Loss of circulation during drilling is a pressing problem that typically occurs when significant volumes of the drilling fluid leak to the surrounding formation. In fractured media, circulation loss is often a sudden event which may interrupt the drilling operations, increase nonproductive time, and add complexity in controlling the fluid flow and wellbore stability. Developing models to predict the fluid leak dynamics and identify the suitable loss-control-material (LCM) is of significant interest to the industry. In this work, simulation and analytical models are developed to model the dynamics of fluid loss for power-law drilling fluids from the wellbore to a surrounding fractured medium. The model is based on the Cauchy momentum equation and accounts for the key predominant flow mechanisms, including the non-Newtonian behavior of the drilling fluid (mud) using a power-law model. Experiments from the literature are used to assess the behavior of mud loss into a single horizontal fracture, which includes the effects of mud rheology and yield stress. We show that previous unsuccessful attempts to match the experimental data using analytical models are related to the assumption of uniform fracture aperture. Simulations are then used to demonstrate that a slight deformation in the fracture walls leads to a significant deviation in the analytical model predictions, and therefore the dynamics of fracture opening is a critical parameter that should be carefully assessed. In other related experiments, the analytical model also failed to capture the mud invasion distance within the fracture as a function of pressure drop and the mud properties. To address this issue, a modified analytical model that accounts for non-parallel fracture walls is proposed. The new model allows to quickly assessing the impact of nonuniform fracture apertures. It can serve as a screening tool to evaluate mud invasion distance as a function of fracture geometry, mud rheological properties, and pressure drop resulting from the well depth, injection rate, and mud density.