The objective of this research is to help better understand the effects of exit face inclinations on critical gradients. This will lead to more practical ways to predict critical hydraulic gradients and slope stability using soil properties and exit face conditions. Current geotechnical engineering does not consider these factors because they are not thoroughly understood. Despite the attempts to better predict critical gradients, methods developed over 70 years ago to model the heave mechanism are used to model failure mechanisms such as backward erosion (or piping). This critical hydraulic gradient is only calculated by means of the buoyant unit weight of the soil. While modern engineering estimates critical piping gradients near unity, research has shown that calculated critical gradients can largely under-predict actual piping gradients. The results of this thesis research will help provide an empirical, but mechanism-based, grain-scale model that takes into account the effect of non-horizontal exit faces, and soil properties while assessing the potential for piping initiation to occur. This research is expected to help the understanding of the internal erosion mechanism known as piping, and eventually help to develop more practical ways of predicting and preventing conditions which are susceptible to this type of erosion. This research should be used to initiate more research, develop better methods, and eventually increase public safety with regards to designing and improving earth structures such as dams and levees.