Similarity Analysis of Condensing Flow in a Fluid-Driven Fracture

Abstract

Similarity solutions are derived for some fundamental problems of condensing flow in a hydraulically driven fracture. The governing equations describe one-dimensional homogeneous turbulent flow along a wedge-shaped hydraulic fracture in an elastic medium. The instantaneous fracture speed is determined as an analytical function of fracture length, material properties, process parameters, and a single eigenvalue, which is calculated by solving a system of ordinary differential equations for the variation of pressure, energy, velocity, and opening displacement along the fracture. Results are presented for abrupt condensation of a pure substance and for gradual condensation of air/water mixtures. The rate of condensation is controlled by the rate of heat transfer to the fracture wall, which depends upon a single dimensionless parameter. For small and large values of this parameter the present multiphase solutions are in agreement with previous solutions for single-phase flows of vapors and liquids. Although most of the results are presented in dimensionless form, some numerical examples are given for steam-driven fractures emanating from the cavity resulting from an underground nuclear explosion.