We introduce a stochastic microscopic model to investigate the jamming and reorganization of grains induced by an object moving through a granular medium. The model reproduces the experimentally observed periodic sawtooth fluctuations in the jamming force and predicts the period and the power spectrum in terms of the controllable physical parameters. It also predicts that the avalanche sizes, defined as the number of displaced grains during a single advance of the object, follow a power law P(s);s2t , where the exponent is independent of the physical parameters.