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Semianalytical Solution for \(\text{ CO}_{2}\) Plume Shape and Pressure Evolution During \(\text{ CO}_{2}\) Injection in Deep Saline Formations

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Abstract

The injection of supercritical carbon dioxide (\(\text{ CO}_{2})\) in deep saline aquifers leads to the formation of a \(\text{ CO}_{2}\) rich phase plume that tends to float over the resident brine. As pressure builds up, \(\text{ CO}_{2}\) density will increase because of its high compressibility. Current analytical solutions do not account for \(\text{ CO}_{2}\) compressibility and consider a volumetric injection rate that is uniformly distributed along the whole thickness of the aquifer, which is unrealistic. Furthermore, the slope of the \(\text{ CO}_{2}\) pressure with respect to the logarithm of distance obtained from these solutions differs from that of numerical solutions. We develop a semianalytical solution for the \(\text{ CO}_{2}\) plume geometry and fluid pressure evolution, accounting for \(\text{ CO}_{2}\) compressibility and buoyancy effects in the injection well, so \(\text{ CO}_{2}\) is not uniformly injected along the aquifer thickness. We formulate the problem in terms of a \(\text{ CO}_{2}\) potential that facilitates solution in horizontal layers, with which we discretize the aquifer. Capillary pressure is considered at the interface between the \(\text{ CO}_{2}\) rich phase and the aqueous phase. When a prescribed \(\text{ CO}_{2}\) mass flow rate is injected, \(\text{ CO}_{2}\) advances initially through the top portion of the aquifer. As \(\text{ CO}_{2}\) is being injected, the \(\text{ CO}_{2}\) plume advances not only laterally, but also vertically downwards. However, the \(\text{ CO}_{2}\) plume does not necessarily occupy the whole thickness of the aquifer. We found that even in the cases in which the \(\text{ CO}_{2}\) plume reaches the bottom of the aquifer, most of the injected \(\text{ CO}_{2}\) enters the aquifer through the layers at the top. Both \(\text{ CO}_{2}\) plume position and fluid pressure compare well with numerical simulations. This solution permits quick evaluations of the \(\text{ CO}_{2}\) plume position and fluid pressure distribution when injecting supercritical \(\text{ CO}_{2}\) in a deep saline aquifer.

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Abbreviations

\(b\) :

Thickness of the aquifer

\(b_\mathrm{c} \) :

Thickness of the \(\text{ CO}_{2}\) plume at the injection well

\(d\) :

Thickness of a layer

\(g\) :

Gravity

\(h_\alpha \) :

Head of \(\alpha \)-phase (\(\alpha =c,w)\)

\(J_\alpha \) :

Mass flow rate per unit thickness of \(\alpha \)-phase (\(\alpha =c,w)\)

\(k\) :

Intrinsic permeability

\(k_{\mathrm{r}\alpha } \) :

Relative permeability of \(\alpha \)-phase (\(\alpha =c,w)\)

\(n\) :

Number of layers

\(N_g \) :

Gravity number

\(P_0 \) :

Reference fluid pressure corresponding to the hydrostatic pressure at depth \(z_0 \)

\(P_\mathrm{cc} \) :

Capillary entry pressure

\(P_\alpha \) :

Fluid pressure of \(\alpha \)-phase (\(\alpha =c,w\))

\(Q_0 \) :

\(\text{ CO}_{2}\) Volumetric flow rate

\(Q_\mathrm{m} \) :

Prescribed \(\text{ CO}_{2}\) mass flow rate

\(Q_\mathrm{w} \) :

Brine volumetric flow rate

\(\mathbf{q}_\alpha \) :

Volumetric flux of \(\alpha \)-phase (\(\alpha =c,w)\)

\(R\) :

Radius of influence

\(r\) :

Radial coordinate

\(r_\mathrm{i} \) :

Radial position of the interface at the depth \(z\)

\(r_\mathrm{if} \) :

Interface position at the bottom of the \(\text{ CO}_{2}\) plume

\(r_\mathrm{p} \) :

Radius of the injection well

\(S_\alpha \) :

Degree of saturation of \(\alpha \)-phase (\(\alpha =c,w)\)

\(S_\mathrm{rw} \) :

Residual degree of saturation of brine

\(S_\mathrm{s} \) :

Specific storage coefficient

\(t\) :

Time

\(z\) :

Vertical coordinate

\(z_0 \) :

Reference depth

\(\Delta z_j \) :

Thickness of layer \(j\)

\(z_\mathrm{f} \) :

Depth of the bottom of the \(\text{ CO}_{2}\) plume

\(\alpha \) :

Phase index, \(c\) for the \(\text{ CO}_{2}\) rich phase and \(w\) for the aqueous phase

\(\beta \) :

\(\text{ CO}_{2}\) compressibility

\(\zeta \) :

Vertical position of the \(\text{ CO}_{2}\) plume with respect to \(z_0 \)

\(\Phi _\alpha \) :

Potential of \(\alpha \)-phase \((\alpha =c,w)\)

\(\Phi _i \) :

Potential at the interface

\(\Phi _{R} \) :

Potential at the radius of influence

\(\varphi \) :

Porosity

\(\mu _\alpha \) :

Viscosity of \(\alpha \)-phase \((\alpha =c,w)\)

\(\rho _0 \) :

A reference \(\text{ CO}_{2}\) density corresponding to the reference pressure \(P_0 \)

\(\rho _\alpha \) :

Fluid density of \(\alpha \)-phase \((\alpha =c,w)\)

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Acknowledgments

V.V. wishes to acknowledge the Spanish Ministry of Science and Innovation (MCI), through the “Formación de Profesorado Universitario”, and the “Colegio de Ingenieros de Caminos, Canales y Puertos—Catalunya” for their financial support. This study has been funded by Fundación Ciudad de la Energía (Spanish Government) (www.ciuden.es) and by the European Union through the “European Energy Programme for Recovery” and the Compostilla OXYCFB300 project. We also want to acknowledge the financial support received from the ‘MUSTANG’ (www.co2mustang.eu) and ‘PANACEA’ (www.panacea-co2.org) projects (from the European Community’s Seventh Framework Programme FP7/2007-2013 under Grant agreements No. 227286 and No. 282900, respectively).

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Appendices

Appendix 1

Here we develop the mathematical formulation of the problem for the case in which \(\text{ CO}_{2}\) density varies linearly with pressure (Eq. 24), and \(\text{ CO}_{2}\) viscosity and brine properties are constant.

First, we integrate Eq. (4) for the \(\text{ CO}_{2}\) phase, which yields

$$\begin{aligned} h_\mathrm{c} =z-z_0 +\frac{1}{g\beta }\ln \left({1+\frac{\beta }{\rho _0}\left({P_\mathrm{c} -P_0}\right)}\right). \end{aligned}$$
(27)

The inverse of Eq. (27) gives the \(\text{ CO}_{2}\) pressure as a function of the head as

$$\begin{aligned} P_\mathrm{c} -P_0 =\frac{\rho _0 }{\beta }\left({{\text{ e}}^{g\beta \left({h_\mathrm{c}-\left({z-z_0}\right)} \right)}-1} \right). \end{aligned}$$
(28)

Integrating Eq. (12) and using Eq. (28) gives the following expression for the \(\text{ CO}_{2}\) potential

$$\begin{aligned} \Phi _\mathrm{c} =\frac{\pi k\rho _0^2 }{\mu _\mathrm{c} \beta }{\text{ e}}^{-2g\beta \left({z-z_0 } \right)}\left({{\text{ e}}^{2g\beta h_\mathrm{c} }-1} \right). \end{aligned}$$
(29)

Since the exponent \(2g\beta h_\mathrm{c} \) is small, \(\left({{\text{ e}}^{2g\beta h_\mathrm{c}}-1} \right)\) can be approximated as \(2g\beta h_\mathrm{c} \). Therefore, the \(\text{ CO}_{2}\) potential can be expressed as:

$$\begin{aligned} \Phi _\mathrm{c} \approx 2\pi gk\frac{\rho _0^2 }{\mu _\mathrm{c} }h_\mathrm{c} \text{ e}^{-2g\beta \left({z-z_0} \right)}, \end{aligned}$$
(30)

where the potential is composed of a part corresponding to a constant \(\text{ CO}_{2}\) density (\(\rho _0 )\) multiplied by a correction due to \(\text{ CO}_{2}\) compressibility [the exponential in the right-hand side of Eq. (30)]. Combining Eqs. (27) and (29) and operating, yields an expression of \(\text{ CO}_{2}\) pressure as a function of the potential

$$\begin{aligned} P_\mathrm{c} -P_0 =\sqrt{\frac{\mu _\mathrm{c} }{\pi k\beta }\Phi _\mathrm{c} +\frac{\rho _0^2 }{\beta ^{2}}{\text{ e}}^{-2g\beta \left({z-z_0 } \right)}}-\frac{\rho _0}{\beta }. \end{aligned}$$
(31)

Note that the head (Eq. 27) at the interface can be expressed as a function of the \(\text{ CO}_{2}\) density as

$$\begin{aligned} h_{\mathrm{ci}} =z-z_0 +\frac{1}{g\beta }\ln \left({\frac{\rho _0 +\beta \left({P_{\mathrm{ci}}-P_0} \right)}{\rho _0 }} \right)=z-z_0+\frac{1}{g\beta }\ln \left({\frac{\rho _{\mathrm{ci}} }{\rho _0}}\right), \end{aligned}$$
(32)

where the subscript \(i\) indicates interface. Combining Eq. (29) with Eq. (32) yields the following expression for the \(\text{ CO}_{2}\) potential at the interface

$$\begin{aligned} \Phi _\mathrm{i} =\frac{\pi k\rho _0^2 }{\mu _\mathrm{c} \beta }\left({\frac{\rho _{\mathrm{ci}}^2 }{\rho _0^2 }-{\text{ e}}^{-2g\beta \left( {z-z_0 } \right)}} \right). \end{aligned}$$
(33)

In the brine phase, integration of Eq. (4) yields

$$\begin{aligned} h_\mathrm{w} =z-z_0 +\frac{P_\mathrm{w}-P_0 }{\rho _\mathrm{w}g} \end{aligned}$$
(34)

which is the expression of the head for an incompressible fluid. Integrating Eq. (12) gives the potential in the brine phase as

$$\begin{aligned} \Phi _\mathrm{w} =2\pi gk\frac{\rho _\mathrm{w}^2 }{\mu _\mathrm{w} }h_\mathrm{w}. \end{aligned}$$
(35)

Combining Eqs. (34) with (35), gives the following expression for the brine pressure

$$\begin{aligned} P_\mathrm{w} -P_0 =\frac{\mu _\mathrm{w} }{2\pi k\rho _\mathrm{w}}\Phi _\mathrm{w} -\left({z-z_0 }\right)\rho _\mathrm{w}g. \end{aligned}$$
(36)

Note that, the brine pressure varies with the logarithm of the distance to the injection well [see the form of the potential in Eq. (16a)].

Appendix 2

A system of two equations with two unknowns has to be solved in step 4 of the time stepping algorithm when a mass flow rate is prescribed at the injection well. The unknowns are the head at the well and the thickness of the \(\text{ CO}_{2}\) plume at the well. The two equations are Eqs. (23) and (8).

Combining Eq. (23) with Eqs. (18), (29), and (33), after some algebra, gives the following expression for the head at the well as a function of the increment of the \(\text{ CO}_{2}\) plume thickness

$$\begin{aligned} {\text{ e}}^{2g\beta h_\mathrm{c} }=\frac{Q-A_1 \Delta z_\mathrm{f} +A_2 -A_3 }{A_4 }, \end{aligned}$$
(37)

where

$$\begin{aligned} Q&= \frac{\mu _\mathrm{c} \beta }{\pi k\rho _0^2 }Q_\mathrm{m}, \end{aligned}$$
(38a)
$$\begin{aligned} A_1&= \frac{\mu _\mathrm{c} \beta }{\pi k\rho _0^2 }\frac{\bar{{\rho }}_\mathrm{c} \pi r_{\mathrm{if}}^2 \varphi \left( {1-S_{\mathrm{rw}} } \right)}{\Delta t}, \end{aligned}$$
(38b)
$$\begin{aligned} A_2&= \sum _{j=1}^m {\frac{{\rho _{\mathrm{ci},j}^2 }/{\rho _0^2 }}{\ln \left( {{r_j }/{r_\mathrm{p} }} \right)}} \Delta z_j , \end{aligned}$$
(38c)
$$\begin{aligned} A_3&= \frac{\mu _\mathrm{c} \beta }{\pi k\rho _0^2 }\sum _{j=1}^n {\frac{J_{\mathrm{c}z,j+1} \ln \left( {{r_{j+1} }/{r_j }} \right)}{\ln \left( {{r_j }/{r_\mathrm{p} }} \right)}} \Delta z_j , \end{aligned}$$
(38d)
$$\begin{aligned} A_4&= \sum _{j=1}^m {\frac{e^{-2g\beta \left( {z_j -z_0 } \right)}}{\ln \left( {{r_j }/{r_\mathrm{p} }} \right)}} \Delta z_j. \end{aligned}$$
(38e)

Next, assuming hydrostatic conditions at the well (Eq. 8) and that the bottom of the \(\text{ CO}_{2}\) plume coincides where brine pressure equals \(\text{ CO}_{2}\) pressure and using Eqs. (29) and (31) give the second equation of the system of equations

$$\begin{aligned} {\text{ e}}^{g\beta h_\mathrm{c} }=p+\frac{\rho _\mathrm{w} }{\rho _0 }g\beta \Delta z_\mathrm{f}, \end{aligned}$$
(39)

where

$$\begin{aligned} p=\frac{\beta }{\rho _0 }P_\mathrm{w} \left({z_{f-1} }\right)+1, \end{aligned}$$
(40)

where \(P_\mathrm{w} \left({z_{\mathrm{f}-1} } \right)\) is the brine pressure evaluated at the depth reached by the \(\text{ CO}_{2}\) plume in the previous time step. The combination of Eqs. (37) and (39) gives the following quadratic equation

$$\begin{aligned} \Delta z_\mathrm{f}^2 +B\Delta z_\mathrm{f}+C=0, \end{aligned}$$
(41)

where

$$\begin{aligned} B&= \frac{2p\rho _0 }{g\beta \rho _\mathrm{w} }+\frac{\rho _0^2 {\text{ e}}^{-2g\beta \left( {z_\mathrm{f} -z_0 } \right)}}{\rho _\mathrm{w}^2 g^{2}\beta ^{2}}\frac{A_1 }{A_4 }, \end{aligned}$$
(42a)
$$\begin{aligned} C&= \frac{\rho _0^2 }{\rho _\mathrm{w}^2 g^{2}\beta ^{2}}\left( {p^{2}-\frac{Q+A_2 -A_3 }{A_4 }{\text{ e}}^{-2g\beta \left( {z_\mathrm{f} -z_0 } \right)}} \right). \end{aligned}$$
(42b)

Once Eq. (41) is solved and the increment of the \(\text{ CO}_{2}\) plume thickness at the well in a given time step is known, the head at the well can be calculated from Eq. (37) as

$$\begin{aligned} h_\mathrm{c} \left({r_\mathrm{p} } \right)=\frac{1}{2g\beta }\exp \left({\frac{Q-A_1 \Delta z_\mathrm{f} +A_2 -A_3 }{A_4 }} \right). \end{aligned}$$
(43)

Appendix 3

The mean \(\text{ CO}_{2}\) density in a given layer has to be calculated in order to apply Eq. (23). Assuming that \(\text{ CO}_{2}\) density varies linearly with pressure (Eq. 24), and using Eqs. (31), (16c), and (33), after some algebra, the following expression for the \(\text{ CO}_{2}\) density is obtained

$$\begin{aligned} \rho _\mathrm{c} =\sqrt{\rho _{\mathrm{ci}}^2 +\frac{\mu _\mathrm{c} \beta }{\pi k}J_\mathrm{c} \ln \frac{r_j }{r}}. \end{aligned}$$
(44)

The mean \(\text{ CO}_{2}\) density in a layer is obtained from dividing the \(\text{ CO}_{2}\) mass in a given layer by the volume that it occupies

$$\begin{aligned} \bar{{\rho }}=\frac{1}{V}\int \limits _{z_{j-1} }^{z_j } {\int \limits _{r_\mathrm{p} }^{r_j } {2\pi r\varphi \left({1-S_{\mathrm{rw}} } \right)\rho _\mathrm{c} } } {\text{ d}}r{\text{ d}}z. \end{aligned}$$
(45)

Introducing Eq. (44) in Eq. (45) and integrating yields

$$\begin{aligned} \begin{array}{l} \bar{{\rho }}=\frac{1}{\left({r_j^2 -r_\mathrm{p}^2 } \right)}\left[{r^{2}\sqrt{a-b\ln r}-\frac{1}{2}\sqrt{\frac{\pi }{2}}\sqrt{b}{\text{ e}}^{2a/b}{\text{ erf}}\left( {\sqrt{\frac{2}{b}}\sqrt{a-b\ln r}} \right)} \right]_{r_\mathrm{p} }^{r_j } \\ =\frac{1}{\left( {r_j^2 -r_\mathrm{p}^2 } \right)}\left[ {r_j^2 \rho _{\mathrm{ci}} -r_\mathrm{p}^2 \rho _{\mathrm{cp}} +\frac{r_j^2 }{2}\sqrt{\frac{b\pi }{2}}\exp \left({\frac{2\rho _{\mathrm{ci}}^2 }{b}} \right)\left({{\text{ erf}}\left({\sqrt{\frac{2}{b}}\rho _{\mathrm{ci}} } \right)-{\text{ erf}}\left({\sqrt{\frac{2}{b}}\rho _{\mathrm{cp}}} \right)} \right)} \right], \\ \end{array}\nonumber \\ \end{aligned}$$
(46)

where

$$\begin{aligned} a&= \rho _{\mathrm{ci}}^2 +\frac{\mu _\mathrm{c} \beta }{\pi k}J_\mathrm{c} \ln r_j , \end{aligned}$$
(47a)
$$\begin{aligned} b&= \frac{\mu _\mathrm{c} \beta }{\pi k}J_\mathrm{c}. \end{aligned}$$
(47b)

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Vilarrasa, V., Carrera, J., Bolster, D. et al. Semianalytical Solution for \(\text{ CO}_{2}\) Plume Shape and Pressure Evolution During \(\text{ CO}_{2}\) Injection in Deep Saline Formations. Transp Porous Med 97, 43–65 (2013). https://doi.org/10.1007/s11242-012-0109-7

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