# Archie's law

In petrophysics, Archie's law relates the in-situ electrical conductivity of a sedimentary rock to its porosity and brine saturation:

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

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 brine, ${\displaystyle S_{w}}$ is the brine saturation, ${\displaystyle m}$ 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 electrical resistivity, the equation reads

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

with ${\displaystyle R_{t}}$ for the fluid saturated rock resistivity, and ${\displaystyle R_{w}}$ for the brine resistivity.

The factor

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

is also called the formation factor, where ${\displaystyle R_{o}}$ is the resistivity of the rock filled with only water (${\displaystyle S_{w}=1}$).

The factor

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

is also called the resistivity index.

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 assumed not to be present within the rock grains or in fluids other than water.

Archie's law is named after Gus Archie (1907–1978) who developed this empirical quantitative relationship between porosity, electrical conductivity, and brine 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 ${\displaystyle 1-S_{w}}$).

## Parameters

### Cementation exponent, ${\displaystyle m}$

The cementation exponent models how much the pore network increases the resistivity, as the rock itself is assumed to be non-conductive. If the pore network were to be modelled as a set of parallel capillary tubes, a cross-section area average of the rock's resistivity would yield porosity dependence equivalent to a cementation exponent of 1. However, the tortuosity of the rock increases this to a higher number than 1. This relates the cementation exponent to the permeability of the rock, increasing permeability decreases the cementation exponent.

The exponent ${\displaystyle m}$ has been observed near 1.3 for unconsolidated sands, and is believed to increase with cementation. Common values for this cementation exponent for consolidated sandstones are 1.8 < ${\displaystyle m}$ < 2.0. In carbonate rocks, the cementation exponent shows higher variance due to strong diagenetic affinity and complex pore structures. Values between 1.7 and 4.1 have been observed. [1]

The cementation exponent is usually assumed not to be dependent on temperature.

### Saturation exponent, ${\displaystyle n}$

The saturation exponent ${\displaystyle n}$ usually is fixed to values close to 2. The saturation exponent models the dependency on the presence of non-conductive fluid (hydrocarbons) in the pore-space, and is related to the wettability of the rock. Water-wet rocks will, for low water saturation values, maintain a continuous film along the pore walls making the rock conductive. Oil-wet rocks will have discontinuous droplets of water within the pore space, making the rock less conductive.

### Tortuosity factor, ${\displaystyle a}$

The constant ${\displaystyle a}$, called the tortuosity factor, cementation intercept, lithology factor or, lithology coefficient is sometimes used. It is meant to correct for variation in compaction, pore structure and grain size.[2] The parameter a is called the tortuosity factor and clearly is related to the path length of the current flow. The value lies in the range 0.5 to 1.5. a may be different in different reservoirs. However a typical value to start with for a sandstone reservoir might be 0.6, which then can be tuned during log data matching process with other sources of data such as core.

### Measuring the exponents

In petrophysics, the only reliable source for the numerical value of both exponents is experiments on sand plugs from cored wells. The brine conductivity can be measured directly on produced water samples. Alternatively, the brine conductivity and the cementation exponent can also be inferred from downhole electrical conductivity measurements across brine-saturated intervals. For brine-saturated intervals (${\displaystyle S_{w}=1}$) Archie's law can be written

${\displaystyle \log {C_{t}}=\log {C_{w}}+m\log {\phi }\,\!}$

Hence, plotting the logarithm of the measured in-situ electrical conductivity against the logarithm of the measured in-situ porosity (Pickett plot), according to Archie's law a straight-line relationship is expected with slope equal to the cementation exponent ${\displaystyle m}$ and intercept equal to the logarithm of the in-situ brine conductivity.

## Sands with clay/shaly sands

Archie's law postulates that the rock matrix is non-conductive. For sandstone with clay minerals, this assumption is no longer true in general, due to the clay's structure and cation exchange capacity. The Waxman–Smits equation[3] is one model that tries to correct for this.