Flow-induced compaction of a deformable porous medium

Fluid flowing through a deformable porous medium imparts viscous drag on the solid matrix, causing it to deform. This effect is investigated theoretically and experimentally in a one-dimensional configuration. The experiments consist of the downwards flow of water through a saturated pack of small, soft, hydrogel spheres, driven by a pressure head that can be increased or decreased. As the pressure head is increased, the effective permeability of the medium decreases and, in contrast to flow through a rigid medium, the flux of water is found to increase towards a finite upper bound such that it becomes insensitive to changes in the pressure head. Measurements of the internal deformation, extracted by particle tracking, show that the medium compacts differentially, with the porosity being lower at the base than at the upper free surface. A general theoretical model is derived, and the predictions of the model give good agreement with experimental measurements from a series of experiments in which the applied pressure head is sequentially increased. However, contrary to theory, all the experimental results display a distinct and repeatable hysteresis: the flux through the material for a particular applied pressure drop is appreciably lower when the pressure has been decreased to that value compared to when it has been increased to the same value.

Figure 1

The setup, with (a) an impermeable base and no flow, and (b) a base that is permeable to the fluid, but not the solid, and a downwards fluid flow with flux $Q\left(t\right)$ per unit area.

Figure 2

Evolution to the steady state for a step change in the dimensionless fluid pressure difference $\mathcal{H}$. (a–b) Step increase at $t=0$ from $\mathcal{H}=0$ to $\mathcal{H}=0.1$ (black solid), $\mathcal{H}=1$ (blue long dashed), or $\mathcal{H}=10$ (red short dashed): (a) the magnitude of the flux, and (b) the depth of the medium. (c–d) The porosity at times between $t=0.01$ and $t=5.12$, separated by a factor of 2, together with the steady-state solution from (23) (dashed), for (c) a step increase from $\mathcal{H}=0$ to $\mathcal{H}=1$, and (d) a step decrease from $\mathcal{H}=1$ to $\mathcal{H}=0$, starting from the steady-state solution of (c). All solutions have $\mathrm{\Phi }=0.4$.

Figure 3

Steady-state results of the specific model, for $\mathrm{\Phi }=0.3$ (black line), $\mathrm{\Phi }=0.4$ (blue long dashed), and $\mathrm{\Phi }=0.5$ (red short dashed): (a) the magnitude of the flux $Q_s$; (b) the depth $l_s$ of the medium; and (c) the effective permeability $k_{\text{eff}}$ of the medium. (d) The porosity $\varphi (z/l_s)$ for $\mathrm{\Phi }=0.4$ and different values of $\mathcal{H}$ between $\mathcal{H}=0.01$ and $\mathcal{H}=10.24$ separated by a factor of 2. (e) The asymptotic prediction for the minimum depth from (24b) (black solid), and the corresponding theoretical minimum value $l_s=1-\mathrm{\Phi }$ for compression under an arbitrary external load (blue dashed).

Figure 4

A schematic showing the experimental setup, as described in the main text. The tee junction was connected to an adjustable clamp that could be moved up or down to set the desired pressure head $H$. Arrows show the direction of the flow when the exit valve at the base of the tank was opened.

Figure 5

Properties of the swollen beads. (a) Histogram of the radii of a sample of 763 particles, measured by image processing a photograph of a monolayer of saturated particles. The mean radius is $0.380$ mm, which is used to estimate the Kozeny-Carman permeability scale $k^{*}=r^2/45=3.2\times {10}^{-9}{\mathrm{m}}^2$, and the standard deviation is $0.08$ mm. (b) Stress $\sigma$ as a function of strain $e$, measured as described in the main text from tests in which the stress was increased and then decreased as indicated by the arrows. The measurements for increasing stress (circles) deviate appreciably from those for decreasing stress (crosses). The same test was repeated three times, shown as blue, red, and green, consecutively. (c) The same data as in (b), converted into a function of $\varphi$ using (2), with $\mathrm{\Phi }=0.41$. The data for increasing stress are compared to a simple fit of the form (15) with an elastic modulus of ${\sigma }^{*}=1900\mathrm{Pa}$ (dashed).

Figure 6

Experimental results with $l_0=11.4$ cm for a series of experiments in which the height $H$ was increased and then decreased four times. (a) The flux per unit cross-sectional area $Q_s$, (b) the depth $l_s$, and (c) the effective permeability $k_{\text{eff}}$. The first sweep up and down in shown in red with crosses. Arrows show the direction of increasing and decreasing $H$. After the first sweep, the measurements of the flux and the effective permeability collapse onto reproducible hysteresis loops.

Figure 7

Experimental results for three sweeps of increasing and decreasing $H$, scaled by the dimensional scalings of §2, for $l_0=8.9$ cm (black dots), $11.4$ cm (blue pluses), and $13.3$ cm (red crosses). (a) The dimensionless flux $Q_s\mu l_0/k^{*}{\sigma }^{*}$, (b) the dimensionless change in the depth $(l_s-l_0)/l_0$, and (c) the dimensionless effective permeability $k_{\text{eff}}/k^{*}$. Note that all results are taken after one initial sweep of increasing and decreasing $H$, and $l_0$ was chosen to be the height of the medium after this sweep, rather than the initial height, in an attempt to account for any particle rearrangement (and thus changing of $\mathrm{\Phi }$) during the initial sweep. The other parameters were ${\sigma }^{*}=1900\mathrm{Pa}, \mathrm{\Phi }=0.41, \rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$, a measured viscosity of $\mu =9.336\times {10}^{-4}\mathrm{Pa}$ s, and $k^{*}=r^2/45=3.2\times {10}^{-9}{\mathrm{m}}^2$.

Figure 8

Comparison of the experimental results from three sweeps as in Fig. 7, showing only the measurements for increasing $H$ (black), with the predictions of the theoretical model (blue dashed), for experiments with $l_0=8.9$ cm. The other parameters are as in Fig. 7.

Figure 9

Measurements from particle tracking of the dimensionless displacement $\zeta /l_0$. For a particle at depth $z$, the displacement is defined by the vertical distance traveled relative to its initial position $Z$, such that $\zeta =z-Z$. Measurements are shown as a function of the dimensionless initial (Lagrangian) position $Z/l_0$, for different initial depths $l_0=8.9$ cm (black), $11.4$ cm (blue), and $13.3$ cm (red). Each experiment comprised three sweeps of increasing and decreasing $H$, and the displacement in each case was measured relative to the position of the particles at the start of that sweep. All measurements were taken after one initial sweep of increasing and decreasing stress, to allow for any initial rearrangement of particles. (a) Measurements for increasing $H$ at $H=10$ cm (dots) and $H=80$ cm (stars), together with predictions of the theoretical model (green lines) given by (3) and (23) using the parameters as in Fig. 7. (b) Smoothed profiles of the experimental measurements for $l_0=11.4$ cm, for increasing stress (blue solid) and decreasing stress (red dashed) at $H=10$ cm (indicated by a star), $H=20$ cm (indicated by a cross), $H=80$ cm (indicated by $\#$), and $H=120$, which was the highest stress reached (black dotted).

Figure 10

Smoothed measurements of the flux over time for a set of experiments with $l_0=13.3$ cm, following a step change in the height at $t=0$: (a) from $H=10$ cm up to $H=40$ cm, and (b) from $H=130$ cm down to $H=40$ cm. The red dashed lines show the predicted flux using the time-dependent model, with the parameters as in Fig. 7. The insets show the comparisons at early times in more detail. It is evident that the experiment evolves over a much longer time scale than the theory predicts.