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

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Abstract
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.