Robert C. Ransom



What are Archie’s Basic Relationships

The Graphical Model

What is Meant by the Plot of Rt versus Swtϕt

Summary of Equations

Parallel Resistivity Equations Used in Resistivity Interpretation

What is the Formation Resistivity Factor

The m Exponents

How is Exponent n Related to Exponent m

The a Coefficient

The Saturation Evaluation

Challenging Well-Log Examples

Observations and Conclusions from Figure 10 about Exponent n

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




Symbols Defined



All Figures

About the Author

Table of Retrievable Contents:



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In Figure 1, the Y-axis is Rt, the resistivity of rock. The X-axis is Swtϕt , bulk volume water. Bulk volume water is made up of two parts, ϕt and Swt , which when added together on logarithmic scales, become Swtϕt . 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 Rwe at 100% porosity, but Rwe is one of the products of Figure 2. Figure 2 is a detailed view of an interior part of Figure 1 showing how Rw in the presence of dual water becomes Rwe to become the origin of the diagram. The value Rwe , as an equivalent water resistivity, is determined algebraically from Eq.(1b). The proportions of Rw and Rwb that become Rwe in the figure are a function of water saturation as well as clay shale content. Within Figure 2, ϕe and ϕt exist simultaneously, and no matter what are their values, the same or different, when Rw becomes Rwe , R0 becomes R0 corrected .

In Figure 1, Rt is seen plotted on a log-log plot versus the volume Swtϕt , of which ϕt 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 ϕt when Swt = 1.0 and slope m = m1 , or it can be Swtϕt when Swt ≤ 1.0 and slope m = m2 . Slope n represents the case where the two resistivity end points for the slope are the resistivity of a rock completely filled with the equivalent water, and the resistivity of the same rock and the same water after hydrocarbon has displaced some of the water. There is no extrapolation and no interpolation involved in the m or n evaluation when the two end points are known. The resulting slope, be it m or n, is a measure of the difficulty and interference electrical-survey current experiences as it is forced to flow through the water in the rock and represents a resistivity gradient. In Figure 1, the line representing the slope n rotates throughout the range of arc δ shown in the diagram as either or both n and Swt vary. The steeper the slope for either m or n, the more inefficient will be the water path in the rock for conducting electrical-survey current, the greater will be the values of m and n, and the greater will be the value of Rt . Each value applies only to the individual sample of interest, whether in situ or in the laboratory.

The slopes representing values of m or n or m2 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 m2 represent rates of change in resistivity Rt relative to changes in water volume ϕt or saturation Swt , 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 m, n and m2 is derived as trigonometric tangents and calculated from their respective right triangles, and what can be derived from each.

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

n = ( logRt -logR0 corrected ) / ( log1 - logSwt )

                                                    -n ( logSwt ) = ( logRt - logR0 corrected )

Therefore,                                ( Swt ) n = R0 corrected / Rt                                                      . . . (4b)

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 m2 is the tangent

                                                  m2 = logFt / ( log1 - log (Swtϕt ) )

                                                  -m2 ( log (Swtϕt ) ) = logFt

                                                   Ft = 1.0 / (Swtϕt ) m2                                                                    . . . (3b)

Please note the single exponent for the bulk volume of water. The formation factor, Ft, as it is derived here from Figure 1, applies to all values of Swt  ≤  1.0 and is represented on the resistivity axis as the difference between the logarithms of resistivity of the equivalent water ( Rwe ) in the rock and the resistivity of the rock containing that water ( Rt ) . This difference is expressed as ( logRt - logRwe ), or Rt / Rwe . So, when the equivalent Rt / Rwe is substituted for Ft , we have

                                                   Rt / Rwe = 1.0 / ( Swtϕt )m2

or                                               ( Swtϕt )m2 = Rwe / Rt

and further resolved becomes

                                                   Swtm2 = ( 1 / ϕt m2 ) = ( Rwe / Rt )                                       . . . (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 Rwe instead of R0 corrected . Both Rwe and R0 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 Swtϕt as a combined term. Water saturation Swt might be determined indirectly as seen in Eq.(4d) by using well logs recorded from resistivity- and porosity-measuring downhole instruments. Exponent m2 is evaluated from triangle AEG, similarly to the measurement of exponent m or n. The value of m2 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 Rw to Rwe and R0 to R0 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 Ft and Swt .

In addition, in an examination of the line slopes in Figure 1, it can be seen that ( Swtϕt ) m2  has the same function as, and is equivalent to, ( Swt ) n t ) m1 . It is important to remember this in the developments that follow. For proof of this equality, see the APPENDIX (B)  (3).

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