Three-dimensional buoyant hydraulic fractures: finite-volume release

Abstract

In impermeable media, a hydraulic fracture can continue expanding even without additional fluid injection if its volume exceeds the limiting volume of a hydrostatically loaded radial fracture. This limit depends on the mechanical properties of the surrounding solid and the density contrast between the fluid and the solid. We show that two dimensionless numbers characterize self-sustained fracture growth. The first is a buoyancy factor that compares the total released volume to the volume of a hydrostatically loaded radial fracture to determine whether buoyant growth occurs. The second number is the dimensionless viscosity of a radial fracture when buoyant effects become of order one. Notably, this dimensionless viscosity depends on the rate at which the fluid volume is released, indicating that both the total volume and release history impact self-sustained buoyant growth. We identify six well-defined propagation histories based on these two dimensionless numbers. Their growth evolves between distinct limiting regimes of radial and buoyant propagation, resulting in different fracture shapes. Notably, our findings reveal two growth rates depending on the dominant energy dissipation mechanism (viscous flow versus fracture creation) in the fracture head. For finite values of material toughness, the toughness-dominated limit represents a late-time solution for all fractures in growth rate and head shape (possibly reached only at a very late time). The viscosity-dominated limit can appear at intermediate times. Our three-dimensional simulations confirm the predicted scalings. This contribution highlights the importance of the entire propagation and release history for accurate analysis of buoyant hydraulic fractures.