The CPMG Pulse Sequence in Strong Magnetic Field Gradients with Applications to Oil-Well Logging

Abstract

It is shown that transverse relaxation measurements obtained from CPMG echo trains are valid even in the presence of strong, static magnetic field gradients. In the context of in situ measurements for water or oil exploration, low magnetic fields and short echo spacings are utilized to minimize diffusional effects. Under these conditions, it is shown that, for T 1 = T 2, the inverse Laplace transformation of the echo train is essentially independent of field homogeneity. For T 1 ≠ T 2, the error in determining T 2 does not exceed ∼12%, even for high T 1/ T 2 ratios. In most porous media, T 1/ T 2 is less than 3, in which case the error is below 8%. Analytical expressions for the echo amplitudes including relaxation are derived based on the density-matrix formalism. We define recursion relations that give the density matrix at echo i by a simple multiplication of the density matrix at echo i − 1 with a set of operators describing the evolution between consecutive echoes. The echo intensity is shown to be a function of the ratio between the radiofrequency strength and the receiver bandwidth. The optimal signal-to-noise ratio is obtained when this ratio is unity. The paper provides the theoretical framework for interpreting data obtained in situ by a modern NMR logging instrument. Furthermore, the results are directly applicable to magnetic resonance imaging.