Abstract
This study investigates internal erosion in gap-graded gravelly soils, focusing on the critical fine content transition zone (30–35%). By integrating µ-CT imaging, geometric criteria analysis, and a novel predictive model based on the graded erosion principle, we demonstrate that fine particle content governs the transition between suffusion and piping erosion modes. Experimental results reveal that exceeding the 30–35% threshold shifts the soil fabric from a stable coarse-grained skeleton to an “over-filled” structure, where coarse particles float within a fine-dominated matrix, drastically increasing piping susceptibility. A multi-criteria assessment framework validates the limitations of traditional geometric criteria (e.g., Cu, Kenney & Lau) for transitional soils, while the grading entropy criterion offers enhanced robustness. The proposed graded erosion model successfully predicts particle size distribution evolution toward stable Fuller limits and porosity changes by incorporating particle-size-sensitive erosion rates with physical thresholds. This integrated methodology advances the understanding and prediction of internal erosion in gap-graded soils, supporting the design of resilient geotechnical structures and improving infrastructure risk assessment.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (NSFC41462014). We acknowledge the provided µ-CT image scanning assistance by personnel of the Key Lab of New Processing Technology for Nonferrous Metals and Materials Ministry of Education and Guangxi Key Laboratory of Geomechanics and Geotechnical Engineering. Guilin University of Technology, during field instrumentation, troubleshooting and data collection in this study. We thank the Avizo (Version 2020.1) software team for providing us home office licenses to conduct CFD simulation.
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This work was supported by the National Natural Science Foundation of China (NSFC41462014).
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Appendices
Appendix A. Grading entropy criterion
Through application of a pair of derived entropy-based parameters incorporating all of the information of the grading curves of granular soils, the three rules of particle migration, filter and segregation can be interpreted from the structure characteristics of granular soils. The pair of entropy-based parameters are the relative base entropy A and normalized entropy increment B, respectively, corresponding to the base entropy S0 and the entropy increment ΔS. The grading entropy S can be separated into the sum of two parts, comprising the base entropy S0 and entropy increment ΔS:
The first part is the base entropy, which takes into account the relative spread of particle sizes within the entire size distribution, and its relative base entropy are defined by:
where: S0i is the grading entropy of the i-th fraction; xi is the relative frequencies of the i-th fraction of each grading curve, which satisfy the normalization criterion, i.e., sum of all of xi being equal to one; S0max and S0min are the base entropies of the largest and the smallest fractions in the mixture, respectively.
The second part is the entropy increment, which accounts for the relative distribution of particles across all of the defined fractions, and its normalized entropy increment are defined by:
An entropy diagram (Figure A1) is created with parameter A as the abscissa and B as the ordinate. Each point on the diagram represents the complete information about the grain size distribution as shown by the grading curve of a granular soil. For a two-component granular soil (N = 2) with coarse and fine grains, if A is close to 0 or 1, it indicates a dominance of fine or coarse grains, respectively. When A equals 0.5, the entropy increment (ΔS = 1 or B = 1.44) is at its maximum, reflecting an equal proportion of both grain types.
Zones in the Entropy Diagram are listed as follows.
Zone I (A< 2/3):
Characteristics: The coarser grains are not well-structured within the matrix of finer grains. The coarse grains tend to “float” in the finer matrix, leading to an unstable mixture. Implications: This instability makes the mixture prone to erosion problems like suffosion or piping[1,20,42], where water flow can erode the soil structure. The soil may not have a strong supporting framework, which could lead to failure under seepage conditions.
Zone II (A > 2/3)
Characteristics: Coarser grains form a stable skeleton within the soil mixture. The finer particles fit into the voids between the coarse particles, creating a stable and well-supported structure.Implications: In this zone, the soil structure is more resistant to erosion. The coarse grains provide a supporting framework that helps prevent total erosion. The stability of the mixture reduces the likelihood of significant erosion or collapse under seepage.
Zone III:
Characteristics: This zone is not explicitly defined in the given diagram description but generally refers to scenarios where: (1) Finer particles may migrate under seepage conditions, a phenomenon known as "suffusion."; (2) However, this migration does not lead to the collapse of the coarse particle skeleton.Implications: In Zone III, while the finer particles may be transported, the structural integrity provided by the coarse particles remains intact. Thus, the overall stability of the granular matrix is preserved, even if some fines are lost due to seepage.
Internal stability criterion for a gravel soil in a normalized diagram: _I_ piping (unstable), _II_ transition (stable), _III_suffosion (unstable); the boundary line between II and III is approximate (indexed from Imre et al. 2015).
Appendix B. Optimal fine-grain content criterion
where ϕ is the porosity of the original soil (%); Pop is the optimal fine-grain content of the soil (%); and ϕc is the porosity of the coarse grains alone and depends upon the uniformity coefficient Cu’ of the coarse grains; ϕ0 is the porosity of a gravel soil that is assumingly composed by the uniform-sized particles. This is an original formula.
According to the Chinese code GAQSIQ-CHN (2022)5, an improved formula has been proposed based on extensive experiment and computation data consistently supporting that the optimal fine-grain content is approximately 30%, regardless of B2. Practically, a fine-grain content of around 30% is considered the threshold where fine particles begin to contribute to the soil’s skeletal structure. When the fine-grain content is below 30% of ϕc, these particles do not adequately fill the voids between coarse grains, resulting in permeability primarily controlled by the coarse grains. Conversely, if the fine-grain content exceeds 30%, the fine particles start to fill the voids, leading to closer interaction within the mixture. The improved formula is as follow Figure B1.
Based on Pop4,45, the potential for internal instability can be evaluated with the following scenarios:
- 1.
Px < 0.9 Pop: Indicates potential for suffusion mode, corresponding to Zone III in the entropy diagram.
- 2.
Px > 1.1 Pop: Indicates potential for suffusion or suffosion or piping mode, corresponding to Zone I in the entropy diagram.
- 3.
Px = (0.9 ~ 1.1) Pop: Indicates a transitional state between the above two erosion modes, corresponding to Zone II in the entropy diagram.
Discussion on Erosion Sensitivity As is known, for equal spheres, the porosity ϕ0 ranges from 25.95% (in the densest packing, e.g., face-centered cubic or hexagonal close packing) to 47.6% (in the loosest regular packing, i.e., simple cubic packing). This range represents the variation in the compactness of sandy soils. The change in compactness affects the uniformity coefficient Cu′ of the coarse particles (as expressed in Equation B2). It can be inferred that the tighter the particle packing, the smaller the space available for accommodating particles of other sizes; therefore, for coarse particles consisting of two size fractions, Cu′ becomes larger. Conversely, the looser the packing, the smaller the Cu′. For ease of observation, planar diagrams can be used to illustrate the morphological basis for the changes in compactness and Cu′ resulting from these two packing configurations.
Porosity ϕ0 of the densest packing (left) and the loosest packing (right) and the resulting difference in the uniformity coefficient Cu’ values..
Analyzing in conjunction with the formulas: Equation B2 indicates that different packing modes of equal spheres lead to differences in the compactness and uniformity of the coarse particles. Specifically, under dense packing, ϕ0 is smaller and Cu′ is larger, resulting in a smaller porosity ϕc of the coarse fraction. Under loose packing, ϕ0 is larger and Cu′ is smaller, leading to a larger ϕc. From Equation B1, the variation in ϕc directly induces a corresponding change in the optimal fines content Pop. This change can cause an abrupt shift in the erosion mode, making the system highly sensitive to the packing arrangement. The underlying cause of this sensitivity lies in the assumed packing pattern of uniform equal spheres, which is characterized by the porosity ϕ0.In practice, when using the original formula (Equation B1) to assess the erosion mode, we adopted an empirical range of ϕ0=33% to 37%, in order to align with the evaluation results of the Chinese national standard (GAQSIQ-CHN, 2022), which directly takes ϕc=30%.
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Zhu, B., Xie, YF., Su, DR. et al. Gap-graded soil erosion modes predicted by µ-CT informed graded erosion model and multi-criteria assessment. Sci Rep (2026). https://doi.org/10.1038/s41598-026-48213-y
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DOI: https://doi.org/10.1038/s41598-026-48213-y
