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Marine seismic exploration is a method employed by the hydrocarbon industry to find geological structures in the sub-surface with the potential to contain trapped hydrocarbons. A source of seismic energy is towed behind a ship. The energy produced by the source propagates as a sound wave through the sea into the sub-surface. Within the sub-surface the energy is reflected, refracted and diffracted. The ship also tows an array of hydrophones behind the seismic source, and these are used to measure the wavefield. If the source signal is known, then the received signal at each hydrophone can be deconvolved for the source signal to obtain the impulse response of the earth between the source and the hydrophone. These impulse responses can highlight some of the structures in the subsurface. Maps of the subsurface built up from these impulse responses are then interpreted to estimate the locations of trapped hydrocarbons. The most commonly used seismic source is the seismic air gun, which is a canister containing highly compressed air. The air is released into the sea, forming an oscillating bubble. There are two methods used by industry to determine the signal produced by an air gun or air gun array: (1) modelling, and (2) extrapolation from near-field measurements. Traditionally, industry uses the first method. With broader bandwidth data that are being recovered in data processing by removing the sea-surface reflection at the source and receiver (source and receiver ghosts), it has been found that modelling is inferior to extrapolation from near field measurements, although industry has been slow to adopt the second method. Despite this change, modelling remains a valuable tool in the design of air gun arrays, where designs can be optimised by adjusting parameters of the array and using modelling to determine the wavefield of each variation of the array. The aim of this work is to develop methods which can improve on current air gun bubble modelling. In this thesis I develop a novel artificial boundary condition for use in finite volume simulations of oscillating bubbles. The purpose of the work is an improvement to the modelling of seismic air gun bubbles. However, the techniques presented in this thesis are not limited to air gun bubbles, but are applicable to any oscillating bubbles, or indeed any fluid dynamics problem which is spherical in nature, close to spherically symmetric, and produces flow speeds of low (< 0:1) Mach number some distance from the region of interest. The boundary condition is based on an existing approximation to the motion around a spherical bubble, which is derived from the asymptotic solution to the motion in the far field. It is applied as follows: (1) use the solution on the domain boundary to calculate the approximate solution external to the domain; (2) use the approximate external solution to calculate spatial derivatives of properties on the domain boundary, due to the external solution, and (3) use the spatial derivatives to describe characteristic waves incoming to the domain. I develop a finite volume scheme in which I apply this boundary condition. I present the results of one- and two-dimensional of simulations using this scheme, and demonstrate the efficacy of this boundary condition. The boundary condition performs well, allowing finite volume simulations of bubbles to be carried out for long run-times (5 105 time steps with a CFL number of 0:8) on highly truncated domains, in which the boundary condition may be applied within 0:1% of the maximum bubble radius. Conservation errors due to the boundary condition are found to be of the order of 0:1% after 105 time steps. One- and two-dimensional results show a third-order convergence rate of errors due to the boundary condition as the domain is enlarged. The one- and two-dimensional simulations of air gun bubbles I present are, to my knowledge, the first finite volume simulations of air gun bubbles carried out, and the first air gun bubble simulations in which the contents of the bubble are not considered to be homogeneous. Two-dimensional results show non-spherical aspects of air gun bubbles, which may be incorporated into models used by industry. The model captures surface instabilities, bubble translation and deformation due to gravity, and the formation of jets due to asymmetries on collapse. The results indicate that bubble surfaces are unstable throughout collapse. These phenomena are shown to increase the damping of bubble oscillations. The results of the two-dimensional air gun modelling highlight the potential value of my artificial boundary condition, and also the aspects of my computational scheme which require improvement. I extend the numerical scheme to include viscous effects, which I show to have limited impact on the signals emitted by air gun bubbles, although the influence of a boundary layer around the bubble is significant, causing an 18% reduction in rise rates. I extend the scheme to include the effects of the sea surface, and present results which show the impact of the reflection from the sea surface (the ghost wave) on the bubble. This extension shows the reflection of the ghost wave off the bubble, which provides a novel explanation of some of the higher frequencies present in measurements. This extension further increases the practical value of my contribution, and further demonstrates the ability of the boundary condition to handle asymmetrical flow features. |
