The Equations Archie Forgot: Anisotropy of the Rocks
- Carlos F. Haro (Occidental Oil and Gas)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- October 2010
- Document Type
- Journal Paper
- 823 - 836
- 2010. Society of Petroleum Engineers
- 5.4.1 Waterflooding, 1.6.9 Coring, Fishing, 4.3.4 Scale
- tortuosity, Laplace equation, pore geometry, Archie equation
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- 1,375 since 2007
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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.
|File Size||1 MB||Number of Pages||14|
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