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Understanding deformations and fluid flow in fractured rocks is of central importance for many subsurface flow applications. Thus, numerical frameworks are needed that capture the coupled mechanical and hydraulic behaviour, including scenarios with complex fracture networks. This paper employs the extended finite volume method to represent fracture manifolds in a poroelastic matrix domain and to compute shear and tensile displacements of fracture segments. However, using embedded fractures with non‐conforming grids can lead to severe convergence issues while computing shear slip and tensile opening of intersecting and parallel fractures, particularly if they have similar slopes. Our proposed solution of this problem is to slightly deform the fracture geometry by merging critical segments. We discuss and show examples of which attributes the merged segment has to adopt to ensure the correct slip behaviour at an intersection. Thereby it is crucial that the flow topology remains unaltered. Analysing failures of kinked fractures and fracture intersections show the flexibility of the devised algorithm. Moreover, it is demonstrated that deformation and opening patterns of a natural network with more than 200 fractures are qualitatively predicted very well. This method is a simple solution to treat multiple fracture segments in a grid cell, hence it can be used for any model embedding fractures in non‐conforming grids. [ABSTRACT FROM AUTHOR]

Summary: Fluid injection–induced tensile opening is modeled using an extended finite volume method (XFVM). An embedded fracture strategy is used for the flow problem, that is, the fractures are discretized using finite volume segments without resolving the grid around them. Further, the discontinuities across fractures are modeled using special basis functions. The fracture openings due to enhanced fluid pressure and the associated shear slip due to traction free boundary condition on the fracture segments are both modeled using these special discontinuity basis functions. Mass transfer between fractures and matrix is modeled using the pressure difference. The enhancement of fracture storativity due to tensile opening leads to stronger coupling between flow and mechanics. An iterative scheme relying on the fixed‐stress approach for fractures, which conserves the stress dependent terms over each iteration of the flow problem, has been introduced. Tensile opening has been simulated for single fractures embedded in two‐ and three‐dimensional matrices. The convergence criterion for sequentially implicit fixed‐stress scheme for fractures embedded in elastic media is established and has been validated numerically. Further, for 2D simulations, the effect of the matrix permeability for fracture propagation due to tensile opening has been studied. [ABSTRACT FROM AUTHOR]

In reservoir simulation, it is important to understand the mechanical behaviour of fractured rocks and the effect of shear and tensile displacements of fractures on their aperture. Tensile opening directly enhances the fracture aperture, whereas shear of a preexisting rough-walled fracture creates aperture changes dependent on the local stress state. Since fracture dilatation increases reservoir permeability, both processes must be included in a realistic and consistent manner into the mechanical reservoir simulation model. Here, we use the extended finite volume method (XFVM) to conduct flow and geomechanics simulations. In XFVM, fractures are embedded in a poroelastic matrix and are modelled with discontinuous basis functions. On each fracture segment the tractions and compressive forces are calculated, and one extra degree of freedom is added for both the shear and tensile displacement. In this particular XFVM implementation we assume that linear elasticity and steady state fluid pressure adequately constrain the effective stress. In this paper, shear dilation is not calculated a posteriori, but it enters the equations such that aperture changes directly affect the stress state. This is accomplished by adding shear dilation to the displacement gradients and therefore ascertains a consistent representation in the stress-strain relations and force balances. We illustrate and discuss the influence of this extra term in two simple test cases and in a realistic layer-restricted two-dimensional fracture network subjected to plausible in situ stress and pore pressure conditions. [ABSTRACT FROM AUTHOR]

Buoyant incompressible multiphase flow and transport in porous media typically is simulated by coupling an elliptic flow equation with a hyperbolic transport description. The tight coupling between flow and transport either requires a fully implicit solution algorithm or very small time steps, if solved sequentially. In any case, however, a large linear system has to be solved each time step. Here a new solution approach is devised, which relies on solving a coupled hyperbolic system of conservation laws with an explicit finite volume method. Consequently, only local operations have to be performed, which is a great advantage in case of massive parallel simulations and if GPUs are employed. The devised method is based on the isothermal Euler equations with momentum source terms accounting for resistance due to the porous medium and buoyancy. In this system the pressure is proportional to the density and in case of very small Mach numbers and relative density variation, the computed velocity field is divergence free and converges towards the mean total volumetric flow in the porous domain. To account for saturation transport, the system was augmented by an additional hyperbolic equation. In order to obtain the numerical fluxes, a characteristic based approximative Riemann solver was developed and 2nd order accuracy in space and time is achieved by piecewise linear reconstruction. Two-phase flow test cases with buoyant plumes and a lock exchange problem demonstrate that the devised hyperbolic approach recovers the correct incompressible solutions. Numerical experiments also demonstrate that the method is accurate and efficient. Thus the devised method is very promising, and conceptually it also can be applied for flow and transport in heterogeneous media; but it remains to be investigated how well it performs for such cases. • Flow and transport are described by a coupled hyperbolic system of PDEs. • Fully explicit solution scheme for strongly coupled two-phase flow in porous media. • An efficient approximate Riemann solver for the numerical fluxes is devised. • Second order in space and time is achieved by a TVD method. • The proposed method is an ideal candidate for GPUs. [ABSTRACT FROM AUTHOR]

In this paper, a numerical model to predict flow-induced shear failure along pre-existing fractures is presented. The framework is based on a discrete fracture representation embedded in a continuum describing the damaged matrix. A finite volume method is used to compute both flow and mechanical equilibrium, whereas specifically tailored basis functions are used to account for the physics at discontinuities. The failure criterion is based on a maximum shear strength limit, which changes with varying compressive stress on the fracture manifold. The displacements along fracture manifolds are obtained such that force balance is achieved under conditions, where shear stress of the failing fracture segment is constrained to the maximum shear strength at the segment. Simultaneously, the fluid pressure is computed independently of the shear slip. A relaxation model approach is used to obtain the maximum shear limit on the fracture manifold, which leads to grid convergence. [ABSTRACT FROM AUTHOR]

A finite volume-based numerical modeling framework using a hierarchical fracture representation (HFR) has been developed to compute flow-induced shear failure. To accurately capture the mechanics near fracture manifolds, discontinuous basis functions are employed which ensure continuity of the displacement gradient across fractures. With these special basis functions, traction and compressive forces on the fracture segment can be calculated without any additional constraints, which is extremely useful for estimating the irreversible displacement along the fracture (slip) based on a constitutive friction law. The method is further extended to include slip-dependent hydraulic aperture change and grid convergent results are obtained. Further, the change in hydraulic aperture is modeled using an asymptotic representation which respects the experimentally observed behavior of pore volume dilation due to shear slip. The model allows the initial rapid increase in hydraulic aperture due to shear slip and asymptotically approaches a finite value after repeated shearing of a fracture segment. This aperture increase is the only feedback for mechanics into the fluid flow for a linear elastic mechanics problem. The same model is also extended to include poroelastic relations between flow and mechanics solver. The grid convergence result in the case of poroelastic flow-mechanics coupling for flow-induced shear failure is also obtained. This proves the robustness of the numerical and analytical modeling of fracture and friction in the extended finite volume method (XFVM) set-up. Finally, a grid convergent result for seismic moment magnitude for single fracture and fracture network with random initial hydraulic and friction properties is also obtained. The b-value, which represents the slope of seismic moment occurrence frequency decay vs seismic moment magnitude, which is approximately constant in a semi-logarithmic plot, is estimated. The numerical method leads to converged b-values for both single fracture and fracture network simulations, as grid and time resolutions are increased. For the resulting linear system, a sequential approach is used, that is, first, the flow and then the mechanics problems are solved. The new modeling framework is very useful to predict seismicity, permeability, and flow evolution in geological reservoirs. This is demonstrated with numerical simulations of enhancing a geothermal system. [ABSTRACT FROM AUTHOR]

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

In this work, the zonal Multiscale Finite Volume (zMSFV) framework is presented. The framework generalizes previous extensions of the Multiscale Finite Volume (MSFV) method and allows to integrate additional models in a unified way. The new approach is based on splitting the domain of interest into a classical MSFV domain and additional zones. Within each zone, the solution space is spanned by so-called zonal functions. Coupling of the zonal solutions to the surrounding domain is achieved via extended zonal functions. The construction of basis, zonal and extended zonal functions can be performed such that the functions form a partition of unity. The zMSFV framework allows to include the influence of the zonal functions directly in the coarse-scale problem. Similar to previous MSFV method formulations, the zMSFV framework can ensure conservative coarse-scale solutions and allows for the construction of conservative fine-scale solutions. The main features of the zMSFV framework are used to derive robust preconditioners for iterative solutions of high-contrast problems as encountered in subsurface flow and transport simulations. It is shown that the obtained convergence rates are independent of the contrast. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

An adaptive conservative time integration scheme (ACTI) for compressible flow simulations is devised. A finite volume method is used, in which every cell employs a time step size which is some power of two times smaller than a global time step. By first advancing those cells with smaller time steps, this allows to use the already computed accumulated fluxes when the adjacent cells with larger time steps catch up. Conservation and periodic synchronization are thus guaranteed by construction; both important advantages of the new scheme (especially in the presence of shock waves) compared to previously proposed adaptive time stepping methods for compressible flow. Accuracy and computational speedup are demonstrated for challenging 1D and 2D test cases. Comparisons of ACTI results with an analytical solution and with results obtained without ACTI show excellent agreement. Although maximum wave speeds and/or cell sizes vary by several orders of magnitude throughout the computational domains, the CFL numbers employed by ACTI in each cell vary not more than by approximately a factor of two. Without ACTI, on the other hand, the CFL numbers are tiny in most of the cells. Since this results in a dramatic reduction of required flux computations, high speedup gains are achieved due to ACTI; in particular, if only few cells experience a very small time step, while the time step size limitations are less severe for the majority of cells. • Explicit adaptive time integration for compressible flow. • Strictly conservative - is of great importance especially in the presence of shocks. • Piecewise linear reconstruction in space and time - is important for accuracy. • Total number of flux computations and cell value updates can be drastically reduced. • Proposed scheme is robust and can be integrated immediately into existing codes. [ABSTRACT FROM AUTHOR]

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

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

Abstract: The iterative-multi-scale-finite-volume (IMSFV) procedure is applied as an efficient solver for the pressure Poisson equation arising in numerical methods for the simulation of incompressible flows with the immersed-interface method (IIM). Motivated by the requirements of the specific IIM implementation, a modified version of the IMSFV algorithm is presented to allow the solution of problems, in which the varying coefficient of the elliptic equation (e.g. the permeability of the medium in the context of the simulation of flows in porous media) varies over several orders of magnitude or even becomes zero within the integration domain. Furthermore, a strategy is proposed to incorporate the iterative procedure needed by the IIM to converge out constraints at immersed boundaries into the iterative IMSFV cycle. No significant deterioration of performance of the IMSFV method is observed with respect to cases, in which no iterative improvement of the boundary conditions is considered. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

The multiscale finite-volume (MSFV) method has been designed to solve flow problems on large domains efficiently. First, a set of basis functions, which are local numerical solutions, is employed to construct a fine-scale pressure approximation; then a conservative fine-scale velocity approximation is constructed by solving local problems with boundary conditions obtained from the pressure approximation; finally, transport is solved at the fine scale. The method proved very robust and accurate for multiphase flow simulations in highly heterogeneous isotropic reservoirs with complex correlation structures. However, it has recently been pointed out that the fine-scale details of the MSFV solutions may be lost in the case of high anisotropy or large grid aspect ratios. This shortcoming is analyzed in this paper, and it is demonstrated that it is caused by the appearance of unphysical "circulation cells." We show that damped-shear boundary conditions for the conservative-velocity problems or linear boundary conditions for the basis-function problems can significantly improve the MSFV solution for highly anisotropic permeability fields without sensitively affecting the solution in the isotropic case. [ABSTRACT FROM AUTHOR]

• An adaptive and still conservative time integration scheme for finite volume solvers. • In combination with a new flux solver the new sub-time stepping scheme is of almost second order (temporally and spatially). • The new adaptive conservative time integration scheme is robust and much faster (in some of the cases more than 50 times) than conventional time stepping (at almost the same accuracy). This is demonstrated in the paper with numerical studies. Finite volume methods for time dependent problems typically employ time integration schemes with constant time step sizes. The latter, in order to ensure stability and time accurate solutions, have to be chosen small enough, such that the Courant-Friedrichs-Lewy (CFL) number stays below a critical value everywhere. In many cases, however, this time step size limitation leads to huge numbers of very small time steps, which renders simulations very expensive. Motivated by this drawback of conventional time stepping schemes, various sub-time stepping algorithms have been proposed. However, most of them are inherently asynchronous, require small local CFL numbers or are not strictly conservative. In this paper a new adaptive time integration scheme for finite volume methods, which is conservative, of high spatial and temporal order, robust and easy to implement is presented. It relies on local sub-time steps which are fractions of a global time step by powers of two, i.e., the grid cells proceed asynchronously in the order of their termination time, but since they all synchronize at the end of each global time step, it is possible to guarantee continuity of the mean fluxes and thus strict conservation at the global time step resolution. Numerical experiments with 1D and 2D test cases demonstrate that the adaptive conservative time integration (ACTI) scheme can achieve extreme speed-up factors over conventional time integration, while still maintaining high spatial and temporal accuracy. [ABSTRACT FROM AUTHOR]

The time step of an explicit time integration scheme for solving time-dependent hyperbolic partial differential equations in a finite volume method (FVM) framework is restricted by the Courant–Friedrichs–Lewy (CFL) criterion. Conventional time stepping integrates all grid cells with the same time step. This causes unnecessary computational costs when wave speeds and/or grid spacing vary considerably throughout the domain, and a few critical cells dictate the global time step, although most cells could be advanced with a much larger time step. Adaptive time stepping overcomes this issue by allowing different local time step sizes for each grid cell. The adaptive conservative time integration (ACTI) scheme is a recently developed adaptive time stepping method which relies on local time steps that are equal to the largest time step divided by powers of two. This work extends the ACTI scheme to tracer transport in fractured porous media. When fluid velocity within highly permeable fractures is higher than in the rock matrix and local grid refinement is applied around fractures, the CFL restriction would require prohibitively small time steps in the vicinity of fractures. For two-dimensional discrete fracture and matrix models of fracture patterns we demonstrate that ACTI reduces the computational cost by orders of magnitude compared to global time stepping while retaining solution accuracy. Empirically, we show that ACTI is stable in combination with a first-order explicit flux discretization scheme. Since combination with a standard higher-order MUSCL scheme can lead to spurious oscillations in the solution, we propose a modified MUSCL scheme relying on advection of an inclined reconstruction (MUSCL-AIR). • Adaptive time stepping for solving hyperbolic PDEs based on a local CFL criterion. • Highly accurate calculation of tracer transport in fractured porous media. • Sharp tracer front at exit of fully resolved fracture if no mixing occurs within it. • Robust higher-order flux calculation with a modified MUSCL scheme. • Highly efficient: up to 80 times faster compared to conventional global time stepping. [ABSTRACT FROM AUTHOR]

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