Inkjet technology is a versatile and widely used process in various industries, ranging from printing to microfluidic applications. However, the successful simulation of inkjet dynamics is a complex task due to the multiple physical phenomena involved. These include the interaction of ink properties with additives, the intricate behavior of liquid-gas interfaces, and the dynamics of contact lines at the nozzle and substrate. To address these challenges, this work introduces an Enriched Finite Element Method (E-FEM) designed to model these complex phenomena with high accuracy and computational efficiency.