PHYSICAL REVIEW B VOLUME 45, NUMBER 1 1 JANUARY 1992-I Effects of microgeometry and surface relaxation on NMR pulsed-field-gradient experiments: Simple pore geometries Partha P. Mitra' and Pabitra N. Sen Schlurnberger D-oll Research, Old Quarry Road, Ridgegeld, Connecticut 06877 $10-8 (Received 24 May 1991; revised manuscript received 14 August 1991) We derive an expression for the magnetization M(k, b, ) in a pulsed-fiield-gradient experiment for spins diffusing in a con6ned space with relaxation at the pore walls. Here k = pbg, , b = pulse width, g = gradient strength, p = the gyromagnetic ratio, and 6 is the time between gradient pulses. We show that the deviation of — ln[M(k, D)/M(0, b)] from quadratic behavior in k in experiments in porous media can be a more sensitive probe of the microgeometry (size, connectivity, size distribution, shape, etc. ), than either the enhancement of 1/Ti over the bulk water value, or the macroscopic diffusion coefficient, which is derived from the slope of — ln[M(k, b, )/M(0, b, )] at small k, in the limit of large A. We propose some simple models of randomly oriented tubes and sheets to interpret the k dependence of the amplitude beyond the leading small-k quadratic behavior. When the macroscopic diffusion coefficient is unobtainable, due to the decay, the present considerations should be useful in extracting geometrical information. The effective diffusion constant derived from NMR exactly equals that derived from electrical conductivity only when the surface relaxivity is zero, but can be close to each other in favorable circumstances even for finite surface relaxivity. Exact solutions with partially absorbing boundary conditions are obtained for a slab and a sphere to infer that the normalized amplitude M(k, 4, p)/M(0, 4, p) depends only weakly on the surface relaxivity p for rnonodisperse convex-shaped pores in the parameter ranges of interest. We also obtain expressions for the mean lifetime of the amplitude in the geometries considered. I. INTRODUCTION Spin echo measurements are routinely used for study- ing molecular diffusion in fluids. When the diffusion is confined by the presence of obstacles, the experiments yield information about the confining geometry2 as seen by the diffusing particle. In porous media, precisely such information, i.e. , length scales, such as pore and throat sizes, and geometric factors, such as tortuosity and connectivity, determine the transport and other proper- ties. There has been a lot of interest recently in using NMR techniques to obtain information on microgeom- etry, the interest being in part due to the noninvasive nature of these measurements. The main purpose of this paper is to show that in a pulsed-field-gradient (PFG) experiment, the full k ("mo- mentum") dependence contains much more information about the microgeometry than just the usual diffusion coeKcient, which is derived from the k dependence of the logarithm of the normalized echo amplitude at small k. We give examples to show the influences of size, the local anisotropy of the pore space, and of a distribution of sizes on the k" term. The true macroscopic diffusion coefficient (i.e. , the diKusion constant measured in the limit of large 6, for a connected pore space) contains the valuable information of tortuosity, but not the pore size. Furthermore, the method suggested here can give geom- etry information when the enhanced decay forbids a suc- cessful extraction of the macroscopic diffusion coefFicient. The importance of the pulsed-gradient experiment from a theoretical point of view is that it directly measures. the Fourier transform of the diffusion propagator in pore space. In order to interpret data it is a prerequisite to account for the decay at pore walls which dominates the relax- ation in many porous media. We give here the proper for- mulation which takes into account the surface relaxation. The presence of surface relaxers enters as a partially ab- sorbing boundary condition for the magnetization at the pore wall. Considering the importance of surface relaxiv- ity, it is surprising that the partially absorbing boundary condition has previously not been treated properly for field gradient experiments, even in the context of the ef- fective diffusion coefficient. In addition to the investigation of the effective dif- fusion coefIicient using field gradients, there has been much effort in obtaining geometrical information using the enhancement~ in the NMR longitudinal decay rate 1/Ti. The increase in decay rate in rocks gener- ally comes from the paramagnetic impurities on the pore walls. The longitudinal decay rate is enhanced by relax- ers on pore walls in proportion to the surface to volume ratio i.e. 1 bS pS Ti TggV V ' where Tqg is the enhanced relaxation within a distance b, p is a surface relaxivity, S is surface area, and U the pore volume. The principal di%culty in relating pore size to the relaxation rate is the appearance of an unknown 143 1992 The American Physical Society