This work examines methods for modelling reservoir flow in presence of permeability tensor. Usually, the control volume multipoint discretizations are used to handle simultaneously the tensor permeability and complex geometry. Instead, the method used in this work is based on simple extension of conventional finite difference method. It is shown that such method (which results in 9-point approximations with full tensor) cannot accurately predict the behavior of reservoirs in presence of permeability anisotropy. It suffers from what we call a "Tensor Orientation" effect, in addition to the wellknown grid orientation effect. The tensor orientation effect introduces an error in the magnitude and shape of pressure field, which depends on the relative orientation of the grid in relation to the principal axes of the permeability tensor.
This problem has been solved by developing a 13-point extension of the conventional 9-point finite difference method for the tensor permeability, which essentially eliminates the tensor orientation errors. Since this difference scheme is not easily implemented in conventional simulators, an approximate semi implicit method, in which only nine points are in implicit mode, was also developed. The semi-implicit method provides a good match with the 13-point method for the test problem. However, further reduction to a 5-point implicit is associated with accuracy loss.
Comparative evaluation against the Flux Continuous Control Volume Multipoint discretization shows that while both methods are free of the tensor orientation effect, the 13-point method has a lower value for well block pressure. Lack of an analytical solution makes it difficult to determine which method is closer to reality.
In complex reservoirs, orientation and magnitude of principal permeabilities may vary spatially, and due to geomechanical effects also evolve in time. In such cases, formulation with full permeability tensor should be used to model fluid flow. In this paper, we examine methods for modelling fluid flow with permeability tensor, and in particular the effect of the permeability tensor orientation on the results with various numerical methods.
Dependency of simulation results of fluid flow in porous media to the type of the grid mesh is well-known and called the "grid orientation" effect. This problem was first demonstrated for five-point reservoir simulators by Todd et al (Ref. 6) in 1972. They suggested using twopoint upstream mobility method to alleviate this effect. This problem is associated mainly with unfavorable mobility ratios, such as in most EOR isothermal processes and steam and combustion, and can alter very seriously the results and conclusions of simulation studies. The grid orientation effect is severe for simulating miscible displacement. Settari et al. (Ref. 7) have shown that standard five-point approximation gives unacceptable results even for moderate adverse mobility ratios (M=10). Until now, a completely satisfactory solution has not been found for finite difference simulators and the grid orientation remains one of the more difficult numerical research problems. Nine point discretizations are the usual method for solving the problem. However, the nine-point methods still have some orientation error, which depends on the problem solved (Ref. 2).