Unsaturated hydraulic conductivity from nuclear magnetic resonance measurements

Transcription

1 WATER RESOURCES RESEARCH, VOL. 42,, doi: /2006wr004955, 2006 Unsaturated hydraulic conductivity from nuclear magnetic resonance measurements M. A. Ioannidis, 1 I. Chatzis, 1 C. Lemaire, 2 and R. Perunarkilli 1 Received 7 February 2006; revised 19 April 2006; accepted 1 May 2006; published 14 July [1] Gravity-driven drainage of water from a column of glass beads of uniform size is studied using nuclear magnetic resonance (NMR). The evolution of proton magnetization and its spin-spin relaxation time is measured as a function of drainage time at different locations within the column. On the basis of these measurements a model for calculating water relative permeability directly from relaxation time data at various saturations, originally proposed by Chen et al. (1994), is successfully tested for the first time. Citation: Ioannidis, M. A., I. Chatzis, C. Lemaire, and R. Perunarkilli (2006), Unsaturated hydraulic conductivity from nuclear magnetic resonance measurements, Water Resour. Res., 42,, doi: /2006wr Introduction [2] Knowledge of the unsaturated soil hydraulic functions is an essential element for the modeling of vadose zone flow and transport processes of interest to soil science, agricultural engineering, environmental engineering and groundwater hydrology [e.g., Hillel, 1998; Kool et al., 1987]. Flow in the vadose zone is commonly described in terms of the classical Richards equation, obtained by combining the continuity and Darcy equations under the assumption of negligible pressure drop in the gas phase [e.g., Celia et al., 1990; Dullien, ¼ ; ð1þ v w ¼ kk rwðs w r wgx P c ðs w ÞÞ; ð2þ m w in which f is the porosity and v w, S w, m w, r w are the velocity, saturation, viscosity and density of water, respectively. P c (S w ) and k rw (S w ) are hysteretic functions describing the dependence of capillary pressure and water relative permeability, respectively, on water saturation. From a theoretical perspective and with particular emphasis on drainage-type displacements, much effort has been directed toward explaining the form of P c (S w ) and k rw (S w ) functions in terms of soil structure and pore-scale fluid distribution [e.g., Tyler and Wheatcraft, 1990; Rieu and Sposito, 1991; Perrier et al., 1995; Kosugi, 1996; Or and Tuller, 1999; Hunt and Gee, 2002; Chan and Govindaraju, 2003], offering physically meaningful alternatives to commonly used empirical expressions [e.g., Brooks and Corey, 1964; Mualem, 1976; van Genuchten, 1980]. 1 Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada. 2 Department of Physics, University of Waterloo, Waterloo, Ontario, Canada. Copyright 2006 by the American Geophysical Union /06/2006WR [3] Traditional steady state methods for measuring soil hydraulic properties [Dane and Topp, 2002; Stolte et al., 1994] are tedious, time consuming and subject to restrictive simplifying assumptions. Faster and more flexible alternatives have been sought, the most popular of which is inverse modeling of transient flow data to determine the P c (S w ) and k rw (S w ) functions simultaneously by optimization (i.e., least squares estimation of the parameters in Brooks-Corey, van Genugthen-Mualem or other empirical models of the hydraulic functions) [Kool et al., 1987]. This approach has been applied to data obtained from onestep [e.g., van Dam et al., 1992] and multistep outflow [e.g., Eching et al., 1994; Nützmann et al., 1998], evaporation [e.g., Romano and Santini, 1999] and centrifuge experiments [Šimůnek and Nimmo, 2005]. Posed as an inverse problem, however, determination of soil hydraulic functions is complicated by issues of parameter nonuniqueness and identifiability. Perhaps more importantly, every experimental method available to date rests on the validity of description of unsaturated flow by a continuum phenomenological equation (multiphase Darcy s law), which offers little insight into the pore-scale physics of immiscible flow. [4] Geometric confinement strongly affects the nuclear magnetic resonance (NMR) response of fluids through an enhancement of the rate of relaxation of nuclear magnetization caused by interactions of fluid molecules with the pore surface. This fact has found widespread application in laboratory core analysis and borehole logging of hydrocarbon formations (for comprehensive reviews, see Watson and Chang [1997], Kenyon [1997], and Dunn et al. [2002]). Recently, magnetic resonance sounding (MRS) has also emerged as a new, surface geophysical method for the determination of water content and its characteristic NMR relaxation times [Lubczynski and Roy, 2003; Roy and Lubczynski, 2005]. NMR monitoring of gravity drainage of water and light oil in a sand column was first demonstrated by Choi et al. [1997]. These authors determined the evolution of proton magnetization and its spinspin relaxation time as functions of drainage time at one position in the column. Their measurements revealed that the bulk-like fraction of the liquids drains out quickly, while the pendular and surface film fractions drain slowly 1of6

2 IOANNIDIS ET AL.: RAPID COMMUNICATION until equilibrium is reached. More recently, Bird et al. [2005] used stray field NMR to measure the distribution of decay times for spin-lattice relaxation of water in soil, sand, and glass bead packs under drainage conditions. They observed a shortening of the average relaxation time and a shift of the relaxation time distribution to values smaller than are present in the distribution at full saturation. Their observations are in agreement with the experimental findings of Choi et al. [1997] and with pore network simulations of NMR relaxation in partially saturated porous media [Chang and Ioannidis, 2002]. In this communication it will be demonstrated experimentally that the relative permeability (unsaturated hydraulic conductivity) for drainage of water from a granular medium is quantitatively related to the proton nuclear magnetization and its spin-spin relaxation time, in a manner consistent with accepted theoretical concepts of flow through porous media. 2. Theoretical Concepts [5] The permeability of fluid-saturated media has been empirically correlated with average NMR relaxation time, t a,sat,ask / f m 2 t a,sat with 2 m 4[Thompson et al., 1989; Dunn et al., 2002, and references therein]. This correlation is supported by theoretical arguments based on percolation theory [e.g., Thompson et al., 1989] and computations of flow [Liang et al., 2000] and NMR relaxation [Olayinka and Ioannidis, 2004] in stochastic replicas of porous media, which further reveal that t a,sat is associated with the correlation length of the pore space, i.e., with a characteristic pore size. Also on percolation theoretical grounds, Katz and Thompson [1986] have obtained a scaling of permeability in the form k / 2 c /F, where c is a critical pore throat size associated with the breakthrough of the nonwetting phase during drainage (percolation threshold), and F = s w /s (Sw =1) is the formation resistivity factor, defined as the ratio of the electrical conductivity of water to the electrical conductivity of a water-saturated porous medium. The formation factor is related to the tortuosity of fluid-filled pore space [Dullien, 1992]. Since t a,sat and c reflect characteristic pore and throat sizes, respectively, a strong correlation of permeability (which is controlled by pore throat sizes) to average NMR relaxation time implies a strong correlation between pore and throat sizes [Kleinberg, 1996]. This implication is verified by computer simulation [Olayinka and Ioannidis, 2004; Arns et al., 2005]. A strong correlation between pore and throat sizes may be expected of granular media, but would not be universally true. Consistent with percolation arguments, one may therefore write [Thompson et al., 1989]: k / t 2 a;sat sð Sw¼1Þ : ð3þ s w We expect that a similar relationship would also hold for the effective water permeability at partial saturation (i.e., the unsaturated hydraulic conductivity): Consideration of second Archie s law [Dullien, 1992], s (Sw =1)/s (Sw <1) = S w 2, then yields the following expression, first suggested by Chen et al. [1994]: k rw k ef f k ¼ t 2 a Sw 2 t : a;sat Equation (5) is a simple, yet physically sound model of water relative permeability in terms of quantities that are directly accessible by NMR and which characterize not only the amount of water present, but also the average size of pore spaces in which water is confined. In what follows, this model will be tested for the first time and will be shown that it adequately describes gravity drainage of water from a column of glass beads of uniform size. 3. Experimental Methods [6] Experiments were conducted in a long glass column (internal diameter of 0.8 cm) filled to a height of 125 cm with glass beads of average diameter equal to 300 mm. The column was first filled with degassed and deionized water and then glass beads were added gradually, vibrating the column lightly after each addition. The porosity of the bead packs produced in this manner was f = ± Their permeability, measured by the falling head method [Dullien, 1992], was k = 100 ± 1 Darcy. To perform gravity drainage experiments, a valve at the bottom of the column was opened and the water was allowed to drain into a container resting on a digital balance to measure cumulative outflow as a function of time. [7] During gravity drainage, the water saturation at any location in the column, S w (x, t), decreases continuously with time as progressively narrower pore spaces are invaded by air. In the NMR experiments reported here, continuous measurements of the relaxation of transverse proton magnetization were analyzed to provide the time evolution of total magnetization, M o (t), and transverse (spin-spin) relaxation time constants at selected measurement locations. A robust model for the fitting and extrapolation of magnetization decay data is the modified stretched-exponential (MSE) model [Peyron et al., 1996]: Mt ð NMR Þ ¼ M o ðþexp t t NMR t o 1 þ t NMR t c ð5þ " # b 1 ; ð6þ where t o, t c and b are fitting parameters. For gravity drainage, NMR monitoring is possible if the measurement timescale, t NMR, is much smaller than the timescale of saturation change at the measurement location, so that the local total magnetization M o (t) may be considered constant during signal acquisition. Fitting of magnetization decay data, M(t NMR ), by equation (6) provides the parameters t o, t c and b and enables the determination of total magnetization, M o (t), by extrapolation to t NMR = 0. Total magnetization is directly proportional to water content, so that water saturation at the measurement location is readily obtained as k eff / t 2 a sð Sw<1Þ : ð4þ s w S w ðx; t Þ ¼ M oðþ t M o ð0þ ; ð7þ 2of6

3 IOANNIDIS ET AL.: RAPID COMMUNICATION repetition time of 7 min. Coil tuning was checked every time the column was repositioned. All NMR data were interpreted in terms of the MSE model. 4. Results and Discussion [9] The saturation distribution at gravity-capillary equilibrium, measured by NMR in duplicate experiments, is shown in Figure 1 alongside the least squares fit of these data in terms of the Brooks-Corey empirical model [Brooks and Corey, 1964]: S e S w S wr ¼ P l c 1 S wr Pc o : ð8þ Figure 1. Water saturation distribution in a column of glass beads (hdi = 300 mm) at gravity-capillary equilibrium. Points are duplicate experiments, and solid line is the Brooks-Corey model fit (equation (8), parameters given in text). where M o (0) is the local total magnetization at complete saturation. In the short-time regime, t NMR t c, the MSE model exhibits the theoretically expected, single-exponential behavior with characteristic transverse relaxation time t o that is inversely proportional to the product of average pore surface-to-volume ratio and surface relaxation strength. In the long-time regime, t NMR t c, the MSE model exhibits stretched exponential behavior with exponent b and characteristic transverse relaxation time constant t a = t o 1/b (t c ) (b 1)/b [Peyron et al., 1996]. The values of t a and b reflect, in an average sense, the shape and breadth of an underlying distribution of time constants characterizing the state of confinement of water [Thompson et al., 1989; Choi et al., 1997]. [8] A Bruker DMX 500 MHz Ultrashield TM NMR spectrometer was used for all NMR measurements. A 3-cm birdcage coil was employed to ensure a coil quality factor that was independent of the water content of the investigated volume and the temperature was controlled to T = 295 ± 1 K. All measurements employed the Carr-Purcell- Meiboom-Gill pulse sequence with 400 ms spacing between 180 pulses. Gravity drainage was monitored at three different locations in the column (25-cm, 40-cm and 55-cm from the top) in three different experiments. In these tests, 3073 echoes were acquired at a repetition rate of 14 s per scan, providing for a delay sufficiently long to ensure full recovery of longitudinal magnetization, but short enough to ensure minimal saturation change within the investigated volume. Gravity drainage at each location was monitored for a total time of 152 min and coil tuning was checked at the beginning and end of each experiment. At the completion of a drainage experiment the column was sealed to prevent evaporation and left to equilibrate for 24 h. NMR measurements were then performed at several different locations to determine the water saturation distribution, information from which the drainage capillary pressure curve was determined. In these tests, 32 accumulations of 6000 echoes were acquired at a The experimentally determined residual water saturation is S wr = ± and the breakthrough capillary pressure and exponent determined from the fit are P c o = 2,274 Pa and l = 5.07, respectively. An estimate of the breakthrough capillary pressure as P c o =2g air-water /hr t i,wherehr t i =0.21hDi [Ng et al., 1978], g air-water =72mN/mandhDi =300mm, is 2,286 Pa, in excellent agreement with experiment. [10] The decrease of the average transverse relaxation time t a with drainage time in three different locations within the column (see Figure 2a) reflects the confinement of water in pore spaces of gradually decreasing dimensions. The same data are plotted against water saturation in Figure 2b, which also shows t a versus S w from duplicate NMR measurements at gravity-capillary equilibrium. Tight clustering of the t a (S w ) data is observed for either flow (dynamic) or equilibrium NMR measurements, as would be expected for a porous medium of uniform properties, but the dependence of t a on S w is somewhat different between the two. At high values of S w, t a (dyn) < t a (eq) is consistent with an apparent shortening of relaxation time caused by loss of transverse magnetization due to flow of excited protons out of the measurement volume [Fukushima, 1999]. This effect is expected to be significant only at high values of water saturation, since the minimum residence time of protons in the measurement volume (calculated for fully saturated, gravity-driven flow of water through a cylinder of diameter equal to 0.8 cm and height of about 3 cm) is approximately 30 s much greater than the acquisition time of 1.3 s. For 0.2 < S w < 0.8, t a (dyn) > t a (eq) reflects actual differences in the state of water confinement (microscopic distribution) between flow and equilibrium conditions. While at equilibrium the water saturation within the measurement volume is dictated by the balance of gravity and capillary forces and is relatively homogeneous, under flow (dynamic) conditions a displacement front delineates draining pores and waterfilled pores, the presence of the latter giving rise to an average relaxation time t a (dyn) > t a (eq) within the measurement volume. No significant difference is observed between t a (dyn) and t a (eq) for S w < 0.2, as expected. [11] Water relative permeability calculated solely on the basis of NMR data by virtue of equation (5) is shown in Figure 3. Alongside is plotted the water relative permeability calculated according to the empirical Brooks-Corey model [Assouline and Tartakovsky, 2001]: krw ðbcþ S w ð Þ ¼ S w S wr 1 S wr 2þ2:5l ð Þ=l : ð9þ 3of6

4 IOANNIDIS ET AL.: RAPID COMMUNICATION The latter is seen to agree closely with k (eq) rw (S w ) determined from equilibrium NMR data in the range 0.25 < S w <1.To test whether the data of Figure 3 can predict experimental data of gravity drainage (cumulative outflow and local saturation evolution), numerical simulations of the experiment were carried out using the CompFlow code [Unger et al., 1995]. One simulation was performed using k (dyn) rw (S w ) determined from dynamic NMR data and one using k (BC) rw (S w ) as input. In both cases, the experimentally determined P c (S w ) data were used. Figure 4a compares the cumulative outflow measured in duplicate experiments to the simulation predictions. The prediction based on k (BC) rw (S w ) is reasonable, but the one based on k (dyn) rw (S w ) clearly agrees better with experiment. That the NMRdetermined k (dyn) rw (Sw) closely characterizes the gravity drainage experiment is seen in more detail in Figure 4b, which compares simulation results to measured saturation evolution at three locations within the column. [12] The NMR-based approach may be contrasted to alternative methods of hydraulic conductivity determination, which are based on combination of numerical modeling of transient flow experiments with parameter estimation (optimization) methods. Multistep outflow experiments in a glass bead pack of f = and k = 82 Darcy, and thus similar to ours, have been analyzed in this manner by Nützmann et al. [1998]. Assuming a representation of P c (S w ) and k rw (S w ) functions in terms of the general van Genuchten-Mualem model: S e S w S wr ¼ 1 þ P n m c 1 S wr Pc o ; ð10þ p k rw ðs e Þ ¼ ffiffiffiffi 2; S e Ix ðp; qþ ð11þ Figure 2. Characteristic transverse relaxation time t a (a) as a function of drainage time at three different locations within the column and (b) as a function of water saturation from NMR measurements at three different locations under gravity drainage (solid symbols) and at gravity-capillary equilibrium in duplicate experiments (open symbols). where x S e 1/m, p m +1/n, q 1 1/n and I x (p, q) is the incomplete beta function, they reported best fit values of P c o = 2180 Pa, S wr = 0.137, m = and n = Their k rw (S e ) function compares favorably to our measurements (see Figure 4c), but the water retention function obtained by substitution of their best fit parameter values into Equation (10) is not representative of a glass bead pack. It is emphasized that Nützmann et al. [1998] solved an inverse problem to estimate the hydraulic functions. On the contrary, NMR provides a direct measurement, subject to the validity of Equation (5). Here, numerical simulation using Richards equation is employed only to verify the NMR measurements. [13] Finally, some comments regarding the extension of Equation (3) to partially saturated porous media are in order. Should connectivity of the wetting phase be lost during drainage and parts of the wetting phase become hydraulically isolated, then Equation (4) will not hold. Under conditions of strong wettability, however, even during drainage to water phase saturations so low that bulk water seemingly exists only as isolated pendular structures in pores invaded by the nonwetting phase, the wetting phase remains hydraulically connected through thin surface films. This has been elegantly demonstrated in glass bead packs by Dullien et al. [1986] and should also be the case for 4of6 Figure 3. Water relative permeability as a function of water saturation determined by equation (5) from NMR measurements under gravity-capillary equilibrium (open symbols) and gravity drainage conditions (solid symbols). Solid line is equation (9) with l = 5.07 and S wr =

5 IOANNIDIS ET AL.: RAPID COMMUNICATION not be representative of the length scale controlling the effective permeability of a partially saturated porous medium. While not the case here, this possibility merits theoretical and experimental investigation. 5. Conclusions [14] NMR measurements of proton transverse magnetization and its characteristic relaxation time were carried out under conditions of gravity drainage of water from a column of glass beads of uniform size for the purpose of testing an NMR-based model of water relative permeability that was first suggested by Chen et al. [1994]. Experiments and simulations demonstrated that this model accurately describes the unsaturated hydraulic conductivity under conditions of drainage in a granular medium. This finding has potentially significant implications for the understanding and measurement of unsaturated hydraulic conductivity by NMR methods in the field and laboratory. [15] Acknowledgment. The authors gratefully acknowledge financial support for this research provided by the Natural Sciences and Engineering Research Council (NSERC) of Canada. Figure 4. (a) Cumulative water outflow measured in duplicate experiments (symbols) and simulation predictions using water relative permeability data from NMR measurements (solid line) and from equation (9) (dashed line). (b) Evolution of water saturation at three locations within the column measured by NMR (symbols) and predicted by simulation using water relative permeability data from NMR measurements (solid lines). (c) Comparison of NMRmeasured relative permeability to the parameter estimation results of Nützmann et al. [1998] (see text for details). natural sands and sandstones, the solid surfaces of which exhibit fractal geometry [Stallmach et al., 2002; Radlinski et al., 2004] that enhances wetting phase retention and connectivity. More significant is the possibility that t a might References Arns, C. H., M. A. Knackstedt, and N. S. Martys (2005), Cross-property correlations and permeability estimation in sandstone, Phys. Rev. E, 72(4), Assouline, S., and D. M. Tartakovsky (2001), Unsaturated hydraulic conductivity function based on a soil fragmentation process, Water Resour. Res., 37, Bird, N. R. A., A. R. Preston, E. W. Randall, W. R. Whalley, and A. P. Whitmore (2005), Measurement of the size distribution of water-filled pores at different matric potentials by stray field nuclear magnetic resonance, Eur. J. Soil Sci., 56, Brooks, R. H., and A. T. Corey (1964), Hydraulic properties of porous media, Hydrol. Pap. 3, Colo. State Univ., Fort Collins. Celia, M. A., M. T. Bouloutas, and R. L. Zarba (1990), A general massconservative numerical solution for the unsaturated flow equation, Water Resour. Res., 26, Chan, T. P., and R. S. Govindaraju (2003), A new model for soil hydraulic properties based on a stochastic conceptualization of porous media, Water Resour. Res., 39(7), 1195, doi: /2002wr Chang, D., and M. A. Ioannidis (2002), Magnetization evolution in network models of porous rock under conditions of drainage and imbibition, J. Colloid Interface Sci., 253, Chen, S. H., H. K. Liaw, and A. T. Watson (1994), Measurements and analysis of fluid saturation-dependent NMR relaxation and line broadening in porous media, Magn. Reson. Imaging, 12, Choi, C., J. Bharatam, I. Frola, M. B. Dusseault, M. B. Geilikman, I. Chatzis, and M. M. Pintar (1997), Monitoring of gravity drainage of water and light oil through a sand column by proton nuclear magnetic resonance, Appl. Phys. Lett., 71, Dane, J. H., and G. C. Topp (Eds.) (2002), Methods of Soil Analysis, part 1, Physical Methods, 3rd ed., Soil. Sci. Soc. of Am., Madison, Wis. Dullien, F. A. L. (1992), Porous Media: Fluid Transport and Pore Structure, 2nd ed., Elsevier, New York. Dullien, F. A. L., F. S. Y. Lai, and I. F. Macdonald (1986), Hydraulic continuity of residual wetting phase in porous media, J. Colloid Interface Sci., 109, Dunn, K. J., D. J. Bergman, and G. A. LaTorraca (2002), Nuclear Magnetic Resonance: Petrophysical and Logging Applications, Handb. Geophys. Explor., vol. 32, edited by K. Helbig and S. Treitel, Elsevier, New York. Eching, S. O., J. W. Hopmans, and O. Wendroth (1994), Unsaturated hydraulic conductivity from transient multistep outflow and soil-water pressure data, Soil Sci. Soc. Am. J., 58, Fukushima, E. (1999), Nuclear magnetic resonance as tool to study flow, Annu. Rev. Fluid Mech., 31, Hillel, D. (1998), Environmental Soil Physics, Elsevier, New York. Hunt, A. G., and G. W. Gee (2002), Application of critical path analysis to fractal porous media: Comparison with examples from the Hanford site, Adv. Water Resour., 25, of6

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