We present a numerical method for a one-dimensional parabolic free boundary problem from hydrology about the propagation of the saturated zone in wetted soils. Two difficulties appear in the considered problem. First, the domain on which the corresponding PDE has to be solved is not fixed with respect to time due to the presence of the free boundary. We apply Kirchhoff transformation in order to deal with this issue. On the other hand, the boundary and initial data are incompatible and the solution is singular. We change the space and time variables blowing up the vicinity around the singularity. Then we construct a numerical scheme which appears to be first order convergent.