Nuclear magnetic relaxation in porous media: The role of the mean lifetime tau ( rho,D).

Abstract

We show that the mean lifetime \ensuremath{\tau}(\ensuremath{\rho},D) provides a compact and convenient description of surface-enhanced nuclear magnetic relaxation in fluid-saturated porous media. Here \ensuremath{\rho} is a parameter that measures the relaxation rate at the pore-grain interface and D is the bulk diffusion constant for the fluid in the pore space. In the case of simple pore shapes with uniform magnetization at the interface, e.g., slabs (d=1), cylinders (d=2), or spheres (d=3) of radius a, we derive the equation \ensuremath{\tau}(\ensuremath{\rho},D)=${\mathit{a}}^{2}$/d(d+2)D +a/d\ensuremath{\rho}. For more general pore shapes the relation between \ensuremath{\tau}, ${\mathit{D}}^{\mathrm{\ensuremath{-}}1}$, and ${\mathrm{\ensuremath{\rho}}}^{\mathrm{\ensuremath{-}}1}$ is nonlinear, but is well represented by a Pad\'e approximant based on four parameters that are characteristic of the pore geometry. The utility of this representation is illustrated by numerical calculations on a series of two-dimensional pore geometries. The average lifetime is also of interest because a recently established bound on the permeability of porous media can be recast in terms of \ensuremath{\tau}(\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\infty},D). We show that a modified version of this bound can be expressed in terms of the directly measurable quantity \ensuremath{\tau}(\ensuremath{\rho},D). The limitations of such bounds are illustrated by numerical simulations on simple three-dimensional pore geometries.