Summary
Archie's empirical equation is used extensively to estimate hydrocarbons in
place. This power-laws combination has stood the test of time with few changes.
However, it is still poorly understood and considered an ad hoc relation. Our
original analysis will prove these laws rigorously, show how they must be
amended, and introduce additional accompanying equations. This comprehensive
model, which represents the electrical flow through the intricate conductive
paths of the rock, is confirmed with Archie's and Hamada's core data sets. It
corrects for Archie's inaccuracies.
A thorough appreciation of the pore-scale physics behind the modified
version of Archie's equation is presented. The principles can be applied in
clean and complex formations (shaly sands, thin beds, and vuggy or fractured
carbonates) to obtain enhanced values of water saturation. The theory sheds
light on the role and quantification of anisotropy.
Solving for the elaborate pore geometry, we use the Laplace differential
equation (not Ohm's law), appropriate in the analysis of electrostatic fields
in charge-free regions. Rock morphology dictates its boundary conditions (Jin
2007; Ghous 2005), characterized as corner angles. The corresponding particular
solution (flow around a corner) and modeling tactic delineate the streamlines
throughout the pores. The angles establish strong mathematical links among the
exponents of Archie's equation, the geometry of the rock frame, and the spatial
fluid distribution. This quantitative method is lacking in previous saturation
models.
The solution constitutes the basis to solve more-complicated rock layouts.
It enables the calculation of equivalent resistivities (normalized resistances)
to take advantage of well-established electrical relationships. The extra
equations compute the variable exponents and coefficients of Archie's equation
at every depth. They obtain the saturation exponent in clean rocks as a
function of water saturation, crucial to the quality control of core electrical
data and to the quantification of reservoirs under changing saturation
(waterflooding). Therefore, improved calculations of original and remaining
hydrocarbons are achieved.
© 2010. Society of Petroleum Engineers
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History
- Original manuscript received:
30 May 2009
- Meeting paper published:
5 October 2009
- Revised manuscript received:
7 February 2010
- Manuscript approved:
4 May 2010
- Published online:
11 October 2010
- Version of record:
27 October 2010
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