Robert C. Ransom
What are Archie’s Basic Relationships
What is Meant by the Plot of Rt versus Swtϕt
Parallel Resistivity Equations Used in Resistivity Interpretation
What is the Formation Resistivity Factor
Table of Retrievable Contents:
APPENDIX
The model appearing in Figure 1 incorporates Figure 2. The origin always is Rwe . But, Rwe is the same as Rw in clean rocks and different from Rw in heterogeneous rocks.
(A) Figure 2. In this figure are five variables relative to ϕe and ϕt . They are: Rw , R0 , Rwe , R0 corrected , and a . Input variables to the figure are Rw and ϕe and ϕt . Input value Rw is from the best known source or from Figure 5 after clayiness has been determined by the best clay indicators available. Porosity values, ϕe and ϕt , are determined by traditional methods, or from Ransom (1977, 1995), after employing the appropriate matrix values for the host rock. Variable R0 is from Eq.(2a) with Rw as input, Rwe is from Eq(1a), R0 corrected from Eq.(2a) with Rwe as input, all values are at 100% water saturation in this figure. Coefficient a is shown as a multiplier of Rw but is never calculated independently for use in an interpretation program when it is an integral part of Rwe , as seen in Eqs.(1a) and (1b). Coefficient a may be calculated for comparison purposes only.
(B) Figure 1. Figure 1 illustrates the model that Archie’s relationships obey. It is intended for informative and illustrative use, only. It is not drawn to scale. The inclinations of the straight lines representing slopes are determined by trigonometric tangents. The value of a specific trigonometric tangent is the value of the specific m or n. The steeper is the slope, the greater is the interference and resistance to electrical-current flow through the pores and pore paths in the rock, the greater is the inefficiency for the transmission of electrical-survey current, and the greater will be the value of m or n.
(1) On the X - axis, it is seen that Swtϕt decreases to the right. A study of the logarithmic scales will show, for example, that ϕt = 0.2 and Swt = 0.3 ; and, as a result, their product Swtϕt = 0.06.
(2) There are three right triangles of interest in Figure 1. They are triangles ABC, CDG, and AEG.
In any triangle shown in the figure, the slope or tangent of an angle is described as the side opposite divided by the side adjacent. The trigonometric law pertaining to tangents, that right triangles follow, can be described as:
tangent of acute angle = ( side opposite ) / ( side adjacent )
Because the X-axis is descending in value to the right of the origin, the sign of the tangent is negative. Therefore, on the log-log plot such as Figure 1,
( - ) (tangent of acute angle) ( log10 ( side adjacent ) ) = log10 ( side opposite )
and
1 / ( side adjacent ) tangent of acute angle = ( side opposite )
Trigonometry was created for solving problems. Express these trigonometric functions in equation form and the Formation Resistivity Factor and an improved Archie’s water saturation relationship will emerge.
In Figure 1, the tangent of the acute angle β of the right triangle AEG is represented by
tan β = -m2 = log10 ( EG ) / log10 ( AE )
-m2 ( log10 ( AE ) ) = log10 ( EG ) = log10 Ft
Ft = 1.0 / ( AE ) m2
In equation form, this is
-m2 = log10 Ft / ( log10 ( Swtϕt ) )
-m2 ( log10 ( Swtϕt ) ) = log10 Ft
Ft = 1.0 / ( Swtϕt )m2 . . .(3b)
In Figure 1, the tangent of the acute angle ϒ of the right triangle CDG is represented by
tan ϒ = -n = log10 ( DG ) / log10 ( CD )
-n ( log10 ( CD ) ) = log10 ( DG ) = ( log10 Rt - log10 R0 corrected )
( CD )n = R0 corrected / Rt
In equation form, this is
-n = ( log10 Rt - log10 R0 corrected ) / log10 Swt
-n ( log10 Swt ) = ( log10 Rt - log10 R0 corrected )
( Swt )n = R0 corrected / Rt . . .(4b)
(3) ( Swtϕt )m2 has the same function as and is equal to ( Swt ) n ( ϕt )m1 .
In triangle AEG, -m2 ( log10 ( Swtϕt ) ) = log10 Rt - log10 Rwe
( Swtϕt )m2 = Rwe / Rt
In triangle CDG, -n ( log10 Swt ) = log10 Rt - log10 R0 corrected
In triangle ABC, -m1 ( log10 ϕt ) = log10 R0 corrected - log10 Rwe
adding the two equations that involve n and m1 , yields
-n ( log10 Swt ) + ( -m1 ( log10 ϕt ) ) = log10 Rt - log10 Rwe
Swtnϕt m1 = Rwe / Rt
But, ( Swtϕt )m2 = Rwe / Rt , from above.
Therefore, ( Swtϕt ) m2 = Swtnϕt m1
(4) In addition, in partially oil-wet or oil-wet rocks, the assumption that n = m can lead to calculated water saturations that are too low. In Figure 1, it can be seen that when the default value of n equals m the line representing n will intercept the Rt level far to the right at H. The resulting calculated water saturation will be too low and “unreasonable”. When the corrected n value is used, the slope representing n will intercept the level of Rt at some point G, which will yield a “reasonable” value for Swt depending on the validity of n. Straight lines representing values of n can rotate along the arc δ as either n or Swt varies.
(C) Figure 5. This figure is a plot of Rwa vs Clayiness. It could have been a plot of Rt vs Clayiness where R0 and R0 corrected could have been estimated in the same manner as Rw and Rwz . The ratios R0 / Rt and Rwz / Rwa would still be the same. The plot of Rwa vs Clayiness yields more information.
Assuming that the resistivity values are reliable, the most critical variable in this plot is Clayiness, which should be determined from the best clay-shale-responsive indicators available.
A CLARIFYING CONCEPT OF ARCHIE'S RESISTIVITY RELATIONSHIPS AND PARAMETERS.
A MODEL AND DISCUSSION
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