  Phase field model (PFM) is an efficient fracture modeling method and has high
potential for hydraulic fracturing (HF). However, the current PFMs in HF do not
consider well the effect of in-situ stress field and the numerical examples of
porous media with stress boundary conditions were rarely presented. The main
reason is that if the remote stress is applied on the boundaries of the
calculation domain, there will be relatively large deformation induced on these
stress boundaries, which is not consistent with the engineering observations.
To eliminate this limitation, this paper proposes a new phase field method to
describe quasi-static hydraulic fracture propagation in porous media subjected
to stress boundary conditions, and the new method is more in line with
engineering practice. A new energy functional, which considers the effect of
initial in-situ stress field, is established and then it is used to achieve the
governing equations for the displacement and phase fields through the
variational approach. Biot poroelasticity theory is used to couple the fluid
pressure field and the displacement field while the phase field is used for
determining the fluid properties from the intact domain to the fully broken
domain. In addition, we present several 2D and 3D examples to show the effects
of in-situ stress on hydraulic fracture propagation. The numerical examples
indicate that under stress boundary condition our approach obtains correct
displacement distribution and it is capable of capturing complex hydraulic
fracture growth patterns.
