NMR, POROUS MEDIA, AND FUNCTION ESTIMATION

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1 Proceedings of the 3rd International Conference on Inverse Problems in Engineering June 13-18, 1999, Port Ludlow, Washington, USA NMR, POROUS MEDIA, AND FUNCTION ESTIMATION A. Ted Watson Depar tment of Chemical Engineering and Engineering Imaging Laboratory Texas A&M University College Station, TX ABSTRACT Nuclear magnetic resonance (NMR) is a powerful, noninvasive method which can be used to obtain spatially resolved information about media. New approaches for analyzing NMR data can provide many new opportunities to study a wide variety of physical systems. We discuss the development of experimental and analysis methods, called NMR probes, for determining quantitative estimates of properties in the context of characterizing fluids and flow in porous media. An essential element of these probes is the solution of associated function estimation problems. These problems are solved using regularization or sequential estimation with B-spline representations for the unknown functions. INTRODUCTION Nuclear magnetic resonance, first observed in the mid s, is a phenomenon associated with the magnetic properties of some nuclei, whereby they can absorb electromagnetic radiation of a very distinct energy (or frequency), and re-emit this energy during relaxation back to their equilibrium states. In 1950, it was shown that the precise frequency of radiation absorbed depends on the chemical environment of the nuclei, resulting in the observation of chemical shift. Soon thereafter, commercial spectrometers were available. Today, NMR spectrometers are a standard, essential component of many chemistry and biological laboratories for studying chemical structures and properties. The invention of imaging in the early 1970 s sparked additional developments of NMR. Much of the commercial development of NMR imaging (MRI) has been directed to the medical profession, for which MRI is today a standard tool for diagnostic purposes. While it is clear that NMR has become a powerful tool within the fields mentioned, its use outside those fields has as of yet been somewhat limited. There are two main reasons for this. First, new equipment designs are required for bringing NMR measurements to new applications. An example of this is the development of the NMR well logging device for measuring nuclear magnetic properties of fluids in underground porous sediments (Kleinberg, 1996). While early exploratory implementations utilized the earth s magnetic field, commercial applications followed with the development of permanent magnets that can be lowered into well bores. This inside-out configuration, whereby the sample surrounds the magnet, instead of vice versa, retains little resemblance to laboratory spectrometers. Second, new methods for measuring and interpreting NMR data are necessary in order to assess different properties and systems which fall outside the scope of those fields having well-developed applications. In this paper, we describe a new approach for analyzing NMR data, which differs substantially from methods used in spectroscopy and medical imaging. Here, we are concerned with determining properties of interest for the system under investigation. This is accomplished by posing and solving suitable inverse problems. We refer to the specification of experimental and analysis protocols to determine properties as an NMR probe. This methodology is described in context of the development of NMR probes to study fluids and flow in porous media. A no- 1 Copyright 1999 by ASME

2 table feature of the estimation problems encountered is that the unknown properties to be determined are functions. In the next section, we provide a brief description of the basic measurements of NMR. Then, we provide some background on modeling flow in porous media and function estimation, and finally illustrate the development of NMR probes for fluids and flow in porous media. NMR Background There are three major components in NMR imaging equipment: a static magnetic field, a radio-frequency (rf) antennae, and magnetic field gradient coils. When a substance containing an NMR active nuclei, such as hydrogen, is placed within the static magnetic field, the active nuclei will occupy two (or more) discrete energy states. By switching on the rf transmitter, electro-magnetic energy at the appropriate frequency (the Larmor frequency) will increase the population of nuclei (or spins ) in the higher energy state. When the transmitter is turned off, the antennae serves as the receiver to observe the return of spins to the equilibrium energy state. The gradient coils are used to modify the magnetic field so that position can be encoded. In particular, the Larmor frequency is proportional to the magnetic field, so that frequency can be related to position. In fact, a great variety of molecular events can be investigated with different kinds of experiments, each characterized by a particular set of pulse (transmitter) and gradient sequences. There are four basic types of measurements: spin density, relaxation, molecular motion, and chemical shift. Spin density is the relative number of active spins at equilibrium, and it is the primary method to determine the amounts of observed fluids. It is notable, however, that the equilibrium state is not actually observed, and instead must be inferred from transient measurements. Relaxation is a characteristic of the nuclei and its environment. This property is often exploited in medical imaging. Hydrogen nuclei corresponding to fluids in different tissues exhibit different rates of relaxation. By using an experimental protocol whereby the signal intensity depends on the relaxation property, a map of the signal intensity will indicate the relative positions of different tissues. Certain experimental protocols will yield measurements which are sensitive to molecular motion, both incoherent (diffusion) and coherent (fluid flow). Finally, chemical shift reflects the molecular environment of observed nuclei; it is the primary measurement for chemical structure identification. One of the major strengths of NMR is that it can provide information within objects, noninvasively. All the measurements are made outside of the sample, yet information within the sample is resolved spatially. The value to the medical profession is clear: by performing experiments for which there are contrasts in observed signal intensities for nuclei residing within different materials, intensity maps provide pictures of internal tissues and organs that are valuable for clinical diagnoses. Imaging can similarly be used in science and engineering, and a number of such applications have been reported. But, outside of the medical field, the use of MRI as a visualization tool would seem to be of limited utility, particularly when costs are considered. Over the last ten years, we have conducted a research program to develop the use of NMR for studying issues associated with fluids and flow in porous media. To do so, we have developed a new approach for employing NMR imaging one which can be used for research in a wide variety of fields in science and engineering. Our primary concern is quantitative analysis of NMR data to determine information or properties that are important for characterizing fluids and flow in porous media. To do this, we develop NMR probes: experimental and analysis protocols for determining useful information about an observed system. The procedure is as follows. First, identify the property of interest. Select an experiment with measurements that reflect that property, either directly, or through a mathematical model for the process. Model the measurement process, and observed physical system, if necessary. Pose, and solve, a suitable inverse problem to obtain the property of interest from the measured data. In the next section, we provide a brief background for mathematical models and properties of porous media in order to illustrate the development of NMR probes for such processes. Modeling Flow in Porous Media There are two basic approaches for describing the flow of fluids in porous media. We refer to these as microscopic and macroscopic descriptions. In the microscopic approach, one uses the well-known continuum equations to describe conservation of momentum, energy, and mass within the fluid. Boundary conditions are to be specified at all fluid-solid interfaces. This approach is not realistic for large scale simulations of fluid flow since, even in the unlikely event that the detailed morphology of the porous media were known, the solution of the associated boundary value problems of that magnitude and complexity is beyond our current capabilities. The simultaneous flow of multiple fluid phases, such as oil, water, and gas, is even more daunting. Nevertheless, this approach may be used to provide some insight into the phenomena, such as the effects of certain features on experimental observations, and it can be used to predict macroscopic properties. In the macroscopic approach, a continuum-level description representing a larger scale than that for the fluid is used. Whereas the continuum descriptions expressing the conservation principles for the fluid are based on local volume averages corresponding to tens or hundreds of molecules, those for porous media involve local volume averages of tens or hundreds of pores. The advantage of this approach is that the detailed geometry of the porous media need not be explicitly considered. However, the model equations contain effective properties which must be specified. The key challenge here, as in many endeavors involv- 2 Copyright 1999 by ASME

3 ing mathematical modeling, is the determination of the pertinent properties. Efforts to characterize porous media properties have been severely limited by the lack of methods for measuring fluid states within porous media. Indeed, most research in this area is based on measurements on fluids which are outside of the porous sample. The ability of NMR to noninvasively observe events within samples provides many exciting opportunities for studying flows in porous media. It is remarkable that NMR provides unique methods for characterizing porous media both microscopically and macroscopically. For purposes of presentation, we will restrict our discussion of microscopic characterization to the determination of pore-size distributions that is, a representation of the porous media with a distribution function P(S/V ), which represents the relative number of pores corresponding to each pore-size (represented as the volume-to-surface area ratio). We will discuss macroscopic characterizations within the context of single and two-phase flow in porous media. More extensive discussions of the use of NMR for characterizing porous media can be found in a recent review (Watson and Chang, 1997). The macroscopic models can be derived by averaging the mass and momentum balance equations over local volume elements corresponding to the porous media (Slattery, 1969; Whitaker, 1969). This procedure leads to a set of partial differential equations, in which the fluid states, defined as functions of position and time, represent local volume averages. Within those equations are media properties, which are also functions of position. While the storage properties can be defined independently of the averaging process, the transport properties are effective properties that are essentially defined within those equations. Consequently, they must be determined through a suitable inverse procedure, whereby the properties are inferred from observations of fluid states. The flow of a single fluid phase is modeled by the wellknown Darcy expression that relates the hydraulic gradient to the flow velocity: v = k(z) ( p ρg), (1) µ and an equation of continuity that represents the principle of conservation of mass: [φ(z)ρ] t = (ρ v)+ ψ. (2) ψ represents sources or sinks, bold type denotes vectors, and the functional dependence of media properties is explicitly noted. These equations can be combined, with an equation of state for the fluid, to obtain a single partial differential equation for the fluid pressure (or density), as a function of position z and time. The density ρ and viscosity µ represent fluid properties, which may be determined independently of the porous medium. The porosity φ(z) and permeability k(z) are, respectively, storage and transport properties of the porous medium, and they are functions of position. A generalized Darcy equation and equation of continuity for each fluid phase is used to describe the flow of multiple immiscible fluid phases. The continuity equation for each fluid phase is: [φ(z)ρ i s i ] t = (ρ i v i )+ ψ i (3) and Darcy s equation for multiphase flow can be written as: v i = k(z)k ri (s i ) ( ) p i ρ i g. (4) µ i The relative permeability to phase i, k ri (s i ), is a function of fluid saturation the fraction of the pore space occupied by phase i. For two-phase flow, fluid saturations are related by s 1 + s 2 = 1 (5) Due to surface tension, the pressures associated with multiple fluid phases may be different. Capillary pressure is defined as the pressure difference between the fluid phases: p c (s 1 ) = p 1 p 2 (6) where phase 1 is taken as the nonwetting phase. Specifying equations of state for the fluids, the dependent variables (which represent local volume-averaged fluid states) are taken to be pressure and saturation. The relative permeabilities k ri (s i ) and capillary pressure p c (s 1 ) are medium properties, collectively referred to as multiphase flow functions, which must be specified in addition to those properties used to describe single-phase flow. The multiphase flow properties primarily depend on fluid saturation, and thus are functions of a dependent variable in the model equations. ESTIMATION OF PROPERTIES The mathematical model of a physical process can be represented in a generic fashion as: X = G(X, t;p) (7) 3 Copyright 1999 by ASME

4 where the vector X refers to the state, or dependent variables, and G denotes a computational procedure or mapping which may depend on the state, as well as certain independent variables t and properties P. Consider now the estimation of those properties from observations of the process. Suppose measurements Y of the process are available. These measurements reflect the process states, or some functions of the states, and contain some unknown observation errors, represented as: Y = F(X; t)+ ε. (8) Typically, we desire the estimation of properties P from measured data Y i.e., solution of the inverse problem. An important consideration is the functional dependence of those properties. Three general classes of problems can be identified: 1. Parameters are constants, P = C. 2. Parameters are functions of independent variables, P(t). 3. Parameters are functions of dependent variables, P(X). Procedures for solving the first class of problems are relatively well developed. For example, the solution can be found as those parameters that optimize a performance index specified using a selected statistical principle, such as maximum likelihood theory, and suitable qualifications on the measurement errors. There are robust optimization methods available to determine locally optimal solutions, although further research is desired to develop methods that will always provide the global optimum. In many important problems, the parameters are functions of the independent variable, as in the second class. Note that this is the case for flow in porous media, for which the porosity and permeability are functions of position. The representation of NMR measurements in terms of a distribution of pore-sizes is also a model within this class. On the other hand, the multiphase flow properties relative permeability and capillary pressure functions depend on a dependent variable of the model, and thus belong to the third class. The estimation of properties within the second and third classes is a problem of function estimation. In general, we desire the determination of the entire mapping P(t) or P(X) from a set of discrete measurements. The theory associated with solving such problems is not nearly so well developed as for the first class of problems. In particular, there has been very little theoretical work directed to the third class. Nevertheless, such problems are quite common within engineering, often arising within distributed state models represented by partial differential equations. We have developed and applied several methods to solve these function estimation problems, which are summarized in the next subsections. Microscopic Characterization The ability for NMR to detect information about the microscopic structure stems from the fact that fluids in the immediate vicinity of surfaces undergoing enhanced relaxation compared to the fluid under bulk conditions. Under the fast-exchange approximation (Brownstein and Tarr, 1977), the spin-lattice relaxation rate corresponding to a single isolated pore is given by: 1 = 1 + ηs T 1 T 1b V, (9) where S/V is the pore surface-to-volume ratio, T 1b is the relaxation time associated with the bulk fluid, and η is the surface relaxivity. The latter two properties can be determined independently, so that measurement of the relaxation rate of a pore can provide the pore size (represented here as the pore volume-tosurface area ratio). The basic experiment to measure spin-lattice relaxation is inversion-recovery. The pulse sequence corresponds to inversion of the magnetization intensity vector (which is proportional to the net difference in the spin states), and after a selected experimental time, tipping the magnetization vector by 90 so that the net magnetization intensity at that particular time can be observed. The magnetization evolution of spin-lattice relaxation in a single pore, as probed by the inversion-recovery sequence, is represented as ) M(t) = 1 2 exp ( tt1, (10) where t refers to the experimental inversion time. When performing an experiment on a fluid-saturated porous medium, all the pores are observed at the same time, so that the signal is represented as where the kernel function is τmax M(t) = P(τ)K(t, τ)dτ, (11) τ min ( K(t, τ)= 1 2 exp t ). (12) τ P(τ)dτ represents the relative number of pores having relaxation times between τ and τ + dτ. Once the distribution function P(τ) is estimated from a set of discrete measurements {M(t i ), i = 1,..., n}, it can be scaled to determine the pore-size distribution (Liaw et al., 1996). Notice that this is a function estimation problem. It corresponds to the solution of a Fredholm integral equation, and is 4 Copyright 1999 by ASME

5 known to be an ill-posed problem. We employ regularization to stabilize the solution. A performance index is formulated with a term representing the precision of fit to the data and a regularization term: J = [Y F] T τmax W [Y F]+ λ [ P(τ) ] 2 dτ (13) τ min The data vector Y is comprised of measurements M(t i ), i = 1,..., n,and the vector F represents the corresponding values calculated using eq. 11 with a specified distribution function P(τ). The weighting matrix W is chosen on the basis of maximum likelihood principles (Watson et al., 1990). The regularization term serves to penalize a lack of smoothness in the solution. We employ a B-spline basis to represent the distribution function (Liaw et al., 1996): P(τ) = N c j B m j (τ), (14) j=1 where m is the order of the spline. The partition the number and location of knots is not explicitly noted. The degrees of freedom, N, is given by the sum of the order of the spline and the number of interior knots. B-splines are chosen because they can represent any continuous function arbitrarily accurately (Schumaker, 1981). Using eq. 14, the performance index can be written as: J = Y Aβ 2 W +λ Rβ 2 (15) where β contains the set of coefficients c i, i = 1,..., N. The coefficients are determined by finding the value of β that minimizes eq. 15. We incorporate linear inequality constraints, Hβ d, (16) to insure the estimated distribution is nonnegative. For a given value of λ, the unique, global minimum is determined. The B-spline partition is to be specified. We use a relatively large number of knots, so that the solution is not affected by the number and location of the knots. The regularization parameter λ controls the relative smoothness of the solution. Basically, the regularization parameter should be small enough so that there is insignificant bias error in the estimates, but large enough to dampen the relatively high frequency components that are not represented by the data. We make use of a graphical procedure to select the regularization parameter λ, and are currently investigating statistically based criteria. Determining Fluid Distributions A fundamental measurement desired for studying the flow of fluids is the amount of each fluid phase corresponding to various positions within the sample. Using MRI, submillimeter resolution can be obtained. The intrinsic magnetization intensity (which would correspond to equilibrium resonance conditions) is proportional to the amount of the observed fluid phase within a voxel. However, due to the relatively fast relaxation associated with fluids in porous media, the intensity observed is attenuated due to the relaxation process by an amount that depends on the characteristic relaxation of the fluids as they reside within the media and the particular pulse sequence used. Accurate estimates of the intrinsic magnetization intensity can be obtained by modeling the relaxation process and extrapolating to equilibrium conditions (Chen et al., 1993). This can be done by observing the magnetization intensity under different degrees of relaxation. The analysis described in the previous section forms an integral part of that analysis. Once the amount of fluid is determined, the porosity distribution can be determined from an experiment performed on a sample saturated with a single, observed fluid phase. When two phases are present, the saturation can be determined when only one of the phases is observed, or when the signal can be discriminated, such as by using isotope substitution or chemical shift imaging (Mandava et al., 1990; Chang and Edwards, 1993). The reader is referred elsewhere for more complete information (Chen et al., 1993; Chen et al., 1994; Qin et al., 1995; Kulkarni and Watson, 1997). Permeability Distributions The analyses presented in the previous two subsections were directed to the determination of fluids states and, in the case of the porosity, a storage property which can be defined independently of the volume-averaged equations describing flow. The methods involved modeling the measurement process in order to determine particular information within those models. In this subsection, and the next, we address the estimation of the transport properties which appear in the constitutive equations from data measured during flow experiments. In these cases, we first perform the analyses to determine fluid states from the NMR data. Then, we solve an inverse problem to estimate the chosen properties from the measured fluid states using the pertinent set of equations that model the fluid flow experiment. The permeability is a key property for designing operations in underground reservoirs, such as petroleum recovery and groundwater remediation. It is more elusive than the porosity. Whereas there are several different methods for which the porosity distribution might be determined, the permeability must be inferred through an inverse problem since it is essentially defined within the Darcy equation. Furthermore, it tends to vary over orders of magnitudes in a given field or reservoir, whereas the porosity is much less variable. Still, there is very little infor- 5 Copyright 1999 by ASME

6 mation about how permeability may vary within a given sample, since there are no reliable methods for resolving the permeability distribution. MRI provides an exceptional opportunity for determining permeability distributions within samples. Whereas in previous work discussed we have used observations of spin density or relaxation, we now use the rather unique capabilities of NMR for observing molecular motions to determine velocity distributions within porous media. The reader is referred elsewhere for presentation of methodology for determining reliable measurements of velocity distributions in porous media (Chang and Watson, 1999). Here, we outline the methodology to determine estimates of the permeability distributions from an experiment conducted by flowing water, at a constant velocity, through a porous sample (Chang and Watson, 1999). We formulate the estimation using a performance index similar to that used to estimate the relaxation distribution (Seto et al., 1999): [ J = [Y F] T d 2 ] 2 k(z) W [Y F] + λ dz (17) z Note that in this case the permeability is a function of three independent variables, corresponding to the three coordinate directions. The data vector Y is comprised of values of the velocity corresponding to each imaged pixel and pressure values at the inlet and outlet locations, and at various positions on the periphery of the sample, if measured. The corresponding calculated values are obtained from a finite-difference solution of the model equations (eqs. 1 and 2). The model is somewhat simplified since the flow represents steady state conditions, so that the time derivative is omitted, and the fluid is taken to be incompressible. The unknown permeability distribution depends on three spatial dimensions, so it is represented by tensor product B-splines: k(z) = Ns z1 i Ns z2 j dz 2 i Ns z3 Ci, m j,kb m i (z 1 )B m j (z 2 )B m k (z 3 ) (18) k Again, relatively large numbers of knots are used, and inequality constraints are incorporated to ensure non-negativity of the estimates. The estimates are obtained using an iterative minimization procedure based on adjoint equations to calculate the gradient of the performance index and a quasi-newton method for updating the estimates (Seto et al., 1999). Multiphase Flow Properties In the previous problems discussed, the unknown properties were functions of independent variables. We used B-splines to represent the unknown functions and chose the number of knots to be relatively large. The relative smoothness of the estimated function was controlled through selection of the regularization parameter. The multiphase flow properties are a function of a state variable namely saturation and are the major source of nonlinearities in the model equations. The manner in which they are represented can significantly affect the stability of the associated numerical solution of the model equations. Since it is preferable to avoid unnecessarily nonsmooth representations, we perform the estimation process in a sequential fashion. This methodology, initially reported by Watson et al., 1988, has been further developed and extended through applications to a variety of experimental scenarios (see, e.g., Richmond and Watson, 1990; Nordtvedt et al., 1993; Mejia et al., 1995; Mejia et al., 1996; Nordtvedt et al., 1997; Kulkarni et al., 1998). The unknown properties are again represented with B- splines: k ri (S w ) = P c (S w ) = N i j=1 N c j=1 C i jb m j (S w, y i ), i = w, nw (19) C c j Bm j (S w, y c ). (20) The coefficients can be arranged as a single vector: β = [C w 1,...,Cw N w,c nw 1,...,Cnw N nw,c c 1,...,Cc N c ]. (21) For a given B-spline partition, the parameters are determined by minimizing the performance index: subject to the constraints J = [Y F] T W [Y F] (22) Hβ d. (23) The data vector is comprised of pressures measured at the inlet and outlet of the sample, at various times, and saturations measured by MRI at various locations and times during an experiment in which one fluid phase saturating a porous sample is displaced with a second immiscible phase (Kulkarni et al., 1998). The corresponding calculated values are computed using a numerical solution of eqs A trust-region based, linear inequality constrained, Levenberg-Marquardt algorithm (Richmond, 1988) is used to solve this nonlinear minimization problem. Selection of a suitable spline partition is important for accurate estimation of flow functions. The goal is to eliminate 6 Copyright 1999 by ASME

7 significant bias errors, while limiting unnecessary variance errors (Kerig and Watson, 1986; Watson et al., 1988). This is accomplished by successively increasing the number of knots (and hence the number of unknown parameters, and correspondingly the candidate solution space for the unknown functions) until suitable predictions of the measured quantities are obtained. A variety of methods can be used to select the locations for inserting additional knots (Richmond, 1988; Nordtvedt et al., 1993). The basic idea is to add knots to those regions of the properties which are responsible for the greatest discrepancies between measured and predicted data. We find that by using as the initial guess the function obtained by fitting the latest estimate with a spline corresponding to the new partition, a stable progression to an evident global optimum is obtained. Validation Regardless of the methodology used to estimate properties within mathematical models, the estimates should be validated. A minimal validation is to use the estimates together with the mathematical model of the experiment to simulate the experiment. If the simulation provides a reasonably accurate calculation of the measured values, the elements of the estimation process i.e., the mathematical model and the estimates of the properties are validated. This does not prove the results are correct, since there may be other sets of properties that could also provide accurate simulations. Still, it certainly provides a measure of confidence. An unsatisfactory match means that one of the elements of the estimation has failed, so the validity of the estimates must be questioned. In particular, note that any statistical measures regarding the quality and/or accuracy of the estimates is predicated on having the true mathematical model. In particular, we prefer that the mathematical model is sufficiently complete so that the differences between the measured and calculated quantities can be attributed to be random. Residual analysis is a good way to evaluate this (Watson et al., 1990). Note that in situations for which functions are to be estimated, if the functional form used can not adequately represent the true (but unknown) property for some choice of the coefficients, the mathematical model is not correct, and any subsequent estimates may be of little value. This is a common pitfall when simple functional relations are assumed. Other, more stringent, validations can be devised. For example, one could evaluate assumptions of the mathematical model by using a more complete model for simulation, if those neglected physical effects can be estimated. The mathematical model can be further tested by comparing predictions of measured fluid states with measurements that were not included in the estimation. SUMMARY NMR probes provide a new method for utilizing NMR to investigate problems arising in many different fields of science and engineering. Models such as those that represent the flow of fluids through porous media give rise to function estimation problems. The functions may depend on independent variables, or dependent variables within the model equations. Effective solutions to those inverse problems are obtained using B-spline representations for the unknown properties and regularization, or sequential estimation, with robust methods for solving the associated parameter estimation problems. ACKNOWLEDGMENT I gratefully acknowledgement contributions from many collaborators for work cited here, including: current graduate students, Jeromy Hollenshead, Sang-Joon Lee, Kenji Seto; current research associate: Philip Chang; previous graduate students: Raghu Kulkarni, Phil Kerig, David Liaw, Fangfang Qin, Peyton Richmond, Gerardo Mejia Velazquez; previous research associates: Songhua Chen, Carl Edwards, Kyung-Ho Kim; faculty associates: Jan-Erik Nordtvedt and John Slattery. Support from the following sources is gratefully acknowledged: Agip S.p.A., Conoco Inc., Department of Energy, National Science Foundation, Norges Forskningsråd, Saga Petroleum a.s., and the University-Industry Cooperative Research Program for Petrophysical and Reservoir Engineering Applications of NMR at Texas A&M University. REFERENCES Brownstein, K. R. and Tarr, C. E. (1977). Spin-lattice relaxation in a system governed by diffusion. J. Magn. Reson., 17:17. Chang, C. T. and Edwards, C. M. (1993). 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