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:**

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

**A MODEL AND DISCUSSION**

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**WHAT IS MEANT BY THE PLOT OF R_{t} VERSUS S_{wt}ϕ_{t}**

In **Figure 1**, the Y-axis is *R*_{t}, the resistivity of rock. The X-axis is *S*_{wt}*ϕ*_{t}_{ }, bulk volume water. Bulk volume water is made up of two parts, *ϕ*_{t} and *S*_{wt}_{ }, which when added together on logarithmic scales, become *S*_{wt}*ϕ*_{t}_{ }. On the X-axis, bulk volume water is dimensional and has resistivity. Kindly refer to the **PREFACE** at the beginning of the **APPENDIX** for further explanation.

In **Figure 1**, again, the origin of the diagram is represented by *R*_{we} at 100% porosity, but *R*_{we} is one of the products of **Figure 2**. **Figure 2** is a detailed view of an interior part of Figure 1 showing how *R*_{w}* _{ }* in the presence of dual water becomes

In **Figure 1**, *R*_{t} is seen plotted on a log-log plot versus the volume *S*_{wt}*ϕ*_{t}_{ }, of which *ϕ*_{t}_{ } is a part. In resistivity well-log interpretation, the *m* or *n* slope in every case represents only the slope between two points: the value of the resistivity of the equivalent water, and the value of the resistivity of the total rock volume filled with the same water in whatever fraction and physical distribution. This fractional volume can be *ϕ*_{t} when *S*_{wt} = 1.0 and slope *m* = *m*_{1 }, or it can be *S*_{wt}*ϕ*_{t} when *S*_{wt} ≤ 1.0 and slope *m* = *m*_{2}* _{ }*. Slope

The slopes representing values of *m* or *n* or *m*_{2} are trigonometric tangential ratios of the side opposite (on Y-axis) over the side adjacent (on X-axis) of any right triangle in the figure. Exponents *m* and *n* and *m*_{2} represent rates of change in resistivity *R*_{t} relative to changes in water volume *ϕ*_{t} or saturation *S*_{wt}* _{ }*, and reflect the efficiency or inefficiency of that water volume to conduct electrical-survey current. In the following examples, it will be shown how each exponent

It can be seen from the triangle CDG where *n* is the tangent that slope *n* is equal to

*n* = ( log*R*_{t} -log*R*_{0 corrected} ) / ( log1 - log*S*_{wt} )

*-n* ( log*S*_{wt} ) = ( log*R*_{t} - log*R*_{0 corrected} )

Therefore, ( *S*_{wt} ) * ^{n}* =

This is the derivation of Archie’s dual-water equation that uses the two exponents, *m* and *n*.

Also, it can be seen in the triangle AEG where *m*_{2} is the tangent

* m*_{2} = log*F*_{t }* _{ }*/ ( log1 - log (

* -m*_{2 }( log (*S*_{wt}*ϕ*_{t}_{ }) ) = log*F*_{t}

* F*_{t} = 1.0 / (*S*_{wt}*ϕ*_{t}_{ })^{ }^{m}2 . . . (3b)

Please note the single exponent for the bulk volume of water. The formation factor, *F*_{t}, as it is derived here from Figure 1, applies to all values of *S*_{w}_{t}_{ }≤ 1.0 and is represented on the resistivity axis as the difference between the logarithms of resistivity of the equivalent water ( *R*_{we}_{ )} in the rock and the resistivity of the rock containing that water ( *R*_{t }) . This difference is expressed as ( log*R*_{t} - log*R*_{we} ), or *R*_{t}_{ }/ *R*_{we} . So, when the equivalent *R*_{t}_{ }/ *R*_{we}_{ }is substituted for *F _{t }*, we have

* R*_{t} / *R*_{we} = 1.0 / ( *S*_{wt}*ϕ*_{t} )^{m}2

or ( *S*_{wt}*ϕ*_{t} )^{m}2 = *R*_{we }/ *R*_{t}

and further resolved becomes

*S*_{wt}^{m}2 = ( 1 / *ϕ*_{t} ^{m}2 ) = ( *R*_{we} / *R*_{t} ) . . . (4d)

This is a water saturation equation derived from triangle AEG that uses a single exponent that is equivalent to Archie’s equation in Eq.(4b) above that uses two exponents. The commonality of the exponents can be observed. Furthermore, it can be seen that one of the terms in the equation is *R*_{we} instead of *R*_{0 corrected} . Both *R*_{we} and *R*_{0 corrected} are dual-water dual-porosity derivatives and signify that the water saturation equations (4b) and(4d) are dual-water dual-porosity equations. Equation (4d) is an alternative equation to Archie’s two-exponent equation, Eq. (4b). In Equation (4d), bulk volume water can be, and is, evaluated by core analysis, and sometimes under in situ conditions by downhole well-logging instruments. But it might not be necessary to evaluate *S*_{wt}*ϕ*_{t}_{ } as a combined term. Water saturation *S*_{wt} might be determined indirectly as seen in Eq.(4d) by using well logs recorded from resistivity- and porosity-measuring downhole instruments. Exponent *m*_{2} is evaluated from triangle AEG, similarly to the measurement of exponent *m* or *n*. The value of *m*_{2} will reside between the slopes representing values of exponents *m* and *n* . The fact that neither exponent *m* nor exponent *n* is required or used, in the evaluation of water saturation by the single-exponent method is no small matter.

Archie’s saturation relationship is a straightforward derivation from the model, but is improved for use in dual-water dual-porosity methodology by the correction of *R*_{w} to *R*_{we} and *R*_{0 }to *R*_{0 corrected} . Both corrections can be observed in graphic form in **Figure 2**.

See the **APPENDIX (B)** for a more detailed explanation of Figures 1 and 2 and the derivations of *m* and *n*, and *F*_{t} and *S*_{wt }.

In addition, in an examination of the line slopes in Figure 1, it can be seen that ( *S*_{wt}*ϕ*_{t ) }^{m}2 has the same function as, and is equivalent to, ( *S*_{wt} ) ^{n}*(ϕ*_{t} ) * ^{m}*1 . It is important to remember this in the developments that follow. For proof of this equality, see the