Abstract
The objective of this work is the derivation of the wave equations for describing acoustics in a deformable porous medium saturated by a bubbly fluid, when capillary, thermal and phase change effects are accounted for. This is performed using an homogenisation technique: the macroscopic model is obtained by upscaling the bubble-scale and the pore-scale descriptions. For convenience a bubbly fluid near the bubble point, in the bulk of which a small perturbation can generate small bubbles is considered. Although the derived macroscopic wave equations are similar in their structure to Biot's equations that describe wave propagation in saturated porous media, important differences appear as a result of the presence of bubbles. In effect, gas-liquid phase change considerably decreases the apparent rigidity of the bubbly fluid, and consequently decreases the wave velocity in the porous medium. Moreover, this phenomenon is amplified for very small bubbles, for which the apparent rigidity of the bubbly fluid can be negative. The influence of the bubbly liquid apparent rigidity on the wave velocity and attenuation is highlighted on an illustrative example: it is shown that they strongly differ from wave velocities and attenuations in porous media saturated by a liquid or by a gas.
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