Resistivity Index

In a pore space containing hydrocarbons (gas or oil), both of which are nonconduc-tors of electricity, with a certain amount of water, resistivity is a function of water or brine saturation S_w. For the given porosity, at partial brine saturations, the resistivity of a rock is higher than when the same rock is 100% saturated with brine. Archie¹² determined experimentally that the resistivity factor of a formation partially satu-rated with brine can be expressed as

R

R^o S ⁿ

t

= ( )w (5.24)

where

R_o is the resistivity of the same rock when fully saturated with brine expressed in Ω m

R_t is the resistivity of the rock when partially saturated with brine in Ω m n is the saturation exponent

The resistivity of the rock partially saturated with brine, R_t, is also referred to as true resistivity of formation containing hydrocarbons and formation water.

Comparing Equations 5.23 and 5.24, R_o can be eliminated to obtain a generalized relationship for water saturation:

The ratio of R_t/R_o is commonly referred to as the resistivity index, I. The resistivity index is equal to 1 for a fully brine-saturated rock, whereas I > 1 when the rock is partially saturated with brine or hydrocarbons are present.

Equation 5.25 can also be expressed in terms of the resistivity index:

S R slope  n. For a given core plug sample, after measurement of R_o, at 100% brine saturation, the core plug can be desaturated in several steps by displacing the brine with oil. At each step, voltage drop and water saturation can be measured. The measured voltage drop, the current, and the sample dimensions yield the value of R_t at that particular water saturation. These measurements on each core plug typi-cally continue up to the irreducible water saturation (for definition see Chapter 6).

It should also be noted here that, since overburden also affects the electrical prop-erties, all measurements should be carried out at representative confining pres-sures. Based on the measured data, the saturation exponent can be determined by a straight-line fit for each core plug. An average value of n is normally calcu-lated for a particular rock type on the basis of n values determined for multiple core plug samples. In summary, the saturation exponent and R_o are experimentally

85 Mechanical and Electrical Properties of Reservoir Rocks

determined in the laboratory, whereas the true resistivity can be obtained from the well logs. Therefore, the in situ water saturation can be calculated using Equation 5.26. Finally, based on the material balance equation for the formation, S_w + S_o + S_g = 1.0, the in-place hydrocarbons can be estimated.

5.3.2 EFFECT OFWETTABILITY ONELECTRICAL PROPERTIES

The electrical resistivity of a porous medium can be significantly affected by impor-tant factors such as wettability and saturation history because they control the location and distribution of fluids. The most comprehensive review of the effect of wettability on electrical properties of porous media was presented by Anderson¹⁴ as part of the series of review papers published on the effect of wettability on various rock proper-ties. In fact, the parameter most significantly affected by wettability is the saturation exponent n because of its dependence on the distribution of the conducting phase in the porous medium, which in turn depends on the wettability of the system (see Chapter 7 for discussion on wettability). The uncertainty in the saturation exponent can directly impact the calculated water saturation (Equation 5.25 or 5.26) and will obviously lead to errors in the calculation of hydrocarbons in place.

Anderson’s¹⁴ examination of the effect of wettability on saturation exponent basi-cally resulted in the following major conclusions:

1. The saturation exponent is essentially independent of the system wettability when the brine saturation is sufficiently high to form a continuous film on the grain surfaces of the porous medium. The film provides a continuous path for a current flow.

2. This type of film continuity is common in clean and uniformly water-wet systems.

The saturation exponent in such systems is close to 2 and remains essentially constant as the core sample is desaturated to its irreducible water saturation.

1 10 100

0.1 1

Water saturation, fraction

Resistivity index, dimensionless

I = 0.9984 (S_w)^–1.8484 n = 1.8484

FIGURE 5.9 Log–log plot of resistivity index versus water saturation for a carbonate core sample for the determination of saturation exponent n.

86 Petroleum Reservoir Rock and Fluid Properties

3. These two observations, however, do not apply to uniformly oil-wet systems.

The saturation exponent remains close to a value of 2 up to a certain mini-mum water saturation. However, as the core is desaturated further from this minimum water saturation to its irreducible water saturation, rapid increase in the saturation exponent is observed. Values of n as high as 9 at irreducible water saturation are not uncommon.

4. The rapid increase in the saturation exponent with decreasing brine saturation for oil-wet systems is attributed to an increase in the resistivity of the system.

The increase in resistivity is due to the disconnection and trapping of the portion of the brine (nonwetting but conducting phase) by oil (wetting but nonconducting phase). The disconnected portion of the brine obviously no longer contributes to the flow of current because it is surrounded by oil that is the nonconducting phase, eventually resulting in an increase in the resistivity of the system.

Mungan and Moore¹⁵ studied the effects of wettability on resistivity using a Teflon^® (DuPont Dow Elastomers L.L.C., Wilmington, DE) core. The two fluid pairs they used were air-brine and oil-brine. The brine is then the conducting, nonwetting phase, behaving in a fashion similar to brine in an oil-wet core. The saturation exponents for the two systems are shown in Figure 5.10. An examination of Mungan and Moore’s¹⁵ data shown in Figure 5.10 demonstrates what typically happens in an oil-wet system as the brine saturation is decreased. Above a certain conducting phase saturation, the saturation exponent is fairly constant and is near 2. However, below this saturation, the exponent begins to increase rapidly by a small decrease in the water saturation. For example, the data of Mungan and Moore indicate that for a reduction of water saturation from 34.3% to 33.9% in the case of oil-brine

0 1 2 3 4 5 6 7 8 9 10

20 25 30 35 40 45 50 55 60 65 70

Water saturation, fraction

Saturation exponent

Air-brine system Oil-brine system

FIGURE 5.10 The variation in saturation exponent n, as a function of brine (conducting, nonwetting phase) saturation for an oil-wet system. (Data from Mungan, N. and Moore E.J., J. Can. Petrol. Technol., 7, 20, 1968.)

87 Mechanical and Electrical Properties of Reservoir Rocks

system, the saturation exponent jumps from 4 to 7.15 and eventually reaches a value of 9 at a water saturation of 31%. A similar behavior is observed in the case of an air-brine system.

Anderson¹⁴ also recommends that unless the reservoir is known to be strongly water wet, the saturation exponent should be measured on native or restored state cores. Anderson also states that if a clean core is used to measure the saturation exponent and the reservoir is actually oil wet, the water saturation can be under-estimated when logging. These conclusions by Anderson were based on the work of Moore¹⁶ in which the effects of cleaning on the Archie saturation exponent of the Bradford third sand, known to be oil wet, were examined. Moore studied six pairs of adjacent core plugs; one core plug from each set was extracted with toluene, making it more water wet, while the other core was unextracted and left oil wet.

In case of each core plug, extraction was found to significantly lower the satura-tion exponent. In the case of unextracted core plugs, the saturasatura-tion exponent was observed to be higher than the extracted samples. The saturation exponent data for the six extracted and unextracted core plugs reported by Moore are shown in Figure 5.11, which clearly shows the differences in the saturation exponent. The differences in the calculated water saturations for various iso-resistivity values using the extracted and unextracted saturation exponents for one of the core plugs are shown in Figure 5.12.

Clearly such differences in the calculated water saturation would obviously impact the determination of hydrocarbon saturation.

Moore,¹⁶ however, measured the resistivity of the unextracted cores only for brine saturations greater than 35%. Therefore, it is quite plausible that the saturation expo-nent would have shown a rapid increase at lower brine saturations, as observed by Mungan and Moore.¹⁵

1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Saturation exponent of unextracted core

Saturation exponent of extracted core

45° agreement line

Average of six samples

FIGURE 5.11 The alteration in the values of saturation exponents caused by core extraction with toluene. (Data from Moore, J., Producers Monthly, 22, 30, 1958.)

88 Petroleum Reservoir Rock and Fluid Properties

5.3.3 EFFECT OF CLAY ONELECTRICAL PROPERTIES

The clay minerals present in a reservoir rock act as separate conductors and are referred to as conductive solids. As a matter of fact, the water in the clay and the ions in the water act as the conducting materials. The effect of the clay on the resistivity of the rock is dependent upon the amount, type, and manner of distribution of the clay in the rock. The presence of conductive solids or clays in reservoir rocks thus requires a different approach for calculation of the formation factor.

Regarding the effect of clays on electrical properties, investigations by Wyllie¹⁷ indicated that clays contribute significantly to the conductivity of a rock when the rock is saturated with low conductivity water. The formation factor of a clayey sand increases with decreasing water resistivity and approaches a constant value at a water resistivity of about 0.1 Ω m, whereas the formation factor of a clean (clay-free) sand remains constant throughout the wide range of water resistivities.¹

For the determination of the formation factor of clay-laden rocks, Wyllie¹⁷ proposed that the observed effect of clay minerals was similar to having two electrical circuits in parallel, that is, the conducting clay minerals and the water-filled pores. Therefore,

1 1 1

R_oa = R_c+R_o (5.27)

where

R_oa is the resistivity of the clayey rock 100% saturated with water of resistivity R_w and R_c is the resistivity due to the clay minerals

0 5 10 15 20 25 30 35 40 45

100 10

1

Resistivity index, dimensionless

% difference in calculated water saturation

FIGURE 5.12 The percentage difference in the calculated water saturations based on the altered saturation exponents due to core extraction. Water saturations are calculated from Equation 5.26 using n = 1.91 (extracted) and n = 2.71 (unextracted). (Data from Moore, J., Producers Monthly, 22, 30, 1958.)

89 Mechanical and Electrical Properties of Reservoir Rocks

Substituting the value of R_o from Equation 5.21,

1 1 1

R_oa = R_c +FR_w (5.28)

The apparent formation factor F_a for clayey rock by definition is given by F R

a Roa w

= (5.29)

A plot of 1/R_oa versus 1/R_w thus results in a straight line having a slope of 1/F and an intercept of 1/R_c. Equation 5.28 will reduce to Equation 5.21 for a clay-free clean rock because the intercept will be zero.

Equation 5.28 can be rearranged¹ to express R_oa for developing the expression for F_a:

R R R

R R F

oa c w

w c/

=[ +( )] (5.30)

F R

R R F

a c

w c/

=[ +( )] (5.31)

As R_w approaches zero, F_a equals F as shown in Equation 5.31.

PROBLEMS

5.1 A 10 mm diameter and 50 mm long sandstone core plug is pulled with 1500