Spatial and Temporal Coarse Graining for Dispersion in Randomly Packed Spheres

Abstract

The propagator for molecular displacements P(zeta, t) and its first three cumulants were measured for Stokes flow in monodisperse bead packs with different sphere sizes d and molecular diffusion coefficients D(m). We systematically varied the normalized mean displacement <zeta>/d and diffusion length L(D)=sqrt[2D(m)t]/d. The experimental results map onto each other with this scaling. For L(D)/d<0.2 the propagator remains non-Gaussian, and thus an advection diffusion equation is not obeyed, for mean displacements measured up to <zeta>>10d. A Gaussian shape is approached for large mean displacements when L(D)>0.3d.

A single-scan method for measuring flow along an arbitrary direction

“…An average flow velocity can be measured for a symmetric distribution of velocities. For non-symmetric velocity distributions [16,17], the imaginary part in [Á Á Á] in Eq. (5) becomes non-zero and / 1 is not exactly linear inf Q .…”

“…(5) becomes non-zero and / 1 is not exactly linear inf Q . Then the phase dependence onf Q can be used to extract higher order moments of the velocity distribution [16].…”

Multiple‐modulation‐multiple‐echo magnetic resonance

“…It is worth noting that when the distribution of velocities and displacements is not symmetric about the mean value (50,51), the phase of the echoes may no longer be exactly linear in f Q . Higher order cumulants of the displacement distribution can then in principle be inferred from the phase of the signal.…”

Low magnetic fields for flow propagators in permeable rocks

“…A laboratory technique receiving much current attention is the measurement of the NMR flow propagator [3][4][5][6][7][8][9][10][11][12][13][14][15] and the NMR time-of-flight technique [16][17][18]. This is mainly because the largest length scales that can be probed by diffusion and relaxation measurements [19][20][21] are $100 lm, which can be less than the size of the largest pores in some rocks, especially carbonates (limestones and dolomites).…”

“…These flow processes are controlled by the nature of the interconnections, and the topology of the pore space over length scales equivalent to many pores. Although for highly heterogenous samples, multiple length scales can be important [7,8,12,[24][25][26][27] and not all are accessible by NMR methods, the NMR flow propagators can nevertheless probe displacements of the order of $5 mm. This is almost two orders of magnitude larger than those achievable by the diffusion and relaxation methods currently used, in petroleum industry practice, as probes of the rock pore space.…”