Determining fractal dimension from nuclear magnetic resonance data in rocks with internal magnetic field gradients

Abstract

Pore size distributions in rocks may be represented by fractal scaling, and fractal descriptions of pore systems may be used for prediction of petrophysical properties such as permeability, tortuosity, diffusivity, and electrical conductivity. Transverse relaxation time ([Formula: see text]) distributions determined by nuclear magnetic resonance (NMR) measurements may be used to determine the fractal scaling of the pore system, but the analysis is complicated when internal magnetic field gradients at the pore scale are sufficiently large. Through computations in ideal porous media and laboratory measurements of glass beads and sediment samples, we found that the effect of internal magnetic field gradients was most pronounced in rocks with larger pores and a high magnetic susceptibility contrast between the pore fluid and mineral grains. We quantified this behavior in terms of pore size and Carr-Purcell-Meiboom-Gill (CPMG) half-echo spacing through scaling arguments. We additionally found that the effects of internal field gradients may be mitigated in the laboratory by performing [Formula: see text] measurements with different CPMG half-echo spacings and fitting the apparent fractal dimensions determined by the NMR measurements with a model to determine the true pore system fractal dimension.

Nuclear Magnetic Resonance Investigation of Surface Relaxivity Modification by Paramagnetic Nanoparticles

“…According to Equation 9, the glass bead surface relaxivity 1,Mediumsurface was 10.42 m/sec. The fact that the glass beads had a larger surface relaxivity than the nanoparticles (1.4 m/s) was probably due to the presence of paramagnetic impurities in the glass beads, as has been noted in other studies using these particular beads (e.g., Daigle et al, 2014).…”

An electrical conductivity model for fractal porous media

“…After fractal geometry was introduced by Mandelbrot [1983], it became a powerful tool for the analysis of physicogeometrical properties and processes associated with porous media, including surface roughness [Brown, 1987], permeability [Costa, 2006;Xu and Yu, 2008;Xiao et al, 2014], tortuosity [Ghanbarian et al, 2013b], capillary flow [Cai et al, 2012], and pore structure [Ghanbarian-Alavijeh et al, 2011;Daigle et al, 2014]. It has been shown that the irregular nature of porous media exhibits fractal characteristics, and fractal geometry is an effective means for describing porous media [Sahimi, 1993;Yu and Cheng, 2002;Cihan et al, 2009;Cai and Yu, 2011].…”

An improvement of the fractal theory and its application in pore structure evaluation and permeability estimation

“…It is well established that the pore space of a wide variety of media exhibits fractal structure and the surface of the grains or particles that comprise porous systems can themselves be fractal [ Deinert et al ., ]. A general self‐similar set of the pore size distributions is expressed by the box counting method [ Mandelbrot , ; Daigle et al ., ], where r and N ( R > r ) are pore radius and pore numbers with radius larger than r and D is the fractal dimension.…”

“…Fractal theory [ Mandelbrot , ] has been developed as a popular tool to investigate pore space of sedimentary rocks and other objects [ Tyler and Wheatcraft , ; Sahouli et al ., ; Hunt and Gee , ; Cai and Yu , ; Ghanbarian et al ., ; Daigle et al ., ; Ge et al ., ]. Compared with classical Euclidean geometry which only works with objects in integer dimension, fractal geometry can be adopted to describe irregular geometrical shapes with nonintegral dimensions.…”