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.
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.
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 t = t p that have a maximum amplification at a final time t = t f . We then investigate the sensitivity of the optimal perturbations to the initial time, t p , and the final time, t f , as well as different measures of perturbation amplification. Due to the transient nature of the base state, we demonstrate that there is an optimal initial perturbation time, t o p . 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.
In this paper, we follow a similar procedure as proposed by Koval (SPE J 3(2): [145][146][147][148][149][150][151][152][153][154] 1963) to analytically model CO 2 transfer between the overriding carbon dioxide layer and the brine layer below it. We show that a very thin diffusive layer on top separates the interface from a gravitationally unstable convective flow layer below it. Flow in the gravitationally unstable layer is described by the theory of Koval, a theory that is widely used and which describes miscible displacement as a pseudo two-phase flow problem. The pseudo two-phase flow problem provides the average concentration of CO 2 in the brine as a function of distance. We find that downstream of the diffusive layer, the solution of the convective part of the model, is a rarefaction solution that starts at the saturation corresponding to the highest value of the fractional-flow function. The model uses two free parameters, viz., a dilution factor and a gravity fingering index. A comparison of the Koval model with the horizontally averaged concentrations obtained from 2-D numerical simulations provides a correlation for the two parameters with the Rayleigh number. The obtained scaling relations can be used in numerical simulators to account for the density-driven natural convection, which cannot be currently captured because the grid cells are typically orders of magnitude
We present a novel reaction analogy (RA) based forcing method for generating stationary scalar fields in incompressible turbulence. The new method can produce more general scalar PDFs (e.g. double-delta) than current methods, while ensuring that scalar fields remain bounded. Such features are useful for generating initial fields in non-premixed combustion or for studying non-Gaussian scalar turbulence. The RA method mathematically models hypothetical chemical reactions that convert reactants in a mixed state back into its pure unmixed components. Various types of chemical reactions are formulated and the corresponding mathematical expressions derived such that the reaction term is smooth in the scalar space and is consistent with mass conservation. For large values of the scalar dissipation rate, the method produces statistically steady double-delta scalar PDFs. Quasi-uniform, Gaussian, and stretched exponential scalar statistics are recovered for smaller values of the scalar dissipation rate. The shape of the scalar PDF can be further controlled by changing the stoichiometric coefficients of the reaction. The ability of the new method to produce fully developed passive scalar fields with quasi-Gaussian PDFs is also investigated, by exploring the convergence of the third order mixed structure function to the "four-thirds" Yaglom's law. * dond@lanl.gov † livescu@lanl.gov ‡ jairyu@cau.ac.kr arXiv:1805.11413v2 [physics.flu-dyn]
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