Finite difference based wave simulation in porous media

Project description
The project deals with the development of methodology for seismic modeling; that is, numerical methods and codes for computing seismic data given a model of the subsurface. It is common to use an elastic or even acoustic wave equation (approximation) to describe the propagation of seismic waves in the solid earth. However, solid earth materials are fluid-saturated porous media, suggesting that the use of such acoustic or elastic approximations may sometimes lead to significant errors in the seismic modeling results.

Wave propagation in porous media are described by the equations of Biot that were introduced 50 years ago (Biot, 1962). Biot theory have been used in the oil and gas industry for exploration and monitoring purposes. However, it can also be used for detection of CO2 at Carbon Capture and Storage projects where CO2 is injected into geological formations for permanent storage.

Wave propagation in porous media is more complicated than in elastic media due to the existence of the pore fluid. The relative motion of the fluid with respect to the solid generates a second P-wave called the slow P-wave, which is strongly dissipative and has a velocity close to the velocity of the fluid (Masson et al., 2006).

Numerical simulation of wave propagation in porous media can be performed on the basis of different numerical methods (Carcione et al., 2010). In this project, the focus will be on the finite difference time domain method, which is relatively efficient and easy to implement. More specifically, we propose to use a velocity-stress-pressure staggered-grid 2D finite difference algorithm.

The student shall develop a method and code to solve the Biot equations in two dimensions and perform a series of numerical experiments to investigate the effects of heterogeneities in the seismic and poro-elastic parameters on the observable seismic waveforms. Particular attention should be given to the phenomenon of seismic attenuation due to wave-induced fluid flow at different length-scales.

Although the focus will be on seismic forward modellng, it may also be possible to investigate the corresponding seismic inverse problem; that is, how to estimate the seismic (pore-elastic) model parameters from the observed seismic data. However, this depends on the interests of the student as well as the progress of the project.

References
Biot, M. A., 1962, Mechanics of deformation and acoustic propagation in porous media: Journal of Applied Physics, 33, 1482–1498.
Carcione, J. M., Morency, C., and Santos, J. E., 2010, Computational poroelasticity-a review: Geophysics, 75, No. 5, 75A229–75A243.
Masson, Y. J., Pride, S.R. and Nihei K.T., 2006. Finite difference modeling of Biot's poroelastic equations at seismic frequencies. Journal of Geophysical Research: 111, B10305.

Proposed course plan during the master's degree (60 ECTS)
GEOV218  Rock physics
GEOV276 Introduction to theoretical seismology
GEOV277 Signal analysis and inversion in the earth sciences

GEOF211 Numerical modeling
MAT160 Introduction to numerical methods
MAT260 Numerical solution of differential equations

Prerequisites
This project is associated with VISTA Center for Modeling of Coupled Surface Dynamics (https://www.uib.no/en/vista-csd); which represents a collaboration with the departments of Earth Science
Mathematics. Therefore, it is an advantage of the student has an interest in applied mathematics and computer programming in addition to geophysics and geology. However, there are no formal mathematical prerequisites, suggesting that the courses listed above will provide the required background.

The courses GEOV211, MAT160 are MAT260 are all recommended, but one of these courses may be replaced by a special syllabus on practical numerical methods for geophysicists, depending on the background and interests of the student. The co-supervisor Florin Radu is a Professor of Applied Mathematics from VISTA CSD and the porous media group at the Department of Mathematics.