A solution to the problem of a hydraulic fracture driven by an incompressible Newtonian fluid at a constant injection rate in a permeable rock is presented in this paper. A set of governing equations are formed to obtain the fracture half-length, crack opening, and net fluid pressure. The solution is derived under the assumptions of plane strain, zero lag between fluid front and crack tip, followed by negligible fluid viscosity. The last assumption is related to a toughness-dominated fracture propagation regime therefore leading to a uniform fluid pressure along the crack surface. Early-time and late-time asymptotic solutions are obtained, which correspond to both regimes when the fluid contains within the crack and most of the injected fluid infiltrates into the rock, respectively. It is shown that these asymptotic solutions are in a simple form when the fracture propagation is dominated by the material toughness. The transient solution for the evolution from the early time to the late time is also obtained by a numerical method.

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