Natural H2 emissions from the ground have now been measured in many places worldwide. These emissions can be localized on faults or be more diffuse in some sedimentary basins, usually of Proterozoic age. In such a case, emanation zones are often visible from aerial images or on high-resolution topographic maps since they correspond to slight depressions of circular to elliptic shape. Furthermore, the rounded depressions are covered with a scrubby vegetation which often contrasts with the surrounding vegetation. Although the emission structure displays a very regular shape, the distribution of H2 concentration in the first meter of soil in such a structure does show a clear pattern. For example, the maximum concentration is almost never measured in the center of the structure and the few time-resolved data show that the soil H2 concentration is variable with time. Here, the time and space evolution of H2 concentration is simulated using a 2-D advective-diffusive model of H2 transport in porous media. Several parameters have been tested as the depth and periodicity of the H2 point source (pulsed), bacterial H2 consumption and permeability heterogeneities of the soil. The radius of the structure is linked to the time spent by the H2 in the soil that depends on the soil permeability, the depth of the gas leakage point and the pressure of the bubble. To account for field observations, the case of a shaly, less permeable, heterogeneity in the center of the structures has been modeled. It resulted in an increase of the concentration toward the rim of the structure and a close to zero signal in its center. If the deep signal is periodic with a frequency smaller than a few hours, H2 concentration within the soil is almost constant; in other cases, the near surface concentration wave reflects the concentration periodicity of the source with a delay (in the range of 12 h for 30 m of soil) and so the near surface H2 concentration values will be highly dependent on the time at which the measurement is performed. H2 monitoring through a sensor network is thus mandatory to characterize the H2 dynamics in the soil of fairy circles.