Hybrid Analytical/Numerical Computation of Heat Transfer in a Gas-Driven Fracture

Abstract

In gas-driven hydraulic fractures, as occur in rock blasting and underground nuclear testing, the high-temperature gases (1000 to 30,000 K) are radically cooled by heat transfer to the host material. This significantly reduces both the maximum extent and rate of fracture growth. The coupled processes of fluid flow, heat transfer, and rock deformation governing fracture growth are calculated here by a hybrid analytical/numerical procedure. The gas motion along a fracture of increasing length and aperture is described by a finite-difference form of the one-dimensional transport equations; fluid friction, advective heat transfer, and heat loss to the walls of the fracture are considered. Lateral heat losses are evaluated in a quasi-analytical fashion, based on an integral method that accounts for the convective film resistance between the fluid and fracture wall, as well as the conductive resistance within the surrounding medium. The calculations are performed on a difference grid that expands to maintain a fixed number of points uniformly distributed along the fracture. The present numerical results agree, within appropriate limits, with known similarity solutions. Beyond this, new nonsimilar solutions for early-time fracture growth are presented.