Immiscible three-dimensional fingering in porous media: A weakly nonlinear analysis

We present a weakly nonlinear theory for the development of fingering instabilities that arise at the interface between two immiscible viscous fluids flowing radially outward in a uniform three-dimensional (3D) porous medium. By employing a perturbative second-order mode-coupling scheme, we investigate the linear stability of the system as well as the emergence of intrinsically nonlinear finger branching events in this 3D environment. At the linear stage, we find several differences between the 3D radial fingering and its 2D counterpart (usual Saffman-Taylor flow in radial Hele-Shaw cells). These include the algebraic growth of disturbances and the existence of regions of absolute stability for finite values of viscosity contrast and capillary number in the 3D system. On the nonlinear level, our main focus is to get analytical insight into the physical mechanism resulting in the occurrence of finger tip-splitting phenomena. In this context, we show that the underlying mechanism leading to 3D tip splitting relies on the coupling between the fundamental interface modes and their first harmonics. However, we find that in three dimensions, in contrast to the usual 2D fingering structures normally encountered in radial Hele-Shaw flows, tip splitting into three branches can also be observed.

Figure 1

Plot of the growth rate $\mathrm{\Lambda }$ as a function of $l$ for three values of the unperturbed radius $R$. Here we set $\mathrm{Ca}=110$ and $A=1$. The small open circles locate the most unstable mode $l_{\max }$. It is clear that variations in $R$ do not influence the width of the band of unstable modes and the mode of maximum growth rate $l_{\max }$.

Figure 2

Stability diagram $A-\mathrm{Ca}$ illustrating linearly stable and unstable regions of 3D viscous fingering. The boundary separating the two regions is calculated by setting $\lambda \left(l_{\max }\right)=0$. The 3D patterns illustrate the shape of two typical fluid-fluid interfaces that arise at the linear regime during 3D fingering formation in porous media.

Figure 3

Plot of typical 3D fingering interfaces for which initial conditions only contain modes with $l=3$: (a) and (d) the purely linear interfaces, (b) and (e) the equivalent nonlinear interfaces, and (c) and (f) the same nonlinear interfaces shown in (b) and (e) but with the mode with $l=6$ removed. The patterns illustrated in (a)–(c) used $\mathrm{Ca}=60$ and $R=500$, while the structures displayed in (d)–(f) utilized the parameters $\mathrm{Ca}=80$ and $R=50$.

Figure 4

Plot of characteristic 3D fingering interfaces for which initial conditions only contain modes with $l=4$: (a) and (d) the purely linear interfaces, (b) and (e) the corresponding nonlinear interfaces, Finally, and (c) and (f) the same nonlinear interfaces shown in (b) and (e) but with the mode with $l=8$ filtered out. The patterns illustrated in (a)–(c) used $\mathrm{Ca}=110$ and $R=105$, while the structures displayed in (d)–(f) utilized the parameters $\mathrm{Ca}=130$ and $R=57.5$.

Figure 5

The 3D fingering interfaces for $\mathrm{Ca}=110,R=80,l=4$, and $m=4$: (a) the purely linear interface, (b) the weakly nonlinear interface, and (c) the weakly nonlinear interface, where the mode $a_{88}$ has been removed.

Figure 6

Variation of (a) the fundamental mode cosine amplitude $a_{44}$ and (b) the harmonic mode cosine amplitude $a_{88}$, as $R$ is increased. This is done for two values of the capillary number: $\mathrm{Ca}=110$ (solid curves) and $\mathrm{Ca}=120$ (dashed curves).

Figure 7

The 3D fingering interfaces showing different types of fingertip-splitting patterns: (a) and (b) interfaces having $l=3,\mathrm{Ca}=90$, and the same initial amplitude $|{\zeta }_{lm}|=0.03$ but different initial phases and (c) and (d) interfaces having $l=4,\mathrm{Ca}=150$, and the same initial amplitude $|{\zeta }_{lm}|=0.0009$ but different initial phases. One can identity (a) and (c) traditional two-lobed tip splitting and (b) and (d) a more peculiar three-lobed tip splitting.