Gravity-Driven Hydraulic Fractures
Abstract:This study is motived by a new method for disposing of nuclear waste by injecting it as a dense slurry into a hydraulic fracture that grows downward to great enough depth to permanently isolate the waste. Disposing of nuclear waste using gravity-driven hydraulic fractures is mechanically similar to the upward growth of dikes filled with low density magma. A fundamental question in both applications is how the injected fluid controls the propagation dynamics and fracture geometry (depth and breadth) in three dimensions.
Analog experiments in gelatin [e.g., Heimpel and Olson, 1994; Taisne and Tait, 2009] show that fracture breadth (the short horizontal dimension) remains nearly stationary when the process in the fracture “head” (where breadth is controlled) is dominated by solid toughness, whereas viscous fluid dissipation is dominant in the fracture tail. We model propagation of the resulting gravity-driven (buoyant or sinking), finger-like fracture of stationary breadth with slowly varying opening along the crack length. The elastic response to fluid loading in a horizontal cross-section is local and can be treated similar to the classical Perkins-Kern-Nordgren (PKN) model of hydraulic fracturing. The propagation condition for a finger-like crack is based on balancing the global energy release rate due to a unit crack extension with the rock fracture toughness. It allows us to relate the net fluid pressure at the tip to the fracture breadth and rock toughness.
Unlike the PKN fracture, where breadth is known a priori, the final breadth of a finger-like fracture is a result of processes in the fracture head. Because the head is much more open than the tail, viscous pressure drop in the head can be neglected leading to a 3D analog of Weertman’s hydrostatic pulse. This requires relaxing the local elasticity assumption of the PKN model in the fracture head. As a result, we resolve the breadth, and then match the viscosity-dominated tail with the 3-D, toughness-dominated head to obtain a complete closed-form solution. We then analyze the gravity fracture propagation in conditions of either continuous injection or finite volume release for sets of parameters representative of dense waste injection technique and low viscosity magma diking.