On the shape and velocity of £uid-¢lled fractures in the Earth

In this paper I present a model to estimate the shape and velocity of slowly ascending buoyancy-driven £uid-¢lled fractures. The model considers elastic deformation, linear fracture mechanics and £uid £ow. An attempt is made to incorporate the e¡ect of the 2-D £uid pattern on the viscous pressure drop. Most other models assume that the viscous pressure drop can be approximated by £ow through a channel with constant width, although the form of the fracture is known to deviate from such a simple shape. The 2-D £ow in my model has an important consequence for the mechanism of buoyancy-driven fracture propagationöit predicts a large pressure gradient at the tail of the propagating fracture, indicating that the tail of the fracture is most important in hindering the fracture propagation. A singularity at the tail of the fracture can be avoided when a small amount of £uid trails in the channel left behind the propagating fracture. The trailing and decoupling of £uids at the tail seems to be accompanied by small £ow and shape instabilities, which is indicated by the jerky movement of the tail observed in propagating air-¢lled fractures in solidi¢ed gelatine, and by numerical boundary element solutions of the coupled £ow-deformation^fracturing problem. By comparing the predictions for propagation velocities with laboratory observations of buoyancy-driven fracture propagation in gelatine, I derive a non-dimensional e¡ective thickness at the tail of the fracture for which trailing of £uids may occur. The model is applied to Earth-relevant problems such as oil- and water-¢lled fractures in pressurized sediments or magma-¢lled dykes in the lithosphere, which are discussed in the paper. Key words: cracks, £uids, fracture propagation, rock fracture.