Archie

Summary

The Archie equation gives the Water saturation of a clean, water-wet reservoir rock from its True formation resistivity, the resistivity of its formation water (Formation water resistivity) and its Effective porosity. It is the base relation of resistivity-based saturation, and every shaly-sand model on this site reduces to it when the clay volume is zero. Use it for clean sandstones and for carbonates with simple pore systems.

Inputs and outputs

Equations

Archie's first relation links the Formation factor \(\Fform\) to porosity, and the second links the Resistivity index \(\Ires\) to water saturation:

Eliminating \(\RoWet\) and solving for water saturation gives the working form:

The result is limited to the interval 0 to 1. The hydrocarbon saturation is \(1 - \Sw\). Bulk volume water is \(\phie\,\Sw\).

Single-value calculator

Behavior

Water saturation falls as resistivity rises, and it falls as porosity rises at the same resistivity. With \(a = 1\), \(m = n = 2\) and \(R_w = 0.05\) ohm·m, a rock of 20% porosity has a water-saturated resistivity of 1.25 ohm·m, so a reading of 20 ohm·m gives \(S_w\) = 0.25. Halving the resistivity to 10 ohm·m raises it to 0.354, because with \(n = 2\) saturation goes as the inverse square root of resistivity. At the same 20 ohm·m, a 10% porosity gives 0.50 and a 30% porosity gives 0.17. Each curve is limited to 1 at resistivities below \(R_0\): 5.0, 1.25 and 0.56 ohm·m for the three porosities.

Parameter guidance

Rw must be at formation temperature and is the largest single source of error in a clean sand: see Rw Determination. a, m and n are in Cementation and Saturation Exponents. Without core data the usual starting point is \(a = 1\), \(m = 2\) and \(n = 2\) for clean sandstones, and the Humble form \(a = 0.62\), \(m = 2.15\) is an older sandstone alternative (use one or the other, never a mix). Porosity is Effective porosity from the Porosity step. In a clean rock total and effective porosity are the same. Where they differ, a clay-bound-water correction is needed and Archie is no longer the right model: see Total vs Effective Sw. The step page (Water Saturation) covers the choice between Archie and the shaly-sand models.

Worked example

A clean sand with \(R_t\) = 20 ohm·m, \(R_w\) = 0.05 ohm·m, 20% porosity and Archie's default constants, followed by the sensitivity to each constant:

a, m, n = 1.0, 2.0, 2.0
rw, phie, rt = 0.05, 0.20, 20.0

def sw_archie(rt, rw, phie, a=1.0, m=2.0, n=2.0):
    return min(1.0, (a * rw / (phie ** m * rt)) ** (1.0 / n))

f = a / phie ** m
ro = f * rw
i = rt / ro
sw = i ** (-1.0 / n)
print(f"F  = {a:g} / {phie:g}^{m:g} = {f:.2f}")
print(f"R0 = F x Rw = {f:.2f} x {rw:g} = {ro:.3f} ohm.m")
print(f"I  = Rt / R0 = {i:.1f}")
print(f"Sw = I^(-1/n) = {sw:.3f}   Sh = {1 - sw:.3f}   BVW = {phie * sw:.4f}")
print()
print("Sensitivity (one constant changed at a time):")
cases = [("m = 1.8", dict(m=1.8)), ("m = 2.2", dict(m=2.2)),
         ("n = 1.8", dict(n=1.8)), ("n = 2.2", dict(n=2.2)),
         ("Rw = 0.04", dict(rw=0.04)), ("Rw = 0.06", dict(rw=0.06)),
         ("a = 0.62, m = 2.15", dict(a=0.62, m=2.15))]
for label, kw in cases:
    args = dict(rt=rt, rw=rw, phie=phie, a=a, m=m, n=n)
    args.update(kw)
    print(f"{label:20s} Sw = {sw_archie(**args):.3f}")

Output

F  = 1 / 0.2^2 = 25.00
R0 = F x Rw = 25.00 x 0.05 = 1.250 ohm.m
I  = Rt / R0 = 16.0
Sw = I^(-1/n) = 0.250   Sh = 0.750   BVW = 0.0500

Sensitivity (one constant changed at a time):
m = 1.8              Sw = 0.213
m = 2.2              Sw = 0.294
n = 1.8              Sw = 0.214
n = 2.2              Sw = 0.284
Rw = 0.04            Sw = 0.224
Rw = 0.06            Sw = 0.274
a = 0.62, m = 2.15   Sw = 0.222

Assumptions and limitations

• The rock is clean: the only conductive path is the formation water in the pore space. Clay, pyrite and other conductive minerals add conductivity and make the computed Sw too high.
• The rock is water-wet, so the water forms a continuous film and the saturation exponent is constant. Oil-wet or mixed-wet rock has a higher and variable exponent.
• The formation water has a single, known resistivity at formation temperature, and it does not change across the interval.
• a, m and n are constants for the interval. Pore-system changes, especially in carbonates, break this.
• Porosity is the effective porosity and resistivity is a true, invasion- and shoulder-corrected value.

QC checks

• Sw is 1, or very close to 1, in a known water leg. If it is not, Rw, m or the porosity is wrong.
• In a hydrocarbon zone Sw is not below the irreducible water saturation expected for the rock quality.
• Rwa (apparent Rw, from the water-leg Pickett plot) gives the same Rw that was used. See the Rw Determination page.
• Sw does not track the clay volume. If a shale-affected zone reads lower Sw as Vcl rises, a shaly-sand model is needed.
• Sw compares with core Dean-Stark or capillary-pressure saturation at the same depths, at similar height above the free water level.

Going Deeper

The relation was published by Archie in 1942, from laboratory measurements on clean consolidated sandstones. The formation factor is a property of the rock alone, and the resistivity index is a property of the water saturation and its distribution. The exponents were treated as constants by Archie, and most later work is about when they are not: variable cementation with porosity, wettability effects on \(n\), and the extra conductivity of clay and of conductive minerals. The shaly-sand models on the following pages add terms to the same backbone, and each reduces to this equation when the clay volume is zero. Archie's equation remains the right answer for clean rock, and the default against which every other model is compared.

References

1. Archie, G.E., 1942. The electrical resistivity log as an aid in determining some reservoir characteristics. Transactions of the AIME, 146(1), 54–62.

Python reference implementation

Related methods

• Simandoux
• Modified Simandoux
• Indonesian
• Dual Water
• Waxman-Smits
• Lucia
• Rw Determination
• Cementation and Saturation Exponents