Mixing Rules and Correlations of NMR Relaxation Time With Viscosity, Diffusivity, and Gas/Oil Ratio...

Mixing Rules and Correlationsof NMR Relaxation Time With Viscosity,

Diffusivity, and Gas/Oil Ratio ofMethane/Hydrocarbon Mixtures

Sho-Wei Lo,* SPE, George J. Hirasaki, SPE, Waylon V. House, and Riki Kobayashi, Rice U.

SummaryViscosity, diffusivity, relaxation time, and gas/oil ratio are impor-tant properties in the characterization of reservoirs by nuclear mag-netic resonance (NMR) well logging and in prediction of produc-tion performance. For the past few years, NMR well logging hasbeen used to estimate formation properties and hydrocarbon liquid/vapor characterization. Previous work has shown that pure al-kanes, alkane mixtures, viscosity standards, and stock tank crudeoils have NMR relaxation times that vary linearly with viscosity/temperature and diffusivity on a log-log scale. However, puremethane at some temperatures and pressures does not follow thesame trend. Thus, the linear correlation may not be valid for livecrude oils that contain a significant amount of methane. Therefore,the study of methane-hydrocarbon mixtures is of interest.

An NMR spectrometer equipped with a high-pressure probewas used to study the relationship between NMR T1 relaxationtime and viscosity/temperature, diffusivity, and gas/oil ratio ofmethane-hydrocarbon mixtures. Relaxation time and diffusivitymeasurements of three mixtures were made: methane-n-hexane,methane-n-decane, and methane-n-hexadecane. It was found thatunlike stock tank oil, relaxation times do not depend linearly onviscosity/temperature on a log-log scale. Each of the mixturesforms a different curve.

Generalized correlations between viscosity, diffusivity, gas/oilratio, and NMR relaxation times were developed. First, the relax-ation time mixing rule was developed by studying the theory ofNMR relaxation mechanism. From the mixing rule, it was foundthat departure of relaxation times of methane-n-alkane mixturesfrom linear correlations on a log-log scale can be correlated withthe proton fraction of methane, expressed as gas/oil ratio. Thus, cor-relations between relaxation time, viscosity/temperature, and gas/oilratio were developed. Correlations between relaxation time, diffusiv-ity, and gas/oil ratio were also developed. There is a linear relationbetween diffusivity and viscosity/temperature that is independentof composition. From these correlations, viscosity and gas/oil ratiocan be estimated from NMR T1 relaxation time and diffusivity.

IntroductionThere are existing correlations between NMR relaxation time andviscosity for pure alkanes, alkane mixtures, and crude oils. In1961, Brown made relaxation time measurements on a number ofcrude oils and showed that relaxation time was closely correlatedwith viscosity.1 Recently, measurements of T2 relaxation times ofdead crude oils were made, and it was found that T2 dependslinearly on viscosity on a log-log plot for crude oils.2–4 There wasalso work done on deoxygenated pure alkane and alkane mixtures,and it was found that relaxation times of pure alkane and alkanemixtures could also be linearly correlated with viscosity/

temperature.5–8 However, there were no existing correlations be-tween viscosity, diffusivity, and NMR relaxation time for live oils.

Previous publications of this work have shown that methane-decane mixtures do not follow the same correlation of pure alkanesand alkane mixtures.9,10 The objective of this work was to developcorrelations between transport properties (viscosity, diffusivity), gas/oil ratio, and NMR relaxation time of methane-hydrocarbon mixtures.

EquipmentTwo NMR spectrometers were used to measure relaxation times.One was a low-field spectrometer, which operates at 2 MHz witha permanent magnet, the MARAN-2 (Resonance Instruments, Inc.,Skokie, Illinois). This spectrometer was used for relaxation timemeasurements of pure alkanes at 30°C and ambient pressure.

Relaxation times of methane-hydrocarbon mixtures and purehexane, decane, and hexadecane at elevated temperatures and pres-sures were measured with an integrated superconducting NMRspectrometer. This spectrometer is connected with a high pressurevapor-liquid equilibrium (VLE) apparatus and a temperature-regulated air bath that maintains a constant temperature of the fluidas it is introduced to the NMR probe. The magnet was a super-conducting magnet made by Oxford with a proton frequency of 90MHz. The probe was made specifically for high-pressure fluids byconstructing the sample chamber and sensing coils inside the pres-sure vessel.

Experimental ProcedureSample Preparation. The alkane samples were obtained fromFisher Scientific and Aldrich Chemicals. The purities stated by themanufacturers were 99% for all alkanes. The primary impuritieswere estimated to be other alkanes of similar boiling points, whichwould not be expected to have a significant effect on relaxationtimes. Therefore, no further purification of the alkanes was per-formed except deoxygenation.

Pure methane gas was obtained from Matheson Gas Products.The quality was ultrahigh purity (99.97% minimum). The sum ofimpurities, N2, O2, CO2, C2H4, C3+, and H2O, was less than 300ppm. No further purification of methane was attempted exceptfurther removal of oxygen.

Oxygen presence affects relaxation time significantly becauseit is paramagnetic. Therefore, deoxygenation was performed. Forthe alkane samples, the freeze-thaw method was used to removeoxygen. The sample was frozen in liquid nitrogen, and the solidwas evacuated for fifteen minutes. The sample was then thawedand backfilled with nitrogen gas. According to previous investi-gation,5 oxygen was completely removed after the first cycle.However, the method was performed three or four times for eachsample to ensure complete removal of oxygen. The oxygen con-tained in methane gas was removed by passing methane gasthrough an oxygen absorbing purifier, Matheson Model 6411,which gives an oxygen content of less than 0.1 ppm after purifying.

The samples were introduced to a clean probe. The apparatuswas filled with toluene for one day, and then the toluene wasflushed out. Then the apparatus was heated to 50°C and evacuatedfor at least 8 hours to ensure complete removal of toluene. Thecleaning procedure was performed three times before introducinga new sample.

* Currently with Schlumberger Sugar Land Product Center

Copyright © 2002 Society of Petroleum Engineers

This paper (SPE 77264) was revised for publication from paper SPE 63217, first presentedat the 2000 SPE Annual Technical Conference and Exhibition, Dallas, 1–4 October. Originalmanuscript received for review 2 April 2001. Revised manuscript received 22 October 2001.Manuscript peer approved 20 December 2001.

24 March 2002 SPE Journal

For measurements of mixtures, methane and hydrocarbon wereloaded into the system, and the system was brought to the desiredtemperature and pressure. Methane was then pumped through liq-uid hydrocarbon for more than 90 minutes to ensure that the sys-tem reached an equilibrium state. Either liquid or vapor phase canbe pumped into the sample probe for measurement.

Spin-Lattice Relaxation Measurements. The inversion recoverysequence was used to measure spin-lattice relaxation times. Foreach measurement, 35 to 50 FIDs were acquired over duration timet up to six times the longest T1. The data collected were ts and theircorresponding magnetization M(t). For each measurement, a dis-tribution of relaxation times was obtained according to the equation

M�t� = �i

Mi �0��1 − 2e

− t

T1t�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1)

The T1 fitting algorithm used in this work was developed in theprevious works.11,12 For each sample, log mean relaxation timewas calculated from the distribution.

Self-Diffusion Coefficient Measurements. The pulse gradientspin-echo sequence was used for diffusion measurements.13 Foreach measurement, the gradient duration � was changed. Thirty to40 data points were taken at different �s, and the correspondingecho height was recorded. All other variables (gradient strength gand duration between two gradient pulses �) were kept constantwithin one experiment. For each sample, a distribution of diffusioncoefficients were obtained by fitting the raw data to

M = �i

M0ie

− �2�2g2�� −1

3��Di

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 2)

Log mean diffusion coefficient was calculated from the distribution.

Estimations of Viscosity and Methane Fraction. Methane frac-tion in a mixture was estimated by interpolation of experimentaldata. Viscosity of a mixture was estimated using SUPER-TRAPP,14,15 a software for thermodynamic and transport proper-ties estimations. Details of estimations were reported in Ref. 16.

ResultsT1 Results. Three mixtures were measured: methane-n-hexane,methane-n-decane, and methane-n-hexadecane. For each mixture,both liquid phase and vapor phase were measured. The criticalpressure at our temperature range is approximately 3,000 psia formethane-hexane system; therefore, the highest pressure for thissystem measured is approximately 3,000 psia. For methane-decanesystem, the critical pressure is approximately 5,300 psia, and thehighest pressure measured is approximately 5,200 psia. The criti-cal pressure of methane-hexadecane is above our equipment limit,6,000 psia; thus, methane-hexadecane mixtures were measured atup to 6,000 psia.

The results of all three mixtures were plotted as T1 vs. viscos-ity/temperature, along with pure alkanes and pure methane, to seethe T1 dependence of viscosity/temperature (Fig. 1). T1 of meth-ane-hydrocarbon mixtures do not follow the same correlation aspure alkanes. Methane-hydrocarbon mixtures deviate from thestraight line owing to the methane relaxation. Each of the mixturesforms a separate curve.

Diffusivity Results. Diffusion coefficients were measured for allthree methane-hydrocarbon mixtures. Some pure alkanes were alsomeasured. Fig. 2 is the plot of log mean T1 against log meandiffusion coefficient of both liquid and vapor phases of the threebinary mixtures, along with pure components. The mixtures departfrom the straight line of the pure components. Again, each mixtureforms its own curve.

Diffusion coefficient dependence of viscosity/temperature wasplotted and observed (Fig. 3). Diffusion coefficients have the same

Fig. 1—T1 dependence on viscosity/temperature for pure methane, pure alkanes, and methane-alkane mixtures. All three methane-alkane mixtures, methane-hexane, methane-decane, and methane-hexadecane, depart from the linear correlation of pure alkanes.

25March 2002 SPE Journal

linear relationship with viscosity/temperature for all of the mix-tures and pure alkanes. Diffusion coefficients are proportional toT/�, D�5.05×10−8 T/�, where D is expressed in cm2/sec, viscos-ity is expressed in cp, and T is absolute temperature in degrees K.

Development of CorrelationsMixing Rule of T1 for Methane-Hydrocarbon Mixtures. As afirst step of development of correlations between T1 and transportproperties of fluids, a mixing rule of T1 is developed. For each

methane-alkane mixture, there are two contributions to T1, onefrom protons of methane and the other from protons of higheralkane. The log mean T1 can be described as

log�T1,log mean� = HC1 log�T1,C1� + HA log�T1,A�, . . . . . . . . . . . . ( 3)

where HC1�the proton fraction of methane, and HA�the protonfraction of the higher alkane; T1,C1 and T1,A�the relaxation timesof methane and the higher alkane, respectively. Earlier work shows

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01

Viscosity/Temperature, cp/K

DiffusionCoefficient,cm

2/sec

C1-C10 liq C1-C6 liq C1-C10 vap C1-C6 vap pure C6

pure C10 pure C16 pure C1 C1-C16 liq linear fit

Fig. 3—Diffusivity dependence on viscosity/temperature for pure methane, pure alkanes, and methane-alkane mixtures. Diffusivitydepends linearly on viscosity/temperature, regardless of composition.

1

10

100

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02

D, cm2

/sec

T1

,sec

Pure C6 Pure C10 Pure C16

C1+C6 liquid C1+C16 liquid linear fit for pure alkanes

C1-C10 liq C1-C6 vap C1-C10 vap

Fig. 2—T1 dependence on diffusion coefficient for pure alkanes and methane-hexane, methane-decane, and methane-hexadecanemixtures. T1 for pure alkanes depend linearly on diffusivity, but T1 of methane-alkane mixtures does not depend linearly on diffusivity.


that the dominant relaxation mechanism for pure n-alkanes is in-tramolecular interaction,5 and the dominant relaxation mechanismfor liquid methane is either intermolecular or spin-rotation inter-actions, depending on the density of the fluid.17 Thus, we made theassumptions that protons of higher alkanes relax by intramoleculardipole-dipole interaction, and protons of methane relax by spinrotation interaction and intermolecular dipole-dipole interaction. Itis possible that under some conditions, the higher alkanes mayhave a significant contribution from intermolecular dipole-dipoleinteractions. This relaxation mechanism has a dependence on thedensity of the material or, alternatively, the free volume betweenthe molecules. Contributions from this mechanism may becomeapparent when the density or pressure is changed at constant com-position. All measurements presented here were with saturatedfluids and thus any contribution from intermolecular interactionscould not be distinguished.

Higher Alkane Relaxation. For intramolecular dipole-dipoleinteractions,18

T1 = T2 =3r6DR

�4h̄2I�I + 1�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 4)

where ��gyromagnetic ratio, h̄�Plank’s constant/2�, r�distancebetween two protons in the same molecule, and DR�rotationaldiffusion coefficient.

From Eq. 4, we can conclude that

T2 = T1 � DR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 5)

For a spherical molecule with radius a,18

DR =kT

8�a3�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 6)

Thus,

T2 = T1 � DR �T

a3�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 7)

For a mixture with components A and B,

T1A�

T

aA3 �

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 8)

T1B�

T

aB3 �

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 8)

and

T1m�

T

am3 �

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 10)

where T1m�log mean T1 and am�log mean radius.Thus,

T1A

T1m

=am

3

aA3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 11)

The log mean radius can be calculated

am3 = e �HA log�aA

3�+ HB log�aB

3�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 12)

am = aAHA aB

HB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 13)

For methane-hydrocarbon mixtures, the relaxation time of thehigher hydrocarbon is

T1,dd = T1m

am3

aCn3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 14)

T1m can be obtained from the linear fit for pure alkanes fromthe viscosity/temperature of the mixture. The diameters of themolecules used are the Lennard-Jones diameters estimated fromviscosity data. Lennard-Jones diameters are available for normalalkanes from methane to octane.19 Diameters of decane and hexa-decane were estimated from extrapolation through fitting the avail-able data with carbon number.

This method of estimating the dipole-dipole interaction relax-ation time is tested with hexane-hexadecane mixtures. T2s of threehexane-hexadecane mixtures were measured; 0.33, 0.49, and 0.67weight fraction of hexane. Then the raw data were fitted intobiexponential decay function to obtain the experimental relaxationtimes of hexane and hexadecane in each mixture. The biexponen-tial fit to the raw data resulted in individual component relaxationtimes that compared well with the peaks of the relaxation timedistribution. The two relaxation times were also obtained using themixing rule, and compared with experimental results. The resultsof comparison are plotted in Fig. 4. The log mean T2s of themixtures fall on the linear fit for pure alkanes. The experimentalT2s of hexane and hexadecane were obtained from fitting the rawdata to biexponential decay functions. The estimations of relax-ation times of both components are close to experimental results.

Methane Relaxation. Oosting and Trappeniers17 examined therelaxation mechanisms of methane. They found the spin-rotation tobe the main contribution in the gas phase. At liquid densities andlow temperatures, the intermolecular dipole-dipole interactions arethe dominating contribution. It was discovered that pure methanerelaxation times can be correlated with density and temperature inthe form of20,21

T1

�=

C1

T C2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 15)

where ��density and T�temperature. C1 and C2 can be calcu-lated by fitting literature pure methane T1 data with the previousequation in log-log scale.16,22

logT1

�= logC1 − C2 logT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 16)

C1 was found to be 1.57×105 and C2 1.50. Therefore, T1,SR ofmethane of the mixtures is estimated by

T1,sr =1.57 × 105 �

T 1.50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 17)

1

10

100

1.0E-04 1.0E-03 1.0E-02


RelaxationTim

e,sec

pure C5 to C16 C6+C16 C6 experimetal

C16 experimental C6 estimated C16 estimated

Fig. 4—Comparison of alkane relaxation time in mixture: experi-mental data and estimations from mixing rule. Mixing rule givesgood estimations for both hexane and hexadecane relaxationtimes in mixtures.


The intermolecular dipole-dipole relaxation rate can be de-scribed as (McConnell, 1987)

T1,int er =15kT

32�2N�4h̄2I�I + 1��n�, . . . . . . . . . . . . . . . . . . . . . . . ( 18)

where N�number of spins per molecule and �n�number density.It can be calculated that for hydrogen proton

T1,int er =TMW

450.84N ��, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 19)

where ��density in g/cm3 and MW�molecular weight. In a meth-ane-hydrocarbon mixture, N can be calculated as

N = 4xC1 + NCn xCn , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 20)

where xC1�the mole fraction of methane in the mixture, NCn�thenumber of hydrogen atoms in hydrocarbon, and xCn�the molefraction of the hydrocarbon. The molecular weight M can be cal-culated as

MW = MWC1 xC1 + MWCn xCn . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 21)

MW C1 and MW Cn�the molecular weight of methane and higherhydrocarbon, respectively.

The relaxation model for methane is tested on pure methane.The experimental data of liquid methane relaxation times werecompared with the relaxation times of methane estimated from therelaxation model. In Fig. 5, estimated and measured relaxationtimes were plotted, and the spin-rotation relaxation times and in-termolecular dipole-dipole relaxation times computed from Eqs.17 and 19 were also plotted. Methane liquid of higher densityrelaxes mainly by intermolecular dipole-dipole interaction, whilespin rotation interaction is the dominant relaxation mechanism forless-dense methane liquid and methane vapor. Methane is a spheri-cal molecule with only four hydrogen atoms per molecule. There-fore, the intramolecular dipole-dipole interaction, which involvesthe interactions between protons within the same molecules, isinsignificant.17 Spin-rotation interaction is based on rotational mo-tion of molecules; thus, it is the dominant relaxation for methanegas and low-density methane liquid. Intermolecular dipole-dipoleinteraction involves translational motions and interactions of aproton with protons on other molecules, and it is the main relax-ation mechanism for dense methane liquid. Our relaxation modelfor methane seems to describe the pure methane relaxation behav-ior. There is a qualitative agreement between relaxation time es-timations for pure methane liquid using our model and experimen-tal results. Our model is based on that of Oosting and Trappeniers.17

Methane and Higher Alkane Mixture Relaxation. In sum-mary, the relaxation time models for higher alkane, Cn, and meth-ane, C1, in mixtures are

T1,Cn = T1m

am3

aCn3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 22)

and

T1,C1 =1

1

T1,SR+

1

T1,int er

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 23)

where the calculation of1

T1,SRand

1

T1,int er

are from Eqs. 17 and 19.

The estimated log mean T1 for binary mixtures is

log�T1,log mean� = HC1 log�T1,C1� + HCn log�T1,Cn�. . . . . . . . . . . ( 24)

For some methane-n-hexadecane mixtures, the T1 distributionis bimodal. Thus, the mixing rule was tested on one of thesemixtures at 4,928 psia and 29.5°C. By fitting the raw data tobiexponential decay function, T1 of hexadecane was found to be2.37 seconds, and T1 of methane was found to be 10.84 seconds.From estimations of mixing rule, T1 of hexadecane is 3.31 secondsand T1 of methane is 9.79 seconds. The distribution of this mixtureis shown in Fig. 6.

1

10

100

1.0E-05 1.0E-04 1.0E-03 1.0E-02


T1RelaxationTim

e,sec

T1 experimental-C1 liquid17

T1 experimental-C1 vapor17

T1 Spin Rotation

T1 inter dipolar

estimatedT1

Fig. 5—Comparison of methane relaxation times: experimentaldata and estimations from the relaxation model. The modelgives close estimations for methane liquid and vapor.

0

1

2

3

4

5

6

7

1.0E+02 1.0E+03 1.0E+04 1.0E+05

T1 Relaxation Time, msec

DensityFunction distribution

C1-Mixing rule

C1-Experiment

C16-Mixing Rule

C16-Experiment

C1

C16

Fig. 6—Test of mixing rule on a methane-hexadecane mixture. This is the relaxation time distribution, and the data were fitted intobiexponential decay to obtain relaxation times of methane and hexadecane. The relaxation times of the two components were alsoestimated with the mixing rule. The mixing rule gives close estimations for both components.


The method is used to estimate the log mean relaxation timesof all three mixtures and compare with experimental results.Fig. 7 is the comparison. For all three mixtures, the mixing ruleseems to give close estimations to the experimental results. There-fore, this confirms that the departure of the methane-containingmixture relaxation time from that of the higher alkanes is becauseof the different relaxation mechanism of methane. The mixingrule explicitly expresses the mixture relaxation time as a func-tion of the methane proton fraction and implicitly as a functionof the various parameters in the methane relaxation model. Thisexplicit dependence on methane proton fraction suggests thehypothesis that the dependence of methane on the mixture relax-ation time can be empirically correlated with only the methaneproton fraction.

Correlations of T1, Viscosity/Temperature, Diffusion Coeffi-cient, and Gas/Oil Ratio. Correlations between transport proper-ties and NMR relaxation time were developed. First, the relation-ship of deviation of relaxation time from the linear dependence formethane-free alkanes and proton fraction of methane was ob-served. Deviation of relaxation time on log-log plot is defined asthe difference of T1 and T1,linear for given viscosity/temperature ordiffusion coefficient (Fig. 8):

deviation = Log10�T1,linear� − Log10�T1�

= Log10�T1,linear

T1�. . . . . . . . . . . . . . . . . . . . . . . . . . . ( 25)

From the mixing rule, deviation was found to be a function of protonfraction of methane. Proton fractions of methane and higher hydro-carbons can be expressed as gas/oil ratio, where gas/oil ratio is definedas standard cubic meters of solution gas (methane only) per cubicmeter of stock tank oil. The standard condition is 60°F and 1 atm.

1

10

100

1.E-05 1.E-04 1.E-03 1.E-02


T1RelaxationTim

e,sec

T1 estimated (liq)

T1 experimental (liq)

T1 estimated (vap)

T1 experimental (vap)

linear fit for pure alkanes

C1-C6

1

10

100

1.E-05 1.E-04 1.E-03 1.E-02


T1RelaxationTim

e,sec

T1 estimated (liq)


T1 estimated (vap)


linear fit for pure alkanes

C1-C10

1

10

100

1.E-05 1.E-04 1.E-03 1.E-02


T1RelaxationTim

e,sec

T1 estimated (liq)


T1 estimated (vap)


linear fit from pure alkanes

C1-C16

Fig. 7—The relaxation times of the three mixtures, C1-C6, C1-C10, and C1-C16, were estimated from mixing rule and compared withexperimental data. For all three mixtures, mixing rule gives good estimations.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.0E-04 1.0E-03 1.0E-02


Log,T1

deviation

Mixture T1

T1,linear

(pure alkanes)

Fig. 8—Deviation was defined as the difference between the logof mixture T1 and linear correlation T1 at the same viscosity/temperature. C1-C10 mixtures are used here as an example.


GOR =cubic meters of methane at standard condition

cubic meters of stock tank oil

=xC1MC1

xCnMCn×

�Cn,60°F,latm

�C1,60°F,latm, . . . . . . . . . . . . . . . . . . . . . . . . . . ( 26)

where xC1 and xCn�the mole fraction of methane and n-alkane,MC1 and MCn�the molecular weight of methane and n-alkane, and�C1 and �Cn�the mass density of methane and n-alkane, respec-tively. There is a one-to-one correspondence between gas/oil ratioand proton ratio, where proton ratio is

Proton Ratio

=number of protons from methane in the mixture

number of protons from n− alkane in the mixture

=4xC1

�2n + 2�xCn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 27)

From Eqs. 26 and 27, the conversion between GOR and protonratio is

GOR =�2n + 2�MwC1

4MwCn×

�Cn,60°F,latm

�C1,60°F,latm× �Proton Ratio�. . . . . . ( 28)

Fig. 9 is the plot of deviation of mixture T1 against gas/oil ratio.The relaxation mechanisms and mixing model developed here areindependent of whether the fluid is liquid or vapor. Thus, the vapordata are an extension of the liquid data. The vapor phase data areincluded because they represent mixtures with higher GOR thanthe liquid phase data, thereby extending the correlation to GORhigher than that usually found in liquids.

Deviation of relaxation time from linear dependence on vis-cosity/temperature and diffusion coefficient on a log-log plot canbe correlated with gas/oil ratio by the function fitting curve,f(GOR). Thus,

deviation = Log10�T1,linear� − Log10�T1�

= Log10�f �GOR�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 29)

From the fitting curve,

Log10�deviation� = − 0.127�Log10GOR�2

+ 1.25Log10GOR − 2.80. . . . . . . . . . . . . . . ( 30)

To test this hypothesis, the methane-hydrocarbon mixtureswere mapped onto the linear correlation on log-log plot of T1 vs.viscosity/temperature by adding the deviation to T1. Define T1* asthe projection of T1 onto T1,linear on T1 vs. viscosity/temperature plot.

Log10�T*1� = Log10�T1� + deviation . . . . . . . . . . . . . . . . . . . . . ( 31)

Log10�T*1� = Log10�T1� + Log10�f �GOR�� . . . . . . . . . . . . . . . . ( 32)

T*1 = T1 × f �GOR�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 33)

Fig. 10 is the plot of T1* vs. �/T. T1* falls near T1,linear (methane-free system) on log-log plot of T1 vs. �/T. The average differencebetween T1* and T1,linear for liquids is 16.3% (see Table 1).

Because T1 can be expressed as a function of T1,linear andgas/oil ratio, and T1,linear can be expressed as a function of viscos-ity/temperature, T1 can be expressed as a function of gas/oil ratioand viscosity/temperature.

T1 =T1,linear

f �GOR�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 34)

T1,linear = 0.009558T

�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 35)

1.E-02

1.E-01

1.E+00

1.E+01

1.E+00 1.E+02 1.E+04 1.E+06 1.E+08

Gas/Oil Ratio, m3/m

3

Deviation

C1-C6 liq

C1-C10 liq

C1-C6 vap

C1-C10 vap

C1-C16 liq

Log10[f(GOR)]

C1-C10 liq

C1-C16 liq

C1-C6 liq

C1-C6 vap

C1-C10 vap

C1-C16 vap

Fig. 9—The plot of deviation vs. GOR. All three mixtures fromboth T1 vs. D and T1 vs. µ/T fall on the same curve. A quadraticfunction was fitted through the points.

1

10

100

1000

1.0E-05 1.0E-04 1.0E-03 1.0E-02


T1*,sec

pure C5 to C16

C1+C10 liquid

C1+C10 vapor

linear fit of pure

alkanes

C1+C16 liquid

C1+C16 vapor

C1+C6 liquid

C1+C6 vapor

Fig. 10—As a test of the function of GOR, T1* is plotted against µ/T to see if f(GOR) maps the mixture data to the linear correlationof pure alkanes. f(GOR) works well for all three mixtures for T1 vs. µ/T.


T1 =0.009558

T

�

f �GOR�= F1�GOR,

�

T�. . . . . . . . . . . . . . . . . . . . . . ( 36)

From this correlation, GOR contour lines can be drawn on theplot of T1 vs. �/T. The contour lines are parallel to the linear lineof zero GOR. Fig. 11 is the plot with constant GOR contour lines.

T1 dependence on diffusion coefficient can be treated the sameway. Define T1** as a projection of T1 onto T1,linear on T1 vs.diffusivity plot.

Log10�T**1 � = Log10�T1� + deviation . . . . . . . . . . . . . . . . . . . . ( 37)

Log10�T**1 � = Log10�T1� + Log10�f �GOR�� . . . . . . . . . . . . . . . ( 38)

T**1 = T1 × f �GOR�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 39)

Fig. 12 is the plot of T1** dependence on diffusion coefficient.The average percent difference between T1** and T1,linear for liq-uids is 12.0% (see Table 1).

T1 is a function of T1,linear and gas/oil ratio, and T1,linear is afunction of diffusion coefficient. Thus, T1 can be expressed as afunction of gas/oil ratio and diffusion coefficient.

T1 =T1,linear

f �GOR�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 40)

T1,linear = 2.04 × 105D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 41)

T1 =2.04 × 105D

f �GOR�= F2�GOR,D�. . . . . . . . . . . . . . . . . . . . . . . . ( 42)

The contour lines of constant GOR can also be plotted on thefigure of T1 vs. D. Fig. 13 is the plot of T1 vs. D with constantGOR contour lines.

From Eqs. 36 and 42, the correlation between diffusion coef-ficient and viscosity/temperature can be calculated

D = 4.69 × 10− 8T

�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 43)

The relationship between diffusion coefficient and viscosity/temperature was studied directly in the “Results” section. From theexperimental data in Fig 3, it was found that

D = 5.05 × 10− 8T

�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 44)

These two methods have only about 7% difference.T1 vs. D and T1 vs. T/� can be plotted together with common

contour lines of constant GOR. Fig. 14 shows the plot.Spin-spin relaxation time (T2) can be expressed as a function of

T1, �, gradient strength (g), and echo spacing.

1

T2=

1

T1+ �2g2D

�2

3, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 45)

where � is half of echo spacing.Combining Eqs. 36, 44, and 45,

T2 = F3��

T,GOR,g,��. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 46)

Combining Eqs. 42 and 46,

T2 = F4�D,GOR,g,��. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 47)

T2 can be expressed as a function of �/T, GOR, g, and �, or afunction of D, GOR, g, and �.

The NMR logging tools are capable of measuring T1 and in-ferring the diffusion coefficient from the echo spacing dependenceof the T2 measurements. When T1 or T2 and the diffusion coeffi-cient are known, GOR can be estimated from Eqs. 42 and 46. GORand T1 or T2 can be used in Eqs. 36 or 46 to obtain viscosity.Viscosity can also be estimated directly from the diffusion coef-

TABLE 1ÑAVERAGE ABSOLUTE DEVIATION OF PROJECTED

T1 FROM LINEAR CORRELATION FOR PURE ALKANES

Pure

alkanes C1 Ð C6 C1 Ð C10 C1 Ð C16

Liq. Vap. Liq. Vap. Liq. Vap.

T1* 9% 10% 172% 9% 32% 21% 61%

T1** 3% 16% 208% 5% 31% 13%

1

10

100

1.0E-05 1.0E-04 1.0E-03 1.0E-02


T1RelaxationTim

e,sec GOR=10000 5000 1000 500 100 liquid C1

pure

alkanes

C1+C16

C1+C10

C1+C6

vapor

Fig. 11—Contour lines of constant GOR that are parallel to zero GOR can be plotted on T1 vs. µ/T. From this plot, if two of the threeparameters, T1, µ/T, and GOR, are known, the third one can be estimated. Liquid methane data were taken from Gerritsma et al.,22

and some alkane data were taken from Zega (1989).


ficient, using Eqs. 43 or 44. Recent presentations have recognizedthe importance of the methane content in NMR well logging andfluid sampling.24–26

ConclusionsFor the first time, relaxation times of methane-n-alkane mixtures atelevated pressures and temperatures were measured. Three meth-ane-n-alkane mixtures were measured; methane-n-hexane, meth-ane-n-decane, and methane-n-hexadecane. The linear correlationsfor pure alkanes do not hold for methane-n-alkane mixtures. Themixture data depart from linear correlation of pure alkanes owingto the dissolved methane.

A mixing rule was developed for methane-n-alkane mixtures. Itwas assumed that the higher alkanes relax by intramolecular di-pole-dipole interactions in the mixture, and methane relaxes by

both intermolecular dipole-dipole interaction and spin-rotation in-teraction. The T1 estimations from mixing rules and experimentalmeasurements agree with one another.

Correlations of T1 and transport properties (viscosity/tempera-ture, GOR, and diffusion coefficient) of methane-n-alkane mix-tures were developed. The logarithm of the ratio of T1 and T1,linear

for given viscosity/temperature or diffusion coefficient correlateswith the gas/oil ratio, independent of whether the oil is hexane,decane, or hexadecane. Thus, T1 can be expressed as a function ofviscosity/temperature and gas/oil ratio. It can also be expressed asa function of diffusivity and gas/oil ratio. Diffusivity was found tobe inversely proportional to viscosity/temperature, regardless ofcomposition, gas/oil ratio, and temperature. The correlations arebased on saturated liquid-vapor systems and thus may not apply toundersaturated liquids.

1

10

100

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02

Diffusion Coefficient, cm2/sec

T1RelaxationTim

e,sec

GOR= 100 500 1000 5000 10000

C1+C6

C1+C10

C1+C16

vapor

Fig. 13—Contour lines of constant GOR that are parallel to zero GOR can be plotted on T1 vs. D. From this plot, if two of the threeparameters, T1, D, and GOR, are known, the third one can be estimated.

1

10

100

1000

1.0E-06 1.0E-05 1.0E-04 1.0E-03

Diffusion Coefficient, cm2/sec

T1**

C1-C6 liq

C1-C16 liq

C1-C10 liq

C1-C6 vap

C1-C10 vap

Pure C6

Pure C10

Pure C16

linear fit for

pure alkanes

Fig 12—As a test of the function of GOR, T1** is plotted against D to see if f(GOR) maps the mixture data to the linear correlationof pure alkanes. f(GOR) works well for all three mixtures for T1 vs. D.


Because T2 can be expressed as a function of T1, gradient strength,and echo spacing, T2 can be expressed as a function of viscosity/temperature, gas/oil ratio, gradient strength, and echo spacing, or afunction of diffusivity and gas/oil ratio, gradient strength, and echospacing. Therefore, if any two of these properties [relaxation time(T1 or T2), diffusion coefficient, gas/oil ratio, and viscosity] aregiven, the other two can be estimated from the correlations.

Nomenclaturea � molecular radius, ÅD � diffusivity, cm2/sec

DR � rotational diffusivity, sec−1

g � gradient strength, gauss/cmf(GOR) �function of gas/oil ratio

GOR � gas/oil ratio, m3/m3

h̄ � Plank’s constant/2�, g cm2/secH � proton fractionM � magnetization

M0 � total magnetizationMw � molecular weight, g/mole

N � number of spins per moleculer � distance between two protons in the same moleculeT � temperature, K

T1 � spin-lattice relaxation time, secT2 � spin-spin relaxation time, sec

x � mole fraction� � gyromagnetic ratio, gauss−1sec−1

� � duration of gradient, msec� � duration between two gradient pulses, msec� � viscosity, cp� � mass density, g/cm3

�n � number density, cm−3

� molecular diameter, Å� � half of echo spacing, msec

Subscripts

A � higher hydrocarbonsC1 � methaneCn � n-alkane with n carbonsdd � dipole-dipole interaction

inter � intermolecular dipole-dipole interactionintra � intramolecular dipole-dipole interaction

linear � correlation for methane-free systemsSR � spin-rotation interaction

AcknowledgmentsThe authors wish to express sincere appreciation to the followingorganizations: The Natl. Science Foundation (CTS-9321884), U.S.D.O.E. (DE-AC26–99BC15201), Arco, Baker Atlas, Chevron,Exxon, GRI, Halliburton, Mobil, Norsk Hydro, Phillips, Saga,Schlumberger, and Shell for the financial support, and NIST forthe use of SUPERTRAPP. The authors gratefully acknowledge theadvice of Harold Vinegar.

References1. Brown, R.J.S.: “Proton Relaxation in Crude Oils,” Nature (February

1961) 387.2. Kleinberg, R.L. and Vinegar, H.J.: “NMR Properties of Reservoir Flu-

ids,” The Log Analyst (November–December 1996) 20.3. Morriss, C.E. et al.: “Hydrocarbon Saturation and Viscosity Estimation

from NMR Logging in the Belridge Diatomite,” The Log Analyst(March–April, 1997) 44.

4. Tutunjian, P.N. and Vinegar, H.J.: “Petrophysical Application of Au-tomated High-Resolution Proton NMR Spectrometry,” The Log Ana-lyst (March–April 1992) 136.

5. Zega, J.A.: “Spin-Lattice Relaxation in Pure and Mixed Alkanes andTheir Correlation With Thermodynamic and Macroscopic TransportProperties,” MS thesis, Rice U., Houston (1987).

6. Zega, J.A.: “Spin-Lattice Relaxation in Normal Alkanes at ElevatedPressures,” PhD dissertation, Rice U., Houston (1990).

1

10

100

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02

D (cm2/sec) or 4.69x10-8(T/ µ )

T1

Rel

axat

ion

Tim

e, s

ec

pure alkanes C1+C6 liquid C1+C16 liquid

C1-C10 liq C1-C6 vap C1-C10 vap

linear fit for pure alkanes pure alkanes C1+C10 liquid

C1+C10 vapor C1+C16 liquid C1+C16 vapor

C1+C6 liquid C1+C6 vapor

solid: T1 vs. Dopen: T1 vs. T/µ

GOR= 100 500 1000 5000 10000

Fig. 14—Because T1 vs. T/µ and T1 vs. D share the same f(GOR), the two plots can be plotted together with common contour linesof constant GOR.


7. Zega, J.A., House, W.V., and Kobayashi, R.: “Spin-Lattice Relaxationand Viscosity in Mixtures of n-Hexane and n-Hexadecane,” Ind. Eng.Chem. Res. (1990) 29, 909.

8. Beznik, F.: “Corresponding States Correlations Relating Proton Spin-Lattice Relaxation Rates of Hydrocarbons to Viscosities at AdvancedPressures,” MS thesis, Rice U., Houston (1994).

9. Lo, S. et al.: “Relaxation Time and Diffusion Measurements of Meth-ane and n-Decane Mixtures,” The Log Analyst (November–December1998) 43.

10. Zhang, Q. et al.: “Some Exceptions to Default NMR Rock and FluidProperties,” paper FF presented at the 1998 Annual Logging Sympo-sium of the Society of Professional Well Log Analysts, Keystone,Colorado, 26–29 May.

11. Chuah, T.: “Estimation of Relaxation Time Distribution for NMRCPMG Measurements,” MS thesis, Rice U., Houston (1996).

12. Huang, C.: “Estimation of Rock Properties by NMR Relaxation Meth-ods,” MS thesis, Rice U., Houston (1997).

13. Stejskal, E.O. and Tanner, J.E.: “Spin Diffusion Measurements: SpinEchoes in the Presence of a Time-Dependent Field Gradient,” J. Chem.Phys. (1965) 42, No. 1, 288.

14. NIST Thermodynamic Properties of Hydrocarbon Mixture Database,NIST Standard Reference Data Products Catalog (1995–1996).

15. Huber, M.L. and Hanley, H.J.M.: “The Corresponding-States Principle:Dense Fluids,” Transport Properties of Fluids, J. Millat, J.H. Dymondand C.A. Nieto de Castro (eds.), IUPAC, Cambridge U. Press, NewYork City (1996) 285.

16. Lo, S.: “Correlations of NMR Relaxation Time With Viscosity/Temperatrure, Diffusion Coefficient, and Gas/Oil Ratio of Methane-Hydrocarbon Mixtures,” PhD dissertation, Rice U., Houston (1999).

17. Oosting, P.H., and Trappeniers, N.J.: “Proton-Spin-Lattice Relaxationand Self-Diffusion in Methane, III. Interpretation of Proton-Spin-Lattice Relaxation Experiments,” Physica (1971) 51, 395.

18. McConnell, J.: The Theory of Nuclear Magnetic Relaxation in Liquids,Cambridge U. Press, New York City (1987).

19. Rowley, R.L., Statistical Mechanics for Thermophysical Property Cal-culations, PTR Prentice Hall, Englewood Cliffs, NJ (1994).

20. Rajan, S., Lalita, K. and Babu, S.V.: “Nuclear Spin-Lattice Relaxationin CH4-Inert Gas Mixtures,” J. Magn. Res. (1974) 16, 115.

21. Jonas, J.: “Pressure As an Experimental Variable in NMR Studies ofMolecular Dynamics,” Nuclear Magnetic Resonance Probes of Mo-lecular Dynamics, Tycko R. (ed.), Kluwer Academic Publishers, Bos-ton (1994) 265.

22. Gerritsma, C.J., Oosting, P.H., and Trappeniers, N.J.: “Proton-Spin-LatticeRelaxation and Self-Diffusion in Methanes, II. Experimental Resultsfor Proton-Spin-Lattice Relaxation Times,” Physica (1971) 51, 381.

23. Kashaev, S.K., Le, B., and Zinyatov, M.Z.: “Proton Spin-Lattice Re-laxation, Viscosity, and Vibration of Molecules in the Series of n-Paraffin,” (English Translation) Doklady Akadmii Nauk SSSR (1964)157, No. 6, 1438.

24. Appel, M. et al.: “Reservoir Fluid Study by Nuclear Magnetic Reso-nance,” paper HH presented at the 2000 SPWLA Annual LoggingSymposium, Dallas, 4–7 June.

25. Freedman, R. et al.: “A New NMR Method of Fluid Characterizationin Reservoir Rocks: Experimental Confirmation and Simulation Re-sults,” SPEJ (December 2001) 452.

26. Prammer, M.G., Bouton, J., and Masak, P.: “The Downhole NMRFluid Analyzer,” paper N presented at the 2001 SPWLA Annual Log-ging Symposium, Houston, 17–20 June.

SI Metric Conversion Factorsatm × 1.013 250* E+05 � Pa

cp × 1.0* E–03 � Pas°F (°F−32)/1.8 � °C

in.3 × 1.638 706 E+01 � cm3

psi × 6.894 757 E+00 � kPa

Sho-Wei Lo is a chemical engineer with Schlumberger. e-mail:[email protected]. Lo worked as a postdoctoral research associatein the chemical engineering department at Rice U. for 6months after receiving her PhD and joined Schlumberger inJune 2000. She holds a BA degree in chemistry andmathemat-ics from Wheaton College and a PhD degree in chemical en-gineering from Rice U.George J. Hirasaki had a 26-year careerwith Shell Development and Shell Oil Cos. before joining theChemical Engineering faculty at Rice U. in 1993. e-mail:[email protected]. His research interests are in NMR well logging,reservoir wettability, asphaltene deposition, emulsion coales-cence, and surfactant/foam aquifer remediation. He is amember of the National Academy of Engineering. Hirasakiholds a BS degree in chemical engineering from Lamar U. anda PhD degree in chemical engineering from Rice U. He wasnamed an Improved Oil Recovery Pioneer at the 1998 SPE/DOE IOR Symposium and was the 1989 recipient of the LesterC. Uren Award.Waylon House is an adjunct professor in chemi-cal engineering at Rice U. e-mail: [email protected]. One of thepioneers in NMR imaging, House has spent over a decade inresearch at Rice on the relationships of NMR parameters totransport in liquids and porous media. A physicist, he holdsdegrees from M.I.T. and the U. of Pittsburgh and was a post-doctoral fellow at SUNY-Stony Brook. Riki Kobayashi is theCalder Emeritus Professor in chemical engineering at Rice U.and is a member of the National Academy of Engineering.e-mail: [email protected]. He has been using NMR to measuretransport properties since the 1960s, when he, R. Dawson, andF. Koury measured the diffusivity of methane. Kobayashi holdsa BS degree from Rice U. and MS and PhD degrees from the U.of Michigan.


Mixing Rules and Correlations of NMR Relaxation Time With Viscosity, Diffusivity, and Gas/Oil Ratio...

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