A mathematical model is developed for the flow of water through a channel impregnated with a polymer gel that is treated as an elastic and deformable porous medium. The model uses a Brinkman equation along with an experimentally observed velocity-dependent permeability. Numerical and approximate analytical solutions are given. These results show that the gel intrinsic properties, i.e., gel reference permeability and elastic index, control the water flow. First, the permeability of water flow through the gel increases with an increase of gel reference permeability. Second, the velocity of water decreases when the gel velocity exponent increases. Our theoretical results show that the velocity-dependent permeability of water flow through polymer gels is in fact an intrinsic property of the gel rather than a property of the channel or some interaction between the gel and the pore walls.