Nuclear magnetic resonance (NMR) in porous materials covers the application of using NMR as a tool to study the structure of porous media and various processes occurring in them. This technique allows the determination of characteristics such as the porosity and pore size distribution, the permeability, the water saturation, the wettability, etc.
Theory of relaxation time distribution in porous media Microscopically the volume of a single pore in a porous media may be divided into two regions; surface area S {\displaystyle S} and bulk volume V {\displaystyle V} (Figure 1).
The surface area is a thin layer with thickness δ {\displaystyle \delta } of a few molecules close to the pore wall surface. The bulk volume is the remaining part of the pore volume and usually dominates the overall pore volume. With respect to NMR excitations of nuclear states for hydrogen-containing molecules in these regions, different relaxation times for the induced excited energy states are expected. The relaxation time is significantly shorter for a molecule in the surface area, compared to a molecule in the bulk volume. This is an effect of paramagnetic centres in the pore wall surface that causes the relaxation time to be faster. The inverse of the relaxation time T i {\displaystyle T_{i}} , is expressed by contributions from the bulk volume V {\displaystyle V} , the surface area S {\displaystyle S} and the self-diffusion d {\displaystyle d} :
1 T i = ( 1 − δ S V ) 1 T i b + δ S V 1 T i s + D ( γ G t E ) 2 12 {\displaystyle {\frac {1}{T_{i}}}=\left(1-{\frac {\delta S}{V}}\right){\frac {1}{T_{ib}}}+{\frac {\delta S}{V}}{\frac {1}{T_{is}}}+D{\frac {\left({\gamma Gt_{E}}\right)^{2}}{12}}} with i = 1 , 2 {\displaystyle i=1,2}
where δ {\displaystyle \delta } is the thickness of the surface area, S {\displaystyle S} is the surface area, V {\displaystyle V} is the pore volume, T i b {\displaystyle T_{ib}} is the relaxation time in the bulk volume, T i s {\displaystyle T_{is}} is the relaxation time for the surface, γ {\displaystyle \gamma } is the gyromagnetic ratio, G {\displaystyle G} is the magnetic field gradient (assumed to be constant), t E {\displaystyle t_{E}} is the time between echoes and D {\displaystyle D} is the self-diffusion coefficient of the fluid. The surface relaxation can be assumed as uniform or non-uniform. The NMR signal intensity in the T 2 {\displaystyle T_{2}} distribution plot reflected by the measured amplitude of the NMR signal is proportional to the total amount of hydrogen nuclei, while the relaxation time depends on the interaction between the nuclear spins and the surroundings. In a characteristic pore containing for an example, water, the bulk water exhibits a single exponential decay. The water close to the pore wall surface exhibits faster T 2 {\displaystyle T_{2}} relaxation time for this characteristic pore size.
NMR permeability correlations NMR techniques are typically used to predict permeability for fluid typing and to obtain formation porosity, which is independent of mineralogy. The former application uses a surface-relaxation mechanism to relate measured relaxation spectra with surface-to-volume ratios of pores, and the latter is used to estimate permeability. The common approach is based on the model proposed by Brownstein and Tarr. They have shown that, in the fast diffusion limit, given by the expression:
ρ r / D {\displaystyle \rho r/D}
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