NMR relaxation on porous media: a superstatistical approach with applications to petrophysics

The superstatistics has been applied to interpret Nuclear Magnetic Resonance (NMR) data on transverse relaxation on porous media with application in petrophysics. Porous media are very important kind of structures, and many areas in Science and Technology have studied these materials [1], including mathematics, physics, chemistry, engineering, biology, medicine, and industrial areas such as Oil & Gas. The NMR technique has raised many interesting questions about porous media through the study of the transverse relaxation of uids in these materials [2]. This thesis aims to contribute to the discussion about the ill-posed problem of the inversion of transverse relaxation T2 decay measured with the standard CPMG sequence, by the application of mathematical tools and ideas from superstatistics and nonextensive statistics. The T2 time decay of one single pore is a statistical sum of the time evolution of spins states inside it. One pore yields a well de_ned T2 time that is a solution of di_usion equation with boundary conditions. If the porous media is a statistical sum of pores, the total T2 time will be given by a distribution of values rather than a single value. We show that the NMR transverse decay can be modelled by q-exponential functions, represented by the distribution fi (T2) α T-22 χ2 (T-12). To test the validity of these parametric statistical model, we perform high-field NMR T2 relaxation measurements on porous media built from glass microsphere with difeerent ranges of two diferent companies, the Multiesferas and Cospheric, and three samples of outcrop rocks. The Multiesferas set is of soda lime glass and the spheres radii are in the following ranges: A (425-600 µm); B (250-425 µm); C (106-212 µm); D (75-125 µm); and E (45-90 µm). The Cospheric's microspheres are in the following ranges: A (710-850 µm); B (425-500 µm); C (212-250 µm); D (106-125 µm); and E (45-53 µm). The outcrops rocks used were the Berea and the Buff Berea sandstone, and the Indiana Limestone. This new model for NMR petrophysics includes statistical assumptions for the constitution of rocks and porous media and can be very useful to estimate petrophysical parameters like, water saturation (Sw), clay bound water (CBW), bound volume index (BV I), free fluid (FF), T2cutoff and even a pore size distribution. We show the mathematical formulation to find these petrophysical parameters without inversion and use experimental data to evaluate the applicability of the model.