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:
WHAT IS MEANT BY THE PLOT OF Rt VERSUS Swtϕt
It must be recognized that ϕt and Swtϕt have no units, but are fractions and in themselves are not volumes. Only when ϕt or Swtϕt are multipliers of cross-sectional area (1.0 m2) or total volume (1.0 m3) do these fractions become dimensional. In this paper there will be applications where they will be fractional multipliers of both area and volume. In Figure 1, they are fractions of total volume.
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 becomes Rwe to become the origin of the diagram. The value Rwe , as an equivalent water resistivity, is determined from Eq.(1b) when Swt is 1.0. Within Figure 2, ϕe and ϕt exist simultaneously, and when they have different values, Rw becomes Rwe , and R0 has become R0 corrected .
In Figure 1, Rt is 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 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. 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 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 represent rates of change in resistivity Rt relative to water volume ϕt or Swtϕt , and reflect the efficiency or inefficiency of that water volume to conduct electrical-survey current. In the following examples, it is shown how each m and n is derived as trigonometric tangents and calculated from their respective right triangles, and what can be derived from each.
In the triangle AEG where m2 is the tangent
-m2 = log10 Ft / ( log10 (Swtϕt )
-m2 (log10 (Swtϕt ) ) = log10Ft
Ft = 1.0 / (Swtϕt ) m2 . . . (3b)
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 in the rock (Rwe ) and the resistivity of the rock containing that water ( Rt ) .
Also, it can be seen from the triangle CDG where n is the tangent that
-n = ( log10 Rt - log10 R0 corrected ) / log10 Swt
-n ( log10 Swt ) = ( log10 Rt - log10R0 corrected )
( Swt ) n = R0 corrected / Rt . . . (4b)
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 are from Equations (1a) and (2a), respectively, but 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 (ϕt )m1 . It is important to remember this in the developments that follow. For proof of this equality, see the APPENDIX (B) (3) .
The line representing either slope, m or n, does not represent a locus of points as either porosity or saturation changes, and there should be no reference to boundary or unlikely regions or conditions because these have nothing to do with the conductive efficiency of the specific rock sample of interest, whether in situ or in the laboratory. If the locus of points were shown, it might or might not represent a constant slope. The locus of data points for both m and n for several rock samples can be curvilinear and can represent a number of different slopes as the porosities, saturations, and proportions of minerals vary. The locus of points for either m or n depends entirely on the geometry of the pores and pore paths and the saturation distributions imposed on the conductive water volume by the nonconductive environment, whether that environment is electrically inert rock or hydrocarbon.
A CLARIFYING CONCEPT OF ARCHIE'S RESISTIVITY RELATIONSHIPS AND PARAMETERS.
A MODEL AND DISCUSSION
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