Nematic liquid crystals (NLCs) under micron-range confinement exhibit a rich defect phenomenology that can be used to extract elastic (Frank moduli) material parameters of critical importance for next generation electro-optical devices. In this work we develop a model to predict defect-driven textural transformations that arise when a NLC is confined to a circular capillary. In the initial transformation stage an unstable disclination defect of strength +1 nucleates in the axis of the capillary and quickly branches into two stable +1/2 disclination defects. The model includes: (1) the Kirchhoff branch balance equation which predicts the splitting of a +1 into two +1/2 wedge disclinations; (2) the curvature of the +1/2 disclination lines as a function of elastic properties. This model shows that by increasing the ratio of tension strength to bending stiffness, the branch point angle increases, but the final defect distance decreases; and (3) the aperture branching angle of the +1/2 lines as a function of the elastic properties and the magnitude of the curvature at the branch point. These three predictions form the basis for the evaluation of the Frank elastic moduli on NLCs. The key advantage of the implemented methodology is to use time-dependent textural transformations under micron-range capillary confinement to extract elastic parametric data needed to further develop NLCs in functional and structural application.