A step function density profile model for the convective stability of CO$_2$ 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 possess 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.

Appendix

In the process of conducting the linear stability analysis, the homogeneous linear system of equations obtained through the matching conditions at $Z_0$ is described by

$$\begin{aligned} \left\{ \begin{array}{ll} Q_{11}\,A^- + Q_{12}\,B^-+ Q_{13}\,A^+ + Q_{14}\,B^+ &{}= 0, \\ Q_{21}\,A^- + Q_{22}\,B^-+ Q_{23}\,A^+ + Q_{24}\,D^+ &{}= 0, \\ Q_{31}\,A^- + Q_{32}\,B^-+ Q_{33}\,A^+ + Q_{34}\,B^+ &{}= 0, \\ Q_{41}\,A^- + Q_{42}\,B^-+ Q_{43}\,A^+ + Q_{44}\,B^+ &{}= 0, \\ \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} Q_{11}= & {} \sinh {(\alpha Z_0)}, Q_{12}=Z_0\,\sinh {(\alpha Z_0)} +(\beta /\alpha ) Z_0\,\cosh {(\alpha Z_0)}, Q_{13}=-\sinh {(\alpha (Z_0-1))},\\ Q_{14}= & {} -(Z_0-1)\,\sinh {(\alpha (Z_0-1))} +(\beta /\alpha ) (Z_0-1)\,\cosh {(\alpha (Z_0-1))},\\ Q_{21}= & {} \alpha \,\cosh {(\alpha Z_0)}, Q_{22}=(1+\beta \,Z_0)\,\sinh {(\alpha Z_0)}+(\alpha Z_0-\beta /\alpha )\,\cosh {(\alpha Z_0)},\\ Q_{23}= & {} -\alpha \cosh {(\alpha (Z_0-1))}, \\ Q_{24}= & {} -(1-\beta (Z_0-1))\sinh {(\alpha (Z_0-1)) }- (\alpha (Z_0-1)-\beta /\alpha )\cosh {(\alpha (Z_0-1)},\\ Q_{31}= & {} \alpha ^2 \sinh {(\alpha Z_0)}, Q_{33}= -\alpha ^2 \sinh {(\alpha (Z_0-1))},\\ Q_{32}= & {} (2\alpha +\beta \alpha Z_0)\cosh {(\alpha Z_0)} + (\alpha ^2 Z_0 + 2 \beta )\sinh {(\alpha Z_0)},\\ Q_{34}= & {} -(2\alpha - \beta \alpha (Z_0-1)) \cosh {(\alpha (Z_0-1))} - (\alpha ^2 (Z_0-1)-2 \beta ) \sinh {(\alpha (Z_0-1))}, \\ Q_{41}= & {} -\alpha ^3\,\cosh {(\alpha Z_0)}- R \alpha ^2 \sinh {(\alpha Z_0)}, \\ Q_{43}= & {} \alpha ^3\,\cosh {(\alpha (Z_0-1))},\\ Q_{42}= & {} -(3 \alpha \,\beta + Z_0\,\alpha ^3) \cosh {(\alpha Z_0)} - (3 \alpha ^2 + \beta Z_0 \alpha ^2) \sinh {(\alpha Z_0)}- R \alpha ^2 (Z_0 \sinh {(\alpha Z_0)}\\&\quad +\, Z_0 (\beta /\alpha ) \cosh {(\alpha (Z_0-1)}),\\ Q_{44}= & {} -(3 \alpha \,\beta + (Z_0-1) \alpha ^3) \cosh {(\alpha (Z_0-1)} + (3 \alpha ^2 - \beta (Z_0-1) \alpha ^2) \sinh {(\alpha (Z_0-1))}, \\ P_{41}= & {} -\alpha ^3\,\cosh {(\alpha Z_0)}, P_{42}=-(3 \alpha \,\beta + Z_0\,\alpha ^3) \cosh {(\alpha Z_0)} - (3 \alpha ^2 + \beta Z_0 \alpha ^2) \sinh {(\alpha Z_0)},\\ M_{41}= & {} \alpha ^2 \sinh {(\alpha Z_0)}, M_{42}=\alpha ^2 (Z_0 \sinh {(\alpha Z_0)} + Z_0 (\beta /\alpha ) \cosh {(\alpha (Z_0-1)}).\\ \end{aligned}$$

The matrices $\mathbf{Q}_1$ and $\mathbf{Q}_2$ are given by

$$\begin{aligned} \mathbf{Q_1}=\left[ \begin{array}{cccc} Q_{11} &{} \quad Q_{12} &{} \quad Q_{13} &{} \quad Q_{14} \\ Q _{21} &{} \quad Q_{22} &{} \quad Q_{23} &{} \quad Q_{24} \\ Q_{31} &{} \quad Q_{32} &{} \quad Q_{33} &{} \quad Q_{34} \\ P_{41} &{} \quad P_{42} &{} \quad Q_{43} &{} \quad Q_{44} \end{array} \right] , \qquad \mathbf{Q_2}=\left[ \begin{array}{cccc} Q_{11} &{} \quad Q_{12} &{} \quad Q_{13} &{} \quad Q_{14} \\ Q _{21} &{} \quad Q_{22} &{} \quad Q_{23} &{} \quad Q_{24} \\ Q_{31} &{} \quad Q_{32} &{} \quad Q_{33} &{} \quad Q_{34} \\ M_{41} &{} \quad M_{42} &{} \quad 0 &{} \quad 0 \end{array} \right] . \quad \end{aligned}$$

The eigenfunctions for the linear problem for the case $c=0$ at $z=1$ are given by

$$\begin{aligned}&W^-(z)=(1-1.3783\,z)\sinh {(\alpha _C z)}, \end{aligned}$$

(52)

$$\begin{aligned}&W^+(z)= 0.9482(z-1)\,\cosh {(\alpha _C(z-1))}-0.8876 \sinh {(\alpha _C(z-1))}, \end{aligned}$$

(53)

$$\begin{aligned}&{\mathscr {J}}= \frac{\alpha _C^2\,(1-1.3783 Z_{0C}) \sinh {(\alpha _C Z_{0C})}}{\alpha _C\,\cosh {\alpha _C}\bigl (\cosh {(\alpha _C(1-Z_{0C})}/\sinh {(\alpha _C(1-Z_{0C})} \bigr )-1},\nonumber \\&S^-(z)=-{\mathscr {J}} \cosh {(\alpha _C z)}, \end{aligned}$$

(54)

$$\begin{aligned}&S^+(z) = \frac{{\mathscr {J}} \cosh {(\alpha _C Z_{0C})} }{\sinh {(\alpha _C(1-Z_{0C}))}}\,\sinh {(\alpha _C(1-z))}, \end{aligned}$$

(55)

where $\alpha _C=2.4$ and $Z_{0C}=0.38$.