Convective stability of carbon sequestration in anisotropic porous media

Hill, Antony A.; Morad, Mohammad Reza

doi:10.1098/rspa.2014.0373

Cited by 31 publications

(40 citation statements)

References 14 publications

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“…Except for the dependence of the permeability on depth, our mathematical formulation in Sec. 2 is similar to Hill and Morad's model [15]. For this model, the exchange of instabilities has been shown to hold.…”

Section: Introductionsupporting

confidence: 50%

“…(12) in [6]), this ratio happens to play the same role as our ratio of diffusivities ξ . Hill and Morad [15] considered a linear function of depth which we retrieve with our exponential model when we consider a slowly decaying exponential function. It is also noteworthy to observe that these models not only capture qualitatively the space variation of permeability and diffusivity, but their simple mathematical form allows for the derivation of purely analytical results.…”

Section: Formulationmentioning

confidence: 99%

“…It is also noteworthy to observe that these models not only capture qualitatively the space variation of permeability and diffusivity, but their simple mathematical form allows for the derivation of purely analytical results. With the assumptions that both Darcy's law and the Boussinesq approximation hold, the governing system of non-dimensionalized equations, which consists of Darcy's equation, the conservation of mass and carbon dioxide equations and an equation of state, is described by [15] …”

Section: Formulationmentioning

confidence: 99%

“…These studies have considered Dirichlet boundary conditions at both top and lower walls to obtain an unstably stratified steady equilibrium reference state, the stability of which can then be carried out. The nonlinear behavior of the steady profile considered in [15] for the one-sided model is due solely to the presence of the reaction term in the model, without which the equilibrium state is uniform and thus stably stratified. Other investigations have considered approximations to the unsteady pure diffusion boundary-layer profile (cf.…”

Section: Introductionmentioning

confidence: 99%

“…Some investigations have adopted a linear steady diffusive basic state profile to model the carbon sequestration as a typical convection problem in a fluid saturated medium (e.g. [15]- [20]). These studies have considered Dirichlet boundary conditions at both top and lower walls to obtain an unstably stratified steady equilibrium reference state, the stability of which can then be carried out.…”

Section: Introductionmentioning

confidence: 99%

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A step function density profile model for the convective stability of CO $$_2$$ 2 geological sequestration

Wanstall

Hadji

2017

J Eng Math

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The convective stability associated with carbon sequestration is usually investigated by adopting an unsteady diffusive basic profile to account for the space and time development of the carbon saturated boundary layer instability. The method of normal modes is not applicable due to the time dependence of the nonlinear base profile. Therefore, the instability is quantified either in terms of critical times at which the boundary layer instability sets in or in terms of long time evolution of initial disturbances. This paper adopts an unstably stratified basic profile having a step function density with top heavy carbon saturated layer (boundary layer) overlying a lighter carbon free layer (ambient brine). The resulting configuration resembles that of the Rayleigh-Taylor problem with buoyancy diffusion at the interface separating the two layers. The discontinuous reference state satisfies the governing system of equations and boundary conditions and pertains to an unstably stratified motionless state. Our model accounts for anisotropy in both diffusion and permeability and chemical reaction between the carbon dioxide rich brine and host mineralogy. We consider two cases for the boundary conditions, namely an impervious lower boundary with either a permeable (one-sided model) or poorly permeable upper boundary. These two cases posses neither steady nor unsteady unstably stratified equilibrium states. We proceed by supposing that the carbon dioxide that has accumulated below the top cap rock forms a layer of carbon saturated brine of some thickness that overlies a carbon-free brine layer. The resulting stratification remains stable until the thickness, and by the same token the density, of the carbon saturated layer is sufficient to induce the fluid to overturn. The existence of a finite threshold value for the thickness is due to the stabilizing influence of buoyancy diffusion at the interface between the two layers. With this formulation for the reference state, the stability calculations will be in terms of critical boundary layer thickness instead of critical times, although the two formulations are homologous. This approach is tractable by the classical normal mode analysis. Even though it yields only conservative threshold instability conditions, it offers the advantage for an analytically tractable study that puts forth expressions for the carbon concentration convective flux at the interface and explores the flow patterns through both linear and weakly nonlinear analyses.

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Section: Introductionsupporting

confidence: 50%

Section: Formulationmentioning

confidence: 99%

Section: Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A step function density profile model for the convective stability of CO $$_2$$ 2 geological sequestration

Wanstall

Hadji

2017

J Eng Math

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Dissolution‐driven convection of a power‐law fluid in a porous medium in the presence of chemical reaction

Reddy,

Ragoju,

Reddy

et al. 2023

Heat Trans

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The flow through porous medium accounts for numerous applications in various fields namely, agriculture, geothermal sciences, and engineering. Furthermore, dissolution‐driven convection in porous media has grabbed great attention in recent years due to its practical applications in long‐term geological storage of carbon dioxide, in the production of mineral deposits, and other industrial applications. In this regard, the current numerical analysis focuses on addressing the thermal instability of dissolution‐driven convective phenomena of a power‐law fluid through a porous horizontal domain with a first‐order chemical reaction. For linear stability analysis, the method of normal modes has been employed to solve governing dimensionless equations which give rise to an eigenvalue problem. The bvp4c routine in MATLAB R2020a has been used to solve the raised problem for the onset of convection. The impact of Damköhler number, Péclet number, and power‐law index on the onset of convection has been investigated. The role of these critical parameters is found to be highly significant in stabilizing the system. An increase in the power‐law index causes stabilization or destabilization in the system, depending on the Péclet number. An enhancement in the magnitude of Damköhler number makes the system stable for all values of the Péclet number. Also, Damköhler and critical Rayleigh number are inter‐related, that is, an increment in Damköhler number results in the enhancement of critical Rayleigh numbers, which in turn leads to stabilization of the system. The critical wave number is observed to have a remarkable influence on Damköhler number as well as power‐law index.

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Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

Straughan

2015

Advances in Mechanics and Mathematics

111

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In this chapter we describe work of [366] and of [393] on thermal convection in a Darcy porous material in a vertical layer subject to different temperatures on the vertical walls.For the case of a single temperature the problem where the porous medium occupies a vertical layer subject to differential heating from one side to the other has attracted much attention. This problem has much application to insulation, such as in the area of building design, and is of importance in window double glazing. For the single temperature situation the first proof that a vertical porous slab of Darcy type which is held at fixed but different temperatures on the vertical walls is stable to perturbations from the equilibrium state is due to [156]. His analysis is based on the linear theory and analyses the two-dimensional problem.[409] further analysed this problem but in three-dimensions and he treated the completely nonlinear situation by employing energy-like integral methods. In particular, [409] produced a threshold for the Rayleigh number which guarantees global nonlinear stability, i.e. regardless of how large the initial perturbation may be. He also employed a generalized energy method to demonstrate that the result of [156] is true in the fully three-dimensional case although the initial data must be restricted. Articles dealing with other interesting aspects of convection in a vertical porous slab or pipe involving effects like double diffusion, and LTNE, include the papers by [32, 33, 36, 46, 211, 220, 221, 329, 330, 362] and [460]. The methods of energy stability techniques applied specifically to convection in a vertical porous slab are addressed in [144, 229] and [356], and these are discussed in detail in the book by [414, pp. 126-134].The analogous problem in LTNE to that of [156] was first addressed by [366]. Specifically [366] dealt with the [156] problem of thermal convection in a vertical porous medium, but he employed the theory of local thermal non-equilibrium, allowing for different fluid and solid temperatures. The interesting work of [366] demonstrates that the vertical configuration is always stable according to linear theory, even with the much more complicated LTNE theory. [393] continued the problem of [366] but they analysed the complete nonlinear three-dimensional situation.

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Convective stability of carbon sequestration in anisotropic porous media

Cited by 31 publications

References 14 publications

A step function density profile model for the convective stability of CO $$_2$$ 2 geological sequestration

A step function density profile model for the convective stability of CO $$_2$$ 2 geological sequestration

Dissolution‐driven convection of a power‐law fluid in a porous medium in the presence of chemical reaction

Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

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