Development of unconventional tight oil and gas reservoirs such as shale pays presents a huge challenge to the petroleum industry due to the naturally low permeability of shale formations and thus low productivity of oil and gas wells. Shale formations are also vulnerable to the contamination of the water in the drilling and completion fluids, which further reduces reservoir permeability. Although gas-drilling (drilling with gas) has been used to address the issue, several problems such as formation water influx, wellbore collapse, excessive gas volume requirement and hole cleaning in horizontal drilling, still hinder its application. A new technique called gas-lift drilling has recently been proposed to solve these problems, but the optimal design of drilling operation requires a thorough investigation of fluid flow field below the asymmetric drill bits for evaluating the fluid power needed to clean the bottom hole. Such an investigation is conducted in this work based on the Finite Element Method (FEM) implemented in an open source computational framework, FEniCS. Pressure and flow velocity fields were computed for three designs of drill bit face characterized by radial bit blades and one eccentric orifice of discharge. One of the designs is found superior over the other two because it generates relatively uniform flow velocities between blades and provides a balanced fluid power needed to clean all the bit teeth on each bit blade. To quantify the capability of borehole cleanup presented by three drill bit designs, the energy per unit volume is calculated in each region of drill bit and compared with the required value suggested by the literature. In addition, the developed FEM model under FEniCS framework provides engineers an accurate tool for optimizing drill bit design for efficiently gas-lift drilling unconventional tight oil and gas reservoirs.