An Integrated Approach to the Determination of Permeability Tensors for Naturally Fractured Reservoirs
Authors: Gupta A. , Penuela G. , Avila R. 
Published: March 2010

Abstract
The use of permeability tensors is required when modelling fluid flow in anisotropic and heterogeneous reservoirs presenting multiple zones of directional permeability, or those categorized as naturally fractured reservoirs.
A general procedure for characterizing complex reservoirs utilizing their permeability tensor is being developed by integrating data and methods from different disciplines. Permeability tensors for geologically defined fracture patterns are derived, and finally these small-scale descriptors are incorporated into a reservoir simulation program capable of handling full tensor permeabilities. The application and convenience of the method presented in this paper is illustrated with a field example from a naturally fractured reservoir.

Introduction
For many years, substantial research has been conducted in the areas of geosciences and engineering in order to characterize naturally fractured reservoirs. Numerous approaches have been presented to properly overcome this difficult task. Geoscientists have focused their research towards understanding the process of fracturing (rock mechanics), and the subsequent description of fracture characteristics such as density and orientation. Engineers, on the other hand, have focused their attention to the description of the fluid flow in the fracture systems, and in the development of accurate models (reservoir simulators) to reproduce the history, and predict the hydrocarbon production, for these complex systems.
One of the first matrix-fracture models was presented by Warren and Root(1), who presented an idealized sugar cube model with two classes of porosity, a primary porosity that is intergranular, and a secondary porosity that is induced by fractures. Even though the sugar cube model has been widely accepted as the forerunner of the modern interpretation of dual-porosity systems, its limitations in describing the behaviour of some complex fractured reservoir systems have been observed. It is now well known that almost all fracture systems are much more complicated than the suggested model of three orthogonal sets of uniform fractures.
One of the most important factors that has been identified as a necessary addition to improve the overall description of such complex reservoirs is the definition of a nine-component permeability tensor for the fracture system. This tensor is used to model fluid flow in complex reservoirs with multiple zones of directional permeability, where the orientation and magnitude of the principal permeabilities may vary between different zones in the reservoir.
Snow(2, 3) studied the convenience of using mathematical equivalents of parallel plate openings to simulate fractures dispersed in orientation, distributed in aperture, and of arbitrary spacing. Models for fractured media which contained any number of planar conductors of any orientation and any fine aperture were presented. A key assumption was that all of the conduits have smooth parallel plane walls of indefinite extent (infinite fractures), and an rbitrary aperture. As a result, a permeability tensor could be obtained by superposition of contributions due to the fractures, and due to the permeable matrix.
Long et al.(4, 5) addressed the more realistic scenario of finite or discrete fracture systems, where properties such as shape, orientation and location of the fractures in an impermeable matrix were considered to be random variables.