Stability Criteria for Soils Under Seepage

The detailed geometric information of each soil is summarized in Table 3. The soils range from clay to gravel and most of them belong to broadly-graded soils with values of coefficient of uniformity, C_u, larger than 3. Nearly one third of the soils contain a significant amount of fines. The soils include crushed stone, moraine soil, fluvial soil, sandy gravel, completely decomposed granite (CDG), Loire sand, Leighton Buzzard sand, Fontainebleau sand, silica sand, concrete sand, clayey silt, kaolin, and bentonite, and represent soils from 12 countries (i.e. Australia, Canada, China, France, Germany, Italy, Japan, Russia, Sweden, The Netherlands, U.S., and U.K.). Moreover, the particle-size distributions of the test soils represent well the actual base soils and filters for embankment dams.

Most of the tests collected in the dataset are conducted in cylindrical seepage cells, with diameters ranging from 25 to 580 mm. The ratio of sample height to sample diameter ranges from 0.7 to 2.7. The hydraulic gradients in some tests were kept constant during the testing process (e.g., Aberg, 1993; Chapuis et al., 1996; Honjo et al., 1996; Lafleur et al., 1989; Bendahmane et al., 2008 ;  Wan and Fell, 2008), while the hydraulic gradients in other tests were increased gradually (e.g., Lafleur and Nguyen, 2007; Skempton and Brogan, 1994; Fannin and Moffat, 2006; Cividini et al., 2009; Chang and Zhang, 2011 ;  Chang and Zhang, 2013). Table 2 summarizes the applied hydraulic gradient in each test. Most of the tests were conducted under downward seepage flow. To facilitate escaping of fine particles, 43 tests were assisted with tamping. The duration of flow during the internal erosion tests primarily ranged from 0.5 to 10 h. Most tests followed similar testing procedures, although there were differences in magnitude of hydraulic gradient, flow direction, sample filter size and sample vibration. The maximum hydraulic gradients applied in most of the tests were high to mobilize geometrically unstable particles. Therefore, the test results are comparable.

2.2. Identification of soil internal stability

Within the 131 soils in the dataset, 60 soils were tested to be stable and 71 were found to be unstable, as shown in Table 2. There is no general rule to judge the stability of soils based on the testing results. The three identification methods were based on the following: (I) the fraction loss, (II) the variation of permeability, and (III) observation of piping failure.

In the first method, a sample is regarded as unstable if continuous loss of fine particles or significant settlement of the sample occurs. USACE (1953), Kenney and Lau (1985), Lafleur et al. (1989), Chapuis et al. (1996), Honjo et al. (1996), Moffat (2005), Fannin and Moffat (2006), Lafleur and Nguyen (2007), Li (2008), Wan and Fell (2008), Cividini et al. (2009), Chang and Zhang (2011) and Salehi Sadaghiani and Witt (2011) followed this method. The fraction of loss of fine particles can be reflected either by collecting the eroded soil (e.g., Honjo et al., 1996 ;  Lafleur and Nguyen, 2007) or measuring the post-test grain-size distribution of the soil specimen (e.g., Kenney and Lau, 1986 ;  Wan and Fell, 2008). Particularly, Chang and Zhang (2011) measured the cumulative eroded soil mass and specimen deformation during the entire erosion process, as well as the post-test grain-size distribution. There is no general quantitative rule for identifying internally unstable soils with respect to mass loss or specimen deformation. For instance, the minimum loss of fines of internally unstable soils in Kenney and Lau’s tests (Kenney and Lau, 1985) is about 7%, whereas the minimum value in Wan and Fell’s tests (Wan and Fell, 2008) is about 4%. The specimen deformation is 2–5% associated with 8–13% loss of fine particles in the tests by Moffat (2005), and 1.2% associated with 4% loss of fine particles in the tests by Chang and Zhang (2013). The shear strength of the soil decreases significantly after a loss of about 4% of fine particles (Chang, 2012). Therefore, according to the lost mass of fine particles in the unstable soils during the internal erosion tests and the soil response after loss of fine particles, an approximate boundary value of fraction loss to identify unstable soils can be estimated as 4%. The boundary of specimen deformation for identifying unstable soils during internal erosion testing is about 1.2%.

In the second method, a sample is regarded as unstable if a sudden change in soil permeability occurs during the testing process. This can be recognized from a permeability–time relationship. When the permeability of a soil increases progressively or suddenly, the soil is classified as unstable. Sun (1989), Liu (2005) and Kaoser et al. (2006) adopted this method.

In the third method, unstable results represent piping failure under a hydraulic gradient which is much smaller than its theoretical value. Skempton and Brogan (1994) stated that piping of sand in unstable sandy gravels can occur at one third to one fifth of the theoretical critical hydraulic gradient. Adel et al. (1988) also followed this identification method.

In fact, the first two identification methods are common in nature. A significant loss of fine particles can result in sudden changes in the permeability of the soil sample, and the gradation curves will change eventually. However, the hydraulic conditions in the tests that followed these two identification methods were more severe than those that would be expected in practice, whether the hydraulic gradient was increased gradually or kept constant during the testing process. The tests following the third identification method were conducted under hydraulic gradients between 0 and 1.0.

3. Evaluation of existing criteria based on the dataset

Three criteria [i.e., Istomina’s criterion (Istomina, 1957); Kezdi’s criterion (Kezdi, 1969), and Kenney and Lau’s criterion (Kenney and Lau, 1985)] are used in design specifications to evaluate the internal stability of either granular filters or core materials in embankment dams. Geotechnical Engineering Office (GEO) (1993), U.S. Department of Agriculture (USDA) (1994) and U.S. Society of Dams (USSD) (2011) adopted Istomina’s criterion to evaluate possible segregation of filters using the coefficient of uniformity, C_u, as an indicator. GEO (1993) and Canadian Dam Association (CDA) (2007) used Kenney and Lau’s criterion to assess the internal stability of granular filters, and International Commission on Large Dams (ICOLD) (1994) recommended the same criterion for evaluating base soils. USSD (2011) recommended the use of Kezdi’s criterion in evaluating the internal stability of base soils. Kenney and Lau’s criterion and Kezdi’s criterion are also widely used by researchers to assess the internal stability of granular soils (e.g., Adel et al., 1988; Skempton and Brogan, 1994; Chapuis et al., 1996 ;  Kaoser et al., 2006; Chang and Zhang, 2011). In the following section, the feasibility of the three geometric criteria is evaluated based on the test dataset.

3.1. Istomina’s criterion

The coefficient of uniformity, C_u, of a soil can be used as an indicator for the internal stability of the soil. The diameter of coarse fraction can be represented by d₆₀ (particle diameter at 60% finer by weight), while the diameter of the fine fraction can be represented by d₁₀ (particle diameter at 10% finer by weight) of a soil. C_u=d₆₀/d₁₀ indicates whether the fine particles can pass through the pores formed by the coarse fraction of the soil. According to Istomina (1957), soils with C_u of less than 10 are internally stable, and those with C_u of larger than 20 are likely to be internally unstable. Those with C_u between 10 and 20 fall into a transitional zone. Istomina’s criterion is valid only for sandy gravel.

Fig. 1 shows a plot of d₆₀ against d₁₀ of all the soils in the dataset, where d₆₀ and d₁₀ are the particle diameters at 60% and 10% finer by weight, respectively. For soils with d₁₀>0.5 mm or d₆₀>10 mm, most of those with C_u<10 are stable and those with C_u>20 are unstable. However, for the soils with d₁₀<0.5 mm and d₆₀<10 mm, C_u may not be a good indicator for internal stability: several unstable soils fall into the “stable” zone.

3.2. Kezdi’s criterion

Kezdi (1969) proposed a geometric criterion to assess the internal stability of soils following the filter criterion proposed by Terzaghi (1939). A soil is divided into a finer component and a coarser component at an arbitrary point (with particle size d as shown in Fig. 2) on the grain size distribution curve. The coarser component represents the particles with grain sizes larger than the selected particle size d, whereas the finer component includes the particles with sizes less than the selected particle size d. The coarse component is treated as the filter for the fine component. If the soil satisfies (d_15c/d_85f)_max≤4 (where d_15c represents the diameter at 15% finer by weight in the coarse component and d_85f the diameter at 85% finer by weight in the fine component as shown in Fig. 2), then the soil as a whole is considered as internally stable. Sherard (1979) advocated this approach, but modified the criterion to (d_15c/d_85f)_max≤5.

Fig. 3 shows the testing results for all the soils in the dataset on the (d_15c/d_85f)_max – d₅₀ plot, where d₅₀ is the mean particle diameter. It can be observed that Kezdi’s criterion sets a safe boundary for engineering design. Many stable soils are identified as unstable according to this criterion. Further investigation into the 34 internally stable soils that are assessed as internally unstable ones shows that 19 of them have fines contents more than 10%.

3.3. Kenney and Lau’s criterion

Kenney and Lau (1985) hypothesized that particles finer than grain size d (having a mass fraction, F) would likely erode from a soil if there were not enough particles in the grain sizes ranging from d to 4d (having a mass fraction, H) as shown in Fig. 2. H/F can be obtained from the grain-size distribution curve over a portion of its finer fraction given by F≤20% for soils with a broadly-graded primary fabric (C_u>3), and by F≤30% for soils with a narrowly-graded primary fabric (C_u≤3). If (H/F)_min is larger than 1.3, then the soil is classified as internally stable. Kenney and Lau (1986) later revised the boundary to (H/F)_min>1.0. This criterion is valid only for granular soils.

The application of Kenney and Lau’s criterion to assess the internal stability of the test soils in the dataset is shown in Fig. 4. It can be observed that this criterion also sets a safe boundary for engineering design. Soils with (H/F)_min larger than 1.0 are indeed mostly stable. However, many soils with (H/F)_min smaller than 1.0 are also stable. Especially when the mean particle size is smaller than 1.0 mm, 14 of the 22 stable soils are evaluated as unstable. This means that this criterion is much on the safe side when the soil contains a significant amount of fines.

3.4. Comments on existing criterion

In view of the ability of the aforementioned three criteria to predict the internal stability of the test soils, it seems that these criteria indeed set safe boundaries for engineering design. However, no single criterion can be used to predict the internal stability of all types of soil exclusively, particularly soils containing a significant amount of fines. This is because the control variables for different types of soil may not be exactly the same. The effect of fines content in enhancing the erosion resistance of the soil has not been sufficiently reflected. Therefore, it is pertinent to classify different types of soil and develop criteria for each type of soil and, if possible, develop composite criteria that cover several types of soil.

4. Extended criteria

4.1. Soil microstructures and representative control variables for internal erosion

The internal stability of a soil is controlled by such geometric parameters as gradation and constriction sizes. The grain size distribution of a soil has a significant influence on its internal stability. Narrowly-graded soils usually do not involve internal instability problems, but gap-graded soils are usually susceptible to internal erosion. The control variables for different types of soil may not be the same.

The soils in the collected dataset are divided into two groups: well-graded and gap-graded soils. A well-graded soil is one defined by a broad gradation which has a good representation of all sizes (i.e., soils A and B in Fig. 5). In the dataset, 80 soils are well-graded as shown in Table 3. The internal stability of a well-graded soil depends on whether its coarse particles can prevent the loss of its medium-sized particles, which in turn prevents the loss of its fine particles. The fines content has a significant influence on the internal stability of a broadly-graded soil ( Bendahmane et al., 2008 ;  Wan and Fell, 2008). If the fines content is within a certain margin, the coarse and medium-sized particles may form a matrix. If the fines content is increased so significantly that the coarse grains float within the fines, then the soil behavior may be governed by the matrix of fines. Therefore, referring to the soil microstructures and the classification methods for base soils for no-erosion filters proposed by Sherard and Dunnigan (1989), well-graded soils are classified into three different categories in this paper according to their fines contents, i.e. contents smaller than 0.063 mm (British Standard Institution (BSI), 1990):

(1)

Well-graded soils with fines contents less than 5%;

(2)

Well-graded soils with fines contents more than 20%;

(3)

Well-graded soils with fines contents between 5% and 20%.

In the first category, the fine particles (<0.063 mm) only fill part of the pores formed by the coarse particles as shown in Fig. 6(a); thus the effect of the fine particles on internal stability is not significant. The internal stability of this type of soil is mainly controlled by the characteristics of the grading curves. Wan and Fell (2008) stated that a soil with a steep slope in the grain-size distribution of the coarse fraction and a flat slope in the fine fraction was likely to be internally unstable. Fig. 7(a) illustrates that the control variables for this type of soil are the slopes of the grain size distribution curve. The steeper the slope in the coarse fraction, the easier the fine particles can erode as the pores formed by the coarse particles are less likely to be filled by the medium-sized and fine particles. The flatter the slope in the fine fraction, the easier the fine particles can erode as the controlling constriction sizes (Kenney et al., 1985) are larger.

In the second category, the pores formed by the coarse particles can be largely filled by the medium-sized particles, and the pores formed by the medium-sized particles can be almost fully filled by the fine particles, or the medium-sized particles and coarse particles can only float in a matrix of fines when the fines content is sufficiently large as shown in Fig. 6(c). Therefore, a suitable control variable for the internal stability of such soil is its fines content. In the third category, some of the medium-sized or coarse particles are enclosed by fine particles as shown in Fig. 6(b). The amount of fine particles determines the amount of medium-sized or coarse particles that could be enclosed. The control variables for this type of soil are fines content and characteristics of the medium-sized and coarse fraction. The exact boundaries among the three categories of soil may be found through statistical analysis, an example of which will be presented in Section 4.2.

A gap-graded soil is one defined by a broad gradation in which a portion is significantly under-represented (e.g., soil C in Fig. 5) or completely absent (e.g., soil D in Fig. 5). In the dataset, 51 soils are gap-graded as shown in Table 3. The missing part in most of the gap-graded soils in the set is either sand or silt. The internal stability of a gap-graded soil depends on whether the coarse particles can prevent the loss of the fine particles. If the finer fraction can fully fill the pores formed by the coarser fraction, the soil is usually internally stable. The fines content has a significant influence on the internal stability of gap-graded soils. The soil will change from coarse-matrix controlled behavior to fine-matrix controlled behavior with the increase of fines content. Following a similar classification of well-graded soils, gap-graded soils are also divided into three categories according to the fines content:

(1)

Gap-graded soils with fines contents less than 10%;

(2)

Gap-graded soils with fines contents more than 35%;

(3)

Gap-graded soils with fines contents between 10% and 35%.

In the first category, the fine particles can only fill the local pores formed by the coarse and medium-sized particles as shown in Fig. 8(a). Hence the effect of the fines content is not a major issue. The internal stability of this type of soil could be dominated by the gap ratio as shown in Fig. 7(b). Gap ratio, G_r, is defined as the ratio of the maximum particle size, d_max and the minimum particle size, d_min of the severely under-represented portion on the grain-size distribution curve (Fig. 4):

A large gap ratio means the coarse fraction cannot successfully serve as a filter for the finer fraction, thus the fine particles can erode easily even under low hydraulic gradients [Fig. 7(b)]. In the second category, the medium-sized and coarse particles only float in the fine particles as shown in Fig. 8(c), thus the control variable for this type of soil is its fines content. In the third category, the local medium-sized and coarse particles are enclosed by the fine particles as shown in Fig. 8(b). The amount of enclosed medium-sized and coarse particles depends on the amount of fine particles. The stability of the unenclosed medium-sized and coarse particles is also dominated by the gap ratio. Therefore, the control variables for this type of soil are fines content and gap ratio. The classification thresholds will be presented in Section 4.3.

The microstructures of a soil change significantly with the fines content (Li and Zhang, 2009 ;  Zhang and Li, 2010). The soils in the first category of the two groups are rich of inter-aggregate pores, so the constriction size is relatively large and the fine particles are easy to pass under seepage. The soils in the second category of the two groups are rich of intra-aggregate pores. Hence, the loss of fine particles is less likely. For the soils in the third category, both intra-aggregate and inter-aggregate pores dominate. The fines content will influence the relative percentage of the intra-aggregate and inter-aggregate pores.

4.2. Extended criteria for well-graded soils

4.2.1. Well-graded soils with fines contents (<0.063 mm) less than 5%

If the fines content of a well-graded soil is less than 5%, the effect of fines on the internal stability of the soil can be neglected. The control variables for this type of soil are the characteristics of the grading curve; particularly the slopes of the grading curve that reflect the possibility for the fine particles to go through the constrictions controlled by the coarse matrix. Kenney and Lau’s criterion [(H/F)_min>1.0] described in the previous section is based on the constriction size. A small value of H/F indicates that the slope in the grain-size distribution of the finer fraction is relatively flat or the slope in the grain-size distribution of the coarse fraction is steep. The internal erosion tests conducted by Kenney and Lau (1985) were on well-graded soils without fines content.

Fig. 9 shows the prediction for this type of soil using Kenney and Lau’s criterion. Among the 26 unstable soils, 25 of them are assessed to be internally unstable, and 25 of the 33 stable soils can be successfully predicted. Recently, information of internal stability tests on five well-graded soils with fines contents of less than 5% is available (Andrianatrehina et al. ,2012). These cases are not included in the dataset and hence serve as an independent check of the criterion. Among them, four can be successfully assessed according to the criterion [(H/F)_min>1.0] as shown in Fig. 9. Therefore, (H/F)_min>1.0 could be used as a boundary separating internally stable and unstable soils for well-graded soils with a fines content of less than 5%.

4.2.2. Well-graded soils with fines contents of more than 20%

Fell and Fry (2007) stated that internal instability may not be an issue if a well-graded soil has a large fines content. As shown in Fig. 10, when the fines content is more than 20%, all the soils are tested to be internally stable even at (H/F)_min values smaller than 1.0. This may be due to the easy formation of clusters of soil particles under normal compaction conditions when the fines content is high, especially when the fines contain a high fraction of clay. Hence, the erodibility of the soil is enhanced. The tests conducted by Lafleur and Nguyen (2007) showed that natural moraines with fines contents of more than 25% did not involve internal instability problems. Once the fines content exceeds a certain margin, the soil exhibits fines-controlled behavior as the soil is rich of intra-aggregate pores. The tractive shear stress induced by the seepage flow through the intra-aggregate pores would be small.

Of the six soils falling into this category (four of them with detailed plasticity information as shown in Table 3), three are with low plasticity and one is with medium plasticity. Here high, medium, and low plasticity are defined based on the liquid limit, LL (BSI, 1981): low plasticity, LL<35%; medium plasticity, 35%<LL<50%; high plasticity, LL>70%. For a soil with high plasticity that falls into this category, the erosion resistance is even higher than that of the soils with low and medium plasticity. Thus, this type of soil is internally stable.

4.2.3. Well-graded soils with fines contents between 5% and 20%

The internal stability of a well-graded soil with a fines content of between 5% and 20% depends on both the gradation of the coarse fraction and the fines content. Fig. 10 shows the relation between (H/F)_min and fines content, P. Here, the (H/F)_min value is derived over a portion of its finer fraction given by 5%≤F≤20%. Soils that fall into this category are assessed to be internally stable if they satisfy the following inequality:

The fines enhance the seepage erosion resistance by interlocking and formation of clusters. Hence, the stability range for this type of soil is wider than the range proposed by Kenney and Lau (1985). Therefore, two control variables, fines content and (H/F)_min, should be used to identify the internal stability of this type of soil.

Soil plasticity also plays an important role in the internal stability of the soils in this category. A soil with higher plasticity tends to be more internally stable under the same fines content. If a soil contains a significant amount of fines, the fines may be present in the form of soil aggregates (Zhang and Li, 2010). The structural stability of soil aggregates in water is reinforced with increasing soil plasticity (Sheridan et al., 2000). A higher structural stability of soil aggregates tends to enhance the erosion resistance of the soil. Subsequently, the soil will be more internally stable under seepage. Of the 15 well-graded soils with 5–20% of fines, 8 are non-plastic or of low plasticity, and no plasticity information is available for the rest seven soils. Therefore, the extended criterion is only applicable to low plasticity soils. For soils with medium to high plasticity in this category, the unstable zone could be narrower, because the erosion resistance of the soils may be enhanced by electro-chemical forces.

4.2.4. Composite criterion for well-graded soils

Finally, the proposed composite criterion for assessing the internal stability of well-graded soils is shown in Fig. 11. A soil above the line is deemed to be internally stable, otherwise it is internally unstable. The mathematical expressions for these criteria are summarized in Table 4. The internal stability of a well-graded soil with a fines content of less than 5% can be evaluated using (H/F)_min, while the internal stability of a well-graded low plasticity soil with a fines content between 5% and 20% can be assessed using Eq. (2). A well-graded soil with a fines content of more than 20% is deemed to be internally stable. Of the 44 well-graded stable soils in the dataset, 34 are evaluated stable; of the 36 well-graded unstable soils in the dataset, 35 are evaluated unstable based on the extended criteria in this paper.

4.3. Extended criteria for gap-graded soils

4.3.1. Gap-graded soils with fines contents of less than 10%

Fig. 12 shows the stability of gap-graded soils with fines contents of less than 10% on the (H/F)_min–G_r plot. All the soils are internally stable when the gap ratio is smaller than 3.0 whether the value of (H/F)_min is larger than 1.0 or not. This may be because the representative particle size of the fine fraction is smaller than the controlling constriction size of the coarse fraction under normal compaction conditions if the gap ratio is small enough. Recently, information of internal stability tests on eight additional soils in the category is collected ( Andrianatrehina et al., 2012; Ke and Takahashi, 2012a ;  Ke and Takahashi, 2012b). These soils are out of the dataset. The stability of all the soils is correctly assessed using the criterion (G_r<3.0) as shown in Fig. 12. Therefore, the internal stability of this type of soil can be evaluated by the gap ratio alone.

4.3.2. Gap-graded soils with fines contents of more than 35%

Unlike well-graded soils, more fine particles are needed to fully fill the pores formed by the coarse particles of a gap-graded soil, because a portion of medium-sized particles is significantly under-represented or completely absent in the gap-graded soil. Fig. 13 indicates that a gap-graded soil with a fines-content of more than 35% is internally stable no matter how large or small the gap ratio is. This is because the coarse particles can only float in the matrix formed by the fine particles. Moreover, a high fraction of fines can enhance the seepage erosion resistance of the soil by electro-chemical forces (Chang, 2012). According to Skempton and Brogan (1994), the coarse particles float in the matrix of fine particles if the fine fraction is in excess of 35%. Liu (2005) and Mao (2005) adopted the same boundary of 35% for identifying whether the coarse particles can float in the matrix of fine particles or not.

4.3.3. Gap-graded soils with fines contents between 10% and 35%

Evaluation of 21 gap-graded soils in this category in Fig. 13 shows that a soil that satisfies the following inequality can be regarded as internally stable:

4.3.4. Composite criterion for gap-graded soils

Finally, a composite criterion is proposed for assessing the internal stability of gap-graded soils as shown in Fig. 14. A soil in the upper zone is deemed to be internally unstable, otherwise it is internally stable. The mathematical expressions for the composite criterion are summarized in Table 4 and Fig. 14. The internal stability of a gap-graded soil with a fines content of less than 10% can be assessed based on gap ratio; the internal stability of a gap-graded medium plasticity soil with a fines content between 10% and 35% can be evaluated based on both gap ratio and fines content; and a gap-graded soil with a fines content of more than 35% is deemed to be internally stable. Of the 16 gap-graded stable soils in the dataset, 13 are evaluated stable and all of the 35 gap-graded unstable soils in the dataset are evaluated unstable based on the extended criteria in this paper.

5. Summary and conclusions

The feasibility of three commonly used geometric criteria (Istonima’s criterion, Kezdi’s criterion, and Kenney and Lau’s criterion) in assessing the internal stability of 131 soils in a laboratory test dataset has been evaluated. These criteria set safe boundaries for engineering design. However, when soils contain certain fines, these criteria tend to identify stable soils as unstable ones. The effect of fines content in enhancing the erosion resistance of the soil has not been sufficiently reflected.

The soils from the 131 internal erosion tests are divided into two groups: well-graded soils and gap-graded soils. Each group is further divided into three categories: category I (with fines contents less than 5% for well-graded soils and 10% for gap-graded soils), category II (with fines contents more than 20% for well-graded soils and 35% for gap-graded soils), and category III (with fines contents between 5% and 20% for well-graded soils and between 10% and 35% for gap-graded soils). In the first category, the fine particles (<0.063 mm) only fill part of the pores formed by the coarse particles. In the second category, the pores formed by the medium-sized and coarse particles can be largely filled by the fine particles or the medium-sized and coarse particles float in the fines matrix. In the third category, the medium-sized or coarse particles are locally enclosed by the fine particles. The microstructures of these soils provide basis for selecting geometric control variables for assessing the internal stability of these soils.

The internal stability of a well-graded soil with a fines content of less than 5% can be evaluated using (H/F)_min as a control variable, and the internal stability of a well-graded, low plasticity soil with a fines content between 5% and 20% can be assessed according to a relation between (H/F)_min and fines content. A well-graded soil with a fines content of more than 20% is deemed to be internally stable.

The internal stability of a gap-graded soil with a fines content of less than 10% can be assessed using the gap ratio as a control variable, and the internal stability of a gap-graded, medium plasticity soil with a fines content between 10% and 35% can be evaluated using gap ratio and fines content as two control variables. A gap-graded soil with a fines content of more than 35% is deemed to be internally stable. Finally, of the 60 stable soils in the dataset, 47 are evaluated as stable based on the extended criteria in this paper; of the 71 unstable soils in the dataset, 70 are correctly evaluated as unstable. Thirteen test cases not included in the dataset are also assessed using the proposed criteria and 12 out of the 13 impendent cases are correctly evaluated.