The physical motivation behind this thesis is the phenomenon of fracturing of rocks and other porous media due to ice growth inside pre-existing faults and large pores. My aim is to explain the basic physical processes taking place inside a freezing elastic porous medium and develop a mathematical model to describe the growth of ice and fracturing of ice-filled cavities. There are two physical processes that can potentially cause high pressures inside a cavity of a porous medium. The expansion of the water by 9% as it freezes causes flow away from the freezing front and through the porous medium, resulting in a water pressure rise inside the cavity. Flow of water towards freezing cavities can occur during the later stages of freezing, when cavities are almost ice-filled, with a thin premelted film separating the ice from the medium. The pressure rise in this case is due to the flux of water into the cavities, which then freezes and increases the overall ice mass. The special geometry of a spherical cavity is initially considered, as a means of comparing how the different processes can contribute to pressure rise inside a cavity. Having established that the expansion of water only contributes to the overall pressure rise in limited situations, I focus attention on the premelting regime and develop a model for the fracturing of a 3D penny-shaped cavity in a porous medium. Integral equations for the pressure and temperature fields are found using Green’s functions, and a boundary element method is used to solve the problem numerically. A similarity solution for a warming environment is discussed, as well as a fully time-dependent problem. I find that the fracture toughness of the medium, the size of pre-existing faults and the undercooling of the environment are the parameters determining the susceptibility of a medium to fracturing. I also explore the dependence of the growth rates on the permeability and elasticity of the medium. Thin and fast-fracturing cracks are found for many types of rocks. I consider how the growth rate can be limited by the existence of pore ice, which decreases the permeability of a medium, and propose an expression for the effective “frozen” permeability. An important further application of the theory developed here is the growth of ice lenses in saturated cohesive soils. I present results for typical soil parameters and find good agreement between our theory and experimental observations of growth rates and minimum undercoolings required for fracturing.