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\title{Supporting Information\\
\large Voronoi-entropy characterisation of geometric disorder in two-dimensional
porous networks}
\author{V.~Franz Suxo M., A.~Kirch, C.~R.~Miranda}
\date{\today}

\begin{document}
\maketitle

\section{Network generation}

The obstacle positions were drawn at random within the square domain subject to a
minimum surface-to-surface separation of \SI{0.09}{\micro\metre} between obstacles,
which keeps the pore space connected. For each target configuration, 1{,}000 independent realisations
were generated. For the studies that hold the entropy fixed, only the realisations
whose Voronoi cell-area entropy fell within a narrow tolerance of the target value were
retained, and the flow was solved on this retained ensemble, as in the other studies.

\section{Entropy estimation and bin-width sensitivity}

The Voronoi cell-area Shannon entropy (Eq.~2 of the main text) was computed by scaling
the cell areas by their mean $\langle A\rangle$, discretising the distribution of
$A/\langle A\rangle$ over the interval $[0,5]$ into 50 equal bins (width $0.1$ in
units of $\langle A\rangle$), and normalising the binned frequencies. Because this is a
binned estimate of a continuous distribution, the absolute value of $H$ depends on
the bin width: with the interval fixed, finer bins raise $H$ and coarser bins lower
it. Recomputed on the obstacle coordinates of the five disorder cases for bin counts
from 10 to 80, $H$ rises with disorder under every rule and the rank order of the
five cases is preserved; only the absolute scale shifts, the fully random case
moving from about \num{0.7} at the coarsest binning to about \num{2.3} at the finest,
with the reported value of \num{1.92} corresponding to the 50-bin convention above.
The absolute value of the entropy is therefore convention-dependent and must be
reported together with its bin definition, whereas the increase with disorder and
the rank order on which the analysis relies are not.

\section{Lattice-Boltzmann implementation}

The single-phase flow was solved with the D2Q9 lattice and a single-relaxation-time
collision operator with an explicit-force fluid--solid treatment, as implemented in
the Taxila package. Flow was driven by injection at the left boundary, and the
tortuosity and Darcy permeability were evaluated once the velocity field reached a
steady state. The domain was discretised on a $256\times256$ lattice, so the
spacing is $\Delta x=\SI{5.68}{\micro\metre}/256\approx\SI{22.2}{\nano\metre}$;
the smallest obstacle radius (\SI{0.22}{\micro\metre}) spans about ten nodes and
the minimum throat (\SI{0.09}{\micro\metre}) about four. The permeability is
returned by the solver in lattice units and converted to physical units through
$k=k_{\mathrm{lu}}(\Delta x)^2$; the raw lattice-unit values for every case are
listed in Table~\ref{tab:si-raw}. The collision used a single relaxation time
$\tau_{\mathrm{LB}}=1.23$, set by the brine kinematic viscosity. Flow was driven by a fixed inlet
velocity (\num{0.003} in lattice units, \SI{0.44}{\metre\per\second}) with an open
outlet and periodic lateral boundaries. At the pore scale the Reynolds number is
small ($Re\approx0.25$ on the obstacle scale, $3.17$ on the domain scale), so the
flow is in the linear Stokes regime; the lattice Mach number is of order $10^{-3}$,
which keeps the lattice-Boltzmann compressibility error negligible.

\clearpage
\section{Geometric design of the controlled studies}

The three families used to isolate the geometric factors are designed in the same
way: for a target combination of porosity, obstacle size and entropy, many random
arrangements are generated and the entropy is computed from the Voronoi
tessellation, so that each family is characterised not by a single entropy value
but by a range that reflects the diversity of accessible configurations. The
representative Voronoi diagrams, the cell-area distributions and the entropy ranges
for each family are shown in Figs.~\ref{fig:si-disorder}--\ref{fig:si-porosity}.
The filled symbol marks the representative configuration whose Voronoi diagram is
shown in panel~(a); the lattice-Boltzmann flow in the main text was solved over the
full ensemble of configurations at each point.

\begin{figure}[h]
  \centering
  \includegraphics[width=\linewidth]{fig2.pdf}
  \caption{Geometric design of the positional-disorder study, in five steps from an
  ordered to a fully random arrangement at fixed porosity ($\phi=0.68$) and obstacle
  radius ($r=\SI{0.22}{\micro\metre}$). (a) A single representative Voronoi diagram from the 1{,}000 generated configurations. (b)
  Cell-area distributions for each disorder level. (c) Shannon entropy at each level,
  with the spectrum of values from the 1{,}000 configurations; the labels at the right
  mark the five entropy targets ($H=1.60$ to $2.20$) drawn from the fully random
  ensemble for the entropy study.}
  \label{fig:si-disorder}
\end{figure}

\begin{figure}[h]
  \centering
  \includegraphics[width=\linewidth]{fig3.pdf}
  \caption{Geometric design of the obstacle-size study: five obstacle radii at fixed
  porosity ($\phi=0.68$) and entropy ($H\approx1.90$), obtained by reducing the
  obstacle count from 64 to 8. (a) A single representative Voronoi diagram from the 1{,}000 generated configurations. (b) Cell-area
  distributions. (c) Shannon entropy with the spectrum of values.}
  \label{fig:si-size}
\end{figure}

\begin{figure}[h]
  \centering
  \includegraphics[width=\linewidth]{fig4.pdf}
  \caption{Geometric design of the porosity study: five porosities at fixed obstacle
  radius ($r=\SI{0.22}{\micro\metre}$) and entropy ($H\approx1.90$), obtained by
  reducing the obstacle count from 64 to 8. (a) A single representative Voronoi diagram from the 1{,}000 generated configurations. (b)
  Cell-area distributions. (c) Shannon entropy with the spectrum of values.}
  \label{fig:si-porosity}
\end{figure}

\clearpage
\section{Transport results for all configurations}

Table~\ref{tab:si-raw} lists, for every configuration, the hydraulic tortuosity and
the Darcy permeability produced by the lattice-Boltzmann solver, given both in
lattice units and in physical units. The physical permeability follows from
$k=k_{\mathrm{lu}}(\Delta x)^2$ with $\Delta x=\SI{5.68}{\micro\metre}/256$, a
factor of \num{0.4988} per lattice unit in millidarcy.

\begin{table}[!ht]
\centering
\caption{Transport results for each case, averaged over the 1{,}000 configurations
(standard deviations shown as error bars in Figs.~2--5 of the main text): hydraulic tortuosity $\tau$
(dimensionless), permeability in lattice units $k_{\mathrm{lu}}$, and the physical
permeability $k$ in millidarcy, with $k(\mathrm{mD})=\num{0.4988}\,k_{\mathrm{lu}}$.}
\label{tab:si-raw}
\begin{tabular}{@{}ll S[table-format=1.5] S[table-format=3.5] S[table-format=3.2]@{}}
\toprule
Study & Varied parameter & {$\tau$} & {$k_{\mathrm{lu}}$ (lu)} & {$k$ (mD)} \\
\midrule
Positional disorder & $H=0.00$ (ordered)      & 1.03785 & 9.51708 & 4.75 \\
                    & $H=0.81$ (8 random)     & 1.07017 & 9.71235 & 4.84 \\
                    & $H=1.22$ (16 random)    & 1.09069 & 9.77795 & 4.88 \\
                    & $H=1.65$ (32 random)    & 1.13544 & 9.79912 & 4.89 \\
                    & $H=1.92$ (fully random) & 1.16047 & 10.03535 & 5.01 \\
\midrule
Entropy             & $H=1.60$ & 1.15846 & 9.64100 & 4.81 \\
                    & $H=1.75$ & 1.16003 & 9.61521 & 4.80 \\
                    & $H=1.90$ & 1.16346 & 10.12303 & 5.05 \\
                    & $H=2.05$ & 1.16544 & 10.06057 & 5.02 \\
                    & $H=2.20$ & 1.17395 & 9.79055 & 4.88 \\
\midrule
Obstacle size       & $r=\SI{0.22}{\micro\metre}$ & 1.15261 & 10.14427 & 5.06 \\
                    & $r=\SI{0.25}{\micro\metre}$ & 1.15839 & 13.46764 & 6.72 \\
                    & $r=\SI{0.30}{\micro\metre}$ & 1.16466 & 17.85809 & 8.91 \\
                    & $r=\SI{0.38}{\micro\metre}$ & 1.16598 & 31.16096 & 15.54 \\
                    & $r=\SI{0.64}{\micro\metre}$ & 1.19385 & 69.08401 & 34.46 \\
\midrule
Porosity            & $\phi=0.68$ & 1.15261 & 10.14427 & 5.06 \\
                    & $\phi=0.75$ & 1.14784 & 21.41214 & 10.68 \\
                    & $\phi=0.82$ & 1.12269 & 47.62213 & 23.75 \\
                    & $\phi=0.89$ & 1.09569 & 141.27228 & 70.47 \\
                    & $\phi=0.96$ & 1.05306 & 746.21890 & 372.22 \\
\bottomrule
\end{tabular}
\end{table}

\section{Supporting geometric analysis: pore-throats and the permeability--porosity relation}

\noindent\emph{Note.} The analysis in this section is computed from the obstacle
coordinates of the generated configurations, using the network-generation and Voronoi-analysis scripts described in the Methods;
it supports the main-text statements that the permeability is governed by the
pore-throat scale and that positional disorder, at fixed porosity, does not change
the typical throat and hence the permeability.

\subsection{Pore-throat definition}
The resistance to flow is set by the constrictions between obstacles, which we
characterise directly from the obstacle coordinates. Two obstacles define a pore
throat when they are neighbours in the Gabriel graph of the centres, a subgraph of
the Delaunay triangulation that joins two points only when no third obstacle falls
inside the circle drawn on the segment between them as diameter. This proximity rule
discards the spurious long links that the bare Delaunay triangulation lays across the
diagonals of an ordered array, so a perfectly ordered lattice reduces to a single
throat width. The throat width between two neighbouring obstacles $i$ and $j$ is the
surface-to-surface gap,
\begin{equation}
w_{ij}=\lVert c_i-c_j\rVert-2r,
\end{equation}
the centre-to-centre distance minus the two radii, and we summarise the network by the
median value $w_{\mathrm{med}}$.

\subsection{Permeability follows the pore-throat scale}
Across the porosity, obstacle-size and positional-disorder studies, the Darcy
permeability follows the median throat width as a single power law,
$k\propto w_{\mathrm{med}}^{2.9}$ (fitted exponent \num{2.9}), with a coefficient of
determination of \num{0.99} on the pooled fifteen configurations
(Fig.~\ref{fig:si-throat}a). The exponent is close to the cubic value expected for a
two-dimensional channel, whose conductance grows as the cube of its aperture.
Raising the porosity widens the median throat from \num{0.26} to
\SI{1.05}{\micro\metre}, and enlarging the obstacles widens it from \num{0.26} to
\SI{0.47}{\micro\metre}; in the disorder study, at fixed porosity and obstacle size,
the median stays at \SI{0.26}{\micro\metre}, which is why the permeability remains
near 5 mD. Disorder reshapes the distribution without moving its centre: the ordered
lattice has a single throat width, whereas the random arrangements develop both
narrower and wider throats, the widest growing from \num{0.26} to about
\SI{0.66}{\micro\metre} (Fig.~\ref{fig:si-throat}b). These few wide channels account
for the small (about 5\%) permeability rise but are too rare to move the typical
constriction that sets the bulk conductance.

\begin{figure}[h]
  \centering
  \includegraphics[width=\linewidth]{fig_throat.pdf}
  \caption{Pore-throat geometry and the permeability. (a) The Darcy permeability
  follows the median throat width $w_{\mathrm{med}}$ as a single power law
  $k\propto w_{\mathrm{med}}^{2.9}$ across the three studies (dashed; coefficient of
  determination \num{0.99}); the disorder configurations (triangles) sit at the
  low-throat end at fixed median (inset). (b) Throat-width survival in the disorder
  study: the ordered lattice has one throat width near \SI{0.26}{\micro\metre},
  whereas displacing obstacles spreads the tail to about \SI{0.66}{\micro\metre}
  without moving the median.}
  \label{fig:si-throat}
\end{figure}

\subsection{The entropy and the coefficient of variation of the cell areas}
The same coordinates permit the comparison, quoted in the Discussion of the main
text, between the cell-area entropy and the coefficient of variation (CV) of the
Voronoi cell areas, a bin-independent measure of their spread. The CV was evaluated
for each level of the positional-disorder series and averaged over the available
realisations of that level. It rises monotonically with disorder, through
\numlist{0;0.05;0.08;0.13;0.19}, and tracks the hydraulic tortuosity as closely as
the entropy does: the coefficients of determination are \num{0.99} for the CV and
\num{0.95} for the entropy, and the two descriptors are themselves correlated with
a Pearson coefficient of \num{0.98}.

\subsection{Same porosity, different permeability}
The four studies, plotted together as permeability against porosity
(Fig.~\ref{fig:si-kphi}), recover the observation that motivates the main text. At a
single porosity, $\phi=0.68$, the permeability already spans about sevenfold, from 5.1
to 34 mD: the obstacle-size series raises it as the throats widen, whereas the
positional-disorder and entropy series leave it clustered near 5 mD. The
fixed-porosity spread is set by the pore-throat scale, not by the degree of
positional disorder; its magnitude here, about sevenfold, is smaller than the
several decades seen in natural rock, as expected for a planar medium with a single
obstacle shape and a narrow range of sizes.

\begin{figure}[h]
  \centering
  \includegraphics[width=\linewidth]{fig_kphi.pdf}
  \caption{Same porosity, different permeability. Permeability against porosity for
  the four studies. The porosity series (circles) traces the permeability--porosity
  trend; at $\phi=0.68$ the obstacle-size series (squares) spans 5.1 to 34 mD, whereas
  the positional-disorder (triangles) and entropy (diamonds) series stay clustered
  near 5 mD. The three fixed-porosity studies are drawn at slightly offset $\phi$ for
  clarity.}
  \label{fig:si-kphi}
\end{figure}

\end{document}
