We devise a hybrid MultiScale Finite Element-Finite Volume (h-MSFE-FV) framework to simulate singlephase flow through elastic deformable porous media. The coupled problem is solved based on a two-field fine-scale mixed finite element-finite volume formulation of the governing equations, namely conservation laws of linear momentum and mass, in which the primary unknowns are the displacement vector and pressure. For the MSFE displacement stage, we develop sets of local basis functions for the displacement vector over coarse cells, subject to reduced boundary condition. This MSFE stage is then coupled with the MSFV method for flow, where coarse and dual-coarse grids are imposed to obtain approximate but conservative multiscale solutions. Numerical experiments are presented to demonstrate accuracy and robustness of the proposed h-MSFE-FV method---both as an approximate, non-iterative solver, and a preconditioner. @en

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

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 can be 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. © 2015, The Author(s).@en

We describe an Enriched Algebraic Multiscale Solver (EAMS) that overcomes the deficiency of existing multiscale methods for flow in heterogeneous media with large coherent correlation structures and high contrasts. For a given multiscale method, EAMS enriches the coarse space with local basis-functions specifically aimed at the largest error components in the solution space. For this purpose, the discrete error equation is used to identify the solution modes that are missing from the multiscale operator. The identified error modes, which are complex combinations of a spectrum of wave numbers, are then localized (truncated) and added to the prolongation operator. The enrichment process is repeated iteratively until the desired convergence rate is reached. The identification and enrichment processes are algebraic, and they are performed adaptively during the iterative solution process. Using challenging test cases from the literature, we show that EAMS leads to great improvements in the robustness and efficiency of existing state-of-the-art multiscale linear solvers. In most settings, the convergence rate of AMS is improved significantly by supplementing the standard basis functions with a few basis functions guided by the error equation. Since the enrichment is adaptive and algebraic, it can be integrated into any existing multiscale linear-solver implementation. EAMS is expected to be most useful in modeling evolution multiphase problems in heterogeneous reservoirs, whereby the changes in the character of the linear system - across Newton iterations and time steps - are relatively mild. @en

We present a mixed hybrid finite-element (FE) formulation for modeling subsurface flow and transport for general-purpose compositional reservoir simulation. The formulation is fully implicit in time and employs a hybrid FE method for the spatial discretization of the conservation equations. The hybrid FE formulation is implemented in the Automatic Differentiation General Purpose Research Simulator (ADGPRS); consequently, the new FE-based methodology inherits all the `physics’ capabilities of ADGPRS, including compositional EOR models. The high-order mixed hybrid FE discretization scheme works for many types of finite elements and can handle highly anisotropic material properties. The formulation is locally conservative. The momentum and mass balance equations are solved simultaneously, including Lagrange multipliers on element interfaces. The fully implicit scheme uses the automatic differentiation capability to construct the Jacobian matrix. The hybrid FE approach accommodates unstructured grids, which are needed for honouring the complex geometry of the subsurface, in a straightforward manner. We present compositional test cases with full permeability tensors, and we discuss the accuracy and computational efficiency of the formulation. We also compare the performance of the hybrid FE-based scheme with finite-volume based Multi-Point Flux Approximation (MPFA) methods. @en

In this paper, the Multiscale Finite-Volume (MSFV) method is extended to include compositional displacements in heterogeneous porous media, in which accurate modeling of the mass transfer and associated phase behaviors is critical. A sequential-implicit strategy is employed 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 has weak dependencies 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. Given the discrete forms of the flow and transport problems, the details of the MSFV strategy are then described for the efficient solution of the pressure equation. The only source of error in this MSFV framework is due to the well-known localization assumptions. No additional assumption due to the complex physics is done in this framework. For one-dimensional problems, the proposed sequential strategy results are validated against those obtained by a fully implicit simulator. The accuracy and efficiency of the MSFV method for compositional simulations are then illustrated for different numerical test cases. It is shown that the MSFV results are identical to the fine-scale solutions for one-dimensional problems in transient- and steady- states.@en