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.
© 2008. Society of Petroleum Engineers
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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
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