CPMG relaxation by diffusion with constant magnetic field gradient in a restricted geometry: numerical simulation and application

Carr-Purcell-Meiboom-Gill (CPMG) measurements are the primary nuclear magnetic resonance (NMR) technique used for evaluating formation properties and reservoir fluid properties in the well logging industry and laboratory sample analysis. The estimation of bulk volume irreducible (BVI), permeability, and fluid type relies on the accurate interpretation of the spin-spin relaxation time (T(2)) distribution. The interpretation is complicated when spin's self-diffusion in an inhomogeneous field and restricted geometry becomes dominant. The combined effects of field gradient, diffusion, and a restricted geometry are not easily evaluated analytically. We used a numerical method to evaluate the dependence of the free and restricted diffusion on the system parameters in the absence of surface relaxation, which usually can be neglected for the non-wetting fluids (e.g., oil or gas). The parameter space that defines the relaxation process is reduced to two dimensionless groups: D* and tau*. Three relaxation regimes: free diffusion, localization, and motionally averaging regimes are identified in the (log(10)D*, log(10)tau*) domain. The hypothesis that the normalized magnetization, M*, relaxes as a single exponential with a constant dimensionless relaxation time T*(2) is justified for most regions of the parameter space. The numerical simulation results are compared with the analytical solutions from the contour plots of T*(2). The locations of the boundaries between different relaxation regimes, derived from equalizing length scales, are challenged by observed discrepancies between numerical and analytical solutions. After adjustment of boundaries by equalizing T*(2), numerical simulation result and analytical solution match each other for every relaxation regime. The parameters, fluid diffusivity and pore length, can be estimated from analytical solutions in the free diffusion and motionally averaging regimes, respectively. Estimation of the parameters near the boundaries of the regimes may require numerical simulation.     

Characterization of partially sintered ceramic powder compacts using fluorinated gas NMR imaging.

We use nuclear magnetic resonance (NMR) imaging of C2F6 gas to characterize porosity, mean pore size, and permeability of partially sintered ceramic (Y-TZP Yttria-stabilized tetragonal-zirconia polycrystal) samples. Conventional measurements of these parameters gave porosity values from 0.18 to 0.4, mean pore sizes from 10 nm to 40 nm, and permeability from 4 nm(2) to 25 nm(2). The NMR methods are based on relaxation time measurements (T(1)) and the time dependent diffusion coefficient D(Delta). The relaxation time of C2F6 gas is longer in pores than in bulk gas and it increases as the pore sizes decrease. NMR yielded accurate porosity values after correcting for surface adsorption effects. A model for T(1) dependence on pore size that accounts for collisions between gas molecules and walls as well as surface adsorption effects is proposed. The model fits the experimental data well. Finally, the long time limit of D(Delta)/D(o), where D(o) is the bulk gas diffusion coefficient is useful for measuring tortuosity, while the short time limit was not achieved experimentally and could not be used for calculating surface-area to volume (S/V) ratios.     

基于蒙特卡罗算法的低场核磁共振测量效率提高方法

核磁共振（NMR）的纵向弛豫时间（T₁）、横向弛豫时间（T₂）、自扩散系数（D₀）,以及T₂-T₁、T₂-D₀测量目前广泛应用于石油测井行业.在测量D₀的SGSE序列中,通过逐渐增大90°和180°脉冲之间的时间间隔（T_d）,可以对液体扩散行为产生的影响进行调节.然而T_d的"起点"、"步进数"和"终点"等参数必须设置得当才能准确测量T₁和D₀.目前参数的设置依赖多次的人工调整和测量人员的经验,耗时且使用门槛较高.本文用蒙特卡罗方法进行大量随机模拟,根据前面若干点的测量结果筛选出满足要求的随机值,预测下一个测量点的位置.该算法可以实时更新参数设置,实现自动化测量,达到降低测量门槛、缩短测量时间的目的.经验证,该算法可以适用于T₁、D₀的测量.     

Amide proton relaxation measurements employing a highly deuterated protein

Proton NMR longitudinal and transverse relaxation rates of unlabelled proteins are generally dominated by the many 1H-1H dipolar interactions so that spin diffusion, rather than molecular or internal motions, governs longitudinal relaxation. Here, relaxation measurements of backbone amide proton (1H(N)) magnetisations have been carried out employing the 99% 2H, 98% 15N labelled, small 2F2 protein domain in 10%/90% H(2)O/D(2)O solution. Under these conditions, the longitudinal relaxation rates exhibit time constants, T(1)*=1/R(1)* if described by a mono-exponential, within the range of 3.0 to 18.7s-a wide range which indicates that the phenomenon of spin diffusion has been greatly reduced. The majority of 1H(N) nuclei in this sample (pH 4.0 and 5 degrees C) exhibit chemical exchange with solvent that couples their longitudinal relaxation to that of the solvent. For the subset of 1H(N) nuclei not undergoing detectable solvent chemical exchange, the R(1)* rates correlate well with their individual 1H(N,O)/2H(N,O) structural environments. The correlation for corresponding transverse relaxation rates, R(2)* was found to be less good. Longitudinal relaxation measurements in 1%/99% H(2)O/D(2)O solution identify a further subset of 1H(N) nuclei which exhibit essentially indistinguishable R(1)* rates in both 1% and 10% H(2)O, implying that averaging of rates from spin diffusion processes and different 2F2 isotopomer populations are negligible for these 1H(N) sites. In addition to a high sensitivity to structural parameters, model calculations predict 1H(N) relaxation rates to exhibit pronounced sensitivity to internal dynamics.     

Diffusion and relaxation effects in general stray field NMR experiments

We analyze the evolution of magnetization following any series of radiofrequency pulses in strongly inhomogeneous fields, with particular attention to diffusion and relaxation effects. When the inhomogeneity of the static magnetic field approaches or exceeds the strength of the RF field, the magnetization has contributions from different coherence pathways. The diffusion or relaxation induced decay of the signal amplitude is in general nonexponential, even if the sample has single relaxation times T(1), T(2) and a single diffusion coefficient D. In addition, the shape of the echo depends on diffusion and relaxation. It is possible to separate contributions from different coherence pathways by phase cycling of the RF pulses. The general analysis is tested on stray field measurements using two different pulse sequences. We find excellent agreement between measurements and calculations. The inversion recovery sequence is used to study the relaxation effects. We demonstrate two different approaches of data analysis to extract the relaxation time T(1). Finite pulse width effects on the timing of the echo formation are also studied. Diffusion effects are analyzed using the Carr--Purcell--Meiboom--Gill sequence. In a stray field of a constant gradient g, we find that unrestricted diffusion leads to nonexponential signal decay versus echo number N, but within experimental error the diffusion attenuation is still only a function of g(2)Dt(3)(E)N, where t(E) is the echo spacing.     

Two-dimensional nuclear magnetic resonance petrophysics

Two-dimensional nuclear magnetic resonance (2D NMR) opens a wide area for exploration in petrophysics and has significant impact to petroleum logging technology. When there are multiple fluids with different diffusion coefficients saturated in a porous medium, this information can be extracted and clearly delineated from CPMG measurements of such a system either using regular pulsing sequences or modified two window sequences. The 2D NMR plot with independent variables of T2 relaxation time and diffusion coefficient allows clear separation of oil and water signals in the rocks. This 2D concept can be extended to general studies of fluid-saturated porous media involving other combinations of two or more independent variables, such as chemical shift and T1/T2 relaxation time (reflecting pore size), proton population and diffusion contrast, etc.     

Electron spin relaxation in pseudo-Jahn-Teller low-symmetry Cu(II) complexes in diaqua(L-aspartate)Zn(II).H(2)O crystals.

Low-temperature (4-55 K) pulsed EPR measurements were performed with the magnetic field directed along the z-axis of the g-factor of the low-symmetry octahedral complex [(63)Cu(L-aspartate)(2)(H2O)2] undergoing dynamic Jahn-Teller effect in diaqua(L-aspartate)Zn(II) hydrate single crystals. Spin-lattice relaxation time T(1) and phase memory time T(M) were determined by the electron spin echo (ESE) method. The relaxation rate 1/T(1) increases strongly over 5 decades in the temperature range 4-55 K. Various processes and mechanisms of T(1)-relaxation are discussed, and it is shown that the relaxation is governed mainly by Raman relaxation processes with the Debye temperature Theta(D)=204 K, with a detectable contribution from disorder in the doped Cu(2+) ions system below 12 K. An analytical approximation of the transport integral I(8) is given in temperature range T=0.025-10Theta(D) and applied for computer fitting procedures. Since the Jahn-Teller distorted configurations differ strongly in energy (delta(12)=240 cm(-1)), there is no influence of the classical vibronic dynamics mechanism on T(1). Dephasing of the ESE (phase relaxation) is governed by instantaneous diffusion and spectral diffusion below 20 K with resulting rigid lattice value 1/T(0)(M)=1.88 MHz. Above this temperature the relaxation rate 1/T(M) increases upon heating due to two mechanisms. The first is the phonon-controlled excitation to the first excited vibronic level of energy Delta=243 cm(-1), with subsequent tunneling to the neighbor potential well. This vibronic-type dynamics also produces a temperature-dependent broadening of lines in the ESEEM spectra. The second mechanism is produced by the spin-lattice relaxation. The increase in T(M) is described in terms of the spin packets forming inhomogeneously broadened EPR lines.     

A new approach for simultaneous measurement of ADC and T2 from echoes generated via multiple coherence transfer pathways

The study of rotational and translational diffusion requires the measurement of both T2 and apparent diffusion coefficient (ADC), quantities that are typically measured in separate experiments. The exploitation of echoes generated via multiple coherence transfer pathways offers an opportunity for measuring T2 and ADC values simultaneously in a single experiment. A series of RF pulses can generate multiple echoes via different coherence pathways with each one being uniquely encoded. Here, we demonstrate one pulse sequence that uses an initial theta; RF pulse to generate three coherence orders (C = 0, -1, +1). In the particular version of the method discussed here only two are used (C = 0, +1). Each order is encoded with a different b value from which the ADC is derived. The coherence order echo C = 0 is refocused to quantify T2. The performance of the method--dubbed simultaneous measurement of ADC and relaxation time (SMART)--is demonstrated on a set of samples differing in T2 and ADC achieved by varying the relative volume fractions in mixtures of gadolinium-doped H2O and D2O. The regional SMART derived T2 and ADC agree well with those obtained with conventional double-spin-echo and pulsed gradient spin-echo methods.     

微流燃料电池空气阴极物质传输孔隙尺度模拟

采用格子Boltzmann方法研究了微流燃料电池空气阴极多孔扩散层内多组分物质传输特性。随机重构了扩散层,获得渗透率及有效扩散系数。建立了耦合边界电化学反应的二维模型,研究了过电位、孔隙率对氧气、水蒸气浓度分布及局部反应速率的影响。结果表明,常用的Bruggeman经验关联式会高估氧气有效扩散系数;扩散层孔隙结构对物质传输有重要影响,孔隙率减小使得传质阻力增大,导致局部氧气浓度降低,局部反应速率降低,而水蒸气浓度增大,当孔隙率从0.83降至0.7,催化界面平均氧气浓度从8.472降至8.466 mol·m^-3。     

Spin diffusion in multiple pulse spin-locking in solids containing paramagnetic impurities

Spin diffusion and spin-lattice relaxation in solids containing paramagnetic impurities under influence of a multiple-pulse spin-locking radio-frequency sequence are studied theoretically and experimentally. The diffusion equation obtained provides a clue for determination of the time dependent magnetization. The spin-lattice relaxation time is calculated as a function of the correlation time and multiple-pulse field parameters. From the experimental data the spin diffusion coefficient, the radius of the spin diffusion barrier, and the correlation time for very slow molecular motion in polycrystalline (C(2)F)(n) system are estimated and found to be D～7.1×10(-12)cm(2)/s, r(c)～4.8×10(-10)m, and τ(c)～10.2μs, respectively.     

Diffusion-relaxation correlation in simple pore structures

The effects of independent encoding for relaxation and for diffusion using separate time and gradient dimensions are calculated for spins diffusing in plane parallel and spherical pores with relaxing walls. Two-dimensional inverse Laplace transformation is used to obtain computed (D,T(2)) maps for both geometries, in the regime in which the dimensionless diffusion coefficient is less than unity and the dimensionless relaxation parameter of order unity or greater. It is shown that there exist two distinct branches on the (D,T(2)) maps, one with diffusion and relaxation strongly correlated and one in which the diffusion coefficients vary widely independently of relaxation.     

A global inversion method for multi-dimensional NMR logging

We describe a general global inversion methodology of multi-dimensional NMR logging for pore fluid typing and quantification in petroleum exploration. Although higher dimensions are theoretically possible, for practical reasons, we limit our discussion of proton density distributions as a function of two (2D) or three (3D) independent variables. The 2D can be diffusion coefficient and T(2) relaxation time (D-T(2)), and the 3D can be diffusion coefficient, T(2), and T(1) relaxation times (D-T(2)-T(1)) of the saturating fluids in rocks. Using the contrast between the diffusion coefficients of fluids (oil and water), the oil and water phases within the rocks can be clearly identified. This 2D or 3D proton density distribution function can be obtained from either two-window or regular type multiple CPMG echo trains encoded with diffusion, T(1), and T(2) relaxation by varying echo spacing and wait time. From this 2D/3D proton density distribution function, not only the saturations of water and oil can be determined, the viscosity of the oil and the gas-oil ratio can also be estimated based on a previously experimentally determined D-T(2) relationship.     

In vivo NMR of hyperpolarized 3He in the human lung at very low magnetic fields

We present NMR measurements of the diffusion of hyperpolarized 3He in the human lung performed at fields much lower than those of conventional MRI scanners. The measurements were made on standing subjects using homebuilt apparatus operating at 3mT. O(2)-limited transverse relaxation (T(2) up to 15-35s) could be measured in vivo. Accurate global diffusion measurements have been performed in vivo and in a plastic bag; the average apparent diffusion coefficient (ADC) in vivo was 14.2+/-0.6mm(2)/s, whereas the diffusion coefficient in the bag (3He diluted in N(2)) was 79.5+/-1mm(2)/s. 1D ADC mapping with high SNR ( approximately 200-300) demonstrates the real possibility of performing quality lung imaging at extremely low fields.     

Fast and simultaneous measurement of longitudinal and transverse NMR relaxation times in a single continuous wave free precession experiment

The purpose of this communication is to describe a method for rapid and simultaneous determination of longitudinal (T1) and transversel (T2) relaxation times, based on a single continuous wave free precession (CWFP) experiment which employs RF pulses with a pi/2 flip angle. We analyze several examples, involving nuclei such as 1H, 31P, and 19F, where good agreement with T1 and T2 measurements obtained by traditional methods is apparent. We also compare with the more time-consuming steady-state free precession (SSFP) method of Kronenbitter and Schwenk where several experiments are needed to determine the optimum flip angle. The role of an inhomogeneous magnetic field on the observed decays and its effect upon the accuracy of relaxation times obtained by these methods is examined by comparing numerical simulations with experimental data. Possible sources of error and conditions to minimize its effects are described.     

Longitudinal relaxation of initially straight flexible and stiff polymers

The relaxation mechanism of an initially straight flexible or stiff polymer chain of length N in a viscous solvent is studied through Brownian dynamics simulations covering a broad range of time scales. After the short-time free diffusion, the chain's longitudinal reduction R2(0)-R2 approximately Nt1/2 at early intermediate times is shown to constitute a universal behavior for any chain stiffness caused by a quasisteady T approximately Nt(-1/2) relaxation of tensions associated with the deforming action of the Brownian forces. Stiff chains with a persistence length E > or = N are shown to exhibit a late intermediate-time longitudinal reduction R2(0)-R2 approximately N2E(-3/4)t1/4 associated with a T approximately N2E(-3/4)t(-3/4) relaxation of tensions affected by the deforming Brownian and the restoring bending forces.     

Field gradient CPMG applied on postmortem muscles

As a new approach, Carr-Purcell-Meiboom-Gill (CPMG) experiments were performed in vitro on porcine muscles (n = 10) during the period from 15 min to 85 min postmortem and again at 24 h postmortem in the absence (G = 0) and the presence of an external field gradient (G = 0.5*10(-3) T/m), which was applied throughout the CPMG sequence. The experiments were performed on low-field nuclear magnetic resonance (NMR) equipment (0.47 T). Due to the inclusion of different pre-slaughter treatments (adrenaline treatment and pre-slaughter exercise/electrical stunning), the muscles could be divided into (I) a group (n = 5) characterized by a reduced decrease in pH postmortem and a high water-holding capacity and (II) a group (n = 5) characterized by an increased rate of pH decrease postmortem and a low water-holding capacity. Distributed analysis of the CPMG data revealed two major relaxation populations with relaxation times about 30-40 and 200-500 ms, respectively, and comparison of data obtained with G = 0 and G = 0.5*10(-3) T/m revealed effects of the external gradient on the relaxation time of both the two relaxation populations, which implies that both diffusion and relaxation contributes to the relaxation of the two populations. At 24 h postmortem the effect of the external field gradient on the relaxation time was significantly affected by muscle group (I vs. II), which reveals local differences in water diffusion in the two meat qualities. Finally, the discriminatory power with regard to muscle group (I vs. II) was investigated for data acquired with G = 0 and G = F = 0.5*10(-3) T/m, and both the two types of data were found highly suitable for separation of muscles according to meat quality.     

Surface-induced order and diffusion in 5CB liquid crystal confined to porous glass.

Liquid crystals confined into small cavities are known to have a weak orientational order even above the nematic-isotropic transition temperature. The surface-induced order and molecular dynamics in this temperature range are studied with the aid of deuteron NMR spectra, spin relaxation times T(1) and T(2,) proton dipolar-correlation effect, and direct measurements of the effective diffusion coefficient for the liquid crystal 5CB confined to controlled-pore glasses. Our results show that an arrangement of molecules parallel to the wall is induced by local molecular interactions between the liquid crystal and solid, resulting in a weak and temperature independent surface order parameter, S(0) approximately 0.02 +/- 0.01. There is no indication of a significant slowing-down of molecular diffusion at the wall, neither rotational nor translational. In cavities of nanometer size, where the nematic order evolves gradually upon cooling, a broadening of the NMR linewidths due to dynamic effects should be taken into account.     

Cesium adsorption in hydrated iron oxide particles suspensions: an NMR study

137Cs is an important component of nuclear waste which may pollute water. Its migration in natural environments is slowed down by adsorption on minerals. Cesium adsorption on akaganeite (beta-FeOOH) particles, dextran-coated ferrihydrite (5 Fe(2)O(3)-9H(2)O) particles, and ferritin in aqueous solutions is studied with (133)Cs nuclear magnetic resonance measurements. The longitudinal relaxation time (T(1)) of (133)Cs in the presence of such magnetic particles depends on whether the ions bind to the particle or not. T(1) of (133)Cs ions in aqueous solutions containing the same amount of magnetized particles will not depend on cesium concentration if relaxation is governed by diffusion (when cesium is not able to bind), but it will depend on cesium concentration if exchange governs relaxation (when cesium is able to bind). The method is successfully tested using TEMPO, a nitroxide stable free radical whose relaxation is due to diffusion. (133)Cs relaxation in solutions of ferritin, akaganeite, and dextran-coated ferrihydrite particles is found to result from a cationic exchange of cesium ions between particles surface and bulk ions, owing to adsorption. The effect of pH on (133)Cs relaxation in solutions of the particles is consistent with the adsorption properties of cations on hydrated iron oxides.     

Diffusion and spatially resolved NMR in Berea and Venezuelan oil reservoir rocks.

Conventional and spatially resolved proton NMR and relaxation measurements are used in order to study the molecular motions and the equilibrium and nonequilibrium diffusion of oils in Berea sandstone and Venezuelan reservoir rocks. In the water-saturated Berea a single line with T*2 congruent to 150 microseconds is observed, while the relaxation recovery is multiexponential. In an oil reservoir rock (Ful 13) a single narrow line is present while a distribution of relaxation rates is evidenced from the recovery plots. On the contrary, in the Ful 7 sample (extracted at a deeper depth in a different zone) two NMR components are present, with 3.5 and 30 KHz linewidths, and the recovery plot exhibits biexponential law. No echo signal could be reconstructed in the oil reservoir rocks. These findings can be related to the effects in the micropores, where motions at very low frequency can occur in a thin layer. From a comparison of the diffusion constant in water-saturated Berea, D congruent to 5*10(-6) cm2/sec, with the ones in model systems, the average size of the pores is estimated around 40 A. The density profiles at the equilibrium show uniform distribution of oils or of water, and the relaxation rates appear independent from the selected slice. The nonequilibrium diffusion was studied as a function of time in a Berea cylinder with z axis along H0, starting from a thin layer of oil at the base, and detecting the spin density profiles d(z,t) with slice-selection techniques. Simultaneously, the values of T1's were measured locally, and the distribution of the relaxation rates was observed to be present in any slice.(ABSTRACT TRUNCATED AT 250 WORDS)     

Some evidence for the validity of the noise-temperature inequalityθ ≥T in the relaxation approximation of the one-dimensional electron transport problem in high electric fields

The conjecture that “noise” is always smallest in an equilibrium system is made quantitative for a transport problem by identifying “noise” with the noise temperatureθ. In equilibrium the external fieldF=0, and the fluctuation-dissipation theorem gives θ= T, the temperature. In a strong fieldF the Boltzmann equation in the constant relaxation approximation is used to calculate the driftu(F, T) the diffusion constantD(F, T), and the noise temperatureθ(F, T) for piecewise linear one-dimensional band structuresE(k). The validity of the noise inequalityθ ≥T has been shown for a large variety of band parameters and for all fields and temperatures.