Comparison of methods for modelling the behaviour of bubbles produced by seismic airguns

Three models for the dynamics of seismic airgun-generated bubbles and their associated far-field signals are developed and compared with geophysical data. The first model of an airgun-generated bubble uses a spherical approximation, the second is an approximate Lagrangian model which allows for small deformations from a spherical shape, whilst the final model is an axisymmetric boundary-integral method which permits the bubble to evolve into highly non-spherical geometries. The boundary-integral method also allows both geometric interference and strong dynamic interactions in multi-bubble studies. When comparing the spherical model to experimental data there are three apparent, significant differences: the magnitude of the primary pressure peak, which is greater in the model; the subsequent decay of the pressure peaks and motion - the experimental data demonstrating greater decay and a slower rise rate; and the frequency of oscillation, which is slower in the experimental data. It is believed that the first discrepancy is due to the initial stages of expansion where the compressed air is forced to sparge through the airgun ports. The other differences indicate that there is some other energy-loss mechanism which is not accounted for in the spherical bubble model. Non-spherical bubble behaviour is investigated through the use of two different deformable many-bubble codes and their predictions are compared with the spherical model and experimental data. The Lagrangian model predicts the formation of a buoyancy-driven liquid jet on the first collapse of a typical airgun bubble; however, the model breaks down when the bubble becomes significantly deformed, due to a low-order spherical-harmonic approximation for the potential. The axisymmetric boundary-integral code models the jet shape accurately and it is found that these bubbles evolve to toroidal geometries when the jet impacts on the opposite surface of the bubble. This highly non-spherical behaviour is readily observed on high-speed films of airgun bubbles, and is one key source of energy loss; it damps the pulsations of the bubble and slows its rise speed. Inter-bubble interactions are investigated using the two deformable bubble models, and the predictions are compared to field data. It was found that as the bubbles approach each other, their periods of oscillation increase in accordance with observations, and jets are formed in the direction of motion upon collapse.