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A theoretical model of NMR surface relaxation in porous media Jourabchi, Parisa 2001

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A THEORETICAL M O D E L OF N M R SURFACE R E L A X A T I O N IN POROUS MEDIA by PARISA JOURABCHI B .A.Sc , The University of British Columbia, 1995 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES (Department of Earth & Ocean Sciences; Geophysics Programme) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A May 2001 © Parisa Jourabchi, 2001 U B C Special Collections - Thesis Authorisation Fo rm] In present ing th i s thes is i n p a r t i a l fu l f i lment of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of th i s thes i s for s c h o l a r l y purposes may be granted by the head of my department or by h i s or her representat ives . It i s understood that copying or p u b l i c a t i o n of t h i s thes is for f i n a n c i a l gain s h a l l not be allowed without my wr i t t en permiss ion. Department of (^CaC-H^ -f OogQ-vx S i ^ e r M e S The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada Date 1 of 1 5/28/01 5:11 PM Abstract A model has been developed to express the relaxation time constant of pore water hydrogen protons in porous geological materials. This model is based on the dipolar interactions of the protons with the naturally found paramagnetic ions near the surface of the solid, modulated by the diffusive motion of the pore water molecules. The effect of a uniform non-magnetic coating on the solid, which physically separates the water molecules from the paramagnetic ions, is incorporated in the developed model of surface relaxation rate, 1/T1S-A three-dimensional model pore space is defined as a rectangular prism in a sheet-like pore structure. The motion of the molecules is described as two-dimensional diffusion parallel to the solid pore walls, and one-dimensional diffusion perpendicular to the surfaces. The dipolar interaction is considered between hydrogen protons associated with the water molecules, and a layer of randomly distributed paramagnetic ions on the solid. The protons' distance of closest approach to the layer of paramagnetic ions is determined by the thickness of substance coating the solid surface. The resulting expression for 1/Tis involves a number of parameters, some of which can be eliminated by considering the normalized relaxation rates: l/TiS for the coated sample divided by that of the uncoated sample. Other unknown parameters are used to set bounding limits on the predicted normalized relaxation rates, TiS(plain)/TiS(coated). The predicted trend in Tis(plain)/ TiS(coated) with increasing coating thickness can be approximated by a logarithmic decline for thicknesses up to 2nm. This predicted decay in normalized relaxation rate with increasing coating thickness is more gradual than the trend predicted by previous models. As a result, the application of the developed model to measured relaxation times reported in studies of coated and plain samples is capable of providing realistic estimates of coating thicknesses. ii Table of Contents Abstract ii List of Tables iv List of Figures v Acknowledgements vi Introduction 1 Theory 10 Basic Principles of N M R 10 Ti Relaxation Mechanism 11 Model Description 16 Results and Discussion 23 Summary and Conclusions 34 References 36 iii List of Tables Table 1: References to N M R relaxation rate experiments with different surface states of porous materials 4 Table 2: Relaxation rates and predicted coating thicknesses using models: a) restricted rotational diffusion; and b) 2D planar diffusion! 9 Table 3: Model Parameters 23 Table 4: Model Comparisons 33 iv List of Figures Figure 1: Geometry of a point, P, at a displacement, r, from a pure magnetic dipole at the origin. 12 Figure 2: Cross Section of the sheet-like pore structure 17 Figure 3: Dependence of normalized relaxation rate on the coating thickness 27 Figure 4: The trend in the spectral density functions with respect to angular frequency 28 Figure 5: Dependence of Relaxation rate, 1/Tis, on proton Larmor frequency at different coating thicknesses 28 Figure 6: Dependence of the relaxation rate, l / T i S , on the relative diffusion coefficient, R D 30 Figure 7: Dependence of the relaxation rate, l / T i S , on the thickness of solid layer containing paramagnetic ions, l p 31 Acknowledgements I am grateful for the time, knowledge, stimulating discussions, and other generous support from many people during the course of this research. In particular, I am very thankful to professors Rosemary Knight and Myer Bloom who patiently guided me through this thesis with boundless enthusiasm. I am indebted to professor Rosemary Knight for providing the basis for this research through her many thoughtful suggestions and ideas. Our interactions have been a great learning experience for me. Likewise, many meetings and discussions with Professor Myer Bloom, who is always prepared to answer questions and takes every opportunity to digress to pure physics, have been greatly essential and delightfully pleasant. I would also like to thank Traci Bryar for her eagerness to share valuable experience and knowledge in chemistry, and for all the helpful excitement she brings into this research. I am also very grateful to my friend and fellow student Christina Trotter for her cool, supportive, and cheerful presence, as well as all the useful discussions we had on NMR. In fact, I have very much enjoyed the company of all in Geophysics at UBC. Although it is very difficult to mention all individuals who have provided much useful guidance, I would also like to thank Professor Bruce Buffett. I'm extremely grateful to Maryam and Mohsen, my parents, for their unfailing support and encouragement. I'm thankful also to all dear family and friends. This research was primarily supported by a grant to R. Knight under Grant No. DE-FG07-96ER14711, Environmental Management Science Program, Office of Science and Technology, Office of Environment Management, United States Department of Energy (DOE). However, any opinions, findings, conclusions, or recommendations expressed herein are those of the author and do not necessarily reflect views of DOE. Additional funding was obtained from R. Knight's NSERC Research Grant. V I Introduction Proton nuclear magnetic resonance (NMR) can be used to obtain information about the molecular-scale environment of fluids in porous materials. In a typical proton N M R experiment, the sample is placed in a static magnetic field, which acts on the nuclear magnetic moments of the hydrogen nuclei (protons) associated with the pore fluid, to yield a net macroscopic magnetization at thermal equilibrium. In many experiments, the measured quantity of interest is related to the approach of this macroscopic magnetization to thermal equilibrium following a perturbation. The change in magnetization with time is commonly described as a single exponential with time constant, T i , referred to as the relaxation time. The main uses of N M R in the Earth sciences have involved relaxation time measurements of water or hydrocarbons in porous geological materials. The relaxation mechanism is governed by the diffusive motion of the pore fluid molecules, and the dipolar interactions between the hydrogen protons with other protons or paramagnetic ions. The bulk fluid has a characteristic relaxation time constant associated with this mechanism, which is referred to as T ) B . When the pore fluid is contained in the pore space of a porous material, the chemistry and the physical structure affect the measured relaxation time of the pore fluid. This is due to the diffusive motion of the pore fluid molecules, and the interactions between the hydrogen protons and paramagnetic species such as Fe(III) and Mn(LI) in the solid phase. The relaxation time associated with the effect of the paramagnetics near the solid surface is denoted by Tis. Each source of dipolar interaction leading to relaxation, acts independently to yield the measured relaxation rate, 1/Ti, J — U - L . [i, m r i •. r p L -J M MB MS 1 Most N M R studies use the surface relaxation rate, 1/Tis, in order to obtain information about the molecular-scale environment of the porous geologic material. Of specific interest in my research is the use of N M R to detect, or quantify, the presence of a contaminant adsorbed to, or coating, the solid surface. The basic idea is that a surface coating will affect the interaction between the near surface paramagnetic ions and the hydrogen protons in the pore fluid. What is required is a fundamental understanding of the resulting effect on the measured relaxation rates. Brown and Fatt (1) first demonstrated the effect of silicone coating, on the Ti relaxation rate of water hydrogen protons in the pore space of a sample consisting of sand grains. They concluded that surface relaxation rate, l/T\s, is reduced through physical separation of the paramagnetic ions from the diffusing pore water molecules. Since then, numerous experiments on synthetic and natural materials with various types of coatings have consistently reported a decrease in relaxation rates in the presence of the coating. Table 1 summarizes these experiments in terms of the materials, magnetic field strengths, and the saturating fluids. Although some experiments also involved deuteron T i , Tip, and proton T 2 measurements, we only consider proton Ti in this study. The motivation for many of these experiments was to develop an understanding of the link between the wettability of a geological material and Ti relaxation times. Wettability is defined as the affinity of a fluid to the solid surface. In the analysis of geologic materials, surfaces are thought of as either water-wet or oil-wet. In a water-wet system, the water in the pore space will preferentially form a coating on the pore walls. Thus, water is referred to as the wetting fluid in a water-wet system. In an oil-wet material, water is repelled by the solid surfaces, and is referred to as the non-wetting fluid. Commonly, a layer of hydrophobic substance coating the solid surface of a naturally water-wet system is associated with an oil-wet material. 2 Most studies in Table 1 used a coating substance to change the wettability of their sample porous material. The observed effect of wettability on the surface relaxation rate can thus be attributed to the presence of the coating. Some of the studies considered (1, 3, 4, and 8), attempted to quantify the percentage of each surface type (wettability) using proton relaxation rate measurements of water. Other studies (Table 1: 5-6,9-12, & 14) measured the relaxation rates of protons associated with both the water and oil present in the pore space, as a means to study the surface effects on oil/water saturation. While Kanters, et al. (13) studied the dependence of pore water proton relaxation rates on surface properties through varying the geometry of the pore space (Table 1: 13), Hsu (7) considered the chemistry of the samples and the material coating the solid surfaces (Table 1: 7). Of these studies, only two attempted to correlate the amount of coating on grains with measured Ti relaxation rates (Table 1: 2 & 15). In order to do so, a theoretical model is needed to describe the effect of paramagnetic ions and coating substances on the surface relaxation rate of diffusing pore fluid hydrogen protons. The theoretical basis for the bulk relaxation rate, 1/TIB, of water is well described by Bloembergen, et al. (16). The surface relaxation rate, 1/Tis, in geologic materials is best described by the presence of paramagnetic ions in a layer near the solid surfaces, and diffusing protons in the pore space. There are currently two models which treat such a system: one proposed by Kleinberg et al. (17) specifically for relaxation mechanism in rocks; and the other by Neue (18) for a more general model of dipolar interactions in enhancing relaxation rates. In describing the relaxation mechanism in rocks, Kleinberg et al. (17) used the restricted rotational diffusion model for protons coordinated with paramagnetic ions on the surface of the grains. This model predicts a strong decrease in relaxation rate 1/Tis, as the distance of closest approach, r, of the proton to the paramagnetic ion increases, 1/T ]S x 1/r6. Based on this model, even a thin coating of 5 A, comparable to the size of a single molecule associated with the coating substance, would reduce 1/Tis to less than 1% of its value for the uncoated sample. 3 Table 1: References to N M R relaxation rate experiments with different surface states of porous materials. Reference Study / Year NMR Measurement Field Strength Surface State Saturating Fluid 1. Brown & Fatt (1)/ 1956 proton T] 200 G Sand, some with Dri-film coating Water 2. Devereux (2) / 1967 proton T] 3000 G Clean untreated sandstone; clean sandstone treated with various surfactants; clean sandstone exposed to crude oil Distilled water 3. Kumar, et al. (3) /1969 proton T] 6600 G glass beads and hydrophobic plastic beads (Lucites) Distilled water with paramagnetic salts 4. Williams & Fung (4)/ 1982 proton T] deuteron T l p 0.7 T 4.6 T micro glass beads, some with silicone coating water & deuterium oxide 5. Borgia, et al. (5)/1991 proton T] 0.47 T partial oil coating oil-field & reference Berea cores oil & 5% NaCl brine 6. Howard (6) / 1994 proton Tj 0.23 T chalk samples, some treated with oil-based drilling mud water and decane 7. Hsu (7)/1994 proton T] deuteron T l p 2T glass beads, some with silicone coating and some with crude oil coating; polymer beads, some with silica layer; carbonate core samples, some treated with naphthenic acid Distilled deionized water and deuterium oxide 8. Bonalde, et al. (8)/1995 proton Ti deuteron Ti 2.12T 7T sand and Berea sandstone samples, some with coated phenolic resin water & deuterium oxide 9. Bobroff, et al. (9)/1995 proton Ti & T 2 0.1 T natural & silanated sandstone rocks Dodecane and deuterium oxide 10. Bobroff & Guillot(lO)/ 1996 proton T 2 0.1 T natural & silanated Fontainebleau sandstone, glass capillary tubes Dodecane and deuterium oxide 11. Borgia, et al. (11)/1997 proton Ti 0.47 T well-consolidated clean sandstone cores oil & 5% NaCl brine 12. Howard (12)/ 1998 proton Ti, T 2 400 G chalk samples, some with synthetic mineral oil coating water & synthetic mineral oil 13. Kanters, et al. (13)/1998 proton T] 2.12 T sand, some with crude oil coating Distilled water and NaCl brines 14. Zhang, et al. (14)/1999 proton Ti 400 G sandstone samples, some with crude oil coating refined oil, crude oil, and water 15. Daughney, et al. (15)/2000 proton Ti 2T silica gel, some with sorbed crude oil Distilled de-ionized water 4 Neue (18) provides an explicit numerical expression for the relaxation rate in a generic model of translational diffusion of molecules on a two-dimensional surface. These mobile hydrogen protons interact through dipolar coupling with immobile paramagnetic ions at a fixed distance, r, off the plane of diffusion. If we consider the example of water in the pore space of a geological material, the model would apply to an adsorbed layer of water diffusing on the pore walls, and a random distribution of surface paramagnetic ions with constant density. The separation, z, would correspond to the sum of the coating thickness, and the minimum distance o between the centres of a paramagnetic ion and a water molecule (2.95A). The resulting calculated relaxation rate, l / T i S , varies as 1/z to 1/z according to the field strength of the static magnetic field, the diffusion coefficient as determined by the temperature, and the separation o distance, z. For a coating thickness of 5A, this model would predict a drop in 1/Tis to a maximum of 14% of its value for the uncoated sample at room temperature. Let us evaluate and compare these models using experimental observations. Experiments selected are listed in Table 2, where we give surface coating and the surface relaxation rates for the plain and coated samples. In one case, the reported relaxation times corresponded to the surface relaxation time, Tis (1), while for other studies, we calculated the surface relaxation times using the reported values for Ti and T i B (Eq. [1]). Kanters, et al. (13) reported that the measurements were made at a temperature of 24±1°C, and similarly, Daughney et al. (15) reported a temperature of 25°C. Since the measurement temperature was not mentioned in the other three studies, we assume that those were also conducted at room temperature. Three criteria were considered in the selection process: 1) the substance coating the surfaces contained negligible amount of paramagnetic ions; and 2) the solid surface was uniformly covered by the coating substance; and 3) the geometry, or more specifically the surface area to volume ratio of the pore structure, was not altered by the addition of the coating. 5 It is very difficult to ascertain the above mentioned criteria for all of the studies mentioned in Table 1. It is, however, possible to eliminate those experiments in which the coating process was not well controlled. In consideration of the first criterion regarding paramagnetic ions in the coating substance, we have selected those experiments that used synthetic coating material with known chemical composition, or natural crude oils with available chemical analysis. In the study by Devereux (2), the residual crude oil remaining as surface coating on the clean sandstone after the coating treatment, was analyzed by the author and reported as asphaltene. The dominant sources of paramagnetic ions in asphaltene are vanadium and organic free radicals (19). However, the contribution of these low concentrations of ions in asphaltene to \/T\S is insignificant. Hsu (7) also reports the chemical analysis for two types of crude oil used in that study. One of these, referred to as Kuparuk crude oil, contained < 2ppm by weight of paramagnetic ions. The effect of such low concentrations of paramagnetics in the coating would be insignificant in a Ti measurement. Therefore, the Ti values in the study by Hsu (7) that uses the Kuparuk crude oil is considered in Table 2. Traci Bryar has provided chemical analysis of the crude oil used in the studies by Kanters, et al. (13) and Daughney et al. (15), which also indicated negligible amount of paramagnetic ions. Regarding the second criterion for uniform coverage, studies were selected based on their controlled coating procedure. A dilute solution of the coating substance was applied for uniformly treating all surfaces, while providing appropriate amount of coating in the solution for complete coverage. In the case of Devereux (2), the sample was tested for maximum adsorption of the coating material by measuring the Ti relaxation times for several days during the treatment until an equilibrium value was reached. In two other studies (1, 13), the coated samples floated in water, which is indicative of a strongly oil-wet surface. The sample materials used by Devereux (2) and Daughney et al. (15) had a microporous structure, since their results of relaxation times indicated the sum of two exponentials with 6 distinct time constants, or in other words, they observed a bimodal Ti distribution. Therefore, it is likely that these samples had not been uniformly covered by the coating material. In both cases, the authors attributed the longer relaxation times to the pore fluid contained in the macropores, and thus the values of surface relaxation rates in Table 2 correspond to the longer reported relaxation times. Uniform coverage is a reasonable assumption in the studies considered, but we acknowledge the fact that it is not conclusively established. In consideration of the third criterion, the selected experiments had attempted to remove excess coating using a solvent, and reported leaving only a residue of the coating material. One study reported specific surface area measurements for the coated and plain silica gel samples (15). In this study, Daughney et al. (15) concluded that adsorbed oil did not affect the surface area to volume ratio of their samples. With respect to other studies in Table 2, we assume that a uniform thin coating on a molecular scale does not change the surface area to volume ratio of the pore space, whose dimensions are in the order of microns. In addition, the treatment process in selected experiments involved a dilute solution of the coating substance to ensure a thin and yet uniform coverage. For each of these experiments, it is possible to put a lower limit on the size of the molecules associated with the coating. For the Dri-film material, 11A corresponds to the size of the smallest hydrocarbon chain contained in the coating (7). The heavy crude oil, or asphaltene, can be characterized by hydrocarbon rings, whose thickness is greater than 5 A (20). In the case of synthetic coatings used in the study by Devereux (2), exact values were provided. These estimates of the molecular size of the coating substance impose lower limits on the expected coating thickness if uniform coating is assumed. The dependence of l/T\S on coating thickness, as predicted by the above models, can be assessed using values of relaxation rates reported for coated and plain samples in Table 2. Both the rotational diffusion model by Kleinberg et al. (17), defined as model a, and the two-7 dimensional diffusion model by Neue (18), model b, predict a sharp decay in l / T i S with increasing coating thickness. This effect is attributed to the increasing distance of separation between the interacting water molecules and the paramagnetic ions. As a result, we expect the surface relaxation rates of coated samples to fall considerably by a monolayer of molecules forming the coating substance. According to model a, the relaxation rate of the coated sample would be less than 1 % of its value for the plain sample. Likewise, model b would predict a drop to a maximum of 14%. However, the relaxation rates of coated samples in the studies considered in Table 2, are within 5.7-76% of their corresponding 1/Tis for the plain samples. The reported relaxation rates and the field strength in each study are used to estimate the coating thickness according to models a and b. For this purpose, we have assumed the three criteria regarding the surface states, and used the diffusion coefficient at room temperature for model b. These calculated coating thicknesses are shown in Table 2 along with a lower limit on the molecular size of the coating materials. For uniformly coated samples, the molecular size of the coating substance represents a minimum coating thickness. As shown in Table 2, both models predict coating thicknesses less than the expected minimum. There are two possible explanations for the low estimates of coating thickness by these models. One is that the assumption of uniform coverage does not apply to the studies considered. The second explanation is related to the models themselves. If models a and b are not sufficiently descriptive of the physical system, the resulting trends in relaxation rate with increasing coating thickness would be invalid. In so far as we have no evidence for the non-uniformity of the surface coverage, we plan to develop a more realistic model that would characterize the effect of coating thickness on the surface relaxation rate. A new model is proposed that accounts for the translational diffusion of pore water molecules away from the solid surface. This model considers three-dimensional diffusion of pore water molecules in sheet-like pore structures. It is assumed that paramagnetic ions are 8 uniformly distributed in a layer near the surface of the grains. We consider the presence of a substance uniformly coating the surface as a physical separation between the pore water molecules and the paramagnetic ions. Translational diffusion parallel and perpendicular to the solid surfaces define the motion of the pore water molecules. The purpose of this study is to provide an expression for calculating the surface relaxation rate, l / T i S , in the given three-dimensional pore structure. In particular, we aim to model the dependence of 1/Tis on the coating thickness. The developed model can then be evaluated in comparison to previous models through estimates of coating thickness for experimentally measured values of relaxation times. If successful, this model can be potentially useful for the detection of adsorbed contaminants using Ti relaxation time measurements of fluids in porous geological materials. Table 2: Relaxation rates and predicted coating thicknesses using models: a) restricted rotational diffusion; and b) 2D planar diffusion. Molecular Size Plain 1/T1S Coated 1/T1S Predicted Coating Thickness1: [A] Study / Year Surface Coating c [A] [1/s] [1/s] Model a model b Brown & Fatt (1)/ 1956 Dri-film (SF99 GE) > 11 5.2 1.25 0.8 3.1 Devereux (2) /1967 Methylene blue 4.5 3.61 2.73 0.1 0.5 Dodecylammonium acetate 15 3.61 1.69 0.4 1.5 Stearic acid 23 3.61 1.34 0.5 2.1 asphaltene >5 6.71 0.50 1.6 7.5 Hsu (7)/1994 Dri-film (SF99 GE) > 11 0.53 0.03 1.9 7.7 Kuparuk crude oil >5 0.53 0.12 0.8 3 . 6 Kanters, et al. (13)/ 1998 Crude oil ( for various grain sizes ranging from 115 to 275 pm ) >5 0.88 0.20 0.8 2.9 >5 0.76 0.18 0.8 2.8 >5 0.68 0.13 0.9 3.4 >5 0.34 0.09 0.7 2.5 >5 0.51 0.05 1.4 5.6 Daughney, et al. (15)/2000 Crude oil (sample with highest relative change in relaxation rate due to sorbed crude oil) >5 34.1 19.7 0.3 0.9 1 Distance of closest approach of the pore water protons to the centre of paramagnetic ions in the uncoated samples is estimated to be 2.95A. 9 Theory Basic Principles of NMR This section briefly describes some of the basic principles in N M R Ti relaxation time measurements. In a nuclear magnetic resonance experiment, the sample is placed in a uniform static magnetic field, B 0 . Nuclei in the sample with spin angular momentum, such as hydrogen protons associated with water molecules, H + , precess about the axis of B 0 . The inherent magnetic dipole moment of a spin I particle, fl,, is given by (21) fl, = y,hl, [2] where I is the spin operator with spin quantum number I, h is the Planck constant divided by 2TC, and Yi is the magnetogyric ratio of the spin I particle. A magnetic field exerts a torque on the magnetic moment of nuclear spins, causing them to precess. The angular frequency of precession (Larmor frequency co) is proportional to the magnitude of B 0 , (Oi=-Y,B.- [3] A nucleus of spin 1=1/2 has two spin energy states. The lower energy state is associated with the z-component of spin, I z = +1/2, and points in the direction of the applied field. The higher energy state with I z = -1/2 points in the direction opposite B 0 . In thermal equilibrium, there is a higher population of lower energy states with I z = +1/2. This population imbalance yields a net macroscopic magnetization, <M z> e q, in the z-direction parallel to B 0 . A pulse of an oscillating magnetic field, B i in the x-y plane, and at the protons' Larmor frequency causes <M z > e q to precess about B i . This precession about B i can be described by rotation of <MZ> away from the z-axis. The angle of rotation is linearly proportional to the duration of the pulse, where B i « B 0 . The macroscopic magnetization, <MZ>, returns back to its thermal equilibrium value after the removal of the pulse at time, t = 0. A perturbation caused by Bi for a 180° rotation, can be described by 10 < M z (t) >=< M z >eq (1 - 2e~ t /T'). [4] Many systems exhibit an exponential relaxation to equilibrium characterized by a single time constant, Tj, as in Eq. [4], The inverse of this time constant, 1/Tj, is commonly referred to as the Ti relaxation rate. For the hydrogen protons of the water molecules in a pore space, their instantaneous relaxation is a function of their positions and interactions with the surrounding media. Water molecules diffusing in a pore space sample the entire volume during a Ti measurement. This results in the detection of an average relaxation rate. A typical Tj experiment involves a series of pulse sequences. Each sequence is a 180° rotation followed by a time delay, t, and a 90° rotation, which is referred to as a 180°-t-90° sequence. This is the inversion recovery method, where the 90° pulse brings the z-magnetization into the x-y plane for measurement (22). By varying the time delay, t, in each sequence, the z-component of magnetization can be measured with respect to time. This will allow for the determination of the time constant, T i , as described byEq. [4]. Ti Relaxation Mechanism This section describes the mechanism by which Ti relaxation occurs. Coupling of the nuclear spins (pore water hydrogen protons) with the surrounding medium, historically referred to as the lattice, results in the return of z-magnetization to its thermal equilibrium value. The temperature of the lattice is reflected in the distribution of water and potential energies of the constituent molecules. Therefore, the Ti relaxation rate is also known as the spin-lattice relaxation rate (21). The coupling of the nuclear spins with the lattice is through dipolar interactions that are modulated by the random diffusive motion of the water molecules. The master equation for relaxation of a physical variable due to a random perturbation can be used to derive an expression for the T i S relaxation rate (21). In this case, the physical 11 variable of interest is the macroscopic z-magnetization and the perturbation is due to dipolar interactions. This treatment requires the time dependent dipolar coupling with the lattice to be represented by its Hamiltonian, Hdipolar(t). A description of the interaction energy between two magnetic dipoles, and their relative motion, can be used to derive Hdipular(t). • y Figure 1: Geometry of a point, P, at a displacement, r, from a pure magnetic dipole at the origin. The electronic spins, S, of paramagnetic ions commonly found in natural materials are responsible for these interactions. The following describes the dipolar interaction energy between the magnetic dipoles of two spins, I and S. The spins associated with the hydrogen nuclei of the pore fluid are denoted by I, while S refers to the electronic spin of the paramagnetic ions. The magnetic dipole moment of a spin particle produces a local magnetic field, B i o c . Figure 1 depicts the geometry of a magnetic dipole moment, [is, of a spin S particle at the origin, whose magnetic field at point P is given by (23) B l o c = 0 0 1 [3(ps r ) r -u . s ] , [5] In this equation, |j, 0 is the permeability of free space and r is the displacement vector as shown in Figure 1. The interaction energy, U , between the magnetic dipole moment, of a spin I particle with Bioc is 12 U = -!VB l 0 C = v 4 7 l A r 3 y The next step in deriving an expression for Hdi lar(t) is to introduce the time dependence of the interaction energy. The diffusive motion of pore water proton results in random fluctuations in the relative positions of the interacting spins on a molecular length scale. The paramagnetic ions with electronic spins S are assumed fixed in the solid matrix. As shown in Figure 1, the fluctuating geometrical parameters are r, 0, and <\>. The time dependence of the interaction energy is thus contained in the random diffusive motion of the pore water molecules. It is assumed that each paramagnetic spin S contributes independently to l / T i S . This assumption ignores any correlation between the spins S, which is justified by their fast return to thermal equilibrium relative to the relaxation time of the nuclear spins. The operator form of the interaction energy described above gives the expression for Hdipolar(t). Inserting the expression for magnetic moments in terms of their spin operators (Eq. [2]) into Eq. [6] gives the dipolar coupling Hamiltonian, ^dipolar (t) — ( » V y , Y s / 7 . 2 > V 4 7 1 / [ IS-3(Ir ) (S f)] • [7] Before applying the given expression for the dipolar interaction Hamiltonian to the master equation for relaxation, it is useful to expand it and define its correlation functions. Expansion of this random operator, Hdipolar(t), in polar coordinates yields the sum of five terms2 (21) H ^ W ^ t ^ M A ^ . [8] <7=-2 Each of these terms is the product of the random time varying geometrical component, F(t) ( q ), with the time-independent function of spin operators, A ( q ) . The functions F ( q ) are defined as the ; This is the scalar product of the irreducible rank two tensors: F ( q ) (t) and A (' 13 normalized second order spherical harmonics divided by r3. The remaining time independent terms, A ( q ) , involving spin operators, are then described by A ( 0 ) = 3 + 16TI | l z S z + i ( I + S . + I . S + ) A ' 3 . 8TC {1 Z S + + I + S z } ' A ( 2 ) = 2 + [9] [9a] [9b] [9c] A (q) = A(-q)t _ The spin operators in Eqs. [9] define a complete set of eigenfunctions for systems described by angular momentum. The dipolar coupling of spins, I and S, leading to relaxation, involves the energy transfer between states differing by a spin quantum number 1. Hence, each term in A ( q ) relates to a specific pair of connecting states. It is useful at this point to define the correlation functions, G q q.(T) = F ( q ) (t)F ( ' ! ^(t + T), [10] for the time dependent geometrical components. From an intuitive perspective, any perturbation to the system such as application of B i for rotation, or the dipolar coupling with spins S must involve the resonance frequencies for energy transfer between the spin states as given by Eqs. [9]. These resonance frequencies are described by the Larmor frequency of the protons, (Oi, and that of the electronic spins, CDs. As an example, consider the term A ( 0 ) , which involves an increase in the spin quantum number of S, and a decrease for spin I. This energy difference is denoted by C0s-C0i. For the dipolar coupling to induce relaxation, we expect the time varying geometrical component, Gq q<T), to contain the appropriate resonance frequencies. 14 The master equation for relaxation of a physical variable, such as the z-component of macroscopic magnetization, requires the definition of the correlation functions (Eq. [10]), and their corresponding spectral density functions: <M--2 The spectral density functions, J ( m )(co), are the Fourier transforms of the correlation functions mapped into the laboratory frame. The Wigner rotation matrices, D™(a,/3,y) (24), transform the interaction frame into a laboratory frame with the z-axis parallel to the static magnetic field. The Ti relaxation rate due to dipolar interactions between two unlike spins, I and S, is thus given by (21) [12] where the angular frequencies, co, and cos , represent the Larmor frequencies of the spin systems I and S, respectively. . 3 Note the difference in the coefficients of the spectral density functions arise from a different definition of the geometrical F ( q ) functions by Abragam (1961). 15 Model Description The relaxation rate of pore water hydrogen protons with spin I can be calculated for a given geometry and relative motion of the interacting spins according to Eq. [12]. This would require a description of the diffusive motion of the water molecules in a model representation of the pore space. The time varying position of the pore water molecules relative to that of paramagnetic ions can provide an expression for the correlation functions given by Eq. [10]. Their corresponding spectral density functions defined in Eq. [11] can then be used to calculate the relaxation rate. The model assumes a simple pore structure in three dimensions. Figure 2 shows the cross section of the model pore structure. The pore water molecules diffuse in the rectangular prism between two slabs of the coated solid rock grains. The water thickness, L , is related to the surface area to volume ratio of the pore space and is defined by the following expression: Of particular interest in this model is the parameter l c , which represents the thickness of the substance coating the solid surfaces. The closest distance between the diffusing pore water molecules and the paramagnetic ions is governed by l c . Water molecules, which contain the hydrogen nuclear spins, I, can diffuse to the surface of the solid containing paramagnetic ions with electronic spin S. In case of a metal oxide coating, for example, this model considers the surface to be the centre of the oxygen atoms covering the solid rock grains. The closest distance between I and S, y, is approximated by where l c is the thickness of the coating, n is the bond length between hydrogen and an oxygen atom in water, and d s is the depth of the paramagnetic ion from the solid surface. 16 [13] y = l c + r,+d s' [14] Iron oxides such as goethite and hematite are formed on the surfaces of natural geological materials due to weathering. We have found only one study with reported thickness of the iron-bearing layer, l p on an aquifer sand (25). In this study by Coston et al. (25), various microscopy techniques were used to determine the composition and distribution of oxides on quartz grains. They found through the use of their most sensitive technique, time of flight-secondary ion mass spectroscopy (TOF-SIMS), that all surfaces were uniformly covered by aluminum and iron. Furthermore, they reported thicknesses of these iron-bearing layers to vary from <10nm to more than 30um using the combination of scanning electron microscope (SEM) and TOF-SIMS depth profiles. Coating Material Solid Rock A. T Pore Water Protons With Spin I Layer Containing Paramagnetic Ions With Spin S Figure 2: Cross Section of the sheet-like pore structure. The next step in the derivation of the correlation functions is to express the time varying position of the pore water molecules with respect to a single paramagnetic ion. The correlation functions, Gqq>(T) as defined by Eq. [10], are the ensemble average over the probability distribution of the random functions F(q)(f ,T)F(q)*(fo), and can be expressed as G q q , (T) = J- -JP ( r / f o ,T)p(fJF ( q ) ( f ,x)F ( q r ( fJdf o dr , [15] 17 where, r(t) = r(x),0(x),<t)(x) = r(T),Q(x), and ?(0) = ro,e.,<|>o==roA. .. As previously mentioned, the geometrical components of the Hamiltonian for the interaction energy, F ( q ) and F ( q ) , are related to the second order spherical harmonics, Y 2 ( q ) as in the following expressions: p(q)(ffT) = jfflp^ ) and [16] F W ' ( r „ ) = ^ , [16a] ro where the orientation parameter, Q, represents the polar angles, § and 0, between a diffusing water molecule and a stationary paramagnetic ion with spin S, and r is the distance between them. The conditional probability density, P(r/r0,T), of finding two spins, I and S, with displacement vector, r, given their initial displacement at time zero is rD, can be described by the solution to the diffusion equation for the fluid molecules in a given geometry. In this model, the pore fluid molecules are allowed to diffuse in two-dimensions parallel to the surfaces, and in one-dimension perpendicular to the solid. The probability distribution function is thus expressed in two parts: 1) as the solution to the two-dimensional diffusion equation on a 2D plane parallel to the surfaces; and 2) as the solution to the one-dimensional diffusion equation on a line perpendicular to the surfaces. The former has been expressed by Neue (18) for an infinitesimal layer of diffusing water molecules at a fixed distance of separation, z', from the solid surface containing the paramagnetic ions. The latter solution to the diffusion equation in one-dimension is given by (26) 18 P(z7z ' t) = - j j J = = ^ •p" [ Z ' " ( 2 ' L + Z : ) ] 2 / ( 4 D l T ) + e-[ z'-(21L- Z' 0)-2y] 2/(4D 1 T)J [17] where the pore surfaces represent two reflective barriers at z'=y and z'=y+L. The diffusion coefficient, Dj_, relates to the motion of water molecules perpendicular to the solid surfaces. Eq. [17] is the conditional probability density function for the initial condition, P(z7z',0) = 5 (z ' - z ' ) . The probability density for the initial position is considered to be uniform across the pore space, P(z') = - . H8] L Thus, the probability distribution function of water molecules for the entire pore volume can be expressed as the product of the solution to the two-dimensional diffusion equation (18), with the solution to the one-dimensional diffusion equation in the presence of two reflective barriers (Eq. [17]). The correlation function for a planar uniform distribution of paramagnetic ions on the solid surface is given by Neue (18). Allowing the separation distance, z', between the plane of diffusion and the fixed paramagnetic ions to vary according to the diffusion equation (Eq. [17] & [18]), and defining a planar density of paramagnetic ions at y, N p(y), gives the correlation function due to a planar distribution of paramagnetic ions, '5Np(y)Y 4 V. y+>y+L nl , JO y y V 4 8 A f j j kvD» T k 2- ( z' + z : ) kp(z'/z:,x)p(z:)dz'dz:dk. [19] 2-N The wave number, k, is associated with the spatial Fourier transform in the planar geometry. The diffusion coefficient in the direction parallel to the solid surfaces is expressed as Du. In order to simplify the correlation function Gqq-(T), it is useful to consider only the term with 1=0 in Eq. [17] defining the probability density function in the limit of y « L . This is the 19 solution to the diffusion equation for a single reflective barrier at z'=y. Inserting Eq. [19] into Eq. [11] would give the spectral density functions for z' at an orientation (a,(3,y) with respect to the laboratory frame. Due to symmetry in the x-y plane, the relative orientations can be described by a single angle, (3. In natural geologic materials, the pore surfaces have random orientations, (3. An average over all angles for (3, yields a constant value, (24) D%(a,/3, ytf) => ]\D%(0, /3,0)|2 &in( fi)d0 = \ . [20] Fourier transform of the correlation function and mapping to the laboratory frame using Eq. [20] will give the spectral density functions. The correlation functions can be further simplified before the spectral densities are calculated. Inserting the probability density given by Eq. [18] into the expression for the correlation function expressed in Eq. [19], gives the correlation functions ' 5 N p ( y ) L Y 4 V i i G Q (T) = 48 2 - q -I I fJJk 3e" D | | T k" < L x + L x" + 2 y ) kP(x/x O ,T)dxdx odk [21] V A 1 v 0 0 0 with the change of variables: x = 2 ^ and x„ = ——- [22] It is useful to take the Fourier transform and its inverse with respect to the integrand variable x in order to eliminate one of the integrals: P ( X / X . , T ) = ^ - ?e l s x [fe- I S X P(x/x o , t )dx]ds. [23] Expanding Eq. [23] and inserting it in the expression for the correlation functions given by Eq. [21] yields four integrals with respect to variables: x, x 0, s, and k. The part of the integrand containing x and x 0 can be evaluated analytically and is defined as 20 £(s,u) = Je u x cos(sx) = e u[-ucos(s) + s-sin(s)] + u u 2 + s2 [24] with the change of variable u=kL. The correlation function thus becomes a double integral, and for the limit of one reflecting barrier is described by G Q(X): 5N (y)Y 4 24711" A 2-N f f 3 -(D„u 2+D , s 2)t/L 2-2uy/L e2 / j , , u e " 1 t (s,u)dsdu. — inl J J [25] o o The spectral density functions can be derived by inserting the correlation functions of Eq. [25] into Eq. [11]. The rotation matrices for mapping to the laboratory frame can be represented by an average over all orientations as given by Eq. [20]. The resulting spectral density functions for a single planar density, N p(y), of paramagnetic ions is thus given by ' 4 N ( y)Y f ~ r , _, . .„„. , . . R . u 2 + s 2 37tL2D±J!J y ( R D u 2 + s 2 ) 2 + J ^ duds, [26] where R D is defined as the ratio of the parallel to perpendicular diffusion coefficients, D|/Dj_. It is important to note that the above expression is for a planar distribution of paramagnetic ions, N p(y), at a distance of closest approach, y. The spectral density functions, I(m)(co), can be derived by extending the dipolar interactions to a layer of paramagnetic ions with finite thickness, l p as in the following expression: d+l„ J(m)(co) = - J^m)(co,y)dy. [27] P d A uniform distribution of paramagnetic ions with volume density, N v , is assumed in a layer between y=d to y=lp+d such that d+L Np(y)= JNv(y')dy=Nvlp. [28] 21 The protons' distance of closest approach, d, represents the distance as given by Eq. [14], where d s is taken as the depth of the paramagnetic ion nearest to the solid surface. Thus, inserting Eqs. [26], [28] into [27], yields the spectral density functions for the given geometry of sheet-like pore structure with translational diffusion in three-dimension as follows J ^ ( ( B ) = f - ^ | f u 2 [ e - 2 ' r f / L - e - 2 " < , ' 4 < , ) / L K 2 ( s , u ) VV"'2 | 4 ,duds . [29] The double integrals in the spectral density functions can be evaluated numerically. Consequently, the relaxation rate for hydrogen protons of water in a model pore space of Figure 2, containing a layer of paramagnetic ions, l p , coated by a substance with thickness, l c , can be calculated by insertion of the spectral density functions (Eq. [29]) into the expression for relaxation rate (Eq. [12]). 22 Results and Discussion The developed model can be used to estimate the effect of a coating on the relaxation rate, 1/Tis. Of specific interest is the dependence of 1/Tis on the thickness of the coating, l c . The model incorporates a number of parameters, all of which are listed in Table 3. Accurate values would be required for all of these parameters in order to fully characterize the dependence of relaxation rate on the coating thickness. As a mean of assessing the validity of this model, we will set values for these to compare and explain result reported in the study by Kanters et al. (13). While it is very difficult to obtain some of these parameter values, representative values for all model parameters have been selected and are given in Table 3. The model with these representative values as model parameters is referred to as the standard configuration. Table 3: Model Parameters Model Parameter Description Representative Value 1. Ic thickness of surface coating 0 2. co. 2n x proton Larmor frequency 2 T C X 9 0 M H Z (B0=2.12T) 3. D± diffusion coefficient perpendicular to solid surfaces 2.3x10"9m2/s 4. R D ratio of parallel to perpendicular diffusion coefficients 1 5. Spins S properties of paramagnetic ions FeO(OH) N v number density 2.89x1028m"3 S spin quantum number 5/2 Ys magnetogyric ratio -1.76x1011rad.s"1T1 6. Ip thickness of surface layer containing paramagnetic spins, S 8.9nm 7. Sp/Vp surface area to volume ratio of the porous media 0.36um"1 Some of the parameters in the standard configuration are determined by the experimental conditions such as temperature and the field strength. The temperature dictates the bulk 23 diffusion coefficient of free water, which can be assigned to D i . The experimental data that will be used for comparison were obtained at temperature close to 25°C. The field strength specifies the resonance Larmor frequency of the protons, CO], and the Larmor frequency of spin S, C0s-4 The resonance frequency, C0i, is based on the field strength used in the study by Kanters, et al. (13). The representative surface area to volume ratio, SpA^p=0.36(j,m"1, corresponds to that of a sand sample with average diameter of 115um. This particular type of sand was used in the study by Kanters et al. (13). The specific surface area of this sand was measured by Traci Bryar using the BET Kr-adsorption with a Micromeretics ORR 2100D surface area analyzer. Therefore, three of the parameters are set to known experimental conditions for one of the studies in Table 2 (13). Since these parameters reflect the conditions for Ti measurements of pore water in a sand sample with a particular grain size in the study by Kanters et al. (13), the absolute value of T ] S predicted for this configuration will be compared to their reported measurements of this specific sand sample. The representative values of the other free parameters are based on the following assumptions. The diffusion coefficient, Du=2.3xl0~9m2/s, is that of free bulk water at 25°C (27). This means that the ratio of diffusion coefficients, RD , is set to 1 indicating that diffusion parallel to the surfaces is not restricted. The properties of the paramagnetic spins S, are those of iron(III) associated with the naturally found iron oxide coating, FeO(OH) (goethite), on geological materials. A layer thickness of goethite, l p , can be calculated from a study of goethite coating silica sand (28). This study provides the specific surface area and concentration of goethite coating a silica sand sample. These data, along with the known density of pure goethite (20), yield an estimate of lp=8.9nm. The number density, N v=2.89xl0 2 8m~ 3, is also derived from the density of pure goethite. Free electrons associated with iron(III) give rise to the spin quantum 4 Given the magnetogyric ratio, ys, and the field strength, B0, 005 can be calculated by (Os=-YSB0. 24 number S=5/2, and the magnetogyric ratio, Ys=-1.76xl011rad.s~1T l corresponds to that of an electron. The representative parameter values for the standard configuration are summarized in Table 3. These values allow us to determine the absolute value of the relaxation time for a single configuration of parameters. The predicted surface relaxation time, TiS=0.63s for the standard configuration is in good agreement with the calculated TJS=1.1S corresponding to the measured Tj and T ] B values reported in the study by Kanters et al. (13). For this comparison, the data associated with the 115(im average diameter sand sample with known surface area to volume ratio was chosen. The difference between the two values can be explained by the uncertainties associated with the properties of the paramagnetic ions near the surface and the relative diffusion coefficient R D . The purpose of estimating the Tis relaxation time was simply to verify the validity of the developed model. While the parameter values used in the standard configuration provide an estimate of 1/Tjs, we recognize that there is a high level of uncertainty in some of the selected values that could affect our predicted dependence of l/T\S on the coating thickness. For the purposes of our study, we choose to eliminate the need for some of these parameters by considering the normalized relaxation rate: the ratio of 1/Tis for the coated sample with lc>0, to that of the plain, or uncoated sample with lc=0. This eliminates the need for four of the parameters as follows: N v , y s 2 , S(S+1), and Sp/Vp. The assumption of linear dependence on Sp/V p requires an explanation: this parameter appears in the form of water thickness, L=2(SPA^P)"', in the expression for the spectral density functions. Although it is not explicitly clear from these expressions, their dependence on Sp/V p is effectively linear in the range of interest for all other parameters. This suggests that the model relaxation rate, which is linearly proportional to the spectral density functions, also varies linearly with Sp/Vp. 25 Some of the remaining parameters can be determined from the experimental conditions, while others are unknown. The representative values in Table 3 are used to predict the trend in normalized relaxation rate with respect to the coating thickness. Variation of these parameters can then be used to set bounding limits on the predicted trend of normalized relaxation rate, Tis(plain)/Tis(coated), as l c increases. As the physical separation between the diffusing water molecules and the paramagnetic ions grows with increasing l c , we expect a decrease in the normalized relaxation rate. Figure 3 shows the trend in normalized relaxation rate for increasing coating thickness, l c . The diamonds indicate the predicted normalized relaxation rate for the standard configuration of parameters. A logarithmic trendline using least squares fit to the calculated points (diamonds) is shown by the solid line. In comparison to a power or exponential fit, this trend yields the best correlation coefficient with R = 0.9922. The limiting bounds on the predicted trend are calculated by evaluating the dependence of the normalized relaxation rate on the free parameters: coi, l p , and R D . In the following sections, we consider the role of each parameter in evaluating upper and lower limits on the predicted TiS(plain)/Tis(coated). Subsequently, the limiting values of all parameters are combined to provide the lowest bound on normalized relaxation rate with respect to the coating thickness. Let us first consider the role of (Oi in determining the relaxation rate. Figure 4 shows the variation in the spectral density function, J(co), predicted for the standard configuration on the Larmor frequency, C0i. The frequency spectrum associated with the diffusive motion of the water molecules is described by the spectral density functions. These functions determine the dependence of \fT\s on coi and C0i±C0S (Eq. [12]). These are the transition frequencies of spin energy states for the protons, leading to relaxation. The spectral density functions are characteristic of the particular type of motion involved, and are generally constant at low 26 frequencies. A sudden drop occurs when cox c~l, where the correlation time of the motion, xc, can be intuitively thought of as the system's memory. In other words, the positions of the diffusing particles at times greater than t+xc are independent of their positions at times < t. For our model system, x c is in the order of 100ns with the drop off frequency of ~10MHz (Figure 4). Normalized Relaxation Rate & Coating Thickness 0.8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 lc [nm] Figure 3: Dependence of normalized relaxation rate on the coating thickness. The diamond symbols correspond to the standard configuration, and the solid line is a logarithmic best fit given by the equation shown. The solid and open symbols represent the upper and lower bounds, respectively. The absolute lowest bound with RD=0.005, coI=2Ttx90MHz, and lp=0.3nm is shown by the x symbols. The circles represent the bounds set by frequency (upper for 1MHz and lower corresponding to the standard configuration at 90MHz); the triangles correspond to the bounds for l p (upper at standard configuration with lp=8.9nm and lower at lp=0.3nm); and the square symbols correspond to bounds set by R D (upper at standard configuration with RD=1 and lower at RD=0.005). The variation of l / T i S with frequency is similar to that of the spectral density functions, which are independent of frequency for corXc « 1 , and drop off at C0iXc~l 5 The range of interest for C0i is l-90MF£z. This range was chosen according to the operating frequency of typical field and laboratory instruments. Field instruments operate at frequencies in the order of 1MHz, while the operating frequencies of laboratory instruments range from 10 to 90MHz. The dependence 5 cos is about three orders of magnitude greater than a>i and therefore the contribution of the spectral density functions evaluated at C0i±C0s to the relaxation rate are considerably lower. 27 of 1/Tis on C0i is shown in Figure 5 for coating thicknesses of lc=0, lnm and 2nm. As the proton resonance frequency decreases, the relaxation rate tends to increase for all values of l c . However, this increase is slightly greater at lower coating thicknesses, which leads to lower normalized relaxation rates at higher frequencies. 1.2 1.0 0.8 J 0.6 0.4 0.2 o.o 4 Spectral Density Function vs. Frequency • • • • —i 1 i 1 1 1 " i 1 — 0 1 2 3 4 5 6 7 8 9 10 11 12 13 log co [rad/s] Figure 4: The trend in the spectral density functions with respect to angular frequency. CM 1.0 0.8 0.6 0.4 0.2 H 0.0 Relaxation Rate and Proton Larmor Frequency • A a A • v A D < A2< 2 2 g 5 6 7 8 9 10 11 12 13 log a>, [rad/s] Figure 5: Dependence of Relaxation rate, l /Ti S , on proton Larmor frequency at different coating thicknesses. The calculated values for lc=0 are shown by the diamonds; the square symbols represent values for lc=lnm; and the triangles represent lc=2nm. 28 The circles on Figure 3 represent the limits on predicted normalized relaxation rates for proton Larmor frequencies ranging from 1 to 90 MHz. The lower bound is associated with the highest frequency considered. As expected from the shape of the spectral density functions, this model predicts greater relaxation rates at lower frequencies. The increase in relaxation rate with decreasing frequency is slightly greater for lower l c values (Figure 5). Therefore, for the range of l c = 0.2 - 2nm, the normalized relaxation rates were consistently higher at lower frequencies. Figure 6 shows the trend in predicted relaxation rate of the standard configuration with respect to R D . In choosing a range of possible values for R D , we consider changes in the parallel diffusion coefficient associated with restricted movement of the molecules near the solid surface. We expect the diffusion coefficient corresponding to the motion of the water molecules in the direction perpendicular to the solid surfaces to be that of the free bulk pore fluid at a given temperature. While there are no mechanisms that can physically account for the enhancement of diffusion, we do expect a reduction in Du near the solid surface. If water molecules at the surfaces are adsorbed through hydrogen bonding, their restricted motion could affect the diffusion of water well away from the surfaces. This can be described by a decrease in Du, and hence the ratio parameter, R D -As shown in Figure 6, a reduction in R D from a value of 1 would result in a slight increase in the predicted relaxation rate, which reaches a maximum at the relative diffusion coefficient of RD~0.01 . In selecting a minimum for R D , its value is reduced until the relaxation rate asymptotically reaches a constant. This is achieved for RD<0.001. The predicted increase in relaxation rate with decreasing R D is greater for the plain, or uncoated samples. This can be qualitatively explained by considering the water molecules'closest distance to the paramagnetic spins. Parallel diffusion's contributing to the relaxation rate decays with increasing distance to the spins, S. Therefore, the normalized relaxation rate decreases with decreasing R D - A lower limit for R D indicating the minimum normalized relaxation rate is estimated at RD~0.005. 29 Relaxation Rate & Relative Diffusion • • • i. 4 —i 1 1 1 1 1 1 1 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 log(RD) Figure 6: Dependence of theTelaxation rate, 1/T1S, on the relative diffusion coefficient, R D -The predicted TiS(plain)/TiS(coated) on the limiting values of R D , are shown by the squares on Figure 3. The lower bound, indicated by open squares, corresponds to restricted diffusion in the two dimensional plane parallel to the solid surfaces. This limit is calculated for RD=0.005. Thus, a lower bound would result in the predicted trend of normalized relaxation rates, if diffusion parallel to the solid surfaces were restricted. The upper bound, shown by the solid squares on Figure 3, corresponds to the standard configuration. Another parameter of interest is the thickness of solid layer containing the paramagnetic ions with spin S, l p . In choosing the minimum value of this parameter, we consider the smallest constituent of the substance coating the surface. For an exceedingly thin layer of goethite, which has a crystal structure with orthorhombic symmetry, this thickness would correspond to its lowest length of unit cell, lp=0.3nm. Another commonly found iron oxide is hematite that has rhombohedral symmetry with a greater unit cell length of about 0.5nm. Thus, we choose 0.3nm as the absolute lowest value for l p . Figure 7 shows the trend in relaxation rate of the standard configuration for the range of l p from 0.3 - 30nm. The relaxation rate is found to increase as the 30 total number of paramagnetic spins rise. However, this enhancement of 1/Tis is limited given the decay in relaxation rate as the distance between the interacting spins, S and I, increases. As a consequence, 1/Tis approaches a maximum value and remains constant for lp>~5nm (Figure 7). As expected, the dependence of 1/Tis on the near surface properties of the solid is greatest for lc=0. This results in lower normalized relaxation rates as l p decreases. The bounds on normalized relaxation rate are predicted and indicated by the triangles in Figure 3. The upper bound (solid triangle) corresponds to the predicted TiS(plain)/TiS(coated) for the standard configuration, while the lower bound (open triangles) is found for the lowest value of lp=0.3nm. Figure 7: Dependence of the relaxation rate, l / T i S , on the thickness of solid layer containing paramagnetic ions, l p . In order to predict the absolute bounds on the trend of normalized relaxation rate, we must consider the combination of the limiting values for all three parameters, Cui, RD , and l p . These limiting values for the lower bound correspond to C0i = 27tx90MHz, RD=0.005, and lp=0.3nm. The predicted trend in Tis(plain)/Tis(coated) is shown as x symbols in Figure 3. This is the absolute lowest bound on the trend for normalized relaxation rate predicted in the range of 31 frequencies 1 to 90 MHz. The upper bound is a function of frequency, which is shown in Figure 3 as solid circles for coi= 27txlMHz. A trendline that best describes the predicted normalized relaxation rate for the standard configuration is shown by the solid line on Figure 3. The developed model shows a moderate decrease in the relaxation rate as the coating thickness increases. The equation for the declining logarithmic trendline shown in Figure 3 indicates that the decrease in relaxation rate is more gradual than is predicted by previous models. Models a and b were considered for the estimate of coating thickness of experimental studies in Table 2. The dependence of l / T i S on r=lc+d, as predicted by model a, is a steep decline characterized by 1/r6, where d is the distance of closest approach for the plain sample. This model is based on rotational diffusion of water molecules coordinated to spin S paramagnetic ions. Model b considers two-dimensional diffusion at a fixed distance, z, from a surface distribution of spins S. This model, which better describes the physical system, still 2. 6 predicts a relaxation rate that varies as 1/z' to l/z°. The developed model can be used to obtain an estimate of coating thickness for the studies indicated in Table 2. This can be accomplished by considering the field strength, B 0 , and the normalized relaxation rates, Tis(plain)/Tis(coated), reported in each experiment. The diffusion coefficient, Dj_, for free water at room temperature applies to all experiments considered. A range of estimates is however provided which corresponds to the limiting values of R D and l p . The upper limit is calculated with RD=1 and lp=8.9nm as for the standard configuration, and the lower limit corresponds to RD=0.005 and lp=0.3nm. The ranges of coating thicknesses predicted by each of the models are summarized in Table 4. The lower R D value indicating restricted diffusion in the planar dimension also applies to model b. For this reason, a lower limit in coating thickness for this model is calculated and shown in Table 4. In all cases 32 considered, the developed model consistently predicts greater and more realistic coating thicknesses than predicted by models a and b. Table 4: Model Comparisons Molecular Size i n - 1 s CO, Predicted coating thickness: Study / Year Surface Coating [A] Ratio [2racMHz] model a model b Developed Model Brown & Fatt (1)/1956 Dri-film (SF99 GE) > 11 0.24 1 0.8 2.6 - 3.0 6 .0 -23 Devereux (2)/1967 Methylene blue 4.5 0.75 13 0.1 0.31 -0 .46 0.62-1.8 Dodecylammonium acetate 15 0.45 13 0.4 0.92 - 1.5 1.9 - 6.7 Stearic acid 23 0.34 13 0.6 1.3-2.1 2 .7 -10 asphaltene >5 0.075 13 1.6 3.5 - 7.5 8.5 - 39 Hsu (7)/1994 Dri-film (SF99 GE) > 11 0.057 85 1.9 2.5 - 7.7 7 .5 -23 Kuparuk crude oil >5 0.23 85 0.8 1.2-3.0 3.1 - 9.3 Kanters, et al. (13) /1998 Crude oil (for various grain sizes ranging from 115 to 275 jum) >5 0.23 90 0.8 1.2-2.9 3.0-9.0 >5 0.24 90 0.8 1.1 - 2.8 2.9 - 8.6 >5 0.19 90 0.9 1.3-3.4 3.5-11 >5 0.27 90 0.7 1.0 - 2.5 2.6 - 7.8 >5 0.094 90 1.4 2.0 - 5.6 5 .7-17 Daughney, et al. (15) / 2000 Crude oil (sample with highest relative change in relaxation rate due to sorbed crude oil) >5 0.58 90 0.3 0.41 - 0.88 0.94-2.6 In contrast to the previous models'estimates of coating thicknesses, those predicted by the developed model are in proper agreement with the limits imposed by the size of a single molecule forming the coating substance. Two exceptions are the experimental results of Devereux (2) and Daughney, et al. (15). It should be noted that both cases involved microporous sample materials with bimodal distribution of relaxation times. The higher normalized relaxation rates in these experiments can be explained by the existence of uncoated micropores. Diffusion of water molecules into the micropores can increase the surface relaxation rates reported for the coated larger pores. Such a shift in 1/Tis in both coated and plain samples would increase the normalized relaxation rates. A l l considered, the developed model provides the most realistic estimates of coating thickness in comparison with previous models. Summary and Conclusions Distinct Ti relaxation times of pore water hydrogen protons in plain and coated porous media have been reported in experimental studies since 1956. These studies have consistently shown a lower Ti relaxation rate of pore water hydrogen protons for the coated samples. The observed relaxation rate is described as the sum of bulk water relaxation and surface relaxation, 1/Tis. The surface relaxation rate inside the pore space of geological materials has long been attributed to the dipolar interactions of the pore water hydrogen protons with the electronic spins of the paramagnetic ions near the solid pore walls. A layer of substance coating the solid surfaces physically separates the pore water molecules from the paramagnetic ions near the solid surface. This shielding effect of the surface coating results in the reduced relaxation rates reported for coated samples. There have been no attempts to estimate the amount of substance coating the surfaces of geologic materials, based on the observed change in Ti relaxation times. One model has been proposed to relate the relaxation rate to r=lc+d, where l c is the coating thickness, and d is the distance of closest approach of the diffusing water molecules to the paramagnetic ions in the plain sample. This model of 1/T[S, referred to throughout this thesis as model a, was described by Kleinberg et al. (17) for application in geological materials. A more general model, referred to as model b, is that proposed by Neue (18) for estimating 1/Tis in porous media. For the purpose of evaluating these models, studies were selected with consideration to the following criteria: 1) the coating substance did not contain paramagnetic ions; 2) the surface area of the samples remained constant after the application of the coating; and 3) samples were completely covered by the coating substance. We have assessed the validity of each of these assumptions for each of the studies considered. Application of both models to the measured values of 1/Tis in such studies provided unrealistically low estimates of coating thickness. 34 A new model has been developed for diffusing pore water molecules in a three-dimensional pore structure. The spectral density functions have been derived for coupling of the pore water hydrogen protons with paramagnetic spins S, distributed uniformly in a layer near the surface of the solid sample. In addition to the coating thickness, this model uses a number of parameters, which include the proton Larmor frequency, parameters describing the physical geometry of the pore space, properties of the paramagnetic ions in the solid, and the diffusion coefficients related to the motion of the water molecules parallel and perpendicular to the solid surfaces. The developed model can be used to predict the relaxation rate for a known set of model parameters. In considering the normalized relaxation rate, Tis(plain)/ Tis(coated), we have eliminated some of these parameters for estimating the coating thickness. The extreme values of the unknowns are then used to set bounding limits on the predicted normalized relaxation rates. The developed model shows a gradual decline in Tis(plain)/ Tis(coated) with increasing coating thickness, which is best approximated by a logarithmic decline. Application of this model to measured relaxation rates of coated and plain samples have provided the most realistic estimates of the coating thicknesses. In a given experiment, with known field strength, and a priori knowledge of the paramagnetic properties of the geologic sample, the predicted bounds can be further constrained. In addition, we can remove some of the previous assumptions used for application of models a and b, if the specific surface area of the samples, or the paramagnetic content of the coating are known. If the coating substance contains paramagnetic ions, the developed model can also be used in estimating their contribution to the measured relaxation rate. This model can be potentially useful for quantitative detection of sorbed contaminants in porous geological materials. 35 References 1. R.J.S. Brown, and I. Fatt, Measurements of fractional wettability of oilfield rocks by the nuclear magnetic relaxation method, Pet. Trans. AIME 207, 262-264 (1956). 2. O.F. Devereux, Effect of crude oil on the nuclear magnetic relaxation of water protons in sandstone, Nature 215, 614-615 (1967). 3. J. Kumar, I. Fatt, and D.N. Saraf, Nuclear magnetic relaxation time of water in a porous medium with heterogeneous surface wettability, J. Appl. Phys. 40, 4165-4171 (1969). 4. C.E. Williams, and B. M . Fung, The determination of wettability by hydrocarbons of small particles by deuteron T i p measurement, J. Magn. Reson. 50, 71-80 (1982). 5. G.C. Borgia, P. Fantazzini, and E. Mesini, Wettability effects on oil-water- configurations in porous media: a nuclear magnetic resonance relaxation study, J. Appl. Phys. 70, 7623-7625 (1991). 6. J.J. Howard, Wettability and fluid saturations determined from N M R Ti distributions, Magn. Reson. Imaging 12, 197-200 (1994). 7. W.F. Hsu, "Wettability of porous media by nuclear magnetic resonance relaxation methods," Ph.D. thesis, Texas A & M , College Station, Texas (1994). 8. I. Bonalde, M . Martin-Landrove, and J. Espidel, Nuclear magnetic resonance relaxation study of wettability of porous rocks at different magnetic fields, J. Appl. Phys. 78, 6033-6038 (1995). 9. S. Bobroff, G. Guillot, C. Riviere, L . Cuiec, J.C. Roussel, and G. Kassab, Microscopic arrangement of oil and water in sandstone by N M R relaxation times and N M R imaging, /. Chem. Phys. 92, 1885-1889 (1995). 10. S. Bobroff, and G.Guillot, Susceptibility contrast and transverse relaxation in porous media: simulations and experiments, Magn. Reson. Imaging 14, 819-822 (1996). 36 77. J.J. Howard, Quantitative estimates of porous media wettability from proton N M R measurements, Magn. Reson. Imaging 16, 529-533 (1998). 72. G.C. Borgia, R.J.S. Brown, and P. Fantazzini, Different "average" nuclear magnetic resonance relaxation times for correlation with fluid-flow permeability and irreducible water saturation in water-saturated sandstones, /. Appl. Phys. 82, 4197-4202 (1997). 13. W.A. Kanters, R. Knight, and A. MacKay, Effects of wettability and fluid chemistry on proton N M R Ti in sands, J. Environ. Eng. Geophysics, 3, 197-202, (1998). 14. G.Q. Zhang, C. Huang, G.J. Hirasaki, Interpretation of wettability in sandstones with N M R analysis, in "Proceedings of the 1999 International Symposium of the Society of Core analysts," 1999. 75. C.J. Daughney, T.R. Brayar, and R. Knight, Detecting sorbed hydrocarbons in a porous medium using proton nuclear magnetic resonance, Environ. Sci. Technol. 34, 332-337, (2000). 7t5. N . Bloembergen, E . M . Purcell, and R.V. Pound, Relaxation effects in nuclear magnetic resonance absorption, Phys. Rev. 73, 679-712 (1948). 7 7. R.K. Kleinberg, W.E. Kenyon, and P.P. Mitra, Mechanism of N M R relaxation of fluids in rock, / . Magn. Reson. A 108, 206-214 (1994). 75. G. Neue, Spectral densities for dipolar interactions between two-dimensionally mobile spins and spins localized off-plane, J. Magn. Reson. 78, 555-562 (1988). 79. "AOSTRA Technical Handbook on Oil-Sands, Bitumen, and Heavy Oils," (L.G. Hepler, and C. Hsi, Eds.), AOSTRA Technical Publications Series 6, Edmonton, Canada, 35 (1989). 20. "CRC handbook of chemistry and physics," (R.C. Weast, and M.J . Astle, Eds.), 61 s t ed. CRC Press Inc. Florida (1980-81). 27. A . Abragam, "Principles of Nuclear Magnetism," Clarendon Press, Oxford (1961). 37 22. R. K. Harris, "Nuclear Magnetic Resonance Spectroscopy," Longman Group U K Limited, Great Britain (1986). 23. D.J. Griffiths, "Introduction to Electrodynamics," 2 n d Edition, Prentice Hall, New Jersey (1989). 24. M.E . Rose, "Elementary theory of angular momentum," Wiley, New York (1957). 25. J.A. Coston, C.C. Fuller, and J.A. Davis, Pb 2 + and Z n 2 + adsorption by a natural aluminum-and iron-bearing surface coating on an aquifer sand, Geochimica et Cosmo. Acta 59, 3535-3547 (1995). 26. J. Kevorkian, "Partial Differential Equations: Analytical Solution Techniques," 2 n d Edition, Springer, New York (2000). 27. R. Mills, Self-diffusion in normal and heavy water in the range 1-45°, J. Phys. Chem. 77, 685-688 (1973). 28. A. Scheidegger, M . Borkovec, and H. Sticher, Coating of silica sand with goethite: preparation and analytical identification, Geoderma 58, 43-65 (1993). 38 


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