Fluxes of gas, energy, and fluid (water with dissolved gas) are known to control the distribution of gas hydrate and free gas in gas hydrate reservoirs, but theoretical studies have so far not focused on the effect of sediment properties (e.g., permeability) on these fluxes during the formation of gas hydrate deposits. Using a transient, one-dimensional, two-phase (gas hydrate and fluid) numerical model based on a coupled conservation approach, we examine permeability clogging during gas hydrate formation in end-member porous media and fractures initially lacking hydrate and determine the resulting changes in fluid and advective energy flux. Gas hydrate accumulates most rapidly at depth in both systems, a result consistent with borehole logging and seismic studies on the Blake Ridge. Consideration of the ratio of surface area to fluid volume explains model results that imply more rapid permeability reduction in the porous media system and more uniform hydrate formation as a function of depth in the fracture system. Depending on fluid flux, fracture aperture, or pore sizes, and other factors, the timescale for permeability clogging due to the accumulation of gas hydrate ranges from thousands to millions of years. Regenerating gas hydrate to a concentration of 10--15% of pore space in marine settings with fluid fluxes characteristic of Hydrate Ridge or Blake Ridge requires 103--105 years, meaning that hydrate deposits in these settings cannot be viewed as a naturally renewable resource. These results also imply that significant time is required to reestablish gas hydrate deposits after large-scale dissociation events associated with global-scale climate change.