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Relaxation of Protons in Pores in a Sandstone Alan McCutcheon,† Michael A. Wilson,*,† Birgitta Hartung-Kagi,‡ and Khiam Ooi‡ Department of Chemistry, Materials and Forensic Science, UniVersity of Technology, Sydney, P.O. Box 123, Broadway NSW 2007, Australia, and Geotechnical serVices Pty. Ltd., 41-45 Furnace Road, Welshpool WA 6106, Australia ReceiVed: May 16, 2001; In Final Form: October 19, 2001
Proton nuclear magnetic spin-spin relaxation time distributions (T2 distributions) for a water-saturated cylinder of sandstone (radius 6.5 cm and length 17.55 cm) have been studied as a function of drying time. By freezing the sample to -13.7 °C, a proportionality constant of 38.8 µs/nm was determined that can be used to relate pore size to T2. The results show external water cannot enter some intermediate sized pores of 400 nm diameter; however, these pores are available to water that has been frozen and allowed to thaw. On drying from saturation, water loss commences immediately from pores of the order of 1000 nm and greater and there is a lag period of 51 h before the smaller pores begin to drain. On freezing and then thawing, intermediate sized pores are filled because the large pores are blocked by ice.
Introduction Microgeometry plays an important role in the physical and chemical properties of porous materials. Pores are important in understanding the transport properties of rocks, particularly for oil and water. The methods typically used for measuring pore sizes are gas adsorption and desorption techniques and multipoint mercury injection porosimetry. In addition, it has been known for a long time that information can also be obtained by NMR spectroscopy1-7 although the approach has been more empirical. More recently there is renewed interest in using the NMR technique. This has been driven by the introduction of in situ NMR logging tools for commercial applications to gain a better understanding of rocks as they are drilled. There has been a need to understand the data in a thorough and interpretable way, and this has led to more sophisticated laboratory experimentation. When disturbed from its natural thermal energy by irradiation, a proton may diffuse across a pore enclosure filled with liquid water and relax to a lower energy state by transferring magnetization to the pore surface boundary. If the pore has small dimensions, there is a high probability that the protons will collide with surface boundary or walls of the pore and relax. Thus on average, for a collection of protons, the average relaxation time is shorter in small pores compared with large pores and depends only on the pore size and its relaxation properties. Thus
T1 ) V/(SF) or 1/T1 ) SF/V
(1)
where V is the volume of the pore, S is the surface of the pore, and F is a constant dependent on the surface, termed surface relaxivity. This is the ability of the surface to cause relaxation and has the dimensions of length/time. Since the pore size is characterized (inter alia) by the surface (S) to volume ratio (V) as S/V, NMR can be used to measure pore sizes. For mobile * To whom correspondence should be addressed. E-mail: mick.wilson@ uts.edu.au. † University of Technology, Sydney. ‡ Geotechnical Services Pty. Ltd.
liquids such as water where molecular reorientation is fast, T1 is equal to T2 and hence T2 is also related to pore size and can be substituted for T1 in eq 1. T2 is the preferred measurement since it can be measured rapidly by the Carr-Purcell sequence.8 In the Carr-Purcell experiment, after an initial π/2 pulse, a series of π pulses at τ, 3τ, 5τ, nτ, etc. are applied which produces a series of echos at 2τ, 4τ, 6τ, etc. which decrease exponentially with time constant T2. The effect of diffusion is kept minimal by keeping short the interval between the pulses so that T2 can be derived from the simple Bloch equation9-11 as
M ) M0 exp(-nτ/T2)
(2)
where M is a constant and M0 is the initial signal intensity. Unfortunately, real samples have a number of different pores and water in each different type of pore has a different T2. Thus for a series of T2’s, where t ) nτ
Mt )
∑i Ai exp(-t/T2i)
(3)
where Ai is a constant reflecting the contribution of each type of pore to the relaxation behavior. In principle, if T2i values are known, pore size distributions can be measured by fitting values for Mt to obtain Ai.12,13 In practice, they are best fitted through integration over a continuous T2 from values equal to zero to those for the bulk solution. In these data there is the presence of noise . Thus
M(t) )
∫TT)0(bulk) A(T2 exp(-t/T2)) dT2 + (t) 2
(4)
2
The problem with extracting A values from this equation, which applies generically to all relaxation, including T2, is that there are a wide set of values of A that adequately define the data. This type of problem is termed “ill posed”. However, if the geometry of the structure in which the water is relaxing is known, then a single solution is possible. Thus, Browstein and Tarr14 modeled solutions for cylindrical and spherical geometries and confirmed their results using the practical example of rat
10.1021/jp011884r CCC: $22.00 © 2002 American Chemical Society Published on Web 02/26/2002
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J. Phys. Chem. B, Vol. 106, No. 11, 2002 2929
gastronemius cells. In assuming an annular cylindrical shape, the model predicted the correct cell diameter. More recently, Prammer15 showed that there are ways to obtaining single solutions if weighting functions are applied with least-squares minimization techniques. In effect, this mathematically is a way of constricting the geometry. The important point is that single reproducible values of T2 and A’s can be obtained when computing solutions. That is, a single solution to the problem can be found. Weighting functions thus allow pore sizes to be compared provided the geometry is not too different. Plots of A’s versus T2’s are known as the relaxation distribution. A values are interpreted as the number of pores if the pore geometry is known. T2 is interpreted as pore size. T2 values, however, need to be calibrated by comparing size distributions derived from nitrogen surface measurements and adjusting the weighting function. It would be much better to have a separate independent measurement. One approach, not been fully exploited, is to calibrate T2 measurements by freezing experiments. In small pores, the temperature of water freezing is suppressed. This is often termed the capillary effect and is related to the transmission of surface interaction energy through the bulk liquid to the liquid surface and hence liquid-gas surface tension. The properties of freezing water in pores are thus described by the Kelvin equation.16
∆T ) Tm - Tmx )
4σslTm 1 ∆HfFs x
(5)
where Tm is the normal melting point of bulk liquid, Tmx is the melting point of the liquid in pores with a diameter x, σsl is the surface tension at the liquid-solid interface, ∆Hf is the bulk enthalpy of fusion (per gram of material), and Fs is the density of the solid phase. Since x is pore diameter, the surface to volume ratio can be measured assuming pore geometry. Thus if NMR studies are carried out on freezing solutions, at a given temperature, the water remaining unfrozen should have a relaxation time related to pore size. Before continuing, we should note there are some possible errors in this measurement since Hansen17 suggests some water is totally nonfreezable water and independent of pore size. This is water adsorbed as 1-2 monolayers and occupies 0.2-0. 5 nm on the surface. The size of this layer is not clear because different types of ice have different van de Waals radii.18,19 However, provided the freezing temperature is not reduced to a value where pores of 0.2-0.5 nm are being measured, error should be small. An arbitrary temperature of -13.7 °C corresponding to a pore size of 20 nm is used here, which leaves an error of 2.5% or less, about the accuracy of any NMR methodology. The methodology also assumes similar shape for all pores. The original Kelvin equation was applied to spherical droplets. There has been some related work. Gran and Hansen20 used freezing and thawing cycles to study pastes of different surface areas and showed that line widths narrowed only for samples with low surface areas. Hanson and co-workers17,20 and Overloop et al.21,22 also studied frozen water in porous materials. Akporiaye et al.23 and Schmidt et al.16 have applied a modified Kelvin equation to characterize mesoporous (pores with a diameter 2-50 nm) molecular sieves saturated with water, and Strange and Webber24 have used this equation to characterize the pore size distribution, in the range 4-200 nm, for various sized silicas packed in glass tubes. There have been a number of NMR studies on sandstones as discussed by Liaw et al.25 However, the water freezing approach
has not been employed to determine pore sizes, and in particular for systems with multiple sized pores requiring eq 4. In this work freezing is used to calibrate a sandstone and to investigate the drying phenomenon. In addition, for realistic studies of undisturbed material, large samples should be used in which the number of undisturbed pores significantly outweighs the number of disturbed pores produced by sampling. Hence in this work a very wide gap magnet has been used to accommodate a sample 13 × 17.55 cm. Experimental Section Sample. The whole core section (radius 6.5 cm and length 17.55 cm, volume 559.5 cm3) was solvent extracted in toluene and methanol to remove residual hydrocarbons and salts, respectively. It was allowed to cool to about 25 °C and then injected with helium to have its total pore volume determined. The sample was then evacuated under vacuum and pressure saturated with 10% aqueous potassium chloride solution for storage. For NMR experiments the core was washed, dried at 100 °C, and then soaked in distilled water. Instrumentation. NMR measurements were obtained using a modified Resonance Maran Ultra-2 spectrometer equipped with a large gap magnet capable of taking rock core samples with a maximum diameter of 13 cm and a length >30 cm. The instrument is equipped with a permanent magnet producing a low intensity field with a Larmor frequency of 2.1 MHz. 1H signals from solids are not observed due to the long dead time of the instrument (80 µs). T2 measurements were made using the Carr-Purcell sequence.8 The distribution of the NMR signal as a function of T2 values was obtained using Resonance WinDXP software.26 Weighting coefficients were 0.2133 for saturated and 0.0019 for frozen core. Porosity, Thawing, and Drying. Apart from helium measurements, porosity of the rock core sample was determined by measuring the volume of water contained in the sample using two other methods. In the first method, the weight of water contained in the pore system was measured by taking the weight difference between the dried and water-saturated core samples after saturating with water by immersion for about 12 h. In the second method, the initial amplitude 1H NMR free induction signal was measured. The initial amplitude (M0(t)0)) of the signal is directly proportional to the population of the mobile protons present. To quantify the amount of water contained in the pores of the core sample, the initial amplitude of the NMR signal was compared with the initial amplitude signal measured for a known volume of water. The porosity of the sample was then calculated using
porosity (% ) )
M0*(core) × 100 M0*(water without core)
(6)
where M0*(core) is a corrected M0 value measured for the core saturated with water and M0*(water) is a corrected M0 value measured for a known volume of water. To correct M0
M0* )
M0 Vol × NS
(7)
where Vol is either the volume of the core sample or the volume of water (i.e., the known volume of water) and NS is the number of NMR scans. Porosities calculated by taking the difference in weight between brine-saturated core and dried core and by NMR
2930 J. Phys. Chem. B, Vol. 106, No. 11, 2002
McCutcheon et al.
Figure 3. Shifts in the distribution commencing from -13.7 °C and progressively thawing out with time. Figure 1. Water loss with drying in the magnet at 35 ( 1 °C for an extended period of time.
Figure 2. Distribution of T2 values for different drying times. The sample was dried in the magnet at 35 ( 1 °C.
measurements are 15% and 17%, respectively. Helium injection measurements were an order of 10% higher for reasons discussed below. The rates of drying were observed by commencing with a core sample saturated with water and then recording a decrease in the water content as a function of time in the magnet of the NMR spectrometer at a constant temperature of 35 °C. In a similar way a frozen sample was held at -13.7 °C and the water content measured as the sample warmed at a rate of 13.2 °C/h. Results and Discussion Drying. Figure 1 shows data for drying the saturated rock core for over 500 h. The data in Figure 1 are best fitted as three straight lines although this division is probably arbitrary. Initially the average rate of water loss, up to 79 h, is 107 mmol h-1 (2.55 µg/cm3); then for drying times between 79 and 249 h, the rate is only 31 mmol h-1 (0.740 µg/cm3). Finally, from 249 to 530 h the rate falls to 12 mmol h-1 (0.286 µg/ cm3). At the end of each of these three drying regimes the percentage of water lost, as a percentage of the initial saturated amount, the residual water saturation, is 44%, 67%, and 86%, respectively. Figure 2 shows T2 distributions for an initially saturated core sample and when the core sample decreases in water content as a result of increasing drying time. At the commencement of drying, the saturated core has a bimodal distribution of water. The distribution shows that 64% of the total observable water is in pores with T2 values below 15 000 µs. The remaining portion of water is contained in larger pores, represented by the second distribution with T2’s above this value. The T2 modes are 3900 and 40 000 µs, respectively. There is clearly a discontinuity in the pore distribution in that the overall distribution is not symmetrical. The discontinuity occurs with pores having T2 values of around 15 000 µs. This could be because
pores which would accommodate water with these relaxation times are in low concentration, or because they cannot fill during saturation since they are excluded from the exterior because of a lack of available channels. This may be because these channels are blocked because of narrow throats and hence only filled under vacuum or because they do not exist. From Figure 2 it can be seen that, as the water content diminishes, the distribution of water shifts from bimodal to a single mode distribution with water emptying from large pores. After 51 h there is only the slightest sign of loss of water from the small pores with T2’s of 3900 µs. Thus the large pores drain first. This is probably because the small pores are connected to the large pores and water in the small pores can only escape by draining through the large pores. Thawing. Rock core was saturated with water, frozen for about 12 h with the temperature accurately controlled to -13.7 °C, and then allowed to warm. Saturation was not done under vacuum so that pores which are blocked by a small throat will not fill. Figure 3 provides T2 distribution plots at different intervals of time commencing from a temperature of -13.7 °C and warming at 13.2 °C/h. During warming the distribution does not follow the reverse path of the drying experiment. Water which has not frozen at -13.7 °C is observed initially. After 33 min the intensity of the signal from this water decreases, indicating that, once mobile, some water can refreeze in larger pores. On warming further, the small pores thaw first and water appears in some larger pores of intermediate size in preference to others. This must be water in pores connected closely to small pores and appears to be water with T2 values around 15 000 µs. These pores must be joined to small pores. We note that these are the very same pores which are apparently deficient in saturation experiments. These pores must also be the pores in which the first released water refreezes. One explanation is that in a saturated system prepared from a dried sample, water does not enter these pores because the small pores are plugged by a small throat. This could be because an air bubble is present or the throat is so small that capillary forces at the throat prevent filling. The 10% higher helium injection porosity measurement represents the unfilled pores in the saturation experiment. Helium of course can enter all pores. Nevertheless, as noted above, these can be filled from small pores when water in the largest pores is frozen, that is, when the water in the smaller pores has nowhere else to go or when the flow is so small that the air bubble has time to disperse. Either way the results attest to the presence of narrow throats. A model for this system is given in Figure 4. This describes the three types of pores and flow paths in drying and freezing. In the thawing experiments, the T2 value at the peak of the distribution plot shifts to higher values with an increase in time. This change in the T2 value as a function of time, which has elapsed following warming, is displayed in Figure 5. In this
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J. Phys. Chem. B, Vol. 106, No. 11, 2002 2931
Figure 6. Modal distribution peak for nonfrozen water in the core at -13.7 °C.
Figure 4. Filling and emptying of pores.
Figure 7. Relative distribution of water as a function of pore diameter calculated using the modified Kelvin equation. The core is saturated with water at ambient temperature.
Figure 5. Change in T2 values, at the peak of the distribution plot, as a function of time. At commencement of measurements (t ) 0) the temperature of the core is -13.7 °C.
plot T2 values increase rapidly at first, as a function of time with a maximum value of about 3200 µs. The final T2 value is similar but lower than the 3900 µs peak value measured for the core when saturated with water at ambient temperature. This is because water is now in intermediate pores. The area of the modal distribution peak produced from the first analysis of the sample following its freezing at -13.7 °C represents the amount of liquid-phase water (unfrozen water) contained in the smallest pores. This nonfrozen water represents 16.6% of the total observable water (calculated by comparing the area of peaks) contained in the pores at ambient temperature. The T2 value at the commencement of the peak is shown in Figure 6 and is 775 µs. This corresponds to pores with a particular diameter that can be calculated using the modified Kelvin equation at -13.7 °C. Now
∆T ) Tm - Tmx )
4σslTm 1 ∆HfFs x
(5)
where Tm is the normal melting point of bulk liquid, Tmx is the melting point of the liquid in pores with a diameter x, σsl is the surface tension at the liquid-solid interface, ∆Hf is the bulk enthalpy of fusion (per gram of material), and Fs is the density of the solid phase. Thus, when Tm ) 273.15 K, Tmx ) 13.7, σsl
) 0.075 64 N m-1, ∆Hf ) 333.6 J g-1 ) 333.6 × 103 J kg-1, Fs ) 919.3 kg m-3, and x ) 1.97 × 10-8 m ) 20 nm. These results show the upper limit on water that cannot freeze at -13.7 °C is 20 nm diameter. Equating T2 with pore diameter gives the equation
T2 ) k c d
(8)
where kc is a proportionality constant and d is the diameter of the pores. For this study kc ) 38.8 µs nm-1. Equation 8, with the derived value for kc, enables T2 values in the distribution plot to be reported as a function of pore diameter for the saturated core. This is shown in Figure 7 and can be used to directly convert T2 values shown in previous plots. T2 measurements can be related directly to pore size by freezing the sample. Thus, we can now express the thawing and drying behavior of the sandstone in terms of pore size rather than relaxation times. For the sample studied here, the results demonstrate the lack of access of external water to intermediate sized pores of 400 nm diameter. It is shown that these must be filled through smaller pores and that this only occurs after freezing. On drying from saturation, water loss commences immediately from the largest pores of the order of 1000 nm in diameter or greater and there is a time lag before the smaller pores of the order of 100 nm in diameter begin to drain. Conclusions 1. T2 distributions for an initially saturated sandstone core sample have been measured as a function of drying time. Water loss commences immediately from the largest pores of around 1000 nm or greater. Large pores drain faster than the small
2932 J. Phys. Chem. B, Vol. 106, No. 11, 2002 pores, and large pores drain until they are partly empty before the small pores begin to drain. 2. T2 measurements can be related directly to pore size by freezing the sample. A T2 value of 775 µs corresponds to pores of 20 nm in size. Hence, a simple linear relationship can then be used to calculate actual pore size distributions in nanometers. 3. NMR is a useful method for studying the spatial distribution and flow characteristics of pores. For the sample studied here, the results demonstrate the lack of access of external water to intermediate pores (400 nm) that must be filled through smaller pores. On freezing and then thawing, though, these pores fill because larger pores are blocked by ice. Acknowledgment. We thank ARC for an APAI grant for this work. Dr. Gary Lee assisted in commissioning the NMR equipment. References and Notes (1) Korringa, J.; Seeveers, D. O.; Torrey, H. C. Phys. ReV. 1962, 127, 1143. (2) Resing, H. A.; Thompson, J. K.; Krebs, J. J. J. Phys. Chem. 1964, 7, 1621. (3) Resing, H. A. J. Chem. Phys. 1965, 43, 669. (4) Timur, A. J. Pet. Exploration 1969, 775. (5) Cohen, M. H.; Mendelson, K. S. J. Appl. Phys. 1982, 53, 1127. (6) Schmidt, E. J.; Velasco, K. K.; Nur A. M. J. Appl. Phys. 1986, 59, 2788.
McCutcheon et al. (7) Hurliman, M. D.; Helmer, K. G.; Latlour, L. L.; Soyak, C. H. J. Magn. Reson., Ser. A 1994, 111, 169. (8) Carr, H. Y.; Purcell, E. M. Phys. ReV. 1954, 94, 630. (9) Bloch, F. Phys ReV. 1946, 70, 460. (10) Bloch, F.; Hausen, W. W.; Packard, M. Phys. ReV. 1946, 70, 474. (11) Farrar, T. C.; Becker, E. D. Pulse and Fourier Transform NMR; Academic Press: New York, 1971. (12) Brown, R. J. S. J. Magn. Reson. 1989, 82, 539. (13) Gallegos, D. P.; Smith, D. M. J. Colloid Interface Sci. 1988, 122, 143. (14) Brownstein, K. R.; Tarr, C. E. Phys. ReV. A 1979, 19, 2446. (15) Prammer, M. G. Proceedings Society of Petroleum Engineers; 1994; p 55 (SPE 36980). (16) Schmidt, R.; Hansen, E. W.; Stocker, M.; Akporiaye, D.; Ellestad, O. H. J. Am. Chem. Soc. 1995, 117, 4049. (17) Hansen, E. W.; Sticker, M.; Schmidt, R. J. Phys. Chem. 1996, 100, 2195. (18) Soper, A. K. Chem. Phys. 2000, 258, 121-137. (19) Narten, A. H.; Danford, M. D.; Levy, H. A. Faraday Discuss. 1967, 43, 97-107. Danford, M. D.; Levy, H. A. J. Am. Chem. Soc. 1962, 84, 3965-3966. (20) Gran, H. C.; Hansen, E. W. Cem. Concr. Res. 1997, 27, 1319. (21) Overloop, K.; Van Gerven, L. J. Magn. Reson., Ser. A 1993, 101, 179. (22) Overloop, K.; Van Gerven, L. J. Magn. Reson. 1992, 100, 303. (23) Akporiaye, D.; Hansen, E. W.; Schmidt, R.; Stocker, M. J. Phys. Chem. 1994, 98, 1926. (24) Strange, J. H.; Webber, J. H. Meas. Sci. Technol. 1997, 8, 555. (25) Liaw, H.-K, Kulkarni, R.; Chen, S.; Watson, A. T. AIChE J. 1996, 42, 538. (26) RI WinDXP V2.4 software; Resonance Instruments: Oxfordshire, UK, 1998.
