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УДК 550.31
ЛАБОРАТОРНОЕ ИССЛЕДОВАНИЕ ДИСПЕРСИИ ДИЭЛЕКТРИЧЕСКОЙ ПРОНИЦАЕМОСТИ ПОРИСТОГО ФЛЮИДОНАСЫЩЕННОГО МАТЕРИАЛА НА ОБРАЗЦАХ, ИЗГОТОВЛЕННЫХ МЕТОДАМИ 3D-ПЕЧАТИ
Алексей Владимирович Бондаренко
Новосибирский технологический центр «Бейкер Хьюз», 630090, Россия, г. Новосибирск, ул. Кутателадзе, 4а, кандидат физико-математических наук, научный сотрудник, тел. (383)332-94-43 (доп. 146), e-mail: Alexey.Bondarenko@bakerhughes.com
Николай Николаевич Велькер
Новосибирский технологический центр «Бейкер Хьюз», 630090, Россия, г. Новосибирск, ул. Кутателадзе, 4а, научный сотрудник, тел. (383)332-94-43 (доп. 129), e-mail: Nikolay.Velker@bakerhughes.com
Станислав Вильгельмович Форганг
Хьюстонский технологический центр «Бейкер Хьюз», США, Техас 77073-5114, Хьюстон, 2001 Ранкин Роуд, доктор физико-математических наук, технический руководитель, тел. +1 713 879 3548, e-mail: Stanislav.Forgang@bakerhughes.com
Юлий Александрович Дашевский
Новосибирский технологический центр «Бейкер Хьюз», 630090, Россия, г. Новосибирск, ул. Кутателадзе, 4а, доктор физико-математических наук, директор, тел. (383)332-94-43 (доп. 102), e-mail: Yuliy.Dashevsky@bakerhughes.com
В статье представлены диэлектрические спектры искусственного флюидонасыщенного пористого образца горных пород. Образец был изготовлен с помощью метода 3D-печати, содержал вертикально-соосные прямоугольные поры и был насыщен дистиллированной и пресной водой. Полученные экспериментально спектры диэлектрической проницаемости позволяют сделать вывод о том, что независимо от минерализации поровой жидкости комплексная диэлектрическая проницаемость образца пропорциональна квадратному корню частоты.
Ключевые слова: диэлектрический спектр, пористые флюидонасыщенные образцы, 3D-печать.
LABORATORY STUDY OF DIELECTRIC PERMITTIVITY DISPERSION OF POROUS FLUID-SATURATED MATERIAL IN SAMPLES MANUFACTURED BY 3D PRINTING
Alexey V. Bondarenko
Novosibirsk Technology Center «Baker Hughes», 630090, Russia, Novosibirsk, 4a Kutateladze St., Ph. D., RDD Scientist 3, tel. (383)332-94-43 Ext 146, e-mail: Alexey.Bondarenko@bakerhughes.com
Nikolay N. Velker
Novosibirsk Technology Center «Baker Hughes», 630090, Russia, Novosibirsk, 4a Kutateladze St., RDD Scientist 5, tel. (383)332-94-43 Ext 129, e-mail: Nikolay.Velker@bakerhughes.com
Stanislav W. Forgang
Houston Technology Center, Baker Hughes, USA, Texas 77073-5114, Houston, 2001 Rankin Road, Ph. D., Technical Advisor, tel. +1 713 879 3548, e-mail: Stanislav.Forgang@bakerhughes.com Yuliy A. Dashevsky
Novosibirsk Technology Center «Baker Hughes», 630090, Russia, Novosibirsk, 4a Kutateladze St., Ph. D., Deputy Director Science, tel. (383)332-94-43 Ext 102, e-mail: Yuliy.Dashevsky@bakerhughes.com
The paper presents the dielectric spectra of an artificial fluid-saturated rock sample. The artificial rock was produced by 3D printing and saturated with distilled and fresh water. The sample has vertically oriented coaxial rectangular pores. The experimentally obtained permittivities spectra make it possible to conclude that the complex permittivity of the sample has square root frequency dependence, regardless of mineralization of porous fluids used in the study.
Key words: dielectric permittivity spectrum, fluid-saturated porous samples, 3D printing.
Dispersion of dielectric permittivity is one of important rocks properties [1-6] and the bases of the dielectric borehole logging method, when water-saturated porosity and other formation parameters are evaluated [7]. In study of dielectric spectra of fluid-saturated porous rocks are the following difficulties of interpretation of the experimental data: uncertainty of matrix and porous fluid chemical composition, possible presence of clay in the sample, unknown pore structure.
The objective of our study was to develop the simplest model of fluid saturated porous material and investigate its dielectric permittivity spectrum. This approach has a number of advantages over the classical study of rock samples: known chemical composition of matrix and porous fluid; simple and known pore structure.
The sample of disc shape with the radius of 38 mm and the height of 6.35 mm was produced using a 3D printer. The sample contains vertically oriented coaxial rectangular pores of 1.5 mm thick and the distance between pore centers is 2.0 mm. To print the sample, one used a plastic nonconductive material with the dielectric permittivity value equal to 2. The drawing and scheme of the sample is represented in Fig. 1. The sample's photo is shown in Fig. 2.
Fig. 1. The porous artificial sample produced by a 3D printer. Sizes are given in inches (a); the scheme of water saturated sample (b)
The idea behind the experiment was to create such a fluid-saturated porous sample, where the electric fields induced by a parallel plate capacitor do not intersect the interfacial boundary between the matrix (plastic) and fluid. In this case, the pore surface should remain electrically neutral. Therefore, the described system should get polarized only due to the presence of an electrical double layer [8, 9] at the interfacial boundary.
Dielectric spectrum measurements in the sample saturated with distilled and fresh (10 Qm) water were carried out in a parallel plate capacitor (Fig. 2) within the range of 5 kHz - 10 MHz using the HP4194A impedance analyzer [10].
Fig. 2. The porous artificial sample produced in a 3D printer (a); the parallel plate capacitor used in the permittivity spectra measurements (b)
The measurement results are shown in Fig. 3 and 4.
Fig. 3. Frequency dependence of real (s') and imaginary (s'') parts of dielectric permittivity of the sample saturated with distilled water. The blue line is the measured values; the black line is an interpolation curve
Fig. 4. Frequency dependence of real (s') and imaginary (s'') parts of dielectric permittivity of the sample saturated with fresh (10 Qm) water. The blue line is the measured values; the black line is an interpolation curve
The experimental spectra were approximated by the Havriliak-Negami curve (1), which is a standard equation for approximation of dielectric spectra for various types of dielectric materials [5, 6, 11].
e(ra) = s-is =s^+--—F--, (1)
[ 1 + (/rax)1-" J ®so
where £ denotes complex dielectric permittivity, sm - permittivity in high-frequency limit, A£ - degree of polarization, t - relaxation time, a and p - polarization parameters, o - DC conductivity.
The Havriliak-Negami curve parameters obtained by inversion of the experimental data are shown in Tab. 1. Comparison of the approximating curves and experimental values of measured permittivity is shown in Fig. 3 and 4.
Table 1
The Havriliak-Negami curve parameters
Porous fluid sro As a ß t, ms a, ^S/m
Distilled water 2.30 18.7 0.429 0.859 0.258 0.376
Fresh (10 Qm) water 8.03 690 0.355 0.749 0.185 22.1
The data in the Table 1 allows us to conclude the following: • In the frequency range of 5 kHz-10 MHz the following condition is true:
©T >> 1. (2)
• In both cases (distilled and fresh water saturation) coefficients a and p of the Havriliak-Negami curve satisfy the following condition:
(1 -a)P = 0.5 ± 0.02. (3)
Then the Havriliak-Negami curve can be reduced to the following:
e(ro) = + -—, (4)
yjm ras0
where:
A = As/Vr. (5)
The second term in (4) describes the dielectric permittivity dispersion. Thus, the conducted experiments suggest that the complex permittivity dispersion of the sample has a square root frequency dependence, regardless of porous fluid mineralization.
According to [12] one of possible explanation of the square root frequency dependence is the polarization of the electric double layer at the interfacial boundary due to the motion of diffusion ions along the Stern layer. This effect should lead to the following relation between dielectric permittivity and frequency (equation 34 in [12]):
£(W)~W-(3-d/)/2 (6)
here Df is fractal dimension of the interfacial boundary. The considered sample has identical rectangular pores, thus Df should be equal to 2. This leads to a square root dependence of permittivity on frequency and agrees with the experimentally obtained relation (4).
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© А. В. Бондаренко, Н. Н. Велькер, С. В. Форганг, Ю. А. Дашевский, 2017
