Methods for Modelling Full Tensor Permeability in Reservoir Simulators
Authors: Bagheri M. , Settari A. 
Published: March 2010

Abstract
This work examines methods for modeling 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.

Introduction
In complex reservoirs, orientation and magnitude of principal permeabilities may vary spatially, and also evolve in time due to geomechanical effects. In such cases, a formulation with a full permeability tensor should be used to model fluid flow. In this paper, we examine methods for modeling fluid flow with a 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 5-point reservoir simulators by Todd et al.(1) in 1972. They suggested using 2-point upstream mobility method to alleviate this effect. This problem is associated mainly with unfavourable mobility ratios which occur in most EOR isothermal processes and steam and combustion, and can very seriously alter the results and conclusions of simulation studies. The grid orientation effect is also severe for simulating miscible displacement. Settari et al.(2) have shown that a standard 5-point approximation gives unacceptable results even for moderately 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 9-point method still has some orientation errors, which depend on the problem solved(1).