NMR acceleration mapping in percolation model objects Bogdan Buhai, a Aidar Hakimov, a Ioan Ardelean, b and Rainer Kimmich a, * a Sektion Kernresonanzspektroskopie, Universit€ at Ulm, 89069 Ulm, Germany b Department of Physics, Technical University, 3400 Cluj-Napoca, Romania Received 19 December 2003; revised 18 February 2004 Abstract An NMR microscopy technique is described that permits direct mapping of local accelerations. The method is tested with water flow through a random site percolation model object and compared with computational fluid dynamics simulations. A general formalism, the ‘‘polygon rule,’’ is reported for the design of gradient pulse sequences for phase encoding of higher order motions, or, in other words, for compensation of phase shifts by lower motional orders. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Acceleration; NMR mapping; Percolation; Porous media 1. Introduction In recent papers [1–5] we have reported on micro- scopic NMR mapping experiments and simulations of transport in fluid filled percolation model objects of porous media. The objective was to study and learn the laws governing coherent or incoherent propagation of measurands specifying transport under complex pore space constraints. Phenomena of interest so far were coherent flow characterized by velocity maps, incoher- ent flow, and diffusion leading to hydrodynamic dis- persion, thermal conduction, and convection causing temperature distributions and hydrodynamic flow pat- terns, and electric currents in electrolyte solutions. A comprehensive description of such methods can be found in recent reviews [6,7]. The advantage of percolation model objects first simulated as templates on a computer and then fabri- cated with the aid of milling [8] or lithography [9] techniques is in the first place that pore space topologies based on well-defined generation algorithms can be re- alized. Another advantage is that the boundary condi- tions of the pore space are known and can directly be used for numerical simulations of the transport quanti- ties. Such simulations help to plan and design experi- ments properly so that time and effort consuming failures are avoided. Comparisons of experimental data and their simulated counterparts permit one moreover to judge the reliability both of the numerical simulation method and of the measuring technique. The meaning- fulness of such combined study is thus particularly well founded. In the present paper we report on an NMR method for direct mapping of locally stationary accelerations of liquids flowing through porous systems. Of course, having a velocity map in principle implies the informa- tion on local accelerations as well. However, flow mea- surements with the standard phase encoding technique anticipates [10–12] constant velocities so that the exis- tence of any acceleration leads to additional phase shifts causing experimental velocity artefacts. Even if a ve- locity map were available with the required accuracy, it always refers to average values in the voxels resolved in the experiment. Reconstructing particle trajectories based on such voxel-average velocities and from these local accelerations unavoidably entails strong errors as test experiments demonstrated. The error of such eval- uations depends on the spatial variation of the velocity field and the spatial resolution of the mapping experi- ment. In other words, the correlation length of the ve- locity field must be much larger than the spatial resolution in order to obtain reliable acceleration vec- tors from a velocity vector field. * Corresponding author. Fax: +49-731-5023150. E-mail address: rainer.kimmich@physik.uni-ulm.de (R. Kimmich). 1090-7807/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmr.2004.02.013 Journal of Magnetic Resonance 168 (2004) 175–185 www.elsevier.com/locate/jmr