3026
Ind. Eng. Chem. Res. 2004, 43, 3026-3032
Nuclear Magnetic Resonance Determination of Surface Relaxivity in Permeable Media Jinsoo Uh† and A. Ted Watson*,‡ Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122, and Department of Chemical Engineering, Colorado State University, Fort Collins, Colorado 80523-1370
Nuclear magnetic resonance (NMR) experiments are used to characterize microscopic pore structures in permeable media. NMR relaxation distributions are determined from inversionrecovery experiments, and the surface relaxivity is used to scale relaxation to pore size. The surface relaxivity can be estimated from PFGSTE (pulsed field gradient stimulated echo) experiments. The current methods for determining surface relaxivities from NMR experiments are based on asymptotic approximations to time-dependent apparent diffusivity. However, experimental observations within ranges of time for which such approximations are valid might not be possible. We present a new method for determining surface relaxivity from PFGSTE experimental data that is free from such restrictions on experimental times. 1. Introduction Characterization of permeable media is critical for systems of interest to industry and society. Some examples of processes involving permeable media are petroleum recovery from underground resources, catalytic chemical reactions, and biological metabolism in tissue or bone of the human body. The mathematical simulation of processes within permeable media is usually conducted with continuum-type representations based on volume averaging1,2 or other homogenization techniques. Description of permeable structures at a microscopic level is desired for the prediction of macroscopic transport properties and for an understanding of the physical phenomena occurring within the media. Nuclear magnetic resonance (NMR) spectroscopy is a powerful tool for investigating permeable media. Its use as a probe of microscopic pore structures arises from the contrasting relaxation behavior of NMR-active nuclei, or spins, in the vicinity of a solid surface as compared to bulk fluid. NMR spectroscopy offers a number of advantages compared to other methods that have been used, such as mercury porosimetry, thinsection analysis, and gas adsorption: NMR spectroscopy is noninvasive, it samples the entire structure, and its interpretation does not require geometrical structure assumptions. The information normally determined with NMR spectroscopy is the pore surface-to-volume ratio distribution, which is simply called the pore-size distribution. With the “fast-exchange” approximation and isolated pore assumption,3 the distribution of NMR relaxation rates observed can be directly scaled to the pore-size distribution. The determination of the relaxation rate distribution from measured data has been the subject of numerous studies (see, e.g., Gallegos and Smith,4 Munn and Smith,5 Whittall and MacKay,6 Kroeker and Henkelman,7 Whittall et al.,8 and Liaw et al.9). We use a robust, data-driven method10 based on nonparametric statistical theory.11 The subject of this paper is the * To whom correspondence should be addressed. Tel.: (970)491-5252, Fax: (970)491-7369, E-mail:
[email protected]. † Texas A&M University. ‡ Colorado State University.
determination of the scale that is used to determine the pore-size distribution from NMR relaxation distributions. Several studies have attempted to determine an average pore size from independent experiments, including mercury porosimetry,12 thin-section analysis,13 and gas adsorption,14 to scale the relaxation distribution. However, the advantages of NMR spectroscopy over the conventional experimentation can also be retained here. Pulsed field gradient stimulated echo (PFGSTE) experiments can be used to determine the surface relaxivitysthe NMR relaxation rate at the fluid-solid interfaceswhich provides an intrinsic scale for specifying the pore-size distribution.9,15-18 These reported NMR methods are based on an asymptotic relationship for molecular self-diffusion occurring at vanishingly short times. We have determined that such small times cannot be achieved for many experiments of interest. We present a new method for obtaining NMR surface relaxivity from PFGSTE experiments that is free from restrictions on the experimental times. 2. Estimation of NMR Relaxation Distribution In this section, we describe the method we use to determine relaxation distributions from NMR inversion-recovery experiments and explain the role of surface relaxivity in specifying the pore-size distribution. The determination of the surface relaxivity is addressed in the remaining sections. In an inversion-recovery experiment, the nuclear magnetization M, which represents the ensemble of all observed nuclear magnetic dipole moments in the sample, is initially aligned with the static magnetic field B0. When a 180° radio-frequency (RF) pulse is applied, the magnetization is excited to a higher-energy state and then relaxes back to the initial state. This relaxation is called spin-lattice relaxation. The magnetization evolution at time t of spin-lattice relaxation for a bulk fluid is represented as
[
( )]
M(t) ) M0 1 - R exp -
10.1021/ie030599m CCC: $27.50 © 2004 American Chemical Society Published on Web 03/09/2004
t T1
(1)
Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 3027
Here, M0 is the intrinsic magnetization, and R is a parameter that accounts for inhomogeneities of the RF pulse. R would be equal to 2 if the RF pulse were perfectly homogeneous. The relaxation of fluid within a pore is more complex, but it can be represented as a single-exponential decay at the fast-exchange limit.19 Brownstein and Tarr3 proposed that T1 for a fluid in a pore at the fastexchange limit can be expressed in terms of T1b and T1s, which are the relaxation times in the bulk phase and in the pore surface layer, respectively
1 S 1 S 1 ) 1-η +η T1 V pore T1b V poreT1s
[
() ]
()
(2)
where η is the thickness of the surface layer and (S/V)pore is the surface-to-volume ratio of the pore. Relaxation at the pore surfaces is much faster than that in the bulk phase, because of the interactions between the solid and fluid molecules in the immediate vicinity of surfaces, i.e., T1b . T1s. Then, eq 2 can be approximated as
S η S 1 ) )F T1 T1s V pore V pore
()
()
(3)
Here, η/T1s is defined as the surface relaxivity F. The surface relaxivity represents the strength of the relaxation on pore surfaces, and it depends on the properties specific to the solid surface and the fluid. A uniform value for the surface relaxivity is reasonable regardless of whether the chemical composition of the solid material is uniform within a pore.20 Spatial variations in chemical composition across the medium could lead to spatial variations in the surface relaxivity. One could, in principle, use spatially resolved NMR experiments to determine such spatial variations. Here, we assume that the surface relaxivity is uniform. A porous medium with various sizes of pores exhibits a distribution of relaxation times. The observed magnetization is the sum of the contributions from all of the nuclei within the various pores. The observed magnetization, Mobs(ti), at time ti is expressed as
( )]
[
Np n j
ti
1 - R exp ∑ T j)1N
Mobs(ti) ) M0
+ i
1j
(i ) 1, 2, ..., n) (4)
[
( )] t
∫0∞P(τ) 1 - R exp - τi
3 a) (S/V)pore
(6)
and note that the pore-size distribution P ˜ (a) refers to the distribution of effective pore radii. Combining eqs 3 and 6 gives
a ) 3FT1
(7)
The goal now is to determine the relaxation distribution function P(τ) from the measured magnetization data Mobs(ti), i ) 1, 2, ..., n. We use a nonparametric approach that provides for estimation of the entire function P(τ) without a priori specification of the functional relationship. The distribution function P(τ) is expressed in terms of B-spline basis functions9 as ns
P(τ) )
cjBm ∑ j (τ) j)1
dτ + i
(i ) 1, 2, ..., n) (5)
Equation 3 implies that a T1 distribution can be converted to the corresponding pore-size distribution
(8)
where Bm j (τ) is the mth-order B-spline function, the cj’s are its corresponding coefficients, and ns is the number of knots. The B-spline coefficients cj are then determined by minimizing the difference between the observed data and the calculated values
min J ) (Yobs - Ycalc)TW(Yobs - Ycalc)
(9)
c1,...,cns
The vector Y is composed of observed or calculated values of the magnetization. The weighting matrix W is chosen on the basis of maximum likelihood principles.21 In this work, we assume that the errors are of mean zero and identically distributed, so that W is the identity matrix. Equation 5 is a Fredholm integral equation of the first kind. The recovery of the distribution function from the observed discrete data is an ill-posed problem, i.e., small errors in the data can cause significant differences in the estimate of the distribution function. To stabilize the solution, we use regularization, which imposes smoothness constraints on the distribution function. The regularization term is incorporated into the performance index in eq 9 so that
min J ) (Yobs - Ycalc)TW(Yobs - Ycalc) +
c1,...,cns
Here, Np is the total number of pores, T1j is the relaxation time in pore j, nj is the number of active p nuclei in the pore, N ) ∑N j nj, and i is a random measurement error. We define the T1 distribution function P(τ) as the fraction of nuclei with relaxation times between τ and τ + dτ. Then, eq 4 can be rewritten as
Mobs(ti) ) M0
through surface relaxivity. In this work, we define the effective pore radius as
nλ
∫ττ
1,min
1,max
[ ] d2P(τ) dτ2
2
dτ (10)
Linear equality and inequality constraints are included to provide additional stabilization and to ensure that the estimated distribution is nonnegative. For a specified value of the regularization parameter, a unique, global minimum is determined relatively efficiently.22 This is accomplished by using a numerical least-squares solver capable of incorporating both equality and inequality constraints.23 The unbiased prediction risk criterion11 is used to provide an automatic, data-driven method for the selection of the regularization parameter, λ.10 3. Pulsed Field Gradient Stimulated Echo (PFGSTE) Experiments The surface relaxivity can be related to the sample surface-to-volume ratio, S/V, as follows: The average
3028 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004
Figure 1. Conventional pulsed field gradient stimulated echo sequence.
relaxation time, T av 1 , is defined as the harmonic mean of T1 weighted by the number of nuclei corresponding to each value of T1 Np
1
)
T av 1
1 nj
∑ j)1T
1j
N
)
1
∫0∞ τ P(τ) dτ
(11) Figure 2. Procedure of the proposed method for the determination of surface relaxivity.
Assuming F to be uniform, we obtain the following expression from eq 3 Np
1
1 nj
∑
) F ) F j)1 T1j N T av 1
Np S j
nj
j
N
∑j V
(12)
Here, Sj and Vj represent the surface area and pore volume of pore j, respectively. Given that nj/N ) Vj/V
F
1 T av 1
)
Np S
S
) ∑ V j)1 V j
(13)
Thus, F can be calculated from the sample values S/V and T av 1 . In this section, we describe the PFGSTE experiment and review its use to determine S/V and surface relaxivity. A PFGSTE sequence consists of two identical gradient pulses of amplitude G, pulse width δ, and separation ∆, that are applied during two dephasing and rephasing periods of the stimulated echo sequence24 (Figure 1). The magnetic gradient pulse is actually a vector, but it is written in scalar form here with the understanding that the magnetic gradient is applied only along the axial direction, which is parallel to the direction of the static magnetic field B0. In the PFGSTE sequence, the molecular displacement due to self-diffusion during the time ∆ is encoded by the two magnetic gradient pulses. The apparent diffusivity is defined in terms of the mean-square displacement of the molecules by
Dapp(t) ) 6t
(14)
where r(t) is the position of each spin at time t and the angular brackets indicate the ensemble average. Mitra and Sen25 have shown that the apparent diffusivity can be evaluated from PFGSTE data, M(q,∆), according to
∂ ln[M(q,∆)/M(0,∆)] 1 Dapp(∆) ) - lim ∆ qf0 ∂(q2)
(15)
where q ) δγG and γ is gyromagnetic ratio. In porous media, the apparent diffusivity decreases from the value of bulk diffusivity, D0, as ∆ increases because of restrictions to diffusion posed by the solid surfaces. Mitra et al.26 expanded Dapp(∆) as a polynomial series in terms
of x∆. The “short-time approximation” is given by the first-order term
Dapp(∆) 4 S ≈1xD0∆ D0 9xπ V
()
(16)
The sample surface-to-volume ratio, S/V, has been determined from PFGSTE data using eqs 15 and 16.9,18 To this end, PFGSTE data are measured with a series of different gradient strengths and diffusion times, and the apparent diffusivities are evaluated as a function of diffusion time. The PFGSTE data must be interpolated to find the differential value, as q f 0, in eq 15. Then, S/V is determined by the linear relationship between Dapp(∆) and x∆ given by eq 16. However, the measurements corresponding to the range of time over which the asymptotic representation is valid cannot be achieved in some PFGSTE experiments, particularly when samples exhibit relatively fast relaxation. In an effort to extend the utility of this approach, Hu¨rlimann et al.16 used a two-point Pade´ approximation introduced by Latour et al.27 to interpolate between the short- and long-time asymptotic relations. Consequently, the interpolated equation includes parameters associated with the two asymptotic equations and an additional fitting parameter. The determination of S/V from the interpolated equation involves identification of these parameters, which, in turn, requires additional experiments, such as permeability measurements.15,16 4. Determination of Surface Relaxivity Using PFGSTE Experimental Data In our method, we mathematically model the PFGSTE experiment over the full range of experimental time. Here, the PFGSTE response from a sample with a given pore-size distribution is represented in terms of the strength of the applied gradient pulse and the diffusion time. Once the PFGSTE experiment is modeled, the surface relaxivity is determined from the measured stimulated echo data through an inverse problem. Figure 2 illustrates how this problem is solved. The inversion-recovery experiment is performed, and the T1 relaxation distribution is estimated. The surface relaxivity is initially postulated, and P(τ) is scaled to P ˜ (a). The stimulated echo calculated from the model, Mcalc(q,∆), is compared with the experimental data,
Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 3029
Mobs(q,∆), and the surface relaxivity is updated until it minimizes the performance index
[(
) (
Mobs(q,∆)
J)
Mobs(0,∆)
-
)]
Mcalc(q,∆)
Mcalc(0,∆)
2
(17)
∫0∞P˜ (a) M(a;q,∆) da
(18)
where M(a;q,∆) is the magnetization for pores of size a. The stimulated echo signal M(q,∆) can be expressed as
∫∫Gh (r,r′,∆) exp[-iq‚(r - r′)] dr dr′
M(q,∆) ) M0
∂G h (r,r′, t) h (r,r′,t), ) D0∇2G ∂t
t>0
(20)
with the initial condition
Then, the pore-size distribution can be obtained by the determined surface relaxivity and the relaxation distribution. It should be noted that this approach avoids the need for defining and using time-dependent apparent diffusivities and estimating derivatives from discrete datas often a source of large errorssas is done in previous methods. Furthermore, the entire set of measured data can be used, rather than just those values corresponding to sufficiently “short” times, if they are indeed obtainable. At longer times, however, additional features of the structure of the permeable media, aside from just the value of S/V, become important. One could consider a hierarchy of mathematical models of increasing complexity, depending on the manner in which the structure of the media is represented. We consider here one of the simpler models that still allows for an entire distribution of pore sizes. We assume that the pores are isolated and of spherical shape. Note that current methods used to estimate pore-size distributions from NMR experiments are based on the assumption of isolated pores. To meet this assumption, the data should not be significantly affected by interpore diffusion. The effect of interpore coupling has been investigated by McCall et al.28 They simulated decay rates using an idealized network model containing pores and throats and demonstrated that the spectra of decay rates narrows as the degree of interpore coupling increases. They speculated that interpore coupling is probably not significant for sandstones but could be significant for sol-gel glasses having very large porosities (∼85%). Latour et al.29 investigated whether interpore coupling was significant for various watersaturated rock samples. They showed that the relaxation time distributions are not sensitive to the temperature changes ranging from 25 to 175 °C, implying that the effect of pore connectivity is not significant30 and that use of the isolated pore model is valid in describing such samples. These studies considered the entire range of time associated with spin-lattice relaxation. It should be noted that the diffusion times associated with the PFGSTE experiment can likely be selected sufficiently small to ensure that the isolated pore assumption is always met. The stimulated echo for a sample having the poresize distribution P ˜ (a) is represented by the integral
M(q,∆) )
location r′ during the time ∆. This satisfies the relation
(19)
Here, M0 ) M(q)0,∆)0). The usual short gradient pulse approximation,31 in which the gradient pulse widths are assumed to be much smaller than the diffusion time (δ , ∆), is used. The propagator G h (r,r′,∆) represents the probability that a spin initially at point r moves to
G h (r,r′,0) ) δˆ 3(r - r′)
(21)
and the boundary condition
D0n‚∇G h (r,r′,∆) + FG h (r,r′,∆)|surfaces ) 0
(22)
where δˆ is the Dirac delta function and n is the vector normal to the pore surfaces. The solution of eqs 19-22 can be obtained analytically for simple geometries such as planar, cylindrical, and spherical pores.25,32-34 We use the spherical pore model because it is the most reasonable one to describe the three-dimensional pore space. The stimulated echo of a single spherical pore with radius a is written as25,34 ∞
M(a;q,∆) ) M0
∞
∑∑
6(2l + 1)ζln2 exp(-D0∆ζln2/a2) (ζln2 - q2a2)2
n)0l)0
(
[qajl′(qa) + Fa/D0jl(qa)]2
Fa/D0 -
)
1 2
2
+
ζln2
( )
- l+
1
2
(23)
2
Here, jl is a spherical Bessel function of order l, and the eigenvalue ζln is the nth root of the equation
ζlnj′l(ζln) ) -
Fa j (ζ ) D0 l ln
(24)
We note that numerical solutions could be used if more complex shapes, or even coupled pores systems, were desired. 5. Experiments The methodology for determining surface relaxivity is demonstrated with a cylindrical sandstone sample, 2.5 cm in diameter and 3.9 cm in length. The sample was saturated with deionized water. Experiments were performed using a General Electric 2T Omega CSI system with an Oxford Instruments superconducting magnet having a 31-cm-diameter horizontal bore. The system is equipped with magnetic field gradients of up to 20 G/cm. A 4.4-cm-i.d. birdcage RF resonator was used for RF transmission and reception of the NMR signal. A 180°-delay-90° RF pulse sequence was used in the inversion-recovery experiment. The delay times were carefully selected so that they covered the appropriate range of the spin-lattice relaxation times. For the PFGSTE experiment, we used a pulse sequence based on the “13-interval, condition I” sequence of Cotts et al.35 (Figure 3), which is designed to reduce the errors due to the background magnetic gradient arising from the magnetic susceptibility contrast in the porous media. Compared to the Stejskal and Tanner24 sequence (Figure 1), this pulse sequence has alternating gradient directions with additional 180° RF pulses in dephasing and rephasing stages. As a result, the phase changes due to the background gradient are refocused.
3030 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004
Figure 3. “13-interval, condition I” pulsed field gradient stimulated echo pulse sequence.
Figure 5. Estimated T1 distribution. A logarithmic distribution function, P[log(T1)], is used to represent T1 on the logarithm scale. P[log(T1)] is defined as T1P(T1) ln 10 so that ∫P[log(τ)] d(log τ) ) ∫τP(τ) ln 10 d(log τ) ) ∫τP(τ) d(ln τ) ) ∫P(τ) dτ.
Figure 4. Spin-lattice relaxation data by inversion-recovery experiment.
In stimulated echo sequences, unwanted spin-echoes are also generated and often overlap with the stimulated echoes. This situation becomes more complicated as a greater number of RF pulses are applied. We used phase cycling and spoiler gradients9,18 to mitigate this problem so that the desired stimulated echo was accumulated preferentially. The gradient pulses were designed to be half-sine shapes, which makes q ) 2γδG/π with the maximum amplitude of G and width of δ/2 for each pulse. The PFGSTE experiment was performed with a series of gradient pulses from G ) 1-19 G/cm for each of the four different diffusion times: ∆ ) 23, 38, 58, and 108 ms. 6. Results and Discussion Figures 4 and 5 show the inversion-recovery data and the estimated T1 distribution of the sample, respectively. The area under an incremental part of the T1 distribution curve is directly proportional to the fraction of fluid associated with the corresponding characteristic time T1. The harmonic mean of T1 calculated from the distribution is 74 ms. The surface relaxivity is estimated using PFGSTE data. Figure 6 shows the measured PFGSTE data and the calculated values with the estimated surface relaxivity. The result of the surface relaxivity estimation is summarized in Table 1. Here, the average pore radius, aav, is the effective radius corresponding to the overall S/V evaluated by aav ) 3/(S/V) and S/V ) 1/(FTav 1 ). Figure 6 and the value of root-mean-square error indicate a precise fit to the experimental data, validating the model used in this study. The T1 distribution is converted to the pore-size distribution using the determined surface relaxivity (Figure 7).
Figure 6. PFGSTE data and calculated values. Table 1. Results of Surface Relaxivity Estimation parameter
value
number of data points used performance index, J root-mean-square error surface relaxivity, F average pore radius, aav overall surface-to-volume ratio, S/V
44 0.0145 0.0181 44 µm/s 9.9 µm 0.30 µm-1
It is instructive to investigate the range of time over which the short-time approximation holds. If we suppose that the model and estimated distribution are correct, we can use eq 15 to calculate the apparent diffusivity for any time ∆. This is plotted in Figure 8. Note that these simulated values are consistent with values calculated directly from the measured data (also shown in Figure 8). The short-time approximation given by eq 16, with our estimated value of S/V, is plotted in Figure 8. Note that its slope is consistent with that of the simulated curve as ∆ f 0. The figure indicates that the short-time approximation is valid for xD0∆ < 2 µm. However, that approximation is not useful for the interpretation of the data as no data fall within that range of times.
Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 3031
Nomenclature Symbols
Figure 7. Determined pore-size distribution. The logarithmic distribution function is defined in Figure 5.
a ) effective pore radius (µm) B ) B-spline basis function Dapp ) apparent diffusivity D0 ) bulk diffusivity G ) strength of gradient pulse (G/cm) G h ) propagator of spin jl ) spherical Bessel function of order l J ) performance index M ) magnetization of NMR signal M0 ) intrinsic magnetization N ) total number of nuclei in a sample Np ) total number of pores n ) normal vector n ) number of data points for spin-lattice relaxation experiments nj ) number of nuclei in pore j ns ) number of spline knots P ) relaxation distribution function P ˜ ) pore-size distribution function r ) position of a spin S ) surface area of pore T1 ) characteristic relaxation time of spin-lattice relaxation(s) t ) time (s) V ) pore volume Greek Letters R ) parameter to account for inhomogeneities of RF pulse γ ) gyromagnetic ratio ∆ ) diffusion time (ms) δ ) gradient pulse width η ) thickness of surface layer λ ) regularization parameter F ) surface relaxivity (µm/s) Superscripts
Figure 8. Apparent diffusivities and short-time approximation.
av ) average calc ) calculated m ) order of B-spline obs ) experimentally observed
7. Conclusions
Subscripts
A new method was presented and validated for determining surface relaxivity for use in estimating pore-size distributions. The PFGSTE experiment is mathematically modeled, and the surface relaxivity is found by minimizing the differences between the calculated and measured echo data. The new method is experimentally demonstrated with a sandstone sample. It gave a precise fit to the experimental PFGSTE data, from which the surface relaxivity and the corresponding pore-size distribution are obtained. The new method avoids the restrictions on experimental times that are integral in previous methods. Furthermore, the new method does not require estimates of derivatives of measured data, which can be a significant source of errors. Acknowledgment We gratefully acknowledge contributions from Philip Chang in the experimental work and funding from the U.S. Department of Energy.
b ) bulk fluid pore ) single pore s ) surface layer
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Received for review July 16, 2003 Revised manuscript received January 2, 2004 Accepted January 5, 2004 IE030599M
