Time-dependent injection strategies for multilayer Hele-Shaw and porous media flows

We use linear stability analysis to demonstrate how to stabilize multilayer radial Hele-Shaw and porous media flows with a time-dependent injection rate. Sufficient conditions for an injection rate that maintains a stable flow are analytically derived for flows with an arbitrary number of fluid layers. We show numerically that the maximum injection rate for a stable flow decreases proportional to $t^{-1/3}$ for $t\gg 1$ regardless of the number of fluid layers. However, the constant of proportionality depends on the number of layers and increases at a rate that is proportional to the number of interfaces to the two-thirds power. Therefore, flows with more fluid layers can be stable with faster time-dependent injection rates than comparable flows with fewer fluid layers, even when the additional layers are very thin. We also show that for unstable flows, which may be required to inject a given amount of fluid in a fixed amount of time, an increasing injection rate is less unstable than a constant or decreasing injection rate, and that the inclusion of more fluid layers can overcome poor injection strategies.

Figure 1

The basic solution for ($\mathrm{N}+2$)-layer flow. The radius of the outermost interface is $R_N$.

Figure 2

Plots of the maximum stable injection rate versus time for two-layer flow (solid line) and three-layer flow (dashed line) as well as the lower bound for stabilization derived from Gershgorin's circle theorem and given by Eq. (39) (dotted line). The values of the parameters are ${\mu }_i=0.2$, ${\mu }_o=1$, $T=1$, $R\left(0\right)=1$ for two-layer flow and ${\mu }_i=0.2,{\mu }_1=0.6,{\mu }_o=1,T_0=T_1=1,R_0\left(0\right)=0.8,R_1\left(0\right)=1$ for three-layer flow.

Figure 3

Plots of the maximum stable injection rate versus time for two-layer flow (solid line) and three-layer flow (dashed line) as well as the lower bound Eq. (39) for three values of $R_0$. The values of the parameters are ${\mu }_i=0.2$, ${\mu }_o=1$, $T=1$, $R\left(0\right)=1$ for two-layer flow and ${\mu }_i=0.2,{\mu }_1=0.6,{\mu }_o=1,T_0=T_1=1,R_0\left(0\right)=0.7,0.8,0.9,R_1\left(0\right)=1$ for three-layer flow.

Figure 4

Plots of the maximum stable injection rate versus time for flows with two through six fluid layers. Plot (a) uses the lower bound given by Eq. (48) for flows with three or more layers while plot (b) shows the actual maximum stable injection rate. The values of the parameters are ${\mu }_i=0.2$, ${\mu }_o=1$, $T_j=1$ for all $j$, $R_0\left(0\right)=0.6$, and $R_N\left(0\right)=1$. The interfaces are equally spaced at time $t=0$ in all cases and the viscous jumps are the same at all interfaces.

Figure 5

Plot of the maximum stable injection rate versus time for three-layer flow (solid line) as well as the lower bound derived from Gershgorin's circle theorem [see Eq. (39)] and the decoupled lower bound [see Eq. (48)]. The values of the parameters are ${\mu }_i=0.2,{\mu }_1=0.6,{\mu }_o=1,T_0=T_1=1,R_0\left(0\right)=0.8,R_1\left(0\right)=1$.

Figure 6

Plot (a) is the maximum stable injection rate versus time for several different numbers of interfaces on a log-log scale. Each curve is of the form $Q\left(t\right)=C{t}^{-1/3}$ for $t\gg 1$. Plot (b) shows $C$ versus the number of interfaces on a log-log scale. The values of the parameters are ${\mu }_i=0.2$, ${\mu }_o=1$, $T_j=1$ for all $j$, $R_0\left(0\right)=0.6$, and $R_N\left(0\right)=1$. The interfaces are equally spaced at time $t=0$ in all cases and the viscous jumps are the same at all interfaces.

Figure 7

The maximum growth rate ${\sigma }_{\text{max}}$ versus time for two-interface flows with (a) a relatively low constant injection rate $Q=15$, and (b) a relatively high constant injection rate $Q=200$. The values of the other parameters are ${\mu }_i=0.2$, ${\mu }_1=0.6$, ${\mu }_o=1$, $T_0=T_1=1$, $R_0\left(0\right)=0.5$, and $R_N\left(0\right)=1$.

Figure 8

The top row shows the maximum growth rate ${\sigma }_{\text{max}}$ versus time and the injection rate $Q$ versus time, and the bottom row shows the positions of the interfaces at the beginning, middle, and end times for (a) a constant injection rate with two interfaces, (b) a linearly increasing injection rate with two interfaces, and (c) a linearly increasing injection rate with three interfaces. The values of the parameters are ${\mu }_i=0.2$, ${\mu }_o=1$, $T_0=T_1=1$, $R_0\left(0\right)=0.5$, and $R_N\left(0\right)=1$.

Figure 9

Plot of the wave number $n_{\text{max}}$ of the most unstable wave versus time for three-layer flow with a linearly increasing injection rate. The values of the parameters are the same as Fig. 8 $[{\mu }_i=0.2,{\mu }_1=0.6,{\mu }_o=1,T_0=T_1=1,R_0\left(0\right)=0.5,R_1\left(0\right)=1]$.

Figure 10

The top row shows the maximum growth rate ${\sigma }_{\text{max}}$ versus time and the injection rate $Q$ versus time, and the bottom row shows the positions of the interfaces at the beginning, middle, and end times for (a) a constant injection rate with two interfaces, (b) a decreasing injection rate with two interfaces, and (c) a decreasing injection rate with three interfaces. The values of the parameters are ${\mu }_i=0.2$, ${\mu }_o=1$, $T_0=T_1=1$, $R_0\left(0\right)=0.5$, and $R_N\left(0\right)=1$.