Summary We present a specially designed experimental setup for accurate measurements of the frequency-dependent relative complex dielectric permittivity (RCDP) in porous media. The setup operates on the principles of steady-state flooding and time-domain reflectometry (TDR). The steady-state flooding technique allows for a well-controlled uniform saturation distribution in the sample. The TDR technique enables on-line measurement of the dielectric response. We derive the equations for the propagation and reflection of an electromagnetic (EM) signal along a coaxial transmission line consisting of a standard coaxial cable, a transition unit, and the sample holder. The RCDP at different oil saturations are calculated by means of frequency analysis of the scattered signal.
We present the RCDP obtained from experiments in sand samples at different saturations. We compare the obtained results with those calculated with a number of existing mixing models, including the complex refractive index (CRI) model and the classical Rayleigh formula.
The experimental method has proved to be suitable for on-line measurement of dielectric properties of porous media with varying fluid saturation. It turns out that a small saturation difference can be discerned in the measurements. We have found that the RCDP calculated with the CRI model (refractive index exponent ?=0.75) show the best agreement with those obtained from the experiments.
Introduction The interpretation of data obtained with the electromagnetic propagation tool (EPT) or ground penetrating radar (GPR) requires knowledge of the dielectric properties of water- and oil-saturated porous media. Existing mixing models of dielectric properties give strongly divergent results. To examine the applicability of existing mixing models, it is necessary to have accurate calibration data.
Numerous researchers have performed calibration measurements in laboratory experimental setups. Most of the measurements suffer from an inhomogeneous distribution of fluids. In many cases, the setup had no special construction to reduce fluid inertia effects at the inlet or capillary end effects at the outlet.
In the next section, we explain the principles of time-domain reflectometry (TDR) which we use to measure the dielectric response of porous media. Subsequently, we describe the experimental setup and procedure. Finally, we present and discuss the obtained results.
Appendix A presents the derivation of multiple reflection coefficient for an electromagnetic (EM) signal traveling along a multisection coaxial line. Appendix B describes the procedure we use for the frequency analysis of the measured scatter functions. Appendix C gives a brief summary on the empirical mixing models for multicomponent materials with which we compare the results of our measurements.
The Time-Domain Reflectometry (TDR) Technique Basic Principles. The theoretical background of the TDR technique is described in Ref. 5. When an EM wave is launched into a (coaxial) cable, any change in the electrical and dielectric properties of the cable will cause a partial or total reflection of the wave. Changes in the electrical and dielectric properties cause change in the impedance. Waves reflected on a discontinuous boundary either are in phase or in counter phase with the incoming wave. The voltage amplitude of the reflected waves is a function of the change in impedance which causes the reflection (see Appendix A for detailed description). If the EM wave encounters an increase in impedance, the reflected wave will be in phase with the incoming wave. If the EM wave encounters a decrease in impedance, the reflected wave will be counter phase with the incoming wave.
A standard TDR device consists of a signal generator, a sampler, and an oscilloscope. The signal generator launches the EM wave into the cable. From the measured voltage, it is possible to determine the change in impedance that causes the reflection. In reverse, the amplitude of the reflection can be calculated if the initial voltage step and the impedance on both sides of the reflecting interface are known. The reflection coefficient at such an interface is defined by where Vr and V0 are the amplitude of the reflected and incoming signals, respectively; Z1 is the impedance in the coaxial line where the wave comes from and Z2 is the impedance in the coaxial line where the wave continues.
Open-End Reflection and Its Applications. When a coaxial line is open-ended, the impedance tends towards infinity and the wave is reflected in phase. The voltage amplitude of the reflected wave is theoretically the same as the incoming wave (reflection coefficient R is unity). The reflected wave is in phase with the incoming wave resulting in a voltage amplitude being twice the initial amplitude. By measuring the time needed for the wave to travel from the generator to the open end and back, the propagation velocity v of the wave in the coaxial line can be calculated if the dielectric permittivity is known:
where c0 is the velocity of light in vacuum (3.0×10 m/s) and is the complex dielectric permittivity given by where and are the real and imaginary part of the complex dielectric permittivity, ? dc is the direct current electric conductivity, ? is the circular frequency, and is the dielectric constant of free space. Hence v is also a complex parameter.