Onset of density-driven instabilities in fractured aquifers

Linear stability analysis is conducted to study the onset of density-driven convection involved in solubility trapping of $\mathrm{C}{\mathrm{O}}_2$ in fractured aquifers. The effect of physical properties of a fracture network on the stability of a diffusive boundary layer in a saturated fractured porous media is investigated using the dual porosity concept. Linear stability analysis results show that both fracture interporosity flow and fracture storativity play an important role in the stability behavior of the system. It is shown that a diffusive boundary layer under the gravity field in fractured porous media with lower fracture storativity and/or higher fracture interporosity flow coefficient is more stable. We present scaling relations for the onset of convective instability in fractured aquifers with single and variable matrix block size distribution. These findings improve our understanding of density-driven flow in fractured aquifers and are important in the estimation of potential storage capacity, risk assessment, and storage site characterization and screening.

Figure 1

(a) Idealization of fractured rock with single-size block matrix and (b) geometry and boundary conditions considered in this study [56]. $C^{*}$ is the equilibrium concentration, and C is the concentration of the diffusing species in the aqueous phase. $u$ and $w$ denote the horizontal and vertical components of the Darcy velocity in the fracture domain, respectively.

Figure 2

(a) Base-state concentration profiles versus depth at $\widehat{t}=0.08$ and (b) $\mathrm{C}{\mathrm{O}}_2$ diffusive flux at the upper interface for fracture domain with a constant fracture interporosity flow coefficient of $\lambda ={10}^3$ and different fracture storativity coefficients $\omega =1$, $\omega ={10}^{-1}$, $\omega ={10}^{-2}$, and $\omega ={10}^{-3}$.

Figure 3

Base-state concentration profiles versus depth in fracture domain with fracture storativity $\omega ={10}^{-2}$, and different fracture interporosity flow coefficients of $\lambda ={10}^2,{10}^3$, and ${10}^4$ at $\widehat{t}=0.08$.

Figure 4

Perturbation growth rates versus wave number for systems with different fracture storativity and interporosity flow coefficients at a constant $\mathrm{Ra}=500$ and ${\widehat{t}}_0=18\times {10}^{-3}$.

Figure 5

Onset of instability scaled with Ra for fractured systems with single-size matrix blocks at different fracture storativity and interporosity flow coefficients.

Figure 6

Onset of instability scaled with Ra for fractured systems with single-size matrix blocks at different fracture interporosity flow coefficients.

Figure 7

Neutral stability curves for fractured systems with single-size matrix blocks and a constant fracture interporosity flow coefficient $\lambda =100$, at different fracture storativity coefficients.

Figure 8

Neutral stability curves for fractured systems with single-size matrix blocks and a constant fracture storativity $\omega =0.1$, at different fracture interporosity flow coefficients.

Figure 9

Onset of instability as a function of Ra number for fractured systems with different matrix block sizes represented by exponential probability density function at constant fracture storativity and interporosity flow coefficients. The inset plots show the matrix block size probability distribution.

Figure 10

Scaled onset time of instability for fractured systems with variable matrix block size at different fracture interporosity flow coefficients.