Dynamic-wetting effects in finite-mobility-ratio Hele-Shaw flow

In this paper we study the effects of dynamic wetting on the immiscible displacement of a high viscosity fluid subject to the radial injection of a less viscous fluid in a Hele-Shaw cell. The displaced fluid can leave behind a trailing film that coats the cell walls, dynamically affecting the pressure drop at the fluid interface. By considering the nonlinear pressure drop in a boundary element formulation, we construct a Picard scheme to iteratively predict the interfacial velocity and subsequent displacement in finite-mobility-ratio flow regimes. Dynamic wetting delays the onset of finger bifurcation in the late stages of interfacial growth and at high local capillary numbers can alter the fundamental mode of bifurcation, producing vastly different finger morphologies. In low mobility ratio regimes, we see that finger interaction is reduced and characteristic finger breaking mechanisms are delayed but never fully inhibited. In high mobility ratio regimes, finger shielding is reduced when dynamic wetting is present. Finger bifurcation is delayed, which allows the primary fingers to advance further into the domain before secondary fingers are generated, reducing the level of competition.

Figure 1

(a) Planar view of the 2D radial injection problem, with initial asymmetric boundary. (b) Side view of the Hele-Shaw cell showing trailing film of displaced brine (in blue).

Figure 2

Effect of varying the local capillary number on interfacial displacement. (a) $t=55$; (b) $t=110$. ${\text{Ca}}_g=1000$; $\beta =10$.

Figure 3

Local capillary number variation with time for the dashed line case (${\text{Ca}}_l\text{initial}=1.05\times {10}^{-2}$) in Fig. 2.

Figure 4

Effect of varying ${\text{Ca}}_l$ and ${\text{Ca}}_g$ on interfacial evolution, $t=88$. (a) ${\text{Ca}}_g=2000$; initial ${\text{Ca}}_l=2.19\times {10}^{-4}$. (b) ${\text{Ca}}_g=4000$;initial ${\text{Ca}}_l=4.47\times {10}^{-4}$. (c) ${\text{Ca}}_g=8000$; initial ${\text{Ca}}_l=9.04\times {10}^{-4}$. —, $J_0=0$; , $J_0=3.8$. $\beta =10$.

Figure 5

Asymmetric interfacial displacement with a mobility ratio of 10. (a) $t=20$; (b) $t=45$; (c) $t=70$. The dashed box represents a magnified area for Fig. 6. —, $J_0=0$; , $J_0=3.8$.

Figure 6

Magnified plot of finger interaction in Fig. 5. —, $J_0=0$; , $J_0=3.8$.

Figure 7

Asymmetric interfacial displacement with a mobility ratio of 250. (a) $t=20$; (b) $t=45$; (c) $t=70$. —, $J_0=0$; , $J_0=3.8$.