Flow in fractured porous media occurs in the earth's subsurface, in biological tissues, and in man-made materials. Fractures have a dominating influence on flow processes, and the last decade has seen an extensive development of models and numerical methods that explicitly account for their presence. To support these developments, four benchmark cases for single-phase flow in three-dimensional fractured porous media are presented. The cases are specifically designed to test the methods' capabilities in handling various complexities common to the geometrical structures of fracture networks. Based on an open call for participation, results obtained with 17 numerical methods were collected. This paper presents the underlying mathematical model, an overview of the features of the participating numerical methods, and their performance in solving the benchmark cases.
In this paper we study a numerical method for the simulation of free surface flows of viscoplastic (Herschel-Bulkley) fluids. The approax;h is based on the level set method for capturing the free surface evolution and on locally reflned and dynamically adapted octree cartesian staggered grids for the discretization of fluid and level set equations. A regularized model is applied to handle the non-differentiability of the constitutive relations. We consider an extension of the stable approximation of the Newtonian flow equations on staggered grid to approximate the viscoplastic model and level-set equations if the free boundary evolves and the mesh is dynamically reflned or coarsened. The numerical method is flrst validated for a Newtonian case. In this case, the convergence of numerical solutions is observed towards experimental data when the mesh is reflned. Further we compute several 3D viscoplastic Herschel-Bulkley fluid flows over incline planes for the dam-break problem. The qualitative comparison of numerical solutions is done versus experimental investigations. Another numerical example is given by computing the freely oscillating viscoplastic droplet, where the motion of fluid is driven by the surface tension forces. Altogether the considered techniques and algorithms (the level-set method, compact discretizations on dynamically adapted octree cartesian grids, regularization, and the surface tension forces approximation) result in efflcient approach to modeling viscoplastic free-surface flows in possibly complex 3D geometries.Mathematics subject classification: 65M06, 76D27, 76D99.
We present a new monotone finite volume method for the advection-diffusion equation with a full anisotropic discontinuous diffusion tensor and a discontinuous advection field on 3D conformal polyhedral meshes. The proposed method is based on a nonlinear flux approximation both for diffusive and advective fluxes and guarantees solution non-negativity. The approximation of the diffusive flux uses the nonlinear two-point stencil described in [9]. Approximation of the advective flux is based on the second-order upwind method with a specially designed minimal nonlinear correction [26]. The second-order convergence rate and monotonicity are verified with numerical experiments.The discrete maximum principle (DMP) and local mass conservation are important properties of a numerical scheme for the approximate solution of the steady state advection-diffusion equation. An accurate discretization method satisfying DMP is hard to develop. We address the monotonicity condition as the simplified version of the DMP, which guarantees only solution non-negativity. A number of physical quantities (concentration, temperature, etc.) are non-negative by their nature and their approximations should be non-negative as well. We present a nonlinear finite volume (FV) method on conformal polyhedral meshes that satisfies the monotonicity condition for a wide range of problem coefficients. We admit a jumping diffusion coefficient represented by full anisotropic tensors, a jumping advection coefficient, which may be produced by the Darcy equation in multimaterial media, and both diffusion-dominated and advection-dominated regimes. The presented method is the extension of numerical schemes [9, 26] developed for the 3D diffusion equation [9] and the 2D advection-diffusion equation with continuous coefficients [26].The major difficulty encountered in the design of a monotone numerical scheme is suppressing unwanted spurious (non-physical) oscillations in the numerical solution. These oscillations may appear in advection-dominated problems due to internal shocks and boundary layers, and in diffusion-dominated problems in highly anisotropic media due to inappropriate approximations of the diffusive flux.In the finite element (FE) context, efficient damping of spurious oscillations in advection-dominated regimes has been developed within the streamline upwind
The paper develops a semi-Lagrangian method for the numerical integration of the transport equation discretized on adaptive Cartesian cubic meshes. We use dynamically adaptive graded Cartesian grids. They allow for a fast grid reconstruction in the course of numerical integration. The suggested semi- Lagrangian method uses a higher order interpolation with a limiting strategy and a back-and-forth correction of the numerical solution. The interpolation operators have compact nodal stencils. In a series of experiments with dynamically adapted meshes, we demonstrate that the method has at least the second-order convergence and acceptable conservation and monotonicity properties.
The paper studies a splitting method for the numerical time-integration of the system of partial di erential equations describing the motion of viscous incompressible uid with free boundary subject to surface tension forces. The method splits one time step into a semi-Lagrangian treatment of the surface advection and uid inertia, an implicit update of viscous terms and the projection of velocity into the subspace of divergence-free functions. We derive several conservation properties of the method and a suitable energy estimate for numerical solutions. Under certain assumptions on the smoothness of the free surface and its evolution, this leads to a stability result for the numerical method. E cient computations of free surface ows of incompressible viscous uids need several other ingredients, such as dynamically adapted meshes, surface reconstruction and level set function re-initialization. These enabling techniques are discussed in the paper as well. The properties of the method are illustrated with a few numerical examples. These examples include analytical tests and the oscillating droplet benchmark problem.
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