Abstract.  In this paper, we model water injection through a growing vertical hydrofracture penetrating a low-permeability reservoir. The results are useful in oilfield waterflood applications and in toxic waste disposal through reinjection. Using Duhamel's principle, we extend the Gordeyev and Entov1 self-similar 2D solution of pressure diffusion from a growing fracture to the case of variable injection pressure. The flow of water injected into a low-permeability rock is almost perpendicular to the fracture for a time sufficiently long to be of practical interest. We revisit Carter's model of 1D fluid injection2 and extend it to the case of variable injection pressure. We express the cumulative injection through the injection pressure and effective fracture area. Maintaining fluid injection above a reasonable minimal value leads inevitably to fracture growth regardless of the design of injectors and the injection policy. Therefore, a smart injection controller that can prevent rapid fracture growth is needed.