Multiscale Finite Volume Method Research Articles

A 3-D extension of the Multiscale Control Volume method for the simulation of the single-phase flow in anisotropic and heterogeneous porous media

The level of detail on modern geological models requires higher resolution grids that may render the simulation of multiphase flow in porous media intractable. Moreover, these models may comprise highly heterogeneous media with phenomena taking place in different scales. Scale transferring techniques allow for the solution of such problems in a lower resolution scale at reduced computational cost. Among these techniques, the Multiscale Finite Volume (MsFV) method constructs a set of numerical operators in order to map quantities from the fine-scale mesh to a coarser one and vice-versa while maintaining flux conservation on both scales. However, the MsFV formulation, as originally stated, is only consistent on k-orthogonal grids since it uses a linear Two-point Flux Approximation (TPFA) method and may struggle to generate consistent primal-dual coarse grids pairs on unstructured grids. The Multiscale Restriction Smoothed-Basis method (MsRSB) improves on the MsFV by introducing a new iterative procedure to find the multiscale operators and the concept of support regions which reduces the method's complexity when applied to unstructured fine and coarse grids. The original version was only consistent on k-orthogonal fine grids due to the TPFA discretization, but filtering methods have been developed to also enable consistent multipoint schemes on the fine scale. Meanwhile, the Multiscale Control Volume method (MsCV) replaces the TPFA by the Multipoint Flux Approximation with a Diamond stencil (MPFA-D) scheme on the fine-scale while further enhancing the generation of the geometric entities to allow unstructured grids on the fine and coarse scales for two-dimensional simulations. In this work we propose an extension to three-dimensional geometries of both the MsCV and the algorithm to obtain the multiscale geometric entities based on the concept of a background grid, a coarser grid used as a proxy for the primal coarse grid. We modify the original MPFA-D method to use the very robust Global Least Squares (GLS) interpolation technique to obtain the required auxiliary nodal unknowns. We also introduce an enhanced version of the 3-D MsCV with the incorporation of the enhanced MsRSB (E-MsRSB) to enforce M-matrix properties and improve convergence. Finally, we employ the MsCV operators in a two-stage smoother to show how it can be used as a good iterative procedure to recover the fine-scale solution. We show that the 3-D MsCV method produces good results employing unstructured grids on both scales to handle the simulation of the single-phase flow in anisotropic, and heterogeneous porous media and that the smoothing procedure is able to accurately retrieve the fine-scale solution within a few iterations.

Multiscale simulation of highly heterogeneous subsurface flow with physically adaptive 3D unstructured grids

The multiscale finite volume framework is extended with physical-based grid adaptation (MSFV-GA) for three-dimensional fully unstructured grids. The MSFV method produces large errors for highly heterogeneous and anisotropic problems on standard coarse grids generated based on geometrical features. Adding constraints based on the fine-scale physical properties to construct coarse grids can increase the accuracy and robustness of the MSFV solutions. A novel and automated coarse grid generation strategy adaptable to the reservoir heterogeneity for 3D unstructured grids is presented. Adaptive coarse grids are generated based on local variations of fine scale permeability field. The proposed procedure is very flexible respect to the local physical and geometrical limitations in coarse blocks. Multiscale coarse grid adaptation notably improves the basis functions calculations. The performance of the MSFV-GA in handling challenging problems involving high media contrasts or impermeable shale layers is assessed. The results are very promising and show that the MSFV-GA can accurately approximate complex physics problems. The presented 3D unstructured procedure can notably improve the applicability of the multiscale finite volume method and provide a promising framework for the next generation of reservoir flow simulators.

Algorithmic monotone multiscale finite volume methods for porous media flow

Multiscale finite volume methods are known to produce reduced systems with multipoint stencils which, in turn, could give non-monotone and out-of-bound solutions. We propose a novel solution to the monotonicity issue of multiscale methods. The proposed algorithmic monotone (AM- MsFV/MsRSB) framework is based on an algebraic modification to the original MsFV/MsRSB coarse-scale stencil. The AM-MsFV/MsRSB guarantees monotonic and within bound solutions without compromising accuracy for various coarsening ratios; hence, it effectively addresses the challenge of multiscale methods' sensitivity to coarse grid partitioning choices. Moreover, by preserving the near null space of the original operator, the AM-MsRSB showed promising performance when integrated in iterative formulations using both the control volume and the Galerkin-type restriction operators. We also propose a new approach to enhance the performance of MsRSB for MPFA discretized systems, particularly targeting the construction of the prolongation operator. Results show the potential of our approach in terms of accuracy of the computed basis functions and the overall convergence behavior of the multiscale solver while ensuring a monotone solution at all times.

Two‐level multiscale enrichment finite volume method based on the Newton iteration for the stationary incompressible magnetohydrodynamics flow

In this paper, we propose a stabilized finite volume method based on the Newton's type iteration for the steady incompressible magnetohydrodynamics (MHD) problem. In order to reduce the computational complexity, the lowest order mixed finite element pair ( ‐ ‐ ) is adopted to approximate the velocity, pressure and magnetic fields, and the multiscale enrichment method is introduced to overcome the restriction of discrete inf‐sup (LBB) condition. We firstly solve the incompressible MHD equations by the Newton iterations on a coarse mesh with the mesh size , stability and convergence results of numerical solutions are provided. Secondly, the combination of the two‐level method with the Newton iteration is used to approximate the considered problem, a coupled nonlinear problem is solved on the coarse mesh , then the Stokes and Maxwell equations are considered on a fine mesh with the mesh size , the uniform stability and optimal error estimates of two‐level Newton iterative method are provided. Theoretical findings show that the two‐level method has the same order as the one‐level method in ‐norm as long as the mesh sizes satisfy . However, the two‐level method involves much less work than the one‐level method. Finally, some numerical examples are presented to demonstrate the effectiveness of the considered numerical schemes.

A fixed point multi-scale finite volume method: Application to two-phase incompressible fluid flow through highly heterogeneous porous media

An enhanced Multi-scale Finite Volume (MsFV) method is proposed to efficiently simulate two phase flow through highly heterogeneous porous media. Here, the accuracy of the MsFV method is significantly improved by enhancing its so-called basis functions, while its computational cost remains in the same order of the basic MsFV method. First, a fixed point is defined along each edge of local problems producing the basis functions and a proper basis function value at this point is estimated based on the local absolute permeability data. Then, the variable boundary condition is independently calculated for both sides of each fixed point. The proposed numerical method is validated by using the analytic solution of a heterogeneous problem. In addition, the convergence of the MsFV method is discussed. Considering highly heterogeneous permeability domains derived from 35 top layers of the tenth SPE comparative study problem, the accuracy of different localization schemes is compared for a set of imbibition problems with different global boundary conditions and mobility ratios. Numerical results have indicated that the overall accuracy of multi-scale velocity solutions is increased noticeably (up to 45%) by using the proposed localization scheme in the MsFV method.

A Hybrid Multiscale Finite Cloud Method and Finite Volume Method in Solving High Gradient Problem

Meshfree methods are always the alternatives to be considered besides the finite element method in dealing with complex engineering problems. Among these complex problems, there are problems where discontinuities or high gradients may only emerge at a small region of a domain and the remaining parts of the domain are smooth and continuous. In this case, adding nodes to the critical region is essential for the better accuracy of solutions. It would be effective if a meshfree method is applied to the critical region and a mesh-based method is employed to solve the remaining part of the domain since node refinement is easier for a meshfree method whereas a mesh-based method is simple to develop and capable of producing accurate solutions for smooth regions. In this study, the finite cloud method (FCM) is coupled with the finite volume method (FVM) such that node refinement only takes place at the region where the FCM is applied. A handshaking approach is proposed to enforce the continuity near the interface where both methods are applied. Numerical examples and computation methods are presented in this paper and the numerical results are examined.

Multiscale finite volume method with adaptive unstructured grids for flow simulation in heterogeneous fractured porous media

Multiscale finite volume method with adaptive unstructured grids for flow simulation in heterogeneous fractured porous media

Upscaling numérico baseado em uma metodologia multiescala para a simulação de escoamentos em reservatórios de petróleo heterogêneos e anisotrópicos / Numerical upscaling based on a multiscale methodology for simulating flows in heterogeneous and anisotropic oil reservoirs

This work aims to develop a scale transfer methodology in highly heterogeneous and anisotropic porous media, that makes it possible to solve elliptical problems with accuracy and low computational cost. A scale transfer method is developed that uses the Multiscale Finite Volume Method (MSFVM) to perform a flow based numerical upscaling of absolute permeability. The transmissibility matrix was built in the coarse mesh over the projection and restriction operators obtaining permeability in the coarse scale. This allows to solve only local problems in the coarse mesh volumes and to use pressure multiscale solution in the fine mesh. The permeability field can be obtained by different strategies, and enabling simulations directly on the coarse mesh, controlling the accuracy of the solutions and also with the possibility of updating the permeability field over the recalculated locally conservative multiscale solution.

Simulation of two-phase incompressible fluid flow in highly heterogeneous porous media by considering localization assumption in multiscale finite volume method

Two-phase incompressible fluid flow through highly heterogeneous porous media is simulated by using the Multiscale Finite Volume (MsFV) method. Effects of the localization assumption on the accuracy of the MsFV are investigated by comparing the results associated with different boundary conditions of local problems producing the basis functions. The total number of six boundary conditions of two general types, including Dirichlet and Dirichlet-Neumann types, are compared. For the former, the linear, variable (reduced), and step-type boundary conditions are considered and a modified variable boundary condition is proposed. For the latter, a basic and a step-type Neumann-Dirichlet boundary condition are suggested. To estimate the errors in the MsFV solutions for continuous problems, a heterogeneous two-dimensional problem with continuous permeability field is designed and solved analytically. Synthetic two-scale permeability fields as well as highly heterogeneous random fields are used to assess the accuracy of the MsFV solutions with different localization schemes, in comparison with the fine-scale reference solution. Numerical results indicate that the modified variable boundary condition, with a proper value of its weighting factor, can generally produce the most accurate results, when compared with the other localization schemes.

A benchmark study of the multiscale and homogenization methods for fully implicit multiphase flow simulations

Accurate simulation of multiphase flow in subsurface formations is challenging, as the formations span large length scales (km) with high-resolution heterogeneous properties. To deal with this challenge, different multiscale methods have been developed. Such methods construct coarse-scale systems, based on a given high-resolution fine-scale system. Furthermore, they are amenable to parallel computing and allow for a-posteriori error control. The multiscale methods differ from each other in the way the transition between the different scales is made. Multiscale (finite element and finite volume) methods compute local basis functions to map the solutions (e.g. pressure) between coarse and fine scales. Instead, homogenization methods solve local periodic problems to determine effective models and parameters (e.g. permeability) at a coarser scale. It is yet unknown how these two methods compare with each other, especially when applied to complex geological formations, with no clear scale separation in the property fields. This paper develops the first comparison benchmark study of these two methods and extends their applicability to fully implicit simulations using the algebraic dynamic multilevel (ADM) method. At each time step, on the given fine-scale mesh and based on an error analysis, the fully implicit system is solved on a dynamic multilevel grid. The entries of this system are obtained by using multiscale local basis functions (ADM-MS), and, respectively, by homogenization over local domains (ADM-HO). Both sets of local basis functions (ADM-MS) and local effective parameters (ADM-HO) are computed at the beginning of the simulation, with no further updates during the multiphase flow simulation. The two methods are extended and implemented in the same open-source DARSim2 simulator (https://gitlab.com/darsim2simulator), to provide fair quality comparisons. The results reveal insightful understanding of the two approaches, and qualitatively benchmark their performance. It is re-emphasized that the test cases considered here include permeability fields with no clear scale separation. The development of this paper sheds new lights on advanced multiscale methods for simulation of coupled processes in porous media.

Unstructured grid adaptation for multiscale finite volume method

The multiscale finite volume (MSFV) method produces non-monotone solutions in the presence of highly heterogeneous and channelized permeability fields, or in media with impermeable barriers. The accuracy of the MSFV method can be significantly improved by generating adapted coarse grids according to the fine-scale permeability field. This paper presents efficient algorithms for generating adaptive unstructured primal and dual coarse grids based on permeability features to efficiently improve the MSFV results. The primal coarse grid is generated based on a multilevel tabu search algorithm and its boundaries are adapted to reduced non-physical coarse-scale transmissibilities. Adaptive dual coarse grid is generated based on Dijkstra’s routing algorithm. The performance of the proposed algorithms is assessed for challenging test cases with highly heterogeneous and channelized permeability fields as well as impermeable shale layers. Numerical results show that permeability-adapted coarse grids significantly improve the accuracy of the MSFV method for simulation of multiphase flow in highly heterogeneous porous media.

Multiscale finite volume method for finite-volume-based simulation of poroelasticity

We propose a multiscale finite volume method (MSFV) for simulation of coupled flow-deformation in heterogeneous porous media under elastic deformation (i.e., poroelastic model). The fine-scale fully resolved system of equations is obtained based on a conservative finite-volume method in which the displacement and pore pressure unknowns are located in a staggered configuration. The coupling is treated through a fully-coupled fully-implicit formulation. On this fully-coupled finite-volume system, coarse-scale grids for flow and deformation are imposed. Local basis functions for scalar pore pressure and vectorial displacement unknowns are then solved over their respective local domains at the beginning of the simulation, and reused for the rest of the time-dependent simulations. These local basis functions are then clustered to form the prolongation operator. As for the finite-volume nature of the proposed multiscale system, finite-volume restriction operators for poroelastic systems are utilised. Once the coarse-scale system is solved, its solution is prolonged back to the original fine-scale resolution, providing approximate fine-scale solution. The finite-volume multiscale formulation provides conservative stress and mass flux both at fine and coarse scale. Several numerical test cases are provided first to validate the fine-scale finite-volume discrete fully-implicit simulation, and then to investigate the accuracy of the proposed multiscale formulation. Moreover we compare our fully implicit MSFV method with hybrid multiscale Finite Element-Finite Volume (h-MSFE-FV). Our multiscale method allows for quantification of the elastic geomechanical behaviour with using only a fraction of the fine-scale grid cells, even for highly heterogeneous time-dependent models. As such, it casts a promising approach for field-scale quantification of the mechanical deformation and stress field due to injection and production in a subsurface formation.

A multiscale method for model order reduction in PDE parameter estimation

Estimating parameters of Partial Differential Equations (PDEs) is of interest in a number of applications such as geophysical and medical imaging. Parameter estimation is commonly phrased as a PDE-constrained optimization problem that can be solved iteratively using gradient-based optimization. A computational bottleneck in such approaches is that the underlying PDEs needs to be solved numerous times before the model is reconstructed with sufficient accuracy. One way to reduce this computational burden is by using Model Order Reduction (MOR) techniques such as the Multiscale Finite Volume Method (MSFV). In this paper, we apply MSFV for solving high-dimensional parameter estimation problems. Given a finite volume discretization of the PDE on a fine mesh, the MSFV method reduces the problem size by computing a parameter-dependent projection onto a nested coarse mesh. A novelty in our work is the integration of MSFV into a PDE-constrained optimization framework, which updates the reduced space in each iteration. We also present a computationally tractable way of differentiating the MOR solution that acknowledges the change of basis. As we demonstrate in our numerical experiments, our method leads to computational savings particularly for large-scale parameter estimation problems and can benefit from parallelization.

Comparisons of FV-MHMM with Other Finite Volume Multiscale Methods

Upscaling and multiscale methods in reservoir engineering remain a complicated task especially when dealing with heterogeneities. In this study, we focus on flow field problem with a Darcy’s equation considered at the fine scale. The main difficulty is then to obtain an accurate description of the flow behavior by using multiscale methods available in petroleum engineering. We cross-compare three of the main finite volume formulations: multiscale finite volume method (MsFv), multiscale restriction smoothed (MsRSB) and a new finite volume method, FV-MHMM. Comparisons are done in terms of accuracy to reproduce the fine scale behavior.

Approximate multiscale flow solver for unstructured pore networks

Pore-scale simulations of multiphase flow in porous media are useful in understanding the relations between macroscopic variables relevant in many applications. It is critical that simulation domains are large enough to be considered representative elementary volumes for the relevant macroscopic quantities such as relative permeability, macroscopic capillary pressure and average saturation. However, multiphase pore-scale solvers, such as dynamic flow solvers for pore network models, are typically computationally expensive due to the required solution of a global pressure problem at every time-step. In this paper, an approximate multiscale flow solver is presented, which overcomes this limitation. The solver is applicable to arbitrary unstructured pore networks and consists of three main steps. First, the global pressure problem is solved using the multiscale finite volume (MSFV) method in combination with the multiscale restricted-smoothed basis (MsRSB) prolongation operator. The computed pressure field leads to conservative fluxes across pre-defined subnetwork boundaries. Second, these conservative fluxes are used as boundary conditions to solve local problems on the subnetworks. Third, once these local problems are solved, the subnetworks are synchronized. Validation studies are presented and results obtained using the multiscale solver are compared to results obtained using an existing dynamic network flow solver.

An iterative modified multiscale control volume method for the simulation of highly heterogeneous porous media flow

In this paper, a modification of the multiscale finite-volume method (MsFVM) in its OBMM version (operator-based multiscale method) is introduced. The iterative modified multiscale control volume method (I-MMCVM) is applied to solve the elliptical problem that describes the single-phase fluid flow through heterogeneous porous media. The modifications proposed are accomplished by: (1) resettling the vertices, centroid of the primal coarse volume, of each primal coarse volume laying on the boundary of the computational domain, and by this, avoiding the use of ghost elements; (2) Recalculating the pressure iteratively in each primal coarse volume to improve the accuracy of the solution lost during the process of decoupling the problem in sub-problems. The I-MMCVM achieves efficiently accurate results for flow equations in highly heterogeneous reservoirs, at very low computational cost whilst it recovers physical consistency whenever the multiscale solution fails and produces nonphysical results. The I-MMCVM solution is compared with the fine mesh solution for some highly heterogeneous media. In terms of efficiency, the novel method showed great potential of the multiscale formulations, i.e., computational cost reduction without significant loss of accuracy.

A machine learning approach for efficient uncertainty quantification using multiscale methods

Several multiscale methods account for sub-grid scale features using coarse scale basis functions. For example, in the Multiscale Finite Volume method the coarse scale basis functions are obtained by solving a set of local problems over dual-grid cells. We introduce a data-driven approach for the estimation of these coarse scale basis functions. Specifically, we employ a neural network predictor fitted using a set of solution samples from which it learns to generate subsequent basis functions at a lower computational cost than solving the local problems. The computational advantage of this approach is realized for uncertainty quantification tasks where a large number of realizations has to be evaluated. We attribute the ability to learn these basis functions to the modularity of the local problems and the redundancy of the permeability patches between samples. The proposed method is evaluated on elliptic problems yielding very promising results.

Multiscale finite volume method for discrete fracture modeling on unstructured grids (MS-DFM)

A novel multiscale method for discrete fracture modeling on unstructured grids (MS-DFM) is developed. To this end, the DFM fine-scale discrete system is constructed using unstructured conforming cells for the matrix with lower-dimensional fracture elements placed at their interfaces. On this unstructured fine grid, MS-DFM imposes independent unstructured coarse grids for the fracture and matrix domains. While the conservative coarse-scale system is solved over these coarse-grid cells, overlapping dual-coarse blocks are also formed in order to provide local supports for the multiscale basis functions. To increase the accuracy, but maintaining the computational efficiency, fracture-matrix coupling is considered only for the basis functions inside the matrix domain. This results in additional (enriching) fracture basis functions in the matrix. By construction, basis functions form the partition of unity for both fracture and matrix sub-domains. Furthermore, to enable error reduction to any desired level, a convergent iterative strategy is developed, where MS-DFM is employed along with a fine-scale smoother in order to resolve low- and high-frequency modes in the error. The performance of MS-DFM is assessed for several 2D and 3D test cases. The proposed method achieves accurate results for several test cases even without iterations, and for challenging ones with only a few iterations. MS-DFM is the first of its kind, and thus extends the application of multiscale methods to unstructured discrete fracture models. As such, it provides a promising framework for real-field application of unstructured DFM.

A multi-scale network method for two-phase flow in porous media

Pore-network models of porous media are useful in the study of pore-scale flow in porous media. In order to extract macroscopic properties from flow simulations in pore-networks, it is crucial the networks are large enough to be considered representative elementary volumes. However, existing two-phase network flow solvers are limited to relatively small domains. For this purpose, a multi-scale pore-network (MSPN) method, which takes into account flow-rate effects and can simulate larger domains compared to existing methods, was developed. In our solution algorithm, a large pore network is partitioned into several smaller sub-networks. The algorithm to advance the fluid interfaces within each subnetwork consists of three steps. First, a global pressure problem on the network is solved approximately using the multiscale finite volume (MSFV) method. Next, the fluxes across the subnetworks are computed. Lastly, using fluxes as boundary conditions, a dynamic two-phase flow solver is used to advance the solution in time. Simulation results of drainage scenarios at different capillary numbers and unfavourable viscosity ratios are presented and used to validate the MSPN method against solutions obtained by an existing dynamic network flow solver.

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.