Effective Gradients in Porous Media Due to Susceptibility Differences

In porous media, magnetic susceptibility differences between the solid phase and the fluid filling the pore space lead to field inhomogeneities inside the pore space. In many cases, diffusion of the spins in the fluid phase through these internal inhomogeneities controls the transverse decay rate of the NMR signal. In disordered porous media such as sedimentary rocks, a detailed evaluation of this process is in practice not possible because the field inhomogeneities depend not only on the susceptibility difference but also on the details of the pore geometry. In this report, the major features of diffusion in internal gradients are analyzed with the concept of effective gradients. Effective gradients are related to the field inhomogeneities over the dephasing length, the typical length over which the spins diffuse before they dephase. For the CPMG sequence, the dependence of relaxation rate on echo spacing can be described to first order by a distribution of effective gradients. It is argued that for a given susceptibility difference, there is a maximum value for these effective gradients,g_max, that depends on only the diffusion coefficient, the Larmor frequency, and the susceptibility difference. This analysis is applied to the case of water-saturated sedimentary rocks. From a set of NMR measurements and a compilation of a large number of susceptibility measurements, we conclude that the effective gradients in carbonates are typically smaller than gradients of current NMR well logging tools, whereas in many sandstones, internal gradients can be comparable to or larger than tool gradients.