Application of Analytical Solutions of the Reactive Transport Equation for Underground Methanation Reactors

Fluctuations in the production of renewable-based electricity have to be compensated by converting and storing the energy for later use. Underground methanation reactors (UMR) are a promising technology to address this issue. The idea is to create a controlled bio-reactor system in a porous underground formation, where hydrogen obtained from renewable energy sources by electrolysis and carbon dioxide from industrial sources are fed into the reactor and converted into methane. Microorganisms, known as methanogenic archaea, catalyze the chemical reaction by using the two non-organic substrates as nutrients for their growth and for their respiratory metabolism. The generated synthetic methane is renewable and capable to compete with the fossil methane. Mathematical models play an important role in the design and planning of such systems. Usually, a numerical solution of the model is required since complex initial-boundary problems cannot be solved analytically. In this paper, an existing bio-reactive transport model for UMR is simplified to such an extent that an analytical solution of the advection-dispersion-reaction equation can be applied. A second analytical solution is used for the case without dispersion. The analytical solutions are shown for both the educt (hydrogen) and the reaction product (methane). In order to examine the applicability of the analytical models, they are compared with the significantly more complex numerical model for a 1D case and a 3D case. It was shown that there is an acceptable agreement between the two analytical solutions and the numerical solution in different spatial plots of hydrogen and methane concentration and in the methane concentration in the withdrawn gas. The mean absolute error in the mole fraction is well below 0.015 in most cases. The spatial distribution of the hydrogen concentration in the comparison to the 3D case shows a higher deviation with a mean absolute error of approx. 0.023. As expected, the model with dispersion shows a slightly lower error in all cases, as only here the gas mixing resulting in smeared displacement fronts can be represented. It is shown that analytical modeling is a good tool to get a first estimation of the behavior of an UMR. It allows to help in the design of well spacing in combination with the injection rate and injected gas composition. Nevertheless, it is recommended to use more complex models for the later detailed analysis, which require a numerical solution.