# The Grading Entropy-based Criteria for Structural Stability of Granular Materials and Filters

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

_{x}(or d

_{x}), which represent the particle diameter for which x% of grains (by weight) are smaller.

_{15}/d

_{85}< 9 ([4]; where D and d denote the filter and base soils, respectively) proves to be satisfactory in these cases. However, doubt may arise in their application to broadly-graded filters and also to broadly-graded or silt-clay base materials. However, in many cases, the use of such filters is necessary, and silty-clay base soils are unavoidable.

_{90}.

## 2. Grading Entropy

#### 2.1. Statistical Entropy and Its Application to the Grading Curve

_{i}is the number of the elements in the i-th cell). The statistical entropy S

_{s}is ([16]):

_{i}is the relative frequency of the i-th cell, given by:

#### 2.2. Statistical Cells (Fractions and Elementary Cells)

_{0}(primary statistical cells) as follows (j = 1, 2, …; see Table 1):

_{0}is the the elementary cell width. The number of the fractions N is defined in terms of the difference of the absolute serial numbers of finest and coarsest fractions:

_{0}. It is assumed, that the distribution in these is uniform and it is the same within a fraction. The number of the elementary cells C

_{i}in the fraction i is equal to:

_{i}is the relative frequency of fraction i.

_{1}and α

_{2}= 1−α

_{1}. If the base of the logarithm is set to 2 in Equation (2) then the maximal specific entropy of this system is equal to 1 at the point where the relative frequencies are equal to α

_{1}= α

_{2}= 0.5. This function, being similar to a half-ellipse, (see Figure 1a), will determine all the important lines of the entropy diagrams. The specific entropy is written as follows:

#### 2.3. Simplex Representation

_{min}.

#### 2.4. The Grading Entropy and the Entropy Coordinates

_{i}from Equation (7) into Equation (8) to give:

_{0}is as follows:

_{0i}is the grading entropy of the k-th fraction. The values of S

_{0i}, shown in the bottom row of Table 1, are defined as:

_{0}is the mean value of the entropy of the fractions, weighted by the relative frequencies of the fractions x

_{i}(i = 1, …, N).

_{0min}and S

_{0max}are the entropies of the smallest and largest fractions in the mixture, respectively. The (normalised) base entropy is a linear function which defines parallel hyperplanes in the space generated by the simplex. (Figure 1c,d).

_{0}. For granular soils, the smallest particles are caused by crushing which can produce particles which are limited to the size of some microns. For plastic (clay) soils, the size can be much smaller. The size of the SiO

_{4}tetrahedron (2

^{−22}mm) is adopted here. If the minimum grain diameter d

_{0}is defined differently, the normalized entropy coordinates are unchanged. The range of A and B is independent of N (B varies between 0 and 1/ln2; A varies between 0 and 1). By contrast, the range of the entropy coordinates is dependent on N (ΔS varies between 0 and lnN/ln2; S

_{0}varies between S

_{0min}and S

_{0max}).

#### 2.5. Entropy Diagrams

_{min}. The non-normalized entropy map with coordinates [S

_{o}, ΔS], the normalized entropy map with coordinates [A, B] and the two partly normalized entropy maps with a mixture of normalised and non-normalised coordinates; i.e., [So, B] or [A, ΔS].

_{o}. These ideas are illustrated in the diagrams of Figures 2 and 3.

#### 2.6. The Meaning of the Entropy Coordinates

_{0}is the weighted mean of the fraction entropies. It is monotonically and uniquely related to the mean grain size, since it is basically equal to the mean fraction serial number, i

_{mean}:

_{mean}:

_{j}are equal.

_{0}or i

_{mean}which is the abstract mean grain diameter (or mean fraction number).

## 3. Elaboration of the Grading Entropy-based Rules

#### 3.1. Approach

#### 3.2. General Knowledge Implied in the Rules

_{min}and d

_{max}denote the minimum fraction size of the filter and the maximum fraction size of the base soil, respectively.

#### Geometrical Explanation of Suffosion

_{max}which can pass through the voids between them in each case. The loosest regularly-packed state exists when the unit radius spheres are arranged in a primitive cubic lattice, where spheres are stacked in layers to form a square pattern within each layer, and with spheres perfectly aligned on above the other in successive layers. In this case, the largest spheres that can permeate the space between them have a radius (r

_{max}) of $\sqrt{2}-1=0.4142$ units (the smallest pore is on the face of the cube), and the ratio is R/r

_{max}= 1/0.414 = 2.41 (the largest pore diameter is $\sqrt{3}-1=0.7320$ units on the space diagonal, but whilst such a diameter sphere can occupy the space, it is unable to move from it).

_{max}) of $2\phantom{\rule{0.2em}{0ex}}/\phantom{\rule{0.2em}{0ex}}\sqrt{3}-1=0.1547$ (the smallest pore is on the triangle) and the ratio is R/r

_{max}= 1/0.1547= 6.45. (In the octahedron the largest pore size is $\sqrt{2}-1=0.4142$ situated on the square, in the tetrahedron the largest pore size is $\sqrt{3\phantom{\rule{0.2em}{0ex}}/\phantom{\rule{0.2em}{0ex}}2}-1$, situated on the halving plane generated by an edge and the midpoint of the parallel edge).

_{max}are 2.41 and 6.45, with an average value of R/r

_{max}= 4.4, it is reasonable that in a typical state, R/r

_{max}might be approximately equal to 4. In a fraction size structure based on a multiplication factor of two (the size of the next biggest fraction is twice that of the last) a ratio of 4 is equivalent to two size fraction. Therefore, suffosion may occur typically where there are two empty fractions and the precise limit of the safe and unsafe domains can be studied using gap-graded grading curves.

#### 3.3. Internal Stability and Particle Migration Rule from the Test Data of Lőrincz

#### 3.4. Filter Criterion

_{of}−S

_{ob}), which describes the distance between the mean diameters of the filter and the base soils. The second variable is the sum of the filter and base soil entropy increments, ΔS

_{f}+ ΔS

_{b}, which expresses the sum of the two N values (i.e., the total number of fractions in the two grading curves), since the maximum value of ΔS is dependent on N (i.e., lnN/ln2) and most mixtures of soils with a specified ΔS value map generally close to the maximum point of the maximum ΔS line.

#### 3.5. Segregation Criterion

_{0}, the entropy increment ΔS and the total grading entropy S, using the calculated data given in Appendix A1.

## 4. Applications

#### 4.1. Application to a Large Dam Failure

^{3}. The rock-fill consisted of 4 parts. On the upstream face of the dam there was a thin layer of material with a design particle diameter of 100 mm. Zone I was a transition zone with the design maximum particle diameter of 400 mm. Zones II and III were the main rock-fill with maximum particle diameters d of 600 mm and 800 mm (Figure 14). The likely steps in the failure process were as follows [6]. (1) Rising water level. (2) Water infiltration into rock-fill. (3) Initiation of internal erosion. (4) Progressive development of piping. (5) Washout of slope and falling of pebbles. (6) Dam breach.

#### 4.2. Comparison of Filtering Rules

#### 4.3. Non-segregating Mixtures

## 5. Discussion and Conclusions

#### 5.1. The Grading Entropy Coordinates

#### 5.2. The Internal Stability Rule

#### 5.3. The Filter Rule

#### 5.4. The Segregation Rule

#### 5.5. The Use of the Rules

#### 5.6. Suggestions for Further Research

_{0}and the relative base entropy A have one-to-one relationships with the mean grain diameter and normalized grain diameter. Using the concept of the fraction serial number, an abstract, mean or mean normalized fraction number can be defined in terms of the fraction serial number i, denoted by i

_{mean}and k

_{mean}respectively.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

#### Appendix A1. Data for Segregation Criterion

No | Variation [%] | |||
---|---|---|---|---|

A [−] | S_{0} [−] | ΔS [−] | S [−] | |

42A | 0.1375 | 9.91 | 6.48 | 7.35 |

43A | 0.2625 | 0 | 2.67 | 1.72 |

44A | 0.375 | −4 | −1.03 | −2.09 |

45A | 0.5 | 1.5 | −0.04 | 0.65 |

47A | 0.7375 | 1.36 | 4.24 | 0.84 |

48A | 0.8625 | −1.31 | 4.7 | 0.43 |

49A | 0.9375 | −2.13 | 22.42 | 1.61 |

41B | 0.0625 | 8 | 4.09 | 4.77 |

42B | 0.14 | −1.79 | −0.62 | −0.84 |

43B | 0.3 | −5.83 | −1.97 | −3.35 |

44B | 0.375 | −3.33 | −0.9 | −1.88 |

45B | 0.625 | −0.4 | 0.45 | −0.42 |

46B | 0.79 | −1.9 | 3.29 | 0 |

47B | 0.86 | −0.58 | 2.4 | 0.2 |

48B | 0.9375 | −1.35 | 11.73 | 1.2 |

41C | 0.0625 | 4 | 1.46 | 1.75 |

42C | 0.14 | −3.57 | −1.85 | −2.24 |

43C | 0.3 | 1.67 | −0.63 | 0.19 |

44C | 0.375 | 4 | −0.96 | 1.47 |

45C | 0.4375 | 2.28 | −1 | 0.53 |

46C | 0.7 | −2.14 | 2.42 | −0.67 |

47C | 0.805 | −3.42 | 8.46 | 0 |

48C | 0.9 | −1.87 | 11.37 | 0.59 |

41D | 0.0625 | 8 | 5.47 | 6.42 |

44D | 1.14 | 12.5 | 5.67 | 7.72 |

44D | 0.375 | 10 | −0.57 | 4.17 |

45D | 0.5 | −2 | 0 | −1 |

46D | 0.625 | −3.6 | 0.67 | −1.74 |

44D | 0.7 | −3.21 | 2.1 | −1 |

44D | 0.86 | −1.16 | 3.37 | 0 |

44D | 0.9 | 0 | 0 | 0 |

410D | 0.9375 | 0.53 | 4.28 | 0.22 |

411D | 0.8325 | 2.1 | −6.34 | −0.42 |

41E | 0.0625 | −8 | −6.73 | −7.48 |

42E | 0.1 | 0 | 0.65 | 0.77 |

43E | 0.14 | 10 | 0.1 | 0.13 |

44E | 0.3 | 8.5 | 3.06 | 5.33 |

45E | 0.375 | −2 | −0.46 | −1.03 |

46E | 0.5 | −1 | 0 | −0.61 |

47E | 0.625 | −2 | 0.92 | −0.87 |

48E | 0.7 | 2.5 | 2.2 | −0.93 |

49E | 0.86 | −2.91 | 10.28 | 0 |

410E | 0.9 | −2.13 | 11 | −1.95 |

411E | 0.9375 | −4 | 20 | 0.83 |

412E | 0.8575 | 1.46 | 13 | −0.23 |

#### Appendix A2. Grading Data for Gouhou Dam

d [mm] | I | II | III | Riverbed |
---|---|---|---|---|

0.0625 | 0 | 0 | 0 | 0 |

0.125 | 10 | 3 | 3 | 6 |

0.25 | 20 | 8 | 4 | 8 |

0.5 | 33 | 13 | 7 | 12 |

1 | 43 | 20 | 11 | 17 |

2 | 53 | 30 | 16 | 21 |

4 | 64 | 37 | 24 | 26 |

8 | 72 | 44 | 33 | 30 |

16 | 88 | 57 | 44 | 36 |

32 | 97 | 68 | 58 | 44 |

64 | 99 | 79 | 72 | 52 |

128 | 100 | 90 | 84 | 62 |

256 | 100 | 97 | 93 | 76 |

516 | 100 | 100 | 98 | 100 |

1032 | 100 | 100 | 100 | 100 |

N [−] | 11 | 13 | 14 | 13 |

A [−] | 0.42 | 0.545 | 0.5792 | 0.675 |

B [−] | 1.4193 | 1.4389 | 1.3759 | 1.36 |

S_{0} [−] | 21.21 | 23.54 | 24.53 | 24.1 |

S [−] | 3.26 | 3.57 | 3.5292 | 3.39 |

#### Appendix A3. Data for Optimal Mixtures

**Table A3-1.**The fractal dimension n of optimal mixtures for a fixed N (N = 5) together with parameter a and coordinate B in the function of coordinate A.

A [−] | a [−] | x1 [−] | B [−] | Fractal dimension |
---|---|---|---|---|

0.50 | 1.00 | 0.20 | 1.44 | 3 |

0.60 | 1.23 | 0.13 | 1.41 | 2.70 |

0.67 | 1.42 | 0.09 | 1.34 | 2.49 |

0.70 | 1.54 | 0.07 | 1.29 | 2.38 |

0.71 | 1.60 | 0.06 | 1.27 | 2.32 |

0.76 | 1.79 | 0.05 | 1.19 | 2.16 |

0.79 | 2.00 | 0.03 | 1.11 | 2.00 |

0.82 | 2.22 | 0.02 | 1.04 | 1.85 |

0.84 | 2.46 | 0.02 | 0.97 | 1.70 |

0.90 | 3.44 | 0.01 | 0.75 | 1.22 |

0.95 | 5.98 | 0.00 | 0.48 | 0.42 |

0.4 | 0.81 | 0.13 | 1.41 | 3.30 |

0.33 | 0.70 | 0.09 | 1.34 | 3.51 |

0.3 | 0.65 | 0.07 | 1.29 | 3.62 |

0.29 | 0.63 | 0.06 | 1.27 | 3.68 |

0.24 | 0.56 | 0.05 | 1.19 | 3.84 |

0.21 | 0.50 | 0.03 | 1.11 | 4.00 |

0.18 | 0.45 | 0.02 | 1.04 | 4.15 |

0.16 | 0.41 | 0.02 | 0.97 | 4.30 |

0.1 | 0.29 | 0.01 | 0.75 | 4.78 |

0.05 | 0.17 | 0.00 | 0.48 | 5.58 |

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**Figure 1.**(

**a**) The entropy for two statistical cells and, its numerical approximation with a half-ellipse. (

**b**) The lattice of the continuous sub- simplexes of the simplex (the integers are the serial number of fractions).

**Figure 2.**The normalized entropy map for N = 3. (

**a**) The simplex. (

**b**) The entropy diagram with three points on coordinate line A = 0.5 (B = 1, 1.4 and B

_{max}= 1.44). (

**c**) N = 3, the inverse image in the simplex. (

**d**) The inverse image in the space of the possible grading curves.

**Figure 3.**Some properties of the entropy map. The inverse image of a constant B plane section of the entropy diagram. (

**a**) Normalised entropy diagram. (

**b**) The inverse image in the simplex.

**Figure 4.**Optimal grading curves. (

**a**) N = 7, A varies. (

**b**) A = 2/3, N varies. (

**c**) The constant A sections of the simplex and the optimal line for a 2D simplex and (

**d**) for a 3D simplex.

**Figure 5.**(

**a**) The non-normalized diagram, with the image of the edges (i.e., minimum lines). (

**b**) The simplified normalised entropy diagram with the image of the edges 1 to N (i.e., minimum lines).

**Figure 6.**(

**a–h**) Some grading curves of samples used by Lorincz [1] for suffosion tests. The inset shows the permeameter test arrangement used in the tests.

**Figure 7.**Particle migration zones (

**a**) in half of the partly normalised entropy diagram for mixtures with N = 6 fractions, (

**b**) in the simplified, normalised entropy diagram. The three digit numbers in (a) correspond to the numbers on the grading curves shown in Figure 6.

**Figure 9.**Some grading curves of soils used in the filter tests of Lőrincz [4]. Note that the filters are identified by characters, and the base soils by numbers.

**Figure 11.**Grading curves for the samples used in the segregation tests of Lőrincz [1].

**Figure 12.**Results of the segregation test, indicating the variations of base entropy S

_{0}, the entropy increment ΔS and the entropy S. The grading curves for soils in groups A to E are shown in Figure 11 and the calculated entropy data is given in, Appendix A1.

**Figure 16.**The grading curves for the theoretical soils used in the testing of existing filtering laws.

**Figure 18.**The non-segregating optimal 5-fraction mixtures (see additional data in Appendix A3).

**Table 1.**Definitions of fractions and elementary cells, where d

_{0}is the elementary cell width, based on the smallest particles likely to occur in nature.

j | 1 | … | 23 | 24 |

Limits | d_{0} to 2 d_{0} | … | 2^{22} d_{0} to 2^{23} d_{0} | 2^{23} d_{0} to 2^{24} d_{0} |

S_{0j} [−] | 1 | … | 23 | 24 |

Rule | Criterion |
---|---|

U.S. Bureau of Reclamation for uniform filters and base materials [24] | $\frac{{D}_{50}}{{d}_{50}}=5-10$ |

Sichard for uniform filters and base materials | $\frac{{D}_{50}}{{d}_{50}}=3-4.5$ |

Sherard et al. 1984 [4] | $\frac{{D}_{15}}{{d}_{85}}<9$ |

Derived from Terzaghi [16] broadly-graded | $1\le \frac{{D}_{\mathrm{min}}}{{d}_{\mathrm{max}}}\le 4$ |

Rule | Criterion |
---|---|

Terzaghi [16] for broadly-graded filters and base materials | $\frac{{D}_{15}}{{d}_{85}}\le 4$, $\frac{{D}_{15}}{{d}_{15}}\ge 4$ |

U.S. Bureau of Reclamation [24] | $\frac{{D}_{50}}{{d}_{50}}=12-58$, $\frac{{D}_{15}}{{d}_{15}}=12-4$ |

Bertram [25] | $\frac{{D}_{15}}{{d}_{85}}\le 5$, $\frac{{D}_{15}}{{d}_{15}}=5-9$ |

Cistin [26] | $\frac{{D}_{10}}{{d}_{60}}<5$, ${U}_{D}=\frac{{D}_{60}}{{D}_{10}}<5$ |

D_{50}/d_{50} | D_{15}/d_{85} | D_{10}/d_{60} | D_{15}/d_{15} | S_{0b} | S_{0f} | ΔS_{b} | ΔS_{f} | ||
---|---|---|---|---|---|---|---|---|---|

1 | B1-1^{I} | 7 | 4.58 | 6.5 | 7.86 | 13 | 16 | 0 | 0 |

2 | B1-1^{II} | 14 | 5.67 | 22.5 | 9.71 | 13 | 17 | 0 | 1.585 |

3 | T1-1^{I} | 10.00 | 4.17 | 4.44 | 7.14 | 13 | 16.35 | 0 | 1.44 |

4 | T1-1^{II} | 4 | 2.42 | 3.1 | 4.14 | 13 | 15 | 0 | 0 |

5 | T1-1^{III} | 6.89 | 4.17 | 3.1 | 7.14 | 13 | 15.85 | 0 | 0.61 |

6 | T1-1^{Iv} | 11.11 | 5.17 | 6.3 | 8.86 | 13 | 16.5 | 0 | 1 |

7 | UM1-1^{I} | 13.9 | 7.3 | 0.8 | 12 | 13 | 16.8 | 0 | 0.722 |

8 | UM1-1^{II} | 58 | 7.3 | 0.8 | 12 | 13 | 18.51 | 0 | 2.07 |

9 | UM1-1^{III} | 58 | 25.5 | 2.6 | 40 | 13 | 19. | 0 | 1.585 |

10 | U1-1^{I} | 5.55 | 3.02 | 3.13 | 4.57 | 13 | 15.5 | 0 | 1 |

11 | U1-1^{II} | 7.78 | 5.19 | 5.42 | 7.86 | 13 | 16 | 0 | 0 |

12 | U1-1^{III} | 11.1 | 5.85 | 6.25 | 8.86 | 13 | 16.5 | 0 | 1 |

13 | U1-1^{Iv} | 15.5 | 10.19 | 10.63 | 15.43 | 13 | 17 | 0 | 0 |

Grading curve pair for an Individual filter criteriion | Outcome of grading entropy-based filter rule | Outcome of Individual filter criteriion | Property of Individual filter criteriion | |
---|---|---|---|---|

1 | B1-1^{I} | fails | yes | unsafe |

2 | B1-1^{II} | filters successfully | not | conservative |

3 | T1-1^{I} | filters successfully | not | conservative |

4 | T1-1^{II} | filters successfully | yes | good |

5 | T1-1^{III} | filters successfully | not | conservative |

6 | T1-1^{Iv} | filters successfully | not | conservative |

7 | UM1-1^{I} | fails | yes | unsafe |

8 | UM1-1^{II} | fails | yes | unsafe |

9 | UM1-1^{III} | fails | yes | unsafe |

10 | U1-1^{I} | filters successfully | yes | good |

11 | U1-1^{II} | fails | yes | unsafe |

12 | U1-1^{III} | filters successfully | yes | good |

13 | U1-1^{Iv} | fails | not | good |

A [−] | a [−] | x_{1} [−] | ΔS/ln2 [−] | Structure |
---|---|---|---|---|

0.50 | 1.00 | 0.20 | 1.44 | piping |

0.56 | 1.13 | 0.15 | 1.43 | piping |

0.60 | 1.23 | 0.13 | 1.41 | piping |

0.64 | 1.34 | 0.10 | 1.37 | piping |

2/3 | 1.42 | 0.09 | 1.34 | stable |

0.70 | 1.54 | 0.07 | 1.29 | stable |

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lőrincz, J.; Imre, E.; Fityus, S.; Trang, P.Q.; Tarnai, T.; Talata, I.; Singh, V.P.
The Grading Entropy-based Criteria for Structural Stability of Granular Materials and Filters. *Entropy* **2015**, *17*, 2781-2811.
https://doi.org/10.3390/e17052781

**AMA Style**

Lőrincz J, Imre E, Fityus S, Trang PQ, Tarnai T, Talata I, Singh VP.
The Grading Entropy-based Criteria for Structural Stability of Granular Materials and Filters. *Entropy*. 2015; 17(5):2781-2811.
https://doi.org/10.3390/e17052781

**Chicago/Turabian Style**

Lőrincz, Janos, Emöke Imre, Stephen Fityus, Phong Q. Trang, Tibor Tarnai, István Talata, and Vijay P. Singh.
2015. "The Grading Entropy-based Criteria for Structural Stability of Granular Materials and Filters" *Entropy* 17, no. 5: 2781-2811.
https://doi.org/10.3390/e17052781