This work examines methods for modelling reservoir flow in the presence of a permeability tensor. Usually, control-volume multipoint discretizations are used to simultaneously handle the tensor permeability and complex geometry. Instead, the method used in this work is based on a simple extension of the conventional finite difference method. It is shown that this method (which results in 9-point approximations with a full tensor) cannot accurately predict the behaviour of reservoirs which contain permeability anisotropy. It suffers from what we call a "tensor orientation" effect, in addition to the well known grid orientation effect. The tensor orientation effect introduces an error in the magnitude and shape of the 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 the 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 operator results in a loss of accuracy. Comparative evaluation against the Flux Continuous Scheme technique 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.