Fluid flow in a porous medium with transverse permeability discontinuity

Magnetic resonance imaging (MRI) velocimetry methods are used to study fully developed axially symmetric fluid flow in a model porous medium of cylindrical symmetry with a transverse permeability discontinuity. Spatial mapping of fluid flow results in radial velocity profiles. High spatial resolution of these profiles allows estimating the slip in velocities at the boundary with a permeability discontinuity zone in a sample. The profiles are compared to theoretical velocity fields for a fully developed axially symmetric flow in a cylinder derived from the Beavers-Joseph [G. S. Beavers and D. D. Joseph, J. Fluid Mech. 30, 197 (1967)] and Brinkman [H. C. Brinkman, Appl. Sci. Res. A 1, 27 (1947)] models. Velocity fields are also computed using pore-scale lattice Boltzmann modeling (LBM) where the assumption about the boundary could be omitted. Both approaches give good agreement between theory and experiment, though LBM velocity fields follow the experiment more closely. This work shows great promise for MRI velocimetry methods in addressing the boundary behavior of fluids in opaque heterogeneous porous media.

Figure 1

Diagrams and relevant definitions describing (a) one-dimensional fluid flow and (b) in flow plane cross section of a fully developed axially symmetric fluid flow in a porous medium with a model sample geometry. See the text for further explanations.

Figure 2

Timing diagrams of flow measurements using (a) spin-echo and (b) stimulated-echo MRI protocols. (c) Timing diagram for measuring bulk displacement spectra by NMR. Motion encoding gradients are applied in the direction of flow. Bars indicate that the time axes are not continuous in (b).

Figure 3

Displacement spectra (propagators) for water flow in a bulk reticulated foam at $\text{Re}\approx 13$ for different $\mathrm{\Delta }$ times: (a) measured normalized propagators using NMR protocol displayed in Fig. 2 (solid lines) and (b) measured (symbols) and simulated (dashed lines) with a bimodal Gaussian function and deconvoluted normalized propagator at $\mathrm{\Delta }=1000\mathrm{ms}$.

Figure 4

(a) Example of an individual 2D $\mathrm{X}\mu \mathrm{CT}$ image of the foam with relevant length scales. (b) A 3D visualization of the cylindrical porous foam with a coaxial clear-cut circular channel reconstructed using a series of the segmented 2D $\mathrm{X}\mu \mathrm{CT}$ images displayed in (a). The solid matrix is shown in yellow, while transparent designates pore spaces. The radius of the clear channel in this porous medium produced specifically for $\mathrm{X}\mu \mathrm{CT}$ was approximately 0.5 cm. (c) Azimuthally averaged porosity profile produced from the reconstructed porous medium displayed in (a).

Figure 5

Phased flow images as a function of the applied field gradient $\mathbf{q}$ (a) in the absence of the fluid momentum transport across the interface (the rigid tube is inserted into the channel) and (b) with the momentum transport across the interface (the rigid tube is taken out). Flow encoding was performed by spin-echo protocol in the direction of flow for a few cross sections across the height of the porous medium. The raw matrix size was $256\times 256$ and the FOV was $3\times 3{\mathrm{cm}}^2$ with a slice thickness of 0.1 cm. The flow encoding time $\mathrm{\Delta }$ was kept to 8 ms as no outflow effects influenced the shape of velocity profiles when a volumetric flow rate $Q$ of 1.$4{\mathrm{cm}}^3/\mathrm{s}$ was used.

Figure 6

(a) and (c) Velocity maps and (b) and (d) corresponding azimuthally averaged radial velocity profiles (a) and (b) in the absence and (c) and (d) in the presence of the fluid momentum transport across the interface. The greater error in velocities in (b) is due to the larger bin size used in azimuthal averaging. The increase of errors in velocities near the wall in both experiments is induced by the distortions in image intensity due to the rf inhomogeneity in the coil as mentioned elsewhere in the paper. Larger errors in velocities in (a) in $0.85<r/R<1.14$ are due to the absence of the collected NMR signal from the location of the rigid tube placed inside the channel.

Figure 7

(a) and (c) LBM simulated normalized velocity maps and (b) and (d) corresponding normalized azimuthally averaged radial velocity profiles (a) and (b) in the absence and (c) and (d) in the presence of the fluid momentum transport across the interface. The blue line in (b) represents the analytical solution of the velocity profile for Poiseuille flow. The irregular white shapes in (a) and (c) are the foam.

Figure 8

Effect of spatial averaging on flow in the channel with a permeable wall at the volumetric flow rate $Q=2.25{\mathrm{cm}}^3/\mathrm{s}$. Radial velocity profiles were produced by azimuthal averaging with a bin size of 400 from velocity maps produced by spin-echo flow experiments, acquired in $256\times 256$ matrices for 12 different values of field gradients with $\mathrm{\Delta }=8\mathrm{ms}$ and a FOV of $3\times 3{\mathrm{cm}}^2$.

Figure 9

Effect of temporal averaging on flow in the channel (a) decoupled from and (b) coupled to the flow in the porous medium at a volumetric flow rate $Q=1.4{\mathrm{cm}}^3/\mathrm{s}$. The $\mathrm{\Delta }=8\mathrm{ms}$ profiles were produced from the spin-echo experiments acquired in $256\times 256$ matrices with 12 flow encoding gradients. The $\mathrm{\Delta }=500\mathrm{ms}$ profiles were produced from the stimulated-echo experiments acquired in $128\times 128$ matrices with 8 flow gradients. The FOV in both experiments was $3\times 3{\mathrm{cm}}^2$ with a slice thickness of 1 cm.

Figure 10

Azimuthally averaged radial velocity profiles (a) in the presence of transverse permeability discontinuity and (b) in the absence of the discontinuity in permeability as a function of the adjusted flow rate. The velocity fields were measured using the stimulated-echo protocol, with a raw acquisition matrix size of $128\times 128$, a FOV of $3\times 3{\mathrm{cm}}^2$, and a slice thickness of 0.5 cm at $\mathrm{\Delta }=500\mathrm{ms}$. The measured slip velocities in (a) were $1.0\pm 0.3,1.6\pm 0.5$, and $2.5\pm 0.8\mathrm{cm}/\mathrm{s}$ for flow rates of 1, 2.5, and $5{\mathrm{cm}}^3/\mathrm{s}$, respectively.

Figure 11

Slip and averaged Darcy velocities as a function of the increased mass transport in the presence of transverse permeability discontinuity (red) and in the absence of the discontinuity in permeability (blue). Note the quadratic Darcy velocity dependence on the increased mass transport (dashed black line) in the presence of transverse permeability discontinuity. The quadratic analysis was performed assuming that no slip velocities are observed at zero flow rate. Hence the zero point (black open triangle) has no error bars.

Figure 12

Comparison of radial velocity profile in the presence of fluid momentum transport across the permeable interface at $Q=1.4{\mathrm{cm}}^3/\mathrm{s}$ (open symbols) with Eq. (8) at $r<R$, Eq. (9) at $r>R$ (red), and LBM (blue). Velocity fields were temporally averaged within $\mathrm{\Delta }=8$ ms. The inset shows zoomed in boundary behavior of the exact solution to the problem and LBM modeling. Note a better match to the experiment for LBM.