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\begin{document}
\title{Supporting Information for ``CO$_2$ Dissolution Trapping Rates in Heterogeneous Porous Media"}
\authors{K. A. Gilmore\affil{1}, J. A. Neufeld\affil{1,2,3}, M. J. Bickle\affil{4}}
\affiliation{1}{Bullard Laboratories, Department of Earth Sciences, University of Cambridge, Madingley Rise, Madingley Road, Cambridge, CB3 0EZ, UK}
\affiliation{2}{BP Institute, University of Cambridge, Madingley Road, Cambridge, CB3 0EZ, UK}
\affiliation{3}{Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK}
\affiliation{4}{Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge, CB2 3EQ, UK}
\begin{article}
\section{Scalings of Total Dissolution with Time}
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/FigureS1.eps}
\caption{(a) Length of the CO$_2$ finger $L$ as a function of time $t$ for the single finger case. (b) The total lateral dissolution of CO$_2$ from the high permeability finger into the surrounding water is plotted as a function of time. At early times, the total dissolution scales with $t^{3/2}$. At late times, the total dissolution tends towards being proportional to time. Both graphs are plotted for $\alpha$ = 0, 0.01, 0.1.}
\label{fig:SF_totaldiffusiveloss}
\end{figure}
The length of the finger is governed by the input flux and the amount of dissolution. Figure \ref{fig:SF_totaldiffusiveloss}b shows how the length of the CO$_2$ finger and the total lateral dissolution of CO$_2$ scale with time. At early times ($t \ll 1$), when the dissolution area to input flux ratio of the CO$_2$ finger is small, the amount of dissolution is negligible and so the length evolves as $L \sim t$. The diffusive CO$_2$ profile away from the CO$_2$-water interface (in the $z$ direction) scales like $t^{-1/2}$, hence the total diffusive flux $F_{total}$ scales as,
\begin{equation}
F_{total}(t) \sim \int_{0}^{L(t)} F\, dx \sim t^{1/2}.
\end{equation}
The total lateral dissolution of CO$_2$, $S_{tot}$, into the low permeability layers is the total mass of CO$_2$ that dissolves over time. This is equal to the total CO$_2$ flux across the whole high/low permeability interfacial area over time, and the scaling at early times is
\begin{equation}
S_{tot}(t) \sim \int_{0}^{t} F_{total}\, dt \sim t^{3/2}.
\end{equation}
At late times ($t \gg 1$), the length tends to evolving as $L \sim t^{1/2}$ as diffusive loss dominates. The diffusive flux $F$ at a point $x$ on the finger continues to scale like $F \sim t^{-1/2}$, which means the total diffusive flux tends to a constant. Hence, the total lateral dissolution into the low permeability layers scales as $S_{tot} \sim t$ at late times.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/FigureS2.eps}
\caption{(a) Length of the CO$_2$ fingers $L$ as a function of time $t$ for the multiple finger case. (b) The total lateral dissolution from the CO$_2$ finger is plotted as a function of time. At early times, the total dissolution scales with $t^{3/2}$. At late times, the total dissolution evolves proportional to time, as the diffusive flux becomes constant. Both graphs are plotted for $\alpha$ = 0, 0.01, 0.1 and $\beta=5$ }
\label{fig:MF_totaldiffusiveloss}
\end{figure}
For the multiple finger case, at early times ($t \ll 1$) the scaling for the total dissolution is the same as the single finger case, and goes like $S_{tot} \sim t^{3/2}$. At late times, the CO$_2$ saturation of water in the low permeability layers stops diffusion in the proximal parts of the CO$_2$ layers, and the system enters a steady state with a constant length zone at the front of the CO$_2$ finger in which dissolution of CO$_2$ occurs. The total diffusive flux over this zone is constant, hence the total lateral dissolution into the low permeability layers scales as $S_{tot} \sim t$ at late times (Figure \ref{fig:MF_totaldiffusiveloss}b).
Note that the total dissolution plotted in Figure \ref{fig:SF_totaldiffusiveloss} and Figure \ref{fig:MF_totaldiffusiveloss} only includes lateral CO$_2$ dissolution from the high permeability layer into the surrounding water in the low permeability layers and not dissolution of CO$_2$ into the residual water remaining in the high permeability layer.
\end{article}
\clearpage
\end{document}