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2 Available online at Journal o Magnetic Resonance 90 (2008) Paramagnetic relaxation in sandstones: Distinguishing T and dependence on surace relaxation, internal gradients and dependence on echo spacing q Vivek Anand a, *, George J. Hirasaki b a Schlumberger Product Center, 25 Industrial Boulevard, MD 25-0, Sugar Land, TX 77478, USA b Chemical and Biomolecular Engineering Department, Rice University, Houston, TX 77005, USA Received June 2007; revised 20 September 2007 Available online 4 October 2007 Abstract This work provides a generalized theory o proton relaxation in inhomogeneous magnetic ields. Three asymptotic regimes o relaxation are identiied depending on the shortest characteristic time scale. Numerical simulations illustrate that the relaxation characteristics in the regimes such as the T / ratio and echo spacing dependence are determined by the time scales. The theoretical interpretation is validated or luid relaxation in porous media in which ield inhomogeneity is induced due to susceptibility contrast o luids and paramagnetic sites on pore suraces. From a set o measurements on model porous media, we conclude that when the sites are small enough, no dependence on echo spacing is observed with conventional low-ield NMR spectrometers. Echo spacing dependence is observed when the paramagnetic materials become large enough or orm a shell around each grain such that the length scale o the region o induced magnetic gradients is large compared to the diusion length during the time o the echo spacing. The theory can aid in interpretation o diusion measurements in porous media as well as imaging experiments in presence o contrast agents used in MRI. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Paramagnetic relaxation; Internal ield; Susceptibility; Porous media; Contrast agents. Introduction q Paper was presented at 48th Annual Symposium o Society o Petrophysicists and Well Log Analysts. * Corresponding author. Fax: address: vanand@slb.com (V. Anand). Inhomogeneous magnetic ields are employed or several applications in NMR relaxometry and imaging. For example, ield gradients o the order o 0 00 T/m are applied in Stray Field Imaging or high resolution o systems with broad line widths [ 4]. Functional MRI utilizes the inhomogeneous ield induced by paramagnetic contrast agents or increasing sensitivity and speciicity [5,6]. The ield gradient generated by oil well-logging tools is utilized or identiication o hydrocarbons in earth s ormation [7,8]. In this paper, we provide a theoretical and experimental ramework or understanding proton relaxation in inhomogeneous magnetic ields. Field inhomogeneity can be applied externally or induced internally within the system due to dierence in magnetic susceptibility o constituents. Diusion o luid molecules in inhomogeneous ields leads to additional relaxation o transverse magnetization due to loss o phase coherence. The additional relaxation is called secular relaxation [9] and is deined as the dierence in transverse and longitudinal relaxation rates. ¼ ðþ ;sec T We ocus on secular relaxation in water-saturated porous media although the theory is applicable or any system in which inhomogeneous ields are present. At present, there is no exact theory which explains secular relaxation in porous media in a general inhomogeneous ield. However, two ideal cases o relaxation in a constant gradient and relaxation in an inhomogeneous ield induced by /$ - see ront matter Ó 2007 Elsevier Inc. All rights reserved. doi:0.06/j.jmr

3 V. Anand, G.J. Hirasaki / Journal o Magnetic Resonance 90 (2008) paramagnetic spheres have been described analytically in the past. DeSwiet et al. [0] described three length scales that characterize secular relaxation in a constant gradient g: () Pore structural length, L s (2) Diusion length, L d, deined as p L d ¼ iiiiiiii Ds E ð2þ where D is the diusivity o the luid and s E is the echo spacing or CPMG pulse sequence (3) Dephasing length, L g, deined as the distance over which the spins have to diuse in order to dephase by radian given as siiiii 3 D L g ¼ ð3þ cg where c is the proton gyromagnetic ratio. Depending on the smallest length scale, secular relaxation can be characterized into three relaxation regimes o motionally averaging, ree diusion and localization. Similarly, secular relaxation in the ield induced by paramagnetic spheres has been classiied into regimes o motionally averaging, weak magnetization and strong magnetization depending on three characteristic time scales [9,,2]. Analytical expressions or secular relaxation rates have been proposed or the dierent asymptotic regimes. This study proposes a generalized relaxation theory which extends the deinition o the asymptotic regimes deined or the ideal case o diusion in a constant gradient to general inhomogeneous ields. The regimes are determined by the shortest characteristic time scale [], which can be deined or a general inhomogeneous ield, rather than the smallest length scale. The time scales also quantiy the characteristics o the relaxation regimes such as echo spacing dependence and T / ratio. The paper is organized as ollows. In Section 2, the generalized theory o secular relaxation in inhomogeneous ields is provided. The theory is independent o the particular choice o the inhomogeneous ield distribution. Section 3 illustrates the dependence o characteristic time scales in sedimentary rocks on governing parameters such as the concentration and the size o paramagnetic particles. Section 4 describes random walk simulations o the secular relaxation in the ield induced by paramagnetic spheres to quantitatively illustrate the characteristics o the relaxation regimes. Sections 5 and 6 describe a series o NMR measurements that physically explore relaxation regimes in porous media. 2. Generalized secular relaxation theory 2.. Characteristic time scales or secular relaxation Transverse relaxation due to dephasing in magnetic ield inhomogeneities is characterized by three time scales []: () time taken or signiicant dephasing, s x, deined as the inverse o the spread in Larmor requencies (dx) existing in the system s x ¼ ð4þ dx (2) diusional correlation time, s R, deined as the time taken to diusionally average the inhomogeneities s R ¼ L2 ð5þ D where L is the characteristic length o ield inhomogeneity in the system, and (3) hal echo spacing used in CPMG pulse sequence, s E, given as s E ¼ TE ð6þ 2 where TE is the echo spacing or the CPMG sequence. The length scale o the ield inhomogeneity (L) and spread in Larmor requencies (dx) depend on particular system parameters. For example, let us consider the characteristic time scales or two cases o relaxation in a constant gradient and in the ield induced by paramagnetic spheres.. Restricted diusion in a constant gradient: For the case o restricted diusion in a constant gradient g in a pore, ield inhomogeneity exists over the entire length o the pore. Thereore, the diusional correlation time and time or signiicant dephasing are given as s R ¼ L2 s ð7þ D s x ¼ dx ¼ ð8þ cgl s where L s is the structural length o the pore. 2. Relaxation in the ield induced by a paramagnetic sphere: A paramagnetic particle o magnetic susceptibility dierent rom that o the surrounding medium induces magnetic ield inhomogeneities when placed in an external magnetic ield. Potential theory can be used to estimate the induced ields in idealized geometries. The component o the ield induced by a paramagnetic sphere along the direction o the external ield ~B 0 is given as [3] k B dz ¼ B 0 ð3 cos 2 h Þ R3 0 ð9þ k þ 2 r 3 where R 0 is the radius o the sphere, k =(+v sphere )/ ( + v medium ), v sphere and v medium are magnetic susceptibilities o the sphere and the medium respectively. h is the azimuthal angle rom the direction o ~B 0, and r is the radial distance rom the center o the paramagnetic sphere. The component o the induced ield along the external magnetic

4 70 V. Anand, G.J. Hirasaki / Journal o Magnetic Resonance 90 (2008) ield is considered since this component determines the precession requency. Eq. (9) shows that the induced ield is maximum at the surace o the sphere (positive at poles and negative at equator) and alls as the cube o the radial distance rom the center. Thus, the range o Larmor requencies in the system is the dierence in polar and equatorial ields at the surace o the sphere given as 3ðk Þ dx ¼ ðk þ 2Þ cb 0 ð0þ s x is given by the reciprocal o dx. In addition, ield inhomogeneity extends to distances proportional to the radius o the sphere. Thus, s R is given as s R ¼ R2 0 D 2.2. Asymptotic regimes o secular relaxation ðþ Depending on the shortest characteristic time scale, secular relaxation in inhomogeneous ields can be characterized by three regimes o motionally averaging, ree diusion and localization. A description o the regimes is provided in this section. For each regime, the cases o relaxation in a constant gradient and relaxation in the ield induced by paramagnetic spheres are also included. It is instructive to note that even though the two systems have very dierent ield distribution, the relaxation rates in a particular regime are similar when expressed in terms o characteristic time scales. This correspondence proves that the time scales are the governing parameters that describe relaxation in inhomogeneous ields. Motionally averaging regime: The motionally averaging regime is characterized by ast diusion o protons such that the inhomogeneities in magnetic ield are motionally averaged [9]. This averaging occurs when the diusional correlation time is much shorter compared to the hal echo spacing and the time taken or signiicant dephasing due to ield inhomogeneities. Thus, the conditions o motionally averaging regime are s R s x ð2þ s R s E. Motionally averaging regime in a constant gradient: The conditions or motionally averaging in a constant gradient in a pore o length L s can be obtained in terms o the characteristic length scales (Eqs. (2) and (3)) by substituting Eqs. (7) and (8) in Eq. (2) given as L s L g ð3þ L s L d Eq. (3) imply that the motionally averaging regime is observed when the pore structural length is small compared to the diusion length during time s E (L d ) and the dephasing length (L g ). Thus, the spins typically diuse several times the pore size during the measurement and any magnetic ield inhomogeneities are averaged by their motion. Neumann [4] derived the expression or the secular relaxation rate in the motionally averaging regime by assuming the distribution o phase shits to be Gaussian given as ¼ L4 s c2 g 2 ð4þ ;sec 20D Using Eqs. (7) and (8), Eq. (4) can be expressed in terms o characteristic parameters as shown ¼ ;sec 20 dx2 s R ð5þ Thus, the relaxation rate is independent o the echo spacing and shows a quadratic dependence on ield inhomogeneity (dx) in the motionally averaging regime. This quadratic dependence on ield inhomogeneity was also derived or a non-constant ield gradient in onedimensional restricted geometry by Tarczon et al. [5]. 2. Motionally averaging regime in internal ields induced by paramagnetic spheres: Motionally averaging regime or relaxation by paramagnetic spheres arises when the diusional correlation time (Eq. ()) is the shortest time scale. Thus, the conditions or motionally averaging regime in terms o system parameters are R 2 0 D s E R 2 0 D s x ¼ dx ¼ ðk þ 2Þ 3ðk ÞcB 0 ð6þ The expression or dx in Eq. (6) is substituted rom Eq. (0). The quantum-mechanical outer sphere theory suggests that when conditions or motionally averaging are satisied, the secular relaxation rate is given as [9] ¼ 6 ;sec 405 Udx2 s R ¼ 6 45D 2 ðk Þ U ðk þ 2Þ cb 0R 0 ð7þ where U is the volume raction o the paramagnetic particles. Eq. (7) is derived by solving quantum-mechanical equations or the lip rates o the protons using time dependent perturbation theory [9]. Note the similar unctional orm o the secular relaxation rate predicted by the outer sphere theory (Eq. 7) and Eq. 5. Free diusion regime: This regime is valid when the hal echo spacing is the shortest characteristic time scale. The eect o restriction as well as large ield inhomogeneities is not elt by the spins in the time o echo ormation. Thus, the spins dephase as i diusing in an unrestricted medium. The conditions or the ree diusion regime are s E s x ð8þ s E s R

5 V. Anand, G.J. Hirasaki / Journal o Magnetic Resonance 90 (2008) Free diusion in a constant gradient: For the case o diusion in a constant gradient in a pore, the ree diusion regime may arise or small echo spacing such that the spins do not experience the restriction eects in the time o echo ormation. The conditions or ree diusion inside a pore o length L s can be obtained in terms o the characteristic length scales by substituting Eqs. (7) and (8) in Eq. (8) given as L d L g ð9þ L d L s Thus, the ree diusion regime arises when the diusion distance in s E is much smaller compared to the dephasing length and the pore length. In this regime, the secular relaxation rate is given by the classical expression derived by Carr and Purcell [6] or unrestricted diusion in a constant gradient ¼ c2 g 2 s 2 E D ð20þ ;sec 3 Substituting Eqs. (4) (6) in Eq. (20), the relaxation rate in constant gradient can be written as ¼ dx2 s 2 E ð2þ ;sec 3s R 2. Free diusion in internal ields induced by paramagnetic spheres: The ree diusion regime has been described as the weak magnetization regime (dx Æ s E < ) in the magnetic resonance imaging (MRI) literature mentioned below. The regime arises physically when weakly magnetized paramagnetic particles are used as contrast agents or medical imaging or in biological tissues with iron rich cells or deoxygenated red blood cells [7]. Brooks et al. [] calculated the mean squared gradient o the magnetic ield induced by a paramagnetic sphere given as hg 2 i¼ 4Udx2 ð22þ 5c 2 R 2 0 Substituting Eq. (22) in the expression or the relaxation rate in unrestricted diusion (Eq. (20)) and accounting or restriction rom neighboring particles, the relaxation rate in the presence o weakly magnetized spheres is given as [] ;sec ¼ Udx2 s 2 E 5s R ¼ 9U 5 2 ðk Þ ðk þ 2Þ cb s 2 E D 0 R 2 0 ð23þ The above expression (called the Mean Gradient Diusion Theory, MGDT) shows the similar quadratic dependence o secular relaxation rate on the echo spacing as the expression or unrestricted diusion in a constant gradient (Eq. (2)). However, the relaxation rate shows an inverse squared dependence on the paramagnetic particle size which is opposite to that in the motionally averaging regime (Eq. (7)). This inverse dependence arises because the mean squared ield gradient in Eq. (22) decreases as the square o the particle size. Localization regime: Localization regime o secular relaxation arises when the dephasing time is the shortest time scale. The conditions or the localization regime are s x s R ð24þ s x s E. Localization regime in a constant gradient: Using Eqs. (7) and (8) or dx and s R or restricted diusion in a constant gradient in a pore o length L s, Eq. (24) reduces to the ollowing equations in special cases L g L d ð25þ L g L s Thus, the dephasing length (L g ) is the smallest characteristic length in the localization regime. The spins typically dephase to such an extent during the measurement time that they do not contribute to the total magnetization. The signal comes primarily rom the spins near the boundaries which see smaller change in the magnetic ield due to relection [8]. DeSwiet et al. [0] have shown that at long times, the echo amplitude in the localization regime decays as =3 e a Dc 2 g 2 s 3 Mðg; s E Þ D ¼ c ð M 0 cgl 3 EÞ =3 ð26þ s where a = and c = or parallel plates. 2. Localization regime in internal ields induced by paramagnetic spheres: Secular relaxation in the localization regime has also been discussed in the MRI literature. The regime arises physically or relaxation in presence o strongly magnetized contrast agents such as superparamagnetic particles or which dxs E >. Gillis et al. [2] proposed a semi-empirical model or decay in internal ields induced by strongly magnetized spheres. In their model, the region surrounding the paramagnetic sphere is divided into two regions: an inner region with very strong internal gradients and an outer region with only weak gradients. Since the gradients in the outer region are weak, Gillis et al. [2] suggested that relaxation in the outer region can be described by theory or weakly magnetized particles. Thus, the relaxation rate in the outer region is given by Eq. (23) with a modiication to account or the excluded volume o the inner region as shown below ¼ Udx2 s 2 5=3 E dxs E ð27þ ;sec 5s R a þ budxs E The values o the parameters a (=4.5) and b (=0.99) were calculated by Gillis et al. [2] by itting Eq. (27) to numerical simulations o transverse relaxation in the presence o strongly magnetized spheres.

6 72 V. Anand, G.J. Hirasaki / Journal o Magnetic Resonance 90 (2008) In the inner region, the spins experience strong gradients and dephase very rapidly. Due to rapid dephasing, they do not contribute signiicantly to the macroscopic relaxation rate; thereore Eq. (27) provides an accurate estimate o relaxation rate in presence o strongly magnetized particles. 3. Relaxation regimes in sandstones The previous section described the classiication o secular relaxation in asymptotic regimes depending on the characteristic time scales. We can obtain physical insight o the dierent regimes by considering the example o secular relaxation o luids in sandstones. Sedimentary rocks usually contain paramagnetic minerals such as iron on the suraces o silica grains [9]. Thus, the relaxation o pore luids can be inluenced by ield inhomogeneities induced due to susceptibility dierences between pore luids and the paramagnetic minerals. For the case o paramagnetic relaxation in sedimentary rocks, the characteristic time scales depend not only on the size and susceptibility o paramagnetic particles but also on the concentration o particles and the size o silica grains. The two cases o dilute and high surace concentrations o the paramagnetic particles are described below. 3.. Paramagnetic particles at dilute surace concentration At low concentrations o paramagnetic particles on silica suraces, there is negligible superposition o the ields induced by individual particles. Fig. shows the contour plots o the longitudinal component o the internal ield in the outer region o a silica grain coated with paramagnetic spheres at dilute concentration. The particles are separated ar enough rom each other such that the ield induced by one particle is not signiicantly inluenced by the ields induced by neighboring particles. Due to insigniicant superposition, the protons in the outer region o the grain dephase as i diusing in an internal ield induced by a single paramagnetic sphere. Thus, the asymptotic regimes mentioned or a single sphere are applicable or describing secular relaxation in sandstones with dilute paramagnetic concentration. The spread in Larmor requency (dx) and diusional correlation time (s R ) are given by Eqs. (0) and () deined or a single paramagnetic sphere Paramagnetic particles at high surace concentration At high surace concentration o paramagnetic particles, the internal ields induced by individual particles overlap. Thus, the inhomogeneous ield extends to larger distances than at lower surace concentrations due to superposition. Fig. 2 shows the contour plots o the internal ield in the outer region o a grain coated with paramagnetic spheres at high concentration. Strong ield gradients are induced close to the surace o the particles. In addition, the ields induced by neighboring particles superpose and extend to a length scale comparable to the size o the substrate silica grain. At high concentrations, the particles can be visualized as orming a paramagnetic shell around the silica grain. Fig.. Contour plots o induced magnetic ield in the outer region o a silica grain coated with paramagnetic spheres at dilute concentrations (No. o particles per unit area 3). The ratio o the radius o the silica grain to that o paramagnetic particles is 0. At low concentrations, there is insigniicant superposition o ields induced by neighboring particles. Fig. 2. Contour plots o magnetic ield in the outer region o a silica grain coated with paramagnetic spheres at high concentrations (No. o particles per unit area 30). The ratio o radius o the silica grain to that o paramagnetic particles is 0. At high concentrations, induced ield extends to distance proportional to the radius o the silica grain due to superposition o ields induced by neighboring particles.

7 V. Anand, G.J. Hirasaki / Journal o Magnetic Resonance 90 (2008) Consider a grain o radius R g coated with a paramagnetic shell o thickness e. The internal ield induced by the paramagnetic shell can be calculated by subtracting the ield induced by a sphere o radius R g rom the ield induced by a concentric sphere o radius R g + e. Thus, the internal ield contribution o only the spherical shell survives. Using Eq. (9) or the ield induced by a paramagnetic sphere, the ield distribution or a paramagnetic shell is given as " # B dz ¼ ðk ÞB 0ð3 cos 2 h Þ ðr g þ eþ 3 R3 g ; r > R ðk þ 2Þ r 3 r 3 g þ e ðk ÞB 0ð3 cos 2 h Þ ðk þ 2Þ 3eR 2 g ; e R r 3 g ð28þ Similar to the ield distribution o a solid sphere (Eq. (9)), the ield is maximum at the surace o the shell (positive at poles and negative at equator) and alls as the cube o the radial distance. Thus, the range o Larmor requencies is the dierence in polar and equatorial requencies at the surace given as dx shell ðk ÞcB 0 ðk þ 2Þ 9e R g ð29þ The requency range (dx shell ) is proportional to the ratio o the thickness o the paramagnetic shell to the radius o the silica grain. As a validation o this relationship, consider the limiting case o the ield induced by an ininite paramagnetic sheet (R g i ). A paramagnetic sheet perpendicular to an externally applied ield does not induce any ield gradients since the ield lines still remain parallel. This is also quantitatively veriied by Eq. (29) which shows dx shell i 0asR g i. Eq. (29) shows that the requency range or a paramagnetic shell decreases by a actor 3e/R g in comparison to that or a single solid sphere (Eq. (0)). Thus, the time or signiicant dephasing (s x ) increases by the same actor. In addition, the ield inhomogeneity extends to a length scale comparable to the size o the substrate silica grain. Thus, or a close-packing structure o silica grains in sandstones, the length scale o ield inhomogeneity (L) is proportional to the dimension o the interstitial pore between the grains given as [20] L ¼ 0:225R g ð30þ Thereore, s R increases as the square o the silica grain radius rather than the paramagnetic particle radius. For suiciently coarse grained sandstones, s R can become larger than s x. Hence, the system can experience the localization regime (i s x < s E ) or the ree diusion regime (i s E < s x ) even i silica grains are coated with ine-sized paramagnetic particles. In addition to dx, the paramagnetic volume raction (U) also changes at the transition rom an individual particle at low concentration to a shell at high concentrations. At low concentrations, U is the ratio o the volume o the paramagnetic particles to the total volume. However, at high concentrations, U corresponds to the solid matrix volume raction since the particles cover the surace o silica grains. The condition or transition rom individual particle at low concentration to a paramagnetic shell at high concentration is ðdxuþ particle ¼ðdxUÞ shell ð3þ This condition ensures that the secular relaxation rates vary smoothly at the transition. 4. Random walk simulations The relaxation rates o Section 2 are valid only in asymptotic regions o the parameter space spanned by the three characteristic time scales. To provide a quantitative understanding o secular relaxation in the entire parameter space, random walk simulations are perormed. Secular relaxation in an inhomogeneous ield induced in the annular region outside a paramagnetic sphere is modeled. This model is chosen because relaxation rates in asymptotic regions o the parameter space are known analytically. Few simpliying assumptions are made. Surace relaxation at the inner radius (R 0 ) and the outer radius (R e ) o the annulus is neglected. It is also assumed that spin-echoes are measured by CPMG pulse sequence with ideal pulses. The theory is presented in terms o dimensionless quantities so that the results are invariant under changes in particular system parameters. 4.. Governing Bloch Torrey equations The relaxation o transverse magnetization (M) in an inhomogeneous ield B z ater the application o the irst p/2 pulse is given by the Bloch Torrey equation om ot ¼ icb z M M ;B þ Dr 2 M ð32þ where D is the diusivity o the luid and,b is the bulk relaxation time. By substituting M ¼ M x þ im y and M ¼ m e ix 0tþT t 2;B, Eq. (32) can be expressed as om ot ¼ icb dzm þ Dr 2 m ð33þ Here m represents the transverse magnetization with the precession at the Larmor requency (x 0 = cb 0 ) and the bulk relaxation actored out [2]. B dz (=B z B 0 ) is the component o the internal ield along the direction o the externally applied ield. The ollowing dimensionless variables are introduced to normalize Eq. (33) r ¼ r ; t ¼ t ; m ¼ m R e t 0 M 0 where t 0 is a characteristic time and M 0 is the initial magnetization. Substituting the dimensionless variables and

8 74 V. Anand, G.J. Hirasaki / Journal o Magnetic Resonance 90 (2008) Eq. (9) or the internal ield induced by a paramagnetic sphere in Eq. (33), the ollowing dimensionless equation is obtained om ot ¼ i ð3 cos2 h Þ r 3 þ N D r 2 m cb 0 ðk Þ ðk þ 2Þ! 3 R 0 t 0 R e where dimensionless group N D is deined as m ð34þ N D ¼ Dt 0 ð35þ R 2 e The characteristic time t 0 is chosen such that the coeicient o the second term in Eq. (34) is unity. Thus, ðk þ 2Þ t 0 ¼ ð36þ cb 0 ðk Þ U where U is the volume raction o the paramagnetic particle given as 3 ð37þ U ¼ R 0 R e Using Eq. (0) or the requency range (dx) o the inhomogeneous ield induced by the paramagnetic sphere, t 0 can also be expressed as ) t 0 ¼ 3 ð38þ Udx Eq. (38) is used to normalize secular relaxation rates or the experimental systems as described in Section 6. The dimensionless parameter, N D, can be expressed in terms o system parameters by substituting Eq. (36) in Eq. (35) N D ¼ Dt 0 R 2 e ¼ ðk þ 2ÞDR e ¼ ðk ÞcB 0 R ðdxs R ÞU =3 ð39þ Eq. (39) shows that N D is not an independent dimensionless group but is speciied i two dimensionless parameters dxs R and U are speciied. Thus, the governing Bloch equation in dimensionless orm becomes om ot ¼ i ð3 cos2 h Þ m þ N r 3 D r 2 m ð40þ No surace relaxation at the inner and outer boundary implies om or ¼ 0 at r ¼ and r ¼ R 0 ð4þ R e In addition, the application o p pulse at dimensionless hal echo spacing s E reverses the y component o the magnetization. Thus, m j t ¼ m j t þ ð42þ at t ¼ s E ; 3s E,5s E... where the dimensionless hal echo spacing, s E, is given as s E ¼ s E=t 0 ð43þ A continuous random walk algorithm [22] is applied to numerically integrate the dimensionless equations. Secular relaxation o protons in the ield induced by a paramagnetic sphere o susceptibility 0.2 (SI units) surrounded by a medium o susceptibility (SI units) is simulated. These values o susceptibilities are representative o magnetite and water respectively. The external magnetic ield B 0 corresponds to a proton Larmor requency o 2 MHz. The radius o the paramagnetic sphere is varied rom 0 nm to 25 lm. For each particle size, simulations with several values o s E are perormed to illustrate the echo spacing dependence o secular relaxation in dierent regimes. The dimensionless relaxation rate is calculated rom the slope o exponential it to the simulated echo intensities. However, the magnetization decay is multiexponential or simulations such that dxs R > 000. For such cases, the relaxation rate is calculated rom the slowest component o the multi-exponential decay. The details o the algorithm are mentioned in the Appendix Results and discussion Secular relaxation can be classiied in three asymptotic regimes (Section 2) depending on two dimensionless parameters: normalized diusional correlation time dxs R, and normalized echo spacing dxs E. Fig. 3 shows the plot o simulated dimensionless relaxation rates ð=t D 2;sec Þ as a unction o dxs R or dierent values o dxs E. The superscript D reers to the dimensionless relaxation rate. dx and s R are calculated rom Eqs. (0) and () respectively or the speciied values o radius and susceptibility o the paramagnetic sphere. The plots o theoretical relaxation D /,sec Motionally Averaging =3000 =300 =30 =3 = =0.3 = δωτ R Free Diusion Localization Fig. 3. Plot o simulated relaxation rate (dimensionless) with dxs R as a unction o dxs E. The parameters used in the simulations are U = , k =.2, B 0 = 496 G. The solid, dotted and dash-dotted lines are the theoretically estimated secular relaxation rates according to the outer sphere theory (Eq. (7)), mean gradient diusion theory (Eq. (23)) and modiied mean gradient diusion theory (Eq. (27)) plotted in the respective regimes o validity. The dashed lines are the boundaries dxs R = and dxs E = that delineate the asymptotic regimes.

9 V. Anand, G.J. Hirasaki / Journal o Magnetic Resonance 90 (2008) rates are also shown or comparison in the respective regimes o validity. Thus the solid, dotted and dash-dotted lines are the plots o theoretical relaxation rates (Eqs. (7), (23), (27)) normalized by the characteristic rate /t 0. The dashed lines are the boundaries dxs R = and dxs E = that delineate the asymptotic regimes. Note that the relaxation rates can be made dimensional by using the ollowing equation. ¼ ¼ ðk ÞcB 0U ð44þ ;sec t 0 T D 2;sec ðk þ 2Þ T D 2;sec The characteristics o secular relaxation rate in Fig. 3 can be described in terms o the three regimes:. Motionally averaging regime: Motionally averaging regime exists or dxs R and suiciently large values o dxs E. In this regime, secular relaxation rates are independent o the echo spacing because the ield inhomogeneities are averaged in time much shorter compared to s E. In addition, the rates increase with s R and thus, with the particle size. A physical understanding or the dependence o relaxation rate on s R can be obtained by considering the chemical exchange (CE) model o transverse relaxation [23 25]. The CE model consists o two sites A and B with a requency shit (dx) between them and the protons are chemically exchanging between the sites with an exchange time s ex. The phenomenon o transverse relaxation due to chemical exchange between sites is analogous to secular relaxation due to diusion between continuous spectrum o requencies in the random walk model. The presence o only two discrete chemical shits and no diusion contribution makes the integration o Bloch equation in the CE model easier. Thus, physical insight about secular relaxation can be gained by analyzing the relaxation rates predicted by the CE model. The raction o protons in the two sites is given by F A and F B respectively. When F B, the relaxation rate is given as [24] ðdxþ 2 ¼ F B s ex ð þðdxþ 2 s 2 ex Þ ð45þ In motionally averaging regime (dxs ex ) o the CE model, Eq. (45) reduces to i F B s ex ðdxþ 2 ð46þ When dx /s ex, exchange o protons between the two sites is ast compared to the dephasing rate. Thus, the rate determining step is the amount o dephasing in the exchange time s ex.ass ex increases, the protons in the two sites are dephased by a larger amount and hence, the relaxation rates increase with s ex as shown by Eq. (46). As mentioned earlier, this CE model is analogous to the relaxation model in outer region o a paramagnetic sphere. The protons may be considered either near the sphere (site B) where there is a requency shit o order dx or ar away where there is no shit (site A). The exchange time rom site B to site A is proportional to the diusional correlation time which is the time to diuse the length scale o ield inhomogeneity. Thus, when diusional exchange between the two sites is ast compared to dephasing rate (i.e. motionally averaging, dx /s R ), relaxation rates increase with s R similar to the predictions o the CE model. 2. Localization regime: Localization regime exists or dxs R and large values o dxs E. In this regime, the relaxation rates show strong dependence on the echo spacing. Additionally, the dependence o the relaxation rate on s R (and thus, the particle size) is inversed in contrast to that in the motionally averaging regime. The CE model also provides an explanation or the decrease in relaxation rates with particle size in the localization regime. When the condition dxs ex holds, Eq. (45) reduces to i F B ð47þ s ex Thus, the relaxation rates are inversely proportional to s ex in contrast to Eq. (46) or the motionally averaging regime. For dx /s ex, dephasing rate is much larger compared to the rate at which protons exchange between sites A and B. Due to large dephasing, the protons are lost rom the signal in a single exchange rom site A to site B [26]. Thus, the relaxation rate is determined by the rate at which protons exchange between the two sites i.e. /s ex. For relaxation by paramagnetic spheres, a similar inverse dependence o relaxation rates on s R (which is analogous to s ex ) is observed when dxs R. The simulated relaxation rates in the localization regime o Fig. 3 match well with the predictions o modiied MGDT (Eq. (27)) except at large values o dxs E and dxs R. This deviation is probably because at large values o parameters dxs E and dxs R, the contribution o the inner region to the overall decay becomes substantial. Thus, the assumption that the decay is only governed by outer region in the derivation o Eq. (27) may not be entirely valid. 3. Free diusion regime: Free diusion regime exists or small echo spacings such that dxs E and dxs E dxs R. Due to eective reocusing by p pulses in the ree diusion regime, the simulated relaxation rates are signiicantly smaller than the ones in localization or motionally averaging regimes. For dxs R >, the relaxation rates show inverse dependence on the particle size similar to that in the localization regime. The above mentioned characteristics o the asymptotic regimes can be easily visualized in the (dxs R, dxs E ) parameter space. Fig. 4 shows contour plots (solid curves) o simulated dimensionless relaxation rates in the (dxs R, dxs p iiiii E ) domain. The contour plots dier by a actor o 0.

10 76 V. Anand, G.J. Hirasaki / Journal o Magnetic Resonance 90 (2008) Free Diusion D /,sec Localization δωτ R =36 δωτ =44 R 0 5 δωτ R =576 δωτ =2304 R Fig. 4. Contour plots o dimensionless secular relaxation rate p iiiii in the (dxs R, dxs E ) parameter space. The contours dier by a actor o 0. The parameters used in the simulations are same as or Fig. 3. The dotted lines are the corresponding plots or theoretically predicted relaxation rates given by Eqs. (7), (23) and (27). The bold dashed lines are boundaries between asymptotic regimes. The contours are invariant or dilute volume ractions (U ). Fig. 5. Plot o secular relaxation rates (dimensionless) with dimensionless echo spacing dxs E or dxs R. The dotted and dash-dotted lines are theoretically predicted relaxation rates in the ree diusion and the localization regime given by Eqs. (23) and (27), respectively. The bold dashed line is the boundary dxs E = delineating ree diusion and localization regimes. The dependence o relaxation rates on the echo spacing is quadratic in the ree diusion regime and less than quadratic in the localization regime. Corresponding plots (dotted curves) or the theoretically predicted relaxation rates given by Eqs. (7), (23) and (27) are also shown in respective regions o validity. The boundaries between the asymptotic regimes are shown by dashed lines. In the motionally averaging regime (dxs R, dxs E ), the contour plots are almost parallel to the dxs E axis implying that the relaxation rates are independent o the echo spacing. Also, the rates increase with dxs R showing that the secular relaxation increases with particle size. In localization and ree diusion regimes, the dependence o relaxation on dxs R is reversed and the rates decrease with particle size. In addition, the rates are echo spacing dependent in these regimes. The contours are more closely spaced and have lesser slope in the ree diusion regime than in the localization regime. Thus, the dependence o the relaxation rate on the echo spacing is higher in the ree diusion regime than in the localization regime. The echo spacing dependence o relaxation rates in the ree diusion and the localization regime can also be quantiied. Fig. 5 shows the plot o simulated relaxation rates with dxs E or cases in which dxs R. The dashed line shows the boundary dxs E = between ree diusion and localization regimes. The dotted and dash-dotted lines are theoretically predicted relaxation rates in the two regimes (Eqs. (23) and (27)). Relaxation rates show quadratic dependence on the echo spacing in the ree diusion regime as predicted by Eq. (23). However, the slopes o relaxation rates decrease as the systems transition rom the ree diusion to the localization regime. Power-law its o relaxation rates with dxs E in the localization regime show that the exponent o s E is approximately unity or low values o dxs E ( 6 dxs E 6 5) and decreases to less than 0.6 or larger values o dxs E (0 2 6 dxs E ). Thus, these simulations show that non-quadratic dependence o relaxation rates on the echo spacing can result in the localization regime as has also been observed experimentally by several researchers [27,28]. Since the theory is presented in dimensionless terms, the plots in Figs. 3 5 are invariant under changes in parameters such as magnetic susceptibility o the paramagnetic sphere, diusivity o the luid, external magnetic ield, etc. However, the relaxation rates are invariant under changes in volume raction or only dilute volume ractions (U ). Simulations or dierent volume ractions show that the average absolute deviation is less than 0% or U <0 3. For U = 0.6, the average absolute deviation is greater than 50% [29]. 5. Experimental This section experimentally illustrates the characteristics o secular relaxation in porous media. We irst describe NMR measurements with aqueous dispersions o paramagnetic particles o known sizes. These experiments help to quantitatively estimate the characteristic time scales or the paramagnetic particles. Next we describe NMR measurements with model sandstones synthesized by coating the paramagnetic particles on silica grains. These model systems simulate secular relaxation in sedimentary rocks in which ield inhomogeneities are oten induced due to the presence o paramagnetic minerals and clays. Thus, a conceptual understanding o paramagnetic relaxation in porous media is developed based on theoretical and experimental results.

11 V. Anand, G.J. Hirasaki / Journal o Magnetic Resonance 90 (2008) Paramagnetic particles in aqueous dispersions. Ferric ion: The smallest paramagnetic particle studied was erric ion (hydrated ionic diameter = 0.2 nm [20]). Solutions o erric chloride were prepared at various solute concentrations in 0. N Hydrochloric acid. Acidic ph o the solutions prevents the ormation o erric hydroxide. Proton longitudinal and transverse relaxation o the solutions was measured at 2 MHz and 30 C. Fig. 6 shows T and relaxation rates o the aqueous solution o erric ions as a unction o the solution concentration. The relaxation rates o the solutions increase linearly with the concentration. Furthermore, the T / ratio is unity and no echo spacing dependence o the transverse relaxation is observed. T / ratio o unity suggests that the secular relaxation does not contribute signiicantly to the transverse relaxation or erric ion solutions. This assertion is validated by the calculation o characteristic time scales shown later. 2. Magnetite nanoparticles coated with citrate ion: To explore a larger length scale o ield inhomogeneity, positively charged magnetite nanoparticles o dierent sizes were synthesized in aqueous medium using Massart s Method [30]. The aim o synthesizing positively charged nanoparticles is to adsorb them on the negatively charged silica surace by columbic attraction. Thus, relaxation characteristics o model sandstones with paramagnetic particles adsorbed at dierent concentrations can be studied. The ollowing experimental procedure was used to synthesize aqueous dispersions o magnetite nanoparticles. Ammonia solution was added dropwise to an aqueous mixture o errous and erric chloride till a gelationous precipitate (magnetite) was obtained. The precipitated magnetite was washed with Perchloric acid which makes the particles positively charged due to surace adsorption o protons. The charged particles can then be dispersed in excess water [30]. The size o the particles can be controlled by changing the temperature o the reaction. Larger particles are obtained at higher temperatures due to increased solubility o smaller crystals. Two dispersions were synthesized with /T, / (ms) y=73x 0 3 /T / Conc (gram/ml) Fig. 6. Longitudinal and transverse relaxation rates (s E = 0.2 ms) o erric chloride solutions as a unction o the concentration o the solution. No echo spacing dependence o relaxation is observed and T / ratio is equal to unity. average diameters o the particles equal to 25 and 0 nm. The 25 nm particles were synthesized at 0 C while 0 nm particles were synthesized at room temperature. Positively charged nanoparticles however, agglomerate in external magnetic ields due to interparticle dipole attractions. Thereore, the nanoparticles were stabilized by adding M Sodium Citrate in the volume ratio o %. Stronger columbic repulsion between citrate-coated particles prevents them rom agglomeration. Proton relaxation rates o aqueous dispersions o the citrate-coated particles were measured as a unction o the concentration. Fig. 7 shows the plots o longitudinal, transverse and secular relaxation rates with the concentration o the dispersion or the two particle sizes. Linear best its o the experimental points and the corresponding slopes are also shown. Relaxation rates are linearly dependent on the concentration o the dispersion and no echo spacing dependence o relaxation rates is observed. Additionally, the T / ratio o magnetite dispersions is greater than one and increases with the particle size. The calculations o characteristic time scales mentioned in Section 6 show that dispersions o submicron particles experience the motionally averaging regime. Thus, no echo spacing dependence o relaxation is observed andt / ratio increases with the particle size Paramagnetic particles on silica surace NMR proton relaxation measurements with silica sand coated with paramagnetic particles are described in this section. Coated sand serves as a model to quantitatively understand the paramagnetic relaxation mechanisms in porous media. Sand is coated with paramagnetic particles o various sizes and concentrations to illustrate the transition o the relaxation regimes with the length scale o ield inhomogeneity. /T,2 (ms) /T,2 (ms) / /T /,sec D p =25nm T / =.5 y= 90x y= 8x 0.05 y=72x x D p =0nm y=239x T /T = y= 40x 0.05 y=99x Concentration (g/ml) x 0 4 Fig. 7. Plots o transverse, longitudinal and secular relaxation rates with the concentration o aqueous dispersions o magnetite nanoparticles. relaxation rates are independent o the echo spacing and the T / ratio increases with the particle size.

12 Author's personal copy 78 V. Anand, G.J. Hirasaki / Journal o Magnetic Resonance 90 (2008) Fine sand coated with erric ion: The ollowing experimental protocol was used to coat ine sand (Sigma Aldrich, grain radius 50 lm) with paramagnetic erric ions. Fine sand was repeatedly washed with resh batches o hydrochloric acid to remove paramagnetic particles originally present on grain suraces. The sand was then washed with deionized water and dried. A known quantity o the dried sand was kept in contact with acidic (ph ) solutions o erric chloride o known concentrations or 24 h in a plastic bottle. The solid to liquid ratio (gm sand/ml) o the slurries was kept at :0. Low ph (7) o erric solutions helps to prevent the precipitation o hydroxides on the sand surace. To ensure uniorm coating, the slurries were constantly rotated which prevents the sand rom settling at the bottom. Ater 24 h, the supernatant was removed and the coated sand was repeatedly washed and saturated with deionized water. This last step removes any remaining erric ions in the pore liquid. Thus, the relaxation o the pore liquid can be attributed only to surace relaxation and to diusion in internal ield gradients. The surace concentration o the erric ions can be estimated by measuring the quantity o erric ions deposited on a known surace area o sand as described. BET surace area o sand (0.2 m 2 /g) was determined using N 2 adsorption at 77 K. To estimate the deposited quantity o erric ions, relaxation time o the supernatant was measured. Using the calibration between the concentration and relaxation rate o erric chloride solution (Fig. 6), the concentration o the supernatant can be determined. The dierence in the concentration o the supernatant and the original coating solution multiplied by the volume o the solution gives the quantity o erric ions deposited on the sand surace. Fig. 8 shows T and distributions o the water-saturated sand coated with erric ions at various surace concentrations expressed as surace area/ion. Both T and relaxation times decrease as the surace concentration o Fe 3+ ions increases. However, no echo spacing dependence o relaxation is observed and the T / ratio o coated sand is close to that o washed sand (.26) at all concentrations. Two important conclusions can be deduced rom these observations: I. Surace relaxation increases as the concentration o erric ions on silica surace increases as shown by the corresponding decrease in T and relaxation times. This conclusion is also consistent with the observations that longitudinal and transverse relaxivities o porous media increase with the concentration o paramagnetic particles on pore suraces [9,3]. II. relaxation due to dephasing in inhomogeneous ield induced by Fe 3+ ions is negligible. Secular relaxation is proportional to the square o the size o paramagnetic particle in the motionally averaging regime (valid or Fe 3+, Section 6) and is negligible or angstrom sized erric ions., τ=0.2ms, τ=ms T nm 2 /part. 0.3 nm 2 /part T, T (ms) 2 Fig. 8. T and distributions o ine sand coated with erric ions at dierent surace concentrations. No echo spacing dependence o relaxation is observed and T / is same as that o uncoated sand. These conclusions are however, contingent on the condition that paramagnetic particles are present in dilute surace concentrations. Kenyon et al. [32] showed that surace relaxivities obtain asymptotic values at high concentration o paramagnetic particles. Similarly, secular relaxation can become signiicant at high concentration o particles due to superposition o ields induced by individual particles. 2. Fine sand coated with magnetite nanoparticles: Known quantity o the ine sand was coated with positively charged 25 and 0 nm magnetite particles by ollowing the same procedure as was used or coating erric ions. The concentrations o the coating dispersions were varied to obtain dierent surace concentrations o magnetite. The coated sand was washed and saturated with deionized water. Note that the coating is done with positively charged magnetite not coated with citrate ion. Figs. 9 and 0 show the T and distributions o water-saturated ine sand coated with 25 and 0 nm magnetite at dierent surace concentrations (surace area/particle). At low surace concentrations o either 25 or 0 nm particles no echo spacing dependence o relaxation is observed. At high concentrations however, echo spacing dependence o relaxation is observed or both cases. This transition in echo spacing dependence can be understood in terms o the length scales o ield inhomogeneity. Fig. shows that at low concentration, ield inhomogeneity extends to length scale comparable to the size o paramagnetic particles. Thus, or submicron paramagnetic particles, ield inhomogeneities are motionally averaged in time much smaller than s E (s R < s E, Section 6) and relaxation shows no echo spacing dependence. At high concentrations, the particles orm a shell around silica grains such that the length scale o ield inhomogeneity is comparable to the grain