Minimal Length Elements of Thompson's Group F

Elements of the group are represented by pairs of binary trees and the structure of the trees gives insight into the properties of the elements of the group. The review section presents this representation and reviews the known relationship between elements of F and binary trees. In the main section we give a method of determining the minimal lengths of elements of Thompson's group F in the two generator presentation

Magnetic resonance imaging in laboratory petrophysical core analysis

Magnetic resonance imaging (MRI) is a well-known technique in medical diagnosis and materials science. In the more specialized arena of laboratory-scale petrophysical rock core analysis, the role of MRI has undergone a substantial change in focus over the last three decades. Initially, alongside the continual drive to exploit higher magnetic field strengths in MRI applications for medicine and chemistry, the same trend was followed in core analysis. However, the spatial resolution achievable in heterogeneous porous media is inherently limited due to the magnetic susceptibility contrast between solid and fluid. As a result, imaging resolution at the length-scale of typical pore diameters is not practical and so MRI of core-plugs has often been viewed as an inappropriate use of expensive magnetic resonance facilities. Recently, there has been a paradigm shift in the use of MRI in laboratory-scale core analysis. The focus is now on acquiring data in the laboratory that are directly comparable to data obtained from magnetic resonance well-logging tools (i.e., a common physics of measurement). To maintain consistency with well-logging instrumentation, it is desirable to measure distributions of transverse (_T2$T_2$) relaxation time–the industry-standard metric in well-logging–at the laboratory-scale. These _T2$T_2$ distributions can be spatially resolved over the length of a core-plug. The use of low-field magnets in the laboratory environment is optimal for core analysis not only because the magnetic field strength is closer to that of well-logging tools, but also because the magnetic susceptibility contrast is minimized, allowing the acquisition of quantitative image voxel (or pixel) intensities that are directly scalable to liquid volume. Beyond simple determination of macroscopic rock heterogeneity, it is possible to utilize the spatial resolution for monitoring forced displacement of oil by water or chemical agents, determining capillary pressure curves, and estimating wettability. The history of MRI in petrophysics is reviewed and future directions considered, including advanced data processing techniques such as compressed sensing reconstruction and Bayesian inference analysis of under-sampled data. Although this review focuses on rock core analysis, the techniques described are applicable in a wider context to porous media in general, such as cements, soils, ceramics, and catalytic materials.     

Determining NMR flow propagator moments in porous rocks without the influence of relaxation

Flow propagators, used for the study of advective motion of brine solution in porous carbonate and sandstone rocks, have been obtained without the influence of Nuclear Magnetic Resonance (NMR) relaxation times, T₁ and T₂. These spin relaxation mechanisms normally result in a loss of signal that varies depending on the displacement ζ of the flowing spins, thereby preventing the acquisition of quantitative propagator data. The full relaxation behaviour of the system under flow needs to be characterised to enable the implementation of a true quantitative measurement. Two-dimensional NMR correlations of ζ − T₂ and T₁ − T₂ are used in combination to provide the flow propagators without relaxation weighting. T₁ − ζ correlations cannot be used due to the loss of T₁ information during the displacement observation time Δ. Here the moments of the propagators are extracted by statistical analysis of the full propagator shape. The measured displacements (first moments) are seen to correlate with the expected mean displacements for long observation times Δ. The higher order moments of the propagators determined by this method indicate those obtained previously using a correction were overestimated.