Spatiotemporal Synergetic Control of Roof Stability in Longwall Paste Backfill Mining: An Analytical Model Considering Time-Varying Backfill Characteristics

To address the difficulty of roof control in longwall paste backfill mining and to better capture the backfill’s ‘sequential bearing’ effect, this study proposes an elastic foundation beam model. Based on the Gaohe Coal Mine, the “coal-support-backfill” model incorporates the time-varying characteristics of paste materials. Analytical solutions for roof deflection and bending moment were derived to reveal the spatiotemporal evolutionary laws of key parameters on roof stability. Results show the filling rate is the most sensitive factor controlling ultimate roof subsidence, directly determining the roof’s actual suspended span and breaking risk. The final-set strength limits long-term deformation but exhibits a marginal diminishing effect. Conversely, early strength dominates the initial subsidence rate, preventing excessive early deformation. Additionally, increasing the filling step distance significantly increases overall subsidence, whereas support resistance improves the initial bending moment distribution. Ultimately, surrounding rock control in paste backfill stopes is a dynamic coupling of “spatial support synergy” and “temporal strength matching” providing a theoretical basis for optimizing mining safety, economy, and efficiency, which ultimately fosters sustainable green mining practices.