Relations Between Pore Size Fluid and Matrix Properties, and NML Measurements
Authors: Loren J.D. , Robinson J.D. 
Published: December 2007

Abstract
A theory relating proton relaxation times for water and hydrocarbons confined to rock pores to the pore size distribution of rocks and providing the basis for Nuclear Magnetism Log (NML) interpretation is presented. A model proposed by Senturia and Robinson relates the thermal relaxation time (T1) of the proton spins of molecules in the liquid state to proton spins of molecules in the liquid state to geometrical and physical properties of a liquid-filled porous solid. Four parameters enter their theory, namely, jump distance and correlation time of the liquid molecules, radius of the confining region a, and the probability (1-p) for proton spin-flip at the liquid-solid interface. In this report the theoretical model of Ref. 1 is used to analyze thermal relaxation measurements on a suite of liquid-saturated porcelain samples. The assumption that pore size is inversely proportional to mercury injection capillary pressure proportional to mercury injection capillary pressure (Pc) enables the T1 of the liquid-saturated porous solid to be expressed as
where l/r is the thermal relaxation time of the bulk liquid, and is a parameter containing, p and a constant of proportionality. The suite of porcelain samples have mercury injection displacement pressures ranging from 20 to 200 psi; yet values of pressures ranging from 20 to 200 psi; yet values of derived from the measured values of T1, r and Pc are relatively constant as predicted by the theory. When the identical porcelain samples are saturated with decane rather than water, there is an order of magnitude decrease in; this leads to the conclusion that the T1 of a water-wet, hydrocarbon-saturated rock is relatively insensitive to pore size. Values of determined for a suite of water-saturated sandstones with widely varying pore sizes range from 0.13 psi-1 sec-1 to 0.44 psi-1 sec-1. Measurements on samples with both a water and an oil phase present illustrate the insensitivity of the T1 of the oil phase to the geometry of the confining pore space. pore space. Within carbonates, the observed value of is an order of magnitude less than in sands. This reduction could be brought about by either a change in the factor of proportionality relating pore size to pore entry size, or by a decrease in the probability for relaxation at the surface. Within rocks containing both water and oil, the known values of and the assumption that the oil phase occupies the larger pores permits thermal phase occupies the larger pores permits thermal relaxation curves to be calculated which fit observed data. A method for quantitatively determining residual oil involves use of a paramagnetic aqueous phase to effectively kill the water response and phase to effectively kill the water response and permit the remaining signal to be attributed to the permit the remaining signal to be attributed to the oil phase. This theory provides the basis for NML interpretation. For example, application of the theory permits a pore size histogram to be determined from permits a pore size histogram to be determined from a thermal relaxation curve derived histogram is dependent upon the uncertainty associated with each point on the thermal relation curve; and the fraction of the total proton magnetization observed. Another application is the quantitative determination of residual oil from down-hole thermal relaxation data.

Introduction
The signal recorded on an NML is obtained directly from fluids contained in the pores of a rock. A proper understanding of the relation between the observed response and matrix and fluid properties is essential for the interpretation of the log. In this report, a physical model is presented which provides this relation. Laboratory data from provides this relation. Laboratory data from porcelain and natural cores containing both water and porcelain and natural cores containing both water and hydrocarbon are analyzed in terms of the model.
SPEJ
p. 268