Analytical model of 2D leakoff in waterflood-induced fractures

Waterflood-induced fractures, also known as self-induced fractures, spontaneously form at injection wells during waterflooding. These fractures propagate long distances through rock, allowing injected fluids to travel far away from a well, both within and outside the flooding layer. Essentially, the mechanics of waterflood-induced fracture propagation is similar to that of hydraulic fractures, which are intentionally created for reservoir stimulation. Fracturing models developed for hydraulic fractures can also be applied to waterflood-induced fractures. However, waterflood-induced fractures are typically pumped with much larger volumes of water or brine and grow much longer in time. As a result, fluid leakoff from waterflood fractures into the formation is more extensive and two-dimensional (2D), a characteristic that is often ignored in a majority of modern fracturing simulators, making their application to waterflood fractures unreliable. In this work, we revisit the problem of leakoff for long-growing waterflood-induced fractures and develop a new analytical model for fluid leakoff that provides improved predictions of fracture geometry and can be easily implemented in fracturing simulators. We incorporate the developed solution into the classical Perkins-Kern-Nordgren (PKN) model of fracture growth, which shows that the choice of the Carter or a 2D leakoff model greatly impacts fracture geometry at large time. The conducted parametric study shows while a toughness-dominated regime affects fracture evolution, most of fracture lifetime occurs in a viscosity-and-leakoff-dominated regime. We also develop an asymptotic solution for a leakoff profile in the limiting case of 2D leakoff domination (M˜˜ and K˜˜). Finally, we study 3D fracture growth and out-of-zone injection with three layers and a complex structure of zones. The study shows that ignoring the 2D leakoff during simulation results in a significant overestimation of fracture geometry predictions. The present work, thus, plays an important role in improving waterflood fracture modelling, as it highlights the significance of 2D leakoff in waterflood-induced fractures and provides a reliable analytical model for fluid leakoff that can be incorporated into modern fracture simulators.