The numerical asymptotic solution to initial condition problem of preexisting plane-strain hydraulic fracture with fluid lag

The numerical asymptotic solution to the initial condition problem (early-time solution) of the pre-existing plane-strain hydraulic fracture with a fluid lag is obtained under different levels of in-situ stress. This solution can provide a more accurate initial condition for simulation of hydraulic modelling. The perturbation method is applied to calculate time-derivative terms of the governing equations based on the derivative definition, and the double-iteration method is used to seek more accurate values of these time-derivative terms. Besides, the profiles of dimensionless fracture opening and dimensionless net pressure are obtained under different levels of in-situ stress, and a simplified self-similar model with no time derivative terms is put forward, because time-derivative terms can be neglected under the large in-situ stress.