Mixed-Hybrid Formulation of Multidimensional Fracture Flow

We shall study Darcy flow on the heterogeneous system of 3D, 2D, and 1D domains and we present four models for coupling of the flow. For one of these models, we describe in detail its mixed-hybrid formulation. Finally, we show that Schur complements are suitable for solution of the linear system resulting form the lowest order approximation of the mixed-hybrid formulation

Simulation of rock salt dissolution and its impact on land subsidence

Extensive land subsidence can occur due to subsurface dissolution of evaporites such as halite and gypsum. This paper explores techniques to simulate the salt dissolution forming an intrastratal karst, which is embedded in a sequence of carbonates, marls, anhydrite and gypsum. A numerical model is developed to simulate laminar flow in a subhorizontal void, which corresponds to an opening intrastratal karst. The numerical model is based on the laminar steady-state Stokes flow equation, and the advection dispersion transport equation coupled with the dissolution equation. The flow equation is solved using the nonconforming Crouzeix-Raviart (CR) finite element approximation for the Stokes equation. For the transport equation, a combination between discontinuous Galerkin method and multipoint flux approximation method is proposed. The numerical effect of the dissolution is considered by using a dynamic mesh variation that increases the size of the mesh based on the amount of dissolved salt. The numerical method is applied to a 2D geological cross section representing a Horst and Graben structure in the Tabular Jura of northwestern Switzerland. The model simulates salt dissolution within the geological section and predicts the amount of vertical dissolution as an indicator of potential subsidence that could occur. Simulation results showed that the highest dissolution amount is observed near the normal fault zones, and, therefore, the highest subsidence rates are expected above normal fault zones

Modelling of processes in fractured rock using FEM/FVM on multidimensional domains

AbstractThe paper deals with a new approach to the numerical modelling of groundwater flow in compact rock massifs.Empirical knowledge of hydrogeologists is summarized first. There are three types of objects important for the groundwater flow in such massifs—small fractures, which can be replaced by blocks of porous media, large deterministic fractures and lines of intersection of the large fractures. We solve problem of the linear Darcy's flow on each of these three separated domains and then we join them by coupling the equations. We do not require geometrical correspondence between 1D, 2D and 3D meshes, which simplifies the process of the spatial dicretization. The mixed-hybrid FEM with the lowest-order Raviart–Thomas elements is used for approximation of the solution. The advective mass transfer is solved by the FVM on the same discretization as the flow problem.Results of numerical experiments with the model are shown in the end of the paper

A unified analysis of elliptic problems with various boundary conditions and their approximation

We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue--Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii) several approximation methods. The considered approximations can be conforming, or not (that is, the approximation functions can belong to the energy space of the problem, or not), and include classical as well as recent numerical schemes. Convergence results and error estimates are given. We finally briefly show how the abstract setting can also be applied to other models, including flows in fractured medium, elasticity equations and diffusion equations on manifolds. A by-product of the analysis is an apparently novel result on the equivalence between general Poincar{\'e} inequalities and the surjectivity of the divergence operator in appropriate spaces