# Archie equation

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## What is the Archie equation?

Archie[1] developed his famous equation to calculate, from well log parameters, the water saturation (Sw) of the uninvaded zone in a formation next to a borehole. The Archie equation can be expressed as follows:

${\mbox{S}}_{{{\rm {w}}}}{}^{{{\rm {n}}}}={\frac {{\mbox{R}}_{{{\rm {w}}}}}{(\Phi ^{{{\rm {m}}}}\times {\mbox{R}}_{{{\rm {t}}}})}}$

where:

• Sw = water saturation of the uninvaded zone
• n = saturation exponent, which varies from 1.8 to 4.0 but normally is 2.0
• Rw = formation water resistivity at formation temperature
• Φ = porosity
• m = cementation exponent, which varies from 1.7 to 3.0 but normally is 2.0
• Rt = true resistivity of the formation, corrected for invasion, borehole, thin bed, and other effects

## Limitations of the Archie equation

Even though numerous other relationships have been developed over the years, the Archie equation remains the most flexible and generally useful approach. Yet its proper application requires knowledge of its limitations. The equation was empirical in origin and therefore needs modification in rock–fluid combinations different from Archie's experiments. Modifications need to be made in rocks with the following characteristics:

• Non-Archie pore geometries (i.e., not intergranular or intercrystalline) (See Pore systems.)
• Conductive minerals such as clays and pyrite
• Very fresh (i.e., nonsaline) formation waters

## Caveat

This section discusses the Archie equation in general terms; suggested methods are most useful when dealing with modern log suites of good quality.

## Deriving values for Archie variables

Values for the five Archie variables are relatively easy to derive when a formation is thick, has a clay-free matrix, and/or is dominated by intergranular or intercrystalline porosity (Archie porosity). Formations that are thin bedded (i.e., below limits of logging tool resolution), have clay in their matrix, or have moldic, vuggy, or fracture porosity require adjustments. The table below lists the five variables and methods for deriving or estimating them.

Step Find Use… If… Then…
1 n
• 2.0 for Archie porosity
• 1.8 (or less) for rocks with clayey matrix or fractures
• 4.0 for very strongly oil-wet rocks
Not sure of rock type Use 2.0
2 Rw
• Value calculated from spontaneous potential (SP) log
• Estimated from Rw catalogs
• Estimated from wet zone Ro value
• Measured from water sample
Thin beds, hydrocarbons in zone, or fresh formation waters make SP calculations uncertain Use thin-bed correction or another method
3 Φ Value derived from cores, density, density–neutron, or sonic logs (See Basic open hole tools.) Density–neutron log matrix setting does not match formation matrix Use density– neutron crossplot (See Determining porosity from density-neutron logs.)
4 m
• 2.0 for Archie porosity
• 1.7–2.0 for shaly sandstones
• 2.0–2.5 for porosity with connected vugs
• 2.5–3.0 for nonconnected moldic porosity
• ~1.0 for fractured rocks
Not sure of rock type or pore geometry Use 2.0
5 Rt Value derived from deep resistivity log such as RILD or RLLD Beds are thin, invasion occurred or borehole has washouts Use chartbook corrections