A novel approach to simulate the stress and displacement fields induced by hydraulic fractures under arbitrarily distributed inner pressure

•A new model is built for simulating stress and displacement around fractures under arbitrarily distributed pressure.•We put forward a novel approach to apply the result of complex variable method in theory of elasticity.•The influence of fluctuating geo-stress on induced stress and displacement is studied.•The difference of the computation results of classical model and superposition model is negligible in the far field.

Stress and displacement fields induced by hydraulic fractures have been studied by a lot of researchers since they seriously affect the hydraulic fracture geometry. In previous studies, the distribution of inner pressure was assumed to be uniform or diminishing. Therefore, taking the complexly distributed inner pressures, such as geo-stresses with random fluctuations, into consideration in the classical models was unrealizable. A convenient and stable superposition model is developed to simulate the induced stress and displacement fields around artificial or natural fractures under arbitrarily distributed inner pressures in conjunction with complex variable method in theory of elasticity, which includes Westergaard Stress Function Method. The new model is validated by the analytical solution of induced displacement field around a fracture under linear load. It could be inferred that the superposition model is always convergent with a large enough number of discrete segments since the structure of the superposition model is similar to that of the numerical integrations. Some conclusions are drawn from the simulation results of the stress and displacement fields around fractures, when taking the inner pressure drop, asymmetrical propagation and fluctuating geo-stress into consideration. In consideration of the inner pressure drop in artificial fracture, narrower fracture width will form and the risk of early screen-out will increase significantly. Hence, in this case a greater pad volume or a higher pumping rate is needed. The symmetry of the induced stress and displacement fields would be broken by the asymmetrical propagation of artificial fracture. However, the difference of the results computed by classical model and superposition model is negligible in the far field (where y > a). The influence of fluctuating geo-stress on displacement distribution is non-ignorable since the most important displacement computing data is fracture width, which is the double of the displacement at y = 0.