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In fractured geological formations, as a result of the in-situ stress changes, fractures can propagate or slide. This phenomenon can be efficiently modeled by the extended finite element method (XFEM) when there are only a few fractures present. However, geological reservoirs contain many fractures which can also cross and are densely populated. Therefore, the classical XFEM is too expensive to be applied for the simulation of propagating fractures in geological formations. To reduce the costs, typically, homogenization or upscaling is used. However, they result in inaccurate solutions, since no separation of scales exists in this process. To resolve this challenge, in this work, a multiscale XFEM (MS-XFEM) for propagating fractures is developed and presented. In each time step, given the current geometries of the fractures, local XFEM-based basis functions are constructed or adaptively updated. The adaptive update takes place in certain regions where fracture geometries are changed due to propagation. Using these basis functions, a very efficient FEM-based coarse-scale system is developed since it has no extra degrees of freedom (DOFs). Once the coarse-scale system is solved, its solution is prolonged to the fine-scale original resolution using the basis functions. This approximate fine-scale solution is then used to estimate the group of growing fractures tips and their growing angles. This allows for exploiting the locality of the propagation process fully while solving a global system. To control the error, an iterative procedure is also developed. Proof-of-concept test cases are presented to study the developed MS-XFEM algorithm. It is shown that MS-XFEM results are capable of predicting the propagating paths for complicated fracture patterns. As such, MS-XFEM casts a promising method for field-scale applications. • Adaptive MS-XFEM is developed to simulate multiple fractures propagation in geological formations. • Growth of fractures is captured by updating basis functions only in certain coarse scale elements. • Iterative strategy is used to control the error level. • MS-XFEM is accurate and efficient in simulation of fracture network propagation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The demand for accurate and efficient simulation of geomechanical effects is widely increasing in the geoscience community. High resolution characterizations of the mechanical properties of subsurface formations are essential for improving modeling predictions. Such detailed descriptions impose severe computational challenges and motivate the development of multiscale solution strategies. We propose a multiscale solution framework for the geomechanical equilibrium problem of heterogeneous porous media based on the finite-element method. After imposing a coarse-scale grid on the given fine-scale problem, the coarse-scale basis functions are obtained by solving local equilibrium problems within coarse elements. These basis functions form the restriction and prolongation operators used to obtain the coarse-scale system for the displacement-vector. Then, a two-stage preconditioner that couples the multiscale system with a smoother is derived for the iterative solution of the fine-scale linear system. Various numerical experiments are presented to demonstrate accuracy and robustness of the method. [ABSTRACT FROM AUTHOR]

A smoothed embedded finite-volume modeling (sEFVM) method is presented for faulted and fractured heterogeneous poroelastic media. The method casts a fully coupled strategy to treat the coupling between fault slip mechanics, deformation mechanics, and fluid flow equations. This ensures the stability and consistency of the simulation results, especially, as the fault slip is implicitly found through an iterative prediction-correction procedure. The computational grid is generated independently for embedded faults and rock matrix. The efficiency is further enhanced by extending the finite-volume discrete space by introducing only one degree of freedom per fault element. The embedded approach can lead to an oscillatory stress field at the fault, which damages the robustness of the implicit slip detection strategy. To resolve this challenge, a smoothed embedded strategy is devised, in which the stress and slip profiles are post processed within the iterative loops by fitting the best curve based on a least-square error criterion. The sEFVM provides locally conservative mass flux and stress fields, on staggered grid. Its performance is further investigated for several proof-of-the-concept test cases, including a multiple fault system in a heterogeneous domain. Results indicate that the method develops a promising approach for field-scale relevant simulation of induced seismicity. • A smoothed embedded finite volume method (sEFVM) is developed for faulted and fractured heterogeneous poroelastic media. • sEFVM solves for pore pressure, rock deformation and fault/fracture slip fully-coupled and fully implicitly. • sEFVM is stable and consistent, because the fault slip is implicitly found. • sEFVM provides locally conservative mass flux and stress fields. • Results indicate that sEFVM is a promising approach for field-scale simulations of induced seismicity. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

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]

Abstract: 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. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

• A multiscale (MS) method to study inelastic creep deformation of rocks is developed. • Effects of creep are studied under monotone and cyclic loading. • Fine-scale and MS results are investigated with the experimental data. • It allows for quantification of creep in the subsurface at different timescales. • The developments are made OA at https://gitlab.tudelft.nl/ADMIRE_Public/mechanics. Subsurface geological formations provide giant capacities for large-scale (TWh) storage of renewable energy, once this energy (e.g. from solar and wind power plants) is converted to green gases, e.g. hydrogen. The critical aspects of developing this technology to full-scale will involve estimation of storage capacity, safety, and efficiency of a subsurface formation. Geological formations are often highly heterogeneous and, when utilized for cyclic energy storage, entail complex nonlinear rock deformation physics. In this work, we present a novel computational framework to study rock deformation under cyclic loading, in presence of nonlinear time-dependent creep physics. Both classical and relaxation creep methodologies are employed to analyze the variation of the total strain in the specimen over time. Implicit time-integration scheme is employed to preserve numerical stability, due to the nonlinear process. Once the computational framework is consistently defined using finite element method on the fine scale, a multiscale strategy is developed to represent the nonlinear deformation not only at fine but also coarser scales. This is achieved by developing locally computed finite element basis functions at coarse scale. The developed multiscale method also allows for iterative error reduction to any desired level, after being paired with a fine-scale smoother. Numerical test cases are studied to investigate various aspects of the developed computational workflow, from benchmarking with experiments to analysing the impact of nonlinear deformation for a field-scale relevant environment. Results indicate the applicability of the developed multiscale method in order to employ nonlinear physics in their laboratory-based scale of relevance (i.e., fine scale), yet perform field-relevant simulations. The developed simulator is made publicly available at https://gitlab.tudelft.nl/ADMIRE_Public/mechanics. [ABSTRACT FROM AUTHOR]

• 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]

Abstract 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. Highlights • Fully-implicit conservative discretisation of the poroelasticity formulations. • Locally conservative multiscale method for finite-volume-based poroelasticity. • Real-field relevant method for quantification of the impact of reservoir operations on its surrounding including the subsurface. [ABSTRACT FROM AUTHOR]

An efficient multiscale (MS) gradient computation method for subsurface flow management and optimization is introduced. The general, algebraic framework allows for the calculation of gradients using both the Direct and Adjoint derivative methods. The framework also allows for the utilization of any MS formulation that can be algebraically expressed in terms of a restriction and a prolongation operator. This is achieved via an implicit differentiation formulation. The approach favors algorithms for multiplying the sensitivity matrix and its transpose with arbitrary vectors. This provides a flexible way of computing gradients in a form suitable for any given gradient-based optimization algorithm. No assumption w.r.t. the nature of the problem or specific optimization parameters is made. Therefore, the framework can be applied to any gradient-based study. In the implementation, extra partial derivative information required by the gradient computation is computed via automatic differentiation. A detailed utilization of the framework using the MS Finite Volume (MSFV) simulation technique is presented. Numerical experiments are performed to demonstrate the accuracy of the method compared to a fine-scale simulator. In addition, an asymptotic analysis is presented to provide an estimate of its computational complexity. The investigations show that the presented method casts an accurate and efficient MS gradient computation strategy that can be successfully utilized in next-generation reservoir management studies. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [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]

This paper presents the algebraic dynamic multilevel method (ADM) for compositional flow in three dimensional heterogeneous porous media in presence of capillary and gravitational effects. As a significant advancement compared to the ADM for immiscible flows (Cusini et al., 2016) [33] , here, mass conservation equations are solved along with k-value based thermodynamic equilibrium equations using a fully-implicit (FIM) coupling strategy. Two different fine-scale compositional formulations are considered: (1) the natural variables and (2) the overall-compositions formulation. At each Newton's iteration the fine-scale FIM Jacobian system is mapped to a dynamically defined (in space and time) multilevel nested grid. The appropriate grid resolution is chosen based on the contrast of user-defined fluid properties and on the presence of specific features (e.g., well source terms). Consistent mapping between different resolutions is performed by the means of sequences of restriction and prolongation operators. While finite-volume restriction operators are employed to ensure mass conservation at all resolutions, various prolongation operators are considered. In particular, different interpolation strategies can be used for the different primary variables, and multiscale basis functions are chosen as pressure interpolators so that fine scale heterogeneities are accurately accounted for across different resolutions. Several numerical experiments are conducted to analyse the accuracy, efficiency and robustness of the method for both 2D and 3D domains. Results show that ADM provides accurate solutions by employing only a fraction of the number of grid-cells employed in fine-scale simulations. As such, it presents a promising approach for large-scale simulations of multiphase flow in heterogeneous reservoirs with complex non-linear fluid physics. [ABSTRACT FROM AUTHOR]

We present a scalable collocated Finite Volume Method (FVM) to simulate induced seismicity as a result of pore pressure changes. A discrete system is obtained based on a fully-implicit fully-coupled description of flow, elastic deformation, and contact mechanics at fault surfaces on a flexible unstructured mesh. The cell-centered collocated scheme leads to a convenient integration of the different physical equations, as the unknowns share the same discrete locations on the mesh. Additionally, a generic multi-point flux approximation is formulated to treat heterogeneity, anisotropy, and cross-derivative terms for both flow and mechanics equations. The resulting system, though flexible and accurate, can lead to excessive computational costs for field-relevant applications. To resolve this limitation, a scalable processing algorithm is developed and presented. Several proof-of-concept numerical tests, including benchmark studies with analytical solutions, are investigated. It is found that the presented method is indeed accurate and efficient; and provides a promising framework for accurate and efficient simulation of induced seismicity in various geoscientific applications. • We develop a collocated Finite Volume scheme for faulted poroelastic media. • The scheme works on the cell-centered grid, it supports faults, arbitraty material anisotropy and heterogeneity. • Fully coupled multi-point stress and multi-point flux approximations derived from local balance are extended to faulted media. • We validate the scheme in several benchmarks relevant for induced seismicity. • Block-partitioned preconditioning strategy is applied to facilitate the scalability of the developed framework. [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 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. [ABSTRACT FROM AUTHOR]

Accurate and efficient simulation of multiphase flow in heterogeneous porous media motivates the development of space-time multiscale strategies for the coupled nonlinear flow (pressure) and saturation transport equations. The flow equation entails heterogeneous high-resolution (fine-scale) coefficients and is global (elliptic or parabolic). The time-dependent saturation profile, on the other hand, may exhibit sharp local gradients or discontinuities (fronts) where the solution accuracy is highly sensitive to the time-step size. Therefore, accurate flow solvers need to address the multiscale spatial scales, while advanced transport solvers need to also tackle multiple time scales. This paper presents the first multirate multiscale method for space-time conservative multiscale simulation of sequentially coupled flow and transport equations. The method computes the pressure equation at the coarse spatial scale with a multiscale finite volume technique, while the transport equation is solved by taking variable time-step sizes at different locations of the domain. At each coarse time step, the developed local time-stepping technique employs an adaptive recursive time step refinement to capture the fronts accurately. The applicability (accuracy and efficiency) of the method is investigated for a wide range of two-phase flow simulations in heterogeneous porous media. For the studied cases, the proposed method is found to provide 3 to 4 times faster simulations. Therefore, it provides a promising strategy to minimize the tradeoff between accuracy and efficiency for field-scale applications. • The pressure is solved at coarse-scale in space. • The approximate fine-scale pressure solution is found via multiscale basis functions. • The saturation is solved by developing a conservative multiscale in time method. • The saturation is solved using different time steps at different locations. • The approach casts efficient and accurate simulations for large-scale porous media. [ABSTRACT FROM AUTHOR]