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Abstract: 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. [Copyright &y& Elsevier]

• Multiscale Extended Finite Element method (MS-XFEM) is developed. • Discontinuities are involved only in the multiscale basis functions. • MS-XFEM contains no additional degrees of freedom in the coarse-scale system. • MS-XFEM is directly applicable to heterogeneous highly fractured problems. • MS-XFEM develops an efficient and accurate simulation framework. Deformable fractured porous media appear in many geoscience applications. While the extended finite element method (XFEM) has been successfully developed within the computational mechanics community for accurate modeling of deformation, its application in geoscientific applications is not straightforward. This is mainly due to the fact that subsurface formations are heterogeneous and span large length scales with many fractures at different scales. To resolve this limitation, in this work, we propose a novel multiscale formulation for XFEM, based on locally computed enriched basis functions. The local multiscale basis functions capture heterogeneity of th e porous rock properties, and discontinuities introduced by the fractures. In order to preserve accuracy of these basis functions, reduced-dimensional boundary conditions are set as localization condition. Using these multiscale bases, a multiscale coarse-scale system is then governed algebraically and solved. The coarse scale system entails no enrichment due to the fractures. Such formulation allows for significant computational cost reduction, at the same time, it preserves the accuracy of the discrete displacement vector space. The coarse-scale solution is finally interpolated back to the fine scale system, using the same multiscale basis functions. The proposed multiscale XFEM (MS-XFEM) is also integrated within a two-stage algebraic iterative solver, through which error reduction to any desired level can be achieved. Several proof-of-concept numerical tests are presented to assess the performance of the developed method. It is shown that the MS-XFEM is accurate, when compared with the fine-scale reference XFEM solutions. At the same time, it is significantly more efficient than the XFEM on fine-scale resolution, as it significantly reduces the size of the linear systems. As such, it develops a promising scalable XFEM method for large-scale heavily fractured porous media. [ABSTRACT FROM AUTHOR]

Highlights • Algebraic dynamic multilevel method for multiphase flow in heterogeneous fractured porous media (F-ADM) is developed. • F-ADM generates an automatic framework to accurately represent discrete fractures at multiple scales, based on multilevel operators. • F-ADM develops multi-level embedded discrete fracture model, i.e., independent grids are used at all levels for fractures and matrix. • F-ADM is directly applicable to heterogeneous fractured media with no dependency on upscaled quantities. • F-ADM is a scalable and stable (fully-implicit) multiscale method for multiphase flow simulations in large-scale fractured media. Abstract We present an algebraic dynamic multilevel method for multiphase flow in heterogeneous fractured porous media (F-ADM), where fractures are resolved at fine scale with an embedded discrete modelling approach. This fine-scale discrete system employs independent fine-scale computational grids for heterogeneous matrix and discrete fractures, which results in linear system sizes out of the scope of the classical simulation approaches. To reduce the computational costs, yet provide accurate solutions, on this highly resolved fine-scale mesh, F-ADM imposes independent dynamic multilevel coarse grids for both matrix and lower-dimensional discrete fractures. The fully-implicit discrete system is then mapped into this adaptive dynamic multilevel resolution for all unknowns (i.e., pressure and phase saturation). The dynamic resolution aims for resolving sharp fronts for the transport unknowns, thus constant interpolators are used to map the saturation from coarse to fine grids both in matrix and fractures. However, due to the global nature of the pressure unknowns, local multilevel basis functions for both matrix and fractures with flexible matrix-fracture coupling treatment are introduced for the pressure. The assembly of the full sets of basis functions allows for mapping the solutions up and down between any resolutions. Due to its adaptive multilevel resolution, F-ADM develops an automatic integrated framework to homogenise or explicitly represent a fracture network at a coarser level by selection of the multilevel coarse nodes in each sub-domain. Various test cases, including multiphase flow in 2D and 3D media, are studied, where only a fraction of the fine-scale grids is employed to obtain accurate nonlinear multiphase solutions. F-ADM casts a promising approach for large-scale simulation of multiphase flow in fractured media. [ABSTRACT FROM AUTHOR]

The past decades have witnessed an increasing interest in numerical simulation for flow in fractured porous media. To date, most studies have focused on 2D or pseudo-3D computational models, where the impact of 3D complex structures on seepage has not been fully addressed. This work presents a method for modeling seepage in 3D heterogeneous porous media. The complex structures, typically the stochastic discrete fractures and inclusions, are able to be simulated. A mesh strategy is proposed to discretize the complex domain. In particular, a treatment on the intersected elements is developed to ensure a conforming mesh. Then, numerical discretization is provided, in which the flux interactions of fractures, inclusions and surrounding rock matrix are included. Numerical tests are performed to analyze the hydraulic characteristics of 3D fractured media. First, the developed framework is validated by comparing numerical solutions with the results of embedded discrete fracture model. Next, the effects of orientation, aperture and radius of fractures on fluid flow and equivalent permeability tensor are analyzed. The variations of pressure distribution are studied in heterogeneous and homogeneous media. Finally, the hydraulic properties of a medium with complex structures are investigated to show the difference of hydraulic feature between fractures and inclusions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We present the projection-based embedded discrete fracture model (pEDFM) for hexahedral corner-point grid (CPG) geometries, for the simulation of hydrothermal processes in fractured porous media. Unlike the previously-developed pEDFM for structured box grids, our new development allows for the modeling of complex geometries defined with hexahedral CPG cells. It also advances the pEDFM method to include coupled flow and heat transfer systems. Mass and energy conservation equations are simulated in a fully-coupled manner using a fully-implicit (FIM) integration scheme. This allows for stable simulations, specially when large time steps are taken. Independent corner-point grids are imposed on the rock matrix and all fractures, with conductivities ranging from highly permeable to flow barriers. The connectivities between the non-neighboring grid cells are described such that a consistent discrete representation of the embedded fractures occurs within the corner-point grid geometry, specially as the quadrilateral interfaces are allowed to be fully flexible. Various numerical tests including geologically-relevant and real-field models, which are established in the literature, are conducted to demonstrate the applicability of the developed method. It is shown that pEDFM can accurately capture the physical influence of both highly conductive fractures and flow barriers on the flow and heat transfer fields in complex reservoir geometries. This development is promising for flow simulations of real-field geo-models, increasing the discretization flexibility and enhancing the computational performance for capturing explicit fractures accurately. • Projection-based embedded discrete fracture model (pEDFM) for hexahedral corner-point Grid (CPG) geometries. • Accurate and explicit representation of fractures with generic conductivity contrasts. • Fully coupled mass and heat transport in fractured porous media. • Accurate modeling of geologically-relevant and real field-scale formations. [ABSTRACT FROM AUTHOR]