Analytical modelling of hysteretic constitutive relations governing spontaneous imbibition of fracturing fluid in shale

Understanding spontaneous imbibition of fracturing fluid in shale is critical for hydraulic fracturing design and optimization. In this paper, we present an analytical model for spontaneous imbibition including a hysteretic relative permeability-saturation-capillary pressure (k-S-P) relation and evaluate the relevance of hysteresis effect when modelling fracturing fluid imbibition. In the hysteretic formulations, capillary pressure and relative permeability depend not only on the current saturation but also on the history of saturation in the invaded zone. Herein, we concentrate on fracturing fluid imbibition into shale matrix during shut-in time after refracture treatment, which is driven by the strong capillary forces present in the tight matrix. We demonstrate the importance of accounting for the hysteresis in capillary pressure and relative permeability for predicting the imbibed volume of a fracturing fluid. For a problem that involves drainage and imbibition cycles, a hysteretic k-S-P relation is required to accurately assess the distribution of fluid saturation in the invaded zone. We also demonstrate that although the difference in viscosity between the fracturing fluid and gas is large, the viscosity of the gas cannot be neglected in modelling fracturing fluid imbibition in shale. The results of this investigation are expected to provide a better understanding of fracturing fluid imbibition in shale.