Reservoir: The Equations Archie Forgot: Anisotropy of the Rocks
Archie’s empirical equation
is extensively used to estimate hydrocarbons in place.This power-laws combination has stood the
test of time with little 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
datasets.It corrects for Archie’s
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, vuggy and fractured carbonates) to get 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 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 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 function of water saturation, crucial to quality control core electrical data, and to quantify reservoirs under changing saturation (waterflooding).Therefore, improved calculations of original and remaining hydrocarbons are achieved.
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