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:
THE SATURATION EVALUATION
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 (ϕt )m . In the model it is shown that log10Swt is the length of the projection along the X - axis between log10ϕt and the intercept of slope n with log10Rt . This length also is represented by length CD 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 (2b)
where Ft = 1.0 / (Swtϕt ) m2 (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 , therefore, Eq.(4a) resolves to
Swtn = ( ( 1.0 / ( ϕt m1 ) ) Rwe ) / Rt measured = R0 corrected / Rt measured (4b)
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
For Eq.(4b), the equivalent water resistivity Rwe has been evaluated by Eq.(1b)after substitution for Sweϕe has been made from the volumetric material balance equation for water:
Swtϕt = Sweϕe + ( 1.0 ) ϕne
For the calculation of R0 corrected , Ft was evaluated from Eq.(3a), where Swt = 1.0. 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 )
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 / (ϕt m1 ) because Swt always is 1.0 for this determination. 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 where hydrocarbon saturation is zero. Once the Rwz trend has been established 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.
In an interactive computer program, an equation can be derived for the Rwz trend in the plot, for the evaluation of Rwz in terms of clayiness. A curve for Rwe from Eq.(1a) can be superimposed on the same plot for comparison and analysis. If the analyst is not satisfied with the values for Rw , Rwb , and Rwz , or if Rw from the plot does not agree with an accepted value, the analyst can employ the iteration subroutine as described above for m to vary the value of Rw from the plot until there is agreement. This is another means for the evaluation of Rwe in terms of calculated clayiness. The resulting Rw and Rwb can be used in Eq.(1a) in the calculation of Rwe in terms of porosities.
It is interesting to note, however, that from Figure 5 alone, once an acceptable Rwz trend has been established
Swtn = R0 corrected / Rt = ( Ft Rwz ) / ( Ft Rwa ) = Rwz / Rwa (4c)
It can be seen here that Swtn can be estimated independently of porosity, m , even Rw , in terms of clayiness. 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 Eq.(4b), once exponent n has been established, Swt can be calculated directly from Eq.(4c)and there is no need for the iteration process described above or as seen in Figure 4.
A CLARIFYING CONCEPT OF ARCHIE'S RESISTIVITY RELATIONSHIPS AND PARAMETERS.
A MODEL AND DISCUSSION
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