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:
PARALLEL RESISTIVITY EQUATIONS USED IN RESISTIVITY INTERPRETATIONS
The basic premise is that the host rock is an insulator. Equations will be demonstrated for the case of shaly sand with a uniform distribution of constituents and pore space. Shale is a term used to refer to fine grained, fissile, sedimentary rock. The term shale is a descriptive property of rock, not a mineral, therefore, shale may be referred to as "clay shale" when the predominant mineral constituent is clay. The shale in this presentation is not a source for oil or kerogen, but is shale or mudstone that has been compacted by overburden to the degree that any form of porosity has become noneffective, and migrating oil cannot or has not penetrated the void space.
Connate water is water entrapped within the pores or spaces between the grains or particles of rock minerals, muds, and clays at the time of their deposition. The water is derived from sea water, meteoric water, or ground surface water. Other investigators have shown that as clay shales and mudstones are compacted, a fresh water component of the original connate water is expelled and an ion-concentrated component remains in the voids. Therefore, water saturation in the voids remains 100% and Sw is always 1.0 in ϕne .
The relationship in Eq.(1a) below, presented in Ransom (1977), was developed to show that conductivity in water-filled voids in the host rock and the conductivity in clay shale can be represented by parallel conductivity relationships commonly used in basic physics. The relationship applies to heterogeneous rocks with a uniform distribution of minerals and porosity. In resistivity and conductivity relationships the dimensional units for a reservoir bed (m/m2) must be made to be unity so that bed dimensions will not be a factor.
For shaly sand it was shown that
1 /Rwe = (ϕe /ϕt ) 1 / Rw + (ϕne /ϕt ) 1 / Rwb ( 1a )
where ϕt = ϕe + ϕne .
For 100% water-bearing rock the conductivity relationship in Eq.(1a) yields 1 / Rw (in clean rock) where ϕne is 0.0, and yields 1 / Rwe (in shaly sands) when conductive clay is present and ϕne is greater than 0.0.
It should be noted that in this equation there are no limitations on the values of formation water Rw and bound water in shale, Rwb . However, nature does set limits. The bound water resistivity Rwb in the mudstone and clay shale is related to the original connate water. Waters entrapped at the time of deposition do vary, but usually do not vary greatly from shale to shale. Sea water has an average salinity of about 35,000 ppm, other surface waters probably are significantly lower. However, as indicated above, the ion concentration in clay-bound water can increase as fresh water is expelled from the clays with increasing depth and compaction.
Unlike connate water, interstitial-water resistivity Rw in the reservoir can vary over a wide range. Interstitial water might have undergone many changes through the dissolution and/or precipitation of minerals throughout geologic history. It can vary from supersaturated salt solutions to very fresh potable waters. Waters in the Salina dolomite (Silurian) in Michigan have specific gravities of 1.458 and a reported salinity of 642,798 ppm. In other formations, the waters vary to as little as a few hundred parts per million. Most often Rw will be greater than Rwb ; and Rwe , a mixture of Rw and Rwb , will be lower than Rw . But, it is quite common for Rw to have a lower value than Rwb , in which case Rwe will have some value greater than Rw .
In water-filled dirty sands, Eq.(1a) applies. In clean sands, Eq.(1a) becomes
1 / Rwe = (ϕe /ϕt ) 1 / Rw = 1 / Rw
and the corrected R0 in either event is
R0 corrected = Ft Rwe (2a)
where, from Figure 1,
Ft = 1.0 / ( Swtϕt ) m2 = 1.0 / (ϕtm1 ) (3a)
The value of Swt is 1.0, therefore, the exponent m2 becomes m1 described above and in Figure 1.
However, in the presence of oil, water saturations are lower than 1.0 and the conductive water-filled volumes are reduced by the displacement volume of oil, and Eq.(1a) becomes
1 / Rwe = ( ( Sweϕe) / (Swtϕt ) ) 1 / Rw + (ϕne / (Swtϕt ) ) 1 / Rwb (1b)
and either R0 corrected = Ft Rwe (2a)
or Rt calculated = Ft Rwe (2b)
depending on the water saturation values in Eq.(1b),
where this time Ft = 1.0 / (Swtϕt ) m2 (3b)
and m is a form of m2 .
Equation (1b) converts the basic Archie method from a single-porosity single-water method to the dual-porosity dual-water methodology where the two waters have resistivities of Rw and pseudo Rwb , and Rw from the single-porosity single-water method becomes Rwe , as seen in Eq.(1b). The Rwb has been referred to as a pseudo Rwb because some of the conductivity attributed to the clay shale might not be in the form of water, but the usual resistivity-measuring logging tools do not know the difference.
In dual-porosity dual-water methodology, the numerator in Eq.(3b) always must be 1.0 because the compensation for all electrically conductive influences, along with formation water Rw , should be relegated, as additional terms, in the Rwe equation.
Equation (3b) applies to all cases where Swt ≤ 1.0, and the term ( Swtϕt ) in the denominator of the Ft relationship must be compatible with the same term in the denominator in the calculated Rwe relationship in either Eq.(1a) or (1b) in whatever application either is used. The fractional volume of conductive water represented in the formation factor must always be the same fractional volume of water that exhibits the equivalent resistivity Rwe . Also, note that both Swt and ϕt have the same exponent m . In this case, the graphical model shows this m to be m2 because m2 is the combination exponent equivalent to the individual exponents m1 and n. It is illustrated in the trigonometry of Figure 1 that the same result would be realized if Swt and ϕt were to use their individual exponents n and m1, respectively, because calculated water saturation is related to the resistivity level of Rt and this remains unchanged.
It can be seen that Eq.(1b) is a representation of Eq.(1a) when hydrocarbons are present, and can be used in the calculation of Rt . When Swt is 1.0, the calculated Rt becomes a corrected R0 , or R0 corrected . These relationships will be revisited later.
In Figure 2, adapted from Ransom (1974), it is illustrated that the resistivity value of water-filled rock at any specific depth can be the result of a shift by the influence of a variable a, and the cause of that influence must be explained.
In these equations the accountability or compensation for the influence of clay shale in shaly sands resides in the proportionality terms involving the conductivity counterparts of Rw and Rwb observed in Eq.(1b). In this relationship the proportionality terms are functions of both porosity and saturation as well as clayiness. It further can be seen that for every value of clayiness or “shaliness” and resulting volume of bound water within ϕne , the prevailing values of effective porosity and water saturation can vary the conductive water volume in effective pore space and, therefore, can vary the relative proportions of Rw and Rwb for the mixture Rwe . And, this can and does vary depth by depth.
It is further illustrated in Figure 2 that as Rw is corrected to Rwe , R0 is corrected to R0 corrected , both by the same factor a. Just as ϕe and ϕt are two intrinsic properties of the rock, the m exponent describes an intrinsic property of the rock and produces the parallelism seen in the figure. As a result, the slope represented by m is independent of the conductivity of the waters in the pores. In this figure, Rw is related to Rwe by the factor a, or coefficient a, that varies depth by depth. As a consequence, when these relationships for Rwe are implemented, or their derivatives or equivalent relationships are used in any resistivity-based interpretation, it is important to recognize that the accountability or compensation by the a coefficient must be removed from the modified Formation Resistivity Factor. And, the numerator of the Formation Resistivity Factor must always be equal to 1.0 to prevent duplication in the accountability for the secondary conductivity provided by clay shale or other conductive constituents. This perspective will be discussed in detail below.
A CLARIFYING CONCEPT OF ARCHIE'S RESISTIVITY RELATIONSHIPS AND PARAMETERS.
A MODEL AND DISCUSSION
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