# Ruadhai Dervan Internship Report

**Name:** Ruadhai Dervan

**Current status:** PhD student, DPMMS

**Where I worked:** BP Institute, University of Cambridge

**Supervisors:** Peter Dudfield, PhD student, and Prof. Andy Woods.

**Summary of Work:** My project involved modelling fluid flow in a porous medium, in a two dimensional box. The box was assumed to have two areas with different permeabilities. The fluid flow follows Darcy's law:

where u is the fluid velocity, p is the pressure, mu is the viscosity and k is the permeability. Incompressibility of the fluid then gives

so we are modelling a harmonic function in areas where k is constant. Our model includes a source, where fluid flows in, and a sink, where fluid flows out. We assume a point source, and we get Poisson's equation

where f is non-zero only at the source and sink.

To solve this equation we use a finite element model, by approximating the Laplacian. We solve this equation using numerical methods on a square grid, using the python programming language and Matlab for graphs. Using the video making tool ffmpeg, videos such as the following were created to visualise the fluid flow.

The main question of the project is the following: how much do the permeability and boundary level affect the filling speed? To answer this question numerically, a laboratory experiment was created. Unfortunately, due to time constraints, we were unable to analyse the data.

(video 2)