Energy & Fuels 1995,9, 413-419
413
Dielectric Properties of Partially Saturated Rocks Ali A. Garrouch" and Mukul M. Sharma Department of Petroleum Engineering, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Received October 24, 1994@
A four-electrode setup has been used t o measure the complex impedance of partially saturated Berea sandstone and Ottawa sand-bentonite mixtures saturated with n-decane and NaCl brine solutions. The effects of water saturation, clay content, and wettability on both the real and imaginary parts of the rock impedance are investigated at frequencies between 10 Hz and 1 MHz. A reactivity index exponent (analogous to the resistivity index exponent) is defined and shown to be related to the water saturation by an Archie type relation. The reactivity index exponent is related to Archie's resistivity index exponent. This relationship can be used to estimate water saturations at frequencies above 0.1 MHz where the resistivity index exponent (used in the Archie equation) is found to be inadequate. Both exponents are shown t o increase with wettability and decrease with clay content. The dielectric constant appears to increase linearly with water saturation. The rate of increase of dielectric constant with water saturation is smaller for higher frequencies. The experimentally observed trends are explained qualitatively on the basis of a generalized Maxwell-Wagner theory that accounts for double layer dielectric dispersion.
Introduction For a linear dielectric, the dielectric displacement is proportional to the net electric field, and the constant of proportionality is called the permittivity of the material. The ratio of the permittivity of the material (€1to the permittivity of free space (6") is defined as the dielectric constant K of the material. In a broad sense, the dielectric constant of a material is a measure of the extent to which an applied electric field can polarize the electrical charge in the material.1,2 For fluid-saturated rocks the fluid-mineral interfaces give rise to interfacial polarization behavior. In the absence of an applied electric field, ions form a double layer at the mineral-water interface. When an oscillating electric field, in the frequency range 10 Hz to 10 MHz, is applied, the ions polarize around the mineral grains giving rise to large dipoles (ionic charges separated by a length scale on the order of the grain size) and to large (>lo3)apparent dielectric constants.'J4 As @Abstractpublished in Advance ACS Abstracts, April 1, 1995. (1)Dias, C. A. J. Geophys. Res. 1972, 77, 4945-4956. (2) Fuller, B. D.; Ward, S. H. IEEE Trans. Geosci. Electron. 1970, GE-8, 7-15. (3) Garrouch, A. A,; Sharma, M. M. Geophysics 1994,59,909-917. (4)Garrouch, A. A,; Sharma, M. M. Presented at the SOC.Core Analysts Ann. Conf. June 14-17, Oklahoma City, OK, 1992, Paper 9211. (5) Hasted, J . B. Aqueous Dielectrics; Chapman and Hall: London, 1973. ( 6 )Hoyer, W. A,; Rumble, R. C. Presented a t the SOC.Prof. Well Log Analysts 17th Ann. Logging Symp., June 9-12, Denver, CO, 1976, Paper 0. (7) Kenyon, W. E. J. Appl. Phys. 1984, 55, 3153-3159. (8) Knight, R. J.; Nur, A. Geophysics 1987, 52, 644-654. (9) Knight, R. J.; Endres, A. Geophysics 1990, 55,586-594. (10) Levitskaya, T. M.; Nosova, Y. N. Izu.-Earth Phys. 1986,22, 512-515. (11)Lewis, M. G.; Sharma, M. M.; Dunlap, H. F. SPE-18178. Presented at the SPE Annual Technical Conference and Exhibition Proceedings, Formation Evaluation and Reservoir Geology: Society of Petroleum Engineers, 1988, Omega, 697-703. (12) Lima, 0. A. L.; Sharma, M. M. Geophysics 1992,57, 431-440. (13) Olheft, G. R. Geophysics 1986,50, 2492-2503.
the frequency increases (10 MHz to 10 GHz), ions have shorter travel times, polarize less, and contribute more to the conductivity.13 Over the frequency range 10 Hz to 10 MHz, the dielectric constant is strongly dependent on factors that influence this interfacial polarization, such as brine salinity, fluid saturation, clay content, wettability, specific surface area, and tort~osity.~,*J~ The interpretation of electrical logging tools in this frequency range becomes complicated due to these dependencies. Relatively few studies have investigated interfacial polarization effects in porous media in this frequency range. Knight and NU+ measured the dielectric constant of eight sandstones at various water saturations in the frequency range of 5 Hz to 4 MHz. They observed a power-law dependence of the dielectric constant ( K ) on the frequency ( w ) given by K
=
(1)
where A and a are empirical constants. The power law exponent a was related to the water saturation and to the surface area to volume ratio of the pore space. The two-electrode technique was used and the saturation of the rock discs was varied by drying in air. Knight and Endresg observed a bound water saturation a t which the profile of dielectric constant versus saturation shows a significant decrease in slope. Simple mixing law models were shown t o be adequate for predicting the dielectric constant of saturated rocks when the dry rock dielectric constant was replaced by the dielectric constant of a water coated grain, and the pore space dielectric constant was replaced by the free unbound water response. Hoyer and Rumble6 made complex impedance measurements on sandstone cores and clay-water suspensions in a frequency range 10 Hz to 0.1 MHz using a four-electrode bridge circuit. Their data shows that the (14) Raythatha, R.; Sen, P. N. J. Colloid Interface Sci. 1986, 109, 301-309.
0887-0624/95/2509-0413$09.00/00 1995 American Chemical Society
Garrouch and Sharma
414 Energy &Fuels, Vol. 9, No. 3, 1995
dielectric constant decreases by a factor of 5 when the frequency increases from 100 t o 1000 Hz for a 5 wt % bentonite suspension. Hoyer and Rumble reported that, in the frequency range investigated, the effective dielectric constant varied with water saturation as
where K ~ O Ois the effective dielectric constant at 100% water saturation (S,) and p is approximately equal to 1. No measurements were reported at water saturations below 20%. The effect of bound water reported by Knight and end re^,^ therefore, could not be observed here and a simple mixing law relation, eq 2, was found to be adequate. Scott et al.15 made dielectric constant and conductivity measurements on more than 67 partially saturated (aidfresh water) rock samples ranging from granite t o sandstone and limestone using a twoelectrode method. They developed an empirical relationship that related the dielectric constant of the partially saturated rock to its conductivity at 100 Hz and the frequency,
where u100is the conductivity of the partially saturated rock at 100 Hz in millimhodm andfis the frequency in Hz. In the frequency range of 10 Hz to 10 MHz, the literature is still lacking data relating the dielectric properties of partially saturated rocks t o petrophysical parameters such as porosity, cation exchange capacity (CEC),frequency, and brine salinity. It is the objective of this paper to provide four-electrode complex impedance data for rocks at various brine saturations, clay content, and wettability conditions, to explore quantitative relationships between these variables, and t o explain the observations on the basis of interfacial polarization mechanisms. Care was taken t o ensure that the saturations inside the core were uniform and the measurements were made at a net confining stress. In addition, the four-electrodesetup used in this experiment minimizes the contact polarization effects encountered with two electrodes.
Experimental Setup and Procedures In this study, measurements were performed on Berea cores (having 22% porosity) and Ottawa sand-bentonite mixtures (with porosity ranging from 28 to 35%)using a Hassler type core holder that houses 3 in. long core samples with 1.5 in. diameter. The Ottawa sand was mixed with dry bentonite powder to give 2 and 4 wt % bentonite mixtures. An effective radial stress of 500 psi and an axial stress of 250 psi were applied. The Berea core samples and the sand packs were vacuum saturated with a 10 000 ppm NaCl brine solution. A residual brine saturation was established (to within 11.0%) in the core by dynamically displacing the brine with n-decane. Similarly, a residual decane saturation was established by displacing decane with brine. This flow sequence was repeated to check for saturation hysteresis. As shown in Appendix A, a nonuniform saturation in the core would cause the "apparent" dielectric constant to be controlled by the dielectric constant of the low water saturation region. This can lead t o a large relative error in the measured dielectric constant (as high as 400% at a water saturation of 60%) and can also (15) Scott, J. H.; Carroll, R. D.; Cunningham, D. R. J . Geophys. Res. 1967,72, 5101-5115.
50
60
70
so
90
100
BRINE SATURATION (%)
Figure 1. Dielectric constant versus brine saturation for a 2 wt % bentonite-Ottawa sand mixture saturated with n-decane1 10 000 ppm NaCl brine at different frequencies (four-electrode measurements). change the dielectric constant-water saturation profile to make it appear flatter than the true profile. A computer-assisted tomography scan of Berea sandstone at the end of a drainage experiment indicates that a capillary end effect is evident in the last 3 mm of the core." This zone of variable saturation can be further reduced in length by increasing the flow rate. Since 3 mm is a small fraction of the total sample length, the influence of the end effect is small. An impedance analyzer measures the gain to within 10.1 dB and the phase angle 0 to within 10.5'. These two quantities are then converted into impedance 2 from which the resistance R (12cos 0) and the reactance X (12sin 0) of the core sample are calculated. The circuit data is considered unreliable below 10 KHz since the measured phase angle at this frequency range was in the order of its uncertainty. The experimental apparatus and the electrical measurement procedures are described in more detail by Garrouch and Sharma.3,4
Results and Discussion Variation of Dielectric Constant with Saturation. As can be seen in Figure 1,the dielectric constant increases approximately linearly with saturation for frequencies from 10 KHz to 1 MHz, with an average regression coefficient of 98%. In general, this dependence can be expressed by the empirical relation, eq 2, provided by Hoyer and Rumblee6 It is also possible t o show theoretically the reason such a linear relationship should be obtained. For a partially water saturated shaly sand, the effective complex conductivity 2 0 is given12 as 5" - -* 0 - 0 4
2iFf
+ iFcs- 244a*, - a*cs) + a*,, + $ ( 3 f-
23f
IFcs)
)
(4)
where b*f is the composite fluid complex conductivity, PCs is the composite clay-sand grain complex conductivity, 4 is the porosity, and 30is the effective complex conductivity of the sand pack. Note that each of the complex conductivity functions above can be written as 5*i= ai -jmi
(5)
By substituting expressions for the fluid complex conductivity and the composite shaly-sand grain complex
Energy & Fuels, Vol. 9, No. 3, 1995 415
Dielectric Properties of Partially Saturated Rocks
conductivity, the effective permittivity of the porous medium can be shown (Appendix B)to have the following general dependence on the composite fluid (oil and water) permittivity Ef:
where EO is the effective permittivity of the shaly sand, ecs is the permittivity of the clay coated grain, uf is the composite fluid (oil and water) conductivity, Ef is the composite fluid (oil and water) permittivity, and o is the angular frequency. The linear dependence of permittivity on water saturation is not obvious from eq 6 , since both the fluid conductivity and the permittivity vary linearly with water ~ a t u r a t i o n .This ~ linear dependence is, however, more obvious for the special case where (o~,,)%$ > cf and uf >> conductivities ranging from 0.002 to 2.0 Slm, ccs was at least 1 order of magnitude larger than the water permittivity. Using these two approximations and inserting expressions for the effective complex conductivities, eq 5, into eq B-1, the following equation is obtained:
Partial saturation within core samples is sometimes accomplishedby drying and weighing the samples. This procedure leads to a nonuniform saturation distribution within the core. Higher water saturations exist in the center of the disc whereas lower water saturations exist
By multiplying eq B-2 by the conjugate of its denomina-
. . . .
-." ................^
--_I__
Water conductivity (Slm)
oecs
I ................
10-6 100
lbl
.......
I
1 b2
1 b3
1 b5
1'04
.
106
FREQUENCY (Hz) Figure 6. Comparing the water conductivity to the product of the angular frequency by the dielectric permittivity of a single charged wet grain.
tor and by separating the real and imaginary parts, the following expression for the effective permittivity as a function of the fluid permittivity Ef is obtained:
oil and water, where both phases are continuous, the fluid permittivity and conductivity are shown by Hasted5 to be related to the water permittivity ( E ~ ) ,the oil permittivity ( E d , the water saturation (S,), and the water conductivity a, by Ef
= ( e , - Eh)Sw
+ Eh
(B-4)
and Of= S,OW
where €0 is the shaly sand effective permittivity, cCsis the permittivity of the clay coated grain, uf is the composite fluid (oil and water) conductivity, Ef is the composite fluid (oil and water) permittivity, and w is the angular frequency. For a composite fluid made of
(B-5)
For the special case where WE,,)^^
