We study the linear stability of gravitationally unstable, transient, diffusive boundary layers in porous media using non-modal stability theory. We first perform a classical optimization procedure, using an adjoint-based method, to obtain the perturbations at the initial time . By rescaling the problem, we develop analytical relationships for the optimal initial time and wavenumber in terms of aquifer properties. We also demonstrate that the classical optimization procedure essentially recovers the dominant perturbation structures predicted by a quasi-steady modal analysis. Although the classical optimal perturbations are mathematically valid, we observe that due to physical constraints, they are unlikely to reflect analogous laboratory experiments. Therefore, we propose a modified optimization procedure (MOP) that constrains the optimization to physically admissible initial perturbation fields. We compare the results of the classical and modified optimization procedures with quasi-steady modal analyses and initial value problems commonly used in the literature. Finally, we validate the predictions of the modified optimization scheme by performing direct numerical simulations (DNS) that emulate the onset of convection in physical systems.

The linear stability of transient diffusive boundary layers in porous media has been studied extensively for its applications to carbon dioxide sequestration. The onset of nonlinear convection, however, remains understudied because the transient base state invalidates the traditional stability methods that are used for autonomous systems. We demonstrate that the onset time of nonlinear convection, $t=t_{\mathit{on}}$ , can be determined from an expansion that is two orders of magnitude faster than a direct numerical simulation. Using the expansion, we explore the sensitivity of $t_{\mathit{on}}$ to the initial perturbation magnitude and wavelength, as well as the initial time at which a perturbation is initiated. We find that there is an optimal initial time and wavelength that minimize $t_{\mathit{on}}$ , and we obtain analytical relationships for these parameters in terms of aquifer properties and initial perturbation magnitude. This importance of the initial perturbation time and magnitude is often overlooked in previous studies. To investigate perturbation evolution at late-times, $t>t_{\mathit{on}}$ , we perform direct numerical simulations that reveal two unique features of transient diffusive boundary layers. First, when a boundary layer is perturbed with a single horizontal Fourier mode, nonlinear mechanisms generate a zero-wavenumber response whose magnitude eventually surpasses that of the fundamental mode. Second, when a boundary layer is simultaneously perturbed with many Fourier modes, the late-time perturbation magnitude is concentrated in the zero-wavenumber mode, and there is no clearly dominant, non-zero, wavenumber. These unique results are further interpreted by comparison with direct numerical simulations of Rayleigh–Bénard convection.

The stability of gravitationally unstable, transient boundary layers in heterogeneous saline aquifers is examined with respect to the onset of natural convection. Permeability is assumed to vary periodically across the thickness of the aquifer. We study the interaction between permeability variation and concentration perturbations within the boundary layer. We observe that the instability decreases with an increase in the permeability variance if the boundary layer thickness is large compared with the permeability wavelength. On the other hand, when the boundary layer thickness is smaller than the permeability wavelength, the behaviour of instability as a function of variance depends on the phase of permeability variation. Such behaviours are shown to result from the interaction of two modes of vorticity production related to the coupling of concentration and velocity perturbations with the magnitude and gradient of permeability variation, respectively. We show that these two modes of vorticity production, when coupled with the transient nature of the boundary layer, determine the evolutionary paths followed by the most amplified perturbations that trigger the onset of convection. When the permeability variance is large, we find that small changes in the permeability field can lead to large changes in the onset times for convection.

We study the effect of permeability heterogeneity on the stability of gravitationally unstable, transient, diffusive boundary layers in porous media. Permeability is taken to vary periodically in the horizontal plane normal to the direction of gravity. In contrast to the situation for vertical permeability variation, the horizontal perturbation structures are multimodal. We therefore use a two-dimensional quasi-steady eigenvalue analysis as well as a complementary initial value problem to investigate the stability behavior in the linear regime, until the onset of convection. We find that thick permeability layers enhance instability compared with thin layers when heterogeneity is increased. On the contrary, for thin layers the instability is weakened progressively with increasing heterogeneity to the extent that the corresponding homogeneous case is more unstable. For high levels of heterogeneity, we find that a small change in the permeability field results in large variations in the onset time of convection, similar to the instability event in the linear regime. However, this trend does not persist unconditionally because of the reorientation of vorticity pairs due to the interaction of evolving perturbation structures with heterogeneity. Consequently, an earlier onset of instability does not necessarily imply an earlier onset of convection. A resonant amplification of instability is observed within the linear regime when the dominant perturbation mode is equal to half the wavenumber of permeability variation. On the other hand, a substantial damping occurs when the perturbation mode is equal to the harmonic and sub-harmonic components of the permeability wavenumber. The phenomenon of such harmonic interactions influences both the onset of instability as well as the onset of convection.

We study the interfacial evolution of immiscible two-phase flow within a capillary tube in the partial wetting regime using direct numerical simulation. We investigate the flow patterns resulting from the displacement of a more viscous fluid by a less viscous one under a wide range of wettability conditions. We find that beyond a wettability dependent critical capillary number, a uniform displacement by a less viscous fluid can transition into a growing finger that eventually breaks up into discrete blobs by a series of pinch-off events for both wetting and nonwetting contact angles. This study validates previous experimental observations of pinch-off for wetting contact angles and extends those to nonwetting contact angles. We find that the blob length increases with the capillary number. We observe that the time between consecutive pinch-off events decreases with the capillary number and is greater for more wetting conditions in the displaced phase. We further show that the blob separation distance as a function of the difference between the inlet velocity and the contact line speed collapses into two monotonically decreasing curves for wetting and nonwetting contact angles. For the phase separation in the form of pinch-off, this work provides a quantitative study of the emerging length and timescales and their dependence on the wettability conditions, capillary effects, and viscous forces.