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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.

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... When combined with an iterative multiscale stage, this method has proven to be efficient and robust with results comparable to multigrid methods [18] . Several authors have successfully modified the AMS to incorporate new features such as embedded fracture models [19] , more complex physics [20] , multi-level multiscale simulation [21,22] , and to create a general framework that allows the integration of new models in a unified way [23] . ...

... This framework addresses three main issues required for a proper multiscale simulation on general unstructured grids: i) A background grid is generated that creates the primal and dual grid, simultaneously; ii) we employ a robust Multi-Point Flux Approximation with a Diamond Stencil with full pressure support; and iii) we generalise the AMS operators and modify them to avoid the leaking of the basis functions outside their corresponding support regions. Lastly, we adapt well established iterative procedures [5,19,25] used to improve the quality of the multiscale solution decreasing high-frequency errors. ...

... We denote the fine-scale discretization of the elliptic pressure equation by: (18) where T f , p f and F f represent the respective assembled matrix, pressure solution, boundary conditions including source and sink terms on the fine-scale. The wirebasket segregation [17][18][19] is the grouping of fine-scale volumes into different categories: Nodes, Edges, and Internals in 2D (See Fig. 3 ), here Internals refers to the centres of the fine scale cells enclosed by the (red) edges. ...

... When combined with an iterative multiscale stage, this method has proven to be efficient and robust with results comparable to multigrid methods [18]. Several authors have successfully modified the AMS to incorporate new features such as embedded fracture models [19], more complex physics [20], multi-level multiscale simulation [21,22], and to create a general framework that allows the integration of new models in a unified way [23]. ...

... This framework addresses three main issues required for a proper multiscale simulation on general unstructured grids: i) A background grid is generated that creates the primal and dual grid, simultaneously; ii) we employ a robust Multi-Point Flux Approximation with a Diamond Stencil with full pressure support; and iii) we generalise the AMS operators and modify them to avoid the leaking of the basis functions outside their corresponding support regions. Lastly, we adapt well established iterative procedures [19,25,5] used to improve the quality of the multiscale solution decreasing high-frequency errors. ...

... We denote the fine-scale discretization of the elliptic pressure equation by: (18) where T f , p f and F f represent the respective assembled matrix, pressure solution, boundary conditions including source and sink terms on the fine-scale. The wirebasket segregation [17,18,19] is the grouping of fine-scale volumes into different categories: Nodes, Edges, and Internals in 2D (See Figure 3), here Internals refers to the centres of the fine scale cells enclosed by the (red) edges. ...

We introduce a new multiscale finite volume framework for the simulation of multi-phase flow in heterogenous and anisotropic porous media on quite general unstructured grids that enable geophysical grid defined properties to be used directly on a high definition grid.
A novel background grid strategy is presented, where an auxiliary mesh aids the creation of multiscale primal and dual coarse grids. Due to the unstructured grid connectivity, the dual grids do not retain the natural structured grid decomposition. This may induce errors in the computed basis functions, and contribute to loss of mass conservation. A generalization of the Algebraic Multiscale Solver (AMS) is proposed to prevent the basis function induced leakage outside the support region enabling coupling with a Control Volume Distributed MultiPoint Flux Approximation (CVD-MPFA), ensuring consistency on non k-orthogonal unstructured grids.
We present three different problems in which the accuracy of the framework is validated by comparing the multiscale solver with the direct simulation on the fine-scale. The results obtained show that the method can produce well resolved solutions for two-phase flow in highly heterogeneous and anisotropic porous media using general unstructured grids on all scales. As a consequence, the method captures the most important flow features that result on high-resolution geophysical models while providing suitable discretization for complex geological formations found in current petroleum reservoir problems.
The novelty of this scheme involves three components which are combined to enable consistent multiscale simulation on unstructured grids: i) a new approach to primal and dual coarse grid generation, ii) a novel technique to prevent basis function induced leakage and iii) the coupling of the Algebraic Multiscale Solver (AMS) with a Multipoint Flux approximation with a Diamond Stencil (MPFA-D).

... Similar as the extensive studies for multiscale simulation of flow in porous media (see e.g. [55][56][57]), for deformation simulation, a CPU-based study should be performed to obtain the optimum combination of coarse-grid resolution, type and count of smoothing steps. ...

... Also, from the presented operational-based analyses, one can consider an optimum coarsening ratio, in which the multiscale procedure would perform optimal. Note that, similar as for multiphase flow simulations [57,59], a CPU-based analyses is needed to draw more conclusive remarks about the real speedup of the MS-XFEM. ...

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 promissing scalable XFEM method for large-scale heavily fractured porous media.

... veniently for elliptic [6,7,8,9] and parabolic [10,11] flow equations within 12 sequential [12,13,14,15,16] and fully implicit [17,18] multiphase simulation 13 frameworks. Recent advances include extensions to geothermal flows [19,20] 14 and fractured heterogeneous media [21,22,23,24,25,26]. Note that the 15 difference between the multiscale finite volume and finite element methods 16 is in the coarse scale system solution strategies. ...

... As such, 18 the development of either of them leads to advancement of the alternative 19 method too. An essential feature of this class of multiscale simulation is 20 that it can be formulated algebraically for two-level [27,28,29] and dynamic 21 multilevel [30] simulations, in which the difference between the finite-element 22 and finite-volume approaches is only in the choice of the restriction operator. ...

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 pro-file, 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.

... Multiscale methods for porous media are studied extensively at the continuum (or Darcy) scale. They include multiscale finite element (MsFE) [12,[34][35][36][37][38][39][40][41], multiscale finite volume (MsFV) [42][43][44][45][46][47], and multiscale mortar finite element (MoMsFE) [48][49][50][51], which solve elliptic/parabolic equations of flow and mechanics. At the pore scale, multiscale methods are much less developed and have focused primarily on fluid flow [52][53][54][55][56][57][58][59][60][61]. ...

... The prolongation operator, which includes the basis functions at different coarsening levels, is constructed in an algebraic manner [39]. To compute the basis functions, the finescale system is assembled first. ...

We develop a multiscale simulation strategy, namely, algebraic dynamic multilevel (ADM) method, for simulation of fluid flow and heat transfer in fractured geothermal reservoirs under varying thermodynamic conditions. Fractures with varying conductivities are modeled using the projection-based embedded discrete fracture model (pEDFM) in an explicit manner. The developed ADM method allows the fine-scale system to be mapped to a discrete domain with an adaptive grid resolution via the use of the restriction and prolongation operators. The developed framework is used a) to investigate the impacts of formulations with different primary variables on the simulation results, and b) to assess the performance of ADM in a high-enthalpy reservoir by comparing the simulation results against those obtained from fine-scale grids. Results show that the two formulations produce similar results in the case of single-phase flow, which indicates that the molar formulation is a favorable option that can be applied to varying thermodynamic conditions. Moreover, the ADM can provide accurate solutions with only a fraction of fine-scale grids, e.g., for the studied case, the maximum error is by average 1.3 with only 42% of active cells, thereby improving the computational efficiency. This is promising for applying the developed method to field-scale geothermal systems.

... Multiscale strategy has been implemented in the past for solving porous media flow in the subsurface [61][62][63][64][65][66][67][68] . This strategy has been implemented also for geo-mechanics [69][70][71][72] . ...

Subsurface geological formations can be utilized to safely store large-scale (TWh) renewable energy in the form of green gases such as hydrogen. Successful implementation of this technology involves estimating feasible storage sites, including rigorous mechanical safety analyses. Geological formations are often highly heterogeneous and entail complex nonlinear inelastic rock deformation physics when utilized for cyclic energy storage. In this work, we present a novel scalable computational framework to analyse the impact of nonlinear deformation of porous reservoirs under cyclic loading. The proposed methodology includes three different time-dependent nonlinear constitutive models to appropriately describe the behavior of sandstone, shale rock and salt rock. These constitutive models are studied and benchmarked against both numerical and experimental results in the literature. An implicit time-integration scheme is developed to preserve the stability of the simulation. In order to ensure its scalability, the numerical strategy adopts a multiscale finite element formulation, in which coarse scale systems with locally-computed basis functions are constructed and solved. Further, the effect of heterogeneity on the results and estimation of deformation is analyzed. Lastly, the Bergermeer test case—an active Dutch natural gas storage field—is studied to investigate the influence of inelastic deformation on the uplift caused by cyclic injection and production of gas. The present study shows acceptable subsidence predictions in this field-scale test, once the properties of the finite element representative elementary volumes are tuned with the experimental data.

... Accurate modeling of multiphase fluid flow in subsurface reservoirs is critical in a wide range of environmental and energy applications, including geothermal extraction (Hoteit et al. 2021;Santoso et al. 2019aSantoso et al. , 2019bHe et al. 2021a), oil recovery in fractured reservoirs , and underground carbon storage (Hoteit et al. 2019;Pal et al. 2022). The process requires history matching and model calibration to minimize the mismatch between predictions and observed data (AlAmeri et al. 2020;Omar et al. 2021;Santoso et al. 2019aSantoso et al. , 2019bŢene et al. 2016). To improve model predictability, common procedures often involve adjusting geological characteristics (such as porosity and permeability), and fluid properties (e.g., fluid viscosity and density), and rock-fluid interactions (e.g., capillary pressure and relative permeability) (AlAmeri et al. 2020;Liu and Durlofsky 2020). ...

History matching is a critical process used for calibrating simulation models and assessing subsurface uncertainties. This common technique aims to align the reservoir models with the observed data. However, achieving this goal is often challenging due to the nonuniqueness of the solution, underlying subsurface uncertainties, and usually the high computational cost of simulations. The traditional approach is often based on trial and error, which is exhaustive and labor-intensive. Some analytical and numerical proxies combined with Monte Carlo simulations are used to reduce the computational time. However, these approaches suffer from low accuracy and may not fully capture subsurface uncertainties. This study proposes a new robust method using Bayesian Markov chain Monte Carlo (MCMC) to perform assisted history matching under uncertainties. We propose a novel three-step workflow that includes (1) multiresolution low-fidelity models to guarantee high-quality matching; (2) long-short-term memory (LSTM) network as a low-fidelity model to reproduce continuous time response based on the simulation model, combined with Bayesian optimization to obtain the optimum low-fidelity model; and (3) Bayesian MCMC runs to obtain the Bayesian inversion of the uncertainty parameters. We perform sensitivity analysis on the LSTM’s architecture, hyperparameters, training set, number of chains, and chain length to obtain the optimum setup for Bayesian-LSTM history matching. We also compare the performance of predicting the recovery factor (RF) using different surrogate methods, including polynomial chaos expansions (PCE), kriging, and support vector machines for regression (SVR). We demonstrate the proposed method using a water flooding problem for the upper Tarbert formation of the 10th SPE comparative model. This study case represents a highly heterogeneous nearshore environment. Results showed that the Bayesian-optimized LSTM has successfully captured the physics in the high-fidelity model. The Bayesian-LSTM MCMC produces an accurate prediction with narrow ranges of uncertainties. The posterior prediction through the high-fidelity model ensures the robustness and accuracy of the workflow. This approach provides an efficient and practical history-matching method for reservoir simulation and subsurface flow modeling with significant uncertainties.

... However, the DFM is an accurate tool to describe the flow in fractured porous media. In another approach, the fractures are not resolved by grid but are considered as an overlaying continuum (embedded fracture model, EFM) [31,43,42]. In EFM, matrix and fracture are viewed as two porosity/permeability types co-existing at one spatial location. ...

We consider the coupled system of equations that describe flow in fractured porous media. To describe such types of problems, multicontinuum and multiscale approaches are used. Because in multicontinuum models, the permeability of each continuum has a significant difference, a large number of iterations is required for the solution of the resulting linear system of equations at each time iteration. The presented decoupling technique separates equations for each continuum that can be solved separately, leading to a more efficient computational algorithm with smaller systems and faster solutions. This approach is based on the additive representation of the operator with semi-implicit approximation by time, where the continuum coupling part is taken from the previous time layer. We apply, analyze and numerically investigate decoupled schemes for classical multicontinuum problems in fractured porous media on sufficiently fine grids with finite volume approximation. We show that the decoupled schemes are stable, accurate, and computationally efficient. Next, we extend and investigate this approach for multiscale approximation on the coarse grid using the nonlocal multicontinuum (NLMC) method. In NLMC approximation, we construct similar decoupled schemes with the same continuum separation approach. A numerical investigation is presented for model problems with two and three-continuum in the two-dimensional formulation.

... The interested reader is referred to the paper by Lie et al [239] for an informative discussion of the evolution of the two methods. From the early application to single-phase incompressible subsurface flow problems on Cartesian grids [232], active research allowed to progressively extend the application of Ms methods to advanced models, incorporating the flow of compressible phases with capillarity [234,240], gravity [240,241], components [27,242], sophisticated well controls [243,244], fractured porous media [245][246][247][248] and unstructured grids [249,250]. Still, for these more complex models, Ms methods are typically exploited in sequential solving techniques to address the associated pressure equation. ...

Linear solvers for reservoir simulation applications are the objective of this review. Specifically, we focus on techniques for Fully Implicit (FI) solution methods, in which the set of governing Partial Differential Equations (PDEs) is properly discretized in time (usually by the Backward Euler scheme), and space, and tackled by assembling and linearizing a single system of equations to solve all the model unknowns simultaneously. Due to the usually large size of these systems arising from real-world models, iterative methods, specifically Krylov subspace solvers, have become conventional choices; nonetheless, their success largely revolves around the quality of the preconditioner that is supplied to accelerate their convergence. These two intertwined elements, i.e., the solver and the preconditioner, are the focus of our analysis, especially the latter, which is still the subject of extensive research. The progressive increase in reservoir model size and complexity, along with the introduction of additional physics to the classical flow problem, display the limits of existing solvers. Intensive usage of computational and memory resources are frequent drawbacks in practice, resulting in unpleasantly slow convergence rates. Developing efficient, robust, and scalable preconditioners, often relying on physics-based assumptions, is the way to avoid potential bottlenecks in the solving phase. In this work, we proceed in reviewing principles and state-of-the-art of such linear solution tools to summarize and discuss the main advances and research directions for reservoir simulation problems. We compare the available preconditioning options, showing the connections existing among the different approaches, and try to develop a general algebraic framework.

... Moreover, for impermeable fractures, pEDFM should be considered [35][36][37][38]. We though that the basic framework (loadbalanced domain decomposition and inter-processor communication) of this work is also practicable for parallel pEDFM, but two adjustments are required: one is that when decompose the whole model to different parts, the fracture grids on the boundary of sub-domains should be handled elaborately to make the fracture projections available in parallel solving; and another one is considering the influence of adjusted matrix transmissibility on load-balance. ...

Recently, embedded discrete fracture model (EDFM) using non-conforming staggered 3D unstructured grids was presented for the numerical simulation of fluid-flows in large-scale complex fractured porous media. In this paper, we implement load-balanced parallel simulations for this approach on distributed memory systems. A load-balanced domain decomposition algorithm is presented to partition matrix grids and embedded fracture grids uniformly. We classify all grid connections on the boundary of subdomains to five types and illustrate corresponding inter-processor connections. A packaged inter-processor communication approach is implemented based on message passing interface (MPI) to improve communication efficiency for this simulation method. A two-phase parallel simulator is developed, and block coupled method is adopted to assemble the equations discretized in matrix and grid domains using two-point flux approximation (TPFA) finite volume method. The load-balanced algorithm is tested by nonuniformly distributed fracture networks. The performance and scalability are also tested on 1024 CPU cores by two complex fracture network models with up to 25 million matrix grids and 11 million embedded fracture grids. The results show satisfactory parallel efficiency and scalability.

... In general, both of these two approaches can be used in structured an unstructured grid, but they have different robustness and computational overhead. Moreover, multiscale method is also applied into this EDFM implementation to accelerate simulation [21,26,27]. ...

Numerical simulation in 3D complex fractured media is challenging as the geometric discretization of porous media and fracture domains is on multiple scales. In this paper, we present an improved embedded discrete fracture model (EDFM) for 3D unstructured non-matching grids, and correspondingly efficient and robust domain connectivity algorithms are developed. Unlike traditional EDFM, improved EDFM is capable to simulate flows on 3D staggered overlapping unstructured matrix and fracture grids. We illustrate flux exchange patterns and formulas of improved EDFM. Two types of domain connectivity are built to assemble hybrid dimensional overlapping grids of matrix and fracture domains to a coupled linear system using finite volume method. Building domain connectivity is an important challenge for 3D hybrid-dimensional unstructured non-matching grids. We classify all points of domain connectivity into three types. A face indexed octree is employed for spatial binary search. Two types of grid adjacency information are used to accelerate the calculation of grid intersection points. Fracture-matrix intersection information is adopted to accelerate fracture-fracture connectivity generation. The performance of the algorithms is tested by 8 cases in different scales and compared with traversal method. Improved EDFM is validated against discrete fracture model and traditional EDFM using a two-phase flow solver. A case with highly complex fracture network is simulated to demonstrate the capacity of improved EDFM in real complex physical problems. The effects of grid type and resolution are also analysed in this work.

... The GMsFEM has been studied for a various applications related to poroelasticity problems [9,10,3,40]. The multiscale finite volume method has been applied for the simulation of the flow problems in fractured porous media [23,39]. For the effective numerical solution of such problems different homogenization techniques have been developed [30,29,5,37,6,21]. ...

In this work, we introduce a time memory formalism in poroelasticity model that couples the pressure and displacement. We assume this multiphysics process occurs in multicontinuum media. The mathematical model contains a coupled system of equations for pressures in each continuum and elasticity equations for displacements of the medium. We assume that the temporal dynamics is governed by fractional derivatives following some works in the literature. We derive an implicit finite difference approximation for time discretization based on the Caputo time fractional derivative. A Discrete Fracture Model (DFM) is used to model fluid flow through fractures and treat the complex network of fractures. We assume different fractional powers in fractures and matrix due to slow and fast dynamics. We develop a coarse grid approximation based on the Generalized Multiscale Finite Element Method (GMsFEM), where we solve local spectral problems for construction of the multiscale basis functions. We present numerical results for the two-dimensional model problems in fractured heterogeneous porous media. We investigate error analysis between reference (fine-scale) solution and multiscale solution with different numbers of multiscale basis functions. The results show that the proposed method can provide good accuracy on a coarse grid.

... This method did not allow for independent coarse grids for fracture and matrix domains. An algebraic multiscale method with embedded discrete fractures (F-AMS) was developed on structured grids [15]. The F-AMS relies on primal and dual coarse grids for both the matrix and fracture networks to construct the multiscale prolongation and restriction operators. ...

The multiscale finite volume method for discrete fracture modeling in highly heterogeneous porous media is developed. Multiscale methods are sensitive to the heterogeneity contrasts in both matrix and fracture networks. To resolve this, efficient algorithms for generating adaptive unstructured coarse grids are devised. First, primal coarse grids are independently constructed for the matrix and lower dimensional fractures. Then, flexible dual coarse grids are generated based on the fracture and matrix permeability features. Since the proposed algorithms employ the equivalent graph of unstructured grids, the same coarse grid generation strategy is applied for the fractures and matrix domains. Permeability-adapted coarse grids significantly improve the monotonicity behavior of MSFV method in highly heterogeneous fractured porous media. The performance of the method is assessed through several challenging test cases with highly heterogeneous permeability field in both fractures and matrix domain. Numerical results indicate that the extended MSFV method with adaptive unstructured coarse grids is a significant development for accurate flow simulation in heterogeneous fractured media using DFM approach.

... populated using stochastic methods ( Narr et al., 2006 ). Various upscaling methods are then used to generate the simulation model that could be based on conceptual dual-continua, discrete-fractured, or embeddedmultiscale approaches ( Hajibeygi et al., 2011;Rodriguez et al., 2006;Aene et al., 2016;Karimi-Fard and Durlofsky, 2016 ). Fracture characterization, including well logging, core plugs, and sometimes outcrops, are acquired and analyzed to determine the fracture density, orientation, apertures, and other data, which are critical to constrain the DFN. ...

Hydraulic properties of natural fractures are essential parameters for the modeling of fluid flow and transport in subsurface fractured porous media. The cubic law, based on the parallel-plate concept, has been traditionally used to estimate the hydraulic properties of individual fractures. This upscaling approach, however, is known to overestimate the fractures hydraulic properties. Dozens of methods have been proposed in the literature to improve the accuracy of the cubic law. The relative performance of these various methods is not well understood. In this work, a comprehensive review and benchmark of almost all commonly used cubic law-based approaches in the literature, covering 43 methods is provided. We propose a new corrected cubic law for incompressible, single-phase laminar flow through rough-walled fractures. The proposed model incorporates corrections to the hydraulic fracture aperture based on the flow tortuosity and local roughness of the fracture walls. We identify geometric rules relative to the local characteristic of the fracture and apply an efficient algorithm to subdivide the fracture into segments, accordingly. High-resolution simulations for Navier-Stokes equations, computed in parallel, for synthetic fractures with various ranges of surface roughness and apertures are then performed. The numerical solutions are used to assess the accuracy of the proposed model and compare it with the other 43 approaches, where we demonstrate its superior accuracy. The proposed model retains the simplicity and efficiency of the cubic law but with pronounced improvement to its accuracy. The data set used in the benchmark, including more than 7500 fractures, is provided in open-access.

... The upscaling method for the Richards equation are presented in [29]. In [30][31][32][33][34], the authors present a multiscale methods for filtration problem in fractured porous media. The effective algorithm of generalized multiscale finite element method (GMsFEM) for filtration problems in fractured heterogeneous porous media are developed in [30,35,36]. ...

In this paper, we present a multiscale model reduction technique for unsaturated filtration problem in fractured porous media using an Online Generalized Multiscale finite element method. The flow problem in unsaturated soils is described by the Richards equation. To approximate fractures we use the Discrete Fracture Model (DFM). Complex geometric features of the computational domain requires the construction of a fine grid that explicitly resolves the heterogeneities such as fractures. This approach leads to systems with a large number of unknowns, which require large computational costs. In order to develop a more efficient numerical scheme, we propose a model reduction procedure based on the Generalized Multiscale Finite element method (GMsFEM). The GMsFEM allows solving such problems on a very coarse grid using basis functions that can capture heterogeneities. In the GMsFEM, there are offline and online stages. In the offline stage, we construct snapshot spaces and solve local spectral problems to obtain multiscale basis functions. These spectral problems are defined in the snapshot space in each local domain. To improve the accuracy of the method, we add online basis functions in the online stage. The construction of the online basis functions is based on the local residuals. The use of online bases will allow us to get a significant improvement in the accuracy of the method. We present results with different number of offline and online multisacle basis functions. We compare all results with reference solution. Our results show that the proposed method is able to achieve high accuracy with a small computational cost.

... Russian et al. (2019) used stochastic and global sensitivity analysis to study drilling mud losses in fractured media. Other authors proposed various numerical methods based on higher-order discretizations (Ambartsumyan et al. 2019;Girault and Rivière 2009;Arbogast and Brunson 2007;Shao et al. 2016;Ţene et al. 2016;Hoteit and Firoozabadi 2008). ...

Loss of circulation while drilling is a challenging problem that may interrupt operations and contaminate the subsurface formation. Analytical modeling of fluid flow in fractures is a tool that can be quickly deployed to assess drilling mud leakage into fractures. A new semi-analytical solution is developed to model the flow of non-Newtonian drilling fluid in fractured formation. The model is applicable for various fluid types exhibiting yield-power law (Herschel-Bulkley). We use finite-element simulations to verify our solutions. We also generate type curves and compare them to others in the literature. We then demonstrate the applicability of the proposed model for two field cases encountering lost circulations. To address the subsurface uncertainty, we combine the semi-analytical solutions with Monte Carlo and generate probabilistic predictions. The solution method can estimate the range of fracture conductivity, parametrized by the fracture hydraulic aperture, and time-dependent fluid loss rate that can predict the cumulative volume of lost fluid.

... The multiscale formulation of this type, i.e. local basis function-based, has been applied and developed by various researchers in the past. ( [34][35][36][37][38][39][40][41][42][43]). Researchers have employed multiscale strategy to study flow in porous media [44][45][46], geo-mechanics in porous media [47][48][49][50][51] and other non-linear problems [52][53][54]. ...

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.

... A popular approach is the Discrete Fracture Model (DFM) which uses unstructured grids to place fractures at the interface between matrix cells [39,40,41]. Other approaches are the Embedded DFM [42,43], the multi-continuum model [44,45] and the hierarchical fracture models [46,47,48]. ...

It is well known that domain-decomposition-based multiscale mixed methods rely on interface spaces, defined on the skeleton of the decomposition, to connect the solution among the non-overlapping subdomains. Usual spaces, such as polynomial-based ones, cannot properly represent high-contrast channelized features such as fractures (high permeability) and barriers (low permeability) for flows in heterogeneous porous media. We propose here new interface spaces, which are based on physics, to deal with permeability fields in the simultaneous presence of fractures and barriers, accommodated respectively, by the pressure and flux spaces. Existing multiscale methods based on mixed formulations can take advantage of the proposed interface spaces, however, in order to present and test our results, we use the newly developed Multiscale Robin Coupled Method (MRCM) [Guiraldello, et al., J. Comput. Phys., 355 (2018) pp. 1-21], which generalizes most well-known multiscale mixed methods, and allows for the independent choice of the pressure and flux interface spaces. An adaptive version of the MRCM [Rocha, et al., J. Comput. Phys., 409 (2020), 109316] is considered that automatically selects the physics-based pressure space for fractured structures and the physics-based flux space for regions with barriers, resulting in a procedure with unprecedented accuracy. The features of the proposed approach are investigated through several numerical simulations of single-phase and two-phase flows, in different heterogeneous porous media. The adaptive MRCM combined with the interface spaces based on physics provides promising results for challenging problems with the simultaneous presence of fractures and barriers.

... For two-dimensional structured grids, an analytical solution is available for d ik [57]. For fracture-fracture NNC, star-delta transformation is used to compute the fracturefracture transfer coefficient T f−f (jk) for a fracture intersection [58]: ...

In this article, an open-source code for the simulation of fluid flow, including adsorption, transport, and indirect hydromechanical coupling in unconventional fractured reservoirs is described. The code leverages cutting-edge numerical modeling capabilities like automatic differentiation, stochastic fracture modeling, multicontinuum modeling, and discrete fracture models. In the fluid mass balance equation, specific physical mechanisms, unique to organic-rich source rocks, are included, like an adsorption isotherm, a dynamic permeability-correction function, and an Embedded Discrete Fracture Model (EDFM) with fracture-to-well connectivity. The code is validated against an industrial simulator and applied for a study of the performance of the Barnett shale reservoir, where adsorption, gas slippage, diffusion, indirect hydromechanical coupling, and propped fractures are considered. It is the first open-source code available to facilitate the modeling and production optimization of fractured shale-gas reservoirs. The modular design also facilitates rapid prototyping and demonstration of new models. This article also contains a quantitative analysis of the accuracy and limitations of EDFM for gas production simulation in unconventional fractured reservoirs.

... Several applications of EDFM can be found in the literature. [16][17][18] More recently, Ţene et al. 19 revealed that EDFM techniques are most applicable in cases where the permeability of fractures is higher than the surrounding porous bulk. To address this issue, a projection-based EDFM (pEDFM) was introduced which is capable of capturing the effect of fracture conductivities ranging from high (e.g. ...

A new discrete fracture model is introduced to simulate the steady-state fluid flow in discontinuous porous media. The formulation uses a multi-layered approach to capture the effect of both longitudinal and transverse permeabil-ity of the discontinuities in the pressure distribution. The formulation allows the independent discretisation of mesh and discontinuities, which do not need to conform. Given that the formulation is developed at the element level, no additional degrees of freedom or special integration procedures are required for coupling the non-conforming meshes. The proposed model is shown to be reliable regardless of the permeability of the discontinuity being higher or lower than the surrounding domain. Four numerical examples of increasing complexity are solved to demonstrate the efficiency and accuracy of the new technique when compared with results available in the literature. Results show that the proposed method can simulate the fluid pressure distribution in fractured porous media. Furthermore, a sensitivity analysis demonstrated the stability regarding the condition number for wide range values of the coupling parameter.

... As the localization achieved through these boundary conditions, they are named as the localization scheme , which is the only assumption used in the MsFV framework. Different types of the localization assumption have been proposed and used in the related literature, such as the Linear Boundary Condition (LBC) [9][10][11][12][13][14][15]24,33,34] , the Variable (reduced) Boundary Condition (VBC) [16][17][18][19][20][34][35][36] . Based on the localization assumption, a set of local and independent problems in each dual block is constructed and solved to produce the basis and correction functions. ...

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.

... Discrete fracture model (DFM) is associated with unstructured grids with explicit meshing of the fracture geometry, where the fractures are located at the interfaces between matrix cells (Hoteit and Firoozabadi 2008;Karimi-Fard et al. 2003;Karimi-Fard and Firoozabadi 2001;Garipov et al. 2016;Bosma et al. 2017). In embedded discrete fracture model (EDFM), the fracture mesh is not conformed with porous matrix mesh and structured grid can be used (Lee et al. 2001;Li et al. 2006;Hajibeygi et al. 2011;Tene et al. 2016bTene et al. , 2017Tene et al. 2016a). Fig. 1, where the fine grid mesh T h is depicted by a blue color and the fracture mesh E γ by a red color). ...

In this work, we present an upscaled model for mixed dimensional coupled flow problem in fractured porous media. We consider both embedded and discrete fracture models (EFM and DFM) as fine scale models which contain coupled system of equations. For fine grid discretization, we use a conservative finite-volume approximation. We construct an upscaled model using the non-local multicontinuum (NLMC) method for the coupled system. The proposed upscaled model is based on a set of simplified multiscale basis functions for the auxiliary space and a constraint energy minimization principle for the construction of multiscale basis functions. Using the constructed NLMC-multiscale basis functions, we obtain an accurate coarse grid upscaled model. We present numerical results for both fine-grid models and upscaled coarse-grid models using our NLMC method. We consider model problems with (1) discrete fracture fine grid model with low and high permeable fractures; (2) embedded fine grid model for two types of geometries with differnet fracture networks and (3) embedded fracture fine grid model with heterogeneous permeability. The simulations using the upscaled model provide very accurate solutions with significant reduction in the dimension of the problem.

... To reduce the size of the discrete system, various multiscale methods have been developed [21,22,23,24,25,26]. In [27,28], the multiscale finite volume method was presented for solution of the flow problems in fractured porous media. Multiscale finite volume method for solution of the poroelasticity problem is presented in [29]. ...

In this paper, we consider a poroelasticity problem in heterogeneous multicontinuum media that is widely used in simulations of the unconventional hydrocarbon reservoirs and geothermal fields. Mathematical model contains a coupled system of equations for pressures in each continuum and effective equation for displacement with volume force sources that are proportional to the sum of the pressure gradients for each continuum. To illustrate the idea of our approach, we consider a dual continuum background model with discrete fracture networks that can be generalized to a multicontinuum model for poroelasticity problem in complex heterogeneous media. We present a fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) approach for two and three-dimensional formulations. The coarse grid approximation is constructed using the Generalized Multiscale Finite Element Method (GMsFEM), where we solve local spectral problems for construction of the multiscale basis functions for displacement and pressures in multicontinuum media. We present numerical results for the two and three dimensional model problems in heterogeneous fractured porous media. We investigate relative errors between reference fine grid solution and presented coarse grid approximation using GMsFEM with different numbers of multiscale basis functions. Our results indicate that the proposed method is able to give accurate solutions with few degrees of freedoms.

... Upscaling method for the Richards equation is presented in [16]. Multiscale methods for solution of the flow problems in fractured porous media are presented in [17,18,19,20,21]. In our previous works, we developed multiscale model reduction techniques based on the Generalized Multiscale Finite Element Method (GMsFEM) for flow in fractured porous media [17,6,22]. ...

In this paper, we present a multiscale method for simulations of the multicontinua unsaturated flow problems in heterogeneous fractured porous media. The mathematical model is described by the system of Richards equations for each continuum that coupled by the specific transfer term. To illustrate the idea of our approach, we consider a dual continua background model with discrete fractures networks that generalized as a multicontinua model for unsaturated fluid flow in the complex heterogeneous porous media. We present fine grid approximation based on the finite element method and Discrete Fracture Model approach. In this model, we construct an unstructured fine grid that take into account a complex fracture geometries for two and three dimensional formulations. Due to construction of the unstructured grid, the fine grid approximation leads to the very large system of equations. For reduction of the discrete system size, we develop a multiscale method for coarse grid approximation of the coupled problem using Generalized Multiscale Finite Element Method. In this method, we construct a coupled multiscale basis functions that used to construct highly accurate coarse grid approximation. The multiscale method allowed us to capture detailed interactions between multiple continua. We investigate accuracy of the proposed method for the several test problems in two and three dimensional formulations. We present a comparison of the relative error for different number of basis functions and for adaptive approach. Numerical results illustrate that the presented method provide accurate solution of the unsaturated multicontinua problem on the coarse grid with huge reduction of the discrete system size.

In this paper, we consider the poroelasticity problem with a time memory formalism that couples the pressure and displacement, and we assume this multiphysics process occurs in multicontinuum media. A coupled system of equations for pressures in each continuum and elasticity equations for displacements of the medium are included in the mathematical model.
Based on the Caputo’s time fractional derivative, we derive an implicit finite difference approximation for time discretization. Also, a Discrete Fracture Model (DFM) is used to model fluid flow through fractures and treat the complex network of fractures. Further, we develop a coarse grid approximation based on the Generalized Multiscale Finite Element Method (GMsFEM), where we solve local spectral problems for construction of the multiscale basis functions. The main idea of the proposed method is to reduce the dimensionality of the problem because our model equation has multiple fractional powers, there multiple unknowns with memory effects. Consequently, the solution is on a coarse grid, which saves some computational time.
We present numerical results for the two-dimensional model problems in fractured heterogeneous porous media. After, we investigate error analysis between reference (fine-scale) solution and multiscale solution with different numbers of multiscale basis functions. The results show that on a coarse grid, the proposed approach can achieve good accuracy.

In this paper, a meshless numerical modeling method named mesh-free discrete fracture model (MFDFM) of fractured reservoirs based on the newly developed extended finite volume method (EFVM) is proposed. First, matching and nonmatching point cloud generation algorithms are developed to discretize the reservoir domain with fracture networks, which avoid the gridding challenges of the reservoir domain in traditional mesh-based methods. Then, taking oil/water two-phase flow in fractured reservoirs as an example, MFDFM derives the EFVM discrete scheme of the governing equations, constructs various types of connections between matrix nodes and fracture nodes, and calculates the corresponding transmissibilities. Finally, the EFVM discrete scheme of the governing equations and the generalized finite difference discrete scheme of various boundary conditions form the global nonlinear equations, which do not increase the degree of nonlinearity compared with those in the traditional finite volume method (FVM)-based numerical simulator. The global equations can be solved by the existing nonlinear solver in the FVM-based reservoir numerical simulator by only adding the linear discrete equations of boundary conditions, which reduce the difficulty of forming a general purpose MFDFM-based fractured reservoir numerical simulator. Several numerical test cases are implemented to illustrate that the proposed MFDFM can achieve good computational performance under matching and nonmatching point clouds, and for heterogeneous reservoirs, complex fracture networks, complex boundary geometry, and complex boundary conditions, by comparing the computational results of MFDFM with embedded discrete fracture model (EDFM). Thus, MFDFM retains the computational performances of the traditional mesh-based methods and can avoid the difficulties of handling complex geometry and complex boundary conditions of the computational domain, which is the first meshless numerical framework to model fractured reservoirs in parallel with the mesh-based discrete fracture model (DFM) and EDFM.

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.
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Natural or induced fractures are typically present in subsurface geological formations. Therefore, they need to be carefully studied for reliable estimation of the long-term carbon dioxide storage. Instinctively, flow-conductive fractures may undermine storage security as they increase the risk of CO2 leakage if they intersect the CO2 plume. In addition, fractures may act as flow barriers, causing significant pressure gradients over relatively small regions near fractures. Nevertheless, despite their high sensitivities, the impact of fractures on the full-cycle storage process has not been fully quantified and understood. In this study, a numerical model is developed and applied to analyze the role of discrete fractures on the flow and transport mechanism of CO2 plumes in simple and complex fracture geometries. A unified framework is developed to model the essential hydrogeological trapping mechanisms. Importantly, the projection-based embedded discrete fracture model is incorporated into the framework to describe fractures with varying conductivities. Impacts of fracture location, inclination angle, and fracture-matrix permeability ratio are systemically studied for a single fracture system. Moreover, the interplay between viscous and gravity forces in such fractured systems is analyzed. Results indicate that the fracture exhibits differing effects regarding different trapping mechanisms. Generally speaking, highly-conductive fractures facilitate dissolution trapping while weakening residual trapping, and flow barriers can assist dissolution trapping for systems with a relatively low gravity number. The findings from the test cases for single fracture geometries are found applicable to a larger-scale domain with complex fracture networks. This indicates the scalability of the study for field-relevant applications.

In this work, the scalability of two key multiscale solvers for the pressure equation arising from incompressible flow in heterogeneous porous media, namely, the multiscale finite volume (MSFV) solver, and the restriction-smoothed basis multiscale (MsRSB) solver, are investigated on the graphics processing unit (GPU) massively parallel architecture. The robustness and scalability of both solvers are compared against their corresponding carefully optimized implementation on the shared-memory multicore architecture in a structured problem setting. Although several components in MSFV and MsRSB algorithms are directly parallelizable, their scalability on the GPU architecture depends heavily on the underlying algorithmic details and data-structure design of every step, where one needs to ensure favorable control and data flow on the GPU, while extracting enough parallel work for a massively parallel environment. In addition, the type of algorithm chosen for each step greatly influences the overall robustness of the solver. Thus, we extend the work on the parallel multiscale methods of Manea et al. (2016) to map the MSFV and MsRSB special kernels to the massively parallel GPU architecture. The scalability of our optimized parallel MSFV and MsRSB GPU implementations are demonstrated using highly heterogeneous structured 3D problems derived from the SPE10 Benchmark (Christie and Blunt 2001). Those problems range in size from millions to tens of millions of cells. For both solvers, the multicore implementations are benchmarked on a shared-memory multicore architecture consisting of two packages of Intel® Cascade Lake Xeon Gold 6246 central processing unit (CPU), whereas the GPU implementations are benchmarked on a massively parallel architecture consisting of NVIDIA Volta V100 GPUs. We compare the multicore implementations to the GPU implementations for both the setup and solution stages. Finally, we compare the parallel MsRSB scalability to the scalability of MSFV on the multicore (Manea et al. 2016) and GPU architectures. To the best of our knowledge, this is the first parallel implementation and demonstration of these versatile multiscale solvers on the GPU architecture.
NOTE: This paper is published as part of the 2021 SPE Reservoir Simulation Conference Special Issue.

CO2 injection into deep saline aquifers has shown to be a feasible option, as for their large storage capacity under safe operational conditions. Previous studies have revealed that CO2 can be trapped in the subsurface by several mechanisms. Despite the major advances in studying these trapping mechanisms, their dynamic interactions in different periods of a full-cycle process have not been well understood; i.e., they are studied independently at their so-called ‘separate time scales of importance’. These mechanisms, however, are dynamically interconnected and influence each other even outside of their main time scale of importance. Besides, previous studies on field-scale simulations often choose grid cells which are too coarse to capture flow dynamics especially in post-injection period. To this end, we develop a comprehensive framework to analyze the flow dynamics and the associated hydrodynamic trapping process, in which the CO2 injection, migration and post-migration period are all considered in a unified manner. Through illustrative models with sufficient grid resolution, we quantify the impact of different trapping mechanisms and uncertain reservoir properties through a full-cycle process. We demonstrate that the time scale associated with each trapping mechanism indeed varies, yet their dynamic interplay needs to be considered for accurate and reliable predictions. Results reveal that residual trapping is governed by the advective transport in the injection period, and its contribution to the overall trapped amount becomes more significant in systems with lower permeability. Dissolution trapping operates under varying driving forces at different stages. In the injection period, the dissolution process is controlled by advective transport, and later enhanced by the gravity-induced convection in the post-injection period. Such convective transport diminishes the contribution from residual trapping. Our study sheds light on the impact of the coupled reservoir and fluid time-dependent interactions in estimation of the securely trapped CO2 in saline aquifers.

The conventional uniaxial compression test is used to investigate the interaction of two preexist-ing cracks with an internal hole in some typical gypsum specimens in the laboratory. The gypsum specimens are specially provided with 150 mm Â 150 mm Â 50 mm in dimensions and containing one internal hole and two cracks of 2 cm in lengths. These notches are located at varying distances , 2 cm, 3 cm and 4 cm from the hole and their inclination angles range from 0 to 90 degrees with an increment of 30 degrees. By this procedure, 12 different specimens can be provided for the experimental tests. On the other hand, an Extended Finite Element analysis of these tests is also provided in Abaqus software by assuming a plane strain state for the same specimens and taking the model geometry dimensions as 15 cm Â 15 cm. In this analysis, the length of the cracks is 2 cm and their distances with the hole changes as 2 cm, 3 cm and 4 cm. However, in the two dimensional numerical simulation of the tests, the inclination angles of the cracks ranges from 0 to 90 degrees but the increment is taken as 15 degrees to provide 21 model samples and make a more detail analysis of the geo-mechanical problem. The results of laboratory tests and numerical models are very similar as far as the failure mechanisms and fracture patterns are conserened. The failure strengths are also very close for both analyses. It is concluded that the compressive strength of the specimens is highly affected by the fracture pattern and failure mechanism of the specimens assuming different configurations of the cracks and the hole locations. ARTICLE HISTORY

It is well known that domain-decomposition-based multiscale mixed methods rely on interface spaces, defined on the skeleton of the decomposition, to connect the solution among the non-overlapping subdomains. Usual spaces, such as polynomial-based ones, cannot properly represent high-contrast channelized features such as fractures (high permeability) and barriers (low permeability) for flows in heterogeneous porous media. We propose here new interface spaces, which are based on physics, to deal with permeability fields in the simultaneous presence of fractures and barriers, accommodated respectively, by the pressure and flux spaces. Existing multiscale methods based on mixed formulations can take advantage of the proposed interface spaces, however, in order to present and test our results, we use the newly developed Multiscale Robin Coupled Method (MRCM) (Guiraldello et al., 2018), which generalizes most well-known multiscale mixed methods, and allows for the independent choice of the pressure and flux interface spaces. An adaptive version of the MRCM (Rocha et al., 2020) is considered that automatically selects the physics-based pressure space for fractured structures and the physics-based flux space for regions with barriers, resulting in a procedure with improved accuracy. The features of the proposed approach are investigated through several numerical simulations of single-phase and two-phase flows, in different heterogeneous porous media. The adaptive MRCM combined with the interface spaces based on physics provides promising results for challenging problems with the simultaneous presence of fractures and barriers.

In a literature review of the recent advancements in mathematical hydrologic models applied in fractured karstic formations, we highlight the necessary improvements in the fluid dynamic equations that are commonly applied to the flow in a discrete fracture network (DFN) via channel network models. Fluid flow and pollutant transport modeling in karst aquifers should consider the simultaneous occurrence of laminar, nonlaminar, and turbulent fluxes in the fractures rather than the laminar flow by the cubic law that has been widely applied in the scientific literature. Some simulations show overestimations up to 75% of the groundwater velocity when non-laminar flows are neglected. Moreover, further model development is needed to address the issues of tortuosity of preferential saturated fluid flow in fractures suggesting adjustments of the size of the mean aperture in DFN models. During the past decade, DFN mathematical models have been significantly developed aimed at relating the three-dimensional structure of interconnected fractures within rocky systems to the specific fracture properties measurable on the rock outcrops with the use of reliefs, tracer/pumping tests, and geotechnical field surveys. The capabilities and limitations of previous reported hydrological models together with specific research advancements and findings in modeling equations are described herein. New software is needed for creating three-dimensional contour maps in fractured aquifers corresponding to the outputs of particle tracking simulations. Existing software based on the equivalent continuum or multiple-interacting continua cannot delineate the spread of pollutant migrations affected by the tortuous preferential flow pathways that occur in DFNs.

In this paper, we propose a multiscale method for solving the Darcy flow of a single-phase fluid in two-dimensional fractured porous media. We consider a discrete fracture-matrix (DFM) model that treats fractures as one-dimensional objects, and flows in both the matrix and fractures respect the Darcy’s law. A multipoint flux mixed finite element (MFMFE) method with the broken Raviart–Thomas (RT12) element and the trapezoidal quadrature rule is employed to approximate the matrix velocity and pressure, which results in a block diagonal, symmetric and positive definite mass matrix for the matrix velocity on general quadrilateral grids; the one-dimensional RT0 mixed finite element method with the one-dimensional trapezoidal quadrature rule is exploited to approximate the fracture velocity and pressure, which leads to a diagonal and positive definite mass matrix for the fracture velocity in each single fracture. All these features of the obtained mass matrices allow for velocity elimination. Multiscale basis functions are constructed for the two-dimensional matrix pressure following the generalized multiscale finite element method (GMsFEM) framework to capture the fine-scale information of heterogeneous fractured porous media and effectively reduce the degrees of freedom for the matrix pressure, while fine-grid basis functions are utilized for the one-dimensional fracture pressure in fractures. Various numerical tests with the oversampling technique for different fracture distributions are performed to show that the proposed multiscale method is effective and able to provide good approximations for the fine-grid solution.

Rock fractures are crucial conduits for fluid flow in fractured rock masses. Three-dimensional (3D) discrete fracture network (DFN) modeling approach, which can explicitly characterize the fracture and its network arrangement, has been increasingly utilized in various engineering applications. This paper presents a review of the development and application of 3D DFNs for modeling naturally fractured rocks. The issues relating to generation, discretization and flow calculation of 3D DFNs are investigated. Extensions to modeling multiphase flow in discrete-fractured media using DFNs are also considered. This work can help researchers to recognize previous achievements on 3D DFNs and identify potential future works.

Loss of circulation while drilling is a challenging problem that may interrupt operations, reduce efficiency, and may contaminate the subsurface. When a drilled borehole intercepts conductive faults or fractures, lost circulation manifests as a partial or total escape of drilling, workover, or cementing fluids, into the surrounding rock formations. Loss control materials (LCM) are often used in the mitigation process. Understanding the fracture effective hydraulic properties and fluid leakage behavior is crucial to mitigate this problem. Analytical modeling of fluid flow in fractures is a tool that can be quickly deployed to assess lost circulation and perform diagnostics, including leakage rate decline, effective fracture conductivity, and selection of the LCM. Such models should be applicable to Newtonian and non-Newtonian yield-stress fluids, where the fluid rheology is a nonlinear function of fluid flow and shear stress. In this work, a new semi-analytical solution is developed to model the flow of non-Newtonian drilling fluid in a fractured medium. The solution model is applicable for various fluid types exhibiting yield-power-law (Herschel-Bulkley). We use high-resolution finite-element simulations based on the Cauchy equation to verify our solutions. We also generate type-curves and compare them to others in the literature. We demonstrate the applicability of the proposed model for two field cases encountering lost circulations. To address the subsurface uncertainty, we combine the developed solutions with Monte-Carlo and generate probabilistic predictions. The solution method can estimate the range of fracture conductivity, parametrized by the fracture hydraulic aperture, and time-dependent fluid loss rate that can predict the cumulative volume of lost fluid. The proposed approach is accurate and efficient enough to support decision-making for real-time drilling operations.

This paper presents a mixed finite element framework for coupled hydro-mechanical-chemical processes in heterogeneous porous media. The framework combines two types of locally conservative discretization schemes: (1) an enriched Galerkin method for reactive flow, and (2) a three-field mixed finite element method for coupled fluid flow and solid deformation. This combination ensures local mass conservation, which is critical to flow and transport in heterogeneous porous media, with a relatively affordable computational cost. A particular class of the framework is constructed for calcite precipitation/dissolution reactions, incorporating their nonlinear effects on the fluid viscosity and solid deformation. Linearization schemes and algorithms for solving the nonlinear algebraic system are also presented. Through numerical examples of various complexity, we demonstrate that the proposed framework is a robust and efficient computational method for simulation of reactive flow and transport in deformable porous media, even when the material properties are strongly heterogeneous and anisotropic.

This paper presents a numerical model based on the projection-based embedded discrete fracture model (pEDFM) to appraise CO2 sequestration capacity in shale gas reservoirs with complex boundary shape. Moreover, the proposed novel model can be effectively and efficiently adaptive to complex geological conditions of the 3-Dimensional (3D) shale reservoir which is the actual situation in the practice. The performances to handle a simple case from the proposed model are compared with those from commercial software CMG, and a good agreement between these results indicates the validness of the proposed model. Then, a field case from the New Albany Shale is used to illustrate the practical application of the proposed model to simulate the process of CO2 sequestration and evaluate the CO2 sequestration capacity. In all, this proposed model is currently the only one numerical model to effectively and efficiently appraise CO2 geological-sequestration capacity in actual 3D shale gas reservoirs with complex geological conditions.

An algebraic dynamic multilevel (ADM) method for fully-coupled simulation of flow and heat transport in heterogeneous fractured geothermal reservoirs is presented. Fractures are modeled explicitly using the projection-based embedded discrete method (pEDFM), which accurately represents fractures with generic conductivity values, from barriers to highly-conductive manifolds. A fully implicit scheme is used to obtain the coupled discrete system including mass and energy balance equations with two main unknowns (i.e., pressure and temperature) at fine-scale level. The ADM method is then developed to map the fine-scale discrete system to a dynamic multilevel coarse grid, independently for matrix and fractures. To obtain the ADM map, multilevel multiscale coarse grids are constructed for matrix as well as for each fracture at all coarsening levels. On this hierarchical nested grids, multilevel multiscale basis functions (for flow and heat) are solved locally at the beginning of the simulation. They are used during the ADM simulation to allow for accurate multilevel systems in presence of parameter heterogeneity. The resolution of ADM simulations is defined dynamically based on the solution gradient (i.e. front tracking technique) using a user-defined threshold. The ADM mapping occurs algebraically using the so-called ADM prolongation and restriction operators, for all unknowns. A variety of 2D and 3D fractured test cases with homogeneous and heterogeneous permeability maps are studied. It is shown that ADM is able to model the coupled mass-heat transport accurately by employing only a fraction of fine-scale grid cells. Therefore, it promises an efficient approach for simulation of large and real-field scale fractured geothermal reservoirs. All software developments of this paper is publicly available at https://gitlab.com/DARSim2simulator.

Simulation of mass transfer in shale porous media continues to challenge the hydrocarbon recovery industry due the involved complex porous media transport mechanisms and complicated porous structures. In this paper, we present a new multiscale approach to improve the simulation of gas flow in shale porous media based on a coupled triple-continuum and discrete fracture model. Gas transport in shale formations entails porous media flow through matrix (which contains organic and inorganic matters) and fractures. The fractures consist of small-scale natural fractures and large-scale artificial fractures from hydraulic fracturing. In the triple-continuum model, kerogen matrix, inorganic matrix and natural fractures are modeled as three superimposed continua. The hydraulic fractures are handled by discrete fracture model. Using a framework via the multiscale mixed finite element method, our methodology couples the triple-continuum model and discrete fracture model in a rigorous and systematic approach. The method can capture the small-scale features of gas flow through multiscale basis functions calculated based on the triple-continuum background. We go beyond multiple-continuum modeling by enabling explicit discrete fracture representation in our multiscale framework. As a result, our approach could avoid some oversimplified assumptions made in conventional triple-continuum models and enjoy explicit fracture treatment with high computational efficiency. Numerical aspects of the new approach are detailed with examples demonstrating its efficiency and accuracy for simulating gas flow in shale porous media with multiscale porous features.

In this work, we present a novel nonlocal nonlinear coarse grid approximation using a machine learning algorithm. We consider unsaturated and two-phase flow problems in heterogeneous and fractured porous media, where mathematical models are formulated as general multicontinuum models. We construct a fine grid approximation using the finite volume method and embedded discrete fracture model. Macroscopic models for these complex nonlinear systems require nonlocal multicontinua approaches, which are developed in earlier works [8]. These rigorous techniques require complex local computations, which involve solving local problems in oversampled regions subject to constraints. The solutions of these local problems can be replaced by solving original problem on a coarse (oversampled) region for many input parameters (boundary and source terms) and computing effective properties derived by nonlinear nonlocal multicontinua approaches. The effective properties depend on many variables (oversampled region and the number of continua), thus their calculations require some type of machine learning techniques. In this paper, our contribution is two fold. First, we present macroscopic models and discuss how to effectively compute macroscopic parameters using deep learning algorithms. The proposed method can be regarded as local machine learning and complements our earlier approaches on global machine learning [39], [38]. We consider a coarse grid approximation using two upscaling techniques with single phase upscaled transmissibilities and nonlocal nonlinear upscaled transmissibilities using a machine learning algorithm. We present results for two model problems in heterogeneous and fractured porous media and show that the presented method is highly accurate and provides fast coarse grid calculations.

In this paper, we consider a poroelasticity problem in heterogeneous multicontinuum media that is widely used in simulations of the unconventional hydrocarbon reservoirs and geothermal fields. Mathematical model contains a coupled system of equations for pressures in each continuum and effective equation for displacement with volume force sources that are proportional to the sum of the pressure gradients for each continuum. To illustrate the idea of our approach, we consider a dual continuum background model with discrete fracture networks that can be generalized to a multicontinuum model for poroelasticity problem in complex heterogeneous media. We present a fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) approach for two and three-dimensional formulations. The coarse grid approximation is constructed using the Generalized Multiscale Finite Element Method (GMsFEM), where we solve local spectral problems for construction of the multiscale basis functions for displacement and pressures in multicontinuum media. We present numerical results for the two and three dimensional model problems in heterogeneous fractured porous media. We investigate relative errors between reference fine grid solution and presented coarse grid approximation using GMsFEM with different numbers of multiscale basis functions. Our results indicate that the proposed method is able to give accurate solutions with few degrees of freedoms.

pEDFM is a new numerical model which is proposed in the last three years to handle the flow simulation with complex fracture networks. Previous works show that classical EDFM has a good performance only for production or injection simulation of multistage fractured well, but pEDFM can effectively handle more general cases than EDFM. Although pEDFM has been validated by by some works, this paper points out that there are still some limitations in previous pEDFM need to be resolved, lying on lack of a practical method to select projected faces, the proficiency in the transmissibility formula of f-m connections, and poor performances in some cases. Then, corresponding modifications are presented to resolve these limitations, and these modifications include A practical algorithm called “micro-translation method” to select projected-faces combinations, the physical transmissibility formula of additional fracture-matrix (f-m) connections and an extended pEDFM framework by considering additional fracture-fracture (f-f) connections. Some test cases are implemented to illustrate the necessity and validity of these modifications. Besides, two examples are implemented to illustrate the robustness of the modified pEDFM workflow for practical application of a fractured reservoir.

Multiscale methods aim to address the computational cost of elliptic problems on extremely large grids, by using numerically computed basis functions to reduce the dimensionality and complexity of the task. When multiscale methods are applied in uncertainty quantification to solve for a large number of parameter realizations, these basis functions need to be computed repeatedly for each realization. In our recent work (Chan et al. in J Comput Phys 354:493–511, 2017), we introduced a data-driven approach to further accelerate multiscale methods within uncertainty quantification. The basic idea is to construct a surrogate model to generate such basis functions at a much faster speed. The surrogate is modeled using a dataset of computed basis functions collected from a few runs of the multiscale method. Our previous study showed the effectiveness of this framework where speedups of two orders of magnitude were achieved in computing the basis functions while maintaining very good accuracy, however the study was limited to tracer flow/steady state flow problems. In this work, we extend the study to cover transient multiphase flow in porous media and provide further assessments.

In this paper, we present a multiscale method for simulations of the multicontinua unsaturated flow problems in heterogeneous fractured porous media. The mathematical model is described by the system of Richards equations for each continuum that is coupled by the specific transfer term. To illustrate the idea of our approach, we consider a dual continua background model with discrete fractures networks that is generalized as a multicontinua model for unsaturated fluid flow in the complex heterogeneous porous media. We present fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) approach. In this model, we construct an unstructured fine grid that takes into account complex fracture geometries for two and three dimensional formulations. Due to construction of the unstructured grid, the fine grid approximation leads to the very large system of equations. For reduction of the discrete system size, we develop a multiscale method for coarse grid approximation of the coupled problem using Generalized Multiscale Finite Element Method (GMsFEM). In this method, we construct coupled multiscale basis functions that are used to construct highly accurate coarse grid approximation. The multiscale method allowed us to capture detailed interactions between multiple continua. The adaptive approach is investigated, where we consider two approaches for multiscale basis functions construction: (1) based on the spectral characteristics of the local problems and (2) using simplified multiscale basis functions. We investigate accuracy of the proposed method for the several test problems in two and three dimensional formulations. We present a comparison of the relative error for different number of basis functions and for adaptive approach. Numerical results illustrate that the presented method provides accurate solution of the unsaturated multicontinua problem on the coarse grid with huge reduction of the discrete system size.

A wide variety of multiscale methods have been proposed in the literature to reduce runtime and provide better scaling for the solution of Poisson-type equations modeling flow in porous media. We present a new multiscale restricted-smoothed basis (MsRSB) method that is designed to be applicable to both rectilinear grids and unstructured grids. Like many other multiscale methods, MsRSB relies on a coarse partition of the underlying fine grid and a set of local prolongation operators (multiscale basis functions) that map unknowns associated with the fine grid cells to unknowns associated with blocks in the coarse partition. These mappings are constructed by restricted smoothing: Starting from a constant, a localized iterative scheme is applied directly to the fine-scale discretization to compute prolongation operators that are consistent with the local properties of the differential operators.

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., JCP, 2014], 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.

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

Complex processes in perforated domains occur in many real-world
applications. These problems are typically characterized by physical processes
in domains with multiple scales (see Figure 1 for the illustration of a
perforated domain). Moreover, these problems are intrinsically multiscale and
their discretizations can yield very large linear or nonlinear systems. In this
paper, we investigate multiscale approaches that attempt to solve such problems
on a coarse grid by constructing multiscale basis functions in each coarse
grid, where the coarse grid can contain many perforations. In particular, we
are interested in cases when there is no scale separation and the perforations
can have different sizes. In this regard, we mention some earlier pioneering
works [14, 18, 17], where the authors develop multiscale finite element
methods. In our paper, we follow Generalized Multiscale Finite Element Method
(GMsFEM) and develop a multiscale procedure where we identify multiscale basis
functions in each coarse block using snapshot space and local spectral
problems. We show that with a few basis functions in each coarse block, one can
accurately approximate the solution, where each coarse block can contain many
small inclusions. We apply our general concept to (1) Laplace equation in
perforated domain; (2) elasticity equation in perforated domain; and (3) Stokes
equations in perforated domain. Numerical results are presented for these
problems using two types of heterogeneous perforated domains. The analysis of
the proposed methods will be presented elsewhere.

A novel cell-centred control-volume distributed multi-point flux approximation (CVD-MPFA) finite- volume formulation is presented for discrete fracture-matrix simulations in three-dimensions. The grid is aligned with the fractures and barriers which are then modelled as lower-dimensional interfaces located between the matrix cells in the physical domain. The three-dimensional(3D) pressure equation is solved in the matrix domain coupled with a two-dimensional(2D) pressure equation solved over fracture networks via a surface CVD-MPFA formulation. The CVD-MPFA formulation naturally handles fractures with anisotropic permeabilities on unstructured grids. Matrix-fracture fluxes are expressed in terms of matrix aijd fracture pressures and must be added to the lower-dimensional flow equation (called the transfer function). An additional transmission condition is used between matrix cells adjacent to low permeable fractures to link the velocity and pressure jump across the fractures. Numerical tests serve to assess the convergence and accuracy of the lower-dimensional fracture model for highly anisotropic fractures having different apertures and permeability tensors. A transport equation for tracer flow is coupled via the Darcy flux for single and intersecting fractures. The lower-dimensional approach for intersecting fractures avoids the more restrictive CFL condition corresponding to the equi-dimensional approximation with explicit time discretisation. Lower-dimensional fracture model results are compared with equi-dimensional model results. Fractures and barriers are efficiently modelled by lower-dimensional interfaces which yield comparable results to those of the equi-dimensional model. Highly conductive fractures are modelled as lower-dimensional entities with continuous pressure across these leading to reduced local degrees of freedom for the cluster of cells. Moreover, we present 3D simulations involving geologically representative complex fracture networks.

Advances in reservoir characterization and modeling have given the industry improved ability to build detailed geological models of petroleum reservoirs. These models are characterized by complex shapes and structures with discontinuous material properties that span many orders of magnitude. Models that represent fractures explicitly as volumetric objects pose a particular challenge to standard simulation technology with regard to accuracy and computational efficiency.
We present a new simulation approach based on streamlines in combination with a new multiscale mimetic pressure solver with improved capabilities for complex fractured reservoirs. The multiscale solver approximates the flux as a linear combination of numerically computed basis functions defined over a coarsened simulation grid consisting of collections of cells from the geological model. Here, we use a mimetic multipoint flux approximation to compute the multiscale basis functions. This method has limited sensitivity to grid distortions. The multiscale technology is very robust with respect to fine-scale models containing geological objects such as fractures and fracture corridors. The methodology is very flexible in the choice of the coarse grids introduced to reduce the computational cost of each pressure solve. This can have a large impact on iterative modeling workflows.

An efficient Two-Stage Algebraic Multiscale Solver (TAMS) that converges to the fine-scale solution is described. The first (global) stage is a multiscale solution obtained algebraically for the given fine-scale problem. In the second stage, a local preconditioner, such as the Block ILU (BILU), or the Additive Schwarz (AS) method is used. Spectral analysis shows that the multiscale solution step captures the low-frequency parts of the error spectrum quite well, while the local preconditioner represents the high-frequency components accurately. Combining the two stages in an iterative scheme results in efficient treatment of all the error components associated with the fine-scale problem. TAMS is shown to converge to the reference fine-scale solution. Moreover, the eigenvalues of the TAMS iteration matrix show significant clustering, which is favorable for Krylov-based methods. Accurate solution of the nonlinear saturation equations (i.e., transport problem) requires having locally conservative velocity fields. TAMS guarantees local mass conservation by concluding the iterations with a multiscale finite-volume step. We demonstrate the performance of TAMS using several test cases with strong permeability heterogeneity and large-grid aspect ratios. Different choices in the TAMS algorithm are investigated, including the Galerkin and finite-volume restriction operators, as well as the BILU and AS preconditioners for the second stage. TAMS for the elliptic flow problem is comparable to state-of-the-art algebraic multigrid methods, which are in wide use. Moreover, the computational time of TAMS grows nearly linearly with problem size.

Fractured-reservoir relative permeability, water breakthrough, and recovery cannot be extrapolated from core samples, but computer simulations allow their quantification through the use of discrete fracture models at an intermediate scale. For this purpose, we represent intersecting naturally and stochastically generated fractures in massive or layered porous rock with an unstructured hybrid finite-element (FE) grid. We compute two-phase flow with an implicit FE/finite volume (FV) method (FE/FVM) to identify the emergent properties of this complex system.
The results offer many important insights: Flow velocity varies by three to seven orders of magnitude and velocity spectra are multimodal, with significant overlaps between fracture- and matrix-flow domains. Residual saturations greatly exceed those that were initially assigned to the rock matrix. Total mobility is low over a wide saturation range and is very sensitive to small saturation changes. When fractures dominate the flow, but fracture porosity is low (10–3 to 1%), gridblock average relative permeabilities, kr, avg, cross over during saturation changes of less than 1%. Such upscaled kr, avg yield a convex, highly dispersive fractional-flow function without a shock. Its shape cannot be matched with any conventional model, and a new formalism based on the fracture/matrix flux ratio is proposed.
Spontaneous imbibition during waterflooding occurs only over a small fraction of the total fracture/matrix-interface area because water imbibes only a limited number of fractures. Yet in some of these, flow will be sufficiently fast for this process to enhance recovery significantly. We also observe that a rate dependence of recovery and water breakthrough occurs earlier in transient-state flow than in steady-state flow.
Introduction
Oil is difficult to recover from fractured reservoirs; however, approximately 60% of the world's remaining oil resources reside in heterogeneously deformed formations (Beydoun 1998). The production dilemma is reflected in complex pressure and production histories, unpredictable couplings of wells independent of their spatial separation, rapidly changing flow rates and the risks of rapid water breakthrough, and low final recovery (Kazemi and Gilman 1993).
Qualitatively, the main production obstacle is simple to conceptualize (Barenblatt et al. 1990): while the oil resides in the pores of the rock matrix, production-induced flow will occur predominantly in the fractures. However, they typically contribute less than 1% to the total fluid-saturated void space and are therefore rapidly invaded by the injected fluid. Once short-circuited by the injectant, the injection/production stream entrains only the oil that enters the fractures as a consequence of countercurrent imbibition (CCI) (Lu et al. 2006). The efficiency of this process is relatively well constrained by experimental work (Morrow and Mason 2001) and reproduced accurately by transfer functions (Lu et al. 2006). Rate predictions for fractured reservoirs require a further estimate of the area of the fracture/matrix interface captured by a shape factor (Kazemi et al. 1992). However, in cases where this measure is relatively well-constrained, predicted transfer rates appear to greatly exceed actual values. This observation suggests that, at any one time in the production history, transfer occurs over only a small part of the fracture/matrix interface. Furthermore, as is indicated by packer tests and temperature logs, only a small number of fractures contribute to the flow during production (Long and Billaux 1987, Barton 1995). This is confirmed by field-data-based numerical flow models (Matthäi and Belayneh 2004, Belayneh et al. 2006), highlighting that viscous flow in the rock matrix is usually significant, even if the fractures are well interconnected. All these findings conflict with the simple conceptual model, even qualitatively. How shall we replace it with something more accurate for the prediction of the behavior of fractured reservoirs?

Recently, multiscale methods have been developed for accurate and efficient numerical solution of large-scale heterogeneous reservoir problems. A scalable and extendible Operator Based Multiscale Method (OBMM) is described here. OBMM is cast as a general algebraic framework of the multiscale method. It is very natural and convenient to incorporate more physics in OBMM for multiscale computation.
In OBMM, two multiscale operators are constructed: prolongation and restriction. The prolongation operator can be constructed by assembling basis functions, and the specific form of the restriction operator depends on the coarse-scale dis-cretization formulation (e.g., finite-volume or finite-element). The coarse-scale pressure equation is obtained algebraically by applying the prolongation and restriction operators on the fine-scale flow equations. Solving the coarse-scale equation results in a high quality coarse-scale pressure. The fine scale pressure can be reconstructed by applying the prolongation operator to the coarse-scale pressure. A conservative fine-scale velocity field is then reconstructed to solve the transport equation.
As an application example, we study multiscale modeling of compressible flow. We show that the extension of modeling from incompressible to compressible flow is really straightforward for OBMM. No special treatment for compressibility is required. The efficiency of multiscale methods over stan dard fine-scale methods is retained by OBMM. The accuracy of OBMM is demonstrate by several challenging cases including highly compressible multiphase flow in a strongly heterogeneous permeability field (SPE 10).

Recent advances in multiscale methods have shown great promise in modeling multiphase flow in highly detailed heterogeneous domains. Existing multiscale methods, however, solve for the flow field (pressure and total velocity) only. Once the fine-scale flow field is reconstructed, the saturation equations are solved on the fine scale. With the efficiency in dealing with the flow equations greatly improved by multiscale formulations, solving the saturation equations on the fine scale becomes the relatively more expensive part. In this paper, we describe an adaptive multiscale finite-volume (MSFV) formulation for nonlinear transport (saturation) equations. A general algebraic multiscale formulation consistent with the operator-based framework proposed by Zhou and Tchelepi (SPE Journal, June 2008, pages 267-273) is presented. Thus, the flow and transport equations are solved in a unified multiscale framework. Two types of multiscale operators-namely, restriction and prolongation-are used to construct the multiscale saturation solution. The restriction operator is defined as the sum of the fine-scale transport equations in a coarse gridblock. Three adaptive prolongation operators are defined according to the local saturation history at a particular coarse block. The three operators have different computational complexities, and they are used adaptively in the course of a simulation run. When properly used, they yield excellent computational efficiency while preserving accuracy. This adaptive multiscale formulation has been tested using several challenging problems with strong heterogeneity, large buoyancy effects, and changes in the well operating conditions (e.g., switching injectors and producers during simulation). The results demonstrate that adaptive multiscale transport calculations are in excellent agreement with fine-scale reference solutions, but at a much lower computational cost.

Advances in reservoir characterization and modeling have given the industry improved ability to build detailed geological models of petroleum reservoirs. These models are characterized by complex shapes and structures with discontinuous material properties that span many orders of magnitude. Models that represent fractures explicitly as volumetric objects pose a particular challenge to standard simulation technology with regard to accuracy and computational efficiency.
We present a new simulation approach based on streamlines in combination with a new multiscale mimetic pressure solver with improved capabilities for complex fractured reservoirs. The multiscale solver approximates the flux as a linear combination of numerically computed basis functions defined over a coarsened simulation grid consisting of collections of cells from the geological model. Here, we use a mimetic multipoint-flux approximation to compute the multiscale-basis functions. This method has limited sensitivity to grid distortions. The multiscale technology is very robust with respect to fine-scale models containing geological objects such as fractures and fracture corridors. The methodology is very flexible in the choice of the coarse grids introduced to reduce the computational cost of each pressure solve. This can have a large impact on iterative modeling workflows.

We review and perform comparison studies for three recent multiscale methods for solving elliptic problems in porous media
flow; the multiscale mixed finite-element method, the numerical subgrid upscaling method, and the multiscale finite-volume
method. These methods are based on a hierarchical strategy, where the global flow equations are solved on a coarsened mesh
only. However, for each method, the discrete formulation of the partial differential equations on the coarse mesh is designed
in a particular fashion to account for the impact of heterogeneous subgrid structures of the porous medium. The three multiscale
methods produce solutions that are mass conservative on the underlying fine mesh. The methods may therefore be viewed as efficient,
approximate fine-scale solvers, i.e., as an inexpensive alternative to solving the elliptic problem on the fine mesh. In addition,
the methods may be utilized as an alternative to upscaling, as they generate mass-conservative solutions on the coarse mesh.
We therefore choose to also compare the multiscale methods with a state-of-the-art upscaling method – the adaptive local–global
upscaling method, which may be viewed as a multiscale method when coupled with a mass-conservative downscaling procedure.
We investigate the properties of all four methods through a series of numerical experiments designed to reveal differences
with regard to accuracy and robustness. The numerical experiments reveal particular problems with some of the methods, and
these will be discussed in detail along with possible solutions. Next, we comment on implementational aspects and perform
a simple analysis and comparison of the computational costs associated with each of the methods. Finally, we apply the three
multiscale methods to a dynamic two-phase flow case and demonstrate that high efficiency and accurate results can be obtained
when the subgrid computations are made part of a preprocessing step and not updated, or updated infrequently, throughout the
simulation.

We analyze measurements, conceptual pictures, and mathematical models of flow and transport phenomena in fractured rock systems. Fractures and fracture networks are key conduits for migration of hydrothermal fluids, water and contaminants in groundwater systems, and oil and gas in petroleum reservoirs. Fractures are also the principal pathways, through otherwise impermeable or low permeability rocks, for radioactive and toxic industrial wastes which may escape from underground storage repositories. We consider issues relating to (i) geometrical characterization of fractures and fracture networks, (ii) water flow, (iii) transport of conservative and reactive solutes, and (iv) two-phase flow and transport. We examine the underlying physical factors that control flow and transport behaviors, and discuss the currently inadequate integration of conceptual pictures, models and data. We also emphasize the intrinsic uncertainty associated with measurements, which are often interpreted non-uniquely by models. Throughout the review, we point out key, unresolved problems, and formalize them as open questions for future research.

In this paper, an extension of the multi-scale finite-volume (MSFV) method is devised, which allows to simulate flow and transport in reservoirs with complex well configurations. The new framework fits nicely into the data structure of the original MSFV method and has the important property that large patches covering the whole well are not required. For each well, an additional degree of freedom is introduced. While the treatment of pressure-constraint wells is trivial (the well-bore reference pressure is explicitly specified), additional equations have to be solved to obtain the unknown well-bore pressure of rate-constraint wells. Numerical simulations of test cases with multiple complex wells demonstrate the ability of the new algorithm to capture the interference between the various wells and the reservoir accurately.

The multiscale finite volume (MSFV) method is introduced for the efficient solution of elliptic problems with rough coefficients in the absence of scale separation. The coarse operator of the MSFV method is presented as a multipoint flux approximation (MPFA) with numerical evaluation of the transmissibilities. The monotonicity region of the original MSFV coarse operator has been determined for the homogeneous anisotropic case. For grid-aligned anisotropy the monotonicity of the coarse operator is very limited. A compact coarse operator for the MSFV method is presented that reduces to a 7-point stencil with optimal monotonicity properties in the homogeneous case. For heterogeneous cases the compact coarse operator improves the monotonicity of the MSFV method, especially for anisotropic problems. The compact operator also leads to a coarse linear system much closer to an M-matrix. Gradients in the direction of strong coupling vanish in highly anisotropic elliptic problems with homogeneous Neumann boundary data, a condition referred to as transverse equilibrium (TVE). To obtain a monotone coarse operator for heterogeneous problems the local elliptic problems used to determine the transmissibilities must be able to reach TVE as well. This can be achieved by solving two linear local problems with homogeneous Neumann boundary conditions and constructing a third bilinear local problem with Dirichlet boundary data taken from the linear local problems. Linear combination of these local problems gives the MSFV basis functions but with hybrid boundary conditions that cannot be enforced directly. The resulting compact multiscale finite volume (CMSFV) method with hybrid local boundary conditions is compared numerically to the original MSFV method. For isotropic problems both methods have comparable accuracy, but the CMSFV method is robust for highly anisotropic problems where the original MSFV method leads to unphysical oscillations in the coarse solution and recirculations in the reconstructed velocity field.

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.

In enhanced geothermal systems, cold working fluid (usually water) is injected into a fractured reservoir to enhance the efficiency of the power plant by increasing the conductivity of existing fractures and by creating new ones. For the modeling of single phase flow and transport in such dynamically changing fractured reservoirs, a new approach is presented. It is based on a hierarchical fracture representation that results in a network of multiple dominant fractures through which most of the mass flow occurs. These large fractures have a discrete representation, i.e., each fracture is represented by a lower dimensional continuum. A single continuum representation is also employed for the damaged matrix, which consists of many small and medium sized fractures accounted for by appropriate effective properties. The main advantage of the new approach is that no expensive remeshing of the domain is required whenever new fractures are added. Here, discretization and coupling between the different continua (damaged matrix with discrete dominant fractures) are explained. In order to remove singularities, kernel functions have been introduced in the governing equations to capture discontinuities like fractures and intersections thereof. This way transfer coefficients are well defined. Numerical verification studies involving flow and heat transport are presented and discussed. It has to be emphasized that geomechanics, rock chemistry, and upscaling are not topics of this paper.

Many naturally fractured reservoirs around the world have depleted significantly, and improved-oil-recovery (IOR) processes are necessary for further development. Hence, the modeling of fractured reservoirs has received increased attention recently. Accurate modeling and simulation of naturally fractured reservoirs (NFRs) is still challenging because of permeability anisotropies and contrasts. Nonphysical abstractions inherent in conventional dual-porosity and dual-permeability models make them inadequate for solving different fluid-flow problems in fractured reservoirs. Also, recent technologies for discrete fracture modeling may suffer from large simulation run times, and the industry has not used such approaches widely, even though they give more-accurate representations of fractured reservoirs than dual-continuum models. We developed an embedded discrete fracture model (DFM) for an in-house compositional reservoir simulator that borrows the dual-medium concept from conventional dual-continuum models and also incorporates the effect of each fracture explicitly. The model is compatible with existing finite-difference reservoir simulators. In contrast to dual-continuum models, fractures have arbitrary orientations and can be oblique or vertical, honoring the complexity of a typical NFR. The accuracy of the embedded DFM is confirmed by comparing the results with the fine-grid, explicit-fracture simulations for a case study including orthogonal fractures and a case with a nonaligned fracture. We also perform a grid-sensitivity study to show the convergence of the method as the grid is refined. Our simulations indicate that to achieve accurate results, the embedded discrete fracture model may only require moderate mesh refinement around the fractures and hence offers a computationally efficient approach. Furthermore, examples of waterflooding, gas injection, and primary depletion are presented to demonstrate the performance and applicability of the developed method for simulating fluid flow in NFRs.

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

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

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 precon- ditioning 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 as the last step of the solution process. 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 and improvements are proposed. For a wide range of test cases, including the highly heterogeneous (with over a million cells) SPE 10 permeability field, 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.

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

Discrete fracture models are an attractive alternative to upscaled models for flow in fractured media, as they provide a more accurate representation of the flow characteristics. A major challenge in discrete fracture simulation is to overcome the large computational cost associated with resolving the individual fractures in large-scale simulations. In this work, two characteristics of the fractured porous media are utilized to construct efficient preconditioners for the discretized flow equations. First, the preconditioners are tailored to the fracture geometry and presumed flow properties so that the dominant features are well represented there. This assures good scalability of the preconditioners in terms of problem size and permeability contrast. For fracture dominated problems numerical examples show that such geometric preconditioners are comparable or preferable when compared to state-of-the-art algebraic multigrid preconditioners. The robustness of the physics-based preconditioner for less favourable fracture conditions is further demonstrated by a systematic degradation of the fracture hierarchy. Secondly, the preconditioners are physics-preserving in the sense that conservative fluxes can be computed even for an inexact pressure solutions. This facilitates a scheme where accuracy in the linear solver can be traded for efficiency by terminating the iterative solvers based on error estimates, and without sacrificing basic physical modeling principles. With the combination of these two properties a novel preconditioner is obtained which bridges the gap between multiscale approximations and iterative linear solvers.

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.

Two discrete-fracture models (DFMs) based on different, independent numerical techniques have been developed for studying the behavior of naturally fractured reservoirs. One model is based on unstructured gridding with local refinement near fractures, while in the second model fractures are embedded in a structured matrix grid. Both models capture the complexity of a typical fractured reservoir better than conventional dual-permeability models, leading to a more accurate representation of fractured reservoirs.
The accuracy of the DFM approaches is confirmed by their match with a structured, grid-aligned, explicit-fracture model in tests involving capillary imbibition during water flooding and gravity drainage in oil-gas systems. The DFMs are insensitive to grid orientation. Simulations also show consistency and agreement of results of the DFM methods in synthetic models with complex fracture patterns. Our simulations indicate that conventional dual-permeability approaches are appropriate when the fracture system is very sparse relative to the grid spacing. In these situations a DFM can be used as the basis for defining dual-permeability model parameters. However, conventional dual-permeability approaches are inadequate in the presence of high localized anisotropy and preferential channeling. When used with general purpose reservoir simulators, both DFMs show computational performance that is comparable to that of dual-permeability models.

This paper presents an overview of parallel algorithms and their implementations for solving large sparse linear systems which arise in scientific and engineering applications. Preconditioners constitute the most important ingredient in solving such systems. As will be seen, the most common preconditioners used for sparse linear systems adapt domain decomposition concepts to the more general framework of “distributed sparse linear systems”. Variants of Schwarz procedures and Schur complement techniques are discussed. We also report on our own experience in the parallel implementation of a fairly complex simulation of solid-liquid flows.

This paper describes a hybrid finite volume method, designed to simulate multiphase flow in a field-scale naturally fractured reservoir. Lee et al. (WRR 37:443-455, 2001) developed a hierarchical approach in which the permeability contribution from short fractures is derived in an analytical expression that from medium fractures is numerically solved using a boundary element method. The long fractures are modeled explicitly as major fluid conduits. Reservoirs with well-developed natural fractures include many complex fracture networks that cannot be easily modeled by simple long fracture formulation and/or homogenized single continuity model. We thus propose a numerically efficient hybrid method in which small and medium fractures are modeled by effective permeability, and large fractures are modeled by discrete fracture networks. A simple, systematic way is devised to calculate transport parameters between fracture networks and discretized, homogenized media. An efficient numerical algorithm is also devised to solve the dual system of fracture network and finite volume grid. Black oil formulation is implemented in the simulator to demonstrate practical applications of this hybrid finite volume method.

When modeling reservoir behavior by numerical methods, inevitably the horizontal dimensions of any grid block containing a well are much larger than the wellbore radius of that well. It long has been recognized that the pressure calculated for a well block will be greatly different from the flowing bottom-hole pressure of the modeled well, but the literature contains few specific guides as to how to make the correction. In this study, the authors confine their attention to single-phase flow in two dimensions. When only a single buildup pressure is observed at a different shut-in time, an adjustment to the observed pressure can be made for matching with the simulator well-block pressure.

A simplified discrete fracture model suitable for use with general purpose reservoir simulators is presented. The model handles both two and three-dimensional systems and includes fracture-fracture, matrix-fracture and matrix-matrix connections. The formulation applies an unstructured control volume finite difference technique with a two-point flux approximation. The implementation is generally compatible with any simulator that represents grid connections via a connectivity list. A specialized treatment based on a "star-delta" transformation is introduced to eliminate control volumes at fracture intersections. These control volumes would otherwise act to reduce numerical stability and time step size. The performance of the method is demonstrated for several example cases including a simple two-dimensional system, a more complex three-dimensional fracture network, and a model of a strike-slip fault zone. The discrete fracture model is shown to provide results in close agreement with those of a reference finite difference simulator in cases where direct comparisons are possible.

Discrete-fracture modeling and simulation of two-phase flow in realistic representations of fractured reservoirs can now be used for the design of improved-oil-recovery (IOR) and enhanced-oil-recovery (EOR) strategies. Thus far, however, discrete-fracture simulators usually do not include a third compressible gaseous phase. This hinders the investigation of the performance of gas gravity drainage, water alternating gas injection, and blowdown in fractured reservoirs.
We present a new numerical method that expands the capabilities of existing black-oil models for three-component, three-phase flow in three ways:It uses a finite-element/finite-volume discretization generalized to unstructured hybrid element meshes.It employs higher-order accurate representations of the flux terms.Flash calculations are carried out with an improved equation of state allowing for a more realistic treatment of phase behavior.
We illustrate the robustness of this numerical method in several applications. First, quasi-1D simulations are used to demonstrate grid convergence. Then, 2D discrete-fracture models are used to illustrate the effect of mesh quality on predicted production rates in discrete-fracture models. Finally, the proposed method is used to simulate three-component, three-phase flow in a realistic 2D model of fractured limestone mapped in the Bristol Channel, UK, and create a 3D stochastically generated discrete-fracture model.