# Volume Contraction in Shallow Sediments: Discrete Element Simulation
## 2 Methodology  
We use the two-dimensional discrete element code PFC-2D (ITASCA Consulting Group, Inc.) to create the simulation environment (details in Table 1). The grain size distribution is uniform and ranges between d\({}_{\text{min}}\) = 0.8 mm and d\({}_{\text{max}}\) = 1.2 mm; the uniform size distribution is selected to avoid computational instability caused by very small particles generated in a normal distribution and to avoid preferential particle aligning along boundaries in mono-sized particles. We select the linear grain contact model, the available model in PFC2D for 2D disks [33], and apply an interparticle friction of 0.5, a typical value used in the literature to produce a reasonable macro-scale friction angle of granular soils [34]. We impose no friction between grains and walls, a common boundary assumption adopted to simulate field conditions in a cell environment. Grain angularity and interlocking, which imply rotational resistance, can be numerically implemented by imposing rolling resistance [35; 36; 37; 38] or by introducing non-spherical particles and granular clusters [39; 40; 41]. Hindered particle rotation (HR) is a computationally efficient approach to account for grain angularity [42; 43]. Numerical results are physically inconsistent when all particles have hindered rotation; in this study, we hinder the rotation of an HR-fraction of randomly located particles. Numerical simulations show that the macroscale friction angle increases with interlocking: \(\phi=22^{\circ}\) for HR = 0%, \(\phi=35^{\circ}\) for HR = 40%, and \(\phi=48^{\circ}\) for HR = 80% [44]. We create normally compacted granular assemblies (\"clean sand\") with two markedly different initial densities. The loose packing is formed by air pluviation, in which particles are poured from a low height, gradually filling the container bottom-up under 1 g gravity. We form the dense packing by loading granular packing with zero interparticle friction and no gravity; then, we reinstate friction and gravity after unloading. Finally, we select particles with hindered rotation at random before dissolution starts. The dissolvable granular inclusion has a D = 10 mm diameter and is buried at various depths z (Figure 1). The vertical walls impose zero-lateral deformation, and the upper \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Properties** & **Values** \\ \hline \multirow{5}{*}{Particles (disks)} & \multirow{2}{*}{Diameter} & Uniform size distribution \\  & & (d\({}_{\text{min}}\) = 0.8 mm, d\({}_{\text{max}}\) = 1.2 mm) \\ \cline{2-3}  & Number of disks & 10,567 \\ \cline{2-3}  & Density of disks & 2650 kg/m\({}^{3}\) \\ \cline{2-3}  & & Normal stiffness k\({}_{n}\) = 10\({}^{8}\) N/m \\ \cline{2-3}  & & Shear stiffness k\({}_{s}\) = 10\({}^{8}\) N/m \\ \cline{2-3}  & Inter-particle friction & 0.5 \\ \cline{2-3}  & Local damping & 0.7 \\ \hline Inclusion & Diameter & D = 10 mm \\ \hline \multirow{5}{*}{Boundary conditions} & Initial cell size & 100 mm \(\times\) 100 mm \\ \cline{2-3}  & & Free upper surface \\ \cline{1-1} \cline{2-3}  & & Rigid, zero lateral strain \\ \cline{1-1} \cline{2-3}  & Particle-to-wall friction & 0 \\ \cline{1-1} \cline{2-3}  & Gravity & 1 g \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Two-dimensional discrete element simulation environment. sediment surface is stress-free. Altogether, we run 18 cases to explore loose (e = 0.242) and dense (e = 0.184) initial packings, three burial depths (z/D = 2, 5, and 9), and three levels of granular interlocking (HR = 0%, 40%, and 80%) to vary the macroscopic friction angle. We simulate subsurface volume contraction by removing the granular inclusion. We test two dissolution modes: quasi-static gradual grain contraction (each run requires several weeks to keep the inertia number I \(<10^{-5}\)) and instantaneous removal of the grains (each run requires only a day, but inertia arises in early particle flow). In both cases, the final sediment conditions are very similar (Figure 2). All results reported next are obtained by instantaneous removal.