Accurate and Consistent Lagrangian Transport Simulations for Finite-Element-Models of Thermo-Hydro-Mechanical Processes in Porous Media

Trajectory-based simulations of transport in porous and fractured media are computationally fast and straightforward to parallelize. They neither induce spurious oscillations nor do they introduce numerical dispersion. Such simulation techniques rely on consistent and mass-conservative particle tracking schemes posing an attractive alternative to traditional solutions via an Eulerian discretization of the transport equation possibly including chemical reactions or radioactive decay chains as well. An accurate simulation of many processes in geological media requires a coupled solution of fluid flow, heat transport, and mechanical deformation. Integrated simulation platforms like OpenGeoSys typically rely on the finite element method in different variations for solving the resulting coupled equation system. Reasons for this are the relative ease to implement the coupling of different physcial processes via finite elements, their ability to natively handle full material tensors and unstructured grids, the small number of degrees of freedom compared to other discretization techniques, as well as their matureness and common usage in solving problems from structural mechanics. However, finite element solutions of the transport equation may suffer from spurious oscillations or numerical diffusion, if grids and time-stepping are not appropriate. Particle-tracking circumvents these issues but relies on a consistent velocity field originating from the flow solution. While finite elements yield a continuous solution of the primary unknown and conserve mass in the nodes, unfortunately, they yield Darcy velocity fields in the elements which are neither conforming nor element-wise mass conservative leading to a jump of the Darcy velocity normal to an element interface. Such velocity fields do not meet the requirements for accurate and consistent particle tracking. To overcome this challenge, we adapted the flux projection of Selzer and Cirpka (2020), initially presented for steady-state groundwater flow on simplices, to coupled thermo-hydro-mechanical models based on the standard Galerkin finite element method on triangular prisms, thus yielding a conforming and element-wise mass-conservative Darcy velocity field via postprocessing. Based on this, we used the semi-analytical particle-tracking scheme presented by Selzer et al. (2021) to compute trajectories. We coupled this framework to OpenGeoSys, which is an open-source multi-field simulation platform based on finite elements, and applied it to a three-dimensional thermo-hydro-mechanical model including several geological layers simulating the fate of a conceptually simplified deep geological repository for high-level nuclear waste in clay stone as host-rock formation over one million years including the effects of glacial cycles.