Models of hydraulic fractures in conventional reservoirs assume that Carter’s leak-off law —the leak-off rate is proportional to the inverse of the square root of the time elapsed since exposure to fracturing fluid — is applicable. The validity of Carter’s leak-off law stems from the cake-building properties of the fracturing fluid. In some situations where water is essentially the fracturing fluid, Carter’s leak-off law can also be justified (through an reinterpretation of the leak-off coefficient) as an early-time solution of the diffusion equation. However, in water flooding operations of very permeable reservoirs, the fracture propagates in a region where the pore pressure perturbations caused by injection of water is quasi-stationary. The talk will present the construction of a new class of solutions for hydraulic fractures propagating under these asymptotic conditions. We will first present a KGD-type model of an hydraulic fracture created by injecting fluid in weak, poorly consolidated rocks. By further assuming a “small” or negligible toughness (with the consequence that the crack aperture is “small”), we prove that the system is characterized by two asymptotic fracture propagation regimes: rock-flow dominated at small time and fracture-flow dominated at large time. The timescale that legislates the transition between the small and large time asymptotic regimes is shown to be a strongly nonlinear function of a dimensionless injection rate. The rock-flow dominated regime is characterized by an increasing injection pressure while the fracture-flow dominated regime is associated with an injection pressure decreasing with time. The peak injection pressure takes place during the transition between the two regimes. The KGD model collapses, however, when the total crack length (two wings) becomes larger than the thickness of the reservoir layer, assumed to be bounded by impermeable strata. The changing geometric ratio of the constant crack height over its length affects the fracture compliance, i.e. the relationship between fracture aperture and net pressure. As this ratio decreases, the non-local elastic interaction characteristic of the KGD model progressively vanishes and for ratio approximately larger than 5 the compliance becomes essentially local as in the PKN model. It will be shown that evolution of the fracture from a KGD to a PKN mode causes an unexpected reversal of the regime of propagation, with the rock-flow dominated regime in the PKN geometry becoming the long-term solution.