Nuclear Spin-Lattice Relaxation of Liquids Confined in Porous Solids
Authors: Senturia Stephen D. , Robinson J.D. 
Published: December 2007

Abstract
A random-walk, diffusion model is developed to account for the observed decrease in nuclear spin-lattice relaxation time in liquids confined in porous solids. For the system (water-Selas) and porous solids. For the system (water-Selas) and for a single pore size in the range of petrophysical interest, the form of the spin-lattice relaxation is shown to be a simple exponential with the decay constant being a sum of two terms. The first term is the reciprocal of the relaxation time in the bulk liquid, while the second term represents the rate at which molecules diffuse to the pore surfaces and subsequently undergo surface relaxation.

Introduction
The spin-lattice relaxation time, T1, of the nuclear spins in a bulk liquid is usually thought of as a property of the bulk liquid depending on temperature on the presence of impurities or solute ions, and perhaps, on any applied magnetic field, but not generally depending on the size of the liquid sample. However, as Brown reported, the protons in water confined in the pores of a solid matrix have a shorter T1 than protons in a large water sample. Since 1956, observations too numerous to list have been made of this effect.
We have developed a model which accounts for this reduction in T1. It is based on a random-walk picture in which molecules of a liquid at the picture in which molecules of a liquid at the surface of the confining region experience an enhanced relaxation probability per unit time, arising either from an increase in the correlation time at the surface or from the presence of paramagnetic sites at the surface. This model paramagnetic sites at the surface. This model has proved adequate both for the analysis of the data on proton relaxation in water and hydrocarbons contained in the pores of solids and for the analysis of the Nuclear Magnetism Log (NML) data presented by Loren and Robinson.

EXPERIMENTAL
We shall need some experimental dam on samples of known pore size in order to make some simplifications toward the end of the next section. Accordingly, the relaxation data we actually used to guide the theoretical development are tabulated in Table 1. These data were obtained in the laboratory in 1960 with the Carr-Purcell spin-echo technique. Cylinders 1/2 in. long and 1/2 in. in diameter were cut from discs of Selas porcelain, saturated with water, and sealed. No heating and outgassing as done, so the T1, value for bulk water listed in Table 1 is for water at equilibrium with air. The measurements were made at a Larmor frequency of 28.7 MHz. A 180 degrees pulse was applied at time zero and a 90 degrees pulse at a time t later. The time t was varied until no tail was observed following the 90 degrees pulse. The value of t is the time for the proton magnetization to attain one-half its equilibrium value. If the proton relaxation is simply exponential in time, then the half-time and T1 are related by t 1/2 = T1 ln 2. For purposes of analysis in the present paper, we have assumed that the pore size of each of the Selas samples is uniform and equal to the average value given by the manufacturer. (The mercury-injection capillary pressure curves of Loren and Robinson show that this is only an approximation.)
SPEJ
P. 237