Arrest of a radial hydraulic fracture upon shut-in of the injection

We investigate the propagation and arrest of a radial hydraulic fracture upon the end of the injection. Depending on the regime of propagation at the time of shut-in of the injection, excess elastic energy may be stored in the surrounding medium. Once the injection has stopped, the hydraulic fracture will arrest when the energy release rate falls under the material fracture energy. Fluid leaking-off to the surrounding medium acts as an energy sink such that the available excess energy for fracture growth decreases faster and as a result impacts the arrest (actually controls it in the zero toughness limit). Under the assumption of a homogeneous elastic medium and the Carter’s leak-off model, we show that the post shut-in propagation of the hydraulic fracture depends on the dimensionless toughness and leak-off coefficient at the time of shut-in. Our investigation highlights that for an impermeable rock, the arrest radius is independent of the dimensionless toughness at shut-in. In the limit of a permeable rock with zero fracture toughness, the arrest radius is independent of the dimensionless leak-off coefficient only for . For larger values of , the radius of arrest reduces with increasing . We delineate the limit above which the arrest is immediate upon shut-in. This limit is given by a critical leak-off coefficient at shut-in for the large leak-off/small toughness cases and by the relation for small leak-off/large toughness (where is equivalently the critical dimensionless toughness at shut-in). Immediate arrest in the impermeable limit is observed for . If both ( and ) are smaller than their critical value for immediate arrest, post shut-in propagation occurs and a self-similar pulse viscosity storage solution emerges. Scaling arguments combined with numerical simulations, show that the propagation post shut-in scales as in the impermeable and small leak-off cases, and as in the zero toughness limit. The growth post shut-in can be significant in impermeable rocks - with a final radius up to twice larger than the radius at shut-in for realistic material and injection parameters.