An apparatus is developed to study the interaction forces between two spheres under an external ac electric field. The interaction forces of a pair of spheres as the function of interspherical spacing, electric field strength, and electric field frequency are measured precisely. The results reveal that much stronger interaction can be obtained for metal spheres and high-dielectric spheres compared with the spheres with a very low dielectric constant, such as glass and polymer. The measured forces increase very steeply as the gap of the spheres decreases, and become much larger than those of available theoretical predictions when the two single-crystalline spheres of ${\mathrm{SrTiO}}_3$ are closely spaced. The frequency dependence of the interaction force also shows an anomalous behavior. Our measurements indicate that a more accurate theoretical calculation should be performed to explain the experimental results.

During hydraulic fracturing, the aperture of hydraulic fractures will shrink by the in-situ stress, but will not fully close because of the existence of proppant inside the fracture. In previous studies, few people noticed the existence of proppant, which has resulted in the inaccuracy of simulation results. In this study, based on the boundary element method, a numerical simulation model for sequential fracturing was established, which respectively considered the influence of proppant in staged fracturing and zipper fracturing. In addition, the influence mechanism of proppant on fracture morphology is then revealed. Simulation results show that the residual aperture of the previous hydraulic fracture, which was produced by proppant, may increase with the increase of proppant stiffness and fracture spacing and may also be shrunk by the dynamic propagation of subsequent hydraulic fracture. However, the residual aperture will rebound after hydraulic fracturing construction is finished. The shrinkage and rebound values of residual aperture of hydraulic fracture are usually less than 1 mm. In addition, at the same time, the residual aperture of previous hydraulic fracture may also influence the propagation of subsequent hydraulic fracture. These influences are represented by the bend of fractures in multistage fracturing and the intersection in zipper fracturing. With the increase of well spacing, the influence degree of residual aperture on subsequent fracture propagation is reduced. The previous hydraulic fracture cannot have a significant effect on the deflection of subsequent hydraulic fracture when fracture spacing is between 10 and 30 m. The above research has important guiding significance for controlling fracture morphology in hydraulic fracturing.

The morphology of hydraulic fracture is affected by many factors. In previous studies, due to the heterogeneity of rock samples and the limitations of sample size, influence degree of various factors on fracture deflection angle has not been well distinguished in laboratory experiments. Based on the boundary element method, we established a mathematical model to study the factors controlling the morphology of hydraulic fracture. Simulation results show that with increasing injection pressure, the radius of the fracture curvature increases. When the difference between the injection pressure and the maximum principal stress is 5 times the ground stress difference, the influence of in situ stress on hydraulic fracture deflection can be ignored. Hydraulic fracture deflection angle was relatively larger when considering the viscosity of the fracturing fluid. When stress difference is too small or the injection pressure is too big, the deflection angle of fracture is easy to fluctuate during initial propagation. Too large Young’s modulus and too small Poisson’s ratio will inhibit the fracture deflection and cause a narrower width of fracture. The effect of Poisson’s ratio on fracture aperture is less than 1 mm. When perforation angle is perpendicular to maximum horizontal principal stress, the fracture width first increases rapidly and then gradually decreases from heel to tip. The influence degree of each factor on fracture deflection is ranked: stress difference of in-situ stress is the biggest, followed by injection pressure and perforation angle. This study is of great significance for the control of hydraulic fracture morphology and the further improvement of fracturing effect.

With the widespread application of hydraulic fracturing technology, the corresponding numerical simulation methods have been rapidly developed, and we have a deep understanding about the fracture propagation trajectory during hydraulic fracturing at the same time. However, the fracture interference mechanism during hydraulic fracturing is still insufficiently understood. Based on linear elastic fracture mechanics and relevant fracture criterion, a simulation model was established to simulate the fracture propagation by using boundary element method. The interaction mechanisms between fractures are studied. Results show that hydraulic fractures will change distribution direction of formation stresses during propagation process, and asymmetric fractures tend to merge together during propagation. The reverse propagation phenomenon between two fractures in numerical simulation rarely exists in the actual formation. The analysis and research on the mechanism of crack propagation and interaction can provide a theoretical basis for the trajectory control of hydraulic cracks in the later period.

During the hydraulic-fracturing process, stress interference occurs among multiple wells and fractures, potentially affecting the trajectory of hydraulic fracture propagation. Previous studies have largely overlooked the influence of proppant support stresses on the trajectories of fracture propagation. This paper establishes a mathematical model, grounded in the boundary element method, designed to compute the propagation of multiple fractures, considering both proppant support on the fracture surface and dynamic perturbations within the local stress field. The findings of this research reveal that the stress field induced by hydraulic fracturing exhibits dynamic evolution characteristics, necessitating a comprehensive study of the fracture initiation and extension across the entire fracturing time domain. The effect of the residual fracture width under proppant action on the in situ stress field cannot be ignored. During simultaneous fracturing, hydraulic fractures are inclined to propagate in the direction of the maximum horizontal principal stress, particularly as the in situ differential stress escalates. Staggered fracturing between wells has been proven to be more effective than head-to-head fracturing. Simply increasing the well spacing cannot solve the problem of inter-well fracture interference. In zipper fracturing, adjusting the fracturing sequence can inhibit the fracture intersections between wells, thereby controlling the trajectory of fracture propagation. The aforementioned research has considerable significance in guiding the control of fracture morphology during hydraulic-fracturing processes.