Fluid flow in porous systems

doi:10.1016/S0730-725X(98)00073-3

Magnetic Resonance Imaging

Volume 16, Issues 5–6, June–July 1998, Pages 451-454

https://doi.org/10.1016/S0730-725X(98)00073-3 Get rights and content

Abstract

Nuclear magnetic resonance (NMR) measurements of water velocity flowing through glass bead packs with a bead diameter of 10 mm have been made using the π echo-planar imaging (PEPI) sequence. These results indicate that for various flow rates the flow variance is proportional to the mean flow velocity in agreement with the Mansfield-Issa equation. The velocity distributions are approximately Gaussian. Investigation of the slopes of the variance vs. velocity curves as a function of slice thickness indicate some coherence effects in the connectivity of the glass bead system. An extension of an earlier intervoxel coupling model is presented, which seems to explain the observed coherence effects.

Introduction

The π echo-planar imaging (PEPI) method of flow imaging1 has recently been applied successfully to measure fluid flow and in particular the velocity distribution across a transaxial slice in Bentheimer and Clashach sandstone, and more recently in glass bead phantoms. In all cases the velocity distributions obtained are characterised by an approximate Gaussian distribution curve and a velocity variance proportional to mean flow velocity, which agrees with theoretical predictions based on a stochastic model of flow.2

In a microscopic model of flow a complementary description of the coupling process between adjacent pixels is used to evaluate the velocity variance. This model also leads to the velocity variance being proportional to mean flow velocity, i.e., the Mansfield–Issa equation.3 In all flow velocity distributions the mean velocity of course obeys Darcy’s Law.

The change in variance with mean flow velocity has been attributed to the presence of transverse flow orthogonal to the main flow pressure gradient. This transverse flow has been ascribed to an intervoxel coupling mechanism comprising Bernoulli flow channels, which allow non-dissipative transverse flow across the slice. Using this model, the coupling mechanism is further investigated for an isolated voxel pair. It is shown that the characteristic way in which the fluid velocity in a particular voxel changes with slice thickness depends on the dimensions of the voxel.

Section snippets

Theoretical background

We have shown elsewhere3 that, for a set of isolated pairs of voxels of cross-sectional area dA each with a single nondissipative flow channel or Bernoulli pathway of cross-sectional area Δa_j forming the coupling path for fluid transfer between voxels, the velocity variance calculated for all pairs is given by: $a^{2} =k ∑ j Δa_{j} dA Δk_{j} k v ̄ /ρN=k v ̄ ρN ∑ j C_{j},$ where v̄ is the mean flow velocity, N is the number of voxels in the slice, k is the sample resistivity, ρ the fluid density, Δk_j a small resistive

Results

In natural porous media such as Bentheimer sandstone the pore size is typically 20 μm. In order to investigate the connectivity variation with slice thickness, it would be necessary in natural materials to have extremely thin slice thicknesses and very high-resolution flow maps. At the present time this is not possible with our equipment. In order to test the theory, therefore, we have constructed model systems comprising glass bead packs with various macroscopic-sized beads ranging from 3–10

Discussion and conclusions

The sample container consists of a cylindrical volume with diameter 75 mm and length 150 mm. One of the difficulties with bead packs comprising large diameter beads contained in a finite sample volume is that there are relatively few beads in the statistical assembly. There is, therefore, a clear possibility that the beads will form into regular, close-packed arrays rather than a random arrangement as called for by the theoretical model. Going to a smaller bead size would obviate this

References (3)

D.N. Guilfoyle et al.
Fluid flow measurement in porous media by echo-planar imaging
J. Magn. Reson.
(1992)

There are more references available in the full text version of this article.

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Invited LecturesFluid flow in porous systems