Hamiltonian formulation towards minimization of viscous fluid fingering

A variational approach has been recently employed to determine the ideal time-dependent injection rate $Q\left(t\right)$ that minimizes fingering formation when a fluid is injected in a Hele-Shaw cell filled with another fluid of much greater viscosity. However, such a calculation is approximate in nature, since it has been performed by assuming a high capillary number regime. In this work, we go one step further, and utilize a Hamiltonian formulation to obtain an analytical exact solution for $Q\left(t\right)$, now valid for arbitrary values of the capillary number. Moreover, this Hamiltonian scheme is applied to calculate the corresponding injection rate that minimizes fingering formation in a uniform three-dimensional porous media. An analysis of the improvement offered by these exact injection rate expressions in comparison with previous approximate results is also provided.

Figure 1

Sketch of the injection rate as a function of time, for the exact injection rate $Q_e\left(t\right)$ given by Eq. (16), for the approximate injection rate $Q_a\left(t\right)\propto c_1^{\prime }+c_2^{\prime }t$ obtained in Ref. [33] [see their Eq. (7)], and for the constant injection rate $Q_0$ given by Eq. (18). Notice that the total volume of injected fluid (area under the curves) in the interval $0\le t\le t_f$ is the same for all these pumping rate cases.

Figure 2

Plot of the interfacial perturbation amplitudes ${\zeta }_n\left(t\right)$ [as given by Eq. (4)], as a function of Fourier mode $n$ at $t=t_f$, when the three injection rates are used: $Q_0,Q_a\left(t\right)$, and $Q_e\left(t\right)$. The resulting interfaces obtained for each of these injection rate schemes are also shown.

Figure 3

Amplitude ratio ${\zeta }_{\max }^a\left(t_f\right)/{\zeta }_{\max }^e\left(t_f\right)$ as a function of $\mathrm{Ca}$, for $R_0=0.01,0.02$, and 0.04. Here ${\zeta }_{\max }^a\left(t_f\right) \left[{\zeta }_{\max }^e\left(t_f\right)\right]$ denotes the maximum amplitude for approximate [exact] injection at $t=t_f$.

Figure 4

Injection rate as a function of time for the exact injection rate $Q_e$ given by Eq. (31), for the approximate injection rate $Q_a$ obtained in Ref. [34] [see their Eq. (23)], and for the constant injection rate $Q_0$ given by Eq. (32).

Figure 5

Log-linear plot of the interfacial perturbation amplitudes ${\zeta }_{\ell }\left(t\right)$ [as given by Eq. (20)], as a function of mode $\ell$ at $t=t_f$, when the three injection rates are used: $Q_0,Q_a\left(t\right)$, and $Q_e\left(t\right)$. The resulting 3D interfaces generated by utilizing each of these three injection strategies are also presented.

Figure 6

Amplitude ratio ${\zeta }_{\max }^a\left(t_f\right)/{\zeta }_{\max }^e\left(t_f\right)$ as a function of $\mathrm{Ca}$, for $R_0=0.01,0.02$, and 0.04. Here ${\zeta }_{\max }^a\left(t_f\right) \left[{\zeta }_{\max }^e\left(t_f\right)\right]$ denotes the maximum amplitude for approximate [exact] injection at $t=t_f$.