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

Abstract

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.

A Simple Analytical Model for Estimating the Dissolution-Driven Instability in a Porous Medium

“…where coefficients A − and A + are expressions that are displayed in [8], α is the scaled wavenumber, and z c is the critical thickness of the boundary layer. For a small interface deflection, we have:…”

“…The mathematical model consists of the conservation equations of mass, momentum, and species within the Boussinesq approximation, which incorporate horizontal-to-vertical diffusion anisotropy, height-dependent permeability change, and a first-order reaction between the mixture and the host mineralogy [7,8,17]:…”

“…The other symbols are defined as follows: q c is the coefficient of solutal expansion, d is the separation between the two horizontal boundaries, K is the permeability coefficient, C 0 is a reference CO 2 concentration level, ϕ p is the porosity, κ v is the vertical component of the diffusion tensor, β is the reaction rate, and the parameter > 0 numerically quantifies the value of the permeability variation. As detailed in either VH or WH, the model is simplified by considering the poloidal component of the divergence free velocity vector field, φ, and upon removing the pressure term, we obtain [7,8]:…”

“…The question that has been posed, and which this article addresses, deals with the determination of the stability of a state of convection, the existence of which is known to have a positive impact on improving the dissolution rate of CO 2 into the brine. This article is devoted to a more detailed discussion of an alternative view at tackling the stability problem put forth by Vo and Hadji [7] and Wanstall and Hadji [8], henceforth referred to as VH and WH, respectively. Note that VH imposed a Neumann boundary condition at the top boundary by modeling a physical situation wherein the injection site is assumed closed after a carbon-rich layer has formed at the top.…”

“…WH, however, considered the more physical Dirichlet boundary condition at the top boundary, the so-called one-sided model [9]. Furthermore, due to the way in which the base state is introduced in [7,8], the method is expected to yield only conservative estimates for the instability threshold values. However, one great advantage of this approach is that the problem at hand can be viewed as a stationary Rayleigh-Bénard problem, the stability of which is mathematically well-defined and can be treated by a variety of existing mathematical tools.…”