# SPE Reservoir Evaluation & Engineering Volume 11, Number 3, June 2008, 565-568

SPE-102343-PA

### Directional Permeability

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DOI  10.2118/102343-PA http://dx.doi.org/10.2118/102343-PA

### Citation

• Fanchi, J.R. 2008. Directional Permeability. SPE Res Eval & Eng11 (3): 565-568. SPE-102343-PA.

### Discipline Categories

• 6.5 Reservoir Simulation
• 6.3.1 Flow in Porous Media

### Summary

A relationship between permeability tensor and coordinate orientation is used to estimate the error that occurs when some of the terms in the permeability tensor are neglected. The formula for calculating the errors that appear in the magnitude and direction of flow rate are presented. The results are applicable to any reservoir system that is influenced by directional permeability.

### Introduction

Reservoir management experience has demonstrated that anisotropic permeability is needed to correctly solve fluid-flow problems in a variety of realistic settings. Permeability anisotropy in a plane is usually represented using two directions: the direction of maximum permeability, and the direction that is transverse to the direction of maximum permeability. This procedure establishes a natural coordinate system for describing directional permeability. The coordinate system is considered the diagonalized coordinate system. If flow-rate calculations are not aligned with the diagonalized coordinate system, then additional terms should be included in the form of Darcy’s law, which is used in flow calculations. All of the permeability terms are considered the elements of the permeability tensor.

Most commercial reservoir simulators solve fluid-flow equations that have been formulated on the basis of the assumption that the permeability tensor has been diagonalized (Fanchi 2006b; Ertekin et al. 2001). As a rule, the off-diagonal permeability terms are not included in flow calculations, and an error occurs. Engineers usually assume without justification that the error can be neglected. Research in naturally fractured reservoir modeling (Gupta et al. 2001), geomechanics (Settari et al. 2001), and upscaling (Young 1999) has demonstrated that the full permeability tensor is needed to correctly solve fluid-flow problems in a variety of realistic settings. The purpose of this paper is to show how to assess the magnitude of the error that occurs when the off-diagonal terms are not included in reservoir flow calculations.

The directional dependence of permeability and the permeability tensor are introduced in the section titled "Directional Dependence of Permeability." A relationship between the diagonalized-permeability-tensor assumption and coordinate orientation is discussed in the section titled “Permeability Tensor and Coordinate Orientation.” This relationship is used in the section titled "Error Analysis" to estimate the error that occurs when some of the terms in the permeability tensor are neglected. We show that the error depends on orientation of the coordinate system, the permeability aspect ratio, and the pressure gradient. The formulas for calculating the errors that appear in the magnitude and direction of the flow rate are presented. Concluding remarks are then presented.

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### History

• Original manuscript received: 31 May 2006
• Meeting paper published: 24 September 2006
• Revised manuscript received: 3 January 2008
• Manuscript approved: 3 January 2008
• Version of record: 20 June 2008