Forced imbibition in stratified porous media: Fluid dynamics and breakthrough saturation

Imbibition, the displacement of a nonwetting fluid by a wetting fluid, plays a central role in diverse energy, environmental, and industrial processes. While such flows are typically studied in homogeneous porous media with uniform permeabilities, in many cases, the media have multiple parallel strata of different permeabilities. How such stratification impacts the fluid dynamics of imbibition, as well as the fluid saturation after the wetting fluid breaks through to the end of a given medium, is poorly understood. We address this gap in knowledge by developing an analytical model of imbibition in a porous medium with two parallel strata, combined with a pore network model that explicitly describes fluid crossflow between the strata. By numerically solving these models, we examine the fluid dynamics and fluid saturation left after breakthrough. We find that the breakthrough saturation of nonwetting fluid is minimized when the imposed capillary number Ca is tuned to a value ${\text{Ca}}^{*}$ that depends on both the structure of the medium and the viscosity ratio between the two fluids. Our results thus provide quantitative guidelines for predicting and controlling flow in stratified porous media, with implications for water remediation, oil/gas recovery, and applications requiring moisture management in diverse materials.

Figure 1

(a) Schematic of forced imbibition in a stratified medium with crossflow. Blue indicates wetting fluid and white indicates nonwetting fluid. The subscripts $\{c,f\}$ denote the coarse and fine stratum, respectively. $Q$ represents the total imposed volumetric flow rate. $Q_c$ and $Q_f$ represent the volumetric flow rate into the coarse and fine stratum, respectively. $q_{cf}$ and $q_{fc}$ represent the volumetric crossflow rate per length. $x_c\left(t\right)$ and $x_f\left(t\right)$ represent the position of the wetting/nonwetting fluid interface in each stratum. $\{a_i,A_i,k_i,{\varphi }_i\}$ represent structural descriptors of each stratum $i$. (b) Schematic of pore network model. Each stratum is modeled as a series of nodes separated by a distance $\mathrm{\Delta }x$ and connected by resistors. The black resistors represent longitudinal flow, while red resistors represent transverse crossflow. Our theory and simulations do not rely on a specific pore morphology; however, Ref. [59] provides experimental images of an example stratified medium.

Figure 2

Influence of pore throat ratio on imbibition dynamics (a) without crossflow and (b) with crossflow. Plots show the normalized wetting/nonwetting fluid interface position as a function of normalized time. The open colored circles show the numerical solutions to Eqs. (1, 2, 3) and (6) to obtain the wetting/nonwetting fluid interface in each stratum $x_i\left(t\right)$. The connected solid lines show the results of the network model. Green and magenta indicate the fine and coarse stratum, respectively. The interface position $x$ is normalized by the length of the medium $\ell$ and time $t$ is normalized by the characteristic invasion time $\tau \equiv \ell A\varphi /Q$. Diamonds indicate fine-preferential invasion. Circles indicate transitional invasion when the difference between the initial interface velocity in each stratum is $\le 20\%$ the speed of the faster-moving interface. Squares indicate coarse-preferential invasion. The solid blue line indicates the value of (a) ${\text{Ca}}_0^{*}$ or (b) ${\text{Ca}}^{*}$ for each row. The normalized medium length is $\ell /\sqrt{A}=8.3$, the cross-section area ratio is $A_c/A_f=1$, and the viscosity ratio is ${\mu }_{nw}/{\mu }_w=6.2$.

Figure 3

Influence of pore throat ratio on (a), (b) breakthrough time and (c), (d) breakthrough nonwetting fluid saturation. Plots show both metrics for imbibition (a), (c) without crossflow and (b), (d) with crossflow. (a), (b) Breakthrough time plotted as a function of imposed Ca and $a_c/a_f$. The color scale indicates the normalized breakthrough time $t_b/{\tau }_{ch}$ on a logarithmic scale, where ${\tau }_{ch}$ is fixed to be $\tau$ evaluated at $\text{Ca}=4.0\times {10}^{-5}$. (c), (d) Breakthrough nonwetting fluid saturation plotted as a function of imposed Ca and $a_c/a_f$. The color scale indicates the breakthrough saturation on a linear scale. For all panels, the normalized medium length is $\ell /\sqrt{A}=8.3$, the cross-section area ratio is $A_c/A_f=1$, and the viscosity ratio is ${\mu }_{nw}/{\mu }_w=6.2$. The symbols are as described in the caption to Fig. 2. The solid blue line indicates the value of ${\text{Ca}}_0^{*}$ for (a), (c) and of ${\text{Ca}}^{*}$ for (b), (d).

Figure 4

Influence of cross-section area ratio on imbibition dynamics (a) without crossflow and (b) with crossflow. Plots show the normalized wetting/nonwetting fluid interface position as a function of normalized time. The symbols, colors, and lines are as described in the caption to Fig. 2. The normalized medium length is $\ell /\sqrt{A}=8.3$, the pore throat ratio is $a_c/a_f=8.1$, and the viscosity ratio is ${\mu }_{nw}/{\mu }_w=6.2$.

Figure 5

Influence of cross-section area ratio on (a), (b) breakthrough time and (c), (d) breakthrough nonwetting fluid saturation. Plots show both metrics for imbibition (a), (c) without crossflow and (b), (d) with crossflow. (a), (b) Breakthrough time plotted as a function of imposed Ca and $A_c/A_f$. (c), (d) Breakthrough nonwetting fluid saturation plotted as a function of imposed Ca and $A_c/A_f$. The color scales are described in the caption to Fig. 3. For all panels, the normalized medium length is $\ell /\sqrt{A}=8.3$, the viscosity ratio is ${\mu }_{nw}/{\mu }_w=6.2$, and the pore throat ratio is $a_c/a_f=8.1$. The symbols are as described in the caption to Fig. 2. The solid blue line indicates the value of ${\text{Ca}}_0^{*}$ for (a), (c) and of ${\text{Ca}}^{*}$ for (b), (d).

Figure 6

Influence of medium length on imbibition dynamics (a) without crossflow and (b) with crossflow. Plots show the normalized wetting/nonwetting fluid interface position as a function of normalized time. The symbols, colors, and lines are as described in the caption to Fig. 2. The cross-section area ratio is $A_c/A_f=1$, the viscosity ratio is ${\mu }_{nw}/{\mu }_w=6.2$, and the pore throat ratio is $a_c/a_f=8.1$.

Figure 7

Influence of medium length on (a), (b) breakthrough time and (c), (d) breakthrough nonwetting fluid saturation. Plots show both metrics for imbibition (a), (c) without crossflow and (b), (d) with crossflow. (a), (b) Breakthrough time plotted as a function of imposed Ca and $\ell /\sqrt{A}$. (c), (d) Breakthrough nonwetting fluid saturation plotted as a function of imposed Ca and $\ell /\sqrt{A}$. The color scales are described in the caption to Fig. 3. For all panels, the cross-section area ratio is $A_c/A_f=1$, the viscosity ratio is ${\mu }_{nw}/{\mu }_w=6.2$, and the pore throat ratio is $a_c/a_f=8.1$. The symbols are as described in the caption to Fig. 2. The solid blue line indicates the value of ${\text{Ca}}_0^{*}$ for (a), (c) and of ${\text{Ca}}^{*}$ for (b), (d).

Figure 8

Influence of viscosity ratio on imbibition dynamics (a) without crossflow and (b) with crossflow. Plots show the normalized wetting/nonwetting fluid interface position as a function of normalized time. The symbols, colors, and lines are as described in the caption to Fig. 2. The length of the medium is $\ell /\sqrt{A}=8.3$, the cross-section area ratio is $A_c/A_f=1$, and the pore throat ratio is $a_c/a_f=8.1$.

Figure 9

Influence of viscosity ratio on (a), (b) breakthrough time and (c), (d) breakthrough nonwetting fluid saturation. Plots show both metrics for imbibition (a), (c) without crossflow and (b), (d) with crossflow. (a), (b) Breakthrough time plotted as a function of imposed Ca and ${\mu }_{nw}/{\mu }_w$. (c), (d) Breakthrough nonwetting fluid saturation plotted as a function of imposed Ca and ${\mu }_{nw}/{\mu }_w$. The color scales are described in the caption to Fig. 3. For all panels, the normalized medium length is $\ell /\sqrt{A}=8.3$, the cross-section area ratio is $A_c/A_f=1$, and the pore throat ratio is $a_c/a_f=8.1$. The symbols are as described in the caption to Fig. 2. The solid blue line indicates the value of ${\text{Ca}}_0^{*}$ for (a), (c) and of ${\text{Ca}}^{*}$ for (b), (d).

Figure 10

Influence of averaging permeability between strata on imbibition dynamics (a) without crossflow and (b) with crossflow. Plots show the normalized wetting/nonwetting fluid interface position as a function of normalized time. The open colored circles show the numerical solutions to Eqs. (1, 2, 3) and (6) to obtain the wetting/nonwetting fluid interface in each stratum $x_i\left(t\right)$. The connected solid lines show the results of the network model. Green and magenta indicate the fine and coarse stratum, respectively. The black line indicates a homogeneous medium with a permeability that is the area-weighted average of the two strata [i.e., $k_{\mathrm{avg}}=(A_f{k}_f+A_c{k}_c)/A$].The interface position $x$ is normalized by the length of the medium $\ell$ and time $t$ is normalized by the characteristic invasion time $\tau \equiv \ell A\varphi /Q$. Diamonds indicate fine-preferential invasion. Circles indicate transitional invasion when the difference between the initial interface velocity in each stratum is $\le 20\%$ the speed of the faster-moving interface. Squares indicate coarse-preferential invasion. The solid blue line indicates the value of (a) ${\text{Ca}}_0^{*}$ or (b) ${\text{Ca}}^{*}$ for each row. The normalized medium length is $\ell /\sqrt{A}=8.3$, the cross-section area ratio is $A_c/A_f=1$, and the viscosity ratio is ${\mu }_{nw}/{\mu }_w=6.2$.