Spontaneous droplet movement has gained increased interest in many applications, including microfluidics and microfabrication. This study focuses on the numerical investigation of driving mechanisms of spontaneous droplet motion. The numerical model using the phase-field method was validated by available experimental data. In this study, a heterogeneous wettability condition is implemented to reproduce contact angle hysteresis for the accurate prediction of spontaneous droplet dynamics. Through analysing the capillary pressure within the droplet, the driving mechanism is identified as being governed by the pressure difference between the two interfaces which depends on channel configuration, wettability, and contact angle hysteresis. The impact of channel deformability was further studied, revealing that channel deformability leads to significant changes in velocity or even reversed droplet movement direction. This study provides a novel numerical framework for controllable spontaneous droplet movement in flexible channels.

After fracturing operations, a large amount of fracturing fluid is retained in shale fracture network, resulting in low flowback efficiency. This has been attributed to the imbibition of fracturing fluid into matrix pores. However, it is unclear how the imbibition mechanism is involved, what are its governing laws and controling parameters in fracture networks? Based on the three-dimensional water imbibition theory of matrix blocks, a fracture network model is established, and a number of dimensionless controling parameters are proposed and analyzed for flowback efficiency. The results show that the imbibition characteristics of fracturing fluid in fracture network are mainly determined by two dimensionless numbers; namely, dimensionless imbibition time, fracture width, and imbibition capacity. The dimensionless imbibition time characterizes the contact time between the fracturing fluid and shale formation, which negatively correlates to the flowback efficiency. The dimensionless fracture width is the ratio of the fracture width to the rock length, which is inversely proportional to the flowback efficiency. Smaller value of the dimensionless fracture width corresponds to larger contact area of fracturing fluid and shale, leading to a lower flowback efficiency. The dimensionless imbibition capacity depicts the capacity of shale reservoirs to imbibe fracturing fluid, which has a negative linear correlation with flowback efficiency. In addition, dimensionless time and fracture width are related to the fracturing operations, and are enhanced by increasing the shut-in periods and proppant concentration. Therefore, the flowback efficiency can be controlled by changing fracturing operations. The predictive method of the flowback efficiency established here is of great significance for reservoir damage analysis and flowback regime optimization.