Course content

Motivation in terms of ground water flows, biological tissue, hydrocarbon management, fuel cells, electrophoresis, building materials and the quest for the governing equations. Contents 1. Geometry of porous media - Porosity and the packing of spheres - Real Rocks (porosity distributions, correlations, sedimentrary processes) - Fractals (basic theory, examples from mathematics and box-counting) 2. Hydrodynamics - Navier Stokes equation - Examples of low Raynold number flows (Pouiseuille, Couette and Batchelors lubrication theory) - Darcy's law - Karman-Kozeny - Capillarity, droplets and Laplace law (water is adhesive and supports tension) - Youngs law and wetting - Examples of multi-phase flows (Washburn equaton and the Saffman-Taylor instability) - Capillary dominated flow in porous media (application of box-counting) - Viscous fingering (applicationg of box-counting for fractal dimension) Steady states and the justification of REV approaches (when can we assume that the result of averaging is independent of REV size?) 3. Statistical mechanics - Diffusion and the Langevin equation (leading up to the Einstein relation) - Green-Kubo relations (for the measurement of diffusivity and viscosity via MD. Derive for D, generalize to viscosity) - Percolation and invasion percolation (Could be left entirely for the next chapter?) 4. Simulation methods - Random walks and the advection diffusion equation - Basic principles of molecular dynamics (Newton, Lennard Jones and the celocity Verlet algorithm) - Lattice Boltzmann methods (Basic algorithm fir Navier Stokes and the additions that introduce diffusive tracers, surface tension and thermal gradients/buoyancy) - Network models (Basix algorithm for the flow of fluids or electric currents as well as the use of Washburn equation) - Invasion percolation: Basix model coded efficiently as well as the added feature of gradients/gravity

Learning outcome

After completing the course, the candidate will have the following knowledge, skills and general competence.

Knowledge: (i) Masters the relevant theory, problem formulations and methodologies for description of transport in porous media (ii) Is able to evaluate when it is appropriate to use one vs. another method.

Skills: Can plan and perform a project at an advanced level using the course toolbox. General competence: (i) Can perform research at a high international level. (ii) Has knowledge of recent enabling technologies that meets the needs of society, when the field of transport in porous media is concerned.

Learning methods and activities

Dicsussion groups, problemsolving, lectures and video lectures.

Compulsory assignments

• Exercises
• Project report

Recommended previous knowledge

Assumed background of the students: Equilibrium statistical mechanics. Some students may know about diffusion/Langevin equations/basic theory of Onsager reciprocity relations, others will know the Boltzmann equation.

Required previous knowledge

A basic course in thermodynamics and knowledge corresponding to mathematics 1-3 are required for participation. The course will serve as a link to the experimental course in PoreLab. But it does not depend on this course.