Hadi Hajibeygi

Iterative multiscale finite-volume method

The multiscale finite-volume (MSFV) method for the solution of elliptic problems is extended to an efficient iterative algorithm that converges to the fine-scale numerical solution. The localization errors in the MSFV method are systematically reduced by updating the local boundary conditions with global information. This iterative multiscale finite-volume (i-MSFV) method allows the conservative reconstruction of the velocity field after any iteration, and the MSFV method is recovered, if the velocity field is reconstructed after the first iteration. Both the i-MSFV and the MSFV methods lead to substantial computational savings, where an approximate but locally conservative solution of an elliptic problem is required. In contrast to the MSFV method, the i-MSFV method allows a systematic reduction of the error in the multiscale approximation. Line relaxation in each direction is used as an efficient smoother at each iteration. This smoother is essential to obtain convergence in complex, highly anisotropic, heterogeneous domains. Numerical convergence of the method is verified for different test cases ranging from a standard Poisson equation to highly heterogeneous, anisotropic elliptic problems. Finally, to demonstrate the efficiency of the method for multiphase transport in porous media, it is shown that it is sufficient to apply the iterative smoothing procedure for the improvement of the localization assumptions only infrequently, i.e. not every time step. This result is crucial, since it shows that the overall efficiency of the i-MSFV algorithm is comparable with the original MSFV method. At the same time, the solutions are significantly improved, especially for very challenging cases.

A hierarchical fracture model for the iterative multiscale finite volume method

An iterative multiscale finite volume (i-MSFV) method is devised for the simulation of multiphase flow in fractured porous media in the context of a hierarchical fracture modeling framework. Motivated by the small pressure change inside highly conductive fractures, the fully coupled system is split into smaller systems, which are then sequentially solved. This splitting technique results in only one additional degree of freedom for each connected fracture network appearing in the matrix system. It can be interpreted as an agglomeration of highly connected cells; similar as in algebraic multigrid methods. For the solution of the resulting algebraic system, an i-MSFV method is introduced. In addition to the local basis and correction functions, which were previously developed in this framework, local fracture functions are introduced to accurately capture the fractures at the coarse scale. In this multiscale approach there exists one fracture function per network and local domain, and in the coarse scale problem there appears only one additional degree of freedom per connected fracture network. Numerical results are presented for validation and verification of this new iterative multiscale approach for fractured porous media, and to investigate its computational efficiency. Finally, it is demonstrated that the new method is an effective multiscale approach for simulations of realistic multiphase flows in fractured heterogeneous porous media.

Multiscale finite-volume method for parabolic problems arising from compressible multiphase flow in porous media

The multiscale finite-volume (MSFV) method was originally developed for the solution of heterogeneous elliptic problems with reduced computational cost. Recently, some extensions of this method for parabolic problems have been proposed. These extensions proved effective for many cases, however, they are neither general nor completely satisfactory. For instance, they are not suitable for correctly capturing the transient behavior described by the parabolic pressure equation. In this paper, we present a general multiscale finite-volume method for parabolic problems arising, for example, from compressible multiphase flow in porous media. Opposed to previous methods, here, the basis and correction functions are solutions of full parabolic governing equations in localized domains. At the same time, to enhance the computational efficiency of the scheme, the basis functions are kept pressure independent and do not have to be recalculated as pressure evolves. This general approach requires no additional assumptions and its good efficiency and high accuracy is demonstrated for various challenging test cases. Finally, to improve the quality of the results and also to extend the scheme for highly anisotropic heterogeneous problems, it is combined with the iterative MSFV (i-MSFV) method for parabolic problems. As one iterates, the i-MSFV solutions of compressible multiphase problems (parabolic problems) converge to the corresponding fine-scale reference solutions in the same way as demonstrated recently for incompressible cases (elliptic problems). Therefore, the proposed MSFV method can also be regarded as an efficient linear solver for parabolic problems and studies of its efficiency are presented for many test cases.

Adaptive iterative multiscale finite volume method

The multiscale finite volume (MSFV) method is a computationally efficient numerical method for the solution of elliptic and parabolic problems with heterogeneous coefficients. It has been shown for a wide range of test cases that the MSFV results are in close agreement with those obtained with a classical (computationally expensive) technique. The method, however, fails to give accurate results for highly anisotropic heterogeneous problems due to weak localization assumptions. Recently, a convergent iterative MSFV (i-MSFV) method was developed to enhance the quality of the multiscale results by improving the localization conditions. Although the i-MSFV method proved to be efficient for most practical problems, it is still favorable to improve the localization condition adaptively, i.e. only for a sub-domain where the original MSFV localization conditions are not acceptable, e.g. near shale layers and long coherent structures with high permeability contrasts. In this paper, a space-time adaptive i-MSFV (ai-MSFV) method is introduced. It is shown how to improve the MSFV results adaptively in space and simulation time. The fine-scale smoother, which is necessary for convergence of the i-MSFV method, is also applied locally. Finally, for multiphase flow problems, two criteria are investigated for adaptively updating the MSFV interpolation functions: (1) a criterion based on the total mobility change for the transient coefficients and (2) a criterion based on the pressure equation residual for the accuracy of the results. For various challenging test cases it is demonstrated that iterations in order to obtain accurate results even for highly anisotropic heterogeneous problems are required only in small sub-domains and not everywhere. The findings show that the error introduced in the MSFV framework can be controlled and improved very efficiently with very little additional computational cost compared to the original, non-iterative MSFV method.

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.

Algebraic multiscale method for flow in heterogeneous porous media with embedded discrete fractures (F-AMS)

This paper introduces an Algebraic MultiScale method for simulation of flow in heterogeneous porous media with embedded discrete Fractures (F-AMS). First, multiscale coarse grids are independently constructed for both porous matrix and fracture networks. Then, a map between coarse- and fine-scale is obtained by algebraically computing basis functions with local support. In order to extend the localization assumption to the fractured media, four types of basis functions are investigated: (1) Decoupled-AMS, in which the two media are completely decoupled, (2) Frac-AMS and (3) Rock-AMS, which take into account only one-way transmissibilities, and (4) Coupled-AMS, in which the matrix and fracture interpolators are fully coupled. In order to ensure scalability, the F-AMS framework permits full flexibility in terms of the resolution of the fracture coarse grids. Numerical results are presented for two- and three-dimensional heterogeneous test cases. During these experiments, the performance of F-AMS, paired with ILU(0) as second-stage smoother in a convergent iterative procedure, is studied by monitoring CPU times and convergence rates. Finally, in order to investigate the scalability of the method, an extensive benchmark study is conducted, where a commercial algebraic multigrid solver is used as reference. The results show that, given an appropriate coarsening strategy, F-AMS is insensitive to severe fracture and matrix conductivity contrasts, as well as the length of the fracture networks. Its unique feature is that a fine-scale mass conservative flux field can be reconstructed after any iteration, providing efficient approximate solutions in time-dependent simulations.

The multiscale restriction smoothed basis method for fractured porous media (F-MsRSB)

A novel multiscale method for multiphase flow in heterogeneous fractured porous media is devised. The discrete fine-scale system is described using an embedded fracture modeling approach, in which the heterogeneous rock (matrix) and highly-conductive fractures are represented on independent grids. Given this fine-scale discrete system, the method first partitions the fine-scale volumetric grid representing the matrix and the lower-dimensional grids representing fractures into independent coarse grids. Then, basis functions for matrix and fractures are constructed by restricted smoothing, which gives a flexible and robust treatment of complex geometrical features and heterogeneous coefficients. From the basis functions one constructs a prolongation operator that maps between the coarse- and fine-scale systems. The resulting method allows for general coupling of matrix and fracture basis functions, giving efficient treatment of a large variety of fracture conductivities. In addition, basis functions can be adaptively updated using efficient global smoothing strategies to account for multiphase flow effects. The method is conservative and because it is described and implemented in algebraic form, it is straightforward to employ it to both rectilinear and unstructured grids. Through a series of challenging test cases for single and multiphase flow, in which synthetic and realistic fracture maps are combined with heterogeneous petrophysical matrix properties, we validate the method and conclude that it is an efficient and accurate approach for simulating flow in complex, large-scale, fractured media.

Algebraic Multiscale Linear Solver for Heterogeneous Elliptic Problems

An Algebraic Multiscale Solver (AMS) for the pressure system of equations arising from incompressible flow in heterogeneous porous media is developed. The algorithm allows for several independent preconditioning stages to deal with the full spectrum of errors. In addition to the fine-scale system of equations, AMS requires information about the superimposed (dual) coarse grid to construct a wirebasket reordered system. The primal coarse grid is used in the construction of a conservative coarse-scale operator and in the reconstruction of a conservative fine-scale velocity field. The convergence properties of AMS are studied for various combinations including (1) the MultiScale Finite-Element (MSFE) method, (2) the MultiScale Finite-Volume (MSFV) method, (3) Correction Functions (CF), (4) Block Incomplete LU factorization with zero fill-in (BILU), and (5) point-wise Incomplete LU factorization with zero fill-in (ILU). The reduced-problem boundary condition, which is used for localization, is investigated. For a wide range of test cases, the performance of the different preconditioning options is analyzed. It is found that the best overall performance is obtained by combining MSFE and ILU as the global and local preconditioners, respectively. Comparison between AMS and the widely used SAMG solver illustrates that they are comparable, especially for very large heterogeneous problems.

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.

Algebraic dynamic multilevel (ADM) method for fully implicit simulations of multiphase flow in porous media

This paper presents the development of an algebraic dynamic multilevel method (ADM) for fully implicit simulations of multiphase flow in homogeneous and heterogeneous porous media. Built on the fine-scale fully implicit (FIM) discrete system, ADM constructs a multilevel FIM system describing the coupled process on a dynamically defined grid of hierarchical nested topology. The multilevel adaptive resolution is determined at each time step on the basis of an error criterion. Once the grid resolution is established, ADM employs sequences of restriction and prolongation operators in order to map the FIM system across the considered resolutions. Several choices can be considered for prolongation (interpolation) operators, e.g., constant, bilinear and multiscale basis functions, all of which form partition of unity. The adaptive multilevel restriction operators, on the other hand, are constructed using a finite-volume scheme. This ensures mass conservation of the ADM solutions, and as such, the stability and accuracy of the simulations with multiphase transport. For several homogeneous and heterogeneous test cases, it is shown that ADM applies only a small fraction of the full FIM fine-scale grid cells in order to provide accurate solutions. The sensitivity of the solutions with respect to the employed fraction of grid cells (determined automatically based on the threshold value of the error criterion) is investigated for all test cases. ADM is a significant step forward in the application of dynamic local grid refinement methods, in the sense that it is algebraic, allows for systematic mapping across different scales, and applicable to heterogeneous test cases without any upscaling of fine-scale high resolution quantities. It also develops a novel multilevel multiscale method for FIM multiphase flow simulations in natural subsurface formations.