Robert C. Ransom

What are Archie’s Basic Relationships

What is Meant by the Plot of *R*_{t} versus *S*_{wt}*ϕ*_{t}

Parallel Resistivity Equations Used in Resistivity Interpretation

What is the Formation Resistivity Factor

How is Exponent *n* Related to Exponent *m*

Observations and Conclusions from Figure 10 about Exponent *n*

Are There Limitations to Archie's Relationships Developed in this Model?

**Table of Retrievable Contents:**

**THE a COEFFICIENT**

Historically, the *a* coefficient always has appeared in the numerator of the modified Formation Resistivity Factor, and has been perpetuated in industry with no defined purpose. In this model there is no support for the appearance of an *a* with a constant value in the formation factor. The formation factor is intrinsic to the rock at 100% water saturation. The *a* coefficient is not an intrinsic property, but is dependent on variables not related to the electrically inert rock or the physical geometries of its pores. The *a* is related neither to volume nor shape imparted to the network of electrically-conductive water paths. The *a* coefficient contributes nothing to the efficiency of the conversion of water resistivity to resistance, or vice versa. In the model, the shift in resistivity corresponding to *a* always is found in the resistivity axis, and the shift varies from data to data with depth. Coefficient *a* emerges as an inherent and inseparable factor of the complex resistivity relationship, as in Eqs.(1a) and (1b), and is related to the proportions of *all* secondary electrically-conductive constituents and influences that vary the relative proportions of *R*_{w} and *R*_{wb }.

It can be seen in the graphics of **Figure 2 ** that *R*_{we} is the product of *a* and *R*_{w }, therefore *a**R*_{w}= *R*_{we }. *R*_{we} varies from depth to depth, and so does coefficient* a*. It can be seen in this relationship that a varying *a* coefficient technically can be used in single-water single-porosity methods where *R*_{we} is not calculated. But, in dual-water dual-porosity methods where *R*_{we } is calculated, the *a* coefficient cannot be used as an independent parameter. If the *a* is used in single-water single-porosity methods, it must be calculated properly. And that is difficult to do when the correct value of *a* varies with rock constituents and depth, and only one water and one porosity is available to work with. That is one of the reasons why dual-water dual-porosity methods were conceived.

When *a* is associated with *R*_{w }, as in the equation developments of (1a) and (1b), and as a multiplier or reduction factor for *R*_{w }, the *a* coefficient more readily can be recognized as being an indispensable factor that accounts for those heterogeneities that produce additional electrical conductivity. Coefficient *a* is the composite factor that relates *R*_{w} to the resistivity value of the combination of *R*_{w} and *R*_{wb} , a*nd all other natural electrically-conductive influences*, in such proportions to result in *R*_{we} .

In the modified formation factor used in single-water single-porosity methods, the *a* coefficient appeared in the numerator. In the modified formation factor the *a* became a multiplier of the *R*_{w } in Archie’s saturation equation. The *aR*_{w } became a single-water single-porosity equivalent to *R*_{we} .

Secondary conductive influences, inherent to the rock, that produce coefficient *a* can be: clay shale, surface conductance, or solid semi- conductors such as pyrite and siderite as separate influences or in concert. One influence that is neither electrically conductive nor intrinsic is hydrocarbon saturation. These in turn, and perhaps others, can produce a change in rock resistivity.

The occurrence of one additional electrically-conductive influence would add one more term to Eqs.(1a) and (1b), or their equivalent, and if there were no change in *S*_{wt}*ϕ*_{t }, the value of the compound coefficient *a* would be reduced. These influences, all, can change *R*_{w } to an apparent or pseudo value *R*_{we} with a corresponding change in *R*_{0} to *R*_{0}_{ corrected} at their respective porosities.

Technically, in dual-water dual-porosity methods, the coefficient *a* should be transposed from the modified Formation Resistivity Factor term to the water resistivity relationship. For example:

*R*_{0}_{ corrected} = *F*_{t }*R*_{we} = ( a / ( *ϕ*_{t}^{m} ) ) *R*_{w} = ( 1.0 / ( *ϕ*_{t}^{m} ) ) *aR*_{w} = ( 1.0 / ( *ϕ*_{t}^{m} ) ) *R*_{we}

The question might be asked, "What is the difference?" There are a number of reasons, four of which are:

1. The *a* coefficient is a required proportionality factor, related to all factors influencing the electrical conductivity of the rock, that convert resistivity *R*_{w} to *R*_{we }, and is calculated along with *R*_{we} at each depth increment.

2. To prevent duplication by the user in the correction for conductive influences. Duplication occurs when both *a* and *R*_{we} appear in the same water-saturation equation, or when *a* appears as an independent variable in dual-water dual-porosity methods. Any saturation relationship involving resistivity can employ either coefficient a or *R*_{we }, but not both.

3. To prevent the use of a constant artificial and unwarranted correction factor.

4. Although the formation factor is meaningless at 100% porosity, the value of *F*_{t } always must be 1.0 when both porosity and water saturation in the *F*_{t } equation are 1.0.

As explained above, and in Figure 2,

* * *a* = *R*_{we} / *R*_{w }* _{ }* (1c)

After substituting Eq.(1c) into Eq.(1b), a relationship is derived showing the proportionality terms in coefficient *a* for the usual shaly sand:

1 / *a* = ( *S*_{we}*ϕ*_{e} ) / ( *S*_{wt}*ϕ*_{t} ) + ( *ϕ*_{ne} / ( *S*_{wt}*ϕ*_{t} ) ) ( *R*_{w} / *R*_{wb} )

It can be seen here that coefficient *a* is a function of water saturation as well as porosities. See **APPENDIX (A)** for further explanation. This is the evaluation for the *a* coefficient appearing in **Figure 2**.

This relationship is shown for comparison purposes or informative purposes only. It is not to be calculated and used independently in dual-water dual-porosity methods because it already has been incorporated in the calculation of *R*_{we} as can be seen in Eq.(1b).

**A CLARIFYING CONCEPT OF ARCHIE'S RESISTIVITY RELATIONSHIPS AND PARAMETERS.**

**A MODEL AND DISCUSSION**

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