A MECHANISM FOR THE TEMPERATURE DEPENDENCE OF THE SURFACE RELAXATION RATE IN CARBONATES

Transcription

1 SCA /12 A MECHANISM FOR THE TEMPERATURE DEPENDENCE OF THE SURFACE RELAXATION RATE IN CARBONATES Christian Straley, Schlumberger-Doll Research, Rigefiel, Connecticut ABSTRACT Unexlaine temerature eenence in some carbonate reservoir rock was reorte recently, in which the mean log T 2 for the brine-saturate material increase with temerature [8]. For many years significant temerature eenence was not observe an was consiere unlikely in rock materials base on a variety of evience [5]. Some of the recently announce temerature eenence was exlaine. More efficient iffusive couling between ores, which shoul occur with increase temerature, can shift the T 2 relaxation to shorter times. The examles that were reorte to islay an increase in relaxation time with increasing temerature were roblematic an suggeste that the surface relaxation rate weakene with temerature. Last year, new theory an exeriments were introuce to exlain temerature-eenent T 2 -increases base on the surface iffusion in the vicinity of aramagnetic relaxation centers [4]. Our aer emonstrates that a comlementary or ossibly alternative mechanism for temerature eenence shoul be consiere, too. Brine in rock relaxes faster than in bulk. This augmente relaxation has been attribute to the resence of aramagnetic imurities in the rock, near the rock-water interface. Prior to the general accetance of surface relaxation by aramagnetic centers as the ominant NMR relaxation mechanism for fluis in rock, hinere motion on the rock surface that cause enhance hh (hyrogen-hyrogen) (iole-iole) relaxation was a cometitive theory for the accelerate relaxation. The ossibility of hh relaxation, which was rovie for in the early theory for the relaxation of rock materials, eventually fae away because it i not seem relevant. Inee, it was the failure to observe temeratureeenence in rock that le to the conclusion that only aramagnetic relaxation was imortant an that it was reasonable to exclue consieration of surface hh relaxation an further that only the temerature-ineenent scalar term for relaxation was imortant in the electron-roton interaction. In this aer, examination of a single carbonate rock, which has a reviously ocumente temerature-eenence, inicates that the surface hh relaxation rate is comarable to the surface relaxation rate ue to aramagnetic centers. Significant surface hh relaxation, relative to the total rate, coul account for the observe temerature eenence of relaxation times in carbonates.

2 SCA /12 INTRODUCTION Recent stuies [8] have reorte significant temerature eenence for the NMR relaxation of brine in some limestone reservoir rock 1, whereas an early stuy by Latour [7] observe weak temerature eenence for carbonates. Although Ramakrishnan et al. [8] quantifie iffusive couling between large an small ores to exlain the ecrease in relaxation time with increasing temerature, they i not exlain a few observe cases in which the relaxation time increase with increasing temerature. This temerature eenence is not exlaine by the wiely accete mechanism of relaxation in rock [3, 5]. In fact, the icture of brine relaxation in rock assumes, base on a limite number of exerimental observations, that the temerature eenence of the relaxation times for brine-saturate rock is insignificant. Kleinberg et al. [5] state, ``The lack of strong temerature eenence of the rates inicates that iffusive motion of the flui molecules or the on-off motion at a surface site o not enter into a etermination of T 2 an T 1.'' This statement reflects the unerstaning that the aramagnetic relaxation of water at the rock surface is ominate by the abunance of aramagnetic sites an the electron's correlation time, not the water's correlation or resience times. Aitionally, the aramagnetic relaxation has to take lace rimarily through scalar couling to be ineenent of temerature. The results of the recent carbonate stuies an the inability to reconcile those results using accete theory require re-examination of the unerlying assumtion of omination by aramagnetic centers an re-evaluation of the significance of the water s correlation time or resience time in these unusual cases. Theory of Korringa, Seevers an Torrey, 1962 Most theoretical iscussions of relaxation in rock begin with the funamental work by Korringa, Seevers an Torrey (KST) [6]. The KST moel was very general an allowe relaxation at non-aramagnetic surface sites as well as relaxation at aramagnetic centers an a variable resience time on the surface. The anticiate relaxation mechanism at the non-aramagnetic centers was by hh (hyrogen-hyrogen iole-iole) interaction in the system, rimarily the water. This mechanism is common. That relaxation is attribute to the inter-molecular an intra-molecular (iole-iole) interaction of the hyrogen nuclei uring translation an rotation [1]. Furthermore, the hh relaxation rate is enhance on surfaces because the motion of the absorbe molecule is slowe. For examle, Woessner [10] foun that the rotational rate of water on the surface of hectorite was about 1/5 of the value in the bulk flui. KST s general exression for relaxation in rock is 1 1 Nm 1 N n 1 = + +, (1) T 1, r T1, b N T1, e + τ N T1, n +τ in which, T 1,r is brine s relaxation time in rock, T 1,b is the brine s bulk relaxation, N m is the number of water molecules in the shere of influence of a aramagnetic surface site, 1 This articular issue is iscusse on age 8 of their text uner the heaing Temerature Deenence an illustrate with NMR logs, Figure 8. The author was rovie aitional ata for those samles by Mr. Mark Flaum, a co-author of the reference aer.

3 SCA /12 N is the total number of water molecules, T 1,e is relaxation ue to the interaction of the water s hyrogen an the aramagnetic center, τ is the average resience time of a water molecule in the shere of influence, N n is the number of water molecules in the shere of influence on non-magnetic sites an T 1,n is the relaxation time for a brine molecule at a non-magnetic site. Each relaxation time an the resience time in this equation are otentially temerature eenent. Aatation of KST by Kleinberg, Kenyon an Mitra, 1994 Because the observe temerature eenence in rocks was not consiere significant an because hh relaxation is inherently weak, the terms for the non-aramagnetic sites in KST s equation were eventually roe [5]. For the same reason, that it woul necessarily be temerature eenent, the term for the resience time in the aramagnetic shell isaeare. It was also argue that the scalar term of the electron-roton aramagnetic relaxation interaction must ominate the iolar term because the iolar art woul be temerature eenent. T 1,b, which is temerature eenent, remaine but it is easily measure an remove from the calculation. The generally accete functional form of KST became 1 1 N m 1 = +. (2) T1, r T1, b N T1, e ( scalar) Goefroy, Fleury, Deflanre an Korb, 2001 Contemorary stuies have been resente very recently that re-consier some of the basic issues that le to Equation (2). Goefroy et al. [4] emonstrate the temerature eenence of Lavoux limestone. They constructe a relaxation moel to account for the temerature eenence. Their moel allows for temerature eenent relaxation at aramagnetic surface sites in a manner similar to Korringa et al. but they o not incororate the ossibility of relaxation at non-aramagnetic sites. Consieration of Surface Hyrogen-Hyrogen Interaction The temerature eenence in carbonates observe by Ramakrishnan et al. [8] is significant when the carbonate s relaxation time is long, imlying that the overall relaxation rate is small. It is robable that hh relaxation at the surface can only be a cometitive mechanism uner the conition of weak relaxation because hh relaxation is inherently limite to a small rate. To evaluate whether hh relaxation contributes to the relaxation of brine in carbonates, a simle test can be evise. The relaxation of samles saturate with a small concentration of H 2 O iserse in D 2 O (euterium oxie) can be comare to the relaxation of the same samles saturate with ure H 2 O. The relaxation ue to DH (euteriumhyrogen) interactions is roughly 1/40 th of the comarable HH interactions. If the surface relaxation of hyrogen is urely ue to interactions with aramagnetic imurities, the relaxation rate shoul remain unchange in the two exeriments; there shoul be no eenence on the concentration of hyrogen sins. On the other han, if the relaxation

4 SCA /12 rate slows significantly for the H 2 O/D 2 O-saturate samles, hh interactions must have been imortant in the H 2 O-saturate exeriment. Extracting r 2, an r 2, The observe rate, 1 T 2,obs to the secific interactions, 1 T,, in a uniform magnetic fiel shoul be the sum of the rates ue T = + +. (3) T T T 2, obs 2, 2, 2, b In this exression, 2 is the surface aramagnetic relaxation time-constant, 2 is the surface hh relaxation time-constant, an T 2, b is the bulk relaxation time-constant for the flui. There are no irect questions about temerature eenence of the relaxation times in this test because the two exeriments are run at 30 o C. The test is simly to etermine whether hh relaxation is significant. If confirme as a mechanism, hh relaxation must be temerature eenent. In general, it is necessary to remove the flui s bulk rate because the relaxation times for our samles are long (smaller but comarable to those for water itself) an the bulk rate can affect the result. The bulk rate is easily measure for the two saturating fluis an eliminate from the equation leaving a correcte rate, 1 T 2,corr T,. This correcte rate inclues the elimination of bulk flui relaxation ue to issolve oxygen, which shoul be the same for all samles because they were in equilibrium with air. After correction for the bulk flui relaxation, = +. (4) T T T 2, corr 2, 2, When consiering surface relaxation, the rate of iffusion to the surface an the rate of relaxation at the surface can lea to several ifferent outcomes. Rock materials are ρv generally consiere to be in the fast iffusion limit ( << 1) for aramagnetic SD relaxation when the efinitions of Brownstein an Tarr [2] are alie. In this limit, V T2, = ; (5) Sρ V is the ore volume, S is the ore surface area an ρ is the surface relaxation velocity ue to aramagnetic relaxation. By analogy, since T 2, is a surface relaxation henomenon, it shoul exhibit similar behavior if ρ an ρ are comarable, because Vρ the criterion for fast iffusion, << 1, woul still be satisfie. In this exression, D SD

5 SCA /12 is the iffusion constant for the flui an woul be unchange (or only slightly change), just as V an S woul be unaffecte changing fluis from H 2 O to HOD/D 2 O. We can write, V T2, =. (6) Sρ The average value of S V for Equations 5 an 6 can be foun from available hysical measurements. When ensity, an φ is orosity, S g is the secific surface area measure by BET, g S g ρ g is the grain V 1 S = 1 φ ρ 1 φ. (7) 3.82 s was taken as the relaxation time of oxygen-free water at 30 o C [8] an, exerimentally, the oxygen-correcte bulk relaxation time for the HOD/D 2 O was foun to be 13.1 s. The hh surface rate was assume to vary as the bulk rates. This is more realistic than assuming that the hh interaction is comletely eliminate because 10% of the original hyrogen remains, enough that the hh rate cannot be zero. The HOD/D 2 O is a mixture of isotoes of water an the iffusion shoul be essentially the same as water in bulk or on the rock s surface because the imortant van er Waals forces shoul be similar although the molecular masses are about 10% greater. If the effect of aing or removing hyrogen is a matter of suerosition of magnetic effects, the surface rates shoul vary as the bulk rates, ρ T, HOD 2. b, H 2O =, or, ρ, HOD = 0. 29ρ, H 2O. (8) ρ T, H 2O 2, b, HOD Combining Equations 4, 5, 6 an 8 we can write two equations, 1 ρ S ρ, S H2O = +, an (9a) T2, obs V V 1 ρ S 0.29ρ, S H2O = +, (9b) T2, obs V V for the water-saturate an DOH/D 2 O-saturate samles, resectively. These equations can be solve simultaneously to fin the two surface rates. r 2, an Consequences for Temerature Deenence Consier what effect temerature variation shoul have on surface hh relaxation. With increasing temerature, the bulk water relaxation above ambient temerature is such that the relaxation time constant increases, that is, the hh mechanism is less effective because the water molecules are more energetic an the correlation times are shorter. Because the frequencies associate with the correlation times for the water molecules are further from

6 SCA /12 the Larmor frequency require for resonant interaction an relaxation rate is slowe. The temerature eenence for hh surface relaxation shoul be similar to the bulk relaxation using the same arguments. If the ominant relaxation rocess is aramagnetic, the weak surface relaxation an changes ue to temerature may go unnotice. If the relaxation is comarable or ominates, the measure relaxation times shoul increase with increasing temerature. EXPERIMENTAL METHOD Only two rocks were use in this stuy. One was from the Mile East, to be referre to as samle S, an the other was Leuer's limestone, samle L. The measure temerature eenences for samles S an L from a revious stuy [8] are shown in Figures 1 an 2, resectively. Samle S shows a substantial shift of the major eak to longer relaxation times when the temerature is increase, while the shift for L is more moest an aears to be a shift to slightly shorter relaxation time. These contrasting samles were chosen as a reresentatives of a temerature eenent limestone, S, an a limestone aarently lacking temerature eenence, L. The T 2 -istributions are erive from the results of CPMG measurements, which were mae using a Maran2 sectrometer oerating at 2 MHz an observing the hyrogen NMR signal from the brine. The magnetic fiel for the Maran2 is relatively uniform an the echo sacing was chosen to accommoate the comlete relaxation of the samle in 4096 echoes. The mean log T 2 of the istributions is use for the analysis. The orosity, grain ensity, BET surface area, an NMR T 2 relaxation times are liste in Table 1. A 10% solution of istille water in D 2 O (99.9% urity) was mae in a grauate cyliner. To work at low fiel with lower concentrations woul have been extremely time consuming. The hyrogen in water exchanges raily an shoul result in a flui in which 20% of the molecules are HOD, an isomeric form of water with just one roton. The mass of HOD is slightly higher than for H 2 O an the iffusion constant is robably slightly less than for H 2 O but the intermolecular interactions are very similar. Of course, there is a 90% reuction in the total amount of hyrogen in the saturating flui relative to ure water. The NMR bulk relaxation times for the water an the HOD solution were 2.7 an 5.4 sec resectively. Each of these fluis is in equilibrium with air an contains issolve aramagnetic oxygen. The relaxation rate ue to issolve oxygen was measure as 0.11 s, which amounts to a relaxation time of 9.1 s for both fluis. The rocks were vacuum saturate each time they were imregnate with flui. T 2 -relaxations for the HOD/ D 2 O-saturate samles are shown in Figures 3 an 4. If the H 2 O equilibrate with D 2 O to form HOD, one in five molecules woul have hyrogen. At that concentration all hyrogen woul have one or more hyrogen neighbors an inter-molecular hh relaxation coul still be effective. If the H 2 O i not equilibrate, every hyrogen atom woul have a neighbor in the same molecule an intramolecular hh woul be imortant. In bulk water at room temerature, the self-relaxation of the hyrogen on a molecule ue to iffusional rotation accounts for about halve of the

7 SCA /12 total rate. In any event, the hh interaction is not comletely eliminate esite being stongly reuce even when the amount of hyrogen is reuce by 90%. RESULTS AND DISCUSSION Figures 3 an 4 illustrate that changing the hyrogen concentration affects the relaxation time of the temerature eenent samle S as much as heating it to100 o C. Interestingly, there is a smaller but noticeable shift for Samle L. The rates foun from the alication of Equations 9 an 9a to the exerimental ata are foun in Table 2. Foley et al. [3], rovie an estimate for the maximum theoretical value for ρ that is 0.6µ m / s. The estimate is base on the secon moment of ice, a limiting case for the slow iffusion of water. The two measure values for ρ in this stuy are in the range from µ m / s, accetable values in that they are less than the theoretical limit. Another oint shoul be mae because it is well known that a measure value of ρ can vary eening on the metho chosen to measure the surface area. BET rovies the smallest ruler available for surface area measurements an therefore the largest surface area. Using BET rovies the most aroriate measure of ρ for the comarison to the theoretical limit. It is imortant to note that for samle S, the two estimate surface rates, ρ an ρ are similar, while for samle L, the two rates are ifferent by factor of about ten. The relative values of these two rates exlain why S shows noticeable temerature eenence when L i not. In Samle L the aramagnetic rate ominates an variation in ρ has only a small imact. For Samle S the two surface relaxation rates are comarable at 30 o C an as the temerature increases, the rate shoul ecrease an the overall relaxation time shoul increase noticeably until the relaxation is limite to the weak aramagnetic relaxation. It is seuctive that the measure ρ is comarable for both samles. The similarity coul be an inication that the surface hh relaxation rate of all limestone is similar. In light of this, it is not unreasonable to hoe that only a generalize temerature correction woul be neee. By assuming a constant ρ for all limestone, T 2, can be estimate for the eleven samles from the Ramakrishnan et al. [8] stuy because surface areas, orosities an ensities are available an ρ an T 2, can be calculate similarly. The estimates can be consiere to ascertain whether the calculate T 2, falls below the highest value of T 2,obs foun at the highest temerature. ρ = 0.34 µ, with an average value of about 0.4 m / s For the eleven samles, assuming 0.07 to 0.61 m / s µ m / s, ρ is foun to be in the range from µ (six of the samles are in

8 SCA /12 the range from µ m / s ). Note that this average rate is similar to the assume iolar rate at 30 o C, which means that both relaxation mechanisms are significant an temerature eenence woul be execte. On average, the elimination of iolar relaxation by these calculations accounts for 78% of the T 2 shift seen from 30 o C to 130 o C although for two of the eleven samles, a greater shift is calculate than is measure. For the other nine samles, the inability to account for the entire shift may inicate a limitation of the in the D 2 O / H 2 O exeriment or analysis. Varying the value of ρ use for this calculation, shows that ρ woul have to vary over a range from 0.34 to 0.48 µ m / s at 30 o C to account for all the iniviual temerature eenent changes in T 2 for the members of the suite. The failure to account for all the temerature eenence may inicate that the secific quantitative assumtion about the relationshi between the surface an bulk rates might be unrealistic (although it shoul create an uer boun on ρ 2, ). Accounting for only 80% of variation of ρ in the temerature range from o C also leaves oen the ossibility that the resiual temerature eenence coul be associate with the aramagnetic sites. CONCLUSIONS This stuy leas to several interesting observations. 1) In aition to aramagnetic surface relaxation, surface enhance iole-iole relaxation between hyrogen airs can be a mechanism for the relaxation of brine in the limestone matrix. 2) Surface enhance hh relaxation shoul manifest itself through observable temerature eenence if the aramagnetic surface relaxation rate is small an the surface enhance hh rate is comarable. 3) hh surface relaxation coul account for about 80% the observe increases in relaxation time with temerature or the absence of temerature eenence for this limite suite. 4) The hh surface relaxation rate may be similar for all limestone with a value of about 0.3 µ m / s. It is limite in range to the theoretical limit of 0.6 µ m / s. 5) Ramakrishnan et al. [8] suggeste that ρ might be changing to account for the shifts to longer time with increase temerature that were observe in logs an in the lab. This aears to be the case an the change is reominantly through the reuction of the strength of ρ. This concetion of the mechanism rovies a reasonable hysical exlanation that is consistent with other observations for sanstone an carbonate rock. ACKNOWLEDGEMENTS The extensive temerature eenence stuy carrie out by Dr. Emun Forham an Mr. Mark Flaum, which was artially reorte [7], was funamental to this smaller, comlementary effort. Mr. Mark Flaum was a great hel in selecting the samles an roviing their backgroun ata. Discussions with Dr. Seungoh Ryu were very instructive.

9 SCA /12 BIBLIOGRAPHY 1. Bloembergen, N., Purcell, E. M. an Poun, R. V., Relaxation Effects in Nuclear Magnetic Resonance Absortion, Phy. Rev., 1948, Vol. 73, Brownstein, K. R. an Tarr, C. E., Imortance of Classical Diffusion in NMR Stuies of Water in Biological Cells, Phys. Rev. A, 1979, Vol. 19, Foley, I., Farooqui, S.A. an Kleinberg, R.L., Effect of Paramagnetic Ions on NMR Relaxation of Fluis at Soli Surface, J. Mag. Res. A, 1996, Vol. 123, Goefroy, S., Fleury, M., Deflanre, F. an Korb, J.-P., Temerature Effect on NMR Surface Relaxation, 2001,SPE Kleinberg, R.L., Kenyon, W.E. an Mitra, P.P. Mechanism of NMR Relaxation of Fluis in Rock, J. Mag. Res. A, 1994, Vol. 10, Korringa, J., Seevers, D.O. an Torrey, H.C., Theory of Sin Puming an Relaxation in Systems with a Low Concentration of Electron Sin Resonance Centers, Phy. Rev., 1962, Vol. 127, Num. 4, Latour, L.L., Kleinberg, R.L. an Sezginer, A., Nuclear Magnetic Resonance Proerties of Rocks at Elevate Temeratures, J. Col. an Int. Sci., 1992, Vol. 150, Num. 2, Ramakrishnan, T.S., Forham, E. J., Venkataramanan, L., Flaum, M., Schwartz, L.M., New Interretation Methoology base on Forwar Moels for Magnetic Resonance in Carbonates, Transactions of SPWLA 40 th Annual Logging Symosium, 1999, Paer MMM. 9. Simson, J.H. an Carr, H.Y., Diffusion an Nuclear Sin Relaxation in Water, 1958, The Physical Review, Vol. 111, No Woessner, D. E., An NMR Investigation into the Range of the Surface Effect on the Rotation of Water Molecules, J. Mag. Res., 1968, Vol. 39, NOMENCLATURE D N N m N n S S g iffusion constant number of water molecules in samle number of water molecules in shere of influence of aramagnetic site number of water molecules in shere of influence of non-magnetic site samle surface area secific surface area

10 SCA /12 T 1,r T 1,e T 1,n longituinal NMR relaxation time in a rock matrix T 1 relaxation in the shere of influence of a surface aramagnetic site T 1 relaxation in the shere of influence of a surface non-magnetic site T 2,obs measure value of transverse NMR relaxation time T 2, transverse NMR relaxation time ue to surface aramagnetic sites T 2, transverse NMR relaxation time ue to surface iole-iole relaxation T 2,b transverse NMR relaxation time ue for the bulk flui T 2,corr transverse NMR relaxation time correcte to remove bulk water relaxation V samle ore volume φ fractional orosity ρ ρ ρ g transverse surface relaxation rate ue to aramagnetic sites transverse surface relaxation rate ue to surface iole-iole relaxation ensity Table 1. The measure roerties for the two carbonates use in this stuy. Samle φ fractional ensity g / cc Surface area m 2 /g T 2,HOD s T 2,H2O s S L Table 2. Surface relaxation times an rates for the two rocks. Samle ρ ρ T, s 2 T 2, s µ m / s µ m / s S L

11 SCA /12 Figure 1. T 2 -istributions for Samle S at various temeratures. The ash-ot curve is for 30 o C, the soli curve is at 70 o C, the otte curve is at 100 o C, an the ashe curve is for 130 o C. This ata is from revious a stuy [8]. Figure 2. T 2 -istributions for samle L at various temeratures. The ash-ot curve is for 30 o C, the soli curve is at 50 o C, the otte curve is at 100 o C, an the ashe curve is for 130 o C. This ata is from a revious stuy [8].

12 SCA /12 Figure 3. T 2 -istributions are given for samle S at 100% an 10% H 2 O concentration. If the relaxation were urely aramagnetic, the relaxation of both samles shoul be the same. The soli curve is the istribution for the samle saturate with ure water an the ashe curve is the HOD saturate measurement. Figure 4. T 2 -istributions for samle L at 100% an 10% H 2 O concentration. If the relaxation were urely aramagnetic, the relaxation of both samles shoul have the same istribution of relaxation times. The soli curve is for the water saturate measurement an the ashe curve is for the HOD/H 2 O saturate measurement. The shift of the moe of the istribution is comensate by some variation of the etails an the change in mean log T 2 is small. See Table 2.