Forced Imbibition in Stratified Porous Media

Imbibition in a porous medium plays a central role in diverse energy, environmental, and industrial processes. In many cases, the medium has multiple parallel strata of different permeabilities; however, how this stratification impacts imbibition is poorly understood. We address this gap in knowledge by directly visualizing forced imbibition in three-dimensional porous media with two parallel strata. We find that imbibition is spatially heterogeneous: for a small capillary number Ca, the wetting fluid preferentially invades the fine stratum, while for Ca above a threshold value, the fluid instead preferentially invades the coarse stratum. This threshold value depends on the medium geometry, the fluid properties, and the presence of residual wetting films in the pore space. These findings are well described by a linear stability analysis that incorporates crossflow between the strata. Thus, our work provides quantitative guidelines for predicting and controlling flow in stratified porous media.

Figure 1

An overview of experiments in stratified porous media. (a) A schematic of the experimental setup. We use confocal microscopy to visualize imbibition in 3D stratified media composed of sintered packings of glass beads of different sizes. (b) An optical section acquired within a stratified medium. The pore space is saturated with the fluorescently dyed wetting fluid and the black circles show cross sections of the beads making up the medium. The scale bar denotes 1 mm.

Figure 2

The heterogeneous-invasion behavior during primary imbibition. (a),(c),(e) Optical sections through the entirety of three stratified porous media, obtained a time $t$ after the wetting fluid begins to invade each medium. The red region shows the dyed wetting fluid while the black shows the undyed nonwetting fluid, which initially fully saturates the pore space. The imposed flow direction is from left to right. The scale bars denote 1 mm. (b),(d),(f) Kymographs showing the positions at which the fine and coarse strata are invaded in green and magenta, respectively, while the positions at which both strata are invaded are shown in white, as they vary over time. At a low capillary number Ca, wetting fluid preferentially invades the fine stratum, as shown in (a) and (b). At a moderate Ca, the wetting fluid simultaneously invades both strata, as shown in (c) and (d). At a sufficiently large Ca, the wetting fluid instead preferentially invades the coarse stratum, as shown in (e) and (f). (e) A summary of primary imbibition experiments performed at different Ca; the green and magenta points indicate preferential invasion into the fine and coarse strata, respectively, while the yellow points indicate that both strata are invaded at similar speeds. Specifically, to identify the yellow points, we measure the initial speed of the fluid-fluid interface in each stratum and apply the quantitative criterion that an experiment is in the transitional regime when the speed difference is $\le 20\mathrm{\%}$ the speed of the faster-moving interface. A magenta square and a magenta cross are overlapping. The circles, squares, diamonds, down-pointing triangles, up-pointing triangles, and crosses indicate $0.2\le A_c/A_f<0.7$ and $1.4\le \ell <2.0$ cm, $0.2\le A_c/A_f<0.7$ and $2.0\le \ell <3.0$ cm, $0.7\le A_c/A_f<1.5$ and $1.4\le \ell <2.0$ cm, $0.7\le A_c/A_f<1.5$ and $2.0\le \ell <3.0$ cm, $0.7\le A_c/A_f<1.5$ and $\ell \ge 3.0$ cm, and $A_c/A_f\ge 1.5$ and $2.0\le \ell <3.0$ cm, respectively.

Figure 3

(a) A schematic of the theoretical model of imbibition in stratified porous media. We consider two parallel porous strata $i\in \{f,c\}$ oriented along the $x$ direction with equal length $\ell$ but with different pore-throat sizes $a_i$, absolute permeabilities $k_i$, cross-section areas $A_i$, flow rates $Q_i$, and extent of invasion by the wetting fluid $x_i$. Crossflow upstream and downstream of the wetting-fluid–nonwetting-fluid interface is represented by the flow rates per unit length along the $x$ direction $q_{\mathrm{cf}}$ and $q_{\mathrm{fc}}$. (b) In the absence of crossflow, a linear stability analysis predicts a transition between invasion behaviors at $\mathrm{Ca}={\mathrm{Ca}}_0^{\ast }$; the experiments exceed this prediction by a factor of approximately $5$. (c) Incorporating crossflow, our analysis predicts a transition between invasion behaviors at $\mathrm{Ca}={\mathrm{Ca}}^{\ast }$, where ${\mathrm{Ca}}^{\ast }$ is given by Eq. (2), in good agreement with the experimental data. The symbols are as described in the caption to Fig. 2.

Figure 4

The wetting films alter invasion behavior during secondary imbibition. (a) A 3D reconstruction of the residual wetting films, shown in white, in a fine stratum. The beads are rendered transparent for ease of visualization; the wetting fluid surrounds the nonwetting fluid, also rendered transparent, which fills the rest of the pore space. The residual wetting fluid forms thick pendular rings at the contacts between beads and thin films of thickness $h$ coat the rest of the bead surfaces. The scale bar denotes 50 $\mu \mathrm{m}$. (b),(c) Optical sections through the entirety of two stratified porous media, obtained a time $t$ after the wetting fluid begins to invade each medium. The red region shows the invading wetting fluid, the white shows pre-existing residual wetting fluid and the black shows the undyed nonwetting fluid in the rest of the pore space. The imposed flow direction is from left to right. At a low capillary number Ca, the wetting fluid preferentially invades the fine stratum, as shown in (b), while at a sufficiently large Ca, the wetting fluid instead invades both strata, as shown in (c). The scale bars denote 1 mm. (d) A summary of the secondary-imbibition experiments performed at different Ca; the green points indicate preferential invasion into the fine stratum while the yellow points indicate that both strata are invaded at similar speeds. (e) Our analysis predicts a transition between invasion behaviors at $\mathrm{Ca}={\mathrm{Ca}}^{\ast \ast }$, where ${\mathrm{Ca}}^{\ast \ast }$ is calculated as described in the main text, in good agreement with the experimental data. The symbols are as described in the caption to Fig. 2, with the addition of the “$\times$” indicating $A_c/A_f\ge 1.5$ and $1.4\le \ell <2.0$ cm.

Figure 5

The dependence of the predicted threshold ${\mathrm{Ca}}^{\ast }$ on the geometric characteristics of the strata, as shown by the blue curves. All parameters except the one being varied are held fixed using values representative of our experiments. The orange dashed lines show comparisons to asymptotic power-law scaling behaviors. (a) The dependence on the length of the medium, $\ell$, normalized by the characteristic transverse dimension $\sqrt{A}$. For small $\ell$, ${\mathrm{Ca}}^{\ast }$ only decreases weakly with increasing $\ell$, eventually asymptoting to the approximately ${\ell }^{-1}$ scaling shown by the orange dashed line. Our experiments only probe $\ell /\sqrt{A}$ varying in a narrow range from approximately $5$ to $17$, for which the predicted ${\mathrm{Ca}}^{\ast }$ does not vary appreciably and is in good agreement with the experimentally determined threshold. (b) ${\mathrm{Ca}}^{\ast }$ only varies slightly with the area fraction $A_c/A_f$. Thus, despite our experiments having $A_c/A_f$ varying from approximately $0.2$ to $4$, the predicted ${\mathrm{Ca}}^{\ast }$ does not vary appreciably and is in good agreement with the experimentally determined threshold. (c) ${\mathrm{Ca}}^{\ast }$ decreases with an increasing pore-throat ratio $a_c/a_f$, eventually asymptoting to the approximately $(a_c/a_f{)}^{-1}$ scaling shown by the orange dashed line. (d) Keeping the pore-throat ratio $a_c/a_f$ fixed but proportionally increasing both $a_c$ and $a_f$ increases ${\mathrm{Ca}}^{\ast }$ linearly, as shown by the comparison with the approximately $a_c$ scaling shown by the orange dashed line.

Figure 6

The schematics of a cylindrical pore throat covered with thin films of the wetting fluid. $a$ represents the initial radius of the pore throat, $h$ represents the thickness of the thin wetting films, and $a-h$ represents the radius of the nonwetting fluid.