| Abstract: | Similarity solutions are derived for wedge-shaped hydraulic fractures driven by a constant inlet pressure P0 into a permeable medium under a uniform confining stress σ. These results describe the seepage-dominated regime in which most of the injected fluid is lost into the permeable walls of the fracture; they complement previous results for the capacitance-dominated regime in which seepage is negligible. Fracture propagation velocity is obtained as an analytical function of fracture length, driving pressure, confining stress, material properties and a single separation constant or eigenvalue which is determined numerically. Self-similar profiles of pressure, opening displacement and fluid velocity along the fracture are presented, together with the self-similar isobars of the two-dimensional pressure field within the permeable medium. Comprehensive results are reported for laminar or turbulent flow of a constant-compressibility liquid or an ideal gas driven by overpressures (P0?σ)/σ ranging from 10?2 to 102. |
