On the Calculation and Interpretation of NMR Relaxation Time Distributions

Abstract

Abstract We analyzed NMR relaxation time data using a multi-exponential model. The inversion of NMR data using such a model is an ill-posed problem. The answer is often not unique and requires subjective judgement. We studied two methods of regularization. One uses norm smoothing based on an article by Butler et al. We found that optimal norm smoothing depends on the input of measurement errors. The second method uses curvature smoothing which minimizes variations in the second derivative. The latter is more effective for suppressing fluctuations in the relaxation time distribution, but doesn't directly account for data quality. We studied NMR T1 data at full and partial saturations with desaturation pressures ranging from 15 to 400 psi (air/brine). As a general guideline, we found that the relaxation time cutoff which corresponds to irreducible water saturation is about 33 ms. For a limited number of samples, we found that the T1 relaxation time distribution has very little dependence on the frequency from 200 down to 1 MHz. We also found that T2 could be correlated to permeability. Introduction NMR measurements on reservoir rocks can be used in a wide range of applications. Porosity, producible fluid, fluid viscosity, pore size distribution, surface to volume ratio, and permeability can be estimated. Many of these have been investigated extensively in the past several years. In the present study, we concern ourselves specifically about the problems associated with inverting NMR data using multi-exponential model and the applications of the NMR relaxation time distributions. It has been well documented that the NMR relaxation of a fluid-saturated rock can be treated as a sum of exponentials (1) where fj is proportional to the proton population of pores which have a relaxation time of Tj. In the past, two-, three-, stretched and multi-exponential models have been used to study the NMR relaxation time distribution. When many pore scales are present, we believe that the multi-exponential model (where many relaxation times are equally spaced logarithmically over several decades) is a better choice because it offers not only a clearer qualitative description of pore size distribution but also a more consistent definition of free fluid index. Multi-Expoential Model One of the problems frequently encountered in using the multi-exponential model to analyze the NMR relaxation data is the question of optimal smoothing. Let us consider a set of NMR relaxation time measurements gi. We want to determine fj by minimizing the following quantity: P. 45^