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:


The model in Figure 1 is a diagram showing that Rt is a function of both the volume of water Swtϕt 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 ( Swt )n t )m . In the model it is shown that logSwt is the length of the projection along the X - axis between logϕt and the intercept of slope n with logRt . 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 Rwe equation, then Eq.(1b) can be used only with Eq.(3b). Therefore, when Swe ( or Swt ) is less than 100%, the product resulting from Eq.(1b) and (3b) is

                         Rt  calculated = Ft Rwe                                                                                                  same as  (2b)

where             Ft = 1.0 / (Swtϕt ) m2                                                                                                        from (3b)

After combining Eq.(2b)and (3b) when either the measured or actual Rt is substituted for the calculated Rt , then

                        Rt  measured = (1.0 / ( ( Swtϕt ) m2 ) ) aRw = ( 1.0 / ( ( Swtϕt ) m2 ) ) Rwe          . . . (4a)

However, it was illustrated in Figure 1 in triangle ACG, and in APPENDIX (B) (3) , that ( Swtϕt )m2 is equivalent to (Swt )n ( ϕt )m1 , as seen in the equivalence equation (3c) therefore, Eq.(4a) resolves to

                        Swtn = ( 1.0 / ( ϕt m1 ) ) ( Rwe / Rt  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

                   Swtm2 = ( 1.0 / ( ϕt m2 ) ) ( Rwe / Rt  measured )                                                       . . .  (4d)

Only one Rwe equation can be used in this calculation. It will be from Eq.(1a) or from Eq.(1b).  At Swt = 1.0, the calculations for R0 are the same in either equation if the water mixture is the same. At Swt < 1.0 only Eq.(1b) can be used for calculating Rt because it is the only equation that allows the calculated Rwe to reflect the changing proportions of Rw and Rwb 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 R0  corrected must be determined from the same water mixture proportions and water geometry that exist at each Rt measurement. These proportions are reflected only in the conductivities shown in Eq.(1b). The efficiencies are reflected in the exponent residing in Ft , and the Ft 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 Sweϕe has been made from the volumetric material balance equation for water,

Swtϕt = Sweϕe + (1.0) ϕne .

After the substitution, Eq.(1b) becomes

1 / Rwe = 1 / Rw + ne / ( Swtϕt ) ) ( 1 / Rwb - 1 / Rw ).       simplified Eq. (1b)

This version of Rwe is used in Eq.(4b).

The Rt  measured in Eq.(4b) must be corrected for environmental conditions and tool-measurement characteristics before water saturation is calculated.

The term Swt in the formation-factor relationship of Eq.(3b) is the key element in the dual-water dual-porosity relationship. The term Swt 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 Rwe . When Rwe 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 Rt with a laboratory value of irreducible water saturation fixes the upper limiting value of saturation exponent n for that specific set of data. However, when irreducible water saturation is known, this upper limit of exponent n should be calculated from the algebraics of Eq.(4b), or Eq.(4c) as will be shown below. The same can be said for the lower limit of n in the same rock which could be estimated by inserting water saturation when oil saturation is irreducible. But, in either exercise, no value of n can be lower than m1. An Rt value is required for each of these procedures, whether derived from the well-log or rock sample.

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 Swt also appears in the proportionality terms within the Rwe equation, Eq.(1b), an algebraic solution for Swt 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 Swt . As Swt is varied, the individual curves for the left and right sides of Eq.(4b) will converge and cross at the Swt 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                    Rw = 0.30

    n = 2.92                  Rwb = 0.08

    ϕt = 0.22                        Rt = 20.00

                                                              ϕne = 0.09

In Figure 4 it can be observed that when values from each side of Eq.(4b) are plotted versus Swt the two curves have a common value at a water saturation of about 0.485. The iteration by subroutine will produce the same Swt 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 Sweϕe goes to zero the absolute minimum value for Swtϕt in this example is 0.09, the value of ϕne . The mathematical minimum water saturation Swt that can exist in this hypothetical reservoir is ϕne / ϕt  =  0.4091 or 40.91% where Rwb becomes 0.08. The minimum saturation of 40.91% is related only to the pseudo bound water in clay shale and tells us nothing about irreducible water saturation in the effective porosity. If this were an actual case in a water-wet sand, at 48.5% water saturation, water-free oil might be produced because the only water in the effective pore space might be irreducible. Grain size and surface area would be a consideration. Any water saturation below 40.91% cannot exist and is imaginary.

For the conversion of Swt to Swe , 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 - Swt ) ϕt = ( 1.0 - Swe ) ϕe

For the calculation of Swe the balance can be re-arranged to read:

                                                    Swe = 1.0 - ( ϕt / ϕe ) ( 1.0 - Swt )                                             . . .  (6)

and Swe 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 Rwe , and Swt in turn, both Rw and Rwb must be known. In the event neither Rw nor Rwb is known, these values most often can be estimated by interactive computer graphics from a crossplot of Rwa ( or Cwa ) versus Clayiness (% clay) as shown in Figure 5 , from Ransom (1995), where clayiness is estimated by appropriate clay-shale indicators. Rwa is determined by dividing Rt by Ft where Ft = 1.0 / ( Swtϕt ) m2 and Ft has resolved to 1.0 / t m1 ) because Swt always is 1.0 for the determination or Rw and Rwb . The value for exponent m1 should be the same value of m that will be used for the final interpretation. The value of Rt may be taken from wireline tools or measured while drilling. As pointed out earlier, the measured Rt should be corrected for environmental conditions and measuring-tool characteristics. In a figure such as Figure 5 , Rwa values from zones known to be or believed to be 100% water-filled often describe a trend or curve identified as an Rwz trend or curve identifiedas an Rwz trend where hydrocarbon saturation is zero. This trend can take virtually any curvature, steep, convex, concave, or flat, depending on the layering and isotropical resistivity relationship between Rw and Rwb . If the clay minerals within the reservoir bed are detrital, they almost always will have similar electrical properties to the surrounding clay shales. However, the clays found within a reservoir bed can be different from the clays surrounding the bed, particularly if the clays within the bed are authigenic, i.e. detrital minerals that have dissolved and re-formed in place as pore-lining crystals. If the difference in electrical properties is measurable, a dog-leg can appear in the described trend. The trend, however, is important for saturation analysis only throughout the reservoir beds. It is important that the electrical behavior and clay-indicator behavior of clays be consistent and repeatable. Once the Rwz trend has been established within the reservoir beds it can be extrapolated to 0% clay for Rw , and extrapolated to 100% clay for Rwb at in situ conditions. These values of Rw , Rwb , and Rwz have been estimated from preliminary formation factor and clay indicator information and are subject to examination by the analyst.

Although, in Figure 5, the end point of the Rwz trend for Rwz 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 Rw and Rwb determined from Figure 5 can be used in Eq.(1b) because they are compatible with the corrected Rt 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 Rwz values, or simulated Rwe values, found between Rw and Rwb are similar to those calculated from Eq. (1a). Be that as it may, Eq. (1b) becomes Eq. (1a) when Swe = Swt = 1.0 at all points along the Rwz trend. The instant that oil or gas becomes present, Swe and Swt become less than 1.0, and Eq. (1a) becomes Eq. (1b). Hydrocarbon saturations decrease the volume of water with resistivity Rw , and influence Rwe by making its value move closer to the resistivity value of Rwb . See Figure 2. If Rwe were to plot, it would produce a departure in the vertical axis from the Rwz trend, either higher or lower depending on whether Rwb is higher or lower than Rw . But, it will not plot as Rwe from Eq. (1b) because it does not exist in nature at 100% water saturation, so there can be no natural data. Rwe from Eq. (1b) becomes another Rwa the instant Swe and Swt become less than 1.0. The Rwe 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 Rw and Rwb the difference between the Rwz and the calculated Rwe increases. See APPENDIX (A) for further explanation.

It is interesting to note, however, that from Figure 5 alone, once an acceptable Rwz trend has been established, that generalizes Rwe , estimated water saturation can be previewed by

Swtn = R0  corrected / Rt = ( Ft Rwz ) / ( Ft Rwa ) = Rwz / Rwa                   . . . (4 c)

It can be seen here that the previewed value of Swtn can be estimated independently of porosity, m , even Rw , and Rwb , 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, Swt 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.



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