This study contributes to: 1) a multi-dimensional model framework of lithium-air (Li-air) battery, 2) incorporation of mechanisms of insoluble precipitates' impacts, and 3) analysis and discussion on oxygen supply channel for Li-air battery. The model consists of a set of partial differential equations of species and charge conservation, in conjunction with the electrochemical reaction kinetics, and takes into account the two major mechanisms of voltage loss caused by insoluble discharge products: namely, electrode passivation and increased oxygen transport resistance. Two-dimensional (2-D) simulation indicates that the pore space in the cathode electrode is not fully utilized for Li compounds storage, particularly under high discharging current. For selected battery designs, considerable variation of quantities is observed only in the thickness direction. Through analysis, we evaluate the oxygen concentration drop along an oxygen supply channel and relate it to the Damköhler (Da) number, and further explore potential cases that yield oxygen starvation.