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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. [ABSTRACT FROM AUTHOR]

• The paper presents the first benchmarking study between homogenization and multiscale methods. • Developments are done in a unified adaptive dynamic multilevel (ADM) framework. • Both methods are employed to simulate multiphase flow in highly heterogeneous porous media. • Fully implicit coupling with finite volume scheme is used at all spatial scales. Accurate simulation of multiphase flow in subsurface formations is challenging, as the formations span large length scales (km) with high-resolution heterogeneous properties. To deal with this challenge, different multiscale methods have been developed. Such methods construct coarse-scale systems, based on a given high-resolution fine-scale system. Furthermore, they are amenable to parallel computing and allow for a-posteriori error control. The multiscale methods differ from each other in the way the transition between the different scales is made. Multiscale (finite element and finite volume) methods compute local basis functions to map the solutions (e.g. pressure) between coarse and fine scales. Instead, homogenization methods solve local periodic problems to determine effective models and parameters (e.g. permeability) at a coarser scale. It is yet unknown how these two methods compare with each other, especially when applied to complex geological formations, with no clear scale separation in the property fields. This paper develops the first comparison benchmark study of these two methods and extends their applicability to fully implicit simulations using the algebraic dynamic multilevel (ADM) method. At each time step, on the given fine-scale mesh and based on an error analysis, the fully implicit system is solved on a dynamic multilevel grid. The entries of this system are obtained by using multiscale local basis functions (ADM-MS), and, respectively, by homogenization over local domains (ADM-HO). Both sets of local basis functions (ADM-MS) and local effective parameters (ADM-HO) are computed at the beginning of the simulation, with no further updates during the multiphase flow simulation. The two methods are extended and implemented in the same open-source DARSim2 simulator (https://gitlab.com/darsim2simulator), to provide fair quality comparisons. The results reveal insightful understanding of the two approaches, and qualitatively benchmark their performance. It is re-emphasized that the test cases considered here include permeability fields with no clear scale separation. The development of this paper sheds new lights on advanced multiscale methods for simulation of coupled processes in porous media. [ABSTRACT FROM AUTHOR]

• The method is capable to capture the effects of high or low permeability fractures. • The strategy is applicable to general unstructured tetrahedral meshes. • The simulation tool can handle heterogeneous and anisotropic permeability fields. The modeling of fluid flow in heterogeneous, anisotropic and fractured porous media is relevant in many applications, including hydrocarbon and groundwater extraction, dispersion of contaminants, hydrogen or carbon dioxide (CO 2) storage. Thus, accurate and scalable simulation of fluid flow through these formations continues to be a great challenge. The presence of fractures that are explicitly modeled, ranging from flow barriers to highly conductive channels, significantly increases the complexity of the numerical simulation. The Embedded Discrete Fracture Model (EDFM) produces results that can be as accurate as those obtained using equidimensional Discrete Fracture Models (DFM) at a much lower computational cost for highly conductive fractures, but it is not adequate for impermeable barriers. In contrast, the pEDFM (projection-based Embedded Discrete Fracture Model) produces accurate results for both, channels and barriers by using additional matrix-fracture connectivities. In its current versions, it is restricted to cartesian and corner-point grid geometries and to the classical Two Point Flux Approximation (TPFA) scheme. In this work, for the first time, the pEDFM is extended to handle unstructured tetrahedral meshes (pEDFM-U). In addition, interface fluxes are approximated by using the Multipoint Flux Approximation method with a Diamond stencil (MPFA-D). This allows for simulation of highly heterogeneous and anisotropic geo-models. The developed method is robust and can deal with full permeability tensors on arbitrary tetrahedral meshes. The advective terms are discretized considering an implicit First Order Upwind (FOU) method. Our method is implemented within the DARSim (Delft Advanced Reservoir Simulation) open-source simulator framework. Through several test cases, the proposed method showed to be robust and capable to accurately capture the effects of both high and low permeability fractures for general tetrahedral meshes, under arbitrary heterogeneous and anisotropic permeability tensors for the rock matrix. [ABSTRACT FROM AUTHOR]

• Iterative multiscale gradient algorithms provide up to fine-scale accuracy gradients. • Gradient accuracy can be controlled via the i-MSFV tolerance. • Accurate gradients are obtained for challenging, high permeability contrast models. • The accuracy control can be used to speed-up the computations. We introduce a semi-analytical iterative multiscale derivative computation methodology that allows for error control and reduction to any desired accuracy, up to fine-scale precision. The model responses are computed by the multiscale forward simulation of flow in heterogeneous porous media. The derivative computation method is based on the augmentation of the model equation and state vectors with the smoothing stage defined by the iterative multiscale method. In the formulation, we avoid additional complexity involved in computing partial derivatives associated to the smoothing step. We account for it as an approximate derivative computation stage. The numerical experiments illustrate how the newly introduced derivative method computes misfit objective function gradients that converge to fine-scale one as the iterative multiscale residual converges. The robustness of the methodology is investigated for test cases with high contrast permeability fields. The iterative multiscale gradient method casts a promising approach, with minimal accuracy-efficiency tradeoff, for large-scale heterogeneous porous media optimization problems. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The past decades have witnessed an increasing interest in numerical simulation for flow in fractured porous media. To date, most studies have focused on 2D or pseudo-3D computational models, where the impact of 3D complex structures on seepage has not been fully addressed. This work presents a method for modeling seepage in 3D heterogeneous porous media. The complex structures, typically the stochastic discrete fractures and inclusions, are able to be simulated. A mesh strategy is proposed to discretize the complex domain. In particular, a treatment on the intersected elements is developed to ensure a conforming mesh. Then, numerical discretization is provided, in which the flux interactions of fractures, inclusions and surrounding rock matrix are included. Numerical tests are performed to analyze the hydraulic characteristics of 3D fractured media. First, the developed framework is validated by comparing numerical solutions with the results of embedded discrete fracture model. Next, the effects of orientation, aperture and radius of fractures on fluid flow and equivalent permeability tensor are analyzed. The variations of pressure distribution are studied in heterogeneous and homogeneous media. Finally, the hydraulic properties of a medium with complex structures are investigated to show the difference of hydraulic feature between fractures and inclusions. [ABSTRACT FROM AUTHOR]

We present the projection-based embedded discrete fracture model (pEDFM) for hexahedral corner-point grid (CPG) geometries, for the simulation of hydrothermal processes in fractured porous media. Unlike the previously-developed pEDFM for structured box grids, our new development allows for the modeling of complex geometries defined with hexahedral CPG cells. It also advances the pEDFM method to include coupled flow and heat transfer systems. Mass and energy conservation equations are simulated in a fully-coupled manner using a fully-implicit (FIM) integration scheme. This allows for stable simulations, specially when large time steps are taken. Independent corner-point grids are imposed on the rock matrix and all fractures, with conductivities ranging from highly permeable to flow barriers. The connectivities between the non-neighboring grid cells are described such that a consistent discrete representation of the embedded fractures occurs within the corner-point grid geometry, specially as the quadrilateral interfaces are allowed to be fully flexible. Various numerical tests including geologically-relevant and real-field models, which are established in the literature, are conducted to demonstrate the applicability of the developed method. It is shown that pEDFM can accurately capture the physical influence of both highly conductive fractures and flow barriers on the flow and heat transfer fields in complex reservoir geometries. This development is promising for flow simulations of real-field geo-models, increasing the discretization flexibility and enhancing the computational performance for capturing explicit fractures accurately. • Projection-based embedded discrete fracture model (pEDFM) for hexahedral corner-point Grid (CPG) geometries. • Accurate and explicit representation of fractures with generic conductivity contrasts. • Fully coupled mass and heat transport in fractured porous media. • Accurate modeling of geologically-relevant and real field-scale formations. [ABSTRACT FROM AUTHOR]

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