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.

The term "connate", con-nate, often misused and ill-defined in petrophysics literature and glossaries, is from the Latin meaning: together at birth. In petrophysics it means together at time of disposition. 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 S_w is 1.0 in ϕ_ne .

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 /R_we = (ϕ_e /ϕ_t ) 1 / R_w + (ϕ_ne /ϕ_t ) 1 / R_wb                     . . . ( 1a )

where ϕ_t = ϕ_e + ϕ_ne .

For 100% water-bearing rock the conductivity relationship in Eq.(1a) yields 1 / R_w (in clean rock) where ϕ_ne is 0.0, and yields 1 / R_we (in shaly sands) when conductive clay is present and ϕ_ne is greater than 0.0.

It should be noted that in this equation there are no limitations on the values of formation water R_w and bound water in shale, R_wb . However, nature does set limits. The bound water resistivity R_wb 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 R_w 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 R_w will be greater than R_wb ; and R_we , a mixture of R_w and R_wb , will be lower than R_w . But, it is quite common for R_w to have a lower value than R_wb , in which case R_we will have some value greater than R_w .

In water-filled dirty sands, Eq.(1a) applies. In clean sands, Eq.(1a) becomes

                                        1 / R_we = (ϕ_e /ϕ_t ) 1 / R_w = 1 / R_w

and the corrected R₀ in either event is

                               R_{0  corrected} = F_t R_we                                                                       . . . (2a)

where, from Figure 1,

                               F_t = 1.0 / ( S_wtϕ_t ) ^m2 = 1.0 / (ϕ_t^m1 )                                        . . .  (3a)

 

For the value of R_{0  corrected} the value of S_wtϕ_t becomes ϕ_t  because S_wt has become equal to 1.0. Therefore, the exponent m₂ becomes m₁ as described above and in Figure 1.  In Eq.(3a), the F_t  that formerly pertained to S_wtϕ_t  now pertains only to ϕ_t .

However, in the presence of oil or gas, water saturations are lower than 1.0 and the conductive water-filled volumes are reduced by the displacement of water volume by the hydrocarbon, and Eq.(1a) becomes

               1 / R_we = ( ( (S_weϕ_e) / (S_wtϕ_t ) ) 1 / R_w + (ϕ_ne / (S_wtϕ_t ) ) 1 / R_wb             . . . (1b)

and                                                 R_{t  calculated} = F_t R_we                                                                . . .  (2b)

where R_we is determined from Eq.(1b), and

                                                        F_t = 1.0 / ( S_wtϕ_t ) ^m2                                                              . . . (3b)

Here S_wt is less than 1.0 and exponent m again becomes m₂ . And, it will be shown in Appendix (B) (3) that

                                                            ( S_wtϕ_t ) ^m2 = ( S_wt )ⁿ ( ϕ_t )^m1                                               . . . (3c)

As it will be seen later that Eq.(3b) is the key in dual-water dual-porosity interpretations. Equation (3b) is a direct progression from the model in Figure 1 and will be further corroborated in the development from the common electrical resistance Eq.(3d), that will be developed later.

Now that the importance of R_we in the above equations has been established, just what is R_we and can waters represented by R_w and R_wb actually be combined as seen in Eq.(1b)?

Water with resistivity R_w is interstitial water. Water with resistivity R_we is not interstitial water and does not exist in nature at 100% water saturation, i.e. as R_{0  corrected} . Resistivity R_{0  corrected}  is a hypothetical water mixture of dual waters, influenced by the presence of oil, that exists at 100% water saturation only in a dual-water concept. In electrical equations the dual waters can be mixed together, physically and in nature they cannot. A dual-water mixture with resistivity R_we cannot be recovered in a production test.

To continue, a very similar relationship to Eq.(2b), involving Eqs.(1b) and (3b), is seen in work published by Best et al (1980) and by Schlumberger (1989).

[An historical retrospection: Equations(1b)and (3b) result from early developments in resistivity well-log interpretation. Equation (1a) above was shown in Ransom (1977), and in the presence of hydrocarbon becomes Eq.(1b). The model in Figure 1 in this paper was first seen and described in Figure 2 on page 7 of Ransom (1974). From that model, F_t in Eq.(3b) above was first developed. Formation factor F_t of Eq.(3b) was shown as F₃ in the explanation of Figure (2), page 7, of Ransom (1974). The equivalence relationship, ( S_wtϕ_t ) ^m2, is helpful for the use of Eq.(3b) in the determination of Archie’s dual-water saturation. Although the proof of the equivalence is found in APPENDIX (B) (3) in this paper, the equivalence was first seen as Equation (8) on page 7 of Ransom (1974). Archie's saturation equation of 1942 was first derived in Ransom (1974) on pages 7 and 8. Also, dual-porosity was suggested in Ransom (1974), where ϕ_e was described as hydrodynamically effective porosity and total porosity, ϕ_t , was described as electrically effective porosity. The difference between the two is hydrodynamically noneffective porosity, ϕ_ne . It was shown how to calculate all three porosities in Ransom (1977). Archie’s relationships of 1942 were derived in Figure 2 of Ransom (1974), again in Ransom (1995), and again, in considerably greater detail, in Figure 1 in this paper.

All equations referred to and all equations appearing in this paper support the fundamental certainty of the influence of the geometry of the bulk volume of water on the electrical resistance of rock. The bulk volume geometry concept was first introduced relative to resistivity analysis in Ransom (1974). This concept is best exemplified by the bulk volume of water term, S_wtϕ_t , in Eqs.(1b) and (3b), and by water geometry represented by exponent m₂ , and therefore by both m₁ and n, observed in Figure 1 and seen in Eq.(3b).]

In Eq.(3b) both the bulk volume of water, S_wtϕ_t , and its electrical efficiency can be seen to change in the term ( S_wtϕ_t ) ^m2 as the saturation distribution of hydrocarbon changes the geometry of water. The basic Archie method is changed from a single-water single-porosity method to the dual-water dual-porosity methodology where the two waters have resistivities of R_w and pseudo R_wb . The former R_w  of the single-water single-porosity method becomes R_we , as seen in Eqs.(1a) and(1b). The R_wb has been referred to as a pseudo R_wb 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 R_w , should be relegated to terms in the R_we equation.

Equation (3b) applies to all cases where S_wt ≤ 1.0, and the term ( S_wtϕ_t ) in the denominator of the F_t relationship must be compatible with the same term in the denominator in the calculated R_we relationship in Eq.(1b) in whatever application it 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 R_we . Also, note that both S_wt and ϕ_t 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 S_wt and ϕ_t were to use their individual exponents n and m₁, respectively, because calculated water saturation is related to the resistivity level of R_t and this remains unchanged. The algebraic proof that ( S_wtϕ_t ) ^m2 = ( S_wt )ⁿ ( ϕ_t )^m1 is shown in APPENDIX (B)(3).

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 R_w and R_wb 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 ϕ_ne , 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 R_w and R_wb for the mixture R_we . And, this can and does vary depth by depth.

It is further illustrated in Figure 2 that as R_w is corrected to R_we , R₀ is corrected to R_{0 corrected} , both by the same factor a. Just as ϕ_e and ϕ_t 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, R_w is related to R_we by the factor a, or coefficient a, that varies depth by depth. As a consequence, when these relationships for R_we 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.

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