Underbalanced drilling (UBD) is a method used to minimize formation damage and enhance the reservoir productivity. But, at the same time UBD greatly raises the risk of mechanical wellbore instability by purposefully keeping the wellbore pressure below the formation pore pressure. This thesis systematically examines the geomechanical response of a vertical wellbore under UBD conditions through numerical analysis. The primary objective of this study is to systematically compare wellbore stability predictions derived from three distinct constitutive models, which are the Linear Elastic, Mohr-Coulomb, and the Generalized Hoek-Brown criterion model. To achieve this, a two dimensional, plane strain numerical model of a vertical wellbore in a sandstone formation was developed using the finite difference software FLAC 2D (Itasca).