Loading–Unloading Stress-Sensitivity Experiments and a Mathematical Model for Self-Propped and Proppant-Filled Shale Fractures

A stepwise loading–unloading pressure path was applied to three self-propped and three proppant-filled shale fracture plugs. The propped plugs were tested sequentially with 40/70-mesh quartz sand and ceramic at nominal gaps of 1, 2, and 4 mm. All nine runs showed permeability hysteresis. Observed terminal loss was 63.9–82.7% for self-propped fractures, 57.8–64.0% for quartz-sand runs, and 14.6–23.5% for ceramic runs. A piecewise stress-path model was fitted to all nine runs using a loading coefficient, a history exponent, and a fitted residual-loss parameter to represent the distinct loading and unloading branches. The results quantify stress-path dependence for these tested fracture configurations and provide a laboratory-scale permeability relation for considering loading history in parent–child well studies.