# Volume Contraction in Shallow Sediments: Discrete Element Simulation
## 3 Results and Analyses  
_Grain flow._ Grain flow is accompanied by successive arch formation and collapse events. Grains slide and flow predominantly in the vertical direction to refill the cavity and a counter-propagating porosity front emerges (Figure 3). The flow stream is narrower in sediments with higher interlocking (Figure 3; full dataset in Figure S1 of the Supplementary Data).  
_Contact force arching and refilled cavity._ Grains in the refilled cavity are loosely packed; the small forces they carry provide transverse support to the granular arches that form around the dissolving zone (see Figure 4 and the complete dataset in Figure S2 of the Supplementary Data). In terms of contact forces, a \"ghost zone\" remains after dissolution and it extends beyond the original inclusion size. The egg-shaped force arches extend towards the free surface and define more prominent ghost zones in cases that involve deeper dissolution, denser sediments and higher interlocking. Clearly, rotational frustration plays an important role in particle rearrangement; thus, a higher coordination number in denser packings and hindered rotation combine to exacerbate rotational frustration, marked force chains and loose refill.  
Figure 3: Granular flow during cavity refill—displacement vectors. Vector magnitudes are equally scaled for all cases. The black circles indicate the original location and size of the dissolving inclusions. Note: hindered rotation HR = 40%.  
_Settlement._ Figure 5 summarizes surface settlement profiles for all 18 cases. We fit a Gaussian function to each dataset to extract the maximum surface settlement \(\delta_{\rm max}\) and the \"characteristic width\" W of the settlement trough. This analysis resembles large-scale field observations of previous studies [12; 45; 46; 47]:  
\[\delta_{\rm x}=\delta_{\rm max}\exp\left(-\frac{\rm x^{2}}{2\Sigma^{2}}\right), \tag{1}\]  
where \(\rm x\) is the horizontal distance from the centerline and \(\Sigma\) the standard deviation. Maximum surface settlement \(\delta_{\rm max}\) decreases as the depth-size ratio \(\rm z/D\) increases. Interlocking and packing density (1) do not affect \(\delta_{\rm max}\) for very shallow inclusions \(\rm z/D\leq 2\), where surface settlement is kinematically controlled, but (2) reduce the peak settlement \(\delta_{\rm max}\) when inclusions are deeper than \(\rm z/D\geq 5\) and granular flow is equilibrium-controlled (Figures 5 and 6a). In all cases, effective width \(\Sigma\) increases as \(\rm z/D\) increases and interlocking decreases.  
_Trough area_ vs. _lost area._ Simulation results allow us to compare the trough area \(\rm A_{tr}\) against the area of contracting zone \(\rm A_{cz}\). In most cases, the ratio is \(\rm A_{tr}/\rm A_{cz}<1\). In fact, surface settlement vanishes, and \(\rm A_{tr}/\rm A_{cz}\
rightarrow 0\) for deep cavities \(\rm z/D\geq 5\) dissolving in dense sediments with high interlocking (Figure 6b). In the limit of incompressible grains, difference \(\rm A_{cz}-A_{tr}\) is equal to the gain in internal voids and porosity, as can be seen in the numerical results (Figure 2, Figure 4 and Figure S2). Note that the trough area exceeds the area of the dissolved inclusion \(\rm A_{tr}/\rm A_{cz}>1\) when dissolution takes place in loose sediments with no rotational resistance; in this case, dissolution triggers the densification of the overlying sediment (Figure 6b).  
Figure 4: Contact force chains after dissolution. Force chain thicknesses reflect the force magnitude and are equally scaled for all cases. The white circles indicate the original location of dissolvable inclusions. Notice large local porosity next to strong force chains. Case: hindered rotation HR = 40%.