Diffusion Coefficients of Carbon Dioxide in Brines Measured Using
13
C Pulsed-Field Gradient Nuclear Magnetic Resonance
Shane P. Cadogan, Jason P. Hallett#, Geoffrey C. Maitland, and J. P. Martin Trusler*
Qatar Carbonates and Carbon Storage Research Centre, Department of Chemical
Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ
#
Department of Chemical Engineering, Imperial College London, South Kensington
Campus, London SW7 2AZ
* Corresponding author e-mail: m.trusler@imperial.ac.uk.
Abstract
Tracer diffusion coefficients of CO2 in several brines were measured by 13C pulsed-field
gradient NMR at a temperature of 298 K and at salt molalities of up to 5 mol∙kg-1. The brines
studied were NaCl(aq), CaCl2(aq), Na2SO4(aq) and a mixed brine prepared from seven salts:
NaCl, CaCl2, MgCl2, KCl, Na2SO4, SrCl2, NaHCO3. The experimental results are compared
with the predictions of a modified Stokes-Einstein relation in which the Stokes-Einstein number
is 4 and the hydrodynamic radius of the CO2 molecule in aqueous solution is taken to be 168
pm, as determined in an earlier study of the (CO2 + H2O) binary system at the same
temperature [Cadogan et al., J. Chem. Eng. Data 59, 519−525 (2014)]. This comparison
shows agreement to within the experimental uncertainty, independent of salt type and molality.
We conclude that the modified Stokes-Einstein relation provides a reliable means of estimating
the tracer diffusion coefficient of CO2 in aqueous electrolyte solutions, based on knowledge of
the brine viscosity and the hydrodynamic radius of CO2 in pure water at the same temperature.
1
1. Introduction
Transport properties of CO2 in brines and/or hydrocarbons are of interest in many fields,
including chemical and petroleum engineering, earth science, and the biomedical sciences. In
particular, diffusion coefficients of CO2 and other species in brines and hydrocarbons are
important physical parameters in the fields of carbon capture, utilization, and storage (CCUS)
and carbon-dioxide enhanced oil recovery (EOR). For carbon storage in deep saline aquifers,
these diffusion coefficients often control the rates of interfacial mass transfer and of
heterogeneous chemical reactions involving species in aqueous solution and the porous
reservoir minerals. The Peclet number, defined as the ratio of the advection to the relevant
diffusion coefficient, is often used to characterise dispersive flow in porous media. The Peclet
number of CO2 is typically low far away from the point of injection and in that regime the
transport of CO2 is controlled by molecular diffusion.1 However, the diffusion coefficients of
CO2 in brines and other systems relevant to CCUS and CO2-EOR are still largely unstudied.
While several investigations have been carried out on the diffusion coefficient of CO2 in water,
only two studies have reported values over a range of temperatures and at elevated pressures
(> 1 MPa).2, 3 In the absence of salts, a single diffusion coefficient fully describes the mutual
diffusion of CO2 and water in the system. When the CO2 is dissolved in a brine, there are at
least three diffusing components: CO2, salt and H2O. Diffusion can be described in the Fickian
formalism by means of the following relations where there is no net molar flux with respect to
the co-ordinate system:
 j1  D11c1  D12c2
(1)
 j2  D21c1  D22c2
(2)
j1Vm,1 + j2Vm,2 + j3Vm,3 = 0
(3)
Here, ji, ci and Vm,i are the molar flux vector, concentration, and partial molar volume of
component i, the solvent (H2O) is designated as component 3, and Dij are a set of Fickian
diffusion coefficients coupling the flux of solute i to the concentration gradient of solute j.
Generalization to a system of n solutes in a single solvent involves (n + 1) equations and an
nn matrix of diffusion coefficients which is not generally diagonal. The determination of such
a potentially-large set of coefficients could be a demanding task, especially in non-dilute
solutions where the diffusion coefficients depend upon concentration. Furthermore, a formally
exact description of a multi-solute system may be computationally expensive to incorporate in
a large-scale model or simulation.
In the present case one of the solutes, CO2, is present only in low concentration. In fact, the
mole fraction of CO2 in a brine at saturation is typically only of order 10-2 even at reservoir
pressure, and in the experiments reported here, which were carried out at ambient pressure,
it was of order 10-4. Under these circumstances, it can be appropriate to consider the tracer
diffusion coefficient of the sparingly-soluble component in an otherwise spatially uniform brine
that is treated as a pseudo-pure solvent. Such a reduction has been discussed by GarciaRatés et al.,4 using arguments based on the Stefan-Maxwell formalism, and also by Cussler.5
Thus in the remainder of this work, we focus on the diffusion coefficient D11 coupling the flux
2
and concentration gradient of component 1, CO2, and denote this simply by D. The main
assumptions in applying such a treatment are that the concentration of solute 1 is so low that
there is negligible diffusional coupling with other solutes and that interactions between
component 1 and the other solutes are negligible, in which case Dij ≈ 0 when i ≠ j. The former
assumption is also made by Leaist.6 It should be noted that this approach does not preclude
the possibility of the tracer diffusion coefficient of the CO2 depending upon the concentration
of the other solutes.
2. Experimental Approach
NMR techniques for the measurement of diffusion coefficients have been used for some 60
years, starting with the early work of Hahn.7 This technique has been utilised to determine the
diffusion coefficient of CO2 in several hydrocarbons.8, 9 However, only one literature source
has been identified in which the technique was applied to measure the diffusion coefficient of
CO2 in a liquid;10 in that work the diffusion of CO2 in various beverages was investigated.
In the current work, we used the pulsed-field gradient NMR technique11 to measure the tracer
diffusion coefficient of 13CO2 in water and brines. This technique takes advantage of the effect
of translational diffusion on the attenuation of an NMR signal when placed in a magnetic field
whose strength varies as a function of position. As the Larmor frequency is a linear function
of the magnetic field experienced by the probe species, applying a gradient to the magnetic
field has the effect of making this frequency, and the observed signal attenuation, a function
of the displacement of the probe species along the magnetic field gradient. These gradient
pulses were applied as square shaped pulses.
From the work of Carr and Purcel 12 and Stejskal and Tanner,13 the attenuation of the observed
NMR signal is related to the tracer diffusion coefficient D as follows:
S
δ

ln   G2γ 2δ 2  Δ  D .
3

 S0 
(4)
Here, S and S0 are the signal intensity and initial signal intensity, respectively, G is the applied
magnetic field strength, γ is the gyromagnetic ratio of the probe species, δ is the duration of
the applied magnetic gradient, and Δ is the time between gradient pluses.
The measured diffusion coefficient is the carbon dioxide intradiffusion coefficient, D1*, relating
the flux of magnetically labelled CO2 to the gradient in the concentration of magnetically
labelled CO2 in solutions where the total CO2 concentration (and the concentrations of water
and salt) are constant and uniform. However, in the current limiting case of very low CO2
concentrations, the mutual diffusion coefficient D is equal to the intradiffusion coefficient D1*.
3. Materials and Methods
3.1 Samples
CO2 was supplied by CK Gases with an isotopic purity ≥ 99% 13C and <1% 18O content. D2O
and the salts used in this study were obtained from Sigma-Aldrich. Three single-salt brines
13
3
were studied [NaCl(aq), CaCl2(aq) and Na2SO4(aq)] as well as a multi-component brine, the
composition of which is detailed in Table 1. To prepare the 13CO2-saturated brine solution, a
quantity of brine was transferred to a 100 mL borosilicate glass bottle containing a magnetic
stirrer bar and closed by an air-tight lid fitted with two gas connection ports. One port was
connected via a valve to a vacuum pump, while the second led via a manifold to a rubber gas
bladder and the valved 13CO2 lecture bottle. Using this apparatus, the brine was first degassed
under vacuum and then saturated with 13CO2 at ambient pressure, both operations occurring
under stirring. Under these conditions, dissolved CO2 is present primarily in molecular form
and the fraction dissociated is < 10-2.14, 15 A sample of the fresh solutions was then used to fill
an NMR tube into which a thin sealed capillary containing D2O was also inserted. The D2O
provided an in-situ calibration lock similar to the method of Holz.16
3.2 Methodology
A Bruker 500 MHz AVANCE III HD NMR spectrometer was used to perform the measurements.
The spectrometer was equipped with a z-gradient tuneable probe and a BSMS GAB 10 amp
gradient amplifier with a maximum gradient output of 0.535 T∙m-1. The spectra were collected
at a frequency of 125.76 MHz (13C) with a spectral width of 503 MHz (centred on 124.6 ppm)
and 8192 data points. A relaxation delay of 200 s was employed along with a diffusion time,
Δ, of 49.9 ms and a longitudinal eddy current delay (LED) of 5 ms. Bipolar gradients pulses,
δ/2, of 2.6 ms and homospoil gradient pulses of 1.1 ms were used. The gradient strengths of
the 2 homospoil pulses were -17.13 % and -13.17 % of the maximum gradient output. In each
run, 16 data sets were collected with linearly increasing bipolar gradient strength starting at
2 % and ending at 95 %. All gradient pulses were smoothed-square shaped (SMSQ10.100)
and after each pulse a recovery delay of 200 μs used. The recorded signals were analysed
using Bruker’s TopSpinTM 3.2 software. The field gradient was calibrated at T = 298.15 K using
a standard solution of (0.99 D2O + 0.01 H2O) doped with 0.1 mg/mL GdCl3 and 1 mg/mL
CH3OH. The diffusion coefficient of H2O in D2O was taken to be 1.90 x 10-9 m2·s-1.16 The
temperature calibration of the instrument was checked by measuring the 1H chemical shifts of
a sample of pure ethylene glycol. The standard temperature uncertainty is taken to be 0.5 K.
The raw data were processed with one order of zero-filling and using an exponential function
with a line broadening of 2 Hz. Further processing was achieved using the Bruker Dynamics
Center software (Version 2.1.7) and uncertainty estimation was performed by Monte Carlo
simulation, also using the Bruker Dynamics Center software. The gyromagnetic ratio of 13C, γ,
was 6.73107 rad∙s-1∙T-1.
4. Results
The measured diffusion coefficients for CO2 in pure water and the investigated brines at
T = 298 K, atmospheric pressure and total molality m are given in Table 2, together with the
estimated relative standard uncertainty ur(D). It can be noted that the standard uncertainty
tends to increase with increasing molality and we attribute this to the decreasing solubility of
CO2 arising from the salting-out effect.
4
Literature values for the viscosities of the solvents at the experimental temperature are also
given in Table 2.17-21 The viscosity given for the synthetic reservoir brine was obtained by
measurement using an Ubbelohde capillary viscometer calibrated with pure water.
4. Discussion
The experimental tracer diffusion coefficients are plotted as a function of the solvent viscosity
in Fig. 1. The results of Sell et al.22 are also shown. The impact of pressure was found to be
negligible in their work. Renner23 has reported the diffusivity of CO2 in a 0.25 molkg-1 NaCl(aq)
within a porous medium. The diffusion coefficient was found not only to be a function of
pressure but also of the orientation of the porous core sample. When the core was held in a
vertical orientation, the values of D obtained varied from (3.07 to 7.35) x 10-9 m2s-1 at
pressures between (1.5 and 5.8) MPa. Based on the results of Sell and of our earlier study on
(CO2 + H2O), we find the dependence upon pressure reported by Renner to be ambiguous
and we do not discuss those results further.
Figure 1 shows the expected dependence of the diffusion coefficient of a dilute solute upon
the viscosity of the solvent, as embodied in the Stokes-Einstein equation:
D  kBT / nSEπη a .
(5)
Here, kB is Boltzmann’s constant, T is absolute temperature, nSE is the Stokes-Einstein number,
taken to be 4, η is the solvent viscosity, and a is the hydrodynamic radius of the solute.
In an earlier study, we reported limiting diffusion coefficients of CO2 in pure water at
temperatures between (298.15 and 423.15) K and at pressures up to 45 MPa.2 The results
were correlated to within ±4 % with equation (5) by making the hydrodynamic radius a weak
linear function of temperature as follows:
a  a298 [1  α (T / K  298)] .
(6)
The best fit parameters so determined were a298 = 168 pm and α = 2.0 x 10-3.
The hypothesis to be tested in the present study was that the hydrodynamic radius of a CO2
molecules in a brine is the same as that in pure water, so that the change in the tracer diffusion
coefficient of CO2 upon adding salt at constant temperature is simply a consequence of the
change in the viscosity of the solvent. To test this hypothesis, we plot in Figure 1, the predicted
diffusion coefficient as a function of brine viscosity and we note that there is generally good
agreement between the predictions and our measurements. This supports our hypothesis, at
least at T = 298 K. An attempt was made to extend the measurements to higher temperatures
but it was found that the lower solubility of CO2 resulted in an unacceptable signal-to-noise
ratio. Consequently, no useful results were obtained, expect for a single point at T = 303.15 K
in a 2.5 mol∙kg-1 NaCl solution, which was found to be in good agreement with a value
computed from equations (5) and (6). However, this is certainly not a fundamental limitation
of the method and it is likely that measurements could be performed at higher temperatures,
for example by pressurising the sample to increase the concentration of dissolved 13CO2.
5
5. Conclusions
The tracer diffusion coefficient of 13CO2 has been measured by an NMR technique in a number
of brines, differing in both salt type and molality, at a temperature of 298 K. The results are
consistent with the Stokes-Einstein equation and a hydrodynamic radius for the CO2 molecule
that is independent of salt type and molality. Calculation of the hydrodynamic radius from our
earlier measurement on the (CO2 + H2O) system resulted in quantitative agreement with the
experimental results for the brines investigated. We suggest that the same approach may be
successful at other temperatures.
Acknowledgements
We gratefully acknowledge the funding of QCCSRC provided jointly by Qatar Petroleum, Shell,
and the Qatar Science & Technology Park. We also thank the Department of Chemistry at
Imperial for granting access to the NMR facility and Peter Haycock for collecting the NMR data.
Finally, we thank Claudio Calabrese for measuring the viscosity of the synthetic reservoir brine
studied in this work.
6
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7
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8
Table 1: Composition of the synthetic reservoir brine, where m denotes molality of salt in water.
Species
NaCl
CaCl2
MgCl2
KCl
Na2SO4
SrCl2
NaHCO3
m/(kg∙mol-1)
1.514
0.276
0.080
0.012
0.011
0.004
0.003
9
Table 2: Tracer diffusion coefficients D of CO2 in brine solutions of molality m at T = 298.15 K, together
with the standard relative uncertainty ur(D) and literature values of the viscosity 17-21.a
Salt
None
NaCl
CaCl2
Na2SO4
Mixed
a
m /(mol∙kg-1)
D /(10-9 m2∙s-1)
ur(D)
η /(mPa∙s)
0.0
2.13
1.3 %
0.89
1.0
2.06
1.5 %
0.97
2.5
1.70
2.1 %
1.14
5.0
1.29
7.7 %
1.53
0.9
1.71
3.1 %
1.17
1.0
1.60
5.6 %
1.25
2.0
1.25
6.8 %
1.62
1.0
1.48
8.6 %
1.37
1.9
1.78
3.2 %
1.10
The standard uncertainty of the temperature is u(T) = 0.5 K.
10
D / (10-9 m2s-1)
2.5
2.0
1.5
1.0
0.5
0.5
1.0
1.5
2.0
η / mPa∙s
Fig. 1. Diffusion coefficients of CO2 in brine as a function of solvent viscosity. This work at T = 298.15
K and p = 0.1 MPa: ▬, pure water; , NaCl; , CaCl2; , Na2SO4; , mixed brine; ———, predicted
by the Stokes-Einstein equation with a298 = 168 pm and nSE = 4. Error bars show ± one standard
uncertainty. Sell et al 22 at T = 299 K and p = 5 MPa: .
11