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Numerical methods for wave propagation in solids containing faults and fluid-filled fractures
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis develops numerical methods for the simulation of wave propagation in solids containing faults and fluid-filled fractures. These techniques have applications in earthquake hazard analysis, seismic imaging of reservoirs, and volcano seismology. A central component of this work is the coupling of mechanical systems. This aspect involves the coupling of both ordinary differential equations (ODE)(s) and partial differential equations (PDE)(s) along curved interfaces.  All of these problems satisfy a mechanical energy balance. This mechanical energy balance is mimicked by the numerical scheme using high-order accurate difference approximations that satisfy the principle of summation by parts, and by weakly enforcing the coupling conditions. 

The first part of the thesis considers the simulation of dynamic earthquake ruptures along non-planar fault geometries and the simulation of seismic wave radiation from earthquakes, when the earthquakes are idealized as point moment tensor sources. The dynamic earthquake rupture process is simulated by coupling the elastic wave equation at a fault interface to nonlinear ODEs that describe the fault mechanics. The fault geometry is complex and treated by combining structured and unstructured grid techniques. In other applications, when the earthquake source dimension is smaller than wavelengths of interest, the earthquake can be accurately described by a point moment tensor source localized at a single point. The numerical challenge is to discretize the point source with high-order accuracy and without producing spurious oscillations.

The second part of the thesis presents a numerical method for wave propagation in and around fluid-filled fractures. This problem requires the coupling of the elastic wave equation to a fluid inside curved and branching fractures in the solid. The fluid model is a lubrication approximation that incorporates fluid inertia, compressibility, and viscosity. The fracture geometry can have local irregularities such as constrictions and tapered tips. The numerical method discretizes the fracture geometry by using curvilinear multiblock grids and applies implicit-explicit time stepping to isolate and overcome stiffness arising in the semi-discrete equations from viscous diffusion terms, fluid compressibility, and the particular enforcement of the fluid-solid coupling conditions. This numerical method is applied to study the interaction of waves in a fracture-conduit system. A methodology to constrain fracture geometry for oil and gas (hydraulic fracturing) and volcano seismology applications is proposed.

The third part of the thesis extends the summation-by-parts methodology to staggered grids. This extension reduces numerical dispersion and enables the formulation of stable and high-order accurate multiblock discretizations for wave equations in first order form on staggered grids. Finally, the summation-by-parts methodology on staggered grids is further extended to second derivatives and used for the treatment of coordinate singularities in axisymmetric wave propagation.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. , 27 p.
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1806
National Category
Computational Mathematics Mathematical Analysis
URN: urn:nbn:se:liu:diva-132550DOI: 10.3384/diss.diva-132550ISBN: 9789176856352 (print)OAI: diva2:1046328
Public defence
2016-12-13, Visionen, Hus B, Campus Valla, Linköping, 13:15 (English)
Available from: 2016-11-14 Created: 2016-11-14 Last updated: 2016-11-17Bibliographically approved
List of papers
1. Simulation of Earthquake Rupture Dynamics in Complex Geometries Using Coupled Finite Difference and Finite Volume Methods
Open this publication in new window or tab >>Simulation of Earthquake Rupture Dynamics in Complex Geometries Using Coupled Finite Difference and Finite Volume Methods
2015 (English)In: Communications in Computational Physics, ISSN 1815-2406, E-ISSN 1991-7120, Vol. 17, no 2, 337-370 p.Article in journal (Refereed) Published
Abstract [en]

We couple a node-centered finite volume method to a high order finite difference method to simulate dynamic earthquake ruptures along nonplanar faults in two dimensions. The finite volume method is implemented on an unstructured mesh, providing the ability to handle complex geometries. The geometric complexities are limited to a small portion of the overall domain and elsewhere the high order finite difference method is used, enhancing efficiency. Both the finite volume and finite difference methods are in summation-by-parts form. Interface conditions coupling the numerical solution across physical interfaces like faults, and computational ones between structured and unstructured meshes, are enforced weakly using the simultaneousapproximation-term technique. The fault interface condition, or friction law, provides a nonlinear relation between fields on the two sides of the fault, and allows for the particle velocity field to be discontinuous across it. Stability is proved by deriving energy estimates; stability, accuracy, and efficiency of the hybrid method are confirmed with several computational experiments. The capabilities of the method are demonstrated by simulating an earthquake rupture propagating along the margins of a volcanic plug.

Place, publisher, year, edition, pages
Global Science Press, 2015
elastic waves, earthquake, high order finite difference finite volume, summation-by-parts, simultaneous approximation term, nonlinear boundary conditions
National Category
Computational Mathematics Mathematics
urn:nbn:se:liu:diva-113745 (URN)10.4208/cicp.111013.120914a (DOI)000353693400002 ()
Available from: 2015-01-29 Created: 2015-01-29 Last updated: 2016-11-17
2. Discretizing singular point sources in hyperbolic wave propagation problems
Open this publication in new window or tab >>Discretizing singular point sources in hyperbolic wave propagation problems
2016 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 321, 532-555 p.Article in journal (Refereed) Published
Abstract [en]

We develop high order accurate source discretizations for hyperbolic wave propagation problems in first order formulation that are discretized by finite difference schemes. By studying the Fourier series expansions of the source discretization and the finite difference operator, we derive sufficient conditions for achieving design accuracy in the numerical solution. Only half of the conditions in Fourier space can be satisfied through moment conditions on the source discretization, and we develop smoothness conditions for satisfying the remaining accuracy conditions. The resulting source discretization has compact support in physical space, and is spread over as many grid points as the number of moment and smoothness conditions. In numerical experiments we demonstrate high order of accuracy in the numerical solution of the 1-D advection equation (both in the interior and near a boundary), the 3-D elastic wave equation, and the 3-D linearized Euler equations.

Singular sources, Hyperbolic wave propagation, Moment conditions, Smoothness conditions, Summation by parts
National Category
Computational Mathematics Mathematical Analysis
urn:nbn:se:liu:diva-132630 (URN)10.1016/ (DOI)
Available from: 2016-11-17 Created: 2016-11-17 Last updated: 2016-11-17

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