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The paths of a two-phase flow are usually non-linear and discontinuous in the production of shale gas development. To research the influence mechanism between shale gas and water, several integer two-phase flow models have been studied but few analytical solutions have been obtained on shale gas and water pressure. This study first developed a local fractional mathematical model for gas and water two-phase flow in shale gas production. The model thus created considers the effects of capillary pressure, the fractal dimension of the flow pipe, and the discontinuity of the flow path. Second, the local fractional traveling wave method and variational iteration method were applied to this model for the development of iterative analytical solutions. Both shale gas and water pressure were analytically derived. Third, the depressurization process of the shale gas and water was analyzed, and a parametric study was conducted to explore the impacts of fractional dimension, entry capillary pressure, and travel wave velocity on shale gas pressure. Finally, our conclusions are drawn, based on the results of these studies.

Heat treatment on shale reservoirs can promote the development of secondary fractures in a matrix on the basis of hydraulic fracturing, forming multi-scale gas–water seepage channels and strengthening the gas desorption. Experimental evidence shows that heat treatment can enhance gas recovery in the same mining life. Heat treatment on a shale gas reservoir is a multi-physical and multi-phase coupling process. However, how the thermal stimulation interacts with nonlinear two-phase flow in heterogeneous shale volume fracturing has not been clear. In this paper, a fully coupled THGM model for heating-enhanced shale-gas recovery in heterogeneous shale reservoirs is proposed. First, the governing equations are formulated for the shale-reservoir deformation involving both gas adsorption and thermal expansion, the permeability evolution model for the cracking process of fractured shale, the gas–water two-phase continuity equation considering the effects of gas solubility and the heat transfer equation for heat conduction and convection. The interactions among stress, temperature and seepage in a heterogeneous shale reservoir were studied. Secondly, a test on shale permeability after 50 °C temperature treatment was conducted. The evolution of temperature, capillary pressure, water and gas saturation and the permeability of shale during the heat treatment of the reservoir were numerically analyzed. Finally, the gas production from a shale gas reservoir was numerically simulated with this THGM model. The numerical results indicated that the thermal-induced fracturing, gas desorption and separation from water make predominant contributions to the evolution of permeability. The heat treatment can enhance cumulative gas production by 58.7% after 27.4 years of heat injection through promoting gas desorption and matrix diffusion.