Rheology and dynamics of elastic waves in fluid-saturated porous media with gas bubbles

Rheology and dynamics of elastic waves in fluid-saturated porous media with gas bubbles

Elastic wave propagation in fluid-saturated porous media presents significant practical and theoretical interest for science and engineering. In such media at least two
types of the waves can propagate: the Frenkel-Biot compressional waves of P1 and P2 type. Usually the waves are modelled by partial differential equations (PDEs)
ranging from the classical wave equation to its nonlinear high-order extensions. A popular extension of this kind is the Nikolaevsky equation (1989). In this thesis we
extend it further to included bubbles.

The thesis consists of seven chapters. Chapter 1 presents the literature review and motivation. Chapter 2 gives general concepts from dynamics. In Chapter 3 for the cases when there are no neutral modes, the wave decays exponentially quickly under the linear dispersion relation (which dynamically expresses dissipation). The wave dynamics depend on the grain rheology, which should take into account bubbles in the fluid-saturated pores. We consider the standard linear solid rheological model to include a special element representing a bubble. We derive the 2nd-order evolution PDEs for the P1-wave governing the velocity of the solid matrix in the moving reference frame. Then we derive the corresponding dispersion relation, and compare it to the case without the bubbles. We observe that the increase of the radius and number of the bubbles leads to the increase in the decay rate.

In Chapter 4 of the thesis we use the full rheological model, based on the model by Nikolaevsky. The full model consists of three segments representing the solid continuum, fluid continuum and a bubble surrounded by the fluid. We derive the 4th-order PDE for the wave without the bubbles and 6th-order PDE for the wave with the bubbles, and obtain the corresponding dispersion relations. We discover that the increase of the radius of the bubbles leads to faster decay, while the increase of the number of the bubbles leads to slower decay of the wave. This latter result manifests the opposite trend to the one observed in the second part. We attribute this result to the more complete and, therefore, more realistic structure of the rheological model used.

In Chapter 5 of the thesis we evaluate the influence on the decay rate by the rheological parameters and other parameters of the medium such as pressure and porosity. Each of them has an appreciable effect on the decay rate, as detailed in the thesis. We discover that the model gives complex wave velocity and/or wave growth, which indicates limitations of the model applicability at extremely large amounts of bubbles. However, we calculate some of acceptable values of the parameters. They belong to finite-size cloud(s) of acceptable values in the multi-dimensional parametric space. Full study of this space is a possible direction of future research.

In Chapter 6 we use the centre manifold theory to describe the dynamics of the elastic wave for the special case when there is one neutral mode. After quickly falling onto the centre manifold, the system then exhibits slow algebraic decay or tends to a steady state depending on the initial conditions and the neutral wave number.

Finally, Chapter 7 gives the conclusions and suggestions for future work.