Simandoux

Summary

The Simandoux equation adds a clay conduction term to the Archie relation: the rock conducts through its formation water and, in parallel, through the clay. It is written here for n = 2, which makes it a quadratic in Water saturation with a closed-form solution that takes Clay volume and the clay resistivity (Clay resistivity). Use it for shaly sands where the clay volume is moderate.

Inputs and outputs

Equations

Simandoux treats the conductivity of the rock as the sum of a water term and a clay term, with \(n = 2\):

This is a quadratic in \(\Sw\). Its positive root is the working form:

The result is limited to the interval 0 to 1. With \(\Vcl = 0\) the clay term vanishes and the equation reduces to Archie with \(n = 2\).

Single-value calculator

Behavior

At a fixed resistivity, Simandoux water saturation falls as clay volume rises, and the gap to Archie widens. The dashed Archie line is flat because it ignores clay. At \(R_t\) = 10 ohm·m, \(R_w\) = 0.05 ohm·m, 18% porosity and a clay resistivity of 2.5 ohm·m, Archie gives 0.393 at every clay volume, while Simandoux gives 0.393, 0.336, 0.288 and 0.249 at clay volumes of 0, 0.2, 0.4 and 0.6. The correction is larger at higher resistivity: at \(R_t\) = 20 ohm·m the same clay volumes give 0.278, 0.223, 0.181 and 0.149 against an Archie value of 0.278. Lowering the clay resistivity from 5 to 1.5 ohm·m at a clay volume of 0.3 lowers Sw from 0.349 to 0.268, so the clay resistivity pick matters as much as the clay volume.

Parameter guidance

Clay resistivity Clay resistivity is the deep resistivity read in a thick, clean shale near the zone. Low values (a conductive, wet shale) increase the correction. Clay volume Clay volume comes from the Clay Volume step, and it must be the quantity the equation expects: a clay volume that includes silt (a shale volume) overcorrects. Porosity is effective porosity from the Porosity step. Rw, a and m are as for Archie: see Rw Determination and Cementation and Saturation Exponents. The saturation exponent is fixed at 2 by the closed form. If core shows an exponent clearly different from 2, use the Indonesian equation, which takes \(n\) as an input.

Worked example

A shaly sand with \(R_t\) = 10 ohm·m, \(R_w\) = 0.05 ohm·m, 18% effective porosity, 30% clay and a clay resistivity of 2.5 ohm·m. The block also solves the quadratic back into the original conductivity equation, and checks the clean-sand limit.

import math

def simandoux(rt, rw, phie, vcl, rcl, a=1.0, m=2.0):
    c = phie ** m / (a * rw)
    b = vcl / rcl
    return min(1.0, max(0.0, (math.sqrt(b * b + 4.0 * c / rt) - b) / (2.0 * c)))

def 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))

rt, rw, phie, vcl, rcl = 10.0, 0.05, 0.18, 0.30, 2.5
sw = simandoux(rt, rw, phie, vcl, rcl)
print(f"Archie    Sw = {archie(rt, rw, phie):.3f}")
print(f"Simandoux Sw = {sw:.3f}")

# the root must satisfy the original equation: 1/Rt = phie^m Sw^2/(a Rw) + Vcl Sw/Rcl
lhs = 1.0 / rt
rhs = phie ** 2 * sw ** 2 / rw + vcl * sw / rcl
print(f"1/Rt = {lhs:.5f}   conductivity sum = {rhs:.5f}")

# clean-sand limit: with Vcl = 0 the equation must equal Archie (n = 2)
print()
print("Clean-sand check (Vcl = 0):")
worst = 0.0
for rt_ in (2.0, 5.0, 20.0, 100.0):
    for phi_ in (0.08, 0.15, 0.25):
        d = abs(simandoux(rt_, rw, phi_, 0.0, rcl) - archie(rt_, rw, phi_))
        worst = max(worst, d)
print(f"largest |Simandoux - Archie| over 12 cases = {worst:.2e}")
assert worst < 1e-12

Output

Archie    Sw = 0.393
Simandoux Sw = 0.311
1/Rt = 0.10000   conductivity sum = 0.10000

Clean-sand check (Vcl = 0):
largest |Simandoux - Archie| over 12 cases = 2.78e-17

Assumptions and limitations

• The saturation exponent is 2. The quadratic form is exact only for \(n = 2\).
• Clay conducts in parallel with the formation water, and the clay conductivity is the same in every shale. A single, constant clay resistivity is assumed.
• The porosity is the effective porosity, so clay-bound water is not in it. Using total porosity double-counts the clay water.
• Clay volume is a clay volume, not a shale volume. Silt in a shale volume is not conductive in the way the clay term assumes.
• At high clay volume in fresh or moderately fresh water the equation gives lower Sw (more hydrocarbon) than the Indonesian equation in the same rock, and it is generally regarded as optimistic there. In saline water the models agree closely. See the step page.

QC checks

• At Vcl = 0 the result equals the Archie value with n = 2. The worked example runs this check.
• Sw is never above the Archie value, and falls monotonically as clay volume rises at fixed resistivity.
• Sw in a known water leg is close to 1, with the same Rw and clay resistivity picks as the hydrocarbon zone.
• Compare with the Indonesian result: a large disagreement at high clay volume means the overcorrection is dominating the answer.
• Computed hydrocarbon in the shaly intervals does not exceed what the core, capillary pressure or test data support.

Going Deeper

The relation was proposed by Simandoux in 1963 from laboratory measurements on artificial shaly sands made with a clay. The form of a water term plus a clay term in parallel became the template for most later shaly-sand equations: they differ in how the clay term is written and what it is weighted by. Simandoux's original had a clean-sand porosity term without a clay-volume weighting. A form with the weighting is the modified Simandoux equation. Quadratic forms in \(S_w\) are popular because they have a closed form for \(n = 2\) and need no iteration. The approach, however, has no basis for the clay conduction other than the laboratory fit, which is why the Waxman-Smits and dual-water models were developed.

References

1. Simandoux, P., 1963. Mesures diélectriques en milieu poreux, application à la mesure des saturations en eau: étude du comportement des massifs argileux. Revue de l'Institut Français du Pétrole, 18 (Supplementary Issue), 193–215.

Python reference implementation

Related methods

• Archie
• Modified Simandoux
• Indonesian
• Dual Water
• Waxman-Smits
• Total vs Effective Sw