Uncertain Nonlinearities in Groundwater Flow Models
PhD Candidate: Simon Boisserรฉe
Supervisors: M. Bachmayr (Mathematics), M Moulas (Geosciences), B. Kaus (Geosciences)
At the core of this project is the analysis of fluid flow in poroviscoelastic media in terms of porosity ๐ and pressure ๐, which can be described by systems of nonlinear partial differential equations of the form
There are significant uncertainties concerning the precise form of the porosity-permeability relationship ๐ and in the function ๐, which can be chosen to describe decompaction weakening that can lead to the formation of channels.
The project comprises the mathematical analysis of equations (1), the development of stable and efficient numerical methods and the application of Bayesian inverse methods for the identification of the nonlinearities in the equations. Simon has obtained proof of local existence and uniqueness of solutions to (1), even for non-smooth initial functions ๐0, which was a completely open problem. The analysis of (1) is thus essentially complete. A corresponding publication is currently in preparation.
Simonโs approach for numerically solving (1) in the presence of strongly localized large gradients in solutions, as typical of channel formation, is based on a space-time adaptive discontinuous Petrov-Galerkin (dPG) method. The space-time adaptivity makes it possible to use locally varying time steps (Fig. 2) and provides a posteriori error bounds. Based on this numerical solver, he will develop multilevel techniques for the quantification of uncertainties in (1) based on measurement data. His numerical experiments have also shown that in the case of nonsmooth solutions, the commonly used small-porosity approximation of (1), where the factor (1-ฯ)^(-1) is omitted, can lead to non-physical solutions even for reasonable starting values. These effects are currently under further investigation.
You can download the original proposal here.
Wave equation, radiative transfer and Bayesian inversions: application to modeling and imaging of volcanoes
PhD Candidate: Yi Zhang
Supervisors: L. De Siena (Geosciences), M. Bachmayr (Mathematics), L. Hartung (Mathematics), T. Baumann (Geosciences), B. Kaus (Geosciences)
This project aims to build a new standard of seismic wave propagation codes linked to geodynamics and especially targeting magmatic systems. Traveltime and attenuation tomography are currently widely used. However, those methods are limited to a smooth model for volcanoes. This project will use a comprehensive procedure to image volcanoes with higher resolution (analogue to exploration seismology), which will help understand the interior of volcanoes and, eventually, what triggers eruptions. Yi has recently finalized his original wave-propagation code, which includes advanced analytical and computational tools for the fast generation of scattered waves in stochastic media, adjoint formulations, and mathematical and computational tools that allow it to link to geodynamic simulations. The code is entirely written in Julia and available on GitHub. The publication draft will be submitted by early 2022 to Geophysics. Yi has already applied the code to model seismic and ultrasonic wavefields at Earth and rock-sample scales. He is a coauthor in a paper on ultrasonic wave propagation from acoustic emissions in deformed rock samples, submitted late 2021 to the Journal of Geophysical Research: Solid Earth. He has developed a tool that allows inverting full-waveforms data to JSWAP forward simulations and applied it to data from Campi Flegrei caldera and the Crati Valley, Italy. The corresponding papers, of which he is lead and second author, respectively, are now in draft format and will be submitted by mid-2022. In the second stage of his PhD, he will focus on Hamiltonian and Monte Carlo inversions. Yi has recently been admitted as a Graduate Student at the Max Planck Graduate Center of JGU. After presenting his results for the second time at the European Geoscience Union, he will visit Prof. Andreas Fichtner, who leads the Geophysics group at ETH Zurich and is the primary expert of the technique and its application to tomography.
You can download the original proposal here.