The fictional forces acting on a drill string in a borehole can be large if the drill string is embedded in a pastelike filter cake formed at the borehole wall by the drilling fluid. In extreme conditions, the drill string can become stuck, leading to expensive delays in the drilling process. The hydraulic pressure within the wellbore is usually higher than the pore pressure within the rock. This excess pressure is responsible both for the growth of the filter cake, and for the normal force that can hold the drill string against the borehole wall. We obtain estimates of the normal and frictional forces acting on the drill string, based on an analysis of the pressure and concentration profile within the filter cake. The forces can be minimized by ensuring that the cake is highly compressible, so that the bulk of the pressure drop between the borehole and porous rock occurs across a thin boundary layer at the base of the cake. The forces acting on a sphere (rather than a cylindrical drill string) are also modeled, and are compared against experiment. The predicted frictional force is smaller than observed experimentally, but the predicted t(3/4) growth of the force with time t agrees with experiment.