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Dielectric relaxation in oil-bearing clayey rocks
Russian Geology and Geophysics 52 (2011) 1028–1034 www.elsevier.com/locate/rgg
Dielectric relaxation in oil-bearing clayey rocks M.I. Epov a,*, P.P. Bobrov b, V.L. Mironov c,d, A.V. Repin b a
A.A. Trofimuk Institute of Petroleum Geophysics and Geology, Siberian Branch of the Russian Academy of Sciences, pr. Akademika Koptyuga 3, Novosibirsk, 630090, Russia b Omsk State Pedagogic University, nab. Tukhachevskogo 14, Omsk, 644099, Russia c Kirensky Institute of Physics, Siberian Branch of the Russian Academy of Sciences, Akademgorodok 50/38, Krasnoyarsk, 660036, Russia d Reshetnev Siberian State Aerospace University, pr. Krasnoyarskii Rabochii 31, Krasnoyarsk, 660014, Russia Received 20 April 2010; received in revised form 1 December 2010; accepted 7 December 2010
Abstract The dielectric properties of the bentonite–oil–salt solution mixtures with different water and oil saturation were experimentally studied in the frequency range of 10 kHz to 4 GHz at 25–60 ºC. It has been established that besides the region of the Debye water relaxation, there are two more relaxation regions resulting from the interfacial interaction of the mixture components in this frequency range. To describe the dielectric permittivity and equivalent conductivity of the mixture, a spectroscopic model taking into account the multifrequency relaxation is proposed. The dependence of the model parameters on the water saturation and temperatures of the samples has been determined. The experimental data are compared with the modeling results, and the error of prediction for the dielectric permittivity and conductivity of mixture is estimated using the proposed model. © 2011, V.S. Sobolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. Keywords: oil-bearing deposits; clayey rocks; dielectric permittivity; multifrequency relaxation
Introduction In our earlier paper (Epov et al., 2009), we presented the measured dielectric permittivity and conductivity of the sand– oil–salt solution and bentonite–oil–salt solution mixtures in the frequency range of 0.05–16 GHz at 22 ºC. It was shown that the generalized refraction dielectric model of the mixture proposed by Mironov et al. (2004, 2007) perfectly describes the spectra in the above frequency range for the dielectric permittivity and equivalent conductivity of the mixtures based on sand and disagrees with the experimental data for the mixtures based on bentonite at the same frequencies. The relative error of prediction of dielectric permittivity increases to 120% as frequency decreases to 0.05 GHz. Today, an equipment working in the frequency range of 1–12 MHz is widely used in electromagnetic logging of wells (Epov and Antonov, 2000; Epov and Glinskikh, 2005). This equipment sounds the clay drilling mud penetration zone rich in clay fraction. For the theoretical modeling of electromagnetic logging and the inversion of conductivity in the penetration zone (Epov and Antonov, 2000), it is necessary to have
* Corresponding author. E-mail address: [email protected] (M.I. Epov)
spectroscopic models for water- and oil-saturated rocks containing a clay fraction. Below we present the results of experimental studies of the dielectric permittivity and equivalent conductivity of oil- and water-containing mixtures based on bentonite in the frequency range of 10 kHz to 4 GHz. We propose a model for the dielectric permittivity agreeing with the experimental data. In this paper we substantiate the technique of measurements with an instrument capacitor cell, estimate their errors, and analyze the experimental spectra for different contents of aqueous salt solution and oil at 25 and 60 ºC. For the spectrum modeling, we used a formula of multifrequency dielectric relaxation (Akhadov, 1972) of fluids present in the clay drilling mud penetration zone.
The measurement technique The complex relative dielectric permittivity (CDP) of the samples was measured in the frequency range of 50 kHz to 4 GHz, using a ZVRE Rohde-Schwarz vector network analyzer. For the measurements at 500 kHz–4 GHz, the sample was placed into a coaxial cell, and a complex transmission coefficient (parameter S12 of the scattering matrix) was
1068-7971/$ - see front matter D 201 1, V . S. S o bolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.rgg.2011.08+.008
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the capacitor and measuring the frequency dependencies of the complex transmission and reflection coefficients. The equivalent impedance ZAB of the measuring capacitor containing the sample to be studied can be determined experimentally either via the complex reflection coefficients (S11 or S22) or via the complex transmission coefficients S12 or S21, which are measured with the vector network analyzer, provided that the measuring capacitor is connected to either a single input or both inputs of the vector network analyzer, respectively. In the first case, the reflection coefficient is expressed as Zin − R0 , from which we obtain S11 = Zin + R0 Fig. 1. Measuring capacitor. ε′, Real part of CDP; σe, equivalent conductivity.
determined. The CDP was calculated by the technique described by Epov et al. (2009). For the measurements in the lower frequency range, the sample was placed into a measuring capacitor cell switched into the break of the central conductor of the coaxial line (Fig. 1). An equivalent circuit of the measuring capacitor containing a mixture sample is shown in Fig. 2, A. The capacitances arising between the capacitor plates and the casing are taken into account by the capacitors C1 and C2, respectively, and the impact of the coaxial-line segments connecting the measuring capacitor to the ends of the broken coaxial line, via the inductivities L1 and L2; R0 is the input wave impedance of the vector network analyzer (50 Ohm), Eg is the emf induced by the vector network analyzer, and ZAB is the equivalent impedance of the interior part of the measuring capacitor. The equivalent scheme of the latter is shown in Fig. 2, where C0 is the capacitance of its part containing the sample and the dielectric loss is taken into account through the equivalent conductivity G. The capacitance Ce is the sum of the edge capacitance and the capacitance of the insulating ring. The equivalent impedance ZAB is calculated as ZAB =
1 , GAB
(1)
where GAB = jωC0 + jωCe + G, j is the imaginary unit, and ω is the angular frequency. The parasitic capacitances and inductances C1, L1, C2, L2, and Ce can be determined with auxiliary measurements by placing solid dielectrics with known dielectric permittivity in
1 + S11 Zin = R0 , 1 − S11
(2)
where Zin is the complex input impedance of the measuring capacitor (Fig. 2). On the other hand, it can be expressed as a continued fraction Zin = jωL1 +
1 jωC1 +
,
1 ZAB +
(3)
1 jωC2 +
1 ZOK
where the value of the output impedance ZOK of the measuring capacitor depends on a load at the other input of it, that is ZOK = ∞ if there is no load connected, ZOK = jωL2 if the input is short circuited, and ZOK = jωL2 + R0 if the load is a perfect match. From (3) we have ZAB =
(jωL1 − Zin) (jωC1ZC + 1) + ZC jωC1Zin − ω2L1C1 − 1
,
(4)
1 . Using the measured S11 value, we jωC2 + 1 / ZOK calculate Zin value from (2) and, substituting it into (4), experimentally obtain the value of ZAB. Then, using (1), we find the capacity and conductivity of the measuring capacitor containing a sample measured. ε0π R 2 πR2 From C0 = ε′ , G = σe , where ε0 = 8.85 × d d 10–12 F/m and R is the radius of the capacitor plates (within where ZC =
Fig. 2. Equivalent scheme of connection of measuring capacitor to ZVRE inlets (A) and schematic structure of its inside (B).
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the insulating ring), we easily find the real part of the CDP, ε′, and the equivalent conductivity σe = ωε″ε0, where ε″ is the imaginary part of the CDP. In the alternative variant, we have to connect a measuring capacitor to both inputs of the vector network analyzer and measure the complex transmission coefficient S12. Let us derive the respective formulas for calculating the permittivity and conductivity of the sample with the use of the values S12 measured. The complex transmission coefficient according to (Mattey et al., 1964) and the equivalent circuit shown in Fig. 2, A can be expressed with the formula U2 u2 =2 . Then, using the rule of Kirchhoff’s S12 = 2 Eg U1 + i1 R0 nodal currents, we obtain the following expression for the complex impedance ZAB: 2R0 2ZOK − , ZAB = 2 k k S12
(5)
where k = 1 + jωCZOK. Since on measuring S12 the other input of the measuring capacitor is connected to the second input of the network analyzer, the output impedance ZOK must be equal to ZOK = jωL + R0. Formula (5) was derived assuming the measuring capacitor to be symmetric, that is, L1 = L2 = L and C1 = C2 = C. The measurement showed that the edge capacitance as to the capacitor plates is negligible because of the closely located outer conductor of the coaxial line and therefore can be ignored. Provided that the value of ZAB is known, the real part of the CDP, ε′, and the conductivity σe values can be derived similarly to those obtained with the use of the first approach, when the reflection coefficient S11 is measured. In the frequency range from 10 kHz to 1 MHz, the impedance ZAB of the capacitor cell containing the same sample was measured with a E7-20 LCR instrument, using a three-terminal scheme. In this case, the central conductors of the coaxial output cables of the measuring capacitor were connected to two terminals of the E7-20 LCR meter, while the outer conductors of the coaxial output cables of the measuring capacitor, having been connected to the capacitor casing, were connected to the third terminal of the E7-20 LCR meter. Accordingly, the equivalent circuit became easier because the capacitances C1 and C2, having been shortcut to the casing, no longer affected the values measured. As a result, the impedance measured by the LCR instrument appeared to be Z = ZAB + jωL, where L is the spurious inductance of the measuring capacitor outputs. The tests showed that the values of permittivity and equivalent conductivity derived when using a capacitor measuring cell being connected consequently to either the ZVRE vector network analyzer or the E7-20 LCR meter deviated from each other, through the overlapping frequency range from 300 kHz to 1 MHz, are of the same order of magnitude as the errors relative to the values initially measured by the network analyzer and LCR meter. The same deviation error
was found in the other overlapping frequency range from 30 MHz to 80 MHz, when the capacitor and coaxial cells were connected to the network analyzer only. The errors of deriving the permittivity and effective conductivity arising as a result of the ZVRE and LCR instrument errors and uncertainties in the sizes of the measuring coaxial cell and capacitor were estimated as follows. The quantities initially measured by either the ZVRE or LCR instruments were varied within the limits of the instrumental errors, according to their respective manuals, and the accuracy of geometrical sizes of the measuring cells was estimated. Then, the varied values were applied to derive the limits of variations for the values of ε′ and σe. The total error was determined as a square root of the sum of squared deviations of ε′ and σe due to variations of the quantities initially measured using the ZVRE and LCR instruments. The measurement error was found to depend on the frequency, fluid content in the sample, and the sample temperature. For the studied samples, with the water solution fraction of the saturating fluid of 0, 10, and 33%, the estimated error for ε′ and σe derived when using the ZVRE instrument increased from 3% at the higher frequency limit of 4 GHz to 6% at the lower frequency limit of 5 MHz. For the sample with a higher water solution fraction of the saturating fluid (66.7%), the error in the specific frequency range of 1 to 50 MHz increased up to 25–30% as a result of the higher ohmic conductivity of the sample. In the range of frequencies from 10 to 100 kHz, the error of deriving the value of ε′ with the use of the E7-20 LCR meter was found to be about 1%, and it increased to 3% for the frequencies higher than 100 kHz. The measurement error of the equivalent conductivity obtained with the use of the E7-20 LCR meter did not exceed 1% in the frequency range from 10 kHz to 1 MHz, and that obtained using the ZVRE network analyzer was found to be no higher than 5% in the frequency range from 300 MHz to 4 GHz, except for the frequency subrange from 1 to 50 MHz, where it varied from 10 to 40%.
Results of experimental studies The samples were prepared as follows. First, a weighted portion of bentonite clay was taken, which had been dried out at 105 ºC for 24 h, and the volume of solids was calculated using the known density of bentonite mineral (2.3 g/cm3). Then, the required volumes of liquid (oil and salt solution) were measured out and successively added to the bentonite. The mixture was thoroughly mixed to obtain a homogeneous mass. Before placing the sample into a measuring cell, it was kept in hermetically sealed container for 4–5 h. The sample density deviation was controlled by weighing the measuring container. Figure 3, A presents the results of measurement of the dielectric permittivity and equivalent conductivity of the sample with the following volume fractions of its components: bentonite solids—85%, oil—9%, and salt solution (17 g/l)—
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Fig. 3. Dependence of the frequency of the real part of dielectric permittivity and equivalent conductivity of mixture containing 85% bentonite, 9% oil, and 6% salt solution (17 g/l) (A), 80% bentonite with 0% (a) and 33.3% (b) salt solution (B), and 80% bentonite with 10% (a) and 66.7% (b) salt solution (C). 1, 3, experimental data obtained at 25 and 60 ºC, respectively; 2, 4, values calculated by the model at 25 and 60 ºC, respectively.
6%, with all the percentage calculated relative to the total volume of these particular components. These results agree with each other in the overlapping frequency regions and with the results of the measurements with a ZVK vector network analyzer in the frequency range from 10 MHz to 16 GHz (Epov et al., 2009). The ε′ value significantly increases in the frequency range lower then 1 GHz. This is typical of moist mixtures containing
fine clay particles (Hipp, 1974). Alternation of slopes and flat areas of the frequency dependence of ε′ evidence the presence of several relaxation processes. As temperature grows, the ε′ and σe values decrease regarding the frequency range of >1 GHz and increase with frequency decreasing in the range of <1 GHz. Figures 3, B and C present the frequency dependencies of ε′ and σe for mixtures containing 80% of bentonite solids and
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20% of oil and salt solution (17 g/l) fluids with a varying volumetric fraction of salt solution relative to the total volume of fluids contained in the sample. The measurements were carried out at 25, 35, 45, and 60 ºC, but the data are presented only for the boundary values of temperature. The same figures present the results of calculations by the model described in the next section. The ε′ and σe values at intermediate temperatures lie between the values corresponding to these boundary temperatures. Analysis of the data in Figs. 3, B and C shows that ε′ of the sample depends most strongly on temperature at the water saturation of 33.3%. The dielectric-loss tangent in the case of a 10% salt solution has a maximum at a frequency of ~10 kHz. With increasing the salt solution percentage, the dielectric-loss tangent increases, and the frequency of the maximum shifts to 0.5– 1 MHz. At the same time, the frequency of the maximum is almost unchanged, with the temperature increasing. The dielectric permittivity increase is mostly noticeable with the salt solution percentage increasing from 10 to 30% (Fig. 4), concerning the lower frequency range. To the best of our knowledge, there are no literature data on dielectric measurements for oil-saturated clayey rocks. Levitskaya and Sternberg (1996) presented the measured CDP of oil-saturated sandstone with porosity of 7.61%. Their ε′ values nearly coincide with our data for the salt-free solution in the frequency range of 10–100 MHz. With the frequency decreasing below 10 MHz, the difference between the values of our measurements and those reported by Levitskaya and Sternberg (1996) drastically increases, so that at 1 MHz, the ε′ values obtained by these authors are three times greater than ours. Talalov and Daev (1996) presented data on the CDP of water-saturated samples of sand, sandy loam (with varying clay fraction), and pure clay samples. In the frequency range of 1–100 MHz, the results for the clay-richest sandy loam are the same as our data for the oil-poorest samples (66.7% salt solution). This is well explicable, since a decrease in the clay portion in the sample, like an increase in its oil portion, leads to a decrease in ε′ values in this frequency region. Note that the above and many other published papers concerning dielectric studies lack measurement error estimates.
Modeling of the electrophysical characteristics of mixtures The complex dielectric permittivity of a heterogeneous . mixture, εm, can be described by the formula first proposed by Birchak et al. (1974). This is the so-called refraction model, as it summarizes the relative volumetric portions of the components comprising the sample, multiplied by their refracε ): tion indices (n = √ . . εp , εm = ∑ Vp √ √ (6) p
Fig. 4. Dependence of the real part of dielectric permittivity on salt solution percentage at 25 ºC for different frequencies.
where Vp is a volumetric portion of the pth component of the . mixture and εp is the complex relative dielectric permittivity (CDP) of this component. In our case, the mixture includes bentonite, oil, salt solution, and air. The CDP of bentonite solids and air does not depend on frequency, and the frequency dependences of CDP for both oil and salt solution can be found by the generalized Debye relaxation model for conducting liquids (Akhadov, 1972): σp εSp − ε∞p . εp = ε∞p + +j . ωε0 1 + jω τp . Here, εp is the CDP of oil or salt solution and εSp, ε∞p, τp, and σp are the parameters of the Debye model (static and high-frequency dielectric constants, relaxation time, and conductivity, respectively). These parameters were determined earlier (Epov et al., 2009). In that study we showed that the measured values of the real part of the CDP of the bentonitebased mixture are significantly higher than those calculated by the model, especially in the low-frequency range. The increase in the measured dielectric permittivity at low frequencies and the frequency dependences of ε′ and σ (Fig. 3) evidence the existence of two additional relaxation processes caused by the interlayer polarization, with the overlapping regions of their run. In accordance with Akhadov (1972), we assume that the CDP mixture can be determined as a sum of the CDPs corresponding to each low-frequency relaxation process and the CDP of the mixture, as given by (6). We described the low-frequency relaxation processes by the Cole–Cole empirical formula taking into account the distribution of relaxation times (Akhadov, 1972): . εi = ε∞i +
∆εSi 1 + (jω τ i)βi
,
(7)
where βi are the distribution coefficients of relaxation times, 0 ≤ β ≤ 1; index i (i = 1, 2) marks the model parameters describing particular relaxation processes. At frequencies higher than 4 GHz, low-frequency relaxation is almost absent; therefore, we took dielectric permittivity determined from (6).
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Fig. 5. Temperature dependence of static dielectric constants (A, B) and distribution coefficients of relaxation times (C, D) for 0% (1), 10% (2), 33.3% (3), and 66.7% (4) salt solution. A, C, First relaxation region; B, D, second relaxation region.
Thus, the final formula for the CDP of the mixture looks as 2 2 ∆εSi . σ + ε = ∑ εp + j , Vp √ ∑ ω ε0 β i = 1 1 + (jω τ i) i p
(8)
i.e., a sum of two terms describing low-frequency relaxation (parameters with index i), which is complemented with the CDP of the sample calculated by (6). The volumetric percentage in this formula was assigned to take into account the presence of air in the sample. In addition, the imaginary part of the CDP related to the ionic conductivity is taken into account by the third term in (8). The second term in (8) was calculated using the parameters determined by Epov et al. (2009), with the values of ionic conductivities of oil and with water solution set equal to zero. The parameters in Cole–Cole equation (8) were derived by minimization of the residual with the experimental data first by the least-squares method and then by the minimax method. The minimization problem was solved repeatedly at different initial values of the parameters derived. After the determination of the model parameters, we developed the regression equations for the temperature dependence and again performed minimization of the residual with the initial parameters found from the regression equations. Thus, we succeeded in reducing the deviation of the calculated data from the experimental ones to 10–12% for ε′ and 5–8% for σe (average values). The maximum deviations for ε′ were observed for the sample with 66.7% salt solution and reached 20–22% at some points.
The determined relaxation time in the first (from the upper frequencies) relaxation region is (1–3) × 10−8 s. The relaxation time increases 1.5–2 times as temperature grows from 25 to 60 ºC and ~1.5 times as the percentage of salt solution increases from 0 to 66.7%. The relaxation time in the second relaxation region is 10–4–10–5 s. The relaxation time decreases 1.5–2 times with temperature growth from 25 to 60 ºC and increases 3–4 times with an analogous increase in the percentage of salt solution. The values of the model parameters εSi and βi are given in Fig. 5. In the first relaxation region, the static dielectric constants of the samples with different water saturation increase about 1.5 times with temperature (Fig. 5, A), and in the second relaxation region they increase for all samples except for one with high water saturation (Fig. 5, B). The distribution coefficients of relaxation times vary from 0.3 to 0.6 in the first region (Fig. 5, C) and from 0.4 to 1 in the second region (Fig. 5, D).
Conclusions In the frequency region from 10 kHz to 4 GHz at 25–60 ºC, experimental spectra have been obtained for the dielectric permittivity and conductivity of clayey rocks saturated with oil and salt solution in different proportions, with their total amount being 20% relatively to the volume of the sample. A model for the complex dielectric permittivity has been elaborated, which takes into account low-frequency relaxation
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processes. The model parameters ensuring agreement of the calculated CDPs with the experimental ones in the entire frequency region have been found, and their dependence on temperature and water saturation has been established. The error of prediction of the dielectric permittivity and conductivity of mixtures by the proposed model has been estimated. This work was supported by budget project 2.5.1.1 “Dielectric spectroscopy of natural media in the radio wave frequency region” and Interdisciplinary Project 6 “The theoretical grounds of the principally new technology of sounding in oil-gas wells using subnanosecond electromagnetic pulses” from the Siberian Branch of the Russian Academy of Sciences.
References Akhadov, Ya.Yu., 1972. The Dielectric Properties of Pure Liquids [in Russian]. Izd. Standartov, Moscow. Birchak, J.R., Gardner, G.G., Hipp, J.E., Victor, J.M., 1974. High dielectric constant microwave probes for sensing soil moisture. Proc. IEEE 62 (1), 93–98. Epov, M.I., Antonov, Yu.N. (Eds.), 2000. The Technology of Study of Oil-Gas Wells by the High-Frequency Logging Method. Methodical Manual [in Russian]. Izd. SO RAN, NITs OIGGM SO RAN, Novosibirsk.
Epov, M.I., Glinskikh, V.N., 2005. Electromagnetic Logging: Modeling and Inversion [in Russian]. Akademicheskoe Izd. “Geo”, Novosibirsk. Epov, M.I., Mironov, V.L., Bobrov, P.P., Savin, I.V., Repin, A.V., 2009. Dielectric spectroscopy of oil-bearing rocks at 0.05–16 GHz. Russian Geology and Geophysics (Geologiya i Geofizika) 50 (5), 470–474 (613–618). Hipp, J.E., 1974. Soil electromagnetic parameters as function of frequency, soil density, and soil moisture. Proc. IEEE 62 (1), 98–103. Komarov, S.A., Mironov, V.L., 2000. Microwave Sounding of Soils [in Russian]. Nauka, Novosibirsk. Levitskaya, T.M., Sternberg, B.K., 1996. Polarization processes in rocks. Radio Science 3 (4), 755–759. Mattey, D.L., Young, L., Jones, E.M.T., 1964. Microwave Filters, ImpedanceMatching Networks and Coupling Structures. McGraw-Hill, New York. Mironov, V.L., Dobson, M.C., Kaupp, V.H., Komarov, S.A., Kleshchenko, V.N., 2004. Generalized refractive mixing dielectric model for moist soils. IEEE Trans. Geosci. Remote Sensing 42 (4), 773–785. Mironov, V.L., Kosolapova, L.G., Fomin, S.V., 2007. The method of compiling a spectroscopic database of the dielectric properties of moist soils in wicrowave frequency range. Izvestiya Vuzov. Radiofizika 50 (4), 339–349. Talalov, A.D., Daev, D.S., 1996. The structural mechanism of the frequency dispersion of the electric properties of heterogeneous rocks. Izv. RAN. Fizika Zemli, No. 8, 56–66.
Editorial responsibility: A.D. Duchkov
Dielectric relaxation in oil-bearing clayey rocks M.I. Epov a,*, P.P. Bobrov b, V.L. Mironov c,d, A.V. Repin b a
A.A. Trofimuk Institute of Petroleum Geophysics and Geology, Siberian Branch of the Russian Academy of Sciences, pr. Akademika Koptyuga 3, Novosibirsk, 630090, Russia b Omsk State Pedagogic University, nab. Tukhachevskogo 14, Omsk, 644099, Russia c Kirensky Institute of Physics, Siberian Branch of the Russian Academy of Sciences, Akademgorodok 50/38, Krasnoyarsk, 660036, Russia d Reshetnev Siberian State Aerospace University, pr. Krasnoyarskii Rabochii 31, Krasnoyarsk, 660014, Russia Received 20 April 2010; received in revised form 1 December 2010; accepted 7 December 2010
Abstract The dielectric properties of the bentonite–oil–salt solution mixtures with different water and oil saturation were experimentally studied in the frequency range of 10 kHz to 4 GHz at 25–60 ºC. It has been established that besides the region of the Debye water relaxation, there are two more relaxation regions resulting from the interfacial interaction of the mixture components in this frequency range. To describe the dielectric permittivity and equivalent conductivity of the mixture, a spectroscopic model taking into account the multifrequency relaxation is proposed. The dependence of the model parameters on the water saturation and temperatures of the samples has been determined. The experimental data are compared with the modeling results, and the error of prediction for the dielectric permittivity and conductivity of mixture is estimated using the proposed model. © 2011, V.S. Sobolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. Keywords: oil-bearing deposits; clayey rocks; dielectric permittivity; multifrequency relaxation
Introduction In our earlier paper (Epov et al., 2009), we presented the measured dielectric permittivity and conductivity of the sand– oil–salt solution and bentonite–oil–salt solution mixtures in the frequency range of 0.05–16 GHz at 22 ºC. It was shown that the generalized refraction dielectric model of the mixture proposed by Mironov et al. (2004, 2007) perfectly describes the spectra in the above frequency range for the dielectric permittivity and equivalent conductivity of the mixtures based on sand and disagrees with the experimental data for the mixtures based on bentonite at the same frequencies. The relative error of prediction of dielectric permittivity increases to 120% as frequency decreases to 0.05 GHz. Today, an equipment working in the frequency range of 1–12 MHz is widely used in electromagnetic logging of wells (Epov and Antonov, 2000; Epov and Glinskikh, 2005). This equipment sounds the clay drilling mud penetration zone rich in clay fraction. For the theoretical modeling of electromagnetic logging and the inversion of conductivity in the penetration zone (Epov and Antonov, 2000), it is necessary to have
* Corresponding author. E-mail address: [email protected] (M.I. Epov)
spectroscopic models for water- and oil-saturated rocks containing a clay fraction. Below we present the results of experimental studies of the dielectric permittivity and equivalent conductivity of oil- and water-containing mixtures based on bentonite in the frequency range of 10 kHz to 4 GHz. We propose a model for the dielectric permittivity agreeing with the experimental data. In this paper we substantiate the technique of measurements with an instrument capacitor cell, estimate their errors, and analyze the experimental spectra for different contents of aqueous salt solution and oil at 25 and 60 ºC. For the spectrum modeling, we used a formula of multifrequency dielectric relaxation (Akhadov, 1972) of fluids present in the clay drilling mud penetration zone.
The measurement technique The complex relative dielectric permittivity (CDP) of the samples was measured in the frequency range of 50 kHz to 4 GHz, using a ZVRE Rohde-Schwarz vector network analyzer. For the measurements at 500 kHz–4 GHz, the sample was placed into a coaxial cell, and a complex transmission coefficient (parameter S12 of the scattering matrix) was
1068-7971/$ - see front matter D 201 1, V . S. S o bolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.rgg.2011.08+.008
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the capacitor and measuring the frequency dependencies of the complex transmission and reflection coefficients. The equivalent impedance ZAB of the measuring capacitor containing the sample to be studied can be determined experimentally either via the complex reflection coefficients (S11 or S22) or via the complex transmission coefficients S12 or S21, which are measured with the vector network analyzer, provided that the measuring capacitor is connected to either a single input or both inputs of the vector network analyzer, respectively. In the first case, the reflection coefficient is expressed as Zin − R0 , from which we obtain S11 = Zin + R0 Fig. 1. Measuring capacitor. ε′, Real part of CDP; σe, equivalent conductivity.
determined. The CDP was calculated by the technique described by Epov et al. (2009). For the measurements in the lower frequency range, the sample was placed into a measuring capacitor cell switched into the break of the central conductor of the coaxial line (Fig. 1). An equivalent circuit of the measuring capacitor containing a mixture sample is shown in Fig. 2, A. The capacitances arising between the capacitor plates and the casing are taken into account by the capacitors C1 and C2, respectively, and the impact of the coaxial-line segments connecting the measuring capacitor to the ends of the broken coaxial line, via the inductivities L1 and L2; R0 is the input wave impedance of the vector network analyzer (50 Ohm), Eg is the emf induced by the vector network analyzer, and ZAB is the equivalent impedance of the interior part of the measuring capacitor. The equivalent scheme of the latter is shown in Fig. 2, where C0 is the capacitance of its part containing the sample and the dielectric loss is taken into account through the equivalent conductivity G. The capacitance Ce is the sum of the edge capacitance and the capacitance of the insulating ring. The equivalent impedance ZAB is calculated as ZAB =
1 , GAB
(1)
where GAB = jωC0 + jωCe + G, j is the imaginary unit, and ω is the angular frequency. The parasitic capacitances and inductances C1, L1, C2, L2, and Ce can be determined with auxiliary measurements by placing solid dielectrics with known dielectric permittivity in
1 + S11 Zin = R0 , 1 − S11
(2)
where Zin is the complex input impedance of the measuring capacitor (Fig. 2). On the other hand, it can be expressed as a continued fraction Zin = jωL1 +
1 jωC1 +
,
1 ZAB +
(3)
1 jωC2 +
1 ZOK
where the value of the output impedance ZOK of the measuring capacitor depends on a load at the other input of it, that is ZOK = ∞ if there is no load connected, ZOK = jωL2 if the input is short circuited, and ZOK = jωL2 + R0 if the load is a perfect match. From (3) we have ZAB =
(jωL1 − Zin) (jωC1ZC + 1) + ZC jωC1Zin − ω2L1C1 − 1
,
(4)
1 . Using the measured S11 value, we jωC2 + 1 / ZOK calculate Zin value from (2) and, substituting it into (4), experimentally obtain the value of ZAB. Then, using (1), we find the capacity and conductivity of the measuring capacitor containing a sample measured. ε0π R 2 πR2 From C0 = ε′ , G = σe , where ε0 = 8.85 × d d 10–12 F/m and R is the radius of the capacitor plates (within where ZC =
Fig. 2. Equivalent scheme of connection of measuring capacitor to ZVRE inlets (A) and schematic structure of its inside (B).
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the insulating ring), we easily find the real part of the CDP, ε′, and the equivalent conductivity σe = ωε″ε0, where ε″ is the imaginary part of the CDP. In the alternative variant, we have to connect a measuring capacitor to both inputs of the vector network analyzer and measure the complex transmission coefficient S12. Let us derive the respective formulas for calculating the permittivity and conductivity of the sample with the use of the values S12 measured. The complex transmission coefficient according to (Mattey et al., 1964) and the equivalent circuit shown in Fig. 2, A can be expressed with the formula U2 u2 =2 . Then, using the rule of Kirchhoff’s S12 = 2 Eg U1 + i1 R0 nodal currents, we obtain the following expression for the complex impedance ZAB: 2R0 2ZOK − , ZAB = 2 k k S12
(5)
where k = 1 + jωCZOK. Since on measuring S12 the other input of the measuring capacitor is connected to the second input of the network analyzer, the output impedance ZOK must be equal to ZOK = jωL + R0. Formula (5) was derived assuming the measuring capacitor to be symmetric, that is, L1 = L2 = L and C1 = C2 = C. The measurement showed that the edge capacitance as to the capacitor plates is negligible because of the closely located outer conductor of the coaxial line and therefore can be ignored. Provided that the value of ZAB is known, the real part of the CDP, ε′, and the conductivity σe values can be derived similarly to those obtained with the use of the first approach, when the reflection coefficient S11 is measured. In the frequency range from 10 kHz to 1 MHz, the impedance ZAB of the capacitor cell containing the same sample was measured with a E7-20 LCR instrument, using a three-terminal scheme. In this case, the central conductors of the coaxial output cables of the measuring capacitor were connected to two terminals of the E7-20 LCR meter, while the outer conductors of the coaxial output cables of the measuring capacitor, having been connected to the capacitor casing, were connected to the third terminal of the E7-20 LCR meter. Accordingly, the equivalent circuit became easier because the capacitances C1 and C2, having been shortcut to the casing, no longer affected the values measured. As a result, the impedance measured by the LCR instrument appeared to be Z = ZAB + jωL, where L is the spurious inductance of the measuring capacitor outputs. The tests showed that the values of permittivity and equivalent conductivity derived when using a capacitor measuring cell being connected consequently to either the ZVRE vector network analyzer or the E7-20 LCR meter deviated from each other, through the overlapping frequency range from 300 kHz to 1 MHz, are of the same order of magnitude as the errors relative to the values initially measured by the network analyzer and LCR meter. The same deviation error
was found in the other overlapping frequency range from 30 MHz to 80 MHz, when the capacitor and coaxial cells were connected to the network analyzer only. The errors of deriving the permittivity and effective conductivity arising as a result of the ZVRE and LCR instrument errors and uncertainties in the sizes of the measuring coaxial cell and capacitor were estimated as follows. The quantities initially measured by either the ZVRE or LCR instruments were varied within the limits of the instrumental errors, according to their respective manuals, and the accuracy of geometrical sizes of the measuring cells was estimated. Then, the varied values were applied to derive the limits of variations for the values of ε′ and σe. The total error was determined as a square root of the sum of squared deviations of ε′ and σe due to variations of the quantities initially measured using the ZVRE and LCR instruments. The measurement error was found to depend on the frequency, fluid content in the sample, and the sample temperature. For the studied samples, with the water solution fraction of the saturating fluid of 0, 10, and 33%, the estimated error for ε′ and σe derived when using the ZVRE instrument increased from 3% at the higher frequency limit of 4 GHz to 6% at the lower frequency limit of 5 MHz. For the sample with a higher water solution fraction of the saturating fluid (66.7%), the error in the specific frequency range of 1 to 50 MHz increased up to 25–30% as a result of the higher ohmic conductivity of the sample. In the range of frequencies from 10 to 100 kHz, the error of deriving the value of ε′ with the use of the E7-20 LCR meter was found to be about 1%, and it increased to 3% for the frequencies higher than 100 kHz. The measurement error of the equivalent conductivity obtained with the use of the E7-20 LCR meter did not exceed 1% in the frequency range from 10 kHz to 1 MHz, and that obtained using the ZVRE network analyzer was found to be no higher than 5% in the frequency range from 300 MHz to 4 GHz, except for the frequency subrange from 1 to 50 MHz, where it varied from 10 to 40%.
Results of experimental studies The samples were prepared as follows. First, a weighted portion of bentonite clay was taken, which had been dried out at 105 ºC for 24 h, and the volume of solids was calculated using the known density of bentonite mineral (2.3 g/cm3). Then, the required volumes of liquid (oil and salt solution) were measured out and successively added to the bentonite. The mixture was thoroughly mixed to obtain a homogeneous mass. Before placing the sample into a measuring cell, it was kept in hermetically sealed container for 4–5 h. The sample density deviation was controlled by weighing the measuring container. Figure 3, A presents the results of measurement of the dielectric permittivity and equivalent conductivity of the sample with the following volume fractions of its components: bentonite solids—85%, oil—9%, and salt solution (17 g/l)—
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Fig. 3. Dependence of the frequency of the real part of dielectric permittivity and equivalent conductivity of mixture containing 85% bentonite, 9% oil, and 6% salt solution (17 g/l) (A), 80% bentonite with 0% (a) and 33.3% (b) salt solution (B), and 80% bentonite with 10% (a) and 66.7% (b) salt solution (C). 1, 3, experimental data obtained at 25 and 60 ºC, respectively; 2, 4, values calculated by the model at 25 and 60 ºC, respectively.
6%, with all the percentage calculated relative to the total volume of these particular components. These results agree with each other in the overlapping frequency regions and with the results of the measurements with a ZVK vector network analyzer in the frequency range from 10 MHz to 16 GHz (Epov et al., 2009). The ε′ value significantly increases in the frequency range lower then 1 GHz. This is typical of moist mixtures containing
fine clay particles (Hipp, 1974). Alternation of slopes and flat areas of the frequency dependence of ε′ evidence the presence of several relaxation processes. As temperature grows, the ε′ and σe values decrease regarding the frequency range of >1 GHz and increase with frequency decreasing in the range of <1 GHz. Figures 3, B and C present the frequency dependencies of ε′ and σe for mixtures containing 80% of bentonite solids and
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20% of oil and salt solution (17 g/l) fluids with a varying volumetric fraction of salt solution relative to the total volume of fluids contained in the sample. The measurements were carried out at 25, 35, 45, and 60 ºC, but the data are presented only for the boundary values of temperature. The same figures present the results of calculations by the model described in the next section. The ε′ and σe values at intermediate temperatures lie between the values corresponding to these boundary temperatures. Analysis of the data in Figs. 3, B and C shows that ε′ of the sample depends most strongly on temperature at the water saturation of 33.3%. The dielectric-loss tangent in the case of a 10% salt solution has a maximum at a frequency of ~10 kHz. With increasing the salt solution percentage, the dielectric-loss tangent increases, and the frequency of the maximum shifts to 0.5– 1 MHz. At the same time, the frequency of the maximum is almost unchanged, with the temperature increasing. The dielectric permittivity increase is mostly noticeable with the salt solution percentage increasing from 10 to 30% (Fig. 4), concerning the lower frequency range. To the best of our knowledge, there are no literature data on dielectric measurements for oil-saturated clayey rocks. Levitskaya and Sternberg (1996) presented the measured CDP of oil-saturated sandstone with porosity of 7.61%. Their ε′ values nearly coincide with our data for the salt-free solution in the frequency range of 10–100 MHz. With the frequency decreasing below 10 MHz, the difference between the values of our measurements and those reported by Levitskaya and Sternberg (1996) drastically increases, so that at 1 MHz, the ε′ values obtained by these authors are three times greater than ours. Talalov and Daev (1996) presented data on the CDP of water-saturated samples of sand, sandy loam (with varying clay fraction), and pure clay samples. In the frequency range of 1–100 MHz, the results for the clay-richest sandy loam are the same as our data for the oil-poorest samples (66.7% salt solution). This is well explicable, since a decrease in the clay portion in the sample, like an increase in its oil portion, leads to a decrease in ε′ values in this frequency region. Note that the above and many other published papers concerning dielectric studies lack measurement error estimates.
Modeling of the electrophysical characteristics of mixtures The complex dielectric permittivity of a heterogeneous . mixture, εm, can be described by the formula first proposed by Birchak et al. (1974). This is the so-called refraction model, as it summarizes the relative volumetric portions of the components comprising the sample, multiplied by their refracε ): tion indices (n = √ . . εp , εm = ∑ Vp √ √ (6) p
Fig. 4. Dependence of the real part of dielectric permittivity on salt solution percentage at 25 ºC for different frequencies.
where Vp is a volumetric portion of the pth component of the . mixture and εp is the complex relative dielectric permittivity (CDP) of this component. In our case, the mixture includes bentonite, oil, salt solution, and air. The CDP of bentonite solids and air does not depend on frequency, and the frequency dependences of CDP for both oil and salt solution can be found by the generalized Debye relaxation model for conducting liquids (Akhadov, 1972): σp εSp − ε∞p . εp = ε∞p + +j . ωε0 1 + jω τp . Here, εp is the CDP of oil or salt solution and εSp, ε∞p, τp, and σp are the parameters of the Debye model (static and high-frequency dielectric constants, relaxation time, and conductivity, respectively). These parameters were determined earlier (Epov et al., 2009). In that study we showed that the measured values of the real part of the CDP of the bentonitebased mixture are significantly higher than those calculated by the model, especially in the low-frequency range. The increase in the measured dielectric permittivity at low frequencies and the frequency dependences of ε′ and σ (Fig. 3) evidence the existence of two additional relaxation processes caused by the interlayer polarization, with the overlapping regions of their run. In accordance with Akhadov (1972), we assume that the CDP mixture can be determined as a sum of the CDPs corresponding to each low-frequency relaxation process and the CDP of the mixture, as given by (6). We described the low-frequency relaxation processes by the Cole–Cole empirical formula taking into account the distribution of relaxation times (Akhadov, 1972): . εi = ε∞i +
∆εSi 1 + (jω τ i)βi
,
(7)
where βi are the distribution coefficients of relaxation times, 0 ≤ β ≤ 1; index i (i = 1, 2) marks the model parameters describing particular relaxation processes. At frequencies higher than 4 GHz, low-frequency relaxation is almost absent; therefore, we took dielectric permittivity determined from (6).
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Fig. 5. Temperature dependence of static dielectric constants (A, B) and distribution coefficients of relaxation times (C, D) for 0% (1), 10% (2), 33.3% (3), and 66.7% (4) salt solution. A, C, First relaxation region; B, D, second relaxation region.
Thus, the final formula for the CDP of the mixture looks as 2 2 ∆εSi . σ + ε = ∑ εp + j , Vp √ ∑ ω ε0 β i = 1 1 + (jω τ i) i p
(8)
i.e., a sum of two terms describing low-frequency relaxation (parameters with index i), which is complemented with the CDP of the sample calculated by (6). The volumetric percentage in this formula was assigned to take into account the presence of air in the sample. In addition, the imaginary part of the CDP related to the ionic conductivity is taken into account by the third term in (8). The second term in (8) was calculated using the parameters determined by Epov et al. (2009), with the values of ionic conductivities of oil and with water solution set equal to zero. The parameters in Cole–Cole equation (8) were derived by minimization of the residual with the experimental data first by the least-squares method and then by the minimax method. The minimization problem was solved repeatedly at different initial values of the parameters derived. After the determination of the model parameters, we developed the regression equations for the temperature dependence and again performed minimization of the residual with the initial parameters found from the regression equations. Thus, we succeeded in reducing the deviation of the calculated data from the experimental ones to 10–12% for ε′ and 5–8% for σe (average values). The maximum deviations for ε′ were observed for the sample with 66.7% salt solution and reached 20–22% at some points.
The determined relaxation time in the first (from the upper frequencies) relaxation region is (1–3) × 10−8 s. The relaxation time increases 1.5–2 times as temperature grows from 25 to 60 ºC and ~1.5 times as the percentage of salt solution increases from 0 to 66.7%. The relaxation time in the second relaxation region is 10–4–10–5 s. The relaxation time decreases 1.5–2 times with temperature growth from 25 to 60 ºC and increases 3–4 times with an analogous increase in the percentage of salt solution. The values of the model parameters εSi and βi are given in Fig. 5. In the first relaxation region, the static dielectric constants of the samples with different water saturation increase about 1.5 times with temperature (Fig. 5, A), and in the second relaxation region they increase for all samples except for one with high water saturation (Fig. 5, B). The distribution coefficients of relaxation times vary from 0.3 to 0.6 in the first region (Fig. 5, C) and from 0.4 to 1 in the second region (Fig. 5, D).
Conclusions In the frequency region from 10 kHz to 4 GHz at 25–60 ºC, experimental spectra have been obtained for the dielectric permittivity and conductivity of clayey rocks saturated with oil and salt solution in different proportions, with their total amount being 20% relatively to the volume of the sample. A model for the complex dielectric permittivity has been elaborated, which takes into account low-frequency relaxation
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processes. The model parameters ensuring agreement of the calculated CDPs with the experimental ones in the entire frequency region have been found, and their dependence on temperature and water saturation has been established. The error of prediction of the dielectric permittivity and conductivity of mixtures by the proposed model has been estimated. This work was supported by budget project 2.5.1.1 “Dielectric spectroscopy of natural media in the radio wave frequency region” and Interdisciplinary Project 6 “The theoretical grounds of the principally new technology of sounding in oil-gas wells using subnanosecond electromagnetic pulses” from the Siberian Branch of the Russian Academy of Sciences.
References Akhadov, Ya.Yu., 1972. The Dielectric Properties of Pure Liquids [in Russian]. Izd. Standartov, Moscow. Birchak, J.R., Gardner, G.G., Hipp, J.E., Victor, J.M., 1974. High dielectric constant microwave probes for sensing soil moisture. Proc. IEEE 62 (1), 93–98. Epov, M.I., Antonov, Yu.N. (Eds.), 2000. The Technology of Study of Oil-Gas Wells by the High-Frequency Logging Method. Methodical Manual [in Russian]. Izd. SO RAN, NITs OIGGM SO RAN, Novosibirsk.
Epov, M.I., Glinskikh, V.N., 2005. Electromagnetic Logging: Modeling and Inversion [in Russian]. Akademicheskoe Izd. “Geo”, Novosibirsk. Epov, M.I., Mironov, V.L., Bobrov, P.P., Savin, I.V., Repin, A.V., 2009. Dielectric spectroscopy of oil-bearing rocks at 0.05–16 GHz. Russian Geology and Geophysics (Geologiya i Geofizika) 50 (5), 470–474 (613–618). Hipp, J.E., 1974. Soil electromagnetic parameters as function of frequency, soil density, and soil moisture. Proc. IEEE 62 (1), 98–103. Komarov, S.A., Mironov, V.L., 2000. Microwave Sounding of Soils [in Russian]. Nauka, Novosibirsk. Levitskaya, T.M., Sternberg, B.K., 1996. Polarization processes in rocks. Radio Science 3 (4), 755–759. Mattey, D.L., Young, L., Jones, E.M.T., 1964. Microwave Filters, ImpedanceMatching Networks and Coupling Structures. McGraw-Hill, New York. Mironov, V.L., Dobson, M.C., Kaupp, V.H., Komarov, S.A., Kleshchenko, V.N., 2004. Generalized refractive mixing dielectric model for moist soils. IEEE Trans. Geosci. Remote Sensing 42 (4), 773–785. Mironov, V.L., Kosolapova, L.G., Fomin, S.V., 2007. The method of compiling a spectroscopic database of the dielectric properties of moist soils in wicrowave frequency range. Izvestiya Vuzov. Radiofizika 50 (4), 339–349. Talalov, A.D., Daev, D.S., 1996. The structural mechanism of the frequency dispersion of the electric properties of heterogeneous rocks. Izv. RAN. Fizika Zemli, No. 8, 56–66.
Editorial responsibility: A.D. Duchkov