Keyhole mode laser welding is a valuable technique for welding thick materials in industrial applications. However, its susceptibility to fluctuations and instabilities poses challenges, leading to defects that compromise weld quality. Observing the keyhole during laser welding is challenging due to bright process radiation, and existing observation methods are complex and expensive. This paper alternatively presents a novel numerical modeling approach for laser spot welding of aluminum through a modified mixture theory, a modified level-set (LS) method, and a thermal enthalpy porosity technique. The effects of laser parameters on keyhole penetration depth are investigated, with a focus on laser power, spot radius, frequency, and pulse wave modulation in pulsed wave (PW) versus continuous wave (CW) laser welding. PW laser welding involves the careful modulation of power waves, specifically adjusting the pulse width, pulse number, and pulse shapes. Results indicate a greater than 80 percent increase in the keyhole penetration depth with higher laser power, pulse width, and pulse number, as well as decreased spot radius. Keyhole instabilities are also more pronounced with higher pulse width/numbers and frequencies. Notably, the rectangular pulse shape demonstrates substantially deeper penetration compared to CW welding and other pulse shapes. This study enhances understanding of weld pool dynamics and provides insights into optimizing laser welding parameters to mitigate defects and improve weld quality.