TD-NMR:
In Time-Domain NMR, the relaxation information, whether longitudinal (T1) or transverse relaxation (T2), can be subject to deconvolution to obtain relaxation distribution. T2 is affected by three underlying relaxation mechanisms: bulk, surface and diffusion relaxation [20,21], as follows:
\(\frac{\mathrm{1}}{\mathrm{T}_{\mathrm{2}}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{T}_{\mathrm{2bulk}}}\mathrm{+}\frac{\mathrm{1}}{\mathrm{T}_{\mathrm{2surface}}}\mathrm{+}\frac{\mathrm{1}}{\mathrm{T}_{\mathrm{2diffusion}}}\)(1)
In aqueous solution, the bulk relaxation time usually ranges from 2,000 to 3,000 ms. Its contribution to the harmonic average, therefore, becomes negligible in comparison. Diffusion relaxation, caused by the Brownian movement of the proton spins in an inhomogeneous magnetic field, can be neglected in the presence of small gradients and short echo spacing during measurements [22,23]. Therefore, the dominant relaxation within the pore space is surface relaxation, which is a function of the surface-to-volume ratio (S /V ) and surface relaxivity (\(\mathrm{\rho}_{\mathrm{2}})\). Therefore, T2 can be approximated in most cases as [23,24]:
\(\frac{\mathrm{1}}{\mathrm{T}_{\mathrm{2}}}\mathrm{\approx}\frac{\mathrm{1}}{\mathrm{T}_{\mathrm{2surface}}}\mathrm{=}\mathrm{\rho}_{\mathrm{2}}\frac{\mathrm{S}}{\mathrm{V}}\mathrm{=2}\mathrm{\rho}_{\mathrm{2}}\frac{\mathrm{F}_{\mathrm{s}}}{\mathrm{d}}\)(2)
where d is the pore diameter and Fs is the pore shape factor. For example, Fs for cylindrical and spherical pores equals 2 and 3, respectively. The converted T2 distribution can be used to estimate the pore-size distribution based on Eq. (2). In this study, T2 measurements for Berea sandstone, Indiana Limestone, and Madison Limestone samples were performed at a resonance frequency of 5.7 MHz and echo spacing of 0.5 ms using a Bruker LF110 instrument at room temperature. The instrument runs a 110-mm diameter, absolute-proton probe with long receiver dead time, but high accuracy. The Carr-Purcell-Meiboom-Grill (CPMG) pulse sequence was used during measurements. The number of scans and the recycle delay in this study were 64 and 4s, respectively. Conversion of relaxation signal into a continuous distribution of relaxation components (T2) was performed using the CONTIN application provided by the equipment vendor. Subsequently, the pore diameter distribution (PDD) is estimated through the transform of T2 distribution using Eq. (2). As surface relaxivity relates to mineralogy and is dependent on the approach used for its determination, this surface property varies among different rock types and even among samples of the same rock type [23, 25-27] (Table 2).