The underground coal field mining is inevitably associated with coal oxidation process accompanied with the thermal energy emission and the formation of coal self-heating spots. The self-heating spots can emerge in both coal dust accumulations deposited in the excavations and in the excavated coal seams. 

It is known that self-heating is caused by both chemical processes accompanied with heat emission and physical processes, primarily the filtration of gas-air mixtures through porous and fractured layers of rocks and coal seams. Although the filtration processes accompanying oil and gas field exploration have been thoroughly studied, similar processes accompanying coal mining have been studied less.

The article focuses on the mathematic problem concerning the filtration process in the coal seam when excavated by long pillars. In this system, both fresh airflows and worked out flows move in opposite directions and under different pressures along the mine roadway and the ventilation opening. This causes so-called air suctions due to air infiltration through the coal seam. In theory, this problem has not been solved yet and thus is considered urgent. Based on the classic filtration theories and assumptions regarding the coal seam isotropy and homogeneity, the one-dimensional initial value problem describing the non-stationary air filtration process through the coal seam was formulated. The solution provided explicit formulas to determine the pressure and rate of air-gas mixture filtration and identified several regularities of the filtration process in the coal seam. Particularly, it was established that the filtration rate increases significantly in the direction towards the ventilation opening, especially at the initial stages of filtration. In process of time, however, the filtration rate aligns and after 5 days, the rate difference is not so significant.