Mathematical modeling and parameter interpretation of dynamic fracture flow during hydraulic fracturing in tight reservoirsMathematical modeling and parameter interpretation of dynamic fracture flow during hydraulic fracturing in tight reservoirs

To address the difficulty of characterizing dynamically evolving fracture parameters in tight sandstone reservoirs from conventional post-fracturing well testing, a mathematical model and interpretation method are developed using full-stage pressure data acquired during hydraulic fracturing operations. The proposed framework targets the coupled pressure response associated with fluid injection, fracture initiation, and fracture propagation in a single stage of a horizontal well. By introducing a half-fracture-length evolution coefficient, the time-dependent growth of fracture geometry is explicitly represented, and a variable-fracture-length flow model is formulated from mass conservation within the fracture system. An analytical expression for bottom-hole pressure is derived in Laplace space, and a quantitative interpretation workflow is established through Stehfest numerical inversion. In contrast to traditional approaches that rely primarily on shut-in pressure-decline data, the present method incorporates the entire pressure history of the fracturing treatment and provides an integrated computational framework for dynamic-fracture modeling, analytical solution, and parameter inversion. Sensitivity analysis indicates that the maximum fracture half-length and pumping rate dominate the bottom-hole pressure response, whereas the half-fracture-length evolution coefficient mainly controls the early-time pressure behavior after fracture initiation. Field application shows that the interpreted fracture half-length is 47.62 m, in close agreement with the field-estimated value of 44.5 m. The proposed method offers a practical tool for post-fracturing evaluation and treatment optimization in tight sandstone reservoirs.