This paper investigates the propagation of a plastic zone resulting from fluid injection in a weakly consolidated porous medium with dilatant behavior. It is assumed that the rock behaves as a Mohr-Coulomb elasto-plastic material with different mobility and diffusivity depending on whether it is elastic or yielding. More specifically, the plane strain problem of a point source injecting fluid, at constant rate, in an infinite domain is considered. Under these assumptions, the absence of any characteristic length implies the existence of a self-similar solution. It is shown that this solution consists of a plastic zone growing as the square root of time and is characterised by a constant pore pressure at the elasto-plastic boundary. Depending on the magnitude of the injection rate, three yielding regimes exists. Finally, the paper concludes by investigating the conditions under which a tensile fracture could initate in the vicinity of the injection point.