# Volume Contraction in Shallow Sediments: Discrete Element Simulation
## 4 Discussion  
_Boundary effects._ Some surface settlement profiles do not die out within the simulated width. This is a clear indication that simulations are affected by boundary effects (worst  
Figure 5: Surface settlement after dissolution. The data—folded at the centerline—are fitted with a Gaussian function. Note the ranges in y-axes are not the same for different z/D as indicated by the green square frames.  
Figure 6: Surface settlement for subsurface cavity dissolution in dense and loose sediments. (**a**) Normalized maximum settlement \(\rm 5_{max}\) to inclusion diameter D. (**b**) Trough area \(\rm A_{tr}\) normalized by the area of contracting zone \(\rm A_{cz}\).  
conditions: deep inclusions in loose packings--Figure 5). An alternative view is to consider these simulations as a representative REV that repeats itself with characteristic L-spacing. Similarly, there is some interaction with the lower boundary, particularly in the case of deep inclusions (z/D = 9--see Figures 3 and 4).  
_Shallow vs. deep dissolving inclusions._ To explore the dissolution of deep inclusions subjected to constant vertical stress, we conducted a complementary study. (Note: In both cases, the lateral boundary imposes zero lateral deformation; however, inclusion size and sediment density show minor differences) [48]. Deep inclusion is simulated by applying constant vertical stress (e.g., 100 kPa for the case shown in Figure 7) using plate boundary loading from the top and the bottom of the specimen. Both top and bottom boundaries move in as the inclusion contracts.  
The results displayed in Figure 7 highlight distinct differences in granular flow and force arching. In particular, while the prevalent gravity-driven vertical flow occurs when shallow inclusions dissolve, horizontal grain displacement prevails in dissolving deep inclusions. Similarly, while gravity and the stress gradient underly the evolution of force chains in shallow inclusions, concentric arches form when dissolution takes place under constant vertical stress (i.e., when a local change in vertical stress is minor compared to the stress level).  
_Post-dissolution sediment properties._ Local volume contraction affects interparticle forces and porosity in an area that is significantly larger than the initial size of the contracting zone (Figure 4 and Figure S2). Based on published numerical and experimental results, we anticipate a more contractive sediment response to subsequent normal and shear loading, lower undrained strength, and lower shear wave velocity within the \"ghost zones\" [49; 50; 51; 52; 53].  
Figure 7: Shallow versus deep volume loss. (**a**) Displacement vectors and (**b**) contact forces after dissolution. The deep inclusion is subjected to constant vertical stress of 100 kPa, and both the top and bottom boundaries move in as the inclusion contracts. The circles show the original size of the contracting zone. Note: HR = 40% in both cases.  
Field implications.Displacements induced by underground volume contraction affect the overlying infrastructure and nearby underground structures. For example, piles experience lateral deflection, bending moments, settlement, and abnormal load distribution [14; 15; 16]; lifelines are sheared and disrupted [18; 19; 20; 21; 54]; buildings develop differential settlements and horizontal displacement [22; 23]; and adjacent tunnels may rotate and undergo asymmetric settlement [25; 26].  
Scaling relations and similarities limit the extrapolation of the numerical results presented here to field conditions where the contracting cavity size is much larger than the grain size, D >> d. Nevertheless, macroscale trends observed in the context of tunnels resemble the results gathered in this study: the settlement volume correlates with the ground loss [55; 56], the maximum surface settlement decreases as the soil stiffness increases [57], and volume loss at greater depths causes less surface settlement but a wider settlement trough [45; 58].