Consolidation is one of the most important processes that are needed to be accounted for in the model of cohesive sediment transport. Previous studies have shown that the consolidation paths for soil elements are not unique at low effective stress (here low effective stress means the effective stress is smaller than a threshold effective stress, which is short as TES and denoted as σb) In this study, a new theoretical constitutive equation for the effective stress of soft mud is derived to describe the non-unique consolidation paths at low effective stress.

Firstly, based on the concept of fractal dimension, the mud flocs are treated as self-similar fractal entities and the relationship between mud floc size and sediment volume fraction is established. Due to the fact that the fractal dimension decreases as the floc size increases, the variation of fractal dimension is accounted for. Based on the self-similar model, the theoretical constitutive equation for the effective stress of soft mud is derived.

The constitutive equation is validated using two data sets obtained from consolidation tests in a settling column for soft clay. At low effective stress, the model results curves follow the data trends and the non-uniqueness of consolidation paths are captured. The model results are also compared with existing numerical results and better agreement is presented especially when η is less than 0.5.

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