Fracture tip advancement and pressure distribution of pressure induced fracture in 2D and 3D fully saturated porous media is investigated in detail because of their peculiar features. A cohesive fracture model is adopted for this purpose together with a discrete crack and without predetermined fracture path. The Rankine criterion is used for fracture nucleation and advancement. One solution method presented is based on standard Finite Elements where in a 2D setting the fracture follows directly the direction normal to the maximum principal stress while in the 3D case the fracture follows the face of the element around the fracture tip closest to the normal direction of the maximum principal stress at the tip. This procedure requires continuous updating of the mesh around the crack tip to take into account the evolving geometry. The updated mesh is obtained by means of an efficient mesh generator based on Delaunay tessellation. The second solution employs the XFEM method. The governing equations for both approaches are written in the framework of porous media mechanics and are solved numerically in a fully coupled manner. Numerical examples dealing with well injection (constant inflow) in a geological setting, hydraulic fracture in 2D and 3D concrete dams (increasing pressure) and a peeling test for a fully saturated porous medium are shown and stepwise tip advancement and pressure oscillations are evidenced.