Scaling vertical drag forces in granular media

The average drag forces on intruders slowly plunging into and withdrawing from shallow beds of monodisperse smooth glass beads and of sifted rough sand scale with the immersion depth and lateral dimensions of the intruder. The withdrawal forces are comparable for both types of media; however, the plunging forces for sand are substantially greater than for smooth glass beads. Furthermore, for glass beads, the rescaled plunging and withdrawal forces have two different power law dependences on the immersion depth, with exponents greater than unity.