![1 Introduction
Debris and hyper-concentrated flows are among the most destructive of all
water-related disasters. They mainly affect mountain areas in a wide range of
morphoclimatic environments and in recent years have attracted more and more
attention from the scientific and professional communities and concern from
public awareness due to the increasing frequency with which they occur and the
death toll they claim. These phenomena do not allow a sufficient early warning,
as they are characterised by a very short time-scale and, therefore, defence
measures should be provided, especially when they are associated with flash
floods or dam failures. To this end, the identification of effective procedures
aimed at evaluating the probability of these extreme events and the triggering
and mobilising mechanism has become an essential component of the water and
land use planning processes. This concept leads to a new integrated risk
management approach, which comprises administrative decisions, organisation,
operational skill and the ability to implement suitable policies. The broadness of
the question requires approaches from various perspectives.
To this end, the dynamic behaviour of these hyper-concentrated water
sediment mixtures and the constitutive laws that govern them plays a role of
paramount importance.
Debris flow modelling requires a rheological pattern (or constitutive
equation) that provides an adequate description of these flows.
One of the main difficulties met by the approaches available is linked to their
validation either in the field or in a laboratory environment. Greater research
needs to be directed towards a thorough investigation of the above mentioned
issues.
Such knowledge is essential in order to assess the potential frequency of these
natural hazards and the related prevention and mitigation measures.
With reference to these issues, this paper aims to provide the state-of-the-art
of debris flow rheology, modelling and laboratory and field investigation, along
with a glance to the direction that debris flow in-depth studies are likely to
follow in future.
2 Debris flow model development
A thorough understanding of the mechanism triggering and mobilising debris
flow phenomena plays a role of paramount importance for designing suitable
prevention and mitigation measures. Achieving a set of debris flow constitutive
equations is a task which has been given particular attention by the scientific
community (Julien and O’Brien [33]; Chen [9]; Takahashi [35]). To properly
tackle this problem relevant theoretical and experimental studies have been
carried out during the second half of the last century.
Research work on theoretical studies has traditionally specialised in different
mathematical models. They can be roughly categorized on the basis of three
characteristics: the presence of bed evolution equation, the number of phases and
the rheological model applied to the flowing mixture (Ghilardi et al. [24]).
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![Most models are based on the conservation of mass and momentum of the
flow, but only a few of them take into account erosion/deposition processes
affecting the temporal evolution of the channel bed.
Debris flows are mixtures of water and clastic material with high coarse
particle contents, in which collisions between particles and dispersive stresses
are the dominant mechanisms in energy dissipation.
The rheological property of a debris flow depends on a variety of factors,
such as suspended solid concentration, cohesive property, particle size
distribution, particle shape, grain friction and pore pressure.
Various researchers have developed models of debris flow rheology. These
models can be classified as: Newtonian models (Johnson [32]), linear and non
linear viscoplastic models (O’Brien et al. [41]), dilatant fluid models (Bagnold
[4]), dispersive or turbulent stress models (Arai and Takahashi [2]), biviscous
modified Bingham model (Dent and Lang [15]), and frictional models (Norem et
al. [40]). Among these, linear (Bingham) or non-linear (Herschel-Bulkey)
viscoplastic models are widely used to describe the rheology of laminar
debris/mud flows (Jan, 1997).
Because a debris flow, essentially, constitutes a multiphase system, any
attempt at modelling this phenomenon that assumes, as a simplified hypothesis,
homogeneous mass and constant density, conceals the interactions between the
phases and prevents the possibility of investigating further mechanisms such as
the effect of sediment separation (grading).
Modelling the fluid as a two-phase mixture overcomes most of the limitations
mentioned above and allows for a wider choice of rheological models such as:
Bagnold’s dilatant fluid hypothesis (Takahashi and Nakagawa [56]), Chézy type
equation with constant value of the friction coefficient (Hirano et al. [27]),
models with cohesive yield stress (Honda and Egashira [28]) and the generalized
viscoplastic fluid Chen’s model (Chen and Ling [10]).
Notwithstanding all these efforts, some phenomenological aspects of debris
flow have not been understood yet, and something new has to be added to the
description of the process to reach a better assessment of the events. In this
contest, the mechanism of dam-break wave should be further investigated. So
far, this aspect has been analysed by means of the single-phase propagation
theory for clear water, introducing in the De Saint Venant (SV) equations a
dissipation term to consider fluid rheology (Coussot [12]; Fread and Jin [23]).
Many other models, the so-called quasi-two-phase-models, use SV equations
together with erosion/deposition and mass conservation equations for the solid
phase, and take into account mixture of varying concentrations. All these models
feature monotonic velocity profiles that, generally, do not agree with
experimental and field data.
2.1 Rheology
The rheological property of debris and hyper-concentrated flows depends on a
variety of factors, such as the suspended solid concentration, cohesive property,
size distribution, particle shape, grain friction, and pore pressure. So, modelling
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![these flows requires a rheological model (or constitutive equation) for sediment-
water mixtures.
A general model which can realistically describe the rheological properties of
debris flow should possess three main features (Chen [9]). The model should:
describe the dilatancy of sediment-water mixtures;
take into account the so-called soil yield criterion, as proposed by
Mohr-Coulomb;
assess the role of intergranular or interstitial fluid.
The earliest of such rheological models was empirically formulated by
Bagnold [4].
On the whole, a rheological model of debris and hyper-concentrated flows
should involve the interaction of several physical processes. The non-Newtonian
behaviour of the fluid matrix is ruled, in part, by the cohesion between fine
sediment particles. This cohesion contributes to the yield stress, which must be
exceeded by an applied stress in order to initiate fluid motion.
In view of theoretical soundness behind the development of different non-
Newtonian fluid models, Bailard [5] and Hanes [25] have questioned the validity
of Bagnold’s empirical relations. Limitations in Bagnold’s model may be
attributed to the ambiguity in the definition of some rheological characteristics as
the grain stresses.
To overcome these problems, Chen [9] developed a new generalised
viscoplastic fluid (GVF) model, based on two major rheological properties (i.e.
the normal stress effect and soil yield criterion) for general use in debris flow
modelling.
The analysis Chen conducted on the various flow regime of a granular
mixture identified three regimes: a quasi-static one, which is a condition of
incipient movement with plastic behaviour, a microviscous one at low shear
rates, in which viscosity determines the mixture behaviour, and finally a granular
inertial state, typical of rapid flowing granular mixtures, dominated by
intergranular interactions.
All the models previously reviewed feature monotonic velocity profiles that,
generally, do not agree with experimental and field data. In many tests
(Takahashi [53]) “S” reversed shaped trends have been observed, where the
maximum shear rate is not achieved near the bed, but rather between the bed and
the free surface. The main discrepancy is derived from the assumption of a debris
flow as a uniform mixture. In fact, the solid concentration distribution is usually
non-uniform due to the action of gravity, so that the lower layer could,
consequently, have a higher concentration than the upper layer. Higher
concentration means higher cohesion, friction and viscosity in the flow.
Wan [58] proposed a multilayered model known as the laminated layers
model that features a stratified debris flow into three regions from the bed to the
surface: a bed layer, in which an additional shear stress is dominant in
momentum exchange; an inertial layer, where the dispersive stress of the grains
is dominant; and an upper viscoplastic layer, which can be represented by the
Bingham’s model.
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![The one-layer models are unable to adequately feature the entire thickness of
the flow and, therefore, it has recently become common to use multi-layers
models that combine two or more constitutive relationships in order to analyse
adequately these phenomena. The coefficients of the rheological models have
wide ranges of variation and, therefore, in evaluating them considerable errors
are committed. On the other hand, some empirical equations of velocity are
necessary in any debris flow disaster-forecasting measure, although the
hydraulics of debris-flow is not theoretically comparable to that of a traditional
water flow.
2.2 Triggering and mobilising processes
Debris flow resulting from flash flood or a sudden collapse of a dam (dam-break)
are often characterised by the formation of shock waves caused by many factors
such as valley contractions, irregular bed slope and non-zero tailwater depth. It is
commonly accepted that a mathematical description of these phenomena can be
accomplished by means of 1D SV equations (Bellos and Sakkas [6]).
During the last Century, much effort has been devoted to the numerical
solution of the SV equations, mainly driven by the need for accurate and
efficient solvers for the discontinuities in dam-break problems.
A rather simple form of the dam failure problem in a dry channel was first
solved by Ritter [46] who used the SV equations in the characteristic form, under
the hypothesis of instantaneous failure in a horizontal rectangular channel
without bed resistance. Later on, Stoker [50], on the basis of the work of Courant
and Friedrichs [11], extended the Ritter solution to the case of wet downstream
channel. Dressler [19] used a perturbation procedure to obtain a first-order
correction for resistance effects to represent submerging waves in a roughing
bed.
Lax and Wendroff [35] pioneered the use of numerical methods to calculate
the hyperbolic conservation laws. McCormack [39] introduced a simpler version
of the Lax-Wendroff scheme, which has been widely used in aerodynamics
problems. Van Leer [57] extended the Godunov scheme to second-order
accuracy by following the Monotonic Upstream Schemes for Conservation Laws
(MUSCL) approach. Chen [7] applied the method of characteristics, including
bed resistance effects, to solve dam-break problems for reservoir of finite length.
Sakkas and Strelkoff [47] provided the extension of the method of the
characteristics to a power-law cross section and applied this method to a dam
break on a dry right channel in the case of rectangular and parabolic cross section
shapes. Strelkoff and Falvey [52] presented a critical review of numerical
methods of characteristics of power-law cross sections.
Hunt [29] proposed a kinematic wave approximation for dam failure in a dry
sloping channel.
Total Variation Diminishing (TVD) and Essentially Non Oscillation (ENO)
schemes were introduced by Harten [26] for efficiently solving 1D gas dynamic
problems. Their main property is that they are second order accurate and
oscillation free across discontinuities.
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![Recently, several 1D and 2D models using approximate Riemann solvers
have been reported in the literature. Such models have been found very
successful in solving open channel flow and dam-break problems.
In the past ten years, further numerical methods to solve flood routing and
dam-break problems, have been developed that include the use of finite elements
or discrete/distinct element methods (Asmar et al. [3]).
Finite Element Methods (FEMs) have certain advantages over finite different
methods, mainly in relation to the flexibility of the grid network that can be
employed, especially in 2D flow problems.
Mambretti et al. [38] and De Wrachien and Mambretti [17, 18] used an
improved TVD-Mc Cormack-Jameson scheme to predict the dynamics of both
mature (non-stratified) and immature debris flow in different dam break
conditions.
3 Laboratory and field studies
To validate both the rheological and dynamic models, herewith described,
comparisons need to be made between their predictions and results of laboratory
and field tests. Agreements between the computational and experimental results
are essential since they allow the assessment of the models’ performance and
suggest feasible development of the research.
The experimental point of view in debris flow research, however, encounters
considerable problems that are yet to be fully overcome, connected largely to the
accuracy of measuring techniques and flow simulation in experimental tests.
Lastly, field studies are probably the most difficult and costly study approach of
debris flow; the difficulties encountered are connected to their considerable
complexity and the difficulty of direct observation. The exceptional and
infrequent conditions in which debris flows occur do not generally permit a
sufficient number of observations for the same type of field reality to deduce the
specific behavioural laws for that area. Reference to different territorial
situations also highlights another problem: that of the homogeneity of data, given
the substantial territorial peculiarity in which the phenomena occur. Besides,
field data are essential in determining the quality of any mathematical model, as
they are especially important for estimating velocity, discharge, concentration,
yield stress, viscosity and grain-size.
This need requires the use of laboratory experimentation when the previous
problems cannot be overcome, and in certain cases it is the only possible path to
follow.
Within this ground, many experiments have been carried out, ranging from
solid transport (little amount of particles in a large environment of clear water) to
dry granular flow, where water is not present.
An empirical picture of debris flow physics can be drawn from a combination
of real-time field observations (Okuda et al. [42]); detailed measurements during
controlled field and laboratory experiments (Takahashi [54]), and analyses of
debris flow paths and deposits (Fink et al. [21]).
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![Few reliable techniques exist to measure properties of flowing debris. Grossly
invasive procedures such as plunging buckets or sensors into the flows
conspicuously change the dynamics of the debris, while their behaviour has
discouraged attempts to use non-invasive techniques such as ultrasonic, X ray,
and others (Lee et al. [36]; Abbott et al. [1]).
With regard to the rheological properties, many experiments (Chen [8]) have
shown that the Herschel-Bulkley equations fit quite well laboratory data. One of
the criticisms that may be moved to these tests is related to the scale effect.
Successful models of debris flows must describe the mechanics of
mobilization as well as the subsequent flow and deposition processes.
Mobilization requires failure of the mass, a quantity of water to saturate the solid
phase, such a change of energy, from gravitational to kinetic, to modify the
motion pattern from sliding along a failure surface to a more widespread solid-
liquid mixture that can be assessed as flow.
On the whole, laboratory and field data are essential in determining the
quality of any mathematical model, as it is especially important for estimating
velocity, discharge, concentration, yield stress, viscosity and grain-size
(Lorenzini and Mazza [37]). However, the achievement of good agreement
between theoretical and experimental results does not justify indiscriminate
extrapolation for the various territorial situations, which have very different
boundary conditions from standard laboratory conditions. Assuming that the
scientific research path cannot exclude an accurate observation and description
of the phenomenon in question, without which the analysis of physical processes,
that generate it, would become extremely artificial and uncertain, it is hoped that
any attempt at improving the interpretation of the phenomenon involves critical
comparison between the theoretical, experimental, and field approaches, as well
as extensive osmosis process between the same approaches.
4 Debris flows generated by slope failures
Debris flows can be the result of some form of landslides. In particular sliding
phenomena in granular soils can turn into flow like movements.
The main difference between slides and flow like landslides concerns the
mechanisms of movement. While a slide advances on the slip surface as a rigid
block or with a small internal deformation, a flow spreads downslope as a
viscous fluid, adapting itself to any morphological change encountered along its
path.
In some conditions shear failure (sliding) can be affected by a rapid increase
of positive pore pressures in excess to the hydrostatic values. The raise of excess
pore pressures decreases the shear resistance of the soil inducing an acceleration
of the movement: under these conditions the process can originate a debris flow.
The triggering of positive excess pore pressure in loose granular materials can
occur if the soil is saturated and the mechanism of slope deformation is
characterized by fast volumetric compression.
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![The occurrence of flow like movements is dependent on the un-drained
behaviour of the soil, which refers to the condition of a saturated soil deforming
at constant volume.
The knowledge of the un-drained response of granular soils is of critical
importance in assessing their susceptibility to liquefaction. The term liquefaction
is frequently used to indicate all phenomena involving excessive deformation in
saturated cohesionless soils and is not limited to the development of 100%
excess pore pressure. Liquefaction can be triggered by either static or cyclic
loading. Liquefaction due to static loading is associated with granular soils
deforming in a strain softening (or limited strain softening) manner that results in
limited or unlimited unidirectional flow deformation (Sivathayalan and Vaid
[48]).
A fundamental understanding of the un-drained response of granular soils has
been derived from controlled laboratory studies. Un-drained triaxial compression
tests on sand specimens mostly reconstituted by moist tamping have formed the
basis for the steady state concepts (Poulos [44]).
Susceptibility of soil to liquefaction mainly depends on grain size and
porosity, but also on stress conditions (Picarelli et al. [43]).
Ishihara et al. [30] presented the results of a series of laboratory tests, using
triaxial apparatus, on saturated samples of Toyoura sand consolidated
anisotropically. They found that with an increasing degree of anisotropy at the
time of consolidation the sample becomes more contractive and susceptible to
triggering flow failure. They found that the major effective principal stress at the
time of anisotropic consolidation is a parameter controlling dilative or
contractive behaviour of the sand. As a result the most appropriate way to
normalise the residual strength of anisotropically consolidated sand is by the use
of major principal stress at consolidation. The quasi steady state strength is then
a function of void ratio and the major effective stress at consolidation.
Other contributions devoted to the assessment of the potential for liquefaction
of a soil are based on the concept of region of instability (Sladen et al. [49]). Soil
instability is a phenomenon that resembles liquefaction in that there is a sudden
decrease in the soil strength under un-drained conditions. This loss of strength is
related to the development of large pore pressures reducing effective stresses in
the soil. Lade [34] showed that there exists a region of instability inside the
failure surface. The loss of strength occurs in un-drained condition as a
consequence of disturbances small but fast enough to prevent water drainage.
Conventional slope stability analysis methods (limit equilibrium methods) are
widely used to investigate landslide problems and to determine the state of stress
in slopes. This type of analysis has been used by Lade [34] for the determination
of the state of stress in finite slope made of loose sand in order to investigate the
region of instability by varying the slope height.
Deangeli [13] presented a study devoted to the assessment of the potential for
liquefaction in all zones of finite slopes from the in situ state of stress. For these
purposes numerical models reproducing different slopes have been set up by
using a finite difference code (FLAC manuals, 2001). The state of stress in
slopes has been evaluated in both elastic and elastic-plastic field. By relating this
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![state of stress to the parameters stated by Ishihara et al. [30] to describe the
potential for liquefaction a chart of susceptibility of debris flow in soil slopes has
been set up. The chart reports curves that establish the limit condition on the
basis of critical combinations of void ratio, slope angle and slope height.
In this context some authors define a slope safety factor against liquefaction.
For instance Poulos et al. [45] proposed the ratio between the residual strength of
the soil Sus (the minimum resistance in un-drained conditions for a contractive
soil with respect to the in situ void ratio) and the shear stress required for static
equilibrium along the potential sliding surface. Ishihara et al. [30] defined the
safety factor as the ratio between the residual strength of the soil (which is
dependent on the major effective principal stress at the time of anisotropic
consolidation) and the maximum shear stress along the potential sliding surface.
Deangeli [13] reported the safety factor against liquefaction along different
surfaces passing through a slope and assessed the volume of soil potentially
involved in debris flow.
The analysis of the propagation of debris flows generated by slope failures
can be performed by taking into account the initial value of excess pore pressure
(after slope failure) and its dissipation along the path.
Significant results have been obtained by instrumented laboratory flume
experiments. In these experiments the role of pore pressure in the flow failure
phase, i.e. the transition from sliding to flow was investigated (Eckersley [20]).
Deangeli [14] set up series of flume experiments to analyze the behaviour of
water sand mixture flows, as a consequence of slope failures induced by water
table raising and rainfall. The flows initially accelerated but at a certain stage of
the process, unsteady deposition of the sand occurred, preceded by the
transformation of the movement from flow to sliding. The phenomenon of
deposition of the soil along the flume occurred at inclination greater than in the
case of Spence and Guymer [51] experiments.
On the basis of the reported results, it is evident the need of further
experimental works investigating the dependence of debris flow behaviour by
the triggering mechanisms and the role and generation of pore pressure during
the propagation phase.
5 Concluding remarks
Debris and hyper-concentrated flow result from the interaction of hydrological
processes with geological processes and are triggered when soils get saturated
and the stability of the slope is no longer maintained. These flows are among the
most destructive of all water-related disasters. In this context, the recognised
need to improve knowledge on the mechanics of these solid-liquid flows,
highlighted by a critical analysis of the current international state-of-the-art,
represent the seeding of the present work.
Although the main aspects that rule the mechanics of these phenomena seem
to be understood, it has to be underlined the relative scarcity of experimental
(laboratory and field) data, the only ones that allow effective check of the models
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![nowadays available in different flow conditions and the estimation of the
rheological parameters they contain.
Greater research needs to be directed towards understanding the nature and
the behaviour of these flows. Such knowledge is essential in order to estimate the
potential frequency of these natural hazards and design suitable prevention and
remediation measures.
The ideal sequence that should be pursued in the approach to the difficult task
of the management and mitigation of hyper-concentrated and debris flow can be
obtained as follows (De Wrachien [16]):
first, a systematic collection of field data should be carried out in order
to provide a large base of reliable data that could allow a better
knowledge of the existing risk trends and a deeper understanding of the
mechanics of the phenomena, along with their general behaviour and
effects;
secondly, effective mathematical models, which strongly depend on
data and measurements collected and performed in the field for their
calibration and design, should be constantly developed, updated when
needed, tested and applied;
hazard mapping techniques and identification of possible scenarios,
which need reliable models to be effective and sound, should then be set
up;
on the basis of the knowledge achieved in the previous steps, the best
mitigation solutions should be identified, designed and built up;
finally a program of systematic observations on the sites, where risk has
been mitigated, should be planned and carried out to detect any
shortcoming and test the efficiency of the investigations.
Each of the above studies and investigations needs improvements and
depends, to achieve them, on improvements in other fields. Improving
measurement and documentation procedures would provide a better knowledge
and ideas for new and more advanced models. The application of existing
models based on the data collected in the field and the development of reliable
new ones would allow, on one hand, to better focus what to observe in field and,
on the other hand, improve mitigation measures and procedures. The field
application of these latter would then identify new parameters to be measured
and introduced in the models.
From all these activities would emerge the best direction to be followed in
future in-depth studies and investigations of debris flows.
References
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1993.
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![fulfill this purpose, the model should have specific features, such as the
treatment of wet-dry fronts, the handling of complex geometries and high bed
slopes and the possibility of changing the model application field from
Newtonian to non-Newtonian fluids, simply by changing the resistance law.
These features have previously been tested applying the model to different
test cases that have been properly chosen [1]. The classic frictionless dam-break
test has been used to verify the correctness of waves speed propagation and the
capability of treating wet-dry fronts. A non-cylindrical frictionless ideal channel
has been used to evaluate the model response to abrupt changes in cross-section
wideness and bed elevation, then the effect of friction terms introduction has
been checked using a mud flow dam-break. The first phase of the model
verification ended with the simulation of laboratory experiments on a mud flow
dam-break over a sloping plane.
In the present phase, the model is applied to two real events that occurred in
North-Eastern Italy. The first is a debris flow that took place in the Upper Boite
Valley, in the proximity of Cortina d’Ampezzo, in 1998, the second is a mud
flow event located in the Stava Creek Valley in 1985.
The proposed model is based on an alternative formulation of conservative
balance equations, which includes a particular mathematical expression of source
terms ideated for natural channels, and which has already demonstrated
important stability features under the numerical point of view [2, 3]. The
numerical implementation is performed using the Godunov finite volumes
scheme. This kind of numerical schemes are largely diffused in mud flow or
debris flow treatment [4–6], together with the Roe’s approximation for the
solution of the Riemann problem. The presented model uses the same approach,
but paying careful attention in conserving the general formulation suitable for
complex geometry channels, in particular for what concerns the expression of the
wave propagation celerity. This term is usually expressed as a function of water
depth and cross-section width, but these hydraulic quantities often need to be
corrected or mediated to be representative of irregular cross-sections. As an
alternative, cross-section shape can be parameterized to be numerically handled
[7]. In this work, celerity is determined referring to cross section wetted area and
static moment, in order to ensure the formulation generality.
Source terms are handled using the splitting technique [8] and evaluated with
the Euler’s method. The pressure source terms, induced by the channel irregular
geometry have been treated as in [2, 3], mathematically transforming the
derivative of the static moment in order to eliminate the explicit dependence on
the channel bed slope. This operation keeps its validity also in case of highly
sloping channels, condition which often occurs in mud flow or debris flow
phenomena. Friction source terms depends on the evaluation of friction slope,
and therefore on the adopted resistance law. Like most of numerical models [5],
the proposed model set up permits to easily change the resistance law and
therefore to use the best fitting rheological model for each test case. It is worth
noting that source terms numerical implementation has been kept as simple as
possible, to put in evidence the stability features coming from the basis
mathematical model.
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![2 Mathematical model
The mathematical model is based on an alternative formulation of shallow water
equations for one-dimensional (1-D) flows in natural channels of complex
geometry [2]. The continuity equation and the momentum balance equation are
written in terms of state variables A and Q, considering no lateral inflows.
0
A Q
t x
(1)
2
1
1
w
f
z
I
Q Q
gI g gAS
t x A x
(2)
where t is time, x is distance along the channel, A the wetted cross-sectional area,
Q the discharge, g the gravitational acceleration, I1 the static moment of the
wetted area, defined as:
1 0
cos , d
h x
I b x z h x z z
(3)
I2 is the variation of the static moment I1 along the x-direction, So = sin, where
is the angle between channel bottom and the horizontal, b is the cross-section
width, h is flow depth.
The system closure equation for the evaluation of the friction term Sf will be
described in detail for each examined test case, but the generally considered
formulation is
f
S
gR
(4)
in which Sf is the slope friction, R is the hydraulic radius, ρ is the mixture or the
fluid density, and the shear stress τ depends on the adopted rheological model.
2.1 The source term
Differently from the commonly used formulation of shallow water equations, the
proposed model does not include in the momentum balance equation source term
a direct dependence on bed slope. Details on the mathematical treatment which
led to eqn. (2) can be found in [3].
The classic momentum equation is
2
1 0 2
f
Q Q
gI gA S S gI
t x A
(5)
Focusing on the source term, the pressure term I2 has the following
expression:
1
2 0
,
cos d
h x
h
b x z
I
I h x z z
x x
(6)
Briefly, the pressure term I2 can be expressed as the sum of two terms, one of
which is the variation of static moment I1 along x considering the water surface
elevation zw as a constant, while the other exactly balances gravitational forces in
the momentum equation, unless the presence of the term cos which arises in
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![case of high slopes, and cannot be neglected when considering mud-flow or
debris-flow phenomena.
1
2 0 cos
w
z
I
I S A
x
(7)
The substitution into (6) produces:
1
0 2 0 1 cos
w
f f
z
I
gA S S gI gAS gAS g
x
(8)
In this case, the term AS0 does not disappear as illustrated in [2, 3], but it
remains and it is multiplied by the factor (1-cos). However, numerical proofs
have demonstrated that this term is little if compared to friction terms, and can
therefore be neglected. Eqn. (2) is therefore valid also for high sloping channel
and debris flow simulation.
3 Numerical model
Shallow water equations have been numerically implemented using the first-
order finite volumes Godunov scheme. Numerical fluxes are computed with
Roe’s method and source terms are evaluated with Euler’s approach and taken
into account adopting the splitting technique. Details on the different
components of the numerical model can be found in Toro [8]. The resultant
scheme is explicit, first-order accurate, and has a very uncomplicated structure,
since it is built choosing the simplest solution technique for every element of the
partial differential equations system. This approach has the intention to illustrate
the intrinsic stability features of the mathematical model, which could otherwise
be hidden by sophisticated numerical schemes.
Referring to shallow water equations in the vector form (eqn. (9)) the splitting
approach for source terms treating, consists in separately solving the
homogeneous partial differential equations system (eqn. (10)) and the ordinary
differential equation (eqn. (11)). In detail, the solution obtained from eqn. (10) is
used as initial condition for eqn. (11).
t x
U F U S U (9)
0
t x
U F U U (10)
t t dt
U S U U (11)
The Roe’s scheme, used to solve eqn. (6), requires the definition of the
Jacobian matrix
2
2 2
1
2
0 1
0 1
2
2
I Q Q
c u u
g
A A
A
F
J
U
(12)
Most of models proposed in the literature about the resolution of shallow
water equations for debris flow or natural channels, based on approximate
Riemann solvers (see for example [4, 5, 9]), adopt the same simplification in the
evaluation of the term ∂I1/∂A, assuming
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![Figure 2: Computational scheme for momentum balance pressure source
term.
Considering no lateral inflows, source terms are present only in the
momentum balance equation. This term can be divided into two parts, that is the
friction term and the pressure term, represented by the static moment variation
along channel, taking the water surface elevation as a constant.
The computational scheme for the pressure term quantification in represented
in Figure 2, and the variation of I1 is computed as:
1 1
2 2
1 1
1 1
w w
i i
z z
I h I h
I I
x x x
(20)
4 Numerical tests
In this work the model has been applied to two real events. The first is a natural
debris flow event, due to intense rainfall, surveyed at the Acquabona site in
Northern Italy. It is of particular interest thanks to the large amount of available
field data. The second is the Stava mud flow, a tragic episode occurred in a little
town of Italian Alps. This event was caused by the collapse of two tailing dams,
which released a huge quantity of water into the Stava Creek channel, causing
the formation of a mud flow wave with an enormous destructive power.
4.1 Acquabona debris flow
The Acquabona debris flow has been widely surveyed and documented in the
context of the “Debris Flow Risk” Project, funded by the EU. In particular, the
UPD (resp. Prof. Rinaldo Genevois) has carried out a research on some debris
flow prone watersheds in the Upper Boite Valley (Eastern Dolomites, Southern
Alps) and surroundings, included in the municipality of Cortina d’Ampezzo [10].
A large quantity of field data is therefore available since an automatic, remotely
controlled monitoring system has been installed at Acquabona on June 1997. The
Acquabona site in characterized by one or more debris flow every year, which
usually occur in summer and in early autumn and are associated to intense,
spatially limited rainfall events. The monitoring system installed at Acquabona
was fully automatic and remotely controlled. It consisted of three on-site
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![monitoring stations and an off-site master collection station. Every station was
equipped with a geophone, while at Station 3 also a superficial pressure
transducer and an ultrasonic sensor were present.
In this work we refer to the event of the August 17th
in 1998. The event was
originated by a very intense rainstorm: 25.4 mm of rain were measured during 30
min by the rain-gage at Station 1. The volume of the deposits available for debris
flow generation has been estimated to be around 8000-9000 m3
. The overall
duration of the event was of approximately 38 min and more than 20 different
surges have been surveyed at Station 3.
The geometry of the channel is available thanks to 19 surveyed transversal
cross-sections, for a global channel length of 1120 m and a difference in height
of 245 m. The longitudinal slope ranges from 10% to 55%. For model
application a constant spatial step of 1 m has been adopted. Numerical
simulations were performed adopting the rainfall hydrograph reconstructed by
Orlandini and Lamberti [11], which has an extension of about 2.5 hours and a
peak discharge of 2.3 m3
/s. An open boundary type condition is imposed at the
downstream end. For the debris flow the bulk concentration is assumed to be 0.6
and mixture density 1850 kg/m3
, according to [7].
The rheological model adopted in the simulations is the Herschel-Bulkley
model, which, for simple shear conditions may be written as:
c K
(21)
in which K and η are rheological parameters. Referring to the simulations carried
out by Fraccarollo and Papa [12] on the same event, K is assumed to be 150
Pa·s1/3
, τc is equal to 925 N/m2
, and η has been empirically set equal to 1/3.
In Figure 3 computed flow height is compared with the measured data
collected by the ultrasonic sensor at Station 3. The model satisfactorily captures
wave height and shape, but it underestimates their duration, overestimating as a
consequence their number. Results arte however encouraging and comparable to
those obtained by Fraccarollo and Papa [12] and Zanuttigh and Lamberti [7].
The average velocity of the different flow surges has been estimated through
geophone log recordings. Available data refer to two 100 m channel reaches
Figure 3: Comparison between the flow depth measured and calculated at
Station 3.
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![located in the lower part of the channel before and after Station 3, which
corresponds to the surveyed cross-section 8. Comparison is showed in Figure 4.
In the upstream reach computed velocities compare well with field data, while in
the downstream reach they are generally overestimated.
It is interesting noting that the flow regime is mainly characterized by the
formation of roll waves, as it is evident observing the longitudinal distribution of
discharges and flow depths at two subsequent time steps. Nevertheless,
numerical solution is not affected by relevant numerical instabilities.
4.2 Stava mud flow
In July 19th
1985, two tailing dams suddenly collapsed in Tesero, a little town in
the Italian Alps. The stored water, together with the dam body material flowed
down to the Stava River as a big mud flow, claiming 268 human lives and
destroying 47 houses. As reported by Takahashi [13], the Stava River before the
disaster flowed with an approximately uniform slope of 5°. Although the mud
flow had such an intensive destructive power, as well as fluidity, the Stava River
channel itself had not suffered much erosion or deposition, and it can therefore
be simulated as a fixed bed stream. In his report Takahashi gives important
references also about mud flow solids concentration which was as high as 0.5,
while the particle size was so fine that the relative depth, R/d, had a value of the
order of 105
. In this condition the resistance to flow is similar to that of a plain
water flow and the Manning’s equation can be applied. Takahashi obtained a
Manning’s roughness coefficient in each section by reverse calculation from the
data on velocity computed with the Lenau’s formula applied to measured flow
superelevations at bends.
The channel description is also taken from Takahashi [13]. It includes 24
surveyed cross-sections, their planimetric location and the longitudinal profile. In
this case bed slope ranges from 5% to 12%. The simulated reach is 3500 m long
and a constant spatial step of 1.25 m has been used.
In Figure 6, discharge and depth computed hydrographs are compared with
Takahashi numerical results obtained with the kinematic wave theory [13].
Referring to cross-section 10, located about 3000 m downstream the dams, there
is a good accordance between the computed peak discharge and the value
estimated by Takahashi (3500 m3
/s) as a result of product between the wetted
cross-section area measured in situ (about 500 m2
) and the maximum velocity
derived by the flow superelevation at the nearest bend (7 m/s)
The initial water profile condition reproduces the same hypothesis adopted by
Takahashi, which is a uniform slide of the mud mass until Section 4, from which
the mud flow is assumed to develop.
Figure 8 shows the comparison between computed and measured front arrival
times at different locations. The measured values are estimated on the basis of a
seismograph located in Cavalese, a nearby town. The computed times are in
good agreement with the estimated ones along the entire channel.
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![Figure 8: Comparison between computed and measured front arrival times at
different locations.
5 Conclusions
A numerical model for the simulation of mud flow and debris flow natural events
is presented. It is based on a mathematical model which main features are
concerned with the propagation of the wet-dry fronts, the treatment of irregular
and variable cross sections shape, and the applicability to highly sloping
channels. Two real events have been chosen to test the model. The first is a
natural debris flow event at Acquabona site. In this case a large quantity of field
data was available and model results compared well with wave peak height and
propagation velocities. The second test case refers to the Stava mud flow tragic
event, originated by the collapse of two tailing dams. Also in this case good
accordance between observed data and mud front propagation speed has been
obtained. Simulation results have also been compared with the Takahashi
analysis of the same event, showing good accordance for what concerns peak
discharge estimation at different cross sections.
References
[1] Schippa., L. & Pavan, S. 1-D finite volume model for dam-break induced
mud-flow. River Basin Management V, 07-09 September 2009, Malta, pp.
125-136, ed. C.A. Brebbia, Wit Press, Southampton, Boston, 2009.
[2] Schippa., L. & Pavan, S., Analytical treatment of source terms for complex
channel geometry. Journal of Hydraulic Research, 46(6), pp. 753-763,
2008.
[3] Schippa., L. & Pavan, S., Bed evolution numerical model for rapidly
varying flow in natural streams. Computer & Geosciences, 35, pp. 390-402,
2009.
[4] Garcia-Navarro, P. & Vazquez-Cendon M.E., On numerical treatment of
the source terms in the shallow water equations. Computer & Fluids, 29,
pp. 951-979, 2000.
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WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press
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![[5] Brufau, P., Garcia-Navarro, P., Ghilardi, P., Natale, L. & Savi, F., 1D
Mathematical modelling of debris flow. Journal of Hydraulic Research,
38(6), pp. 435-446, 2000.
[6] Naef, D., Rickenmann, D., Rutschmann, P. & McArdell, B.W., Comparison
of flow resistance relations for debris flow using a one-dimensional finite
element simulation model., Natural Hazards and Earth System Sciences, 6,
pp.155-165, 2006.
[7] Zanuttigh, B. & Lamberti, A., Analysis of debris wave development with
one-dimensional shallow-water equations, Journal of Hydraulic
Engineering, 130(4), pp. 293-303, 2004.
[8] Toro, E.F., Riemann Solvers and Numerical Method for Fluid Dynamics,
Springer-Verlag Berlin Heidelberg New York, 1999.
[9] Ying, X. & Wang, S.S.Y., Improved implementation of the HLL
approximate Riemann solver for one-dimensional open channel flows.
Journal of Hydraulic Research, 46(1), pp. 21-34, 2008.
[10] Berti, M., Geneovis, R., Simoni, A. & Tecca, P.R., Field observations of a
debris flow event in the Dolomites., Geomorphology, 29, pp. 265-274,
1999.
[11] Orlandini, S. & Lamberti A., Effect of wind precipitation intercepted by
steep mountain slopes. Journal of the hydrologic engineering, 5(4), pp.
346-354, 2000
[12] Fraccarollo, L., & Papa, M., Numerical simulation of real debris-flow
events. Physics and Chemistry of the Earth, 25(9), pp. 757-763, 2000.
[13] Takahashi T., Debris flow, IAHR Monograph Series, A.A. Balkema
Rotterdam Brookfield, 165 pp, 1991.
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![1 Introduction
Debris flow is a frequent phenomenon in mountainous regions. It occurs when
masses of poorly sorted sediments, rocks and fine material, agitated and mixed
with water, surge down slopes in response to water flow and gravitational
attraction. A typical surge of debris flow has a steep front or “head” with the
densest slurry, the highest concentration of boulders and the greatest depth. A
progressively more dilute and shallower tail follows this head.
Reviews presented by Iverson [1], exhaustively describe the physical aspects
of debris flow motion and clearly divide previous debris flow research into two
distinct categories. The first, based upon the pioneering work of Johnson [2],
assumes that debris flow behaves as a viscoplastic continuum. This model
describes a single-phase material that remains rigid unless stresses exceed a
threshold value: the plastic yield stress. Various rheological models have been
proposed, derived from experimental results or from theoretical considerations,
such as the Bingham model [3], the Cross model [4], and the quadratic model
proposed by O’Brien and Julien [5]. The Bingham plastic model is the most
commonly used in practice.
The second approach has focus on the mechanics of granular materials. Based
upon the findings of Bagnold [6], two-phase models have been developed by
several authors, such as Takahashi [7] and Iverson [1]. These models explicitly
account for solid and fluid volume fractions and mass changes respectively.
Despite of the considerable progress over the past few years, the flow
dynamics and internal processes of debris flows are still challenging in many
aspects. In particular, there are many factors related to the movement and
interaction of individual boulders and coarse sediments that have not been fully
addressed in previous works. Asmar et al. [8] introduced the Discrete Element
Method (DEM) to simulate the motion of solid particles in debris flows. DEM is
a numerical method to model dry granular flows where each particle is traced
individually in a Lagrangian frame of reference by solving Newton’s equation of
motion.
This paper describes the development of a quasi three-dimensional model to
simulate stony debris flows, considering a continuum fluid phase, and large
sediment particles, such as boulders, as a non-continuum phase. Large particles
are treated in a Lagrangian frame of reference using DEM, and the fluid phase
composed by water and fine sediments is modelled with an Eulerian approach
using the depth-averaged Navier–Stokes equations in two dimensions. Bingham
and Cross rheological models are used for the continuum phase. Particle’s
equations of motion are fully three-dimensional. The model is tested with
laboratory experiments and with a real application.
2 Governing equations
The flow domain is divided in computational cells with triangular base and depth
H, as shown in Figure 1.
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![Figure 1: Schematic representation of debris flow with large solid particles.
Assuming non-Newtonian and incompressible fluid phase, the depth averaged
continuity and momentum equations in Cartesian coordinates can be written as
follows:
0
)
(
)
(
y
H
v
x
H
u
t
H
(1)
0
1
fx
S
g
F
x
y
u
g
v
x
u
g
u
t
u
g
Dx
(2)
0
1
fy
S
g
F
y
y
v
g
v
x
v
g
u
t
v
g
Dy
(3)
where H is the water depth, η is the free-surface elevation, u and v are the depth
averaged velocities in x and y directions respectively, g is the gravitational
acceleration and is fluid density. FD represents the fluid-solid interaction force
exerted on the fluid by particles through the fluid drag force.), this force is
evaluated as:
V
n
i
i
FD
D
1
F
F (4)
where FFD is the fluid drag force on each particle i, V is the volume of the
computational cell and n is the number of particles in the cell. Sfx and Sfy are the
depth integrated stress terms that depend on the rheological formulation used to
model the slurry.
Assuming a Bingham rheological model and Manning’s formula, as proposed
by O’Brien and Julien [5], the stress terms for the fluid can be expressed as
H
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![3
/
4
2
2
2
3
H
u
N
gH
u
gH
y
S
fx
(5)
3
/
4
2
2
2
3
H
v
N
gH
v
gH
y
S
fy
(6)
where N is the Manning roughness coefficient.
The fluid dynamic viscosity and yield stress y, are determined as functions
of the volume sediment concentration Cv, using the relationships proposed by
O’Brien and Julien [9]:
c
e 1
1
(7)
c
e
y
2
2
(8)
in which 1, 1, 2 and 2 are empirical coefficients obtained by data correlation
in a number of experiments with various sediment mixtures. Using a quadratic
formulation combined with the Cross rheological model, the stress terms for the
fluid are expressed as
3
/
4
2
2
H
u
N
gH
eff
S
fx
with
H
u
3
(9)
3
/
4
2
2
H
v
N
gH
eff
S
fy
with
H
v
3
(10)
whereeff is the effective viscosity of the fluid defined by:
B
B
K
K
eff
1
0 (11)
with
y
B
K
0
,
and
3
10
0
In the solid phase, spherical particles of different diameters are considered.
Particle trajectories are tracked using Newton’s second law and the considering
gravity, buoyancy, fluid drag and collision forces.
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![
T
N
E
dt
d
i
m F
F
F
v
(12)
The external force FE is given by
FD
B
E F
F
F
(13)
The expression to compute the net force acting on the particle due to
gravitational effects is
g
F )
(
3
3
4
p
R
B
(14)
where R is the particle radius and p is the particle density.
The expression for the drag on particles in viscous fluid is given by
v
u
v
u
F
d
C
R
FD
2
2
1
(15)
where Cd is the drag coefficient, u is the fluid velocity vector at the particle
location, and v is the particle velocity vector.
The last two terms in equation (12) represent the collision forces or contact
forces among particles. Based on the simplified model that uses a spring-
dashpot-slider system to represent particle interactions [8], the normal contact
force and the tangential contact force are evaluated as
ND
NC
N F
F
F
(16)
TD
TC
T F
F
F
(17)
The normal contact force FNC is calculated using a Hook’s linear spring
relationship,
N
N
NC K
F (18)
where KN is the normal contact stiffness and N is the displacement (overlap)
between particles i and j.
The normal damping force FND is also calculated using a linear relation
given by
N
N
ND v
C
F (19)
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![where vN is the normal component of the relative velocity between particles and
CN is the normal damping coefficient. This constant CN is chosen to give a
required coefficient of restitution defined as the ratio of the normal component
of the relative velocities before and after collision.
The tangential contact force, FTC, represents the friction force and it is
constrained by the Coulomb frictional limit, at which point the particles begin to
slide over each other. Prior to sliding, the tangential contact force is calculated
using a linear spring relationship,
T
T
TC K
F (20)
where KT is the tangential stiffness coefficient, and T is the total tangential
displacement between the surfaces of particles i and j since their initial contact.
When KTT exceeds the frictional limit force f FNC, particle sliding occurs. The
sliding condition is defined as
NC
f
TC F
F
(21)
where f is the dynamic friction coefficient.
The tangential damping force FTD is not included in this model, since it is
assumed that once sliding occurs, damping is accounted for from friction.
Also, particle rotation is not considered. Fluid governing equations (1-3) are
solved by the Galerkin Finite Element method using three-node triangular
elements. To solve the resulting system of ordinary differential equation, the
model applies a four-step time stepping scheme and a selective lumping method,
as described by Garcia-Martinez et al. [10].
Forces on each particle are evaluated at each time step, and the acceleration of
the particle is computed from the particle governing equation, which is then
integrated to find velocity and displacement of each particle.
3 Results
A series of experiments were carried out in a laboratory flume, using
homogeneous fluid and fine sediment mixtures for the continuum phase and
spherical marbles for the discrete phase. The experiments were performed in a
1.9 m long, 0.19 m wide, Plexiglas walled flume, with adjustable slope. The
downstream part of the flume was connected to a wood horizontal platform, 0.75
m long and 0.95 m wide. A dam-break type of flow was initiated by an abrupt
removal of a gate releasing mixtures from a 0.40 m long reservoir situated on the
upstream part of the flume. Water-clay mixtures were used in all the
experiments, with volume sediment concentration 23.5% and 26.5%. For
preparation of the mixtures, kaolinite clay with specific unit weight of 2.77 was
used. Fluid density was measured in the laboratory and rheological parameters
and y were determined using equations (7) and (8) in which parameters are 1 =
0.621x10-3
, 1 = 17.3, 2 = 0.002 and 2 = 40.2.
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Monitoring Simulation Prevention And Remediation Of Dense And Debris Flow Iii Wit Transactions On Engineering Sciences 1st Edition C A Editors Brebbia
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- 7. THIRD INTERNATIONAL CONFERENCE ON MONITORING, SIMULATION, PREVENTION AND REMEDIATION OF DENSE AND DEBRIS FLOWS D. de Wrachien University of Milan, Italy C.A. Brebbia Wessex Institute of Technology, UK Organised by University of Milano, Italy Wessex Institute of Technology, UK INTERNATIONAL SCIENTIFIC ADVISORY COMMITTEE Sponsored by WIT Transactions on Engineering Sciences EurAgEng: European Society of Agricultural Engineers CIGR: International Commission of Agricultural Engineering Supported by The Lombardy Region, Italy DEBRIS FLOW III CONFERENCE CHAIRMEN R. Garcia-Martinez F. Gentile G.P. Giani J. Hubl M.A. Lenzi G. Lorenzini S. Mambretti T. Moriyama F. Wei
- 8. WIT Transactions Editorial Board Transactions Editor Carlos Brebbia Wessex Institute of Technology Ashurst Lodge, Ashurst Southampton SO40 7AA, UK Email: carlos@wessex.ac.uk B Abersek University of Maribor, Slovenia Y N Abousleiman University of Oklahoma, USA P L Aguilar University of Extremadura, Spain K S Al Jabri Sultan Qaboos University, Oman E Alarcon Universidad Politecnica de Madrid, Spain A Aldama IMTA, Mexico C Alessandri Universita di Ferrara, Italy D Almorza Gomar University of Cadiz, Spain B Alzahabi Kettering University, USA J A C Ambrosio IDMEC, Portugal A M Amer Cairo University, Egypt S A Anagnostopoulos University of Patras, Greece M Andretta Montecatini, Italy E Angelino A.R.P.A. Lombardia, Italy H Antes Technische Universitat Braunschweig, Germany M A Atherton South Bank University, UK A G Atkins University of Reading, UK D Aubry Ecole Centrale de Paris, France H Azegami Toyohashi University of Technology, Japan A F M Azevedo University of Porto, Portugal J Baish Bucknell University, USA J M Baldasano Universitat Politecnica de Catalunya, Spain J G Bartzis Institute of Nuclear Technology, Greece A Bejan Duke University, USA M P Bekakos Democritus University of Thrace, Greece G Belingardi Politecnico di Torino, Italy R Belmans Katholieke Universiteit Leuven, Belgium C D Bertram The University of New South Wales, Australia D E Beskos University of Patras, Greece S K Bhattacharyya Indian Institute of Technology, India E Blums Latvian Academy of Sciences, Latvia J Boarder Cartref Consulting Systems, UK B Bobee Institut National de la Recherche Scientifique, Canada H Boileau ESIGEC, France J J Bommer Imperial College London, UK M Bonnet Ecole Polytechnique, France C A Borrego University of Aveiro, Portugal A R Bretones University of Granada, Spain J A Bryant University of Exeter, UK F-G Buchholz Universitat Gesanthochschule Paderborn, Germany M B Bush The University of Western Australia, Australia F Butera Politecnico di Milano, Italy J Byrne University of Portsmouth, UK W Cantwell Liverpool University, UK D J Cartwright Bucknell University, USA P G Carydis National Technical University of Athens, Greece J J Casares Long Universidad de Santiago de Compostela, Spain M A Celia Princeton University, USA A Chakrabarti Indian Institute of Science, India A H-D Cheng University of Mississippi, USA
- 9. J Chilton University of Lincoln, UK C-L Chiu University of Pittsburgh, USA H Choi Kangnung National University, Korea A Cieslak Technical University of Lodz, Poland S Clement Transport System Centre, Australia M W Collins Brunel University, UK J J Connor Massachusetts Institute of Technology, USA M C Constantinou State University of New York at Buffalo, USA D E Cormack University of Toronto, Canada M Costantino Royal Bank of Scotland, UK D F Cutler Royal Botanic Gardens, UK W Czyczula Krakow University of Technology, Poland M da Conceicao Cunha University of Coimbra, Portugal A Davies University of Hertfordshire, UK M Davis Temple University, USA A B de Almeida Instituto Superior Tecnico, Portugal E R de Arantes e Oliveira Instituto Superior Tecnico, Portugal L De Biase University of Milan, Italy R de Borst Delft University of Technology, Netherlands G De Mey University of Ghent, Belgium A De Montis Universita di Cagliari, Italy A De Naeyer Universiteit Ghent, Belgium W P De Wilde Vrije Universiteit Brussel, Belgium L Debnath University of Texas-Pan American, USA N J Dedios Mimbela Universidad de Cordoba, Spain G Degrande Katholieke Universiteit Leuven, Belgium S del Giudice University of Udine, Italy G Deplano Universita di Cagliari, Italy I Doltsinis University of Stuttgart, Germany M Domaszewski Universite de Technologie de Belfort-Montbeliard, France J Dominguez University of Seville, Spain K Dorow Pacific Northwest National Laboratory, USA W Dover University College London, UK C Dowlen South Bank University, UK J P du Plessis University of Stellenbosch, South Africa R Duffell University of Hertfordshire, UK A Ebel University of Cologne, Germany E E Edoutos Democritus University of Thrace, Greece G K Egan Monash University, Australia K M Elawadly Alexandria University, Egypt K-H Elmer Universitat Hannover, Germany D Elms University of Canterbury, New Zealand M E M El-Sayed Kettering University, USA D M Elsom Oxford Brookes University, UK A El-Zafrany Cranfield University, UK F Erdogan Lehigh University, USA F P Escrig University of Seville, Spain D J Evans Nottingham Trent University, UK J W Everett Rowan University, USA M Faghri University of Rhode Island, USA R A Falconer Cardiff University, UK M N Fardis University of Patras, Greece P Fedelinski Silesian Technical University, Poland H J S Fernando Arizona State University, USA S Finger Carnegie Mellon University, USA J I Frankel University of Tennessee, USA D M Fraser University of Cape Town, South Africa M J Fritzler University of Calgary, Canada U Gabbert Otto-von-Guericke Universitat Magdeburg, Germany G Gambolati Universita di Padova, Italy C J Gantes National Technical University of Athens, Greece L Gaul Universitat Stuttgart, Germany A Genco University of Palermo, Italy N Georgantzis Universitat Jaume I, Spain P Giudici Universita di Pavia, Italy F Gomez Universidad Politecnica de Valencia, Spain R Gomez Martin University of Granada, Spain D Goulias University of Maryland, USA K G Goulias Pennsylvania State University, USA F Grandori Politecnico di Milano, Italy W E Grant Texas A & M University, USA S Grilli University of Rhode Island, USA
- 10. R H J Grimshaw Loughborough University, UK D Gross Technische Hochschule Darmstadt, Germany R Grundmann Technische Universitat Dresden, Germany A Gualtierotti IDHEAP, Switzerland R C Gupta National University of Singapore, Singapore J M Hale University of Newcastle, UK K Hameyer Katholieke Universiteit Leuven, Belgium C Hanke Danish Technical University, Denmark K Hayami National Institute of Informatics, Japan Y Hayashi Nagoya University, Japan L Haydock Newage International Limited, UK A H Hendrickx Free University of Brussels, Belgium C Herman John Hopkins University, USA S Heslop University of Bristol, UK I Hideaki Nagoya University, Japan D A Hills University of Oxford, UK W F Huebner Southwest Research Institute, USA J A C Humphrey Bucknell University, USA M Y Hussaini Florida State University, USA W Hutchinson Edith Cowan University, Australia T H Hyde University of Nottingham, UK M Iguchi Science University of Tokyo, Japan D B Ingham University of Leeds, UK L Int Panis VITO Expertisecentrum IMS, Belgium N Ishikawa National Defence Academy, Japan J Jaafar UiTm, Malaysia W Jager Technical University of Dresden, Germany Y Jaluria Rutgers University, USA C M Jefferson University of the West of England, UK P R Johnston Griffith University, Australia D R H Jones University of Cambridge, UK N Jones University of Liverpool, UK D Kaliampakos National Technical University of Athens, Greece N Kamiya Nagoya University, Japan D L Karabalis University of Patras, Greece M Karlsson Linkoping University, Sweden T Katayama Doshisha University, Japan K L Katsifarakis Aristotle University of Thessaloniki, Greece J T Katsikadelis National Technical University of Athens, Greece E Kausel Massachusetts Institute of Technology, USA H Kawashima The University of Tokyo, Japan B A Kazimee Washington State University, USA S Kim University of Wisconsin-Madison, USA D Kirkland Nicholas Grimshaw & Partners Ltd, UK E Kita Nagoya University, Japan A S Kobayashi University of Washington, USA T Kobayashi University of Tokyo, Japan D Koga Saga University, Japan S Kotake University of Tokyo, Japan A N Kounadis National Technical University of Athens, Greece W B Kratzig Ruhr Universitat Bochum, Germany T Krauthammer Penn State University, USA C-H Lai University of Greenwich, UK M Langseth Norwegian University of Science and Technology, Norway B S Larsen Technical University of Denmark, Denmark F Lattarulo Politecnico di Bari, Italy A Lebedev Moscow State University, Russia L J Leon University of Montreal, Canada D Lewis Mississippi State University, USA S lghobashi University of California Irvine, USA K-C Lin University of New Brunswick, Canada A A Liolios Democritus University of Thrace, Greece S Lomov Katholieke Universiteit Leuven, Belgium J W S Longhurst University of the West of England, UK G Loo The University of Auckland, New Zealand D Lóránt Károly Róbert College, Hungary J Lourenco Universidade do Minho, Portugal
- 11. J E Luco University of California at San Diego, USA H Lui State Seismological Bureau Harbin, China C J Lumsden University of Toronto, Canada L Lundqvist Division of Transport and Location Analysis, Sweden T Lyons Murdoch University, Australia Y-W Mai University of Sydney, Australia M Majowiecki University of Bologna, Italy D Malerba Università degli Studi di Bari, Italy G Manara University of Pisa, Italy B N Mandal Indian Statistical Institute, India Ü Mander University of Tartu, Estonia H A Mang Technische Universitat Wien, Austria G D Manolis Aristotle University of Thessaloniki, Greece W J Mansur COPPE/UFRJ, Brazil N Marchettini University of Siena, Italy J D M Marsh Griffith University, Australia J F Martin-Duque Universidad Complutense, Spain T Matsui Nagoya University, Japan G Mattrisch DaimlerChrysler AG, Germany F M Mazzolani University of Naples “Federico II”, Italy K McManis University of New Orleans, USA A C Mendes Universidade de Beira Interior, Portugal R A Meric Research Institute for Basic Sciences, Turkey J Mikielewicz Polish Academy of Sciences, Poland N Milic-Frayling Microsoft Research Ltd, UK R A W Mines University of Liverpool, UK C A Mitchell University of Sydney, Australia K Miura Kajima Corporation, Japan A Miyamoto Yamaguchi University, Japan T Miyoshi Kobe University, Japan G Molinari University of Genoa, Italy T B Moodie University of Alberta, Canada D B Murray Trinity College Dublin, Ireland G Nakhaeizadeh DaimlerChrysler AG, Germany M B Neace Mercer University, USA D Necsulescu University of Ottawa, Canada F Neumann University of Vienna, Austria S-I Nishida Saga University, Japan H Nisitani Kyushu Sangyo University, Japan B Notaros University of Massachusetts, USA P O’Donoghue University College Dublin, Ireland R O O’Neill Oak Ridge National Laboratory, USA M Ohkusu Kyushu University, Japan G Oliveto Universitá di Catania, Italy R Olsen Camp Dresser & McKee Inc., USA E Oñate Universitat Politecnica de Catalunya, Spain K Onishi Ibaraki University, Japan P H Oosthuizen Queens University, Canada E L Ortiz Imperial College London, UK E Outa Waseda University, Japan A S Papageorgiou Rensselaer Polytechnic Institute, USA J Park Seoul National University, Korea G Passerini Universita delle Marche, Italy B C Patten University of Georgia, USA G Pelosi University of Florence, Italy G G Penelis Aristotle University of Thessaloniki, Greece W Perrie Bedford Institute of Oceanography, Canada R Pietrabissa Politecnico di Milano, Italy H Pina Instituto Superior Tecnico, Portugal M F Platzer Naval Postgraduate School, USA D Poljak University of Split, Croatia V Popov Wessex Institute of Technology, UK H Power University of Nottingham, UK D Prandle Proudman Oceanographic Laboratory, UK M Predeleanu University Paris VI, France M R I Purvis University of Portsmouth, UK I S Putra Institute of Technology Bandung, Indonesia Y A Pykh Russian Academy of Sciences, Russia F Rachidi EMC Group, Switzerland M Rahman Dalhousie University, Canada K R Rajagopal Texas A & M University, USA T Rang Tallinn Technical University, Estonia J Rao Case Western Reserve University, USA A M Reinhorn State University of New York at Buffalo, USA
- 12. A D Rey McGill University, Canada D N Riahi University of Illinois at Urbana- Champaign, USA B Ribas Spanish National Centre for Environmental Health, Spain K Richter Graz University of Technology, Austria S Rinaldi Politecnico di Milano, Italy F Robuste Universitat Politecnica de Catalunya, Spain J Roddick Flinders University, Australia A C Rodrigues Universidade Nova de Lisboa, Portugal F Rodrigues Poly Institute of Porto, Portugal C W Roeder University of Washington, USA J M Roesset Texas A & M University, USA W Roetzel Universitaet der Bundeswehr Hamburg, Germany V Roje University of Split, Croatia R Rosset Laboratoire d’Aerologie, France J L Rubio Centro de Investigaciones sobre Desertificacion, Spain T J Rudolphi Iowa State University, USA S Russenchuck Magnet Group, Switzerland H Ryssel Fraunhofer Institut Integrierte Schaltungen, Germany S G Saad American University in Cairo, Egypt M Saiidi University of Nevada-Reno, USA R San Jose Technical University of Madrid, Spain F J Sanchez-Sesma Instituto Mexicano del Petroleo, Mexico B Sarler Nova Gorica Polytechnic, Slovenia S A Savidis Technische Universitat Berlin, Germany A Savini Universita de Pavia, Italy G Schmid Ruhr-Universitat Bochum, Germany R Schmidt RWTH Aachen, Germany B Scholtes Universitaet of Kassel, Germany W Schreiber University of Alabama, USA A P S Selvadurai McGill University, Canada J J Sendra University of Seville, Spain J J Sharp Memorial University of Newfoundland, Canada Q Shen Massachusetts Institute of Technology, USA X Shixiong Fudan University, China G C Sih Lehigh University, USA L C Simoes University of Coimbra, Portugal A C Singhal Arizona State University, USA P Skerget University of Maribor, Slovenia J Sladek Slovak Academy of Sciences, Slovakia V Sladek Slovak Academy of Sciences, Slovakia A C M Sousa University of New Brunswick, Canada H Sozer Illinois Institute of Technology, USA D B Spalding CHAM, UK P D Spanos Rice University, USA T Speck Albert-Ludwigs-Universitaet Freiburg, Germany C C Spyrakos National Technical University of Athens, Greece I V Stangeeva St Petersburg University, Russia J Stasiek Technical University of Gdansk, Poland G E Swaters University of Alberta, Canada S Syngellakis University of Southampton, UK J Szmyd University of Mining and Metallurgy, Poland S T Tadano Hokkaido University, Japan H Takemiya Okayama University, Japan I Takewaki Kyoto University, Japan C-L Tan Carleton University, Canada M Tanaka Shinshu University, Japan E Taniguchi Kyoto University, Japan S Tanimura Aichi University of Technology, Japan J L Tassoulas University of Texas at Austin, USA M A P Taylor University of South Australia, Australia A Terranova Politecnico di Milano, Italy E Tiezzi University of Siena, Italy A G Tijhuis Technische Universiteit Eindhoven, Netherlands T Tirabassi Institute FISBAT-CNR, Italy S Tkachenko Otto-von-Guericke-University, Germany N Tosaka Nihon University, Japan T Tran-Cong University of Southern Queensland, Australia R Tremblay Ecole Polytechnique, Canada I Tsukrov University of New Hampshire, USA
- 13. R Turra CINECA Interuniversity Computing Centre, Italy S G Tushinski Moscow State University, Russia J-L Uso Universitat Jaume I, Spain E Van den Bulck Katholieke Universiteit Leuven, Belgium D Van den Poel Ghent University, Belgium R van der Heijden Radboud University, Netherlands R van Duin Delft University of Technology, Netherlands P Vas University of Aberdeen, UK W S Venturini University of Sao Paulo, Brazil R Verhoeven Ghent University, Belgium A Viguri Universitat Jaume I, Spain Y Villacampa Esteve Universidad de Alicante, Spain F F V Vincent University of Bath, UK S Walker Imperial College, UK G Walters University of Exeter, UK B Weiss University of Vienna, Austria H Westphal University of Magdeburg, Germany J R Whiteman Brunel University, UK Z-Y Yan Peking University, China S Yanniotis Agricultural University of Athens, Greece A Yeh University of Hong Kong, China J Yoon Old Dominion University, USA K Yoshizato Hiroshima University, Japan T X Yu Hong Kong University of Science & Technology, Hong Kong M Zador Technical University of Budapest, Hungary K Zakrzewski Politechnika Lodzka, Poland M Zamir University of Western Ontario, Canada R Zarnic University of Ljubljana, Slovenia G Zharkova Institute of Theoretical and Applied Mechanics, Russia N Zhong Maebashi Institute of Technology, Japan H G Zimmermann Siemens AG, Germany
- 14. Monitoring, Simulation, Prevention and Remediation of Dense Debris Flows III Editors D. de Wrachien State University of Milan, Italy C.A. Brebbia Wessex Institute of Technology, UK
- 15. WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail: witpress@witpress.com http://www.witpress.com For USA, Canada and Mexico Computational Mechanics Inc 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: infousa@witpress.com http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: 978-1-84564-442-0 ISSN: 1746-4471 (print) ISSN: 1743-3533 (online) The texts of the papers in this volume were set individually by the authors or under their supervision. Only minor corrections to the text may have been carried out by the publisher. No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. The Publisher does not necessarily endorse the ideas held, or views expressed by the Editors or Authors of the material contained in its publications. © WIT Press 2010 Printed in Great Britain by Martins the Printers. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publisher. D. de Wrachien State University of Milan, Italy C.A. Brebbia Wessex Institute of Technology, UK Published by
- 16. Preface This book contains papers presented at the Third International Conference on Debris Flow including all aspects of Debris Flow Monitoring, Modelling, Hazard Assessment, Mitigation Measures, Extreme Events, Erosion, Slope Instability and Sediment Transport, held in Milano, Italy, in 2010. The Conference was jointly organised by the State University of Milano and theWessex Institute of Technology, UK, with the co-sponsorship of EurAgEng (European Society of Agricultural Engineers) and CIGR (International Commission ofAgricultural Engineering) and the support of the Lombardy Region, Italy. This successful series of Conferences first started in Rhodes, Greece (2006) and continued in New Forest, UK (2008). Debris and hyper-concentrated flows are among the most destructive of all water related disasters. They affect both rural and urban areas in a wide range of morpho- climatic environments, and in recent years have attracted more and more attention from the scientific and professional communities and concern from the public due to the death toll they claim. The increased frequency of these natural hazards, coupled with climatic change predictions and urban development, suggests that they are set to worsen in the future. The Conference brought together engineers, scientists and managers from across the globe to discuss the latest scientific advances in the field of dense and hyper- concentrated flows, as well as to improve models, assess risk, develop hazard maps based on model results and to design prevention and mitigation measures. The book contains Sections on the following topics: - Debris Flow Modelling - Debris Flow Triggering - Risk Assessment and Hazard Mitigation - Sediment Transport and Debris Flow Monitoring & Analysis
- 17. The Editors would like to thank all the Authors for their excellent contributions as wells as the members of the International Scientific Advisory Committee for their help in reviewing both the abstracts and the papers included in this book. The quality of the material makes this volume a most valuable and up-to-date tool for professionals, scientists and managers to appreciate the state-of-the-art in this important field of knowledge The Editors Milano, 2010
- 18. Contents Section 1: Debris flow modelling Mechanical and fluid-dynamic behaviour of debris and hyper-concentrated flows: overview and challenges D. De Wrachien, S. Mambretti & C. Deangeli .................................................... 3 One-dimensional finite volume simulation of real debris flow events L. Schippa & S. Pavan....................................................................................... 17 Debris flow modelling accounting for large boulder transport C. Martinez, F. Miralles-Wilhelm & R. Garcia-Martinez ................................. 29 New formulas for the motion resistance of debris flows D. Berzi, J. T. Jenkins & E. Larcan................................................................... 41 Rheological behaviour of pyroclastic debris flow A. M. Pellegrino, A. Scotto di Santolo, A. Evangelista & P. Coussot...................................................................................................... 51 Section 2: Debris flow triggering The triggered mechanism of typhoon-induced debris flows and landslides over mainland China G. P. Zhang, J. Xu, F. W. Xu, L. N. Zhao, Y. M. Li, J. Li, X. D. Yang & J. Y. Di......................................................................................... 65 Debris flow occurrences in Rio dos Cedros, Southern Brazil: meteorological and geomorphic aspects M. Kobiyama, R. F. Goerl, G. P. Corrêa & G. P. Michel ................................. 77 Soil moisture retrieval with remote sensing images for debris flow forecast in humid regions Y. Zhao, H. Yang & F. Wei ................................................................................ 89
- 19. Debris flow induced by glacial lake break in southeast Tibet Z. L. Cheng, J. J. Liu & J. K. Liu..................................................................... 101 Experience with treatment of road structure landslides by innovative methods of deep drainage O. Mrvík & S. Bomont ..................................................................................... 113 Technical protection measures against natural hazards taken by the Austrian Federal Service for Torrent, Erosion and Avalanche Control F. J. Riedl ........................................................................................................ 125 Section 3: Risk assessment and hazard mitigation The distribution of debris flows and debris flow hazards in southeast China F. Wei, Y. Jiang, Y. Zhao, A. Xu & J. S. Gardner............................................ 137 Evaluation of sediment yield from valley slopes: a case study F. Ballio, D. Brambilla, E. Giorgetti, L. Longoni, M. Papini & A. Radice...................................................................................................... 149 Shallow landslide full-scale experiments in combination with testing of a flexible barrier L. Bugnion & C. Wendeler............................................................................... 161 Landslide in a catchment area of a torrent and the consequences for the technical mitigation concept F. J. Riedl ........................................................................................................ 175 Regional methods for shallow landslide hazard evaluation: a comparison between Italy and Central America D. Brambilla, L. Longoni & M. Papini............................................................ 185 Section 4: Sediment transport and debris flow monitoring and analysis Special session organised by Daniele De Wrachien, Gian Battista Bischetti, Francesco Gentile & Luca Mao Erosion and sediment transport modelling in Northern Puglia watersheds F. Gentile, T. Bisantino & G. Trisorio Liuzzi .................................................. 199
- 20. Restoration of a degraded torrential stream by means of a flood control system: the case of Arroyo del Partido stream (Spain) J. A. Mintegui Aguirre, J. C. Robredo Sánchez, C. De Gonzalo Aranoa & P. Huelin Rueda..................................................... 213 The effects of large wood elements during an extreme flood in a small tropical basin of Costa Rica L. Mao & F. Comiti ......................................................................................... 225 Rheological properties and debris-flow modeling in a southern Italy watershed T. Bisantino, P. Fischer, F. Gentile & G. Trisorio Liuzzi................................ 237 Formation, expansion and restoration of a sedimentation fan: the case of the Arroyo del Partido stream (Spain) J. A. Mintegui Aguirre, J. C. Robredo Sánchez, L. Mao & M. A. Lenzi................................................................................................... 249 Dynamics of changes of bed load outflow from a small glacial catchment (West Spitsbergen) W. Kociuba, G. Janicki & K. Siwek................................................................. 261 Author Index.................................................................................................. 271
- 21. Section 1 Debris flow modelling
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- 23. Mechanical and fluid-dynamic behaviour of debris and hyper-concentrated flows: overview and challenges D. De Wrachien1 , S. Mambretti2 & C. Deangeli3 1 Department of Agricultural Engineering, State University of Milan, Italy 2 DIIAR, Politecnico di Milano, Italy 3 DITAG, Politecnico di Torino, Italy Abstract Debris and hyper-concentrated flows are among the most destructive of all water-related disasters. They mainly affect mountain areas in a wide range of morphoclimatic environments and in recent years have attracted more and more attention from the scientific and professional communities and concern from public awareness, due to the increasing frequency with which they occur and the death toll they claim. In this context, achieving a set of debris and hyper-concentrated flow constitutive equations is a task that has been given particular attention by scientists during the second half of the last century. In relation to these issues, this paper reviews the most updated and effective geotechnical and fluid-dynamic procedures nowadays available, suitable to predict the triggering and mobilising processes of these phenomena, and proposes a mathematical model that is able to assess the depth of the wave and the velocities of the liquid and solid phases of both non-stratified (mature) and stratified (immature) flows following flash-floods and dam-break events in one and two dimensional cases. Different experimental cases of dam-break situations in a square section channel were considered for the purpose of comparing results. These tools will allow, on one hand, to better focus on what to observe in the field and, on the other hand, to improve both mitigation measures and hazard mapping procedures. Keywords: debris flow, rheological behaviour of the mixture, slope failure, numerical models, laboratory and field tests. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 3 doi:10.2495/DEB100011
- 24. 1 Introduction Debris and hyper-concentrated flows are among the most destructive of all water-related disasters. They mainly affect mountain areas in a wide range of morphoclimatic environments and in recent years have attracted more and more attention from the scientific and professional communities and concern from public awareness due to the increasing frequency with which they occur and the death toll they claim. These phenomena do not allow a sufficient early warning, as they are characterised by a very short time-scale and, therefore, defence measures should be provided, especially when they are associated with flash floods or dam failures. To this end, the identification of effective procedures aimed at evaluating the probability of these extreme events and the triggering and mobilising mechanism has become an essential component of the water and land use planning processes. This concept leads to a new integrated risk management approach, which comprises administrative decisions, organisation, operational skill and the ability to implement suitable policies. The broadness of the question requires approaches from various perspectives. To this end, the dynamic behaviour of these hyper-concentrated water sediment mixtures and the constitutive laws that govern them plays a role of paramount importance. Debris flow modelling requires a rheological pattern (or constitutive equation) that provides an adequate description of these flows. One of the main difficulties met by the approaches available is linked to their validation either in the field or in a laboratory environment. Greater research needs to be directed towards a thorough investigation of the above mentioned issues. Such knowledge is essential in order to assess the potential frequency of these natural hazards and the related prevention and mitigation measures. With reference to these issues, this paper aims to provide the state-of-the-art of debris flow rheology, modelling and laboratory and field investigation, along with a glance to the direction that debris flow in-depth studies are likely to follow in future. 2 Debris flow model development A thorough understanding of the mechanism triggering and mobilising debris flow phenomena plays a role of paramount importance for designing suitable prevention and mitigation measures. Achieving a set of debris flow constitutive equations is a task which has been given particular attention by the scientific community (Julien and O’Brien [33]; Chen [9]; Takahashi [35]). To properly tackle this problem relevant theoretical and experimental studies have been carried out during the second half of the last century. Research work on theoretical studies has traditionally specialised in different mathematical models. They can be roughly categorized on the basis of three characteristics: the presence of bed evolution equation, the number of phases and the rheological model applied to the flowing mixture (Ghilardi et al. [24]). www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 4 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 25. Most models are based on the conservation of mass and momentum of the flow, but only a few of them take into account erosion/deposition processes affecting the temporal evolution of the channel bed. Debris flows are mixtures of water and clastic material with high coarse particle contents, in which collisions between particles and dispersive stresses are the dominant mechanisms in energy dissipation. The rheological property of a debris flow depends on a variety of factors, such as suspended solid concentration, cohesive property, particle size distribution, particle shape, grain friction and pore pressure. Various researchers have developed models of debris flow rheology. These models can be classified as: Newtonian models (Johnson [32]), linear and non linear viscoplastic models (O’Brien et al. [41]), dilatant fluid models (Bagnold [4]), dispersive or turbulent stress models (Arai and Takahashi [2]), biviscous modified Bingham model (Dent and Lang [15]), and frictional models (Norem et al. [40]). Among these, linear (Bingham) or non-linear (Herschel-Bulkey) viscoplastic models are widely used to describe the rheology of laminar debris/mud flows (Jan, 1997). Because a debris flow, essentially, constitutes a multiphase system, any attempt at modelling this phenomenon that assumes, as a simplified hypothesis, homogeneous mass and constant density, conceals the interactions between the phases and prevents the possibility of investigating further mechanisms such as the effect of sediment separation (grading). Modelling the fluid as a two-phase mixture overcomes most of the limitations mentioned above and allows for a wider choice of rheological models such as: Bagnold’s dilatant fluid hypothesis (Takahashi and Nakagawa [56]), Chézy type equation with constant value of the friction coefficient (Hirano et al. [27]), models with cohesive yield stress (Honda and Egashira [28]) and the generalized viscoplastic fluid Chen’s model (Chen and Ling [10]). Notwithstanding all these efforts, some phenomenological aspects of debris flow have not been understood yet, and something new has to be added to the description of the process to reach a better assessment of the events. In this contest, the mechanism of dam-break wave should be further investigated. So far, this aspect has been analysed by means of the single-phase propagation theory for clear water, introducing in the De Saint Venant (SV) equations a dissipation term to consider fluid rheology (Coussot [12]; Fread and Jin [23]). Many other models, the so-called quasi-two-phase-models, use SV equations together with erosion/deposition and mass conservation equations for the solid phase, and take into account mixture of varying concentrations. All these models feature monotonic velocity profiles that, generally, do not agree with experimental and field data. 2.1 Rheology The rheological property of debris and hyper-concentrated flows depends on a variety of factors, such as the suspended solid concentration, cohesive property, size distribution, particle shape, grain friction, and pore pressure. So, modelling www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 5
- 26. these flows requires a rheological model (or constitutive equation) for sediment- water mixtures. A general model which can realistically describe the rheological properties of debris flow should possess three main features (Chen [9]). The model should: describe the dilatancy of sediment-water mixtures; take into account the so-called soil yield criterion, as proposed by Mohr-Coulomb; assess the role of intergranular or interstitial fluid. The earliest of such rheological models was empirically formulated by Bagnold [4]. On the whole, a rheological model of debris and hyper-concentrated flows should involve the interaction of several physical processes. The non-Newtonian behaviour of the fluid matrix is ruled, in part, by the cohesion between fine sediment particles. This cohesion contributes to the yield stress, which must be exceeded by an applied stress in order to initiate fluid motion. In view of theoretical soundness behind the development of different non- Newtonian fluid models, Bailard [5] and Hanes [25] have questioned the validity of Bagnold’s empirical relations. Limitations in Bagnold’s model may be attributed to the ambiguity in the definition of some rheological characteristics as the grain stresses. To overcome these problems, Chen [9] developed a new generalised viscoplastic fluid (GVF) model, based on two major rheological properties (i.e. the normal stress effect and soil yield criterion) for general use in debris flow modelling. The analysis Chen conducted on the various flow regime of a granular mixture identified three regimes: a quasi-static one, which is a condition of incipient movement with plastic behaviour, a microviscous one at low shear rates, in which viscosity determines the mixture behaviour, and finally a granular inertial state, typical of rapid flowing granular mixtures, dominated by intergranular interactions. All the models previously reviewed feature monotonic velocity profiles that, generally, do not agree with experimental and field data. In many tests (Takahashi [53]) “S” reversed shaped trends have been observed, where the maximum shear rate is not achieved near the bed, but rather between the bed and the free surface. The main discrepancy is derived from the assumption of a debris flow as a uniform mixture. In fact, the solid concentration distribution is usually non-uniform due to the action of gravity, so that the lower layer could, consequently, have a higher concentration than the upper layer. Higher concentration means higher cohesion, friction and viscosity in the flow. Wan [58] proposed a multilayered model known as the laminated layers model that features a stratified debris flow into three regions from the bed to the surface: a bed layer, in which an additional shear stress is dominant in momentum exchange; an inertial layer, where the dispersive stress of the grains is dominant; and an upper viscoplastic layer, which can be represented by the Bingham’s model. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 6 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 27. The one-layer models are unable to adequately feature the entire thickness of the flow and, therefore, it has recently become common to use multi-layers models that combine two or more constitutive relationships in order to analyse adequately these phenomena. The coefficients of the rheological models have wide ranges of variation and, therefore, in evaluating them considerable errors are committed. On the other hand, some empirical equations of velocity are necessary in any debris flow disaster-forecasting measure, although the hydraulics of debris-flow is not theoretically comparable to that of a traditional water flow. 2.2 Triggering and mobilising processes Debris flow resulting from flash flood or a sudden collapse of a dam (dam-break) are often characterised by the formation of shock waves caused by many factors such as valley contractions, irregular bed slope and non-zero tailwater depth. It is commonly accepted that a mathematical description of these phenomena can be accomplished by means of 1D SV equations (Bellos and Sakkas [6]). During the last Century, much effort has been devoted to the numerical solution of the SV equations, mainly driven by the need for accurate and efficient solvers for the discontinuities in dam-break problems. A rather simple form of the dam failure problem in a dry channel was first solved by Ritter [46] who used the SV equations in the characteristic form, under the hypothesis of instantaneous failure in a horizontal rectangular channel without bed resistance. Later on, Stoker [50], on the basis of the work of Courant and Friedrichs [11], extended the Ritter solution to the case of wet downstream channel. Dressler [19] used a perturbation procedure to obtain a first-order correction for resistance effects to represent submerging waves in a roughing bed. Lax and Wendroff [35] pioneered the use of numerical methods to calculate the hyperbolic conservation laws. McCormack [39] introduced a simpler version of the Lax-Wendroff scheme, which has been widely used in aerodynamics problems. Van Leer [57] extended the Godunov scheme to second-order accuracy by following the Monotonic Upstream Schemes for Conservation Laws (MUSCL) approach. Chen [7] applied the method of characteristics, including bed resistance effects, to solve dam-break problems for reservoir of finite length. Sakkas and Strelkoff [47] provided the extension of the method of the characteristics to a power-law cross section and applied this method to a dam break on a dry right channel in the case of rectangular and parabolic cross section shapes. Strelkoff and Falvey [52] presented a critical review of numerical methods of characteristics of power-law cross sections. Hunt [29] proposed a kinematic wave approximation for dam failure in a dry sloping channel. Total Variation Diminishing (TVD) and Essentially Non Oscillation (ENO) schemes were introduced by Harten [26] for efficiently solving 1D gas dynamic problems. Their main property is that they are second order accurate and oscillation free across discontinuities. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 7
- 28. Recently, several 1D and 2D models using approximate Riemann solvers have been reported in the literature. Such models have been found very successful in solving open channel flow and dam-break problems. In the past ten years, further numerical methods to solve flood routing and dam-break problems, have been developed that include the use of finite elements or discrete/distinct element methods (Asmar et al. [3]). Finite Element Methods (FEMs) have certain advantages over finite different methods, mainly in relation to the flexibility of the grid network that can be employed, especially in 2D flow problems. Mambretti et al. [38] and De Wrachien and Mambretti [17, 18] used an improved TVD-Mc Cormack-Jameson scheme to predict the dynamics of both mature (non-stratified) and immature debris flow in different dam break conditions. 3 Laboratory and field studies To validate both the rheological and dynamic models, herewith described, comparisons need to be made between their predictions and results of laboratory and field tests. Agreements between the computational and experimental results are essential since they allow the assessment of the models’ performance and suggest feasible development of the research. The experimental point of view in debris flow research, however, encounters considerable problems that are yet to be fully overcome, connected largely to the accuracy of measuring techniques and flow simulation in experimental tests. Lastly, field studies are probably the most difficult and costly study approach of debris flow; the difficulties encountered are connected to their considerable complexity and the difficulty of direct observation. The exceptional and infrequent conditions in which debris flows occur do not generally permit a sufficient number of observations for the same type of field reality to deduce the specific behavioural laws for that area. Reference to different territorial situations also highlights another problem: that of the homogeneity of data, given the substantial territorial peculiarity in which the phenomena occur. Besides, field data are essential in determining the quality of any mathematical model, as they are especially important for estimating velocity, discharge, concentration, yield stress, viscosity and grain-size. This need requires the use of laboratory experimentation when the previous problems cannot be overcome, and in certain cases it is the only possible path to follow. Within this ground, many experiments have been carried out, ranging from solid transport (little amount of particles in a large environment of clear water) to dry granular flow, where water is not present. An empirical picture of debris flow physics can be drawn from a combination of real-time field observations (Okuda et al. [42]); detailed measurements during controlled field and laboratory experiments (Takahashi [54]), and analyses of debris flow paths and deposits (Fink et al. [21]). www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 8 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 29. Few reliable techniques exist to measure properties of flowing debris. Grossly invasive procedures such as plunging buckets or sensors into the flows conspicuously change the dynamics of the debris, while their behaviour has discouraged attempts to use non-invasive techniques such as ultrasonic, X ray, and others (Lee et al. [36]; Abbott et al. [1]). With regard to the rheological properties, many experiments (Chen [8]) have shown that the Herschel-Bulkley equations fit quite well laboratory data. One of the criticisms that may be moved to these tests is related to the scale effect. Successful models of debris flows must describe the mechanics of mobilization as well as the subsequent flow and deposition processes. Mobilization requires failure of the mass, a quantity of water to saturate the solid phase, such a change of energy, from gravitational to kinetic, to modify the motion pattern from sliding along a failure surface to a more widespread solid- liquid mixture that can be assessed as flow. On the whole, laboratory and field data are essential in determining the quality of any mathematical model, as it is especially important for estimating velocity, discharge, concentration, yield stress, viscosity and grain-size (Lorenzini and Mazza [37]). However, the achievement of good agreement between theoretical and experimental results does not justify indiscriminate extrapolation for the various territorial situations, which have very different boundary conditions from standard laboratory conditions. Assuming that the scientific research path cannot exclude an accurate observation and description of the phenomenon in question, without which the analysis of physical processes, that generate it, would become extremely artificial and uncertain, it is hoped that any attempt at improving the interpretation of the phenomenon involves critical comparison between the theoretical, experimental, and field approaches, as well as extensive osmosis process between the same approaches. 4 Debris flows generated by slope failures Debris flows can be the result of some form of landslides. In particular sliding phenomena in granular soils can turn into flow like movements. The main difference between slides and flow like landslides concerns the mechanisms of movement. While a slide advances on the slip surface as a rigid block or with a small internal deformation, a flow spreads downslope as a viscous fluid, adapting itself to any morphological change encountered along its path. In some conditions shear failure (sliding) can be affected by a rapid increase of positive pore pressures in excess to the hydrostatic values. The raise of excess pore pressures decreases the shear resistance of the soil inducing an acceleration of the movement: under these conditions the process can originate a debris flow. The triggering of positive excess pore pressure in loose granular materials can occur if the soil is saturated and the mechanism of slope deformation is characterized by fast volumetric compression. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 9
- 30. The occurrence of flow like movements is dependent on the un-drained behaviour of the soil, which refers to the condition of a saturated soil deforming at constant volume. The knowledge of the un-drained response of granular soils is of critical importance in assessing their susceptibility to liquefaction. The term liquefaction is frequently used to indicate all phenomena involving excessive deformation in saturated cohesionless soils and is not limited to the development of 100% excess pore pressure. Liquefaction can be triggered by either static or cyclic loading. Liquefaction due to static loading is associated with granular soils deforming in a strain softening (or limited strain softening) manner that results in limited or unlimited unidirectional flow deformation (Sivathayalan and Vaid [48]). A fundamental understanding of the un-drained response of granular soils has been derived from controlled laboratory studies. Un-drained triaxial compression tests on sand specimens mostly reconstituted by moist tamping have formed the basis for the steady state concepts (Poulos [44]). Susceptibility of soil to liquefaction mainly depends on grain size and porosity, but also on stress conditions (Picarelli et al. [43]). Ishihara et al. [30] presented the results of a series of laboratory tests, using triaxial apparatus, on saturated samples of Toyoura sand consolidated anisotropically. They found that with an increasing degree of anisotropy at the time of consolidation the sample becomes more contractive and susceptible to triggering flow failure. They found that the major effective principal stress at the time of anisotropic consolidation is a parameter controlling dilative or contractive behaviour of the sand. As a result the most appropriate way to normalise the residual strength of anisotropically consolidated sand is by the use of major principal stress at consolidation. The quasi steady state strength is then a function of void ratio and the major effective stress at consolidation. Other contributions devoted to the assessment of the potential for liquefaction of a soil are based on the concept of region of instability (Sladen et al. [49]). Soil instability is a phenomenon that resembles liquefaction in that there is a sudden decrease in the soil strength under un-drained conditions. This loss of strength is related to the development of large pore pressures reducing effective stresses in the soil. Lade [34] showed that there exists a region of instability inside the failure surface. The loss of strength occurs in un-drained condition as a consequence of disturbances small but fast enough to prevent water drainage. Conventional slope stability analysis methods (limit equilibrium methods) are widely used to investigate landslide problems and to determine the state of stress in slopes. This type of analysis has been used by Lade [34] for the determination of the state of stress in finite slope made of loose sand in order to investigate the region of instability by varying the slope height. Deangeli [13] presented a study devoted to the assessment of the potential for liquefaction in all zones of finite slopes from the in situ state of stress. For these purposes numerical models reproducing different slopes have been set up by using a finite difference code (FLAC manuals, 2001). The state of stress in slopes has been evaluated in both elastic and elastic-plastic field. By relating this www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 10 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 31. state of stress to the parameters stated by Ishihara et al. [30] to describe the potential for liquefaction a chart of susceptibility of debris flow in soil slopes has been set up. The chart reports curves that establish the limit condition on the basis of critical combinations of void ratio, slope angle and slope height. In this context some authors define a slope safety factor against liquefaction. For instance Poulos et al. [45] proposed the ratio between the residual strength of the soil Sus (the minimum resistance in un-drained conditions for a contractive soil with respect to the in situ void ratio) and the shear stress required for static equilibrium along the potential sliding surface. Ishihara et al. [30] defined the safety factor as the ratio between the residual strength of the soil (which is dependent on the major effective principal stress at the time of anisotropic consolidation) and the maximum shear stress along the potential sliding surface. Deangeli [13] reported the safety factor against liquefaction along different surfaces passing through a slope and assessed the volume of soil potentially involved in debris flow. The analysis of the propagation of debris flows generated by slope failures can be performed by taking into account the initial value of excess pore pressure (after slope failure) and its dissipation along the path. Significant results have been obtained by instrumented laboratory flume experiments. In these experiments the role of pore pressure in the flow failure phase, i.e. the transition from sliding to flow was investigated (Eckersley [20]). Deangeli [14] set up series of flume experiments to analyze the behaviour of water sand mixture flows, as a consequence of slope failures induced by water table raising and rainfall. The flows initially accelerated but at a certain stage of the process, unsteady deposition of the sand occurred, preceded by the transformation of the movement from flow to sliding. The phenomenon of deposition of the soil along the flume occurred at inclination greater than in the case of Spence and Guymer [51] experiments. On the basis of the reported results, it is evident the need of further experimental works investigating the dependence of debris flow behaviour by the triggering mechanisms and the role and generation of pore pressure during the propagation phase. 5 Concluding remarks Debris and hyper-concentrated flow result from the interaction of hydrological processes with geological processes and are triggered when soils get saturated and the stability of the slope is no longer maintained. These flows are among the most destructive of all water-related disasters. In this context, the recognised need to improve knowledge on the mechanics of these solid-liquid flows, highlighted by a critical analysis of the current international state-of-the-art, represent the seeding of the present work. Although the main aspects that rule the mechanics of these phenomena seem to be understood, it has to be underlined the relative scarcity of experimental (laboratory and field) data, the only ones that allow effective check of the models www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 11
- 32. nowadays available in different flow conditions and the estimation of the rheological parameters they contain. Greater research needs to be directed towards understanding the nature and the behaviour of these flows. Such knowledge is essential in order to estimate the potential frequency of these natural hazards and design suitable prevention and remediation measures. The ideal sequence that should be pursued in the approach to the difficult task of the management and mitigation of hyper-concentrated and debris flow can be obtained as follows (De Wrachien [16]): first, a systematic collection of field data should be carried out in order to provide a large base of reliable data that could allow a better knowledge of the existing risk trends and a deeper understanding of the mechanics of the phenomena, along with their general behaviour and effects; secondly, effective mathematical models, which strongly depend on data and measurements collected and performed in the field for their calibration and design, should be constantly developed, updated when needed, tested and applied; hazard mapping techniques and identification of possible scenarios, which need reliable models to be effective and sound, should then be set up; on the basis of the knowledge achieved in the previous steps, the best mitigation solutions should be identified, designed and built up; finally a program of systematic observations on the sites, where risk has been mitigated, should be planned and carried out to detect any shortcoming and test the efficiency of the investigations. Each of the above studies and investigations needs improvements and depends, to achieve them, on improvements in other fields. Improving measurement and documentation procedures would provide a better knowledge and ideas for new and more advanced models. The application of existing models based on the data collected in the field and the development of reliable new ones would allow, on one hand, to better focus what to observe in field and, on the other hand, improve mitigation measures and procedures. The field application of these latter would then identify new parameters to be measured and introduced in the models. From all these activities would emerge the best direction to be followed in future in-depth studies and investigations of debris flows. References [1] Abbott J., Mondy L.A., Graham A.L., Brenner H. Techniques for analyzing the behaviour of concentrated suspensions, in Particulate Two-Phase Flow, edited by M. C. Roco, pp. 3-32, Butterworth-Heinemann. Newton, Mass., 1993. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 12 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 33. [2] Arai M., Takahashi T., The Karman constant of the flow laden with high sediment in Proc. of the 3rd International Symposium on River Sedimentation University of Mississippi, 1986, pp. 824-833 [3] Asmar B.N., Lanston P.A., Ergenzinger Z., The potential of the discrete method to simulate debris flow in Proceeding of the First International Conference on Debris Flow Hazard Mitigation: Mechanics, Prediction and Assessment, Eds. Chen, New York, 1997 [4] Bagnold R.A., Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear in Proceedings of the Royal Society of London, Series A, 225, 1954, pp. 49 – 63 [5] Bailard J.A. An experimental study of granular-fluid flow Thesis presented to University of California at San Diego, Calif., 1978 [6] Bellos V., Sakkas J.G., 1D dam – break flood propagation on dry bed Journal of Hydraulic Engineering, 1987, ASCE 113(12), pp. 1510 – 1524 [7] Chen C.J., Laboratory verification of a dam – break flood model Journal of Hydraulic Division ASCE, 106(4), 1980, pp. 535 – 556 [8] Chen C.L. Bingham plastic or Bagnold dilatant model as a rheological model of debris flow? Proc. of Third Int. Sympos. on river sedimentation, University of Mississippi, 31st March – 4th April 1986 [9] Chen L.C., Generalized visco-plastic modelling of debris flow Journal of Hydraulic Engineering, 1988, 114, pp. 237 – 258 [10] Chen C.L., Ling C.H., Resistance formulas in hydraulics based models for routing debris flow in Debris Flow Hazard Mitigation: Mechanics, Prediction and Assessment, Eds. Chen, New York, 1997, pp. 360 – 372 [11] Courant R., Friedrichs K.O., Supersonic flow and shock wave Interscience Publisher Inc., New York, 1948 [12] Coussot P. Steady, laminar, flow of concentrated mud suspensions in open channel, Journal of Hydraulic Research, Vol. 32, n. 4, pp.535-559, 1994 [13] Deangeli C., The Role of Slope Geometry on Flowslide Occurrence, American Jou. of Environmental Sciences, Scipub, New York, 3 (3), 2007, pp. 93-97 [14] Deangeli C., Laboratory Granular Flows generated by Slope Failures, Rock Mechanics Rock Engineering, Springer, Netherlands, 41 (1) 2008, pp. 199– 217 [15] Dent J.D., Lang T.E., A biviscous modified Binghman model of snow avalanche motion Annals of Glaciology, 4, 1983, pp. 42 – 46 [16] De Wrachien D. Debris and hyper-concentrated flows, in G. Lorenzini, C.A. Brebbia, D.E. Emmanouloudis (eds) Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flow, Rhodes, Greece, 2006 [17] De Wrachien D., Mambretti S. Dam-break shock waves: A two-phase model for mature and immature debris flow Second International Conference on Debris Flow, 18 – 20 June 2008, The New Forest, United Kingdom [18] De Wrachien D., Mambretti S. Dam break with floating debris: a 1D, two- phase model for mature and immature flow propagation International www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 13
- 34. Conference on Agricultural Engineering and Industry Exhibition, 23 – 25 June 2008, Hersonissos, Crete, Greece [19] Dressler R.F. Hydraulic resistance effect upon the dam-break functions Proc. of Royal Society of London A(257), 1952, pp. 185 – 198 [20] Eckersley J.D., Instrumented laboratory flowslides, Geotechnique, 40, N. 3, 1990, 489-502. [21] Fink J.H., Malin M.C., D’Alli R.E., Greeley R. Rheological properties of mudflows associated with the spring 1980 eruptions of Mount St. Helens volcano, Washington Geophys. Res. Lett., 8, 43-46, 1981. [22] FLAC manuals, 2001, Version 4, ITASCA Consulting group, Minneapolis, USA [23] Fread D. L., Jin M., One-dimensional Routing of Mud/Debris flows using NWS FLDWAV Model, in Proc. of First International Conference on Debris Flow Hazards Mitigation: Mechanics, Prediction and Assessment, San Francisco, California, 7-9 August 1997 [24] Ghilardi P., Natale L., Savi F., Debris flow propagation and deposition on urbanized alluvial fans, Excerpta, 14, 2000, pp. 7 – 20 [25] Hanes D.M. Studies on the mechanics of rapidly flowing granular-fluid materials, Thesis presented to Univ. of California at San Diego, Calif., 1983 [26] Harten A. High resolution schemes for hyperbolic conservation laws Journal of Computational Physics, 49, 1983, pp. 357-394 [27] Hirano M., Hasada T., Banihabib M.E., Kawahasa K., Estimation of hazard area due to debris flow in Debris Flow Hazard Mitigation: Mechanics, Prediction and Assessment, Eds. Chen, New York, 1997, pp. 697-706 [28] Honda N., Egashira S., Prediction of debris flow characteristics in mountain torrents in Debris Flow Hazard Mitigation: Mechanics, Prediction and Assessment, Eds. Chen, New York, 1997, pp. 707-716 [29] Hunt B., Asymptotic solution for dam-break problems Journal of Hydraulic Division ASCE, 108(1), 1982, pp. 115-126 [30] Ishihara, K., Tsukamoto Y., Shibayama T., Evaluation of slope stability against flow in saturated sand. Reports on Geotechnical engineering, Soil mechanics and Rock engineering, Jubilee volume of Terzaghi Brandl 2000. Wien, 2000-2001, Vol. 5, Institut fur Grundbau und Bodenmechanik- Technische Universitat Wien Ed., 2003, pp. 41-54. [31] Jan C.D., A study on the numerical modelling of debris flow in Debris Flow Hazard Mitigation: Mechanics, Prediction and Assessment, Eds. Chen, New York, 1997, pp. 717-726 [32] Johnson A.M. Physical processes in geology Freeman Ed., San Francisco, 1970 [33] Julien P.Y., O’Brien J.S., Physical properties and mechanics of hyperconcentrated sediment flows in Proceeding Spec. Conference on Delineation of Landslides, Flash Flood and Debris Flow Utah, USA, 1985, pp. 260-279 [34] Lade P. Static instability and liquefaction of loose fine sandy slopes J. Geotech. Engng Div ASCE 118, 1, 1992, 51-71. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 14 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 35. [35] Lax P., Wendroff B., Systems of conservation laws Comp. on Pure and Applied Mathematics 13, 1960, pp. 217-237 [36] Lee J., Cowin S.C., Templeton III J.S. An experimental study of the kinematics of flow through hoppers. Trans. Soc. Rit., 18, 247-269, 1974. [37] Lorenzini G., Mazza N. Debris flow. Phenomenology and Rheological Modelling WIT Press, Ashurst Lodge, Southampton, UK, 2004 [38] Mambretti S., Larcan E., De Wrachien D. 1D modelling of dam – break surges with floating debris Biosystems Engineering, Vol. 100(2), June 2008, pp. 297-308 [39] McCormack R.W., The effect of viscosity in hypervelocity impact cratering AIAA Paper, 1969, 75-1 [40] Norem H., Locat J., Schieldrop B., An approach to the physics and the modelling of the submarine flowslides Marine Geotechnical 9, 1990, pp. 93-111 [41] O’Brien J.S., Julien P.J., Fullerton W.T., Two-dimensional water flow and mudflow simulation, Jou. of Hydraulic Engineering, 1993, 119, pp. 244- 261 [42] Okuda S., Suwa H., Okunishi K., Yokoyama K., Nakano M. Observations on the motion of a debris flow and its geomorphological effects, J. Geomorphol., suppl. 35, 142-163, 1980 [43] Picarelli L., Olivares L., Comegna L., Damiano E. Mechanical Aspects of Flow-Like Movements in Granular and Fine Grained Soils, Rock mechanics rock engineering Springer, Netherlands, 41 (1) 2008, pp. 179-197. [44] Poulos S.J., The steady state of deformation. Jou. of Geotechnical Eng. Div., ASCE, 107, 1981, pp. 553-561 [45] Poulos S.J., Castro G., France J.W., Liquefaction evaluation procedure, Jou. Geotechnical. Eng. Div. ASCE, 111(6), 1985, pp. 772-792. [46] Ritter A. Die Fortplanzung der Wasserwellen Zeitschrift des Vereines Deutscher Ingenieure 36(3), 1892, pp. 947 – 954 (in German) [47] Sakkas J.G., Strelkoff T. Dam break flood in a prismatic dry channel J. Hyd. Div. ASCE 99(12) 2195-2216, 1973 [48] Sivathayalan, S., Vaid, Y. P. (2002): Influence of generalized initial state and principal stress rotation on the undrained response of sands. Can. Geotech. Jou., 39, 63-76. [49] Sladen J.A., d’Hollander R.D., Krahm J., The liquefaction of sands, a collapse surface approach, Can Geotech Jou., 22, 1985, pp. 564-578. [50] Stoker J.J. The breaking of waves in shallow water Annuals New York Academy of Science 51(3), 1949, pp.360-375 [51] Spence K.J., Guymer I., Small scale laboratory flowslides, Geotechnique, 47, 5, 1997, pp. 915-932. [52] Strelkoff T., Falvey H.T. Numerical methods used to model unsteady canal flow J. Irrig. and Drain. Engrg, ASCE, 119(4), 637-655, 1993 [53] Takahashi T. Debris flow Rev. Fluid Mechanics, 13, pp. 57-77, 1981 [54] Takahashi T. Debris Flow, 165 pp., A. A. Balkema, Brookfìeld. Vt. 1991. [55] Takahashi T, Initiation of flow of various types of debris flow Proceeding Second International Conference on Debris Flow Hazard Mitigation: www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 15
- 36. Mechanics, Prediction and Assessment, Eds. Wieczorak and Naeser, Rotterdam, 2000, pp. 15-25 [56] Takahashi T., Nakagawa H., Flood / debris flow hydrograph due to collapse of a natural dam by overtopping Journal of Hydroscience and Hydraulic Engineering, 1994, 12, pp. 41-49 [57] Van Leer B., Towards the ultimate conservative difference scheme Journal of Computational Physics 23, 1977, pp. 263-275 [58] Wan Z. Hyperconcentrated flow Monograph Series of IAHR, Rotterdam, 290 pp., 1994 www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 16 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 37. One-dimensional finite volume simulation of real debris flow events L. Schippa & S. Pavan Department of Engineering, Ferrara University, Italy Abstract A numerical model for the simulation of mud flow and debris flow is presented. It is based on an alternative formulation of conservative balance equations, in which source terms are mathematically reorganized in order to guarantee an improved computational stability over complex geometry channels. For numerical implementation, the first order Godunov scheme with Roe’s approximation is used. Source terms are computed with Euler’s method and added by splitting. Such a simple basic scheme has been chosen to underline that the improved numerical stability depends on the proposed mathematical formulation, and not on a sophisticated numerical scheme. The correct wet-dry front velocity and propagation mechanism have been verified with standard dam-break test cases, and particular attention has been directed to the celerity computation inside the Roe’s scheme when dealing with irregularly shaped cross-sections. The numerical model has already been verified with analytical tests and laboratory experiments. In this work, the model is applied to two real events that occurred in North-Eastern Italy. The first is a debris flow that took place in the Upper Boite Valley, in the proximity of Cortina d’Ampezzo, in 1998, the second is a mud flow event located in the Stava Creek Valley in 1985. These events have been chosen thanks to the wide documentation and significant amount of field data available, which include topographical surveys, flow velocity measures and flow depth estimations. Keywords: mud flow, debris flow, wave propagation, source terms. 1 Introduction The aim of the present work is to check a numerical model that is suitable for the simulation of mud flows and debris flows in channels of complex geometry. To www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 17 doi:10.2495/DEB100021
- 38. fulfill this purpose, the model should have specific features, such as the treatment of wet-dry fronts, the handling of complex geometries and high bed slopes and the possibility of changing the model application field from Newtonian to non-Newtonian fluids, simply by changing the resistance law. These features have previously been tested applying the model to different test cases that have been properly chosen [1]. The classic frictionless dam-break test has been used to verify the correctness of waves speed propagation and the capability of treating wet-dry fronts. A non-cylindrical frictionless ideal channel has been used to evaluate the model response to abrupt changes in cross-section wideness and bed elevation, then the effect of friction terms introduction has been checked using a mud flow dam-break. The first phase of the model verification ended with the simulation of laboratory experiments on a mud flow dam-break over a sloping plane. In the present phase, the model is applied to two real events that occurred in North-Eastern Italy. The first is a debris flow that took place in the Upper Boite Valley, in the proximity of Cortina d’Ampezzo, in 1998, the second is a mud flow event located in the Stava Creek Valley in 1985. The proposed model is based on an alternative formulation of conservative balance equations, which includes a particular mathematical expression of source terms ideated for natural channels, and which has already demonstrated important stability features under the numerical point of view [2, 3]. The numerical implementation is performed using the Godunov finite volumes scheme. This kind of numerical schemes are largely diffused in mud flow or debris flow treatment [4–6], together with the Roe’s approximation for the solution of the Riemann problem. The presented model uses the same approach, but paying careful attention in conserving the general formulation suitable for complex geometry channels, in particular for what concerns the expression of the wave propagation celerity. This term is usually expressed as a function of water depth and cross-section width, but these hydraulic quantities often need to be corrected or mediated to be representative of irregular cross-sections. As an alternative, cross-section shape can be parameterized to be numerically handled [7]. In this work, celerity is determined referring to cross section wetted area and static moment, in order to ensure the formulation generality. Source terms are handled using the splitting technique [8] and evaluated with the Euler’s method. The pressure source terms, induced by the channel irregular geometry have been treated as in [2, 3], mathematically transforming the derivative of the static moment in order to eliminate the explicit dependence on the channel bed slope. This operation keeps its validity also in case of highly sloping channels, condition which often occurs in mud flow or debris flow phenomena. Friction source terms depends on the evaluation of friction slope, and therefore on the adopted resistance law. Like most of numerical models [5], the proposed model set up permits to easily change the resistance law and therefore to use the best fitting rheological model for each test case. It is worth noting that source terms numerical implementation has been kept as simple as possible, to put in evidence the stability features coming from the basis mathematical model. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 18 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 39. 2 Mathematical model The mathematical model is based on an alternative formulation of shallow water equations for one-dimensional (1-D) flows in natural channels of complex geometry [2]. The continuity equation and the momentum balance equation are written in terms of state variables A and Q, considering no lateral inflows. 0 A Q t x (1) 2 1 1 w f z I Q Q gI g gAS t x A x (2) where t is time, x is distance along the channel, A the wetted cross-sectional area, Q the discharge, g the gravitational acceleration, I1 the static moment of the wetted area, defined as: 1 0 cos , d h x I b x z h x z z (3) I2 is the variation of the static moment I1 along the x-direction, So = sin, where is the angle between channel bottom and the horizontal, b is the cross-section width, h is flow depth. The system closure equation for the evaluation of the friction term Sf will be described in detail for each examined test case, but the generally considered formulation is f S gR (4) in which Sf is the slope friction, R is the hydraulic radius, ρ is the mixture or the fluid density, and the shear stress τ depends on the adopted rheological model. 2.1 The source term Differently from the commonly used formulation of shallow water equations, the proposed model does not include in the momentum balance equation source term a direct dependence on bed slope. Details on the mathematical treatment which led to eqn. (2) can be found in [3]. The classic momentum equation is 2 1 0 2 f Q Q gI gA S S gI t x A (5) Focusing on the source term, the pressure term I2 has the following expression: 1 2 0 , cos d h x h b x z I I h x z z x x (6) Briefly, the pressure term I2 can be expressed as the sum of two terms, one of which is the variation of static moment I1 along x considering the water surface elevation zw as a constant, while the other exactly balances gravitational forces in the momentum equation, unless the presence of the term cos which arises in www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 19
- 40. case of high slopes, and cannot be neglected when considering mud-flow or debris-flow phenomena. 1 2 0 cos w z I I S A x (7) The substitution into (6) produces: 1 0 2 0 1 cos w f f z I gA S S gI gAS gAS g x (8) In this case, the term AS0 does not disappear as illustrated in [2, 3], but it remains and it is multiplied by the factor (1-cos). However, numerical proofs have demonstrated that this term is little if compared to friction terms, and can therefore be neglected. Eqn. (2) is therefore valid also for high sloping channel and debris flow simulation. 3 Numerical model Shallow water equations have been numerically implemented using the first- order finite volumes Godunov scheme. Numerical fluxes are computed with Roe’s method and source terms are evaluated with Euler’s approach and taken into account adopting the splitting technique. Details on the different components of the numerical model can be found in Toro [8]. The resultant scheme is explicit, first-order accurate, and has a very uncomplicated structure, since it is built choosing the simplest solution technique for every element of the partial differential equations system. This approach has the intention to illustrate the intrinsic stability features of the mathematical model, which could otherwise be hidden by sophisticated numerical schemes. Referring to shallow water equations in the vector form (eqn. (9)) the splitting approach for source terms treating, consists in separately solving the homogeneous partial differential equations system (eqn. (10)) and the ordinary differential equation (eqn. (11)). In detail, the solution obtained from eqn. (10) is used as initial condition for eqn. (11). t x U F U S U (9) 0 t x U F U U (10) t t dt U S U U (11) The Roe’s scheme, used to solve eqn. (6), requires the definition of the Jacobian matrix 2 2 2 1 2 0 1 0 1 2 2 I Q Q c u u g A A A F J U (12) Most of models proposed in the literature about the resolution of shallow water equations for debris flow or natural channels, based on approximate Riemann solvers (see for example [4, 5, 9]), adopt the same simplification in the evaluation of the term ∂I1/∂A, assuming www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 20 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 41. 1 or I A A h c g c gh A B b (13) In the present model, in order to keep the formulation generality and to ensure the applicability to natural and complex channel geometries, the static moment derivative is explicitly computed as the variation of I1 relative to the variation of A in the water depth variation range h ± Δh 1 1 1 I h h I h h I A A h h A h h (14) The celerity c is therefore defined as 1 I c g A (15) Another important aspect of the Godunov finite volume method application to natural geometries is the quantification of cell water volume V and the definition of the relation between the state variable A and V. For every computational cell, A is defined as 1 2 1 2 1 , d i i x i i x V A A x t x x x (16) Vi is computed as the volume of a pyramid which bases are irregular polygons, since the water profile is assumed to be parallel to channel bed. 1 1 1 1 2 2 2 2 3 i i i i i A A A A x V (17) 3.1 Source terms numerical treatment Source terms are numerically included in computations by splitting, and they are simply computed by Euler’s method , t dt t t t U U S U (18) in which 1 0 w f z I g gAS x S (19) Figure 1: Computational scheme for Vi. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 21
- 42. Figure 2: Computational scheme for momentum balance pressure source term. Considering no lateral inflows, source terms are present only in the momentum balance equation. This term can be divided into two parts, that is the friction term and the pressure term, represented by the static moment variation along channel, taking the water surface elevation as a constant. The computational scheme for the pressure term quantification in represented in Figure 2, and the variation of I1 is computed as: 1 1 2 2 1 1 1 1 w w i i z z I h I h I I x x x (20) 4 Numerical tests In this work the model has been applied to two real events. The first is a natural debris flow event, due to intense rainfall, surveyed at the Acquabona site in Northern Italy. It is of particular interest thanks to the large amount of available field data. The second is the Stava mud flow, a tragic episode occurred in a little town of Italian Alps. This event was caused by the collapse of two tailing dams, which released a huge quantity of water into the Stava Creek channel, causing the formation of a mud flow wave with an enormous destructive power. 4.1 Acquabona debris flow The Acquabona debris flow has been widely surveyed and documented in the context of the “Debris Flow Risk” Project, funded by the EU. In particular, the UPD (resp. Prof. Rinaldo Genevois) has carried out a research on some debris flow prone watersheds in the Upper Boite Valley (Eastern Dolomites, Southern Alps) and surroundings, included in the municipality of Cortina d’Ampezzo [10]. A large quantity of field data is therefore available since an automatic, remotely controlled monitoring system has been installed at Acquabona on June 1997. The Acquabona site in characterized by one or more debris flow every year, which usually occur in summer and in early autumn and are associated to intense, spatially limited rainfall events. The monitoring system installed at Acquabona was fully automatic and remotely controlled. It consisted of three on-site www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 22 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 43. monitoring stations and an off-site master collection station. Every station was equipped with a geophone, while at Station 3 also a superficial pressure transducer and an ultrasonic sensor were present. In this work we refer to the event of the August 17th in 1998. The event was originated by a very intense rainstorm: 25.4 mm of rain were measured during 30 min by the rain-gage at Station 1. The volume of the deposits available for debris flow generation has been estimated to be around 8000-9000 m3 . The overall duration of the event was of approximately 38 min and more than 20 different surges have been surveyed at Station 3. The geometry of the channel is available thanks to 19 surveyed transversal cross-sections, for a global channel length of 1120 m and a difference in height of 245 m. The longitudinal slope ranges from 10% to 55%. For model application a constant spatial step of 1 m has been adopted. Numerical simulations were performed adopting the rainfall hydrograph reconstructed by Orlandini and Lamberti [11], which has an extension of about 2.5 hours and a peak discharge of 2.3 m3 /s. An open boundary type condition is imposed at the downstream end. For the debris flow the bulk concentration is assumed to be 0.6 and mixture density 1850 kg/m3 , according to [7]. The rheological model adopted in the simulations is the Herschel-Bulkley model, which, for simple shear conditions may be written as: c K (21) in which K and η are rheological parameters. Referring to the simulations carried out by Fraccarollo and Papa [12] on the same event, K is assumed to be 150 Pa·s1/3 , τc is equal to 925 N/m2 , and η has been empirically set equal to 1/3. In Figure 3 computed flow height is compared with the measured data collected by the ultrasonic sensor at Station 3. The model satisfactorily captures wave height and shape, but it underestimates their duration, overestimating as a consequence their number. Results arte however encouraging and comparable to those obtained by Fraccarollo and Papa [12] and Zanuttigh and Lamberti [7]. The average velocity of the different flow surges has been estimated through geophone log recordings. Available data refer to two 100 m channel reaches Figure 3: Comparison between the flow depth measured and calculated at Station 3. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 23
- 44. Figure 4: Comparison between measured and computed wave speeds upstream and downstream from Station 3. Figure 5: Longitudinal discharge distribution and flow depth profile. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 24 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 45. located in the lower part of the channel before and after Station 3, which corresponds to the surveyed cross-section 8. Comparison is showed in Figure 4. In the upstream reach computed velocities compare well with field data, while in the downstream reach they are generally overestimated. It is interesting noting that the flow regime is mainly characterized by the formation of roll waves, as it is evident observing the longitudinal distribution of discharges and flow depths at two subsequent time steps. Nevertheless, numerical solution is not affected by relevant numerical instabilities. 4.2 Stava mud flow In July 19th 1985, two tailing dams suddenly collapsed in Tesero, a little town in the Italian Alps. The stored water, together with the dam body material flowed down to the Stava River as a big mud flow, claiming 268 human lives and destroying 47 houses. As reported by Takahashi [13], the Stava River before the disaster flowed with an approximately uniform slope of 5°. Although the mud flow had such an intensive destructive power, as well as fluidity, the Stava River channel itself had not suffered much erosion or deposition, and it can therefore be simulated as a fixed bed stream. In his report Takahashi gives important references also about mud flow solids concentration which was as high as 0.5, while the particle size was so fine that the relative depth, R/d, had a value of the order of 105 . In this condition the resistance to flow is similar to that of a plain water flow and the Manning’s equation can be applied. Takahashi obtained a Manning’s roughness coefficient in each section by reverse calculation from the data on velocity computed with the Lenau’s formula applied to measured flow superelevations at bends. The channel description is also taken from Takahashi [13]. It includes 24 surveyed cross-sections, their planimetric location and the longitudinal profile. In this case bed slope ranges from 5% to 12%. The simulated reach is 3500 m long and a constant spatial step of 1.25 m has been used. In Figure 6, discharge and depth computed hydrographs are compared with Takahashi numerical results obtained with the kinematic wave theory [13]. Referring to cross-section 10, located about 3000 m downstream the dams, there is a good accordance between the computed peak discharge and the value estimated by Takahashi (3500 m3 /s) as a result of product between the wetted cross-section area measured in situ (about 500 m2 ) and the maximum velocity derived by the flow superelevation at the nearest bend (7 m/s) The initial water profile condition reproduces the same hypothesis adopted by Takahashi, which is a uniform slide of the mud mass until Section 4, from which the mud flow is assumed to develop. Figure 8 shows the comparison between computed and measured front arrival times at different locations. The measured values are estimated on the basis of a seismograph located in Cavalese, a nearby town. The computed times are in good agreement with the estimated ones along the entire channel. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 25
- 46. Figure 6: Depth and discharge hydrograph at different cross sections. Figure 7: Initial conditions and flow profiles along the channel during the simulation. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 26 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 47. Figure 8: Comparison between computed and measured front arrival times at different locations. 5 Conclusions A numerical model for the simulation of mud flow and debris flow natural events is presented. It is based on a mathematical model which main features are concerned with the propagation of the wet-dry fronts, the treatment of irregular and variable cross sections shape, and the applicability to highly sloping channels. Two real events have been chosen to test the model. The first is a natural debris flow event at Acquabona site. In this case a large quantity of field data was available and model results compared well with wave peak height and propagation velocities. The second test case refers to the Stava mud flow tragic event, originated by the collapse of two tailing dams. Also in this case good accordance between observed data and mud front propagation speed has been obtained. Simulation results have also been compared with the Takahashi analysis of the same event, showing good accordance for what concerns peak discharge estimation at different cross sections. References [1] Schippa., L. & Pavan, S. 1-D finite volume model for dam-break induced mud-flow. River Basin Management V, 07-09 September 2009, Malta, pp. 125-136, ed. C.A. Brebbia, Wit Press, Southampton, Boston, 2009. [2] Schippa., L. & Pavan, S., Analytical treatment of source terms for complex channel geometry. Journal of Hydraulic Research, 46(6), pp. 753-763, 2008. [3] Schippa., L. & Pavan, S., Bed evolution numerical model for rapidly varying flow in natural streams. Computer & Geosciences, 35, pp. 390-402, 2009. [4] Garcia-Navarro, P. & Vazquez-Cendon M.E., On numerical treatment of the source terms in the shallow water equations. Computer & Fluids, 29, pp. 951-979, 2000. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 27
- 48. [5] Brufau, P., Garcia-Navarro, P., Ghilardi, P., Natale, L. & Savi, F., 1D Mathematical modelling of debris flow. Journal of Hydraulic Research, 38(6), pp. 435-446, 2000. [6] Naef, D., Rickenmann, D., Rutschmann, P. & McArdell, B.W., Comparison of flow resistance relations for debris flow using a one-dimensional finite element simulation model., Natural Hazards and Earth System Sciences, 6, pp.155-165, 2006. [7] Zanuttigh, B. & Lamberti, A., Analysis of debris wave development with one-dimensional shallow-water equations, Journal of Hydraulic Engineering, 130(4), pp. 293-303, 2004. [8] Toro, E.F., Riemann Solvers and Numerical Method for Fluid Dynamics, Springer-Verlag Berlin Heidelberg New York, 1999. [9] Ying, X. & Wang, S.S.Y., Improved implementation of the HLL approximate Riemann solver for one-dimensional open channel flows. Journal of Hydraulic Research, 46(1), pp. 21-34, 2008. [10] Berti, M., Geneovis, R., Simoni, A. & Tecca, P.R., Field observations of a debris flow event in the Dolomites., Geomorphology, 29, pp. 265-274, 1999. [11] Orlandini, S. & Lamberti A., Effect of wind precipitation intercepted by steep mountain slopes. Journal of the hydrologic engineering, 5(4), pp. 346-354, 2000 [12] Fraccarollo, L., & Papa, M., Numerical simulation of real debris-flow events. Physics and Chemistry of the Earth, 25(9), pp. 757-763, 2000. [13] Takahashi T., Debris flow, IAHR Monograph Series, A.A. Balkema Rotterdam Brookfield, 165 pp, 1991. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 28 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 49. Debris flow modelling accounting for large boulder transport C. Martinez 1 , F. Miralles-Wilhelm 1 & R. Garcia-Martinez 2 1 Department of Civil and Environmental Engineering, Florida International University, USA 2 Applied Research Center, Florida International University and FLO-2D Software, Inc., USA Abstract We present a quasi three-dimensional numerical model to simulate stony debris flows, considering a continuum fluid phase of water and fine sediments, and a non-continuum phase of large particles, such as boulders. Large particles are treated in a Lagrangian frame of reference using the Discrete Element Method in three dimensions. The fluid phase is governed by the depth-averaged Navier–Stokes equations in two horizontal dimensions and is solved by the Finite Element Method. The model simulates particle-particle collisions and wall-particle collisions, taking into account that particles are immersed in the fluid. Bingham and Cross rheological models are used for the continuum phase. Both formulations provide stable results, even in the range of very low shear rates. The Bingham formulation is better able to simulate the stopping stage of the fluid. The results of the numerical simulations are compared with data from laboratory experiments on a flume-fan model. The results show that the model is capable of simulating the motion of big particles moving in the fluid flow, handling dense particulate flows that avoid overlapping among particles. An application to simulate a debris flow event that occurred in Northern Venezuela in 1999 shows that the model replicates well the main observed boulder accumulation areas. Keywords: debris flow, mud flow, boulders transport, Eulerian and Lagrangian formulation, finite element method, discrete element method. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 29 doi:10.2495/DEB100031
- 50. 1 Introduction Debris flow is a frequent phenomenon in mountainous regions. It occurs when masses of poorly sorted sediments, rocks and fine material, agitated and mixed with water, surge down slopes in response to water flow and gravitational attraction. A typical surge of debris flow has a steep front or “head” with the densest slurry, the highest concentration of boulders and the greatest depth. A progressively more dilute and shallower tail follows this head. Reviews presented by Iverson [1], exhaustively describe the physical aspects of debris flow motion and clearly divide previous debris flow research into two distinct categories. The first, based upon the pioneering work of Johnson [2], assumes that debris flow behaves as a viscoplastic continuum. This model describes a single-phase material that remains rigid unless stresses exceed a threshold value: the plastic yield stress. Various rheological models have been proposed, derived from experimental results or from theoretical considerations, such as the Bingham model [3], the Cross model [4], and the quadratic model proposed by O’Brien and Julien [5]. The Bingham plastic model is the most commonly used in practice. The second approach has focus on the mechanics of granular materials. Based upon the findings of Bagnold [6], two-phase models have been developed by several authors, such as Takahashi [7] and Iverson [1]. These models explicitly account for solid and fluid volume fractions and mass changes respectively. Despite of the considerable progress over the past few years, the flow dynamics and internal processes of debris flows are still challenging in many aspects. In particular, there are many factors related to the movement and interaction of individual boulders and coarse sediments that have not been fully addressed in previous works. Asmar et al. [8] introduced the Discrete Element Method (DEM) to simulate the motion of solid particles in debris flows. DEM is a numerical method to model dry granular flows where each particle is traced individually in a Lagrangian frame of reference by solving Newton’s equation of motion. This paper describes the development of a quasi three-dimensional model to simulate stony debris flows, considering a continuum fluid phase, and large sediment particles, such as boulders, as a non-continuum phase. Large particles are treated in a Lagrangian frame of reference using DEM, and the fluid phase composed by water and fine sediments is modelled with an Eulerian approach using the depth-averaged Navier–Stokes equations in two dimensions. Bingham and Cross rheological models are used for the continuum phase. Particle’s equations of motion are fully three-dimensional. The model is tested with laboratory experiments and with a real application. 2 Governing equations The flow domain is divided in computational cells with triangular base and depth H, as shown in Figure 1. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 30 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 51. Figure 1: Schematic representation of debris flow with large solid particles. Assuming non-Newtonian and incompressible fluid phase, the depth averaged continuity and momentum equations in Cartesian coordinates can be written as follows: 0 ) ( ) ( y H v x H u t H (1) 0 1 fx S g F x y u g v x u g u t u g Dx (2) 0 1 fy S g F y y v g v x v g u t v g Dy (3) where H is the water depth, η is the free-surface elevation, u and v are the depth averaged velocities in x and y directions respectively, g is the gravitational acceleration and is fluid density. FD represents the fluid-solid interaction force exerted on the fluid by particles through the fluid drag force.), this force is evaluated as: V n i i FD D 1 F F (4) where FFD is the fluid drag force on each particle i, V is the volume of the computational cell and n is the number of particles in the cell. Sfx and Sfy are the depth integrated stress terms that depend on the rheological formulation used to model the slurry. Assuming a Bingham rheological model and Manning’s formula, as proposed by O’Brien and Julien [5], the stress terms for the fluid can be expressed as H www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 31
- 52. 3 / 4 2 2 2 3 H u N gH u gH y S fx (5) 3 / 4 2 2 2 3 H v N gH v gH y S fy (6) where N is the Manning roughness coefficient. The fluid dynamic viscosity and yield stress y, are determined as functions of the volume sediment concentration Cv, using the relationships proposed by O’Brien and Julien [9]: c e 1 1 (7) c e y 2 2 (8) in which 1, 1, 2 and 2 are empirical coefficients obtained by data correlation in a number of experiments with various sediment mixtures. Using a quadratic formulation combined with the Cross rheological model, the stress terms for the fluid are expressed as 3 / 4 2 2 H u N gH eff S fx with H u 3 (9) 3 / 4 2 2 H v N gH eff S fy with H v 3 (10) whereeff is the effective viscosity of the fluid defined by: B B K K eff 1 0 (11) with y B K 0 , and 3 10 0 In the solid phase, spherical particles of different diameters are considered. Particle trajectories are tracked using Newton’s second law and the considering gravity, buoyancy, fluid drag and collision forces. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 32 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
- 53. T N E dt d i m F F F v (12) The external force FE is given by FD B E F F F (13) The expression to compute the net force acting on the particle due to gravitational effects is g F ) ( 3 3 4 p R B (14) where R is the particle radius and p is the particle density. The expression for the drag on particles in viscous fluid is given by v u v u F d C R FD 2 2 1 (15) where Cd is the drag coefficient, u is the fluid velocity vector at the particle location, and v is the particle velocity vector. The last two terms in equation (12) represent the collision forces or contact forces among particles. Based on the simplified model that uses a spring- dashpot-slider system to represent particle interactions [8], the normal contact force and the tangential contact force are evaluated as ND NC N F F F (16) TD TC T F F F (17) The normal contact force FNC is calculated using a Hook’s linear spring relationship, N N NC K F (18) where KN is the normal contact stiffness and N is the displacement (overlap) between particles i and j. The normal damping force FND is also calculated using a linear relation given by N N ND v C F (19) www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III 33
- 54. where vN is the normal component of the relative velocity between particles and CN is the normal damping coefficient. This constant CN is chosen to give a required coefficient of restitution defined as the ratio of the normal component of the relative velocities before and after collision. The tangential contact force, FTC, represents the friction force and it is constrained by the Coulomb frictional limit, at which point the particles begin to slide over each other. Prior to sliding, the tangential contact force is calculated using a linear spring relationship, T T TC K F (20) where KT is the tangential stiffness coefficient, and T is the total tangential displacement between the surfaces of particles i and j since their initial contact. When KTT exceeds the frictional limit force f FNC, particle sliding occurs. The sliding condition is defined as NC f TC F F (21) where f is the dynamic friction coefficient. The tangential damping force FTD is not included in this model, since it is assumed that once sliding occurs, damping is accounted for from friction. Also, particle rotation is not considered. Fluid governing equations (1-3) are solved by the Galerkin Finite Element method using three-node triangular elements. To solve the resulting system of ordinary differential equation, the model applies a four-step time stepping scheme and a selective lumping method, as described by Garcia-Martinez et al. [10]. Forces on each particle are evaluated at each time step, and the acceleration of the particle is computed from the particle governing equation, which is then integrated to find velocity and displacement of each particle. 3 Results A series of experiments were carried out in a laboratory flume, using homogeneous fluid and fine sediment mixtures for the continuum phase and spherical marbles for the discrete phase. The experiments were performed in a 1.9 m long, 0.19 m wide, Plexiglas walled flume, with adjustable slope. The downstream part of the flume was connected to a wood horizontal platform, 0.75 m long and 0.95 m wide. A dam-break type of flow was initiated by an abrupt removal of a gate releasing mixtures from a 0.40 m long reservoir situated on the upstream part of the flume. Water-clay mixtures were used in all the experiments, with volume sediment concentration 23.5% and 26.5%. For preparation of the mixtures, kaolinite clay with specific unit weight of 2.77 was used. Fluid density was measured in the laboratory and rheological parameters and y were determined using equations (7) and (8) in which parameters are 1 = 0.621x10-3 , 1 = 17.3, 2 = 0.002 and 2 = 40.2. www.witpress.com, ISSN 1743-3533 (on-line) WIT Transactions on Engineering Sciences, Vol 67, © 2010 WIT Press 34 Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows III
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