The course addresses the mathematical and numerical modeling of some physical phenomena in the geosciences, with a focus on subsurface flow and deformation. The goal is to introduce numerical complexities gradually, by example, in parallel with the mathematical modeling of relevant physics.
We will address the following topics, progressing in parallel with mathematical and numerical modeling:
- Single phase flow in porous media: introduction, definitions and governing equation. Discretization methods: the advantages of mixed methods vs primal methods, with a focus on heterogeneities in material properties. Exercise session: simple numerical simulations with the software PorePy.
- Discretization of complex domains (layered, heterogeneous media and presence of thin inclusions such as fractures). Coupled hybrid dimensional PDEs. Flexible numerical methods for non-standard grids (such as VEM). Exercise session: numerical experiments with PorePy.
- Transport phenomena in porous media (passive transport, two-phase flow). Implicit and explicit discretization schemes, time step restrictions, choice of the numerical flux and its effects.
- Poro-elasticity: a primer on linear elasticity, introduction to Biot’s theory, coupled mathematical model. Invited seminar on solution strategies (monolithic vs iterative splittings) by J. Both (University of Bergen). Exercise session: numerical tests in PorePy.
- Reactive transport: mathematical modeling by advection-diffusion-reaction PDEs. Solution strategies: fully coupled vs iterative and non-iterative splittings; choice of the time advancing scheme; stiffness due to the coexistence of time scales. Exercise session: comparison of the strategies on a simple one-dimensional problem.
At the end of the course the students should have a basic knowledge of the mathematical models of some relevant phenomena in porous media flow. Moreover, the course provides an overview of the numerical challenges in the simulation of these phenomena: thanks to the exercise sessions the student will develop the ability to understand and comment numerical results in a discerning way.
Written notes and research papers on the aforementioned topics will be made available.