capture total mass influx and penetration depth with excellent accuracy. This demonstrates the need and the effectiveness of modeling local details during the instability onset to capture large-scale features of the concentration patterns at later times.

The Multiscale Finite-Volume Method on Stratigraphic Grids

Summary Finding a pressure solution for large and highly detailed reservoir models with fine-scale heterogeneities modeled on a meter scale is computationally demanding. One way of making such simulations less compute-intensive is to use multiscale methods that solve coarsened flow problems by use of a set of reusable basis functions to capture flow effects induced by local geological variations. One such method, the multiscale finite-volume (MsFV) method, is well-studied for 2D Cartesian grids but has not been implemented for stratigraphic and unstructured grids with faults in three dimensions. We present an open-source implementation of the MsFV method in three dimensions along with a coarse partitioning algorithm that can handle stratigraphic grids with faults and wells. The resulting solver is an alternative to traditional upscaling methods, but can also be used for accelerating fine-scale simulations. To achieve better precision, the implementation can use the MsFV method as a preconditioner for Arnoldi iterations using the generalized minimal residual (GMRES) method or as a preconditioner in combination with a standard inexpensive smoother. We conduct a series of numerical experiments in which approximate solutions computed by the new MsFV solver are compared with fine-scale solutions computed by a standard two-point scheme for grids with realistic permeabilities and geometries. On the one hand, the results show that the MsFV method can produce accurate approximations for geological models with pinchouts, faults, and nonneighboring connections, but on the other hand, they also show that the method can fail quite spectacularly for highly heterogeneous and anisotropic systems in a way that cannot efficiently be mitigated by iterative approaches. Thus, the MsFV method is, in our opinion, not yet sufficiently robust to be applied as a black-box solver for models with industry-standard complexity. However, extending the method to realistic grids is an important step on the way toward a fast and accurate multiscale solution of large-scale reservoir models. In particular, our open-source implementation provides an efficient framework suitable for further experimentation with partitioning algorithms and MsFV variants.

Algebraic multiscale solver for flow in heterogeneous porous media

An Algebraic Multiscale Solver (AMS) for the pressure equations arising from incompressible flow in heterogeneous porous media is described. In addition to the fine-scale system of equations, AMS requires information about the superimposed multiscale (dual and primal) coarse grids. AMS employs a global solver only at the coarse scale and allows for several types of local preconditioners at the fine scale. The convergence properties of AMS are studied for various combinations of global and local stages. These include MultiScale Finite-Element (MSFE) and MultiScale Finite-Volume (MSFV) methods as the global stage, and Correction Functions (CF), Block Incomplete Lower–Upper factorization (BILU), and ILU as local stages. The performance of the different preconditioning options is analyzed for a wide range of challenging test cases. The best overall performance is obtained by combining MSFE and ILU as the global and local preconditioners, respectively, followed by MSFV to ensure local mass conservation. Comparison between AMS and a widely used Algebraic MultiGrid (AMG) solver [1] indicates that AMS is quite efficient. A very important advantage of AMS is that a conservative fine-scale velocity can be constructed after any MSFV stage.

Compositional Multiscale Finite-Volume Formulation

Summary The multiscale finite-volume (MSFV) method is extended to include compositional processes in heterogeneous porous media, which require accurate modeling of the mass transfer and associated phase behaviors. A sequential-implicit strategy is used to deal with the coupling of the flow (pressure) and transport (component overall concentration) problems. In this compositional formulation, the overall continuity equation is used to formulate the pressure equation. The resulting pressure equation conserves total mass by construction and depends weakly on the distributions of the phase compositions. The transport equations are expressed in terms of the overall composition; hence, phase-appearance and -disappearance effects do not appear explicitly in these expressions. The details of the MSFV strategy for the pressure equation are described. The only source of error in this MSFV framework is the localization assumption. No additional assumptions related to the complex physics are used. For 1D problems, the sequential strategy is validated against solutions obtained by a fully implicit simulator. The accuracy of the MSFV method for compositional simulations is then illustrated for different test cases.

A Multilevel Multiscale Finite-Volume Method

The Multiscale Finite-Volume (MsFV) method has been developed over the last decade to efficiently solve large reservoir models. The method projects the original pressure problem onto a second coarser grid, on which it is less expensive to solve, and then prolongates the approximated coarse solution back to the fine-scale grid. One characteristic of the MsFV algorithm is to allow the reconstruction of an approximate but fully conservative velocity field from the prolongated pressure. This makes the method particularly attractive for applications involving the solution of transport problems. Here, we present an extension of the MsFV method (MMsFV) that can employ multiple levels of coarsening instead of the single coarse level used in the standard algorithm. Whereas the coarse problem and the prolongation operators can be easily obtained by recursive application of the MsFV method, formulating an efficient reconstruction of the conservative velocity is not trivial. We devise a nested reconstruction procedure that is novel and has computational cost comparable with the MsFV reconstruction. By analyzing the computational complexity of the algorithm we show that the MMsFV method allows obtaining a conservative approximation of the fine-scale velocity more efficiently than the MsFV method. However, the accuracy of the solution deteriorates and MMsFV errors are larger than MsFV errors. By means of numerical test cases we demonstrate that, when the MMsFV operator is used as preconditioner in GMRES, the number of iterations necessary to achieve the same accuracy is larger than with the MsFV operator. For highly-heterogeneous permeability fields more than several hundreds of iterations might be required. Such a large number of iterations might be practically intractable. Therefore, we propose two ideas that can be used to control the number of iterations. The first is to combine the MMsFV coarse operator with an appropriate smoother in a two-step preconditioner; the second is to modify the localization assumptions used to solve edge problems. Our numerical tests show that both strategies lead to a significant reduction of the iterations and suggest that efficient MMsFV methods can be obtained by identifying optimal smoothers or devising better localizations for the edge problems.

Effects of using altered coarse grids on the implementation and computational cost of the multiscale finite volume method

In the present work, the multiscale finite volume (MsFV) method is implemented on a new coarse grids arrangement. Like grids used in the MsFV methods, the new grid arrangement consists of both coarse and dual coarse grids but here each coarse block in the MsFV method is a dual coarse block and vice versa. Due to using the altered coarse grids, implementation, computational cost, and the reconstruction step differ from the original version of MsFV method. Two reconstruction procedures are proposed and their performances are compared with each other. For a wide range of 2-D and 3-D problem sizes and coarsening ratios, the computational costs of the MsFV methods are investigated. Furthermore, a matrix (operator) formulation is presented. Several 2-D test cases, including homogeneous and heterogeneous permeability fields extracted from different layers of the tenth SPE comparative study problem are solved. The results are compared with the fine-scale reference and basic MsFV solutions.