We present a linear stability analysis of density-driven miscible flow in porous media in the context of carbon dioxide sequestration in saline aquifers. Carbon dioxide dissolution into the underlying brine leads to a local density increase that results in a gravitational instability. The physical phenomenon is analogous to the thermal convective instability in a semi-infinite domain, owing to a step change in temperature at the boundary. The critical time for the onset of convection in such problems has not been determined accurately by previous studies. We present a solution, based on the dominant mode of the self-similar diffusion operator, which can accurately predict the critical time and the associated unstable wavenumber. This approach is used to explain the instability mechanisms of the critical time and the long-wave cutoff in a semi-infinite domain. The dominant mode solution, however, is valid only for a small parameter range. We extend the analysis by employing the quasi-steady-state approximation (QSSA) which provides accurate solutions in the self-similar coordinate system. For large times, both the maximum growth rate and the most dangerous mode decay as $t^{1/4}$. The long-wave and the short-wave cutoff modes scale as $t^{1/5}$ and $t^{4/5}$, respectively. The instability problem is also analysed in the nonlinear regime by high-accuracy direct numerical simulations. The nonlinear simulations at short times show good agreement with the linear stability predictions. At later times, macroscopic fingers display intense nonlinear interactions that significantly influence both the front propagation speed and the overall mixing rate. A dimensional analysis for typical aquifers shows that for a permeability variation of 1—3000 mD, the critical time can vary from 2000 yrs to about 10 days while the critical wavelength can be between 200 m and 0.3 m.

[1] Geological carbon dioxide (CO 2 ) storage is a means of reducing anthropogenic emissions. Dissolution of CO 2 into the brine, resulting in stable stratification, increases storage security. The dissolution rate is determined by convection in the brine driven by the increase of brine density with CO 2 saturation. We present a new analogue fluid system that reproduces the convective behaviour of CO 2 -enriched brine. Laboratory experiments and high-resolution numerical simulations show that the convective flux scales with the Rayleigh number to the 4/5 power, in contrast with a classical linear relationship. A scaling argument for the convective flux incorporating lateral diffusion from downwelling plumes explains this nonlinear relationship for the convective flux, provides a physical picture of high Rayleigh number convection in a porous medium, and predicts the CO 2 dissolution rates in CO 2 accumulations. These estimates of the dissolution rate show that convective dissolution can play an important role in enhancing storage security. [2] The storage of carbon dioxide (CO 2 ) in geological formations has been proposed as a technological means to reduce anthropogenic emissions of this greenhouse gas [Orr, 2009;Benson and Cook, 2006]. The positive buoyancy of supercritical CO 2 relative to the ambient brine filling the pore spaces may lead to leakage along imperfections in the geological seal, which is of considerable concern for the security of long-term storage [Gasda et al., 2004;Pruess, 2005;Neufeld et al., 2009]. One of the primary mechanisms for stable long-term geological storage of CO 2 is the dissolution of injected CO 2 within ambient brine. Under typical conditions injected CO 2 dissolves into the ambient brine thereby increasing the density of the brine [Teng et al., 1997]. This layer of dense, saturated brine forms by the processes of diffusion, dispersion and mechanical mixing during injection and, once it has reached sufficient thickness, becomes rapidly unstable to convective overturning [Ennis-King et al., 2005;Riaz et al., 2006]. The process of convective dissolution of CO 2 has recently been imaged at ambient conditions in a Hele-Shaw cell [Kneafsey and Pruess, 2009], and enhanced mass transfer has been measured at reservoir conditions [Yang and Gu, 2006;Farajzadeh et al., 2007]. Convection is therefore expected in most sequestration sites, and controls the dissolution rate and hence the long-term risk of leakage. Geochemical observations in natural CO 2 reservoirs require large amounts of CO 2 dissolution into the ambient brine and provide field evidence for sustained convective transport of dissolved CO 2 [Gilfillan et al., 2008[Gilfillan et al., , 2009. Convective dissolution of CO 2 is therefore expected in most natural and anthropogenic CO 2 reservoirs, and controls the mobility of carbon in the subsurface. It is therefore an important mechanism in the deep carbon cycle [Sherwood and Ballentine, 2009], and controls the long-term risk of leakage of CO 2 from geological storage.[3] Despi...

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We investigate the effect of viscosity contrast on the stability of gravitationally unstable, diffusive layers in porous media. Our analysis helps evaluate experimental observations of various diffusive (boundary) layer models that are commonly used to study the sequestration of CO2 in brine aquifers. We evaluate the effect of viscosity contrast for two basic models that are characterized with respect to whether or not the interface between CO2 and brine is allowed to move. We find that diffusive layers are in general more unstable when viscosity decreases with depth within the layer compared to when viscosity increases with depth. This behavior is in contrast to the one associated with the classical displacement problem of gravitationally unstable diffusive layers that are subject to mean flow. For the classical problem, a greater instability is associated with the displacement of a more viscous, lighter fluid along the direction of gravity by a less viscous, heavier fluid. We show that the contrasting behavior highlighted in this study is a special case of the classical displacement problem that depends on the relative strength of the displacement and buoyancy velocities. We demonstrate the existence of a critical viscosity ratio that determines whether the flow is buoyancy dominated or displacement dominated. We explain the new behaviors in terms of the interaction of vorticity components related to gravitational and viscous effects.

High-accuracy three-dimensional numerical simulations of miscible displacements with gravity override in homogeneous porous media are carried out for the quarter five-spot configuration. Special emphasis is placed on describing the influence of viscous and gravitational effects on the overall displacement dynamics in terms of the vorticity variable. Even for neutrally buoyant displacements, three-dimensional effects are seen to change the character of the flow significantly, in contrast to earlier findings for rectilinear displacements. At least in part this can be attributed to the time dependence of the most dangerous vertical instability mode. Density differences influence the flow primarily by establishing a narrow gravity layer, in which the effective Péclet number is enhanced owing to the higher flow rate. However, buoyancy forces of a certain magnitude can lead to a pinch-off of the gravity layer, thereby slowing it down. Overall, an increase of the gravitational parameter is found to enhance mostly the vertical perturbations, while larger Pe values act towards amplifying horizontal disturbances. The asymptotic rate of growth of the mixing length varies only with Péclet number. For large Péclet numbers, an asymptotic value of 0.7 is observed. A scaling law for the thickness of the gravity layer is obtained as well. In contrast to immiscible flow displacements, it is found to increase with the gravity parameter.

Nonlinear evolution of viscous and gravitational instability in two-phase immiscible displacements is analyzed with a high-accuracy numerical method. We compare our results with linear stability theory and find good agreement at small times. The fundamental physical mechanisms of finger evolution and interaction are described in terms of the competing viscous, capillary, and gravitational forces. For the parameter range considered, immiscible viscous fingers are found to undergo considerably weak interaction as compared to miscible fingers. The wave number of nonlinear fingers decreases rapidly due to the shielding mechanism and scales uniformly as t−1 at long times. We have observed that even a small amount of density contrast can eliminate viscous fingers. The dominant feature for these flows is the gravity tongue, which develops a “ridge instability” when capillary and gravity effects are of similar magnitude.

Gravitationally unstable, transient, diffusive boundary layers play an important role in carbon dioxide sequestration. Though the linear stability of these layers has been studied extensively, there is wide disagreement in the results, and it is not clear which methodology best reflects the physics of the instability. We demonstrate that this disagreement stems from an inherent sensitivity of the problem to how perturbation growth is measured. During an initial transient period, the concentration and velocity fields exhibit different growth rates and these rates depend on the norm used to measure perturbation amplitude. This sensitivity decreases at late times as perturbations converge to dominant quasi-steady eigenmodes. Therefore, we characterize the linear regime by measuring the duration of the initial transient period, and we interpret the convergence process by examining the growth rates and non-orthogonality of the quasi-steady eigenmodes. To judge the relevance of various methodologies and perturbation structures to physical systems, we demonstrate that every perturbation has a maximum allowable initial amplitude above which the sum of the base-state and perturbation produces unphysical negative concentrations. We then perform direct numerical simulations to demonstrate that optimal perturbations considered in previous studies cannot support finite initial amplitudes. Consequently, convection in physical systems is more likely triggered by “sub-optimal” perturbations that support finite initial amplitudes.

Linear stability analysis of immiscible displacements is carried out for both viscously and gravitationally unstable two-phase flows in porous media with very large adverse viscosity ratios. Capillary dispersion is the proper dissipative mechanism in this case which sets both the preferred length scale and the band width of the spectrum of unstable length scales. The growth rate, the most dangerous and the cutoff wavenumbers, all scale linearly with the capillary number. We show that the instability is governed by fluid properties across the shock rather than those across the full Buckley–Leverett profile. The shock total mobility ratio provides a sufficient condition for the onset of instability; however, it is not an appropriate criterion for predicting the magnitude of the growth rate, particularly for large viscosity ratios. The details of the relative permeability functions are observed to have a significant influence on the stability characteristics. For neutrally buoyant flows the maximum growth rate scales linearly with the viscosity ratio while the most dangerous and the cutoff wavenumbers scale with the square root of the viscosity ratio. In the case of displacements with density contrast, the maximum growth rate scales with the square of the unstable gravity number while the most dangerous and the cutoff wavenumbers scale with an exponent of 1.2, for all viscosity ratios. A marginal stability curve is computed for stable and unstable regions in the parameter space of the viscosity ratio and the gravity number. It is found that flows with unstable viscosity contrasts are more readily stabilized with buoyancy as compared to the viscous stabilization of gravitationally unstable flows.