The results are presented of an investigation of the multiplying properties of lava-formed fuel-containing masses (LFCM); also, the possibility of developing ignition and dynamics of a self-sustaining chain reaction (SCR) in the LFCM of the destroyed Unit 4 of the Chernobyl nuclear power plant (the so-called Shelter) is discussed. The SCALE 4.3 computer code was used to calculate the multiplication factor, the neutron energy spectrum, the spatial distribution of the neutron flux density, etc., as functions of the water content in the LFCM for different system models. These results can help to determine the optimum placement of detectors in the rooms under the reactor. In addition, the dynamic of an SCR under the hypothetical condition that the filling of the LFCM by water leads to an excess multiplication factor over unity was considered. Such a treatment was performed for a simple model that takes into account the evaporation of water and an increase in temperature due to an energy release in the LFCM. The different modes of the LFCM behavior depending on the velocity of water filling are discussed.