A dynamic fracture model combining with Laplace transform and synchronous iteration method for transient behavior analysis of a four-zone system

•A semi-analytical model for transient flow analysis with the capture of dynamic fracture behavior under long-term pumping rate conditions is developed in this four-zone fractured aquifer system.•The transformation in Laplace domain with predetermined time discretization and synchronous iteration techniques are deployed to obtain the solutions of pressure and pumping rate.•The typical features of fracture closure can be identified and the effect dynamic fracture behavior on pressure and pumping rate has been analyzed. A robust and generic semi-analytical model with the capture of dynamic fracture behaviors under long-term pumping rate condition is developed for transient flow analysis in this four-zone system. To characterize the variations of fracture length, the improved pressure-dependent dynamic fracture factor is introduced and incorporated into this model. The Laplace domain with predetermined time discretization and synchronous iteration technique are deployed to obtain the solutions of pressure and pumping rate with the least computational burden. Meanwhile, the stability and convergence of this semi-analytical model can be met within the discrete time. Furthermore, the pressure and pumping rate solutions of this model under the static fracture condition are in good agreement with those of the constant length fracture, which verifies the reliability of this model. The dynamic fracture model with the four -zone system develops five flow stages on the dimensionless pressure and its derivative curves: fracture bilinear flow, dynamic fracture flow, inner area linear flow, transitional flow and boundary-dominated flow. The dynamic fracture behavior is mainly reflected in the obviously positive upwards and pressure derivative curve presents a higher than unit slope. Moreover, the significant decrease of flow rate indicates that fracture closure could result in a rapid decline in pumping rate, which might be the reason why the pumping rate decrease quickly during the long-term pumping period. As the shrinkage coefficient increases, fractures are closed from abruptly to gradually, and typical signatures of fracture closure may be hidden if the shrinkage factor is too small, which may be caused by the instantaneous pressure depletion in the fractures. With the decrease of retained fracture coefficient, there is greater in the positive upward of transient pressure behavior and a lower in pumping rate. As a result, the flow rate will be seriously overestimate