Proterozoic sedimentary basins very often emit natural hydrogen gas that may be a valuable part of a non-carbon energy infrastructure. Vents in the Sao Francisco Basin in Brazil release hydrogen to the atmosphere mainly during the daylight half of the day. Daily temperature and the regular daily tidal atmospheric pressure variations have been suggested as possible causes of the pulsing of H₂ venting. Here, we analyze a ~550 m-diameter depression that is barren of vegetation and venting hydrogen mainly at its periphery. We show that daily temperature changes propagated only ~1/2 m into the subsurface and are thus too shallow to explain the H₂ variations measured at 1-m depth. Pressure changes could propagate deeply enough, and at the depth at which the cyclic variations are measured hydrogen concentration will have the observed phase relationship to atmospheric pressure changes provided: (1) the pressure wave is terminated by geologic barriers at about 25% of its full potential penetration distance, and (2) the volume of gas in the vents is very small compared to the volume of gas tapped by the venting. These constraints suggest that there is a shallow gas reservoir above the water table under the ~550 m-diameter barren-of-vegetation depression. The 1D-analytical and finite-element calculations presented in this paper help define the hydrogen system and suggest the further steps needed to characterize its volume, hydrogen flux and resource potential.