In the paper study of properties of relative phase permeabilities for media with anisotropic filtration properties is carried out using numerical-analytical method. The anisotropy may be related to fracturing caused by brittle crushing. The analysis of relations between phase and absolute permeability tensors is made in assumption of capillary equilibrium. Samples of relative phase permeability functions are obtained for media with orthotropic and monocline symmetry of filtration properties. The influence of connectivity property on the functions is shown, and the saturation dependence of direction of principal axes for phase permeability tensor is investigated. Thereby the misalignment of phase and absolute permeability tensors is shown. The results qualitatively coincide with experimental data.
Mathematical Modeling, V. 23, N. 5, P. 3-15, 2011

The paper presents a critical review of contemporary up-scaling methods with the main focus on up-scaling techniques for the absolute permeability tensor. We propose a new algorithm, which allows to obtain physically accurate absolute permeability tensor for coarse grids with arbitrary geometry. In particular, the resulting tensor is symmetrical and positive defined; moreover dissipative energy integral and flux through the faces of the cells are both approximated, the second is valid in weak sense. The developed algorithm is quite general, i.e. it is valid in cases where the coefficients are piecewise-continuous functions, and the discontinuity lines can be approximated by linear segments in 2D. In addition, the approach allows for down-scaling, i.e. the solution on the fine grid can be obtained using up-scaled coarse grid solution. Numerical experiments results demonstrate significant reduction in computation time while maintaining the quality of solution.
Herald of the CDC Rosnedra, N. 4, P. 23-30, 2010.