Relation and Correlation between NMR Relaxation Times, Diffusion

Article pubs.acs.org/JPCC

Relation and Correlation between NMR Relaxation Times, Diffusion Coefficients, and Viscosity of Heavy Crude Oils Jean-Pierre Korb,*,† Nopparat Vorapalawut,‡ Benjamin Nicot,‡ and Robert G. Bryant§ †

J. Phys. Chem. C 2015.119:24439-24446. Downloaded from pubs.acs.org by DURHAM UNIV on 08/08/18. For personal use only.

Physique de la Matière Condensée, Ecole Polytechnique-Centre National de la Recherche Scientifique (CNRS), 91128 Palaiseau, France ‡ TOTAL EP, Centre Scientifique et Technique Jean Feger (CSTJF), 64018 Pau, France § Department of Chemistry, University of Virginia, Post Office Box 400319, Charlottesville, Virginia 22904-4319, United States ABSTRACT: We present a theory and experiments that relate the NMR longitudinal T1 and transverse T2 relaxation times to the viscosity η for heavy crude oils with different asphaltene concentrations. The nuclear magnetic relaxation equations are based on a one-dimensional (1D) hydrocarbon translational diffusion in a transient porous network of slowly rotating asphaltene macroaggregates containing paramagnetic species VO2+. For heavy crude oils with viscosity η above a certain threshold ηc, the effective 1D confinement causes a transition from the usual Stokes−Einstein relation for the translational diffusion coefficient D ∝ 1/η below ηc to a wetting behavior D ∼ Cte close to the asphaltene aggregates above ηc. The theory is compared successfully with the universal viscosity dependencies of relaxation times T1 and T2 observed over a large range of viscosities. The theory reproduces the relaxation features of the 2D correlation spectra T1−T2 and D−T2 for heavy crude oils when varying the asphaltene concentration. This foundation is important because these measurements can be performed down-hole, thus giving a valuable tool for investigating in situ the molecular dynamics of petroleum fluids. slightly dependent on the Larmor frequency ω0/2π, and (iii) T1 does not depend on the viscosity but depends strongly on the Larmor frequency (T1 ∝ √ω0). There are also anomalous features in the observed T1−T2 and D−T2 correlation plots. For heavy crude oils, the anomalous features may be related to the presence of asphaltene, polynuclear aromatic molecules substituted with alkane chains on the periphery. Asphaltenes have been shown to associate with increasing concentration making complex particles that in turn may aggregate to make macroaggregates. Thus, oils with considerable asphaltene concentrations may have considerable local structure that impacts differently macroscopic properties such as viscosity and microdynamic properties such as molecular reorientation and translation. For instance, at high viscosities T2 becomes significantly shorter than T1, and the translational diffusion coefficient D becomes progressively independent of T2 instead of the usual linear relation D ∝ T2.8,9 Last, the T1−T2 spectrum for short T1 and T2 values presents an upward bent away for T1. We seek a physical and theoretical foundation for understanding these effects. Here, we aim at explaining the universal relation observed between the logarithmic averages ⟨T2,LM⟩ and ⟨T1,LM⟩ and the viscosity η and asphaltene concentration for a large amount of experimental data on various crude oils.3,10 We propose

I. INTRODUCTION Knowing the viscosity of a crude oil as early as possible is vital to the oil industry because it impacts the productivity and the choice of recovery strategies. Nuclear magnetic resonance (NMR) may be used to estimate the viscosity of crude oils, even down-hole, through measurements of nuclear spin−lattice or longitudinal (T1) and transverse (T2) relaxation times provided that the relationship between these time constants and the viscosity is firmly established.1,2 For bulk light oils (low viscosity) the molecular motions of all saturated and aromatic hydrocarbon components are fast enough to average the magnetic dipolar fields of neighboring spin systems (extreme narrowing limit) which results in the usual relation T1 = T2 ∝ 1/η, which is consistent with the standard calculation of relaxation times based on the Stokes−Einstein relation between the translational diffusion coefficient and viscosity.2 However, for bulk heavy crude oils (high viscosity), the molecular motions of asphaltenes, resins, and other high molecular weight structures are not fast enough to average completely the local dipolar fields of neighboring spin systems which results in large distributions of T2 and T1 that reflect the large diversity of molecular sizes. The situation is interesting because the observed viscosity dependencies of the logarithmic averages ⟨T2,LM⟩ and ⟨T1,LM⟩ of these relaxation time distributions follow an universal master curve3 that is different from that given by the well-known BPP relaxation theory.1 In particular, the anomalous nuclear spin relaxation features for heavy crude oil include3−7 (i) T2 is shorter than T1, (ii) T2 ∝ 1/√η and is © 2015 American Chemical Society

Received: August 3, 2015 Revised: September 23, 2015 Published: September 24, 2015 24439

DOI: 10.1021/acs.jpcc.5b07510 J. Phys. Chem. C 2015, 119, 24439−24446

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The Journal of Physical Chemistry C

lated as the cube root of the volume per aggregate ⟨d⟩ = [(cmol (mol/L)NA/1000]−1/3, where NA is the Avogadro number. We have displayed in Figure 1a a schematic diagram representing

relaxation equations for the longitudinal and transverse nuclear relaxation rates constants 1/T1 and 1/T2 of the observed protons of saturated hydrocarbon chains diffusing among the clusters of asphaltenes containing the paramagnetic species VO2+. Consideration of all these experiments suggests a transition for the translational diffusion coefficient from the usual Stokes−Einstein relation D ∝ 1/η which is conserved at low asphaltene concentrations below a threshold viscosity ηc to wetting behavior where the diffusion coefficient does not vary with the viscosity close to the asphaltene aggregates at higher asphaltene concentration above ηc. This model is consistent with the observed behaviors for crude oils in that T2(η) = T1(η) ∝ 1/η for low viscosity, while at high asphaltene concentration and high viscosity T2(η) ∝ 1/√η with a weak Larmor frequency dependence. In addition, T1 is quasi-independent of the viscosity while it is proportional to the square root of the Larmor frequency.3,7 The theory can also reproduce the general relaxation behaviors of the 2D correlation maps T1−T2 and D− T2 for crude oils when varying the asphaltene concentration.8,9 These results provide valuable tools for the down-hole NMR characterization of petroleum fluids.

II. THEORY, DISCUSSION, AND COMPARISON WITH EXPERIMENTS 1. Structural Model for Bulk Crude Oils in the Presence of Asphaltene. The asphaltenes are polar molecules representing a solubility class defined as the npentane insoluble and toluene-soluble fraction of petroleum fluid. Asphaltenes are polynuclear aromatic ring assemblies peripherally substituted with alkyl side chains that incorporate heteroatoms (such as O, N, and S).11 Depending on the type of oil being considered, asphaltene molecules are partly responsible for plugging the pores of oil reservoirs and catalytic networks.12 The tendency of asphaltene to self-aggregate distinguishes them from other oil constituents. Recent X-ray (SAXS) and neutron (SANS) small-angle scattering studies in asphaltene solutions have shown that asphaltenes form discoidal nanoaggregates made of nasph ∼ 2 stacked asphaltene molecules of total radius 3.2 nm with 30% polydispersity and a height of 0.67 nm.13 Eyssautier has also shown that these nanoaggregates form clusters (macroaggregates) of about nnano ∼ 12 nanoaggregates with a fractal structure.14 Mullins et al. have reported that these macroaggregates have gyration radii around Ragg = 2.6 nm in reservoirs.15 The structure of asphaltene macroaggregates in heavy crude oil thus appears as a transient porous network16 that affects the dynamics of other constituents. To characterize such a transient porous structure with the goal of interpreting the viscosity dependence of nuclear spin-relaxation times, we consider a bulk crude oil provided by Total EP, Pau, France, of density ρoil = 0.85 g/cm3 where the asphaltene concentration can vary in the range 0 ≤ casph ≤ 15% wt. To estimate the average distance ⟨d⟩ between the different asphaltene macroaggregates, we assume that these clusters are composed of nnano ∼ 12 nanoaggregates.14 The molar concentration cmol (mol/L) of asphaltene clusters per liter of crude oil thus becomes cmol (mol/L) = casph (wt %, i.e., gsolute/100 gsolution)ρoil (g/cm3) (1000 cm3/L)/[nasphnnanoMwt (g/mol)]. A consensus is forming on the mean molecular weight distribution about Mwt = 750 g/mol for an asphaltene structure corresponding to a fused ring system of 7 benzene rings per petroleum asphaltene molecule including a small number of aliphatic chains.15 These assumptions yield an average distance between asphaltene macroaggregates, calcu-

Figure 1. (a) Schematic diagram representing the transient porous network of asphaltene macroaggregates characterized by a radius Ragg and a rotational correlation time τrot. The paramagnetic VO2+ ions are indicated schematically within the macroaggregates. The tortuous dashed line represents the quasi-1D translational diffusion of hydrocarbons characterized by a molecular size Rmol and a correlation time τ1D. (b) Calculated variation of the average distance ⟨d⟩ of separation between these macroaggregates with the asphaltene concentration casph (wt %). In the inset, we have plotted the calculated variation of the effective porosity with casph (wt %).

the transient porous structure of these asphaltene macroaggregates in heavy crude oils. We show in Figure 1b that ⟨d⟩ ∝ (casph)−1/3 and decreases from 32.8 to 5.8 nm when the asphaltene concentration casph increases from 0.1 to 18 wt %. For instance, we find that ⟨d⟩ = 7.3 nm for our studied native crude oil casph = 9 wt %. We note that this distance is only sligthly larger than the diameter (2Ragg = 5.2 nm) of the macroaggregates. This calculation thus justifies our model of a transient porous network for the crude oil with asphaltene (Figure 1a). An effective porosity can thus be introduced similarly to the case of a granular packing of nonporous grains:17 Φ (%) = 100[⟨d⟩/(2Ragg)]3/{1 + [⟨d⟩/(2Ragg)]3}. The variation of Φ with casph is displayed in the inset of Figure 1b. We note that this effective porosity takes the values 58% ≤ Φ ≤ 24440


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The Journal of Physical Chemistry C 100% in the concentration range generally observed 0.1 ≤ casph ≤ 18 wt %. Of course, the value Φ = 100% corresponds to the case of a pure maltene without asphaltene and Φ = 58% corresponds to the highest asphaltene concentration casph ∼ 18 wt % encountered for heavy native crude oils. For instance, we find Φ = 73% in the case of a native crude oil with casph = 9 wt %. 2. Dynamical Model for Hydrocarbons Diffusing in Bulk Crude Oils in the Presence of Asphaltene. Semiclassical treatments are always needed for relating the measured macroscopic relaxation times T1 and T2 to the molecular correlation times.2 For a crude oil of viscosity η and presuming an isotropic environment and approximately spherical molecular species, the rotational τrot and translational τ1D correlation times can be inversely proportional to the rotational diffusion of asphaltene macroaggregates of radius Ragg and the translational diffusion of hydrocarbon molecules of radius Rmol estimated from the Stokes−Einstein relations τrot = 4πR agg 3η /(3kBT )

(1a)

τ1D = 12πR mol 3η /(kBT )

(1b)

equilibrium function, u(x) = tanh(x/(2a)1/2), where x denotes a locally defined normal coordinate to the interface and √a corresponds to the length of the transition regions between the domains. As expected, this function tends to u = ±1 for sufficiently large x compared to √a indicating separated domains. In the equilibrium situation of Figure 1a, this analogy leads to ⟨d⟩ ∼ √a. One thus notes that the thickness of the layer between the macroaggregates and the hydrocarbon phases becomes exponentially thin when the asphaltene concentration increases. Assuming that the viscosity increases with the asphaltene concentration as shown recently,21 the rotational dynamics of the macroaggregates becomes increasingly slow. Similarly, the average distance ⟨d⟩ as well as the concentration layer thickness √a becomes very small because ⟨d⟩ ∝ casph−1/3. As shown above, we have an exponential concentration profile u(x) in the proximity of the separated phases. One can thus assume a similar form for a viscosity gradient on a limited range of distance x from the surface of the macroaggregates that will influence the translational dynamics of hydrocarbons. For these reasons, we introduce the following a priori exponential form: τ1D(η) = [12πR mol 3ηc /(kBT )][1 − exp( −η /ηc)]

For a native crude oil of viscosity η = 40 cP, we find at room temperature τrot ∼ 0.72 μs with Ragg = 2.6 nm and τ1D ∼ 21.3 ns for Rmol = 0.39 nm corresponding to C8−C12 hydrocarbon chain lengths. The fact that τ1D ≪ τrot justifies the assumption of a very slow rotation of the asphaltene macroaggregates in comparison to the relatively fast translational dynamics of the hydrocarbons. For an asphaltene concentration of casph = 9 wt %, we found that the distance between macroaggregates to be approximately ⟨d⟩ = 7.3 nm, which gives a very small pore space about 2 nm ensuring a quasi-1D translational diffusion for most of hydrocarbon distribution within the low porosity of the transient porous network of quasi immobile asphaltene macroaggregates (Figure 1a). The situation is similar to the case of a percolation network in disordered porous media. A similar single file diffusion situation has been considered for solvent-added processes associated with heavy oil production.18 In this confinement, eq 1a can still be fulfilled for a quasispherical macroaggregate. However, eq 1b becomes doubtful for τ1D when the saturated hydrocarbons have molecular sizes comparable to the average pore sizes of the transient porous network ⟨d⟩ (Figure 1a), and the hydrocarbon dynamics becomes strongly dependent on the confinement as well as the interaction between hydrocarbons and asphaltene. To justify a functional form for τ1D in such confinement, we consider an analogy between the equilibrium structure achieved with the approximately binary fluid of quasi-immobile asphaltene macroaggregates and hydrocarbons with the case of phase separation by which the two components of a diffusive binary fluid spontaneously separate and form domains pure in each component (like in spinodal decomposition). Previous results obtained by cryo-scanning electron microscopy on asphaltene solutions in toluene have shown a kind of heterogeneous system with regions of high and low asphaltenes concentration, suggesting spinodal decomposition behavior.19 It is known that the nonlinear Cahn−Hilliard equation20 describes such a phase separation. At equilibrium and very far from the second-order phase transition where the two particles seek their own kind, the Cahn−Hilliard equation can be linearized and the segregation of such binary mixture is characterized by thin transition layers between the segregated domains with a normal concentration profile given by the

(2)

showing that τ1D(η) tends to a constant value that is independent of the viscosity above a threshold viscosity ηc and gives τ1D(η) ∝ η below ηc. In other words, the dynamics of the highly confined hydrocarbons becomes similar to that for a wetting fluid close to a solid surface. For instance, this behavior is encountered in the 2D correlation map of the translational diffusion constant, D−T2, for water in porous sandstone rocks where the observed D ∝ 1/τ1D is independent of T2 over the whole distribution of the T2 values.9,22 The tortuosity of the diffusion in the transient porous medium (Figure 1a) also favors this assumption. The observation of a quasi-logarithmic behavior for the Larmor frequency dependence of the nuclear spin−lattice relaxation rate constant, 1/T1, for crude oils in the presence of various asphaltene concentrations affirms this surface dynamical affinity.23 We have displayed in Figure 2 the viscosity dependencies of τrot and τ1D calculated from eqs 1a

Figure 2. Calculated variations with eqs 1a and 2 of the correlation times τrot and τ1D as a function of the viscosity (cP). We have indicated the threshold viscosity ηc ∼ 300 cP between the usual behaviors τ1D ≪ τrot ∝ η and the oil-wetting relation τ1D ∼ Cte. 24441


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The Journal of Physical Chemistry C and 2 on a large range 1 ≤ η ≤ 106 cP, where we choose a threshold viscosity ηc ∼ 300 cP. The validity of eq 2 will be demonstrated by comparison with the viscosity dependencies of the observed relaxation times T1(η) and T2(η) of various crude oils (see section II.4).3,7 3. Nuclear Magnetic Relaxation Equations. From the basic properties of the nuclear diffusion-relaxation model in a low dimensional system24 with paramagnetic VO2+ (S = 1/2) embedded in the asphaltene nanoaggregates, the pairwise dipolar correlation function G1D(τ) describing the quasi-1D translational diffusion of these proton−hydrocarbon species (I) in between the slowly rotating asphaltene macroaggregates (Figure 1a) may be written as9,16 G1D(τ ) ∝

e−|τ| / τrot τ /τ1D

magnetization to the liquid−proton magnetization at equilibrium.26 ρoil is the crude oil density, and ⟨δIS⟩ represents the average distance of minimal approach between I and S spins in the local geometry of Figure 1a. We show in Figure 3 that the theoretical expressions of T2 and T1 obtained only with eqs 5 and 6 are mainly dependent on

(3)

Equation 3 is valid at long times when τ ≫ τ1D. Here τ1D depends on the viscosity through eq 2. It represents the hydrocarbon translational correlation time for the local 1D translational diffusion. τrot (≫τ1D), which depends on the viscosity through eq 1a, and represents the rotational correlation time of the macroaggregates necessary to lose all the pairwise dipolar correlations between the I and S spins at long times (Figure 1a). To normalize eq 3 when τ < τ1D, one can introduce the form G1D(τ) ∝ e−|τ|/τrot/(1 + τ/τ1D)1/2 giving G1D(0) = 1. However, this latter form gives a difference in the frequency dependence of the spectral density only in the high Larmor frequency range not studied here. The spectral density J1D(ω) is obtained from the Fourier transform of G1D(τ) given in eq 3 as J1D(ω) =

2π τ1Dτrot

1+

Figure 3. Typical examples of theoretical relaxation times T1 and T2 calculated only with eqs 5 and 6 at 2, 23, and 80 MHz as a function of τrot (s) for a given value of τ1D = 3 ns. The dashed and continuous lines are for T1 and T2 dependencies at the three studied frequencies. The different parameters used in these relaxation equations are described in the text, and we remind the range τrot ≫ τ1D for the validity of the theory.

1 + ω 2τrot 2

1 + ω 2τrot 2

(4)

τrot while τ1D, which is fixed here (τ1D = 3 ns), induces only a scaling effect. In the proton frequency range studied (2 ≤ ωI/ 2π ≤ 80 MHz) and for τrot ∼ 0.72 × 10−6 s, one has 11 ≤ ωIτrot ≤ 362 and the electronic contributions varying with ωs (∼659ωI) in eqs 5 and 6 are almost negligible. One notes that T2 ∝ 1/(τ1Dτrot)1/2 in almost the whole range of τrot studied and T1 ∝ (ωI/τ1D)1/2 for large values of τrot (≫τ1D). These behaviors are compatible with the slight frequency dependence of T2 and the strong frequency dependence of T1 observed in the experiments.3,7,27 However, one sees in Figure 3 that eqs 5 and 6 do not exhibit the observed viscosity dependencies of T1 and T2 for large viscosities.3,7 For instance, when substituting eqs 1a and 1b in eqs 5 and 6, one has T2 = T1 ∝ 1/(τ1Dτrot)1/2 ∝ 1/η as observed at low viscosities, but at large viscosities we find that T2 ∝ 1/(τ1Dτrot)1/2 ∝ 1/η instead of the observed dependence ∝1/√η and T1 ∝ (ωI/τ1D)1/2 ∝ 1/√η instead of the observed constant values.3,7 We show in section II.4 how to modify these equations for reproducing the correct viscosity dependence of T1 and T2. Chen et al.27 introduced equations similar to eqs 5 and 6, aside from the dimensioned prefactor A that depends on the samples, based on somewhat different physical reasoning and the so-called “porous asphaltene model”.16 Though their equations exhibit the strong desired frequency behavior for T1 ∝ √ωI and the slight one for T2 in the limit of large frequency, we saw above that these equations do not predict the correct viscosity dependencies for large viscosities. The main difference with our model schematically displayed in Figure 1a is that they assumed that the non-asphaltene

For the electron−nuclear (I−S) dipolar relaxation process mediated by the 1D translational diffusion, 1/T2 and 1/T1 are proportional to the following linear combinations of the spectral densities at the nuclear (ωI) and electronic (ωS = 659ωI) Larmor frequencies: ⎡ ⎤ 1 3 13 = 2A⎢J(0) + J(ωI ) + J(ωS)⎥ ⎣ ⎦ T2(ωI ) 4 4 ⎡ ⎤ 3 2 13 2 = 4 π A τ1Dτrot ⎢1 + F(ωI , τrot) + F(ωS , τrot)⎥ ⎣ ⎦ 8 8 (5)

1 = A[3J(ωI ) + 7J(ωS)] T1(ωI ) =

2π A τ1Dτrot [3F(ωI , τrot) + 7F(ωS , τrot)]

(6)

Utilizing a rapid exchange condition, one can add to eqs 5 and 6 the contribution of the frequency independent bulk relaxations that is almost negligible in most cases. In eqs 5 and 6, one has A= σSSP,NMRρoil(γ1γSℏ)2⟨δIS−3⟩S(S + 1) and F(ωI, τrot) = (1 + (1 + ωI2τrot2)1/2)1/2/(1 + ωI2τrot2)1/2. σS is the surface density of paramagnetic species of spin S = 1/2 which has been obtained from calibrated electronic spin resonance measurements (ESR).9 SP,NMR = SPF is a NMR-based specific surface area25 of the asphaltene aggregates that appears to be proportional to the true specific surface area SP of asphaltene and the ratio F = msol,eq/mliq,eq ≪ 1 of the solid−proton 24442


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The Journal of Physical Chemistry C hydrocarbon components in the oil (maltenes) undergo correlated one-dimensional diffusion while they are entangled within the porous asphaltene nanoaggregates that tumble slowly through the oil.27 This description might be possible when one considers the maltene diffusion within the asphaltene macroaggregates that present some loosely organized fractal structure of dimensions df ∼ 2.1 for solution of asphaltene in toluene.14 Such a situation is in fact already considered in our model (see Figure 1a). However, the strong π−π interaction between two (or three) related polar hard cores of asphaltene molecules inside a nanoaggregate limits strongly the possibility of a maltene penetration within the asphaltene nanoaggregates. Moreover, the restriction of spatial diffusion of maltenes within asphaltene aggregates would still have to preserve the translational diffusion coefficients, D = 0.4 and 1.5 × 10−10 m2/s, that we found from 2D NMR DOSY experiments (300 MHz) and D−T2 correlation spectrum at 23 MHz (D = 0.6 and 2 × 10−10 m2/s) for the hydrocarbon dynamics in crude oils in the presence of various concentration of asphaltene.21 Both NMR techniques used a diffusion delay Δ = 24 ms that is sufficiently large for the maltenes to explore micrometer rather than nanometer diffusion ranges. This gives arguments in favor of our unbounded one-dimensional diffusion of hydrocarbons in the transient porous network for the crude oil with asphaltenes (Figure 1a). The consideration of the hydrocarbon diffusion within the percolation network of asphaltene macroaggregates (Figure 1a) agrees also with the single file diffusion evidence observed for solvent-added processes associated with heavy oil production.18 In summary, the primary difference in the physical picture underlying the two theoretical approaches is that in the earlier approach a penetration of a porous nanoaggregate particle is assumed to cause the restricted diffusion, while in the present case, the close proximity of the macroaggregates is assumed to create a locally ordered liquid and the primary interactions affecting the one-dimensional hydrocarbon dynamics, in between these macroaggregates, are close to the particle surfaces. 4. Comparison with the Observed Viscosity Dependencies of T1 and T2. It is thus absolutely necessary to introduce the form of τrot(η) and τ1D(η) proposed in eqs 1a and 2 in eqs 5 and 6 for a direct comparison with the viscosity dependence of experimental data of ⟨T2,LM⟩ and ⟨T1,LM⟩ collected at 2.5, 23, and 80 MHz for a large set of different alkanes and crude oils at room temperature.3,7 Though one has an extreme diversity in the experimental oil data, we choose the following parameters for this comparison, on the basis of fundamendal reasons previously introduced in similar situations.21,23 The oil density is ρoil = 0.85 g/cm3. We consider that the distance of minimal approach between I and S spins, ⟨δIS⟩ = 0.655 nm for T1 and 0.855 nm for T2 correspond to the average molecular sizes of a C8−C12 species. We choose ηC = 250 cP for T2 and 160 cP for T1 except for T1 at 2 MHz ηC = 3500 cP, where one clearly sees that the crossover viscosity ηC is enhanced at low frequency. SP,NMR = SPF ∼ 0.86 m2/g when F = 1/100. We have reported previously quantitative calibrated ESR spectra of crude oil and found a surface density of σS = 1.7 × 1013 VO2+/cm2 with spins S = 1/2.9 This value gives an average distance between two S-spins within asphaltene aggregates 1/√σS ∼ 2.55 nm of the order of the average radius of a macroaggregate. Finally, we choose Ragg = 2.6 nm15 and Rmol = 0.39 nm corresponding to C8−C12 chain lengths. The good agreement between experiments and theory on a very large range of viscosity, shown in Figure 4, demonstrates

Figure 4. Experimental and theoretical logarithmic mean relaxation times ( and labeled T1 and T2) as a function of viscosity (cP), obtained at 2, 23, and 80 MHz. Experimental data come from refs 3−7, 10, 21, and 30. The dashed and continuous lines are for T1 and T2 dependencies at the three studied frequencies. Theoretical values have been obtained from eqs 1a, 2, 5, and 6 with the following parameters: Ragg = 2.6 nm, Rmol = 0.39 nm, δIS = 0.855 nm for T2 and 0.655 nm for T1, ηC = 250 cP for T2 and 160 cP for T1 (except at T1 at 2 MHz where ηC = 3500 cP).

the following expected behaviors: T2(η) = T1(η) ∝ 1/η for weak viscosity and T 2(η) ∝ 1/√η and the viscosity independence and strong Larmor frequency dependence T1 = (ω0/τ1D)1/2α √ω0 because τ1D is constant for high viscosity (see Figure 2). This agreement shows that one has relations given by eqs 1a and 2 and eqs 5 and 6 and not only correlations between NMR relaxation times T1 and T2 and the viscosity. The good agreement of the theoretical curves with the experimental dependencies of ⟨T2,LM⟩ and ⟨T1,LM⟩ on the viscosity3,7 (Figure 4) is thus a key signature of the 1D molecular dynamics in between the percolation network of asphaltene macroaggregates in heavy crude oils. 5. Comparison with the Observed 2D Correlation Maps T1−T2 for Crude Oils with Asphaltene. Figure 5 shows an example of our previous T1−T2 data performed at 2.5 MHz for a native crude oil with a concentration of 9 wt % asphaltene and 6 wt % resin.21 We have displayed in Figure 5 the calculated 2D T1−T2 data for different sizes of Ragg, by correlating the T1 and T2 values obtained through eqs 5 and 6 with the parameters listed in section II.3 and the form of τrot(η) and τ1D(η) proposed in eqs 1a and 2, respectively. As expected, when Ragg decreases in the range Ragg ∈ (1.0, 1.5, 2.0, and 2.5 nm), the 2D T1−T2 data become progressively closer to the T1 = T2 line. The best agreement with the experiments is obtained for Ragg = 2.6 nm, justifying again the hypothesis of a 1D translational diffusion of hydrocarbon chains in between the asphaltene macroaggregates (Figure 1a). Basically, the theory reproduces the main relaxation features of the experiments. (i) For the long relaxation times T1 and T2, one has T1 ∼ T2 (∝1/ η) which corresponds to the relaxation induced by the fast reorientational dynamics of small hydrocarbon chains. (ii) For the short relaxation times T1 and T2, one reproduces the 24443


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Figure 5. Experimental and theoretical T1−T2 correlation plots at 2.5 MHz. Experimental data are from ref 21. Theoretical values have been obtained from eqs 5, 6, 1a, and 2 for different radii of the asphaltene macroaggregates Ragg. Both experiments and theory display an upward bent away from the T1 = T2 dashed line at short T1 and T2.

Figure 6. Calculated variations of the translational diffusion coefficient D1D, described in the section II.6, as a function of the viscosity (cP) for different values of the threshold viscosities ηc, from (5 cP) to bottom (50 cP) continuous lines. One clearly observed the transition between the Stokes−Einstein behavior D ∝ 1/η (dashed line) when η < ηc and the regime D ∼ Cte when η ≥ ηc.

systematic upward trend away from the constant line of the T1 ∼ T2 caused by the different viscosity behavior of T2(η) ∝ 1/ √η and the constant values of T1 induced by the slow translational diffusion of the hydrocarbon chains in between the slowly rotating asphaltene macroaggregates. 6. Comparison with the Observed 2D Correlation Maps D−T2 for Crude Oils with Asphaltene. One of the main interests of the 2D NMR spin correlation D−T228 is to probe the molecular dynamics on different length scales. The diffusion editing PGSE sequence probes the translational diffusion in the micrometer range with a diffusion delay of 24 ms, while the CPMG T2 sequence with short interpulse delays explores the molecular dynamics only on the nanometer range. Here, we aim at interpreting the anomalous features of the D− T2 data previously observed in crude oils for various asphaltene concentrations8,9 in terms of our proposed relaxation theory. Because the translational diffusion coefficient is inversely proportional to the translational correlation time D1D ∝ 1/ τ1D, one obtains from eq 2 the following form D1D(η) = (kBT)/ [6πRmolηc[1 − exp(−η/ηc)]] for taking into account the transition from the Stokes−Einstein relation D1D ∝ 1/η to a constant value D1D ≈ Cte above the threshold viscosity ηc. We have displayed in Figure 6 the theoretical variations of D1D(η) with the viscosity when varying the threshold viscosity ηc ∈ {5−50 cP}. Figure 6 shows clearly the transition from the Stokes−Einstein relation D1D ∝ 1/η to a constant value D1D ≈ Cte above ηc for the dynamics of wetting crude oil close to a solid surface. Figure 7 shows the superposition of our previous D−T2 data performed at 2.5 MHz for a crude oil diluted by its maltene at three different asphaltene concentrations ranging between 0 wt % (pure maltene) and 9 wt % (native crude oil).21 When the asphaltene concentration casph increases, one notes a systematic shift of the whole D−T2 data toward the short T2 values away from the usual linear D1D ∝ T2 relation and that the D values level off. This behavior has been observed previously and indicates an enhancement of the surface wettability of the crude oil macroaggregates.29 We show in Figure 8 our experimental data performed at 2.5 and 23 MHz for the 9 wt % asphaltene (native crude oil). One has clearly a

Figure 7. Experimental (from ref 21) and theoretical D−T2 correlation plots obtained at 2.5 MHz. Theoretical values have been obtained from eqs 1a, 2, and 5 and rescaling T2 as described in section II.6 and with the follwing parameters: Rmol = 0.39 nm, δIS = 0.855 nm. One clearly observes the upward bent away of the diffusion coefficient D for short T2,resc values.

better signal-to-noise ratio at 23 MHz and the leveling off of the diffusion coefficient D at short T2 values is clearly observed (Figure 8). We also observe a bimodal distribution of diffusion coefficients for long T2 values confirming our previous NMR DOSY experiment.21 To reproduce the experimental data of Figures 7 and 8, we have correlated the form of D1D(η) given above with a rescaled T2,resc(η,ηc) obtained by multiplying eq 5 by the pseudo-porosity Φ(casph) introduced in section II.1. This rescaling of the T2 values is necessary to take into account the observed shift of the D−T2 data toward shorter T2 values with increasing asphaltene concentration.29 These calculated D− T2,resc data obtained for ηc ∈ {5−80 cP} reproduce the main 24444


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features of the 2D correlation spectra T1−T2 and D−T2 for various asphaltene concentrations, including (i) the systematic upward bent away of T1−T2 data from the T1 = T2 line induced by the slow translational diffusion of the long hydrocarbon chains in between the slowly rotating asphaltene macroaggregates and (ii) the leveling off of the D data with a systematic shift of the whole D−T2 data toward the short T2 values away from the usual linear D1D ∝ T2 relation. In summary, these new foundations give valuable tools for investigating in situ the dynamics of petroleum fluids in a variety of environments.

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*E-mail [email protected], Tel 33 1 69 33 47 39 (J.-P.K.). Notes

The authors declare no competing financial interest.

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Figure 8. Experimental D−T2 correlation maps performed at 2 and 23 MHz for a native crude oil of 9 wt % asphaltene. The red dashed line is the theoretical D−T2,resc plots calculated from eq 5 including eqs 1a and 2 with the same parameters as in Figure 7.

ACKNOWLEDGMENTS J.-P.K. thanks P. Ligneul (Schlumberger, Mems Technology Center, Elancourt, France) for stimulating discussions about Cahn−Hilliard, H. Zhou (Total EP) for stimulating discussions about the structure of asphaltenes, and the scientific direction of Total for financial support.

experimental features: (i) In the low asphaltene concentration domain, one obtains the usual behavior D1D ∝T2∝ 1/η typical of the Stokes−Einstein relation with a threshold viscosity ηc reaching the highest values in our range of viscosity. (ii) When increasing the asphaltene concentration, we note that the calculated data (Figures 7 and 8) reproduce the corresponding experiments as the threshold viscosity ηc decreases. (iii) We note that the diffusion coefficients D as well as the T2 values agree with the experimental values. (iv) The transition from the usual relation D ∝ T2 to the anomalous relation D ∝ √T2 occurs in a very limited range of short T2 values. Even in the experimental data, it is not sure that this transition exists rather than a leveling off for D that better agrees with the observed data for short T2 values (Figure 8). Unfortunately, the short values of T2 < 10 ms are absent in the experimental D−T2 data because of the diffusion editing sequence that is made prior to the CPMG T2 sequence. The good agreement between the aforementioned theory and the D−T2 experiments performed for different asphaltene concentrations is shown more clearly at 23 MHz (Figure 8). This is a very important issue since the D− T2 correlation can be used down-hole allowing an in situ quantitative characterization of the crude oil wettability and asphaltene concentration.

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REFERENCES

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III. CONCLUSION We have proposed a NMR relaxation theory and experiments relating the longitudinal T1 and transverse T2 nuclear magnetic relaxation times to the viscosity η of bulk heavy crude oils with different asphaltene concentrations. The theory is mainly based on the fluctuations of dipole−dipole interactions mediated by the 1D translational diffusion of hydrocarbons in a transient porous network of slowly rotating asphaltene macroaggregates in which the VO2+ paramagnetic ions are located. This theory is fully consistent with the universally observed behaviors for heavy crude oils showing that T2(η) = T1(η) ∝ 1/η for low viscosity and T2(η) ∝ 1/√η with a weak Larmor frequency dependence and viscosity independence but strong Larmor frequency dependence T1 ∝ √ω0 for high viscosity. Key points of this theory are the possibility of reproducing the general 24445


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Relation and Correlation between NMR Relaxation Times, Diffusion

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