On leakage and seepage of CO₂ from geologic storage sites into surface water

Abstract

Geologic carbon sequestration is the capture of anthropogenic carbon dioxide (CO₂) and its storage in deep geologic formations. The processes of CO₂ seepage into surface water after migration through water-saturated sediments are reviewed. Natural CO₂ and CH₄ fluxes are pervasive in surface-water environments and are good analogues to potential leakage and seepage of CO₂. Buoyancy-driven bubble rise in surface water reaches a maximum velocity of approximately 30 cm s⁻¹. CO₂ rise in saturated porous media tends to occur as channel flow rather than bubble flow. A comparison of ebullition versus dispersive gas transport for CO₂ and CH₄ shows that bubble flow will dominate over dispersion in surface water. Gaseous CO₂ solubility in variable-salinity waters decreases as pressure decreases leading to greater likelihood of ebullition and bubble flow in surface water as CO₂ migrates upward.

Appendices

Appendix A

Equations for bubble rise in porous media from Corapcioglu et al. (2004). The equations governing bubble rise in coarse porous media are derived by balancing forces due to buoyancy given by

$$F_{\rm b} = {\left({\rho _{\rm f} - \rho _{\rm g}} \right)}\,g\,\frac{4}{3}\pi\,R_{\rm b} ^{3}, $$

(7)

the surface tension force given by

$$F_{{\rm {st}}} = 2\,\pi\,R^\prime\sigma \,\sin \theta, $$

(8)

where R′ is an equivalent pore throat radius, and the drag force given by

$$F_{\rm d} = A{\left[ {\frac{{150\,\mu _{\rm b} u_{\rm b} {\left({1 - n} \right)}^{2}}}{{d_{\rm p} ^{2} \,n^{3}}} + \frac{{1.75\rho _{\rm g} u_{\rm b} ^{2}\,{\left({1 - n} \right)}}}{{d_{\rm p} \,n^{3} }}} \right]}\,\frac{4}{3}\pi\,R_{\rm b} ^{3} $$

(9)

(variables are defined in Nomenclature). The first term in brackets in Eq. 9 is the Kozeny term, accounting for viscous drag in laminar flow, while the second term is the Burke–Plummer term, accounting for turbulent losses. Summing these three forces and allowing for acceleration of the bubble, we have the balance relation

$$F_{\rm b} - F_{\rm d} - F_{{{\rm st}}} = A_{\rm d} \,\rho _{\rm g} \,\frac{4}{3}\pi\,R_{\rm b} ^{3} {\left({\frac{{\partial u_{\rm b}}}{{\partial t}} +u_{\rm b} \frac{{\partial u_{\rm b}}}{{\partial x}}} \right)}$$

(10)

where the A _d term accounts for entrained liquid ahead of the bubble and is defined as

$$A_{\rm d} = 1 + C_{\rm M} \frac{{\rho _{\rm f}}}{{\rho _{\rm g}}}$$

(11)

Substituting the individual force equations and grouping terms by the powers of bubble rise velocity (u _b), we obtain

$$ - {\left({C_{1}\,u^{2}_{\rm b} + C_{2}\,u_{\rm b} + C_{3}} \right)} = \frac{{\partial u_{\rm b}}}{{\partial t}} + u_{\rm b} \frac{{\partial u_{\rm b}}}{{\partial x}}$$

(12)

where

$$C_{1} = \frac{{1.75\,{\left({1 - n} \right)}A}}{{d_{\rm p} \,n^{3} \,A_{\rm d}}}$$

(13)

$$C_{2} = \frac{{150\,{\left({1 - n} \right)}^{2} A\,\mu _{\rm b}}}{{d_{\rm p} ^{2} \,n^{3} \,\rho _{\rm g} \,A_{\rm d}}}$$

(14)

$$C_{3} = \frac{1}{{\rho _{\rm g} \,A_{\rm d}}}{\left[ {\frac{3}{2}\frac{{R^\prime\sigma\,\sin \theta}}{{R^{3}_{\rm b}}} - {\left({\rho _{\rm f} - \rho _{\rm g}} \right)}\,g} \right]}.$$

(15)

The rise velocity (u _b) can be calculated using the coefficients of Eqs. 13–15 in the quadratic equation (12) for which we assume steady state and zero inertia, i.e., right-hand side of Eq. 12 is set to zero.

Appendix B

Equations for ebullition and diffusion rates are given from Morel and Herring (1993). For each species, the rate of ebullition E _i (mol cm⁻² s⁻¹) is proportional to its partial pressure at the sediment surface:

$$E_{i} = \frac{{P_{i}}}{{P_{z}}}E$$

(16)

where E is the total rate of ebullition of all species together. A steady-state mass balance equation is written for each species at the sediment surface where the sum of its transport by diffusion and ebullition is equal to its rate of formation at depth:

$$\frac{D}{z}{\left({{\left[ {{\rm CO_{2}}} \right]}_{\rm b} - {\left[ {{\rm CO_{2}}} \right]}_{\rm s}} \right)} + EK^{{- 1}}_{{{\rm CO_{2}}}} {\left[ {{\rm CO_{2}}} \right]}_{\rm b} P^{{- 1}}_{z} = m,$$

(17)

$$\frac{D}{z}{\left({{\left[ {{\rm CH_{4}}} \right]}_{\rm b} - {\left[ {{\rm CH_{4}}} \right]}_{s}} \right)} + EK^{{- 1}}_{{CH4}} {\left[ {CH_{4}} \right]}_{\rm b} P^{{- 1}}_{z} = m,$$

(18)

$$\frac{D}{z}{\left({{\left[ {\rm {air}} \right]}_{\rm b} - {\left[ {\rm {air}} \right]}_{\rm s}} \right)} + \frac{{E{\left[ {\rm {air}} \right]}_{\rm b}}}{{K_{{\rm {air}}} P_{z}}} = 0$$

(19)

where the subscripts b and s refer to the bottom and surface concentrations (mol cm⁻³), respectively, and K _i is the Henry’s law constant for each gas species (mol cm⁻³atm⁻¹), and the bottom air flux is assumed to be zero. The diffusive flux is assumed to be driven by the concentration gradient across the entire depth of the surface-water body.

The aqueous concentrations of species at the surface are calculated to be in equilibrium with the atmosphere at 10°C, where $P_{{\rm CO_{2}}}^{{\rm atm}}$, P ^atm_air , and $P_{{\rm CH_{4}}}^{{\rm atm}}$ are 3.12×10⁻⁴, 1.97×10⁻⁶, and 9.87×10⁻¹ atm, respectively (see Table 4 for K _i values):

$$[\hbox{CO}_{2}]_{\rm s} = P_{{\rm CO}_{2}}^{\rm atm} K_{{\rm CO}_{2}}$$

(20)

$$[\hbox{CH}_{4}]_{\rm s} = P_{{\rm CH}_{4}}^{\rm atm} K_{{\rm CH}_{4}}$$

(21)

$$[\hbox{air}]_{\rm s} = P_{\rm air}^{\rm atm} K_{\rm air}.$$

(22)

Unknowns in the mass balance equations are now bottom concentrations and E, while the pressure condition at the sediment surface is:

$$P_{\rm air} + P_{\rm CH_{4}} + P_{\rm CO_{2}} + P_{\rm H_{2}O} = P_{z}.$$

(23)

$P_{{\rm H_{2}O}}$ in Eq. 23 can be neglected relative to the other volatile components and substitute Henry’s law expressions to obtain

$$\frac{{{\left[ {{\rm CO_{2}}} \right]}_{\rm b}}}{{K_{{{\rm CO_{2}}}}}} + \frac{{{\left[ {{\rm CH_{4}}} \right]}_{\rm b}}}{{K_{{{\rm CH_{4}}}}}} + \frac{{{\left[ {{\rm air}} \right]}_{\rm b}}}{{K_{{{\rm air}}}}} = P_{z} $$

(24)

where K _i values are for the P _z considered (Table 4). With the approximations [CO₂]_s << [CO₂]_b and [CH₄]_s << [CH₄]_b, bottom concentrations from Eqs. 17–19 are substituted into Eq. 24 to yield:

$$\frac{m}{{K_{{{\rm CO_{2}}}} D/z + E/P_{z}}} + \frac{m}{{K_{{{\rm CH_{4}}}} D/z + E/P_{z}}} + \frac{{(D/z){\left[ {{\rm air}} \right]}_{\rm s}}}{{K_{{{\rm air}}} D/z + E/P_{z}}} = P_{z}.$$

(25)

By neglecting the first term in Eq. 25 and replacing $K_{{\rm CH_{4}}}$ and K _air with an average K value, an approximate solution for E can be obtained:

$$E = m + {\left({\frac{D}{z}} \right)}{\left[ {{\rm air}} \right]}_{\rm s} - \frac{{P_{z}\,\,KD}}{z}$$

(26)

Substitution of Eq. 26 into Eqs. 17 –19 gives the bottom concentrations of species:

$${\left[ {{\rm CO_{2}}} \right]}_{\rm b} \cong \frac{m}{{D/z + EK^{{- 1}}_{{{\rm CO_{2}}}} P^{{- 1}}_{z}}}$$

(27)

$${\left[ {{\rm CH_{4}}} \right]}_{\rm b} \cong \frac{m}{{D/z + EK^{{- 1}}_{{{\rm CH_{4}}}} P^{{- 1}}_{z}}}$$

(28)

$${\left[ {{\rm air}} \right]}_{\rm b} \cong \frac{{{\left({D/z} \right)}{\left[ {{\rm air}} \right]}_{\rm s}}}{{D/z + EK^{{- 1}}_{{{\rm air}}} P^{{- 1}}_{z}}}$$

(29)

The diffusive and ebullition fluxes, F ^D and F ^E, respectively, of CO₂ and CH₄ can then be calculated:

$$F^{\rm D}_{{{\rm CO_{2}}}} = \frac{D}{z}{\left({{\left[ {{\rm CO_{2}}} \right]}_{\rm b} - {\left[ {{\rm CO_{2}}} \right]}_{\rm s}} \right)}$$

(30)

$$F^{\rm D}_{{{\rm CH_{4}}}} = \frac{D}{z}{\left({{\left[ {{\rm CH_{4}}} \right]}_{\rm b} - {\left[ {{\rm CH_{4}}} \right]}_{\rm s}} \right)}$$

(31)

$$F^{\rm E}_{{{\rm CO_{2}}}} = EK^{{- 1}}_{{{\rm CO_{2}}}} {\left[ {{\rm CO_{2}}} \right]}_{\rm b} P^{{- 1}}_{z} $$

(32)

$$F^{\rm E}_{{{\rm CH_{4}}}} = EK^{{- 1}}_{{{\rm CH_{4}}}} {\left[ {{\rm CH_{4}}} \right]}_{b} P^{{- 1}}_{z} $$

(33)