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
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, 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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