A dynamic multilevel method for fully-coupled simulation of flow and heat transfer in heterogeneous and fractured geothermal reservoirs is presented (FG-ADM). The FG-ADM develops an advanced simulation method which maintains its efficiency when scaled up to field-scale applications, at the same time, it remains accurate in presence of complex fluid physics and heterogeneous rock properties. The embedded discrete fracture model is employed to accurately represent fractures without the necessity of unstructured complex grids. On the fine-scale system, FG-ADM introduces a multi-resolution nested dynamic grid, based on the dynamic time-dependent solution of the heat and mass transport equations. The fully-coupled implicit simulation strategy, in addition to the multilevel multiscale framework, makes FG-ADM to be stable and efficient in presence of strong flow-heat coupling terms. Furthermore, its finite-volume formulation preserves local conservation for both mass and heat fluxes. Multi-level local basis functions for pressure and temperature are introduced, in order to accurately represent the heterogeneous fractured rocks. These basis functions are constructed at the beginning of the simulation, and are reused during the entire dynamic time-dependent simulation. For several heterogeneous test cases with complex fracture networks we show that, by employing only a fraction of the fine-scale grid cells, FG-ADM can accurately represent the complex flow-heat solutions in the fractured subsurface formations.

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

An Algebraic Dynamic Multilevel (ADM) method for simulations of multiphase flow in heterogeneous porous media with an adaptive enriched multiscale formulation for saturation unknowns is presented. ADM maps the fine-scale fully-implicit (FIM) discrete system of equations to a dynamic multilevel system, the resolution of which is defined based on the location of the fluid fronts. The map between the dynamic multilevel resolutions is performed algebraically by sequences of restriction and prolongation operators. While finite-volume restriction operators are necessary to ensure mass conservation at all levels, different interpolation strategies can be considered for each main unknown (e.g., pressure and saturation). For pressure, the multiscale basis functions are used to accurately capture the effect of fine-scale heterogeneities at all levels. In previous works, all other unknowns (e.g., saturation) were interpolated with piece-wise constant functions. Hence, the multiscale nature of saturation equation was not fully exploited. Here, an adaptive interpolation strategy, thus a multiscale transport formulation, is employed for the saturation unknowns that allows to preserve most details of the fine-scale saturation distribution even in regions where a coarser resolution is employed. In regions where the ratio between the coarse and the fine-scale saturation updates is detected to be constant throughout the time-dependent simulation, such ratio is stored and employed as interpolator for subsequent time-steps in which a coarser resolution is employed. Numerical results are presented to study the accuracy and efficiency of the method and the advantages of such interpolation strategy for test cases including challenging non-linear physics, i.e. gravitational and capillary effects.        @en

Field-scale simulation of flow in porous media in presence of incomplete mixing demands for high-resolution computational grids, much beyond the scope of state-of-the-art simulators. Hence, the upscaling-based Todd and Longstaff (TL) approach is typically used, where coarse grid cells are employed with effective mixing fluid properties and parameters found by matching results obtained with fully resolved reference simulations. Dynamic local grid refinement (DLGR) techniques, on the other hand, only employ fine-scale grid resolution where the fully mixed assumption is not valid. The rest of the domain is then solved at coarser resolutions, where the fully mixed assumption is valid. Here, we assess the accuracy and the robustness of DLGR- and TL-based simulations of miscible displacements in homogeneous and heterogeneous porous media. Due to the intrinsic uncertainty within the unstable displacement nature of the studied incomplete mixing processes, the performance of the methods is also investigated based on a range of acceptable solutions rather than relying only on a single reference one. Systematic numerical results illustrate that the DLGR method is much more robust and accurate than the upscaling-based TL approach, and employs only a small fraction of fine-scale reference grids. Especially, the TL upscaling results (though history matched with computationally expensive fine-scale results) are very sensitive to the change of the simulation parameters. Based on this study, we propose a dynamic multilevel simulation strategy for efficient and reliable large-scale simulation of the complex incomplete mixing processes.@en

We present an algebraic dynamic multilevel method for multiphase flow in heterogeneous fractured porous media (F-ADM), where fractures are resolved at fine scale with an embedded discrete modelling approach. This fine-scale discrete system employs independent fine-scale computational grids for heterogeneous matrix and discrete fractures, which results in linear system sizes out of the scope of the classical simulation approaches. To reduce the computational costs, yet provide accurate solutions, on this highly resolved fine-scale mesh, F-ADM imposes independent dynamic multilevel coarse grids for both matrix and lower-dimensional discrete fractures. The fully-implicit discrete system is then mapped into this adaptive dynamic multilevel resolution for all unknowns (i.e., pressure and phase saturation). The dynamic resolution aims for resolving sharp fronts for the transport unknowns, thus constant interpolators are used to map the saturation from coarse to fine grids both in matrix and fractures. However, due to the global nature of the pressure unknowns, local multilevel basis functions for both matrix and fractures with flexible matrix-fracture coupling treatment are introduced for the pressure. The assembly of the full sets of basis functions allows for mapping the solutions up and down between any resolutions. Due to its adaptive multilevel resolution, F-ADM develops an automatic integrated framework to homogenise or explicitly represent a fracture network at a coarser level by selection of the multilevel coarse nodes in each sub-domain. Various test cases, including multiphase flow in 2D and 3D media, are studied, where only a fraction of the fine-scale grids is employed to obtain accurate nonlinear multiphase solutions. F-ADM casts a promising approach for large-scale simulation of multiphase flow in fractured media.

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Accurate numerical simulation of coupled fluid flow and heat transfer in heterogeneous geothermal reservoirs demand for high resolution computational grids. The resulting fine-scale discrete systems--though crucial for accurate predictions--are typically upscaled to lower resolution systems due to computational efficiency concerns. Therefore, advanced scalable methods which are efficient and accurate for real-field applications are more than ever on demand. To address this need, we present an algebraic dynamic multilevel method for flow and heat transfer in heterogeneous formations, which allows for different temperature values for fluid and rock. The fine-scale fully-implicit discrete system is mapped to a dynamic multilevel grid, the solution at which are connected through local basis functions. These dynamic grid cells are imposed such that the sub-domain of sharp gradients are resolved at fine-scale, while the rest of the domain remains at lower (coarser) resolutions. In order to guarantee the quality of the local (heat front) components, advanced multiscale basis functions are employed for global (fluid pressure and rock temperature) unknowns at coarser grids. Numerical test cases are presented for homogeneous and heterogeneous domains, where ADM employs only a small fraction of the finescale grids to find accurate complex nonlinear thermal flow solutions. As such, it develops a promising scalable framework for field-scale geothermal simulations.@en

A dynamic multilevel compositional solver (C-ADM) is introduced for fully- (and sequentially-) implicit systems arising from compositional displacements in natural porous media. The fully (or sequential) implicit system is first described at a fine-scale resolution, where phases are allowed to consist of different components (based on thermodynamics equilibrium). In addition, heterogeneous capillary functions (defined based on Leverett’s J-function) and gravitational effects are both considered, adding significantly to the non-linear complexity of the processes. Given this complex fine-scale system for a heterogeneous reservoir, C-ADM defines a dynamic multilevel system, based on an error criterion, where the grid resolution is defined based on the physics of the process as well as geological complexities and location of wells. Once this multilevel grid is defined, sequences of prolongation and restriction operators are employed to obtain an accurate and efficient multilevel system. CADM allows for a general set of prolongation operators, e.g., constant, bilinear (or polynomial), and multiscale basis functions. The restriction operators, however, are constructed based on a mass-conservative finite-volume formulation at all levels. For several challenging test cases it is shown that C-ADM employs only a small fraction of the fine-scale grids to provide an accurate description of the process. C-ADM casts a promising approach in the application of dynamic grid refinement methods for real-field applications. @en

An algebraic dynamic multilevel method (ADM) is developed for fully-implicit (FIM) simulations of multiphase flow in heterogeneous porous media with strong non-linear physics. The fine-scale resolution is defined based on the heterogeneous geological one. Then, ADM constructs a space-time adaptive FIM system on a dynamically defined multilevel nested grid. The multilevel resolution is defined using an error estimate criterion, aiming to minimize the accuracy-cost trade-off. ADM is algebraically described by employing sequences of adaptive multilevel restriction and prolongation operators. Finite-volume conservative restriction operators are considered whereas different choices for prolongation operators are employed for different unknowns. The ADM method is applied to challenging heterogeneous test cases with strong nonlinear heterogeneous capillary effects. It is illustrated that ADM provides accurate solution by employing only a fraction of the total number of fine-scale grid cells. ADM is an important advancement for multiscale methods because it solves for all coupled unknowns (here, both pressure and saturation) simultaneously on arbitrary adaptive multilevel grids. At the same time, it is a significant step forward in the application of dynamic local grid refinement techniques to heterogeneous formations without relying on upscaled coarse-scale quantities @en