Archie's law

In petrophysics, Archie's law relates the in-situ electrical conductivity (C) of a porous rock to its porosity (${\displaystyle \phi \,\!}$) and fluid saturation (${\displaystyle S_{w}}$) of the pores:

${\displaystyle C_{t}={\frac {1}{a}}C_{w}\phi ^{m}S_{w}^{n}}$

Here, ${\displaystyle \phi \,\!}$ denotes the porosity, ${\displaystyle C_{t}}$ the electrical conductivity of the fluid saturated rock, ${\displaystyle C_{w}}$ represents the electrical conductivity of the aqueous solution (fluid or liquid phase), ${\displaystyle S_{w}}$ is the water saturation, or more generally the fluid saturation, of the pores, ${\displaystyle m}$ is the cementation exponent of the rock (usually in the range 1.8–2.0 for sandstones), ${\displaystyle n}$ is the saturation exponent (usually close to 2) and ${\displaystyle a}$ is the tortuosity factor.

Reformulated for the electrical resistivity (R), the inverse of the electrical conductivity ${\textstyle (R={\frac {1}{C}})}$, the equation reads

${\displaystyle R_{t}=aR_{w}\phi ^{-m}S_{w}^{-n}}$

with ${\displaystyle R_{t}}$ for the total fluid saturated rock resistivity, and ${\displaystyle R_{w}}$ for the resistivity of the fluid itself (w meaning water or an aqueous solution containing dissolved salts with ions bearing electricity in solution).

The factor

${\displaystyle F={\frac {a}{\phi ^{m}}}={\frac {R_{t}}{R_{w}}}}$

is also called the formation factor, where ${\displaystyle R_{t}}$ (index ${\displaystyle t}$ standing for total) is the resistivity of the rock saturated with the fluid and ${\displaystyle R_{w}}$ is the resistivity of the fluid (index ${\displaystyle w}$ standing for water) inside the porosity of the rock. The porosity being saturated with the fluid (often water, ${\displaystyle w}$), ${\displaystyle S_{w}^{-n}=1}$.

In case the fluid filling the porosity is a mixture of water and hydrocarbon (petroleum, oil, gas), a resistivity index (${\displaystyle I}$) can be defined:

${\displaystyle I={\frac {R_{t}}{R_{o}}}=S_{w}^{-n}}$

Where ${\displaystyle R_{o}}$ is the resistivity of the rock saturated in water only.

It is a purely empirical law attempting to describe ion flow (mostly sodium and chloride) in clean, consolidated sands, with varying intergranular porosity. Electrical conduction is only performed by ions dissolved in aqueous solution. So, electrical conduction is considered to be absent in the rock grains of the solid phase or in organic fluids other than water (oil, hydrocarbon, gas).

Archie's law is named after Gus Archie (1907–1978) who developed this empirical quantitative relationship between porosity, electrical conductivity, and fluid saturation of rocks. Archie's law laid the foundation for modern well log interpretation as it relates borehole electrical conductivity measurements to hydrocarbon saturations (which, for fluid saturated rock, equals ${\displaystyle 1-S_{w}}$).