In this paper, three-dimensional(3D) Numerical Manifold Method (NNM) based on hexahedron element cover with full first-order cover function is proposed and the shape function of C8 isoparametric element in FEM is used as the cover weight function. All sub-matrices in equilibrium equations, including stiffness matrix, initial stress matrix, point force matrix, surface force matrix, body force matrix, inertia force matrix, contact matrix and friction matrix, are derived. Different with 2D contact, the direction of shear stiffness and friction force can not be easily defined in 3D contact. A new iterative method based on vector theory to detect the contact direction is developed. The application of 3D NMM in underground excavation is also presented and show good agreement with real engineering.