Unexpected scaling of interstitial velocities with permeability due to polymer retention in porous media

Polymer retention from the flow of a polymer solution through porous media results in substantial decrease of the permeability; however, the underlying physics of this effect is unknown. While the polymer retention leads to a decrease in pore volume, here we show that this cannot cause the full reduction in permeability. Instead, to determine the origin of this anomalous decrease in permeability, we use confocal microscopy to measure the pore-level velocities in an index-matched model porous medium. We show that they exhibit an exponential distribution and, upon polymer retention, this distribution is broadened yet retains the same exponential form. Surprisingly, the velocity distributions are scaled by the inverse square root of the permeabilities. We combine experiment and simulation to show these changes result from diversion of flow in the random porous-medium network rather than reduction in pore volume upon polymer retention.

Figure 1

Measurement of the bulk permeability of the porous medium. (a) Schematic of the experimental apparatus. Fluid flow is controlled by syringe pumps and resultant pressure across the medium is monitored by a transducer. (b) Relative changes in permeability $k_f/k_i$ after flow of polymer solution, $V_{\text{polymer}}$. Error bars represent the standard deviation of several measurements.

Figure 2

Heat map of magnitude of the fluid velocity in a 2D plane within the 3D porous medium, for a flow rate of $200\mu \mathrm{l}$/h (a) before and (b) after 12 pore volumes of polymer flow, showing increased local velocities. (c) Difference between the longitudinal components of velocity before and after polymer flow showing heterogeneous changes in local velocity due to polymer retention. Black circles represent glass beads. Scale bars are 1 mm.

Figure 3

Probability density function (PDF) of the magnitude of fluid flow velocities (a) before polymer (blue squares) and after flow of 12 pore volumes of polymer (PV) (red circles), and after 20 pore volumes (green triangles), and (b) scaling of behavior when normalized by $1/\sqrt{k}$.

Figure 4

(a) Probability density function (PDF) of the magnitude of flow velocities, normalized by the $U_0$, for different values of average tube diameter. (b) PDF normalized by the inverse square root of the permeability. (c) The relation between relative average velocity in the network $U/U_0$ and the inverse square root of permeability $\sqrt{k/k_0}$ calculated directly from simulation. (d) Comparison between geometric porosity and equivalent porosity, measured from average velocity, as the diameters of tubes are reduced.

Figure 5

Heat map of magnitude of velocities (a) before and (b) after reduction of tube diameters, $\overline{d}=0.67$. (c) Difference between the longitudinal component of velocities before and after reduction in tube diameter highlighting heterogeneity in changes in local flow. Blocked pores shown in black.