Dynamics of an expanding fluid‐filled crack

The dike intrusion mechanism proposed recently by B. R. Julian and his colleagues for several large earthquakes at Long Valley caldera has stimulated new interest in the mechanics of fluid injection. This study is an attempt to resolve some of the questions raised by this interpretation by numerically simulating the dynamics of a propagating fluid‐filled crack. The computations are based on the two‐dimensional finite difference method applied by Aki and coworkers to a similar problem of a fluid‐driven crack. We extend this earlier study by improving the boundary conditions applied in the plane of the crack and by analyzing a crack containing a viscous fluid that supports acoustic wave propagation. The problem has two time scales: the duration of the rupture, which is proportional to the distance the crack propagates, and the period of the acoustic resonance of the fluid, which is a function of the length of the crack and the acoustic velocity in the fluid. For a small extension of a long crack containing a fluid of low bulk modulus, these time scales are markedly different. The initial motion of the walls near the propagating crack tip is directed outward, so the radiated first motion is compressive everywhere. The increase of the crack tip volume, however, induces a pressure drop in the fluid which propagates over the length of the crack with the velocity of the acoustic wave, causing a partial collapse of the wall radiated as a long‐period dilatation. The dilatation following the short compressional first arrival is well marked in the vicinity of the crack plane, and for a buried vertical crack it is a conspicuous feature of the near‐field vertical ground motion close to the crack trace. These properties of the signal suggest that first motion studies of this frequency‐dependent source may be delicate without broadband instruments. Inhomogeneous waves propagating at the liquid‐solid interface produce high‐frequency vibrations which are observed only in the immediate vicinity of the crack. The source duration depends strongly on the fluid viscosity and associated viscous damping at the crack wall; damping of the motion by the radiation of elastic waves is a comparatively small effect.