R.J.S. Brown

Information available and unavailable from multiexponential relaxation data

Abstract Computations are presented to provide guidelines to the amount of detail in a distribution of relaxation rates that can be derived from multiexponential relaxation data with varying levels of random noise. These can be useful for planning measurements and also in designing and using data inversion programs to compute distributions of rates. The guidelines can also help identify the unwarranted conclusions and interpretations occasionally encountered. A relaxation function corresponding to a narrow distribution of rates can be approximated by a single exponential to within about a tenth the mean-square deviation of the In rate from its mean. Least-maximum absolute-error (LMAE) fits by from one to five discrete exponential components to distribution of rates uniform in 1n rate were made for varying distribution widths. For a rate range of a thousand (e.g., relaxation times from a millisecond to a second) the M -component LMAE values, E M , relative to initial relaxing amplitude, are E 1 = 14.5%, E 2 = 3.3%, E 3 = 0.85%, E 4 = 0.23%, and E 5 = 0.06%.

Nuclear magnetic resonance relaxivity and surface‐to‐volume ratio in porous media with a wide distribution of pore sizes

The simplest model for the contribution of pore surfaces to nuclear magnetic resonance (NMR) relaxation of a pore fluid gives R, the average relaxation rate minus the bulk rate, equal to a constant ρ, the velocity at which nuclear magnetization flows out of the pore fluid at the surfaces, times the pore‐space surface‐to‐volume ratio S/V. Although ρ can vary widely, a great variety of porous media exhibit ρ values of the order of a few μm/s for longitudinal relaxation when S/V is measured by gas adsorption by the Brunauer, Emmett, and Teller (BET) method or high pressure mercury injection. For samples with wide distributions of relaxation rates it is of interest to find what functions of the relaxation data correlate best with S/V measurements and how different relaxation parameters relate to each other. Longitudinal relaxation data were taken for 77 sandstone samples of different origin, which had been cleaned and saturated with brine. After the NMR measurements the samples were dried and surface areas measured by BET. The samples have S/V from 1.5 to 150 (μm)−1, porosity from 3% to 28%, and permeability from less than 0.1 mD to more than 1 D. Longitudinal relaxation data were taken from 400 μs to 6 s and analyzed in many different ways, including stretched‐exponential fits and multiexponential fits up to five components. S/V and ln(S/V) were correlated with various relaxation rates derived from these computed parameters.In principle, the relaxation parameter to use with a ρ value is the average rate, which is initial slope divided by initial amplitude, namely, R(0), where R(t)=(d/dt)ln S(t) at t=0 and S(t) is the relaxing signal. One can extrapolate an n component fit to t=0 to get Rn(0), but very good signal quality is required even to get small short components reliably for t well within the times covered by the data. Over half of the points have ρ’s within a factor of 2 of the minimum value 0.9 μm/s when the average rate of a five‐component fit to the data is used. There are numerous points with ρ up to 7 μm/s, but none of the high‐ρ points are for samples with high S/V. All samples with high S/V have wide distributions of relaxation rates, but not vice versa. The best simple correlation with ln(S/V) was ln(S/V)≊1.81 ln(R33)−5.73, where R33 is the highest rate of a three‐component fit without regard to the corresponding amplitude, and where S/V is in (μm)−1 and rate in s−1. This result was unexpected. This fit does not represent proportionality to a velocity ρ and does not correspond to any obvious physical model, but it can be of practical interest to estimate in a very simple and noninvasive manner S/V at the BET scale in sandstones.The simplest model for the contribution of pore surfaces to nuclear magnetic resonance (NMR) relaxation of a pore fluid gives R, the average relaxation rate minus the bulk rate, equal to a constant ρ, the velocity at which nuclear magnetization flows out of the pore fluid at the surfaces, times the pore‐space surface‐to‐volume ratio S/V. Although ρ can vary widely, a great variety of porous media exhibit ρ values of the order of a few μm/s for longitudinal relaxation when S/V is measured by gas adsorption by the Brunauer, Emmett, and Teller (BET) method or high pressure mercury injection. For samples with wide distributions of relaxation rates it is of interest to find what functions of the relaxation data correlate best with S/V measurements and how different relaxation parameters relate to each other. Longitudinal relaxation data were taken for 77 sandstone samples of different origin, which had been cleaned and saturated with brine. After the NMR measurements the samples were dried and surface areas me...

Models of water imbibition in untreated and treated porous media validated by quantitative magnetic resonance imaging

Fluid imbibition affects almost every activity that directly or indirectly involves porous media, including oil reservoir rocks, soils, building materials, and countless others, including biological materials. In this paper, magnetic resonance imaging (MRI) has been applied to study water imbibition in a porous medium, in which capillary properties are artificially changed. As a model system, samples of Lecce stone, a material of cultural heritage interest, were analyzed before and after treatment with a protective polymer (Silirain-50 or Paraloid PB72). By using MRI, we can visualize the presence of water inside each sample and measure the height z(t) reached by the wetting front as a function of time during experiments of capillary absorption before and after treatment. The sorptivity S, defined as the initial slope of z versus t1/2, has been determined before treatment and through both treated and untreated faces after treatment. Very good fits to the data were obtained with theoretical and empirical m...

Different “average” nuclear magnetic resonance relaxation times for correlation with fluid-flow permeability and irreducible water saturation in water-saturated sandstones

Fluid-flow properties of porous media, such as permeability k and irreducible water saturation Swi, can be estimated from water 1H nuclear magnetic resonance (NMR) relaxation data, but there are basic questions regarding data processing and interpretation. We found that Swi and k are better estimated if different forms of “average” relaxation time are used. NMR longitudinal relaxation data for a suite of 106 water-saturated clean sandstones were used. Sandstones represent a specialized class of porous media, where even for small porosity, substantially all pore space is connected. The sandstones exhibit distributions of relaxation times ranging over factors from at least 10 to more than 103. We tried several forms of “average” relaxation time T. One family of Ts is 〈Tp〉1/p, where lim p→0 gives the geometric mean. The best estimator we found for Swi uses a form of average relaxation time only, rather than relaxation time cutoff. The time used can be any of several forms of T, giving more emphasis to short ...

Effects of hydrophobic treatments of stone on pore water studied by continuous distribution analysis of NMR relaxation times

The effects of protective hydrophobic products applied to porous media such as stone or mortar vary greatly with the product, the porous medium, and the mode of application. Nuclear Magnetic Resonance (NMR) measurements on fluids in the pore spaces of both treated and untreated samples can give information on the contact of the fluid with the internal surfaces, which is affected by all the above factors. Continuous distributions of relaxation times T(1) and T(2) of water in the pores of both synthetic and natural porous media were obtained before and after hydrophobic treatment. The synthetic porous media are ceramic filter materials characterized by narrow distributions of pore dimensions and show that the treatment does not produce large changes in the relaxation times of the water. For three travertine samples most of a long relaxation time component, presumably from the largest pores, remains after treatment, while the amplitude of an intermediate component is greatly reduced. For three pudding-stone samples, treatment leads to a substantial loss from the long component and an even greater loss from the intermediate component.

Developments in core analysis by NMR measurements

For a large suite of consolidated sandstone samples low in shale content we have measured the permeability k, irreducible water saturation Swi, porosity phi, electrical-resistivity formation factor F, porosity by NMR, geometric-mean relaxation times T1g, and stretched-exponential relaxation times T1s. We find that T1g (or T1s) is the decisive parameter for the estimation of k or Swi of porous sandstones by other than direct measurements of these quantities. The additional use of phi or F brings appreciable, but not decisive, improvement. We show isovalue maps of the error factor delta, which show substantial regions of near-minimum values of delta and show basic compatibility of our estimators for permeability with different published estimators. The exponents of T1g (or T1s) in our power-law estimators and those of various published estimators for k are not very far from 2.0 if either or both of phi and F are also used in the estimators.

Water-air saturation changes in restricted geometries studied by proton relaxation

The results are reported of a systemic T1 and T2 investigation of natural (sandstones) and artificial (microporous porcelain) porous media, after each step of a water desaturation process by centrifugation in air. The analysis of the relaxation curves permitted distinguishing well the different behaviour of the natural samples as compared to the artificial ones, which can be explained by the different pore structures. In both kinds of samples the evolution of the relaxation time distributions yielded a clear picture of the changes of the water distribution in the pore framework following the displacement process, until irreducible water saturation was attained. The results are compatible with the assumption of a fixed amount of surface area contributing to the relaxation of decreasing amounts of fluid as SW is reduced.

Problems in identifying multimodal distributions of relaxation times for NMR in porous media

Abstract NMR data for some water-saturated sandstones show distributions of relaxation times covering ranges of a thousand or more. Pore size distributions have been associated with distributions of relaxation times and can also cover wide ranges. Several dozen good sets of NMR relaxation data for water in porous sandstones have been analyzed in terms of continuous distributions of exponential components. About a dozen of these are for sandstones having significant relaxation time components over ranges of factors of a thousand. In all cases adequately good fits to the data could be obtained with distributions of relaxation times that were monomodal when plotted as functions of log-time (or log-rate). Thus, it appears that bimodality (or multimodality) for the logarithmic plots is not demanded by these particular sets of data, although these distributions plotted linearly are not monomodal. On the other hand, many multimodal distributions can always be found giving adequate fits to the data, since excessively sharp detail is not resolvable. Many programs using regularization methods to prevent excessive detail in computed distributions tend to give undershoot at sides of peaks, and noise tends to give not-quite-periodic oscillations. Lack of adequate range and density of either data points or computed points can lead to multimodal computed solutions. Some resolution expressions are used to indicate what level of detail in a computed distribution is meaningful for a given data set.

The effect of diffusion and susceptibility differences on T2 measurements for fluids in porous media and biological tissues

A number of features of T2 measurements for fluids in porous media have shown behavior contrary to that suggested by intuition developed in other areas. For porous media with relatively uniform pore spaces the following have been observed, in each case for certain ranges only of Xv (susceptibility difference times frequency), D (diffusion coefficient), a (a pore dimension), porespace shape and distribution, echo-time t for single echoes and half-echo-spacing tau for CPMG): (1) In S(t) for FID (free induction decay, S for signal) with constant slope after an initial period of increasing slope; (2) In Ss(t) for single (subscript s) echoes linear (instead of cubic) in t after an initial period; (3) for CPMG R(tau) = 1/T2(tau) - 1/T2(tau-->0) linear in tau over a substantial range; (4) slope of R(tau) independent of D and alpha for this range; (5) slope R(s) of In Ss(t) independent of D and a, and (6) R(s)(t) and R(tau) at long times linear (instead of quadratic) in Xv. These features appear to be compatible with the assumption of a truncated Cauchy-Lorentz distribution of the local magnetic fields due to susceptibility differences. The statistics of repeated sampling of local fields in different parts of the porespace during diffusion lead to a suppression, after a short time, of the effects of diffusion on the FID decay rate and the single-echo decay rate over significant ranges of the parameters. Data are presented to extend the range of parameters studied previously.

Diffusion-weighted spatial information from1H relaxation in restricted geometries

SummaryNMR relaxation of water1H confined in restricted geometries, whatever is the nature of the system (porous media saturated by water as well as biological tissues), exhibits common characteristics. Artificial microporous media saturated by water have been chosen as model systems to study the longitudinal and transverse relaxation of1H magnetization of water molecules diffusing in restricted geometries. These systems are very stable, easy to prepare, with well-characterized pore size distribution and connections, and with highly homogeneous surface properties. The response was compared with that from more complex natural porous media. Scanning Electron Microscopy techniques demonstrated spatial characteristics and surface properties of the samples. The information content of longitudinal relaxation curves associated with spatial structure and due to restricted diffusion is shown in these samples. The effect on transverse relaxation of self-diffusion in the presence of spatially varying magnetic fields due to susceptibility differences is shown. A simple linear relationship has been found in all samples between the transverse relaxation rate and the interpulse delay in CPMG experiments, in spite of the variety of pore shapes and sizes. In general, one can say that relaxation curves beardiffusion-weighted information on the pore space framework. The role of the investigated relaxation mechanisms is important also in the response of biological tissues, including in the presence of MR Imaging contrast agents inducing microscopic magnetic-field gradients.