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Abstract In this paper, we investigate the effects of poroelastic properties and permeability on hydraulic fracturing in an anisotropic medium. A hybrid approach combining the EDFM (embedded discrete fracture model) and XFEM (extended finite element method) is proposed. Fractures are embedded into a nonconforming grid used to determine the fluid flow in a porous matrix by the mimetic finite difference method, accounting for an anisotropic permeability tensor. The stress-strain problem is solved by the extended finite element method with the same grid. The fluid flow and rock deformation as well as the fracture propagation are iteratively coupled. The proposed approach is validated against the analytical solutions for Mandel's problem and the KDG model. A series of calculations are performed, and the obtained results are analyzed to investigate the effects of the anisotropy of permeability, that of the elastic modulus and Biot's coefficient on the hydraulic fracturing process. It is found that the anisotropy of permeability has a significant influence on the geometrical parameter of a fracture, while the anisotropy of the elastic modulus has a dominating influence on the propagation direction of a fracture. Biot's coefficient also has an influence on the fracture propagation kinetics. [ABSTRACT FROM AUTHOR]