In situ observations have shown that underground storage of hydrogen behaves like a natural chemical reactor and generates methane. The mechanism of this generation is the metabolic activity of methanogenic bacteria which consume hydrogen and carbon dioxide and transform them into methane and water. The coupled mathematical model of the reactive transport and population dynamics in a storage is suggested in this paper which also takes into account the fact that the population growth rate depends on the structure of the bacterium colony. The suggested system of equations is reduced to the Turing reaction-diffusion model which proves the appearance of non-attenuating self-oscillations in time which are uniform in space. These solutions are unstable and, once perturbed, generate regular spatial stationary waves which correspond to the alternations of zones which are rich in CH4 or CO2. This result predicts the effect of a natural in situ separation of gases, which was observed in practice. If the diffusivity of bacteria is neglected with respect to the effective diffusivity of the injected gas, then only large-scale spatial waves arise. A low but non-zero bacterium diffusivity causes the appearance of additional small-scale linear oscillations whose period is the intrinsic parameter of the process and is proportional to the bacteria gas diffusivity ratio. The analysis is completed with numerical simulations of 2D problems and analytical solutions of 1D problems obtained using the technique of two-scale asymptotic expansion. The estimations for the parameters of the model were obtained.