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URNAL OF MAGNETIC RESONANCE 131, 232–240 ( 1998)TICLE NO. MN981364

Eff ec tive Gradients in Porous Media Due to Suscep t ibility Diff erencesMartin D. Hu ¨rlimann

Schlumberger-Doll Research, Ridgeeld, Connecticut 06877-4108

Received April 7, 1997; revised September 30, 1997

I n p or ous m ed ia , m a gnet ic suscep t ibili t y diff er ences bet wee n samples can have signicant internal gradients, especially ate soli d ph a se a nd t he uid lling t he por e sp a ce le a d t o el d high magnetic elds. Functional MRI ( 1, 2 ) relies onh om oge nei t ies inside t he por e sp a ce. I n m a ny ca ses, d iff usion of changes in the internal gradients that are caused by changese sp ins in t he u id ph a se t h r ough t hese in t er n a l inh om oge nei t ies in susceptibility. Paramagnetic contrast agents are routinelyn t r ols t he t r a nsve r se dec a y r a t e of t he N MR sign a l. I n d iso r der ed used in medical MRI to deliberately create internal eld

or ous m edia such a s sedim en t a r y r ock s, a d et a iled ev a lu a t ion of inhomogeneities ( 3–5 ). In this paper, we will concentrateis p r ocess is i n p r a ct ice not possi ble bec a use t he el d inh om oge -on water-saturated sedimentary rocks. In these systems,ei t ies depend n ot only on t he suscep t ibili t y diff er ence bu t a lsoparamagnetic impurities in the rock grain lead to a suscepti-n t he det a ils o f t he por e geo m et r y. I n t his r epor t , t he m a jor

bility difference that can vary by orders of magnitude froma t u r es of diff usion in in t er n a l gr a dien t s a r e a n a lyzed wit h t he rock to rock, depending on the impurity concentration.ncep t of eff ec t ive g r a dien t s. E ff ec t ive g r a d ien t s a r e r el a t ed t oe el d inh om oge nei t ies ove r t he deph a sing lengt h , t he t yp ic a l Nuclear magnetic resonance measurements on the uidngt h ove r wh ich t he spins diff use bef or e t hey deph a se. For t he phase has become a standard technique for studying the porePMG sequence, t he depend ence of r el a xa t ion r a t e on echo sp a c- geometry of sedimentary rocks ( 6 ). In the most commong c a n be desc r ibed t o r st or der by a dist r ibu t ion of eff ec t ive measurement, the CPMG sequence is used to measure thea dien t s. I t is a r gued t h a t f or a give n suscep t ibili t y diff er ence, surface-induced relaxation rate of the transverse magnetiza-er e is a m a xim um v a lue f or t hese e ff ec t ive g r a d ien t s, g m a x , t h a t tion, a measure of the local ratio of surface area to pore

epend s on only t he diff usion coef cie n t , t he La r m or f r equency, volume. The effects of internal gradients are minimized bynd t he suscep t ibili t y diff er ence. T his a n a lysis is a pp lied t o t heusing low elds and the shortest possible echo spacing, t Ese of wa t er -sa t u r a t ed sedim en t a r y r ock s. Fr om a set of N MRThe dependence of the measured relaxation time on echoea su r em en t s a nd a com p ila t ion of a la r ge nu m ber of susce p t ibil-spacing and Larmor frequency has been studied by Kleinberg

ym

ea

sure

men

ts, we co nclud e

th

a t the eff ec

tive g

r adien

ts in c

a r- and Horseld ( 7, 8 ) and by the group of Brown, Borgia,on a t es a r e t yp ic a lly s m a lle r t h a n gr a dien t s of cu rr en t N MR well

ggi ng t ools, w her ea s in m a ny sa nd st ones, in t er n a l gr a d ien t s ca n and Fantazzini ( 9, 10 ) .com p a r a ble t o or la r ge r t h a n t ool g r a d ien t s. q 199 8 Ac a dem ic P r ess The commercial NMR logging tools currently used in well

logging ( 11, 12 ) have tool-related gradients that are of theorder of a few tens of gauss per centimeter. Recently, Akkurtet al. (13 ) have proposed to take advantage of the toolI NTR ODUCT I ONgradients and to obtain information about the diffusion coef-cient of the uid from the t E dependence of the measuredWhen a uid-saturated porous medium is placed in a ho-relaxation time. The general uid identication by theogeneous magnetic eld for NMR measurements, suscepti-method of diffusion measurements in the tool gradient islity differences between the solid matrix and the pore uidsuccessful only if the total gradient in the pore space isan lead to substantial magnetic eld gradients in the poredominated by the tool gradient and not by the internal gradi-ace. These susceptibility-induced gradients are called ‘‘in-ents. In this report, we discuss the size of typical internalrnal gradients’’ and it is well known that they can affectgradients in sedimentary rocks.MR measurements in various ways. Diffusion of spins in

We rst review the basic physics of restricted diffusion inese inhomogeneities leads to extra transverse relaxation.a gradient for the Hahn echo and identify the key parametershe internal gradients can also interfere with externally ap-governing the NMR decay rate. We then discuss qualitativelyied gradients and cause distortions in imaging or diffusionthe effects on a CPMG sequence. For disordered systems, theeasurements. These internal gradients depend both on thelocal gradient of the magnetic eld in the pore space can beusceptibility difference between the matrix and uid andvery high. It is argued that the relevant quantities for NMRn the pore geometry.measurements are not the local gradients, but some effectiveThere are several types of porous media where susceptibil-

y-induced eld inhomogeneities are relevant. Biological gradients related to the local eld inhomogeneities averaged

23290-7807/98 $25.00pyright q 1998 by Academic Press rights of reproduction in any form reserved.

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34 MARTIN D. HU ¨ RLIMANN

between the p pulses. The analysis presented above was basedon the Hahn echo. It must be adapted to the case of the CPMGsequence, where an additional length scale is entering theproblem, namely lE å q D0t E . We discuss the asymptotic be-havior for the regimes of very large and very small pores.We note that restricted diffusion in the presence of a gradientfor intermediate regimes has not been fully analyzed for theCPMG sequence. Tarczon and Halperin ( 19 ) made an analy-sis based on the Gaussian phase approximation.

In the free diffusion regime (large pores), lE is now theshortest length scale and the signal starts to decay accordingto the well-known expression

M ( t )/ M 0 Å exp{ 0 112 D0g2 g 2t 2E t }. [4

F I G . 1. Asymptotic Hahn spin echo decay for spins diffusing betweeno parallel plates separated by size ls and in a constant gradient g Å Dv / In this expression, t is the time at the k th echo, i.e., t Å kt Els ). The logarithm (base 10) of the signal is plotted as a function of the

In the motional averaging regime (small pores) whengarithm of normalized time, log 10 ( Dv t ), and the logarithm of normalizedlE @ ls , the decay will follow Eq. [2] and show no depen-e, log 10 ( ls /q D0 /Dv ). Only signals larger than 10 0 6 are displayed.dence on t E . When t E becomes shorter than l 2s / D0 , we expectthat the behavior goes over to Eq. [4].

We want to focus on the dependence of the signal decayon the echo spacing, t E . In large pores, when the spins areplot time in units of the inverse linewidth, i.e., to plot predominantly in the free diffusion regime, Eq. [4] indicates

v t and to express the size in units of q D0 / Dv , which is a strong dependence on t E . In small pores, we expect thate typical length a spin diffuses in a time 1/ Dv . the relaxation rate rst increases a bit and then stays constantThe asymptotic expressions are applicable when the short- for values of t E larger than l 2s / D0 . In this motional averaging

t length scale is much shorter than the other lengths. When regime, there is no dependence on t E . The contribution toe two shortest lengths are comparable, there will be some the decay rate of diffusion cannot be distinguished by its t

eviations. This occurs along the two lines running down dependence from other contributions, such as surface or bulke surface of Fig. 1. When lg approaches lD along the line relaxation. However, note that for smaller and smaller pore

Dv t ) 3 Å ( ls /q D0 / Dv ) 2 , the full solution will show a less sizes, Fig. 1 shows that the diffusion contribution to thebrupt change of slope. There is also some extra structure signal decay decreases rapidly and becomes unimportant.ear the line when ls approaches lg , i.e., when ls / q D0 / Dv

1 (18 ). Even with this limitation, Fig. 1 captures the E FF ECT I VE GR AD I ENT S I N SED I MENT AR Y R OCK Sominant features of the Hahn echo decay in this system.

How do we generalize from the simple model discussedhe motional averaging regime appears on the left-hand sideabove to sedimentary rocks? In rocks, the eld variationsFig. 1. The smaller the pores, the better diffusion averagescaused by the susceptibility differences are clearly moree eld inhomogeneities and the slower the decay (evencomplicated than can be described by a constant gradient.ough the gradients become larger). As the size ls is in-However, a given spin does not diffuse very far during theeased, the behavior goes over to the free diffusion regime,NMR measurement. The question arises, whether the localhich at a later time crosses over into the localization re-eld variation can be adequately modeled by some localme. The larger the ls , the smaller the gradient and theeffective eld gradient. We make a heuristic argument belowower the decay. For a given inhomogeneity, Dv , the fastestthat this effective eld gradient is related to the eld varia-ecay occurs when ls á q D0 / Dv . This is also the conditiontions over the local dephasing length and has an upper limit.at the dephasing length lg and the structural length ls coin-The total signal decay is then to rst order a superpositionde: ls á lg .of the signal decay of subsets of spins, each of which experi-ences a local effective gradient and can be in the free diffu-C PMG PU LSE SE QUENCEsion regime or the motional averaging regime, depending onthe pore size. This is somewhat similar to the earlier workIn downhole NMR measurements, the CPMG pulse se-

uence is used instead of the simple Hahn echo in order to by Bendel ( 20 ). He considered the superposition of signalsfrom spins in the local gradient. We will argue that theaximize the ratio of signal-to-noise. This sequence consists

an initial p /2 pulse, followed by a train of p pulses, each relevant gradient is an effective gradient that is a specicaverage of the local gradient.parated by a time t E from the previous p pulse. Echoes form

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23EFFECTIVE GRADIENTS IN POROUS MEDIA

In sedimentary rocks, spins are not conned to individual tially free diffusion. We ignore more subtle effects at longertimes: The small number of spins close to the surfaces thatores; they can diffuse from one pore to the next. It is also

ten impossible to divide the pore space unambiguously have experienced restricted diffusion have decayed less andmight dominate the signal. On the other hand, they are alsoto individual pores, because the whole pore space is in

eneral connected. However, even in such a system, it is more likely to decay due to surface relaxation. This second-order effect depends on the details of pore size, shape, andossible that the motional averaging regime applies, because

was pointed out by Wayne and Cotts ( 21 ) that diffusion surface relaxivity and will not be further discussed.The spins in the large pores will experience some local,an isolated pore with a constant gradient is equivalent to

nbounded diffusion with a periodic gradient. effective gradient geff ( x). This effective gradient is obtainedby averaging the eld inhomogeneities self-consistently overThe following key question must be answered: What is

e proper scale for subdividing the spins into subsets? For the length lg ( x), as discussed above. The echo attenuationdue to diffusion in the susceptibility-induced eld inhomoge-MR diffusion measurements, it is natural to label spins

cording to their local Larmor frequency. Diffusion in the neities is then determined by the distribution of effectivegradients, f ( geff ), for the large pores. To rst order, weeld inhomogeneity leads to an uncertainty in this value.

he time-dependent Larmor frequency of a spin can be mea- obtain for large poresured only to an accuracy dv if the spin stays a time longompared with 1/ dv in the appropriate eld. Therefore, the

M ( t )/ M 0 É *gmax

0dg eff f ( geff )ze of the subset, lc , should be chosen such that all spins

ithin a subset have the same Larmor frequency within thencertainty imposed by diffusion. This implies that across 1 exp

H0

1

12 D0g 2 g 2eff t 2E t

J . [6

e size of the subset, lc , the uncertainty in Larmor fre-uency, Dv Å Év ( x) 0 v ( x / lc )É, becomes equal to

0 / l 2c . We describe the eld inhomogeneity across the We have indicated in [6] that there is an upper limit of geffze of the subset by the local effective gradient geff ( x): The reason is that the total variation in the local eld isv ( x) É g geff ( x) x. The size of the subset lc is then identical effectively bounded by Dx B0 , as was shown by Brown and lg introduced above, using the effective gradient geff : Fantazzini ( 9 ). The effective gradient geff is associated

with a length lg given by Eq. [5]. This implies directly thatgeff lg ° Dx B0 . Using the expression Eq. [5 ] for lg an

lc Å lg ( x) Å S D0g geff ( x) D1 /3

. [5] solving for geff leads to the approximate upper limit gmax :

he eld inhomogeneities must be averaged over the length gmax É S g D0D

1/ 2

( Dx B0 )3 /2

. [7to obtain the effective gradient geff , with the additionalstriction that the two quantities lg and geff are related to

ach other by Eq. [5]. If the real eld inhomogeneity is An effective gradient of gmax denes a scale for structuralescribed by a uniform gradient g , the effective gradient is features of the order of the associated dephasing length,entical to g : g Å geff . Le Doussal and Sen ( 22 ) studied namely,e case of a parabolic eld prole: B Å B0 / g2 ( z 0 z0 ) 2 .t the eld extremum, the size of the subset calculated bye method outlined above becomes lg ( z0 ) Å ( D0 / g g2 ) 1/ 4 , l* Å

Dx B0gmax

Å S D0gDx B0D1 /2

. [8hich is in agreement with their exact calculation.The length scale lg ( x) is typically in the range of a few

The size l* is the relevant size for distinguishing small andtens of micrometers. Structure in the eld prole on alarge pores.ngth scale shorter than lg is not important, because it gets

There is another argument that there must be a maximalveraged out by diffusion.effective gradient that scales like the expression given in Eq.This allows us to classify the pore space into ‘‘large’’ and[7]. For gmax , the signal decay given by Eq. [4] is propor-small’’ pores or subsets. In the large pores, the pore size

large compared to the local dephasing length lg ( x) and the tional to ( gD B0 ) 3t 2E . The maximal value for t E for whichthis expression applies should be of order ( l* ) 2 / D0 Åho dephasing is essentially governed by the free diffusion

gime. In the small pores, the pore size is small compared ( gD B0 )0 1 . For this value of echo spacing, the maximal

decay rate becomes proportional to gD B0 . This is the static lg ( x) and the internal eld inhomogeneities are motionallyveraged. linewidth and the decay rate of the free induction decay. It

is clear that the amplitudes of the spin echoes cannot decayIn large pores, the dephasing length is by denitionmaller than the pore size. Therefore, we can assume essen- any faster than the free induction decay.

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T AB LE 1 is typically nonexponential and can be described by a distri-Mea su r ed Suscep t ibili t y D iff er ences a nd C a lcula t ed Va lues bution of relaxation times, h ( T 2 ), for T 2 values in the range

of M a xim a l E ff ec t ive G r a dien t a nd C r it ic a l Lengt h between about 1 ms and 1 s:

Sample Dx [SI] gmax at 2 MHz l*

M ( t , t E ) É * dT 2 *gmax

0dg eff f ( geff )

9 278 1 10 0 6 1800 G/cm 0.8 mmrea 100 98.9 1 10 0 6 375 G/cm 1.3 mmrea 400/500 33.8 1 10 0 6 75 G/cm 2.3 mm 1 expH0 112 D0g 2 g 2eff t 2E t J7 13.6 1 10 0 6 19 G/cm 3.6 mmdiana limestone 2.99 1 10

0 6

2.0 G/cm 7.6 mmudstone 1.22 1 10 0 6 0.5 G/cm 11.9 mm 1 h ( T 2 )exp{ 0 t / T 2 }. [9

Here f ( geff ) is the distribution of effective gradients andh ( T 2 ) is the distribution of relaxation time T 2 . As discussedEven though Eq. [7] is only a rough estimate, we can above, h ( T 2 ) includes surface relaxation and diffusive con-onclude that in the majority of carbonates, the value of gmax tributions from the small pores. At the shortest echo spacing

smaller than typical tool gradients. Therefore, the vast t E Å 160 ms, the T 2 term dominates the gradient term. If weajority of local effective gradients in carbonates are smaller can make the further assumption that the two terms are notan tool gradients. The situation is different in sandstones. highly correlated, we obtain for the ratio of M ( t , t E )/ M ( tt the peak of the susceptibility distribution for sandstones, t E Å 160 ms) the simple forme estimate that internal gradients can be as large as 1000 G/m for a Larmor frequency of only 2.3 MHz. In sandstones,ternal gradients can therefore frequently be comparable to M ( t , t E )

M ( t , t E Å 160 ms)É *

gmax

0dg eff f ( geff )

larger than tool gradients.At downhole temperatures, the estimated values for gmax

1 expH0 112 D0g 2 g 2eff t 2E t J. [10e reduced by a factor of 2.5 because the diffusion coef-ent is higher and the susceptibilities are reduced ( assumingurie law behavior). Based on this analysis we expect that

This predicts that the ratio M ( t , t E )/ M ( t , t E Å 160 ms) ie effective gradients in most carbonates are smaller thanto rst order only a function of the combined times t 2E t . Ie tool gradients and can mostly be ignored. However, inFig. 3, this ratio is plotted as a function of t 2E t for the sixany sandstones, the effective gradients can be signicantlyrocks listed in Table 1.rger than tool gradients. This clearly complicates the analy-

In Fig. 3, we note rst of all that for all six rocks, theres of diffusion measurements with static applied tool gradi- is a reasonable collapse of the data on a single curve. Tonts in sandstones.

rst approximation, the ratio is indeed only a function of t 2E t , as given in Eq. [10]. This is not completely obvious.E XP ER I MENT AL N MR RE SU LT SOne might argue that for any given time t , there is a differentdistribution of effective gradients. At long time, surface re-CPMG measurements were performed on six water-satu-laxation relaxes the spins in the smaller pores and only theted sedimentary rocks with susceptibility differences rang-spins in the larger pores survive. One would expect that theg from about 10 0 6 to over 2 1 10 0 4 . Table 1 lists thedistribution at long times is dominated by smaller gradients,easured values of Dx and the calculated values of gmaxwhereas at shorter times, spins in both small and large poresEq. [7]) and l* (Eq. [8]). Note that these samples covercontribute. A careful analysis of the data for C9 and thee typical susceptibility differences of sedimentary rocks,Bereas shows that the scatter in Fig. 3 is in fact somewhatown in Fig. 2. The NMR measurements were performedcorrelated with time t ; i.e., at a given value of t 2E t , ratios2 MHz with a laboratory NMR spectrometer that has afor large t tend to be somewhat higher than ratios for smallomogeneous external magnetic eld and the temperaturet . However, this is a rather weak effect and indicates thatas 25 7C. Eight different echo spacings, t E , were used,surface relaxation and diffusion in internal gradients are sen-amely 160, 200, and 400 ms and 1, 2, 4, 10, and 20 ms.sitive to different features of the pore geometries. Diffusionata were collected up to a total time of 655 ms; i.e., theis sensitive to features of size l * , whereas surface relaxationumber of echoes collected was adjusted to between 4095,is sensitive to the ratio of the surface area to pore volume.r t E Å 160 ms, and 33, for t E Å 20 ms.

The arrows in Fig. 3 mark the value of t 2E t whereIn water-saturated sedimentary rocks, the transverse relax-1

12 D0g2 g 2max t 2E t Å 1. Using the expression in Eq. [7], this ision rate at low eld strength is usually dominated by sur-

ce relaxation; relaxation caused by internal gradients only simply given by t 2E t Å 12/( gDx B0 ) 3 . This is the 1/ e poinfor the largest possible effective gradient. We expect noominates for longer echo spacings t E . The surface relaxation

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F I G . 3. Ratio of M ( t , t E ) / M ( t , t E Å 160 ms) versus t 2E t for six different rocks. In all cases, there is a good data collapse for all the different valuest and t E , spanning t 2E t over six orders of magnitude. The arrow indicates the 1/ e point for the largest effective gradient as calculated from the

sceptibility difference between the rock and water.

gnicant deviation of the ratio M ( t , t E )/ M ( t , t E Å 160 ms) In order for the signal to show decay at the maximal gradi-ent, there must be structure on the length scale l* given inom 1 for values of t 2E t smaller than indicated by the arrows.Table 1. The curve for C9 in Fig. 3 appears to decrease onlyhis simple estimate is in good agreement with the observa-

ons. This is especially noteworthy because the prediction at values of t 2E t that are about a decade larger than shown bythe arrow. In this rock, the pore space might not have a lotas no adjustable parameters and depends only on suscepti-

lity and Larmor frequency. Note that it covers several order of structure on the length scale of 0.8 mm. Alternatively, it ispossible that the decay for t E Å 160 ms is already affected bymagnitudes: for a given value of t E , the diffusion-induced

ecay rate varies by six orders of magnitude in these rocks. diffusion in the internal gradients.

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ACKN OW LEDG MENT S

I had helpful discussions with P. Sen, R. Kleinberg, K. Helmer, andB. Halperin. The CPMG measurements were performed by D. Rossini, andP. Dryden and W. Smith assisted me with the susceptibility data.

RE F ERENCE S

1. K. K. Kwong, J . W . Belliveau, D. A. Chesler, I. E. Goldberg , R. MWeissko ff , B. P . Poncele t , D . N . Kennedy, B. E. Hoppel, M. S ., C ohe n , R. Turner , H. -M . Cheng, T. J . Brady, and B. R . Rosen , ProcNatl. Acad. Sci. USA 89, 5675 (1992 ) .

2. S . Ogawa, D. W . Tank, R. Menon, J . M. Ellermann, S . -G . Kim, HF I G . 4. Estimate of the distribution of internal gradients in the two Merkle, and K. Ugurbil, Proc. Natl. Acad. Sci. USA 89, 5952 (1992cks with the largest susceptibility difference.

3. R. N. Muller, P . Gillis, F. Moiny, and A. Roch , Transverse relaxivito f par t iculat e MRI co n tra s t media: From theories to experimen tsMagn. Reson. Med. 22, 178 (1991 ) .

4. P . Hardy and R. M. Henkelman, On the transverse relaxa t ion ratUsing Eq. [10], we can estimate the distribution of effec-enhancemen t induced by diffusion o f spins through inhomogeve internal gradients, f ( geff ), from the data shown in Fig. neous elds , Magn. Reson. Med. 17, 348 (1991 ) .

We use a tting routine based on single value decomposi- 5. R. M. Weissko ff , C. S . Zuo , J . L. Boxerman , and B. R . Rosen , Mon. The routine was implemented by Sezginer and is de-

croscopic suscep tibility variation and transverse relaxa t ion: Theorribed in ( 23 ) where it is applied to relaxation data. The and experimen t , Magn. Reson. Med. 31, 6 01 (1994 ) .sults for the two rocks with the largest susceptibility differ- 6. R. L. Kleinberg, ‘‘Encyclopedia of Nuclear Magnet ic Resonance ,

Vol. 8 , pp . 4960 –4969 , Wiley, Chichest er (1996 ) .nces, C9 and Berea 100, are shown in Fig. 4.7. R. L. Kleinberg and M. A. Hors eld , Transverse relaxa tion proThese distributions are several decades wide and both

cesses in porous sedimen t ary rock , J. Magn. Reson. 88, 9 (1990ave a signicant fraction of gradients above 100 G/cm, as8. R. L. Kleinberg, Pore size dis tributions , pore coupling , an d transxpected.

verse relaxa t ion spec tra of porous rocks , Magn. Reson. Imaging12, 2 71 (1994 ) .

C ONC LU SI ON S 9. R. J . S . Brown and P. Fant azzini, Conditions for initial quasilineaT 0 12 versus t for Carr– Purcell– Meiboom– Gill NMR with diffusioand suscep tibility diff erences in porous media and tissues , PhysIn this report, we have analyzed the signal decay causedRev. B 47, 14823 (1993 ) .

y diffusion in internal gradients. We have shown that the 10. G. C. Borgia, R. J . S . Brown, and P . Fant azzini, Scaling of spinores can be classied into large or small pores, by compar-echo amplitudes with frequency , diffusion coe f cien t , pore size

g the pore size to q D0 / gDx B0 . Only the contributions and suscep t ibility diff erence for the NMR of uids in porous mediom the large pores show a signicant increase of the decay and biological tissues . Phys. Rev. E 51, 2104 (1995 ) .te with echo spacing t E . This decay can be described to 11. Z. Taicher , G. Coat es , Y. Git ar t z , and L. Berman , A com prehensiv

approach to s tudies o f porous media (rocks ) using a labora t orst order by a distribution of effective gradients. It is arguedspec trome t er and logging tool with similar opera ting charac t erisat there is a maximum value for the effective gradient,tics , Magn. Reson. Imaging 12, 285 (1994 ) .

max , that depends only on diffusion coefcient, Larmor fre-12. R. L. Kleinberg, A. Sezginer, and D. D. Grifn , Novel NMR apparuency, and susceptibility difference.

tus for inves t ig ating an ext ernal sample , J. Magn. Reson. 97, 4 6This analysis was then applied to the study of water- (1992 ) .turated sedimentary rocks, relevant in NMR well logging. 13. R. Akkurt , H. J . Vinegar, P . N . Tut unjian , and A. J . Guillory, NMom a compilation of susceptibility measurements, we con- logging of na t ural gas reservoirs , in ‘‘Trans. SPWLA 36th Annuude that the effective gradients in carbonates are typically Logging Symposium , Paris , 1995 , ’’ Paper N.

maller than tool gradients, whereas in many sandstones, 14. S . D. St oller, W. Happer, and F. J . Dyson, Transverse spin relaxa tion in inhomogeneous magne t ic elds , Phys. Rev. A 44, 745ternal gradients can be comparable to tool gradients. This(1991 ) .omplicates the analysis of diffusion measurements in sand-

15. T. M. de Swiet and P . N . Sen , Decay of nuclear magne t iz at ion bones. CPMG measurements with different echo spacingsbounded diff usion in a c ons t an t eld gradient , J. Chem. Phys. 100n six rocks conrm these theoretical predictions. 5597 (1994 ) .

In biological systems, the susceptibility differences en-16. M. D. H ürlimann , K. G . Helmer, T. M. de Swiet , P . N. S en , and C. H

ountered are usually smaller than those in sedimentary So t ak , Spin echoes in a cons t an t gradien t and in the presence ocks, but the measurements are typically performed at simple res tric t ion , J. Magn. Reson. A 113, 260 (1995 ) .armor frequencies much higher than 2 MHz. This analysis 17. C. H . Neuman , Spin echo of spins diffusing in a bounded medium

J. Chem. Phys. 60, 4508 (1974 ) .ould therefore also be relevant for these systems.

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. M. H. Blees, The eff ec t of nit e dura t ion of gradien t pulses on the 21. R. C. Wayne and R. M. Cott s , Nuclear-magne t ic-resonance s tudo f self-d iffusion in a bounde d medium , Phys. Rev. 151, 264 (1966pulsed- eld-gradien t NMR metho d for s tudying res tric t ed diff usion ,

J. Magn. Reson. A 109, 2 03 (1994 ) . 22. P . Le Doussal and P . N. Sen , Decay of nuclear magne tiz ation bdiff usion in a parabolic magne t ic eld: An exactly solvable mode. J . C. Tarczon and W. P . Halperin, In t erpre t a t ion of NMR diff usion

measuremen ts in uniform- and nonuni form- eld proles . Phys. Rev. Phys. Rev. B 46, 3465 (1992 ) .B 32, 2798 (1985 ) . 23. E. J . Fordham , A. Sezginer, and L. D. Hall, Imaging multiexponent i

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