Predicting the dynamics of rising bubbles is a benchmark problem for several multiphase flow industrial applications. It is crucial in these situations to consider each fluid phase well and to treat precisely the interface dynamics illustrated by a liquid-gas mixture (Ref Mehdi). On the one hand, with the combination of the Reynolds number, the Eotvos number, and the Morton number, different regimes for the shape of bubbles were predicted by Grace et al. (1976) in the so-called Grace diagram (Ref Sung Hoon Park). Meanwhile, on the other hand, it has been shown experimentally by R.M. Davies and Sir G.I. Taylor (ref Taylor) that the rising velocity of bubbles can also be predicted. It is related to the radius of curvature in the vertex region, based on the assumption that the pressure over the front of the bubble is the same as that in ideal hydrodynamic flow around a sphere. However, these experiments involve low-surface tension liquids (water or nitrobenzene) in large, impermeable tanks. In this context, there is heightened interest in modeling correctly and in a coupled manner the properties of each phase (liquid and gas) with the ability to deal with (i) large density and viscosity ratios, (ii) significant surface tensions, (iii) narrow and porous domains. Different immersed numerical methods can be found to simplify such problems involving large structural motion and deformation. In this work, we use the level set method combined with an anisotropic mesh adaptation technique to obtain a high-fidelity description of the moving immersed objects (ref Mehdi). This method only requires knowledge of the material properties and the proper mixing laws. With that being said, our numerical simulations predict the formation, the rise, and the evacuation of the bubbles in different situations combining the above-mentioned operating conditions, which leads us to construct a benchmark graph illustrating the regimes that can be involved in such critical conditions.