Multi-exponential Analysis of Water NMR Spin–Spin Relaxation in Porosity/Permeability-Controlled Sintered Glass

Measuring time constants in multi-exponential decay phenomena is crucial in many areas of knowledge. However, besides the inherent difficulties to the mathematical structure of the problem analysis, noisy experimental data can make the task considerably difficult. One important example is nuclear magnetic resonance (NMR) logging data obtained from measurements of water and hydrocarbons in porous rocks from oil fields. To minimize the uncertainties, it is important to design experiments under controlled conditions. In this paper, we report a systematic study of high (500 MHz) and low (15 MHz)-field NMR $T_2$ relaxation times performed on artificial sintered sand-glass samples saturated with water. Porosity and permeability were controlled by selecting the range of grains, and then applying a specific sintering temperature protocol to produce samples with different porous sizes, constant porosity, but varying permeability. The structure of porous was verified by microtomography and scanning electron microscopy techniques. Porosity and permeability were measured, respectively, by the free-gas expansion and steady-state methods. We analyze the NMR data using three different approaches: (1) Laplace inversion with optimized regularization based on measured noise level, (2) bi-exponential, and (3) q-exponential nonlinear least-squares. Upon a careful measurement protocol, we report that all methods yield essentially similar $T_2$ distributions.