1. Introduction
Viscous debris flows are a very complex and important geomechanical process in subaerial and subaqueous environments. Hyperconcentrated sediments are treated in the same way as yielding non–Newtonian fluids when transported, and the rheological behaviour of such sediments has been widely tested using a variety of different experimental equipment including standard rheometric systems and even unusually large rheometers. Nonetheless, their rheological characteristics are still largely ununderstood, since they involve many factors such as soil type, sediment size distribution, grain concentration, and even other physical and chemical properties (e.g., salinity, pH, mineralogy) which are particularly relevant when very fine sediments are present. In the specific case of natural sediment, even grain shape plays a significant role [
1]. With regard to the flow–like behaviour of slurries, the rate of shear deformation systematically increases as shear stress increases, but the relationships between the two differs according to sediment concentration, showing either dilatant or pseudoplastic material behaviour [
2].
The travel distance of debris flow is important in hazard mapping and can be predicted based on the process of halting the slurry. The latter is controlled by the shearing material and by the circumstances under which the material ceases to move (dynamic yield stress), leading to a different condition from the one under which the material started moving (static yield stress). This mechanism has been investigated over the past decades by several authors using both conventional and non–conventional experimental apparatus [
3,
4,
5,
6]. Mostly, previous studies have concerned the effect of fine fraction and bulk sediment concentration. However, so far the role of sediment grain size distribution has been poorly investigated, partly due to the difficulties inherent in experimental apparatus and procedures.
Several authors have focused on the yielding characteristic of noncolloidal suspensions [
7,
8] and dense granular fluid mixtures [
9,
10,
11,
12,
13]. In particular, Ancey and Jorrot [
10] studied the effects of the solid concentration of unimodal and bimodal suspensions of glass beads and quartz sand in a water–kaolin dispersion on the yield stress. They found evidence that the yield stress of a coarse particle suspension within the colloidal dispersions is strongly dependent on the solid concentration of the coarse fraction.
Experimental work by Yu et al. [
14] stressed the influence of clay minerals on the yield stress of debris flows, and later Yu et al. [
15] carried out experiments involving mixtures of clay, and fine and coarse sand, demonstrating the influence of single clay and mixed clay mixture on the threshold stress value. Pantet et al. [
16] studied the effect of coarse particle concentration on the yield stress of silty sand and muddy sand mixtures and found that the yield stress could be moderately or significantly altered by the content of coarser particles. Other authors have considered the flow–like behaviour of fine–grained slurries and demonstrated that plastic viscosity is very sensitive to the ratio between clay–silt and clay–sand fraction (e.g., [
17]). Jeong et al. [
18] focused on the role of soil texture, and proposed a schematic view of rheological behaviour, depending on grain size, showing a remarkable difference in rheological behaviour from fine–grained to coarse–grained slurry [
19].
Coussot and Piau [
20] carried out experiments on debris flow mixtures and found that coarser sediment fraction has a relevant effect on yield stress. Banfill [
21] stressed the influence of fine material in sand on the rheology of fresh mortar, concluding that yield stress increases as the fine sand fraction rises. Ancey and Jorrot [
10] showed that for poorly sorted materials, the rate of yield variation as a function of bulk sediment concentration very much depends on sediment characteristics and on the relative content of coarse–to–fine grains. Yu et al. [
15] reported a selection of works by Chinese authors [
22,
23,
24] who investigated the role of sediment grading on the yield stress of reconstituted debris flows, showing that the yield increased with a decreasing grain diameter, an increase in sediment concentration, and (corresponding to large particle concentration) with an increasing uniformity of particle size distribution. In their work, Yu et al. [
15] suggested considering grain size distribution, particle shape, and type of material of the particles (in addition to sediment concentration) as the main aspects affecting yield stress value. Jan et al. [
25] tested fine sediment slurries mixed with coarse sand at different concentrations, and showed qualitatively that the presence of coarse grains reduces the yield stress of the mixture, concluding that the rheology of the sediment slurry could vary widely depending on the particle size present in the slurry.
The previous scrutiny shows that most of the reported rheological properties of debris flows have been restricted to fine–graded mixtures, which provide the interstitial fluid matrix of viscous debris flow. However, the transition from fine–grained to coarse–grained soils is of paramount importance for the rheology, largely due to the presence of poorly sorted grains in natural slurries.
In a previous work, Pellegrino and Schippa [
13] tested the effect of granular concentration and sediment grading on the rheological properties of reconstituted debris flows. In particular, for fine–grained mixtures, the Herschel–Bulkley generalised model, which expresses the consistent coefficient as a function of sediment concentration, gives reasonable results. Nevertheless, the presence of coarse sediment greatly affects rheometric parameters, showing that even a moderate content of coarse grains may drastically modify the yield stress value. They concluded that the relative concentration of coarse and fine particles is a discriminating factor of rheological behaviour.
This work revisits the earlier inclined plane experiments [
13]. The experimental results are used to examine the effect of fine–to–coarse sediment fraction on yield stress, and a novel yield stress model, based on bulk sediment concentration and fine–coarse sediment fraction, is proposed. Eventually, its performance and the main features of the model are discussed.
2. Materials and Methods
This paper refers to inclined plane experimental activity presented in Pellegrino and Schippa [
13]. A brief summary of the materials, testing procedure and experimental data is then given. For further details refer to Pellegrino and Schippa [
13].
The source areas of the Monteforte Irpino debris flow (which occurred in 1998 in Campania, Italy) provided the soils used to prepare the testing samples. The source areas involve pyroclastic terrains belonging to the deposits generated by volcanic activity. The soils are sandy silt with small clay fraction, having specific gravity
GS = 2.57, dry weight of soil per unit volume
γd = 7.11 KNm
−3, total weight of soil per unit volume γ = 12.11 KNm
−3, and porosity
p = 0.71. The soils are sandy silt with small clay fraction. Representative threshold sediment size is assumed to be 0.5 mm, which corresponds to the limiting range of medium sand (according to the Wentworth scale). Fine–grained sediment and coarse–grained sediment are defined accordingly, and correspond to a grain diameter finer or coarser than 0.5 mm, respectively. The coarser fraction (i.e., having sediment diameter
d > 0.5 mm) is subdivided into four classes: the first two correspond to coarse sand (0.5 mm <
d < 1.0 mm) and very coarse sand (1.0 mm <
d < 2.0 mm). The latter two (2.0 mm <
d < 5.0 mm and 5.0 mm <
d < 10.0 mm) correspond to the maximum grain size diameter of the collected samples (see
Figure 1).
Before any test, organic elements are removed from the sampled soils and they are dried out in an oven at 104 °C for a day. Then, a mixture of the desired total volumetric concentration
ΦT is prepared by mixing the dry, cooled soils, including the chosen fine– and coarse–grained fraction, with an appropriate amount of distilled water. Therefore, the resulting total bulk volume concentration
ΦT is:
where
Φf and
Φg are the solid volumetric concentration for both the fine and coarse–grained mixtures, respectively:
In Equations (1)–(3) the subscripts s, f, g, and w refer to solid, fine–grained, coarse–grained materials and water, respectively. Before starting each test, a sample of about 0.5 10−3 m3 of distilled water and soils is constantly mixed for 15 min at uniform speed (30 rpm at constant temperature of about 23 °C).
To compare different mixtures with different bulk sediment concentrations, it is preferable to use the reduced fraction
ΦT/
ΦM, where
ΦM represents the maximum sediment concentration of the mixture. Even though the actual maximum concentration mainly depends on grain shape and sorting, the proposed model uses a constant value
ΦM = 0.64, which corresponds to a close random packing configuration [
26].
The inclined plane test (see
Table 1) included 14 runs conducted on a mixture of clear water and fine–grained (i.e., mixtures composed of soil fraction with a particle diameter less than 0.5 mm) and coarse–grained suspensions (i.e., mixtures composed of soil fraction with particle diameters ranging up to 10 mm).
The total grain concentration ranged from 25% to 41% in order to include a large variety of conditions under which to evaluate the macro–viscous behaviour of the mixture.
Tests 1–5 correspond to ΦT = 30%, and runs 6–10 were carried out with ΦT = 32%, varying the relative content of fine and coarse grains. In order to investigate the effect of increasing coarse particle content, with the fine fraction constant (Φf = 25%), runs 11–15 were performed using a different concentration of coarse particles Φg.
A typical inclined plane test starts by splitting the suspension on a rough horizontal plane in order to obtain a wide layer of material, and the sample thickness (h0) corresponding to the initial condition at rest is measured at different locations far from the edge (being a distance at least three times the maximum value of h0). Then, the tray is inclined step–by–step until the critical angle (ic) is reached, corresponding to a notable motion of the front edge, and the test goes on until full stoppage of the sliding mass is achieved. Lastly, the thickness (hf) of the deposited mass at rest is measured, following the same procedure used to measure the initial thickness h0. Typically, each test lasted about ten seconds, from the initial spreading to the stoppage of the slurry.
According to the lubrication assumption (i.e., material thickness
h0 is much smaller than its longitudinal extent), a uniform flow condition may be assumed for the flow mixture and, disregarding inertial effects, momentum balance provides shear stress distribution within the mixture [
27]. Threshold stress corresponding to the start of flowing (
τc1) and to the flow stoppage (
τc2) can be interpreted as a measure of static and dynamic yield stress, and they are calculated as follows:
where
g is the gravitational acceleration and
ρ is the density of the fluid.
Table 1 reports the experimental programme and the results in terms of static (
τc1) and dynamic (
τc2) yield stress.
3. Results and Discussion
Figure 2 shows that, assuming a fixed reduced bulk sediment fraction, the static and dynamic yields decrease after increasing the coarse–to–fine grain content ratio (see test 1–5 and test 6–10 in
Table 1). Thus, we may infer a depletion effect on yield stress curve due to coarse–to–fine content, which flattens the steep enhancing of yield stress associated with increasing reduced sediment fraction, as is qualitatively depicted in
Figure 3. Moreover, tests 11–15 show that scaling up bulk concentration leads to a significant enhancement of yield stress, even though the variation in coarse–to–fine sediment content should counterbalance the yield augmentation.
The influence of bulk sediment concentration (at constant coarse–to–fine content) on yield stress may be understood by considering the fine–grained mixture (i.e.,
Φg/
Φf = 0%) plotted in
Figure 4a. In fact, increasing grain content leads to a monotonic increase in static and dynamic yield stress, and this is consistent with findings reported by several authors [
17,
28]. Like Ancey & Jorrot [
10], who experimented with poorly graded sand–clay mixture, we found that the variation of yield stress with sediment concentration is pronounced, and no minimum value of the yield is expected with an increasing rate. Considering the estimated yield stress of the kaolin suspension (i.e., 39 Pa) and the granulometric sand sorting (sediment diameter up to 0.3 mm and 1.2 mm) reported by Ancey and Jorrot [
10], even the yield stress values seem consistent with the experiments carried out by the authors, which ranged from about 70 Pa to 700 Pa, corresponding to sediment concentrations ranging from 0.27 to 0.55, respectively.
Figure 4a suggests asymptotic yield stress behaviour for the solid concentration, approaching a threshold value.
Figure 4b shows yield stress as a function of coarse–to–fine grain content in the case of constant bulk sediment concentration (i.e., tests 1–5,
ΦT = 30%, and tests 6–10
ΦT = 32%). Static and dynamic yield stresses systematically decreased, increasing the relative amount of the coarse grain fraction. In fact, increasing the relative content of coarse–to–fine sediment up to 50% reduces the yield stress related to fine–grained mixture to 1/3, independently of the total grain concentration, and the yield reduction is more evident when the bulk granular concentration is increased.
Figure 4b shows that an asymptotic low yield stress value may be expected, corresponding to very dominant coarse fraction. Moreover, increasing the relative content of coarse grain reduces the difference between static and dynamic yield, and this effect is more relevant given the lower total sediment concentration (
ΦT = 30%).
The former behaviour is confirmed even when the whole set of experiments is considered (
Figure 5). In this case, the role of the total sediment concentration is also appreciable: the higher the total grain concentration (at a fixed coarse–to–fine grain fraction), the larger the difference between static and dynamic yield. Moreover, the yield stress increases, thus increasing the bulk volume concentration of the mixtures. The fine–grained fraction affects the rheological behavior: increasing the smaller grain content (at a fixed coarse grain fraction) enhances the yield of the slurries, as is shown by a comparison between test 2 (
Φf = 22%,
Φg = 8%), test 7 (
Φf = 24%,
Φg = 8%) and test 12 (
Φf = 25%,
Φg = 8%), where the yield stress more than doubled despite a limited change in fine grain content.
The interpretation of the experimental results plotted in
Figure 5 is not trivial. Since yield stress depends considerably on the presence of poorly sorted sediments, it is necessary to consider individual yield value, depending on both bulk concentration and coarse–to–fine grain content. In fact, considering bulk concentration
ΦT = (30%, 32%, 33%) and
Φg/
Φf ≈ (0.36, 0.33, 0.32), the yield stress shows values close to
τc1 = (17.9, 34.4, 17.9) Pa, and
τc2 = (9.6, 24.8, 14.7) Pa. However, a large range of yield stress results is obtained when
ΦT = (40%, 41%),
Φg/
Φf ≈ (0.64, 0.66).
To better understand this behaviour,
Figure 6 plots the experimental dynamic yield stress as a function of total grain concentration and the ratio of the coarse–to–fine–graded mixture. The interpolating surface suggests a functional relationship in terms of both total grain concentration (
ΦT) and coarse–to–fine grain content (
Φg/
Φf), confirming that maximum stress corresponds to the higher bulk concentration and the finer graded sample.
Figure 7 shows static and dynamic yield stress normalised with maximum yield for a fine–grained mixture,
, in case of constant bulk concentration
ΦT = 30% and
ΦT = 32%, as a function of coarse–to–bulk concentration. Both static and dynamic yield markedly reduced when the coarse fraction increased, and the trend is even more evident when the bulk concentration increased. In fact, when it predominates over the finer fraction, the resulting yield is less than about 20% (mixture with
ΦT = 30%) or less than 10% (mixture with
ΦT = 32%) of the yield stress for the fine–graded mixture (
Φg = 0).
Interestingly, for the lowest bulk concentration
ΦT = 30%, the static and dynamic yield trend towards the same value if the proportion of coarser particles is increased until it is comparable to or dominant over the finer fraction, as it is also evident from
Figure 5. However, for a higher bulk concentration
ΦT = 32%, static and dynamic yield still show different values, even if the coarser fraction is dominant. Remarkably, in the latter case the dynamic yield is comparable to the yield stress of the mixture with
ΦT = 30% (
Figure 5).
It may be argued that the presence of the finer fraction not only epitomises the pseudo–viscous behaviour of the flowing mixture [
2], but even increases the yield stress (
Figure 4a). The higher the concentration, the greater the difference between static and dynamic yield. Conversely, the presence of coarse fraction tends to reduce the yield stress irrespective of bulk concentration, and when coarse sediment is the dominant content, the dynamic yield is almost independent of coarse–to–fine sediment content and of sediment bulk concentration (
Figure 4b;
ΦT = 30% and
ΦT = 32%).
Experiments involving reconstituted debris flow samples highlight the importance of grain concentration in determining yield stress. The presence of coarse graded sediments affects the rheological behaviour, and significantly reduces yield stress. The finer graded matrix content increases the yield stress, whereas the presence of a coarser component reduces the threshold stress. Therefore, increasing relative coarse–to–fine content counteracts the effects of increasing bulk concentration in terms of the resulting yield stress. Consequently, defining a yield stress model is not a trivial matter for natural slurries where grains are always poorly sorted, and its dependence on grain sorting is of great relevance when dealing with natural slurries. From now on, in order to illustrate the model, we will focus on dynamic yield stress, which will now be referred as yield stress.
5. Conclusions
The experiments on poorly graded natural sediment mixtures show that the yield stress depends a great deal on bulk–solid concentration and on coarse–to–fine sediment content. Therefore, it is not possible to apply any model predicting yield stress behaviour based on bulk sediment concentration alone.
According to the experimental results, the larger bulk concentration leads to a significant increase in the yield stress value, and this boost is more evident for finer grained mixtures. Static and dynamic yield significantly reduced increasing coarse fraction, and the higher the bulk sediment concentration the more evident the effect.
Even the ratio between static to dynamic yield stress is affected by the presence of coarse particles, and static to dynamic yield shows different behaviour depending on the bulk sediment concentration of the mixture. The higher the concentration, the greater the difference between static and dynamic yield.
The proposed yield-stress model introduces a grading function which considers the coarse–to–fine grain fraction, fitted based on the mixture characteristics. The variation rate of grading function significantly depends on coarse–to–fine content. The lower the coarse grain presence, the higher the function rate. The model performance is satisfactory, and its reliability increases in the case of higher coarse grain content.
In the case of dominant coarse grain content (namely Φg/Φf > 0.5), the yield stress was found to be almost independent of the coarse–to–fine grain fraction, and its value is lower by one order of magnitude than those calculated for fine–grained mixtures. Further investigations may be oriented towards analysing the effects of splitting in different sediment size classes present in the slurry or using characteristic parameters representing the granulometric curve of natural sediments.