Multiscale finite-element method for linear elastic geomechanics

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.Petroleum Engineerin

Simulation of mineral dissolution at the pore scale with evolving fluid-solid interfaces: review of approaches and benchmark problem set

This manuscript presents a benchmark problem for the simulation of single-phase flow, reactive transport, and solid geometry evolution at the pore scale. The problem is organized in three parts that focus on specific aspects: flow and reactive transport (part I), dissolution-driven geometry evolution in two dimensions (part II), and an experimental validation of three-dimensional dissolution-driven geometry evolution (part III). Five codes are used to obtain the solution to this benchmark problem, including Chombo-Crunch, OpenFOAM-DBS, a lattice Boltzman code, Vortex, and dissolFoam. These codes cover a good portion of the wide range of approaches typically employed for solving pore-scale problems in the literature, including discretization methods, characterization of the fluid-solid interfaces, and methods to move these interfaces as a result of fluid-solid reactions. A short review of these approaches is given in relation to selected published studies. Results from the simulations performed by the five codes show remarkable agreement both quantitatively—based on upscaled parameters such as surface area, solid volume, and effective reaction rate—and qualitatively—based on comparisons of shape evolution. This outcome is especially notable given the disparity of approaches used by the codes. Therefore, these results establish a strong benchmark for the validation and testing of pore-scale codes developed for the simulation of flow and reactive transport with evolving geometries. They also underscore the significant advances seen in the last decade in tools and approaches for simulating this type of problem

Incomplete mixing in porous media: Todd-Longstaff upscaling approach versus a dynamic local grid refinement method

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.Petroleum Engineerin