Magnetic field nonuniformities and NMR of protons diffusing in a porous medium

Magnetic field inhomogeneity can arise either because of an externally applied field gradient or because of spatial variations in magnetic susceptibility. The latter are most important when the solid matrix includes paramagnetic substances and when the uniform applied field, and, consequently, also the Larmor precession frequency are very large. Both types of field inhomogeneity add extra phase shifts to the precessing spins. These phase shifts vary with time and position in a complex and random fashion as a result of the diffusive motion of the spins. We have studied these effects by performing detailed calculations for the case of a fluid filled porous medium with a periodic microstructure. Special attention was devoted to the question of whether the statistical distribution of the phase shifts encountered in a Hahn spin echo experiment or in a Carr-Purcell-Meiboom-Gill (CPMG) spin-echo train can be approximated as a Gaussian. The mean square phase shift is measured in such experiments as an enhanced relaxation rate of the precessing transverse magnetization. We determine this mean square phase shift for periodic composites from the diffusion eigenstates, which were calculated using a previously developed Fourier expansion method. The enhanced relaxation rate depends on the echo spacing time tau in a way that can be correlated with important length scales of the porous microstructure. Those correlations can be extended also to disordered microstructures, like the ones that are found in natural rocks. We compare these theoretically predicted correlations with CPMG measurements performed on protons in laboratory samples of brine saturated sandstone.