Definitions of fractions, based on the smallest particles likely to occur in nature.
Some rules for particle migration, filtering, and segregation were elaborated on the basis of some simple laboratory tests and data of well-designed, artificial mixtures of natural sand grains. Use was made of the knowledge available in the field and two pairs of grading entropy parameters. These parameters incorporate all information of the grading curve and are pseudo-metrics in the “space of the possible grading curves.”
- grading entropy
- internal erosion
The internal stability of compacted earth dam materials, granular filters, and soils on natural slopes is essential. The internal erosion involves loss of particles under seepage flow; the matrix of coarse soil particles may or may not be unstable [1–3]. The term “suffosion” is Russian in origin and is used to describe the process of removal and transport of small soil particles through pores [4, 5].
It is desirable that adjacent materials in earth dams or rockfill dams should act as filters for each other and the material should not segregate [6–14]. Broadly graded materials may segregate during the construction process where the particles are able to flow freely, such as tipping and spilling. The likelihood of backward erosion is greater for segregated soil than for non-segregated soil [7, 8].
The inherent stability or proneness to segregation is usually specified in terms of particular diameters
This chapter summarizes three grading entropy-based rules, on the basis of the original work of Lőrincz and some applications [15–22]. The suggested rules differ from most existing rules in that the whole grading curve is used instead of some limited number of grading curve points, without any constraint on the shape of the grading curve. They were elaborated on the basis of the knowledge available, the measured data available in the literature, and data measured for well-designed sand mixtures by Lőrincz. The rules were verified by the examined cases [19–22], an example included.
2. Grading curve characterization
The grading curve is a statistical distribution of logarithm of the diameter with respect to the dry weight. It is a discrete distribution curve with a non-uniform cell system in arithmetic scale. To characterize it, first of all, the statistical cell system—the so-called abstract fraction system—is defined and the space of the grading curve is introduced.
Then the two grading entropy parameter pairs are introduced. The first pair is related to the expected (log diameter) value of the grain size distribution, in non-normalized and normalized forms. The normalized version has a shift symmetry on the log
The second pair is the entropy arisen from the mixing of the fractions, in non-normalized and normalized forms. Its maximum for a fixed value of the first coordinate is related to a single grading curve with finite fractal distribution. The grading curves can be represented in terms of the two parameters in the entropy diagram.
2.1. The fractions
The fraction system is defined on the pattern of the classical sieve hole diameters (where measurements are made), by successive multiplication with a factor of 2, as follows. The diameter range for fraction
|Fraction number ||1||…||23||24||…|
|…||222 ||223 ||…|
|2−22 to 2−21||…||1–2||2–4||…|
or the upper diameter range for fraction
Using log2 form results in an integer increment by each multiplication and fraction as follows:
The fraction serial number variable can be expressed by the diameter:
2.2. The grading curve space
By the measurements of the fractions during sieving, the relative frequencies of the fractions
which can be rewritten as follows:
The space of the grading curves with
2.3. The grading entropy coordinates of the grading curve
The grading entropy concept is an application of the statistical entropy to the grading curve [15, 23], by introducing a uniform cell system for the derivation besides the fractions. It condenses the information of the whole grading curve into two pairs of parameters. The grading entropy
which are called as base entropy
Any decrease in the base entropy
The relative base entropy
The relative base entropy
The entropy increment Δ
The entropy increment Δ
Being a strictly concave function, the normalized entropy increment
The optimal grading curve is concave if
In the linear case, it has a unique maximum, being equal to 1/ln 2, in the center of the simplex where each relative frequencies
2.4. The entropy diagram
Four kinds of maps can be defined between the
These maps are continuous on the closed simplex for fixed
In terms of the original entropy coordinates, the map is continuous if
The images of the optimal lines—the maximum
The inverse image of a regular normalized entropy diagram point [
3. The construction of the grading entropy-based rules
3.1. The methods
The particle migration (or internal stability) rule, the filter rule, and the segregation rule were constructed as follows. For each rule, simple soil testing programs were designed and executed by using artificial mixtures of natural sand grains . Two variables were carefully constructed using the grading entropy concept  for each rule separately.
The entropy variables were used such that the experimental data were plotted on diagrams, differentiating points which exhibited different physical behavior so that domains of particular behaviors could be defined. In addition, some other information and existing data (e.g. the data base related to Ref. ) were used.
The additional information was as follows: one piece of information used was that there could be no more than two empty particle size fractions between the filter and the base soil, before the base soil cannot be retained by the filter . This can be derived using Pure Geometry Theorems  and also by using the Terzaghi filter criterion (i.e., the finer is to be protected) as follows:
3.2. Particle migration rule
For the particle migration (internal stability-suffosion) rule, simple vertical flow tests were designed and executed using artificial mixtures of natural sand grains. The dimensions of the permeameter were 20 cm in height and 10 cm in diameter. It was closed at the bottom by a sieve which was permeable of grains smaller than 1.2 mm but which retained grains larger than 1.2 mm. The downward hydraulic gradient [
The results of the vertical water flow (suffosion) test were represented in the partly normalized entropy diagram, in terms of the relative base entropy and the entropy increment coordinates
The internal stability zone III was separated by the 2/3 vertical line. The division curve between zones II and III connects the maximum entropy points of the mixtures with fraction number less than
The rule can be interpreted such that, in zone I (where A < 2/3), no structure of the larger grains is present, the coarse particles ?float? in the matrix of the fines and become destabilized when the fines are removed by piping. In the zone where A = 2/3 and A > 2/3 the coarse particles form a skeleton and total erosion cannot occur. In zone III, the structure of larger particles is inherently stable, the smaller particles may move by suffosion.
3.3. Filter rule
The filter rule was developed using three series of tests: the filter tests of Sherard , the filter tests of Lőrincz , and the suffosion tests of Lőrincz . The grading of the soils tested by Sherard is shown in Figure 8 and the grading of the soils tested by Lőrincz is shown in Figures 6 and 9. In the filtering test, the filter and base soils are placed into the permeameter (20 cm in height and 10 cm in diameter) in series separated by a sieve. The downward hydraulic gradient [
Two pseudo-metrics were constructed from the grading entropy parameters. The logarithm of the difference between base entropies of the filter and base soils, log(
Plotting the test results in terms of the foregoing variables, the safe and unsafe areas were separated by a straight line: a layer acts as a filter for an adjacent layer (for the base soil) on the condition that:
3.4. Segregation rule
For the segregation rule, the simple
The artificial mixtures of natural sand grains used for the segregation rule were partly continuous mixtures (A, B), partly gap-graded mixtures (C, D, E), as shown in Ref. . The results are shown in Figure 11. The difference in the initial and the poured base entropy
The results showed that the base entropy difference
3.5.1. Non-segregating mixtures
Minimal segregation occurs, and relatively uniformly textured body of soil is achieved for laboratory testing of granular materials or for the earth works, if a non-segregating mixture with 0.4 <
To construct continuous, non-segregating mixtures, some optimal limit curves can be determined. The optimal mixtures computed by a simple algorithm fulfilling Eqs. (15)–(17) for fixed
However, it can be noted that soils with gap-graded grading curves can be non-segregating also. For example, in case of a two-fraction soil with gap-graded grading curve, the segregation is minimal if the quantity of the larger fraction varies between 0.4 and 0.7.
3.5.2. Testing the filter rules
The grading entropy-based filter law was compared with the existing filtering rules available in the literature. Summaries of well-known filter rules [7, 9–15] for uniformly graded filters and broadly graded filters are presented in Appendix A. These different filtering rules were tested by generating soils with the special-shaped grading curves  shown in Figure 13 and parameterized in Table 3.
The 13 combinations listed in Table 3 were represented for the different filtering rules of the literature, some results are shown in Figure 14. If the rule from the literature predicted a successful filtering (i.e., safe behavior), it was plotted with an open circle; where it predicted a failure to filter (i.e., unsafe behavior), it was plotted with a full circle.
The results indicated that (i) the Terzaghi?s filter rule is too conservative, (ii) the Bertram rule is conservative for mixed filters and not acceptable for uniform soils, (iii) the rule of United States Bureau of Reclamation (USBR) for uniform filters is acceptable, and (iv) for mixed filters is not acceptable.
4. Case study
Several applications of the entropy-based rules, by examining the reason of piping, softening, dispersive soil behavior, and the goodness of a leachate collection system, were previously presented [19–22]. Here, a dam failure case study is summarized.
The 71 m high, Gouhou rockfill dam was founded on a sandy gravel base layer (Figure 15). The dam body consisted of the following parts: the upstream face was a thin layer of material with a design particle diameter of 100 mm, zone I was a transition zone with the maximum diameter of 400 mm, zones II and III were the main rockfill with maximum diameters
The dam failed [22, 27, 28], killing 288 people, immediately after the first rising water level, and infiltrating the water into the dam body causing internal erosion, piping, and washout of material (see 1–6, Figure 15).
The relative base entropies of the soils in zones I, II, and III were 0.42, 0.55, 0.58, respectively, all less than 2/3 and non-segregating. This result explains why the rockfill material was incapable of forming a stable skeleton of coarse fragments. It follows that the grading entropy-based soil behavior rules would have been capable of predicting piping failure in the Gouhou dam.
5. Discussion and conclusion
5.1. Some comments on the entropy parameters
The grading entropy parameters are some kind of integrals of the whole grading curve. The same shaped grading curve has the same
The relative base entropy parameter
The entropy increment Δ
The entropy parameters are pseudo-metrics. The difference between base entropies of the filter and base soils,
5.2. Some comments on the rules
The overall soil stability—according to the experimental results—is described by the criterion that
Some questions arise, for example, in regard to the stability of a single fraction which does not lie in a unique position on the entropy diagram. Since the change due to degradation is the appearance of smalls, which causes an increase in the
Another question is related to the probability that an arbitrary
Significant segregation is unlikely to occur, if the relative base entropy A is between the limits of 0.4 and 0.7. It is important to note that the same parameter—the relative base entropy A—is responsible for overall soil stability. Soils which meet both criteria may constitute very small part of the grading curve space and may need careful design in case of broadly graded soils.
The filtration problems are safely solved in the literature for uniform filters and bases (i.e., soils to be protected by the filters). The suggested filter rule can be used to design broadly graded filters (e.g., for clay cores or for leachate collection systems). However, the rule was estimated on the basis of one data point only at the range of very large
The testing of the existing rules known from the literature was possible on the basis of the suggested filter rule and using some theoretical grading curves. According to the results, the Terzaghi filter rule is too conservative. The filtering rule of USBR for uniform filters is acceptable. The mixed filter rule of the USBR is not conservative and is not acceptable. The Bertram rule is be conservative for mixed filters and not acceptable for uniform soil.
5.3. The importance, use, and implementation of the rules
Applications of the derived entropy-based rules were presented by examining the reason of a dam piping failure, dike piping, dispersive soils, leachate collection system case studies [19–22], a dam example is presented here only. On the basis of the case study, it is apparent that the grading entropy-based soil behavior rules would have been capable of predicting piping failure in the Gouhou dam.
The grading entropy-based criteria can easily be implemented into any laboratory test evaluation software. A basic requirement for the use is that the grading curve information is reliable. The simple soil tests presented here were made on coarse material and the rules apply for soils where the solid fraction is composed of non-clay minerals.
For clay minerals, the same criteria may be valid if the grading curve information is reliable and the appropriate degree of particle agglomeration is reflected in the measurements [30–33]. The first results indicate that the same criteria may be valid for silty soils if the grading curve information is reliable (see e.g., the dispersive soil case studies ).
The chapter is related to the work of the Research Project OTKA 1457/86 on river dykes and of the Research Project NKFP B1 2006 08 on MSW landfills.
Rules for uniform filters
U.S. Bureau of Reclamation :
Sherard et al. :
Rules for broadly graded soils
US Bureau of Reclamation :