Archie's law

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

C_t = C_w \phi^m S_w^n

Here, φ denotes the porosity, Ct the electrical conductivity of the fluid saturated rock, Cw represents the electrical conductivity of the brine, Sw is the brine saturation, m is the cementation exponent of the rock (usually in the range 1.8–2.0), and n is the saturation exponent (usually close to 2).

Reformulated for electrical resistivity, the equation reads

R_t = \frac{R_w}{\phi^m S_w^n}

with Rt for the fluid saturated rock resistivity, and Rw for the brine resistivity.

The factor 1 / φm is also called formation factor.

It is a purely empirical law attempting to describe ion flow (mostly sodium and chlorine) in clean, consolidated sands, with varying intergranular porosity. Electrical conduction is assumed not to be present within the rock grains.

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[edit] Cementation and saturation exponents

The exponent 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 < m < 2.0. Similarly, the saturation exponent n usually is fixed to values close to 2.

[edit] 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 Archie's equation can be written

\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 (a so-called Pickett plot), according to Archie's equation a straight-line relationship is expected with slope equal to the cementation exponent m\,\! and intercept equal to the logarithm of the in-situ brine conductivity.

[edit] Origin

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 1 − Sw).

[edit] References

  • Gus Archie: The Electrical Resistivity Log as an Aid in Determining Some Reservoir Characteristics (Transactions of AIME, 1942).
  • M.H.Rider, The Geological Interpretaion of Well logs (1986)