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 SATURATION EVALUATION**

The model in Figure 1 is a diagram showing that *R*_{t} is a function of both the volume of water *S*_{wt}*ϕ*_{t} and the inefficiency with which electrical current passes through that water. The inefficiency of the electrical current flow is related to the distribution of the water and the interference to that flow within the water network reflected in the exponents *m* and *n* of the expression ( *S*_{wt} )^{n} *(ϕ*_{t })^{m} . In the model it is shown that log*S*_{wt} is the length of the projection along the *X* - axis between log*ϕ*_{t } and the intercept of slope *n* with log*R*_{t }. This length also is represented by length CG of triangle CDG in **Figure 1**.

Revisiting Eq.(1b), (2b), and (3b), the reader already might have deduced that water saturations can be estimated from these equations. Keeping faithful to the self-evident truth that the volume of water referred to in the denominator of the formation factor must be the same volume of water that provides electrical conductivity in the *R*_{we} equation, then Eq.(1b) can be used only with Eq.(3b). Therefore, when *S*_{we} ( or *S*_{wt }) is less than 100%, the product resulting from Eq.(1b) and (3b) is

* **R*_{t}_{ }_{calculated} = *F*_{t }*R*_{we} same as (2b)

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

After combining Eq.(2b)and (3b) when either the measured or *actual* *R _{t }* is substituted for the

* * *R*_{t measured} = (1.0 / ( ( *S*_{wt}*ϕ*_{t} ) ^{m}2 ) ) *aR*_{w} = ( 1.0 / ( ( *S*_{wt}*ϕ*_{t} ) ^{m}2 ) ) *R*_{we} . . . (4a)

However, it was illustrated in **Figure 1** in triangle ACG, and in **APPENDIX (B) (3) **, that ( *S*_{wt}*ϕ*_{t} )^{m}2 is equivalent to *(**S*_{wt} )^{n} ( *ϕ*_{t} )^{m}1 , as seen in the equivalence equation (3c) therefore, Eq.(4a) resolves to

*S*_{wt}^{n} = ( 1.0 / (* ϕ*_{t} ^{m}1 ) ) ( *R*_{we} / *R*_{t measured} ) . . . (4b)

Archie's dual-water dual-porosity equation.

It was passed over quickly that Equation (4a) also yields Archie's equivalent single exponent version seen here

* S*_{wt}^{m}2 = ( 1.0 / ( *ϕ*_{t} ^{m}2 ) ) ( *R*_{we} / *R*_{t measured} ) . . . (4d)

Only one *R*_{we} equation can be used in this calculation. It will be from Eq.(1a) or from Eq.(1b). At *S*_{wt} = 1.0, the calculations for* R*_{0} are the same in either equation if the water mixture is the same. At *S*_{wt} < 1.0 only Eq.(1b) can be used for calculating *R*_{t} because it is the only equation that allows the calculated *R*_{we} to reflect the changing proportions of *R*_{w } and *R*_{wb} resulting from the displacement of interstitial water volume by oil or gas. In Eq.(1a) the proportions of the two waters are fixed by the mineral constituents of the rock. But, the relative proportions of the two waters and their electrical efficiencies also change with the change in saturation and distribution of oil and gas, and these change depth by depth. The *R*_{0} *corrected* must be determined from the same water mixture proportions and water geometry that exist at each *R*_{t} measurement. These proportions are reflected only in the conductivities shown in Eq.(1b). The efficiencies are reflected in the exponent residing in *F*_{t }, and the *F*_{t } used must be compatible with each variation in water saturation, and that is Eq.(3b).

For Eq.(4b), Eq.(1b)can be simplified after the substitution for *S*_{we}*ϕ*_{e} has been made from the volumetric material balance equation for water,

*S*_{wt}*ϕ*_{t} = *S*_{we}*ϕ*_{e} + (1.0) *ϕ*_{ne} .

After the substitution, Eq.(1b) becomes

1 / *R*_{we} = 1 / *R*_{w} + *(ϕ*_{ne } / ( *S*_{wt}*ϕ*_{t} ) ) ( 1 / *R*_{wb} - 1 / *R*_{w} ). simplified Eq. (1b)

This version of *R*_{we} is used in Eq.(4b).

The *R*_{t measured} in Eq.(4b) must be corrected for environmental conditions and tool-measurement characteristics before water saturation is calculated.

The term *S*_{wt} in the formation-factor relationship of Eq.(3b) is the key element in the dual-water dual-porosity relationship. The term* **S*_{wt} is an inherent part of the formation factor derived from the model in Figure 1.

It can be seen in Eq.(1b) and (4a) that the *a* coefficient is variable with depth and is included as part of *R*_{we }. When *R*_{we} has been calculated, and is used, the appearance of a constant *a* coefficient in the formation factor, usually as a fraction less than 1.0, would be gratuitous and would artificially increase the calculated hydrocarbon saturation in productive and nonproductive zones alike; and, in this model, would be both logically and mathematically incorrect.

Figure 1, together with Figure 2, is a concept model that has significant informative and educational value. The graphics of the model are meant primarily to illustrate, to develop, or to explain what is calculated blindly by algebraics in computer-program subroutines.

In an interactive computer program irreducible water saturation or other core-derived information can be input for the purposes of examining the plausibility, validity, and integrity of certain parameters. On a well log above the transition zone in oil-bearing reservoir rock, for example, the intersection of *R _{t}* with a laboratory value of irreducible water saturation fixes the upper limiting value of saturation exponent

The water saturation Eq.(4b) has been developed from the trigonometric model in Figure 1 and again corroborated by the algebraic development of Eq.(4b), all, for certain heterogeneous, but uniform, environments. And, each development herein shows that it authenticates Archie's basic relationships presented in 1942, and further refines these relationships in the developments and discussions.

It has been said that Archie's relationships are empirical developments. Whether or not this is true, it has been shown here that Archie's classic relationships and parameters have a mathematical basis, and have forthright and substantive relevance to rock properties that is quite different from many accepted theories and usages in industry literature.

Saturation exponent *n* is the most difficult of all the parameters to evaluate. If a valid value of oil saturation is known, or can be derived, the value of exponent *n* can be estimated by substitution in Eq.(4b) or (4c). When the actual values of *m* and *n* are known, or can be derived, either or both can be important mappable parameters, and the mathematical difference between *m* and *n* not only can be an important mappable parameter, but can be a possible indicator to the degree of wettability to oil or distribution of oil under in situ conditions. This information not only can be important in resistivity log analysis but can be important in the design of recovery operations.

The calculation of Eq.(1b) is required in the solution of Eq.(4b). Because water saturation* S*_{wt} also appears in the proportionality terms within the *R*_{we } equation, Eq.(1b), an algebraic solution for *S*_{wt} in Eq.(4b) is not viable and is not considered. Probably the simplest and best method for all anticipated integer and noninteger values of *n* is an iterative solution performed by a computer-program subroutine. Graphically the iteration process can be demonstrated by a system of coordinates where both sides of Eq.(4b) are plotted versus input values of *S*_{wt }. As *S*_{wt} is varied, the individual curves for the left and right sides of Eq.(4b) will converge and cross at the *S*_{wt} value that will satisfy the equation.

**Figure 4** shows a crossplot of example data to demonstrate the equivalence of the graphical solution to the iterations performed by a computer-program subroutine. The following input values are for illustration purposes only.

*m* = 2.17 *R*_{w} = 0.30

*n* = 2.92 *R*_{wb} = 0.08

*ϕ*_{t} = 0.22 *R*_{t} = 20.00

*ϕ*_{ne} = 0.09

In **Figure 4** it can be observed that when values from each side of Eq.(4b) are plotted versus *S*_{wt} the two curves have a common value at a water saturation of about 0.485. The iteration by subroutine will produce the same *S*_{wt} of about 0.485 or 48.5% for the same basic input data.

It is worthy of note that in the volumetric material balance for water, when *S*_{we}*ϕ*_{e} goes to zero the absolute minimum value for *S*_{wt}*ϕ*_{t} in this example is 0.09, the value of *ϕ _{ne }*. The mathematical minimum water saturation

For the conversion of *S*_{wt} to *S*_{we }, either the material balance for water (shown above) or the material balance for hydrocarbon (Ransom, 1995) can be used. In terms of hydrocarbon fractions, the material balance for the amount of hydrocarbon in one unit volume of rock is:

( 1.0 - *S*_{wt} ) *ϕ*_{t} = ( 1.0 - *S*_{we} ) *ϕ*_{e}

For the calculation of *S*_{we} the balance can be re-arranged to read:

* **S*_{we} = 1.0 - ( *ϕ*_{t} / *ϕ*_{e} ) ( 1.0 - *S*_{wt} ) . . . (6)

and *S*_{we} now can be estimated.

When the material balance equation for hydrocarbon is multiplied by the true vertical thickness of the hydrocarbon-bearing layer, either side of this equation produces the volume of hydrocarbon per unit area at in situ conditions of temperature and pressure.

Finally, for the evaluation of *R*_{we }, and *S*_{wt} in turn, both *R*_{w} and *R*_{wb} must be known. In the event neither *R*_{w} nor *R*_{wb} is known, these values most often can be estimated by interactive computer graphics from a crossplot of *R*_{wa} ( or* C*_{wa }) versus *Clayiness* (% clay) as shown in **Figure 5 **, from Ransom (1995), where clayiness is estimated by appropriate clay-shale indicators. *R _{wa}* is determined by dividing

Although, in Figure 5, the end point of the *R*_{wz} trend for *R*_{wz} is said to be defined at 100% clayiness, 100% clayiness might not exist for the formation. But, when Figure 5 is used as a reconnaissance tool, it is not important to know the actual amount of clay present for the estimation of water saturations. The X-axis could be renamed Clay Index for this work. The relationships in the vertical axis will not change if the scale on the X-axis is changed. It is important, however, that the clayiness measurement methods and resulting clayiness estimations be consistent and repeatable.

The values of *R*_{w} and *R*_{wb} determined from Figure 5 can be used in Eq.(1b) because they are compatible with the corrected *R*_{t} from which they came. These values are compatible because they have been derived by the same method from measurements by the same resistivity measuring device at the same environmental conditions of temperature, pressure, and invasion profile at the same moment in time.

However, the *R*_{wz} values, or simulated *R*_{we} values, found between *R*_{w} and *R*_{wb} are similar to those calculated from Eq. (1a). Be that as it may, Eq. (1b) becomes Eq. (1a) when *S*_{we} = *S*_{wt} = 1.0 at all points along the *R*_{wz} trend. The instant that oil or gas becomes present, *S*_{we} and *S*_{wt} become less than 1.0, and Eq. (1a) becomes Eq. (1b). Hydrocarbon saturations decrease the volume of water with resistivity *R*_{w }, and influence *R*_{we} by making its value move closer to the resistivity value of *R*_{wb} . See **Figure 2**. If *R*_{we} were to plot, it would produce a departure in the vertical axis from the* **R*_{wz} trend, either higher or lower depending on whether *R*_{wb} is higher or lower than *R*_{w }. But, it will not plot as *R*_{we} from Eq. (1b) because it does not exist in nature at 100% water saturation, so there can be no natural data. *R*_{we} from Eq. (1b) becomes another *R*_{wa} the instant *S*_{we} and *S*_{wt} become less than 1.0. The *R*_{we} of Eq. (1b) exists only in mathematical form within a dual-water concept. If there is a significant difference between the end-point values of *R*_{w} and *R*_{wb} the difference between the *R*_{wz} and the calculated *R*_{we } increases. See **APPENDIX (A)** for further explanation.

It is interesting to note, however, that from **Figure 5** alone, once an acceptable *R*_{wz} trend has been established, that generalizes *R*_{we} , estimated water saturation can be previewed by

*S*_{wt}^{n}_{ } = *R*_{0} _{ corrected} / *R*_{t} = ( *F*_{t }*R*_{wz} ) / ( *F*_{t }*R*_{wa} ) = *R*_{wz} / *R*_{wa }* _{ }* . . . (4 c)

It can be seen here that the previewed value of *S*_{wt}^{n} can be estimated independently of porosity, *m *, even *R*_{w }, and *R*_{wb }, all in terms of an estimated input value for clayiness at each specific depth. The value of exponent *n* lies between its minimum value *of m*, corrected or as established above, and its maximum value determined from irreducible water saturation. Furthermore, unlike the more rigorous Eq.(4b), once exponent *n* has been established, *S*_{wt} can be previewed or estimated directly from Eq.(4c). A cautionary note appears in **APPENDIX (C)**.

Figure 5 has other uses than as a preview of water saturations in shaly sands. Figure 5 can be used as a reconnaissance tool to predict the abundance of organic matter or total organic carbon, or the occurrence of oil- or gas-deposits in shales and marlstones for further analysis. Also, it can be used as a means for selecting depths for taking additional measurements on available cuttings samples. It is one of the purposes of this plot to direct attention to zones of special interest for further investigation.

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

**A MODEL AND DISCUSSION**

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