Calibration Measurements of DielectricProperties of Porous Media

Buu-Long Nguyen,* SPE, A.M. Geels,** Johannes Bruining, and E.C. Slob, Delft U. of Technology

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SummaryWe present a specially designed experimental setup for accumeasurements of the frequency-dependent relative complexelectric permittivity~RCDP! in porous media. The setup operaton the principles of steady-state flooding and time-domain refltometry ~TDR!. The steady-state flooding technique allows fowell-controlled uniform saturation distribution in the sample. TTDR technique enables on-line measurement of the dielectricsponse. We derive the equations for the propagation and refleof an electromagnetic~EM! signal along a coaxial transmissioline consisting of a standard coaxial cable, a transition unit,the sample holder. The RCDP at different oil saturations areculated by means of frequency analysis of the scattered sign

We present the RCDP obtained from experiments in ssamples at different saturations. We compare the obtained rewith those calculated with a number of existing mixing modeincluding the complex refractive index~CRI! model and the clas-sical Rayleigh formula.

The experimental method has proved to be suitable for on-measurement of dielectric properties of porous media with vaing fluid saturation. It turns out that a small saturation differencan be discerned in the measurements. We have found thaRCDP calculated with the CRI model~refractive index exponena50.75! show the best agreement with those obtained fromexperiments.

IntroductionThe interpretation of data obtained with the electromagnpropagation tool~EPT! or ground penetrating radar~GPR! re-quires knowledge of the dielectric properties of water- andsaturated porous media. Existing mixing models of dielecproperties give strongly divergent results. To examine the apcability of existing mixing models, it is necessary to have accurcalibration data.

Numerous researchers have performed calibration measments in laboratory experimental setups.1-4 Most of the measure-ments suffer from an inhomogeneous distribution of fluids.many cases, the setup had no special construction to reduceinertia effects at the inlet or capillary end effects at the outlet

In the next section, we explain the principles of time-domareflectometry~TDR! which we use to measure the dielectric rsponse of porous media. Subsequently, we describe the exmental setup and procedure. Finally, we present and discusobtained results.

Appendix A presents the derivation of multiple reflection coficient for an electromagnetic~EM! signal traveling along a mul-tisection coaxial line. Appendix B describes the procedure wefor the frequency analysis of the measured scatter functions.pendix C gives a brief summary on the empirical mixing modfor multicomponent materials with which we compare the resuof our measurements.

*Now with Dames & Moore.** Now with Iwaco B.V.

Copyright © 1999 Society of Petroleum Engineers

This paper (SPE 57930) was revised for publication from paper SPE 38715, presented atthe 1997 SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, 5-8October. Original manuscript received for review 17 October 1997. Revised manuscriptreceived 25 May 1999. Manuscript peer approved 7 June 1999.

SPE Journal4 ~4!, December 1999

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The Time-Domain Reflectometry „TDR… TechniqueBasic Principles.The theoretical background of the TDR tecnique is described in Ref. 5. When an EM wave is launched ia ~coaxial! cable, any change in the electrical and dielectric proerties of the cable will cause a partial or total reflection of twave. Changes in the electrical and dielectric properties cachange in the impedance. Waves reflected on a discontinuboundary either are in phase or in counter phase with the incing wave. The voltage amplitude of the reflected waves is a fution of the change in impedance which causes the reflection~seeAppendix A for detailed description!. If the EM wave encountersan increase in impedance, the reflected wave will be in phasethe incoming wave. If the EM wave encounters a decreaseimpedance, the reflected wave will be counter phase with thecoming wave.

A standard TDR device consists of a signal generator, a spler, and an oscilloscope. The signal generator launches thewave into the cable. From the measured voltage, it is possibldetermine the change in impedance that causes the reflectioreverse, the amplitude of the reflection can be calculated ifinitial voltage step and the impedance on both sides of the refling interface are known. The reflection coefficient at such anterface is defined by

R5Vr

V05

Z12Z2

Z11Z2, ~1!

whereVr andV0 are the amplitude of the reflected and incomisignals, respectively;Z1 is the impedance in the coaxial linwhere the wave comes from andZ2 is the impedance in the coaxial line where the wave continues.

Open-End Reflection and Its Applications.When a coaxial lineis open-ended, the impedance tends towards infinity and the wis reflected in phase. The voltage amplitude of the reflected wis theoretically the same as the incoming wave~reflection coeffi-cientR is unity!. The reflected wave is in phase with the incominwave resulting in a voltage amplitude being twice the initial aplitude. By measuring the time needed for the wave to travel frthe generator to the open end and back, the propagation velocvof the wave in the coaxial line can be calculated if the dielecpermittivity is known:

v5c0 /Ae* , ~2!

wherec0 is the velocity of light in vacuum (3.03108 m/s) ande*is the complex dielectric permittivity given by

e* 5e82 i S e91sdc

ve0D , ~3!

wheree8 and e9 are the real and imaginary part of the compldielectric permittivity,sdc is the direct current electric conductivity, v is the circular frequency, ande0 is the dielectric constant ofree space. Hencev is also a complex parameter.

Application of TDR to the measurement of the dielectric proerties of material is described in Ref. 6. Application of TDRmeasurements in porous media, mostly in soils, has beencussed in detail by several authors.1,7,8A TDR probe is connectedto a standard coaxial cable and installed in the porous mediumsuch a way that the medium forms the dielectric material oftransmission line along the probe. When an electromagnetic~EM!signal is sent into such a transmission line, two reflections woccur. The first reflection occurs at the cable-probe connect

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The second one occurs at the end of the probe. The appadielectric constantea of the soil can be calculated with

ea5~c0Dt/2L !2, ~4!

whereDt is the travel time of the TDR signal in the soil andL isthe length of the probe embedded in the porous medium. Tapproach does not, however, account for the finite rise time ofsignal and therefore the frequency dependence of the TDRtem. Besides, effects of electric loss and the different frequenin the frequency range do influence the measurement to a ceextent. For that reason, the apparent dielectric constantea is notequal to the real parte8 of the complex dielectric permittivity.

A Frequency Approach for Time-Domain ReflectometryTDR measures the electric-voltage as a function of time. Tresponse functionVr(t) is the convolution of the input signal withthe system functionS(t):9

Vr~ t !5E2`

`

V0~ t2t!S~t!dt, ~5!

in which t is the integration variable. Eq. 5 can be transformed,the convolution theorem,10 to the frequency domain to obtain thfrequency-dependent response function. This gives

Vr~v!5V0~v!S~v!, ~6!

where Vr(v), V0(v), and S(v) are the Fourier transforms oVr(t), V0(t), andS(t). The Fourier transform is given by

X~v!5E2`

`

x~ t !exp~2 ivt !dt. ~7!

S(v) andS(t) are functions of the dielectric behavior and proerties of the transmission line including the sample holder~con-taining porous medium!. The expression for reflection in the frequency domain is given in Appendix A.

Experimental ProcedureFeatures of the Experimental Setup.The main part of the ex-perimental setup is the sample holder with a built-in TDR pro~Fig. 1! which is connected with a network analyzer by a stand50 V coaxial cable. For proper calibration measurements,sample holder should satisfy the following requirements:~1! Noadditional reflection due to the transition unit connecting theaxial cable and the cell;~2! the probe length is limited by theeffects of signal attenuation; and~3! a uniform distribution offluids within the measuring volume.

To avoid additional reflections and to adapt the diameters ofcable and the coaxial cell we designed a cone-shaped transunit with PTFE as dielectric material~Fig. 1!. Measurement of theimpedance of the transition unit provided a value of 50.04V,which is adequate. The sample holder—a stainless steel cyliof 3 cm ~ID! diameter—acts as outer conductor~shield!. A stain-less steel wire of 0.16 cm diameter and 10 cm length formsinner conductor. The length of the wire is optimal for a connectcable shorter than 15 m.11

To create a uniform saturation distribution within the cell, wapply the steady-state flooding technique that is widely usedlaboratory relative permeability measurements.12 The primaryconcern in designing the experiment is to reduce the effectstagnation resulting in fluid ‘‘tongues’’ at the inflow side, and t

Fig. 1–Technical drawing of the coaxial sample holder with thebuilt-in TDR probe.

354 Nguyenet al.: Dielectric Properties of Porous Media

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saturation gradient caused by capillary pressure effects at theflow side of the cell. This can be achieved with an optimal injetion rate.12 Also for this reason, the measuring volume~around theinner conductor! in our design begins and ends at some distafrom both the inflow and outflow sides~Fig. 1!. This means thateffects of incomplete mixing and the capillary end effect occlargely outside the measuring volume.

Material and Preparation of Experiment. We use river sand inall experiments. The sand is washed and dried in a laborafurnace, then passed through a 2.0 mm sieve. The porous meis packed into the coaxial cell using a ‘‘particle distributor.’’13

The method gives very uniform lateral and longitudinal depositof sand with high compaction. With this technique we manageget a porosity of 0.34060.005. The porosity of the porous medium is determined for every experiment. The working fluids an-decane and distilled water. The coaxial cell filled with sandfirst brought under vacuum, then saturated with carbon dioxand then again under vacuum. After that, the porous mediumslowly saturated with water or withn-decane, depending on thexperimental scenarios.

Experimental Process. Fig. 2schematically presents the expermental setup. The pumps~1,2! withdraw water and oil respectively from the reservoir~4! and inject them to the coaxial cell~3!.Produced fluids flow back to the reservoir. The balance~5!, onwhich the fluid reservoir is situated, records changes in total flweight during the experiment and transfers them to the comp~6!. When equilibrium is reached, i.e., when the total fluid weigis constant, the computer activates the network analyzer~7!. Thelatter sends an EM signal through the cable to the coaxial cellmeasures theS11 scatter functions~see explanation below!, that isrecorded by the computer. From the initial and final total fluweight, we calculate the fluid saturation in the porous mediuDifferent fluid saturations can be made by varying the ratio ofoil/water injection flow rates.

Measuring the S11 Scatter Function with a Network Analyzer.The network analyzer has two ports. Each port can be used toand to receive a signal. The situation is shown schematicallyFig. 3. Here,a represents incident signals or the square root ofpowers going into the system;b represents signals flowing out thdevice under test~the sample holder in our case!. The network canbe characterized by four independent variable calledSparameters.S11 andS22 are forward and reverse reflection parameters, andS21and S12 are the forward and reverse transmission parametFrom the theory of nodes, it follows that the received signalb

Fig. 2–Schematic presentation of the experimental setup.

Fig. 3– S-Parameters of a two-port device „e.g., a network ana-lyzer ….

SPE Journal, Vol. 4, No. 4, December 1999

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can be expressed in terms of the incident signalsa and theSi jparameters (i , j 51,2):

b15S11a11S12a2 , ~8!

b25S21a11S22a2 . ~9!

In our case, the sample holder is attached only to one port~port Iof the network analyzer!. Hence,a25b250 andS115b1 /a1 . Forthe expression of theS11 scatter function, refer to Appendix B.

Procedure for Optimization of the Scatter Function. We derivein Appendix A the expression for the multiple reflection coefcient at the first interface. We show in Appendix B that theS11scatter function measured with the network analyzer equalsmultiple reflection coefficient. The method is as follows:

• We measure theS11 scatter function of every sample witunknown dielectric properties.

• We use our reflection model to calculate the theoreticaexpectedS11 scatter function. For this purpose, we have to suggsome initial estimates for the so-called Cole–Cole parameterses ,e` , v rel ~these parameters are explained in Appendix B!.

• We apply then a standard complex iteration technique10 tooptimize the suggested Cole–Cole parameters in such a waythe difference between the measured and theoretically calcuS11 scatter function will be minimum.

• The relative complex dielectric permittivity can then be cculated with the Cole–Cole equation including the energy ldue to ionic conductivity~Eq. B-5!.

Survey on the Validity of the Derived Model and theOptimization ProcedureValidity of the Model for the Multiple Reflection Coefficient.In order to verify the validity and applicability of the deriveexpression for the reflection model and theS11 scatter function,we have done measurements in standard media with knownelectric properties.

For standard media we used air~with open end and short circuit!, butanol and distilled water. The values of the Cole–Cparameters are taken from Ref. 11 and are presented here inTable1. We used our model to calculate the scatter functions ofthree media and compared them with the measured ones.

Fig. 4 shows the real parts of the measured and calcula

TABLE 1– COLE–COLE PARAMETERS OF STANDARDMEDIA „REF. 11…

Medium es e` f rel (Hz) b

Air 1.0 1.0 1.0 0.0Butanol 17.70 3.3 0.274E9 0.0Water 79.88 4.22 17.0E9 0.0125

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scatter functions of air, butanol, and water. For air, the moagrees very well with the measured data except for the slignoisy appearance. This noise may be caused by unwanteduncontrolled reflections along the transmission line.

In the case of butanol, the noise is stronger especially atquencies higher than 1 GHz. However, we still observe a gagreement between the modeled and the measured paramet

Another picture is obtained for water. A fairly good agreemecan still be observed up to a frequency of about 750 MHz. Beyothat frequency, the amplitude of the measuredS11 scatter functiondecreases faster in comparison with the modeled one. This caexplained by an occasional occurrence of higher-order wavethe transmission line which causes additional losses of the trmission electron microscope~TEM! waves and significantchanges in the characteristics of the line. High-order waves ocin a coaxial transmission line when the line cross sectiondimensions that are comparable with the operating wavelenFor a coaxial transmission line, the upper limit for the operatfrequency~below which the TEM mode is valid! is given by14

f c52c0

~D1d!pAe r

, ~10!

wherec0 is the velocity of light in vacuum,D is the inner diam-eter of the outer conductor,d is the diameter of the inner conductor, ande r is the relative dielectric constant of the material.

The conductor diameter of our coaxial cell areD530 cm andd50.16 cm. With the relative dielectric constant of watere r579.88, the limit operating frequency for the cell filled with water is f c,w5676 MHz.

Fully water-saturated porous media with a porosity of 0.would have a relative dielectric constant of about 20. Hencelimit frequency, up to which the measurement results are reliais f c,PM51.35 GHz.

For measurements at higher frequencies, we can adjust thmensions of the coaxial cell.

Validation of the Optimization Procedure. We have done measurements in ethanol at 24°C and applied the optimization produre to this data set. The simulation provided the optimized CoCole parameters which are presented inTable 2 together with theCole–Cole parameters found in the literature.9 The parametersshow close agreement.

TABLE 2– COMPARISON BETWEEN THE COLE–COLEPARAMETERS FOR ETHANOL AT 24°C

Source es e` f rel (Hz)

van Gemert (Ref. 9) 24.75 4.55 0.902E9Experiment 25.31 4.76 0.917E9

Fig. 4–Validation of the model: comparison between the measured and modeled multiple reflection coefficients for air „left …,butanol „center …, and water „right ….

SPE Journal, Vol. 4, No. 4, December 1999 355

Fig. 5–Validation of optimization procedure: a comparison between measured, optimized, and theoretical multiple reflectioncoefficients.

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We calculate the scatter function with the two Cole–Colerameter sets. We call them optimized and theoretical scatter ftions, respectively.

Fig. 5 shows on the left the measured and optimized~or simu-lated! S11 scatter function for the investigated ethanol, and onright, the theoretical and the optimized one. There is good agment between the theoretical and optimized parameter.

Results of Measurements in Fluid-Saturated Sand andDiscussionWe have made eight different water and decane saturations.measured scatter functions are used in the described optimiz

Fig. 6–Real and imaginary parts of the relative complex dielec-tric permittivity at different saturation.

356 Nguyenet al.: Dielectric Properties of Porous Media

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procedure to obtain the Cole–Cole parameters. The relative cplex dielectric permittivities~RCDP! are then calculated using EqB-5.

Fig. 6 presents the real and imaginary parts of the sand afunction of frequency at different saturations for the frequenband 500 kHz–750 MHz. As it is can be expected, the eneloss—characterized by the imaginary parts of the RCDP,creases with increasing water saturation as well as with increafrequency.

Fig. 7 presents a comparison between the RCDP calculawith a number of mixing formulas reported in the literature athose calculated following our procedure.

Rayleigh’s and Odelevskyi’s models show a same trend.both cases, the RCDP increases slightly with increasing wsaturation.

Fig. 7–A comparison between RCDP calculated with existingmixing models and the RDCP calculated from experimentaldata.

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Theoretical results obtained with the complex refractive indformula ~refractive index exponenta50.5! and the Berentsveigformula show good agreement. They also show the same incring trend.

The RCDP calculated from our experimental data are sowhat higher than those calculated with existing models. Usinstandard fitting technique, we ascertained that the experimeRCDP are located near the curve of the CRI model takinga50.75. This should be, however, verified with more calibratidata.

ConclusionsThe experimental setup has proved to be very suitable forcalibration measurements of the dielectric properties of pormedia with varying fluid saturation.

The technique has a resolution of a few saturation units. It isimportance in particularly for environmental engineering to detspilled petroleum products which are often present in the subface at low saturation.

We are now working on improvement of the optimization prcedure for porous media with a distribution of relaxation time.g., sandstone containing clay minerals.

The effects of the setup configuration on energy loss ofsignal and the applicability of this technique to layered systeare also being investigated.

Nomenclature

a 5 incident signal, Eqs. 8 and 9b 5 received signal, Eqs. 8 and 9

c0 5 propagation velocity of electromagnetic wavesfree space, m/s

f 5 frequency, s21

f v i 5 volume fraction of componenti ~equations in Ap-pendix C!, volume fraction of inclusion~only Eq.C-6!

G 5 Green’s functionJ(z,v) 5 electric-current function, A

R 5 reflection coefficientS11 5 scatter functionSw 5 water saturationV0 5 amplitude of the incoming EM signal~Eq. 1!, V/mVr 5 amplitude of the reflected EM signal~Eq. 1!, V/m

V(z,v) 5 electric-potential function, V/mY 5 admittance,V21

Z 5 impedance of a coaxial transmission line,Vv 5 propagation velocity of electromagnetic waves, mz 5 coordinate, directed along the transmission line,

Greeks

a 5 curve-fitting coefficient in Eq. C-3g 5 propagation constantb 5 relaxation time spread coefficient

e0 5 permittivity of vacuum, F/mea 5 apparent dielectric constante* 5 complex relative dielectric permittivitye8 5 real part of relative dielectric permittivitye9 5 imaginary part of relative dielectric permittivityes 5 static ~low-frequency! dielectric constante` 5 infinite-frequency dielectric constant

sdc 5 dc conductivity of soil,V/msw 5 dc conductivity of water,V/mt0 5 generalized relaxation timef 5 porosityv 5 cycle frequency, s21

Subscripts

m 5 mixture,h 5 host mediumi 5 componenti

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AcknowledgmentsWe thank Dr. T.J. Heimovaara from IWACO BV for valuabladvice and for many insightful discussions. We also thanktechnical staff of the Dietz Laboratory for Petroleum Engineerfor their assistance in the design, construction and maintenancthe experimental setup, and in the design of the data acquissystem.

References1. Topp, G.C., Davis, J.L., and Annan, A.P.: ‘‘Electromagnetic Det

mination of Soil Water Content: Measurements in Coaxial Transmsion Lines,’’ Water Resour. Res.~1980! 16, 574.

2. Hallikainen, M., Ulaby, F.T., and Dobson, M.C.: ‘‘Microwave Dielectric Behavior of Wet Soil—Part I: Empirical Models and Expemental Observations,’’IEEE Trans. Geosci. Remote Sens.~1985!GE-23, 25.

3. Kutrubes, D.L.: ‘‘Dielectric Permittivity Measurements of Soils Satrated with Hazardous Fluids,’’ MS thesis, Dept. of Physics, ColoraSchool of Mines~1986!.

4. Dirksen, C. and Dasberg, S.: ‘‘Improved Calibration of Time DomaReflectometry Soil Water Content Measurements,’’Soil Sci. Soc. Am.J. ~1993! 57, 660.

5. Nissen, H.H. and Moldrup, P.: ‘‘Theoretical Background for the TDMethodology,’’ Proc., Symposium Time Domain Reflectometry—Applications in Soil Science, Research Centre Foulum, Septem1994.

6. Fellner-Feldegg, J.: ‘‘The Measurement of Dielectrics in the TimDomain,’’ J. Phys. Chem.~1969! 73, 616.

7. Davis, J.L. and Chudobiak, W.J.: ‘‘In-Situ Meter for Measuring Retive Permittivity of Soils,’’ Paper 75-1A, Geol. Surv. Ottawa, Cana~1975!, 77–79.

8. Dalton, F.N. and van Genuchten: ‘‘The Method for the MeasurSoil Water Content and Salinity,’’Geoderma~1986! 38, 237.

9. Gemert van, M.J.C.: ‘‘High-Frequency Time-Domain Methods in Delectric Spectroscopy,’’Philips Res. Rep.~1973! 28, 530.

10. Press, W.H.et al.: Numerical Recipes in C: the Art of Scientific Computing, second edition, Cambridge U. Press, Cambridge~1992!.

11. Heimovaara, T.J.: ‘‘Time Domain Reflectometry in Soil SciencTheoretical Background, Measurements and Models,’’ PhD dissetion, University of Amsterdam, Amsterdam, The Netherlands~1994!.

12. Honapour, M., Koederitz, L., and Harvey, A.H.:Relative Permeabil-ity of Petroleum Reservoir, CRC Press, Boca Raton, Florida~1986!.

13. Wygal, R.J.: ‘‘Construction of Models that Simulate Oil ReservoirsSPEJ~December 1963! 281; Trans.,AIME, 228.

14. Ramo, S., Winnery, J.R., and van Duzer, Th.:Fields and Waves inCommunication Electronics, Wiley, New York City ~1965!.

15. Nguyen, B.L.et al.: ‘‘Calibration Measurement of Dielectric Properties of Porous Media,’’ paper SPE 38715 presented at the 1997Annual Technical Conference and Exhibition, San Antonio, Tex5–8 October, Vol.V-II, 13.

16. Hewlett Packard: HP8753A Network Analyzer—System Operatand Programming Manual.

17. Cole, K.S. and Cole, R.H.: ‘‘Dispersion and Absorption in Dielectr1. Alternating Current Characteristics,’’J. Chem. Phys.~1941! 9, 341.

18. Shutko, A.M. and Reutov, E.M.: ‘‘Mixture Formulas Applied in Estimation of Dielectric and Radiative Characteristics of Soils aGrounds at Microwave Frequencies,’’IEEE Trans. Geosci. RemoteSens.~January 1982! GE-20~1!, 29.

19. Wang, J.R. and Schmugge, T.J.: ‘‘An Empirical Model for the Coplex Dielectric Permittivity of Soils as a Function of Water ContentIEEE Trans. Geosci. Remote Sens.~October 1980! GE-18~4!, 288.

20. Lord Rayleigh: ‘‘On the Influences of Obstacles Arranged in Rectgular Order on the Properties of a Medium,’’Philos. Mag.~1892! 34,481.

21. Odelevskyi, V.I.: ‘‘Raschet Obobschennoi Provodimosti GeterogeSystem’’ ~Calculation of composed conductivity of heterogeneosystems!, Z. Teknich. Fys.~1951! 21, 667.

22. Leshchanskyi, Yu. I. and Ulyanychev, N.V.: ‘‘Calculation of thElectrical Parameters of Sandy-Clay Soils at Meter and CentimWavelengths,’’Izv. Vyssh. Uchebn. Zaved., Radiofiz.~1980! 23, 529.

23. Birchaket al.: ‘‘High Dielectric Constant Microwave Probe for Sensing Soil Moisture,’’ Proc. IEEE~1974! 62, 93.

24. Roth, K.et al.: ‘‘Calibration of Time Domain Reflectometry for Wa-ter Content Measurements Using a Composite Dielectric ApproacWater Resour. Res.~1990! 26, 2267.

25. Dobson, M.C.et al.: ‘‘Microwave Dielectric Behavior of Wet Soil—Part II. Dielectric Mixing Models,’’ IEEE Trans. Geosci. RemoteSens.~1985! GE-23~1!, 35.

SPE Journal, Vol. 4, No. 4, December 1999 357

o

os

,c

d

ors

m

of

e

nt

the

ndthe

26. Sen, P.N., Scala, C., and Cohen, M.H.: ‘‘A Self-Similar Model fSedimentary Rocks with Application to the Dielectric ConstantFused Glass Beads,’’Geophysics~May 1981! 46~5!, 781.

27. Redman, J.D., DeRyck, S.M., and Annan, A.P.: ‘‘DetectionLNAPL Pools with GPR: Theoretical Modeling and Survey of a Cotrolled Spill,’’ Proc., Fifth International Conference on GPR~1994! 3,1283.

28. Geels, A.M.: ‘‘Calibration for TDR Saturation Measurements in SContaminated with DNAPL,’’ MS thesis, Dept. of Petroleum Engneering, Delft U. of Technology~1996!.

Appendix A–Derivation of Multiple ReflectionCoefficient for an Electromagnetic Signal TransmittedAlong a Heterogeneous Coaxial WaveguideWe consider the transmission and reflection of an EM signal ala coaxial waveguide. The latter shows two interrupt changeimpedance atz50 andz5d and is openly ended atz5L1d. Thesignal is generated by a source located atz5zs,0. In the regionz,0 ~region I!, the waveguide has the dielectric permittivitye1 ,electric conductivitys1 and impedanceZ1 . The dielectric permit-tivity, electric conductivity, and impedance in the region 0,z,d ~region II! are e2 , s2 , and Z2 , respectively. Analogouslye3 , s3 , andZ3 denote the dielectric permittivity, electric condutivity, and impedance of the waveguide in the regiond,z,L1d ~region III!.

The general wave equations are

]zV~z,v!1gZJ~z,v!50, ~A-1!

]zJ~z,v!1gYV~z,v!5Vsd~z2zs!, ~A-2!

whereV(z,v) andJ(z,v) are the electric-potential function anelectric-current function, respectively.

The propagation coefficientg is defined as

g5Aivm~s1 ive!. ~A-3!

Z is the impedance andY5Z21 is the wave admittance

Z5ln~b/a!

2pA ivm

s1 ive, ~A-4!

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whereb anda are the diameter of the outer and inner conductof the coaxial line.

Combining Eqs. A-1 and A-2 yields~when ln(b/a) is indepen-dent ofz!

~]z]z2g2!V~z,v!52gZVsd~z2zs!. ~A-5!

We can rewrite Eq. A-5 in term of Green’s function

~]z]z2g2!G~z,v!52d~z2zs!. ~A-6!

The expression ofG is

G5exp~2guz2zsu!

2g. ~A-7!

Hence, the solution forV for a source in a homogeneous mediuwill be

V512ZVs exp~2guz2zsu!. ~A-8!

At the source positionzs,0, the signal has an amplitude of

A0512Z1Vs . ~A-9!

In the region I,z,0, we have to account for the presenceinterfaces, i.e.,

V15A0$exp@2g1~z2zs!#1R1 exp@g1~z1zs!#%, ~A-10!

J15A0Y1$exp@2g1~z2zs!#2R1 exp@g1~z1zs!#%, ~A-11!

whereR1 is the multiple reflection coefficient at the first interfac~at z50!, to be determined.

For the derivation of the electric-potential and electric-currefunctions in the other two regions, 0,z,d andd,z,d1L, aswell as for the boundary conditions at the interfaces alongcoaxial transmission line, refer to Ref. 14.

Substituting the expressions of the electric-potential aelectric-current functions in the boundary conditions yieldsexpression ofR1 :

R15r 1@r 2RL exp~22g3L !11#1@r 21RL exp~22g3L !#exp~22g2d!

r 1 exp~22g2d!@r 21RL exp~22g3L !#1@r 2RL exp~22g3L !11#, ~A-12!

ersi-

the

lec-on-

it

r-

ate

f a

wherer 1 and r 2 are the reflection coefficients caused by theterrupt changes in impedance at the interfacesz50 and z5d,respectively.RL is the reflection coefficient at the open end of tprobe,RL51. These local reflection coefficientsr n are generallydefined as

r n5Zn112Zn

Zn111Zn, ~A-13!

wherer n is the reflection coefficient due to change in impedanat thenth interface;Zn andZn11 are the impedance of thenth andthe (n11)th parts, respectively.

Appendix B–Frequency Analysis of the ExperimentallyMeasured Scatter FunctionsOur experimental setup contains a three-sectional transmissioaxial line. The three sections are: the connecting cable~region I!,the built-in transition unit~region II!, and the coaxial sampleholder ~region III!. Hence, we can use Eq. A-12 for multiple reflection coefficients for our system.

With the network analyzer we measure theS11 scatter functionfor every sand sample. TheS11 scatter function is defined as thratio of the reflected power to the incident power.16 Using thesame denotation as in Appendix A, we obtain from Eqs. A-8 aA-10:

n-

e

ce

co-

-

e

nd

S115V1

2

V05

R1 exp@g1~zr1zs!#

exp~2g1uzr2zsu!, ~B-1!

whereV12 denotes the wave traveling in the negativez direction

andzr is the location of the receiver inside the network analyzwhich is commonly considered to coincide with the source potion, zr5zs .

Because we calibrate the network analyzer at the end ofcable, we havezs5zr50, henceS115R1 ~Eq. A-12!.

The measured scatter function is thus a function of the dietric properties of the whole coaxial transmission line and its cfiguration.

We can calculate the propagation coefficientg~v! and compleximpedanceZ(v) of the connecting cable and the transition unusing Eqs. A-3 and A-4. The single reflection coefficientr 1 at theinterface between these two sections can then be calculated~Eq.A-13!.

Eq. A-12 is a function of the relative complex dielectric pemittivity e* (v). Once the probe lengthL is known, it is possibleto obtaine* (v) from the measured scatter functionS11(v) usinga complex iteration technique. It is also possible to calculS11(v) and the complete wave form oncee* (v) and L areknown.

Cole and Cole17 showed that the dispersion and absorption o

SPE Journal, Vol. 4, No. 4, December 1999

e

.

v

mt

i

i

s

f. 22c-

al

-

nfor

his

tingd in

nt-ted

considerable number of liquids and dielectric are representedthe empirical formula

e* ~v!5S e`1es2e`

11~ ivt0!12bD . ~B-2!

In this equationes and e` are the static~low-frequency! andinfinite-frequency dielectric constants, andt0 is a generalized re-laxation time. The parameterb—the so-called relaxation timespread coefficient—can assume values between 0 and 1 for atribution of relaxation time. It is equal to 0 when the dielectric honly one relaxation time.

Eq. B-2 may be rewritten in the form

S e82es1e`

2 D 2

1~e9!25S es2e`

2 D 2

. ~B-3!

By plotting e9 againste8 one obtains the well-known Cole–Colplot. It is a semicircle with radius (es2e`)/2. Expression B-2requires that in the case of a distribution of relaxation time, i0,b,1, the complex plane locus obtained will be a circular a

By defying the cycle relaxation frequencyv rel52p/t0 , we canrewrite Eq. B-2 as

e* ~ f !5S e`1es2e`

11~ iv/v rel!12bD . ~B-4!

For conductive materials, the energy loss due to ionic conductishould be included:

e* ~ f !5S e`1es2e`

11~ iv/v rel!12bD2

isdc

2p f e0. ~B-5!

We use this equation to describe the frequency dependencrelative complex dielectric permittivity. In this paper, we assuthe fluid-saturated porous media possess only one relaxationproviding b50.

Appendix C–Mixing Models for Dielectric PropertiesIn mixing models the dielectric permittivity of a composite matrial is expressed in terms of the dielectric permittivity of eveconstituting component and their volume fraction. The existmixing models have been examined by several authors.17-19In thispaper, we consider only four models: the classical Raylemodel,20 the complex refractive index~CRI! formula and the sta-tistical Odelevskyi’s model21 and the Berentsveig model used bRef. 22.

The Rayleigh model is given by

em21

em125 f V1

e121

e1121 f V2

e221

e212, ~C-1!

whereem , e1 ande2 are the dielectric constant of the mixture, anits components 1 and 2, respectively;f V1 and f V2 are the volumefractions of the components. This model can be extended ton-component system.

The CRI formula

Aem5 f V1Ae11 f V2Ae2 ~C-2!

was originally derived by Birchaket al.23 to enable calculation ofcomplex dielectric permittivity of dispersed system with the asumption that the size of dispersed particles is much less thanwavelength. That is,d!l. The denotation is the same as in EC-1. The CRI formula for ann-component material has the form

ema 5(

i 51

n

f Vie ia , ~C-3!

whereem and e i are dielectric permittivities of then-componentmaterial and of thei th component, respectively;a is an empiricalparameter;f Vi is the volume fraction of thei th component. Theexponent a is found to be 0.5 for three-component so~solid-water-air!,24 and 0.65 for four-component soils~solid-freewater-bound water-air!.25

Nguyenet al.: Dielectric Properties of Porous Media

by

dis-as

e.,rc.

ity

e ofe

ime,

e-ryng

igh

y

d

the

s-the

q.

ls

The Odelevskyi~statistical! formula can be found in numerouRussian technical papers. It is given by

em5A1An A21Ge i

n, ~C-4!

where

A51

n2 (i 51

n

~3 f Vi21!e i ,

whereem , e i , and f Vi are as in Eq. C-3.The Berentsveig formula has been used successfully by Re

for the frequencies from 100 MHz to 9 GHz to model the dieletric response of sandy clay. The formula has the form

em5 e1

( i 51n f Vi

e i2 e

e i12e

( i 51n f Vi

1

e i12e

, ~C-5!

where

e5(i 51

n

f Vie i

andem , e i , and f Vi are defined as in Eq. C-3.Another mixing model that is often mentioned in geophysic

literature is the Bruggeman–Hanai–Sen formula

eh* 2em*

eh* 2e i*S e i*

em*D 1/3

5 f Vi, ~C-6!

whereem* , eh* , ande i* are the dielectric permittivity of the composite, the host medium, and the inclusion, respectively;f Vi is thevolume fraction of the inclusion. This well-known Bruggemaformula arrived at by the method of iterations and generalizedcomplex dielectric permittivities by Hanai. Senet al.26 used thismodel to develop a self-similar model for sedimentary rocks. Tmodel proved to work well for sand saturated with one fluid3 andhas been used by Ref. 27 for interpretation of ground penetraradar data. The proposed two-step approach—oil is includewater, then the fluid phase~water and oil! is included in soilmatrix—is however doubtful. Besides, results of our receresearch28 show that the dielectric permittivity calculated following the two-step approach strongly diverge from those calculawith other formulas in the saturation range from 0.3 to 0.6.

Buu-Long Nguyen is a consultant with Dames and Moore inThe Netherlands. e-mail: b.l.nguyen@mp.tudelft.nl. His currentresearch interests are multiphase flow aspects of soil contami-nation by DNAPL and dielectric behavior of fluid-saturatedporous media. He holds a MS degree in geology from MoscowState U. and a MS degree in mining and petroleum engineer-ing and a PhD degree in Petrophysics from Delft U. of Technol-ogy. Amanda M. Geels is a consultant at Iwaco B.V. in TheNetherlands. Her research interests are groundwater contami-nation by petroleum products and geophysical methods forthe shallow subsurface. She holds a MS degree in mining andpetroleum engineering from Delft U. of Technology. Jo-hannes Bruining is Associate Professor of Reservoir Engineeringat Delft U. of Technology and a guest professor at the U. ofLeuven. e-mail: j.bruining@ta.tudelft.nl. He previously was avisiting professor in the Dept. of Petroleum and GeosystemsEngineering at the U. of Texas, Austin, and headed the DietzLaboratory for Petroleum Engineering at Delft U. of Technol-ogy. Bruining holds a PhD degree in physical chemistry fromthe U. of Amsterdam. He is a member of the Editorial ReviewCommittee. Evert C. Slob is a research fellow of the RoyalNetherlands Academy of Arts and Sciences at Delft U. ofTechnology. His research interests are electromagnetic meth-ods for geophysical characterization of the shallow surface.He holds a MS degree in mining engineering and a PhD de-gree in electromagnetics, both from Delft U. of Technology.

SPE Journal, Vol. 4, No. 4, December 1999 359