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Loss of circulation while drilling is a challenging problem that may interrupt operations, reduce efficiency, and may contaminate the subsurface. When a drilled borehole intercepts conductive faults or fractures, lost circulation manifests as a partial or total escape of drilling, workover, or cementing fluids, into the surrounding rock formations. Loss control materials (LCM) are often used in the mitigation process. Understanding the fracture effective hydraulic properties and fluid leakage behavior is crucial to mitigate this problem. Analytical modeling of fluid flow in fractures is a tool that can be quickly deployed to assess lost circulation and perform diagnostics, including leakage rate decline, effective fracture conductivity, and selection of the LCM. Such models should be applicable to Newtonian and non-Newtonian yield-stress fluids, where the fluid rheology is a nonlinear function of fluid flow and shear stress. In this work, a new semi-analytical solution is developed to model the flow of non-Newtonian drilling fluid in a fractured medium. The solution model is applicable for various fluid types exhibiting yield-power-law (Herschel-Bulkley). We use high-resolution finite-element simulations based on the Cauchy equation to verify our solutions. We also generate type-curves and compare them to others in the literature. We demonstrate the applicability of the proposed model for two field cases encountering lost circulations. To address the subsurface uncertainty, we combine the developed solutions with Monte-Carlo and generate probabilistic predictions. The solution method can estimate the range of fracture conductivity, parametrized by the fracture hydraulic aperture, and time-dependent fluid loss rate that can predict the cumulative volume of lost fluid. The proposed approach is accurate and efficient enough to support decision-making for real-time drilling operations.