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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
107
DOI: 10.4408/IJEGE.2011-03.B-013
THE UNIFED THEORY OF DEBRIS FLOW INITIATION BY USING
HOMOGENIZATION THEORY
y
inG
-H
sin
wu
(*)
& k
o
-f
ei
LIU
(**)
(*) PhD candidate, Department of Civil engineering, National Taiwan University, Taiwan ROC(**) Professor, Department of
Civil engineering, National Taiwan University, Taiwan ROC
from landslides to debris flows, the controlling phys-
ics also changed. But if we consider these phenomena
from scales of particles, the major difference would
be the interactions between solids and fluid motion in
the pore scale.
From the continuum point of view, there are many
theories used to interpreting the flowing properties of
debris flow. An extensive review was given by a
nCey
(2007). Many theories are validated useful and practi-
cal in certain domains. However, most of these theo-
ries can be used either when bulk material is almost
stationary (such a soil) or has large movement (such
a debris flows and avalanches). i
veRson
et alii (1997)
gave a review for models involving the effect of pore
pressures and granular temperature in the mobilization
of debris-flow. In the same paper, they assessed the re-
lationship between Coulomb failure and liquefaction,
and considered the role of granular temperature and
soil volume change in an infinite-slope formulation.
i
veRson
(2000) also proposed multiple time-scales
together with Richards’ equation to develop a math-
ematical model to evaluate effects of rainfall infiltra-
tion on landslide occurrence, depth, and acceleration.
The model provided a tool to assess the possibility of
landslide triggered by rainfall and post-failure motion.
But this approach still used the continuum concept to
model landslide process macroscopically.
As the continuum motion is actually the result
from small scale motion, there should be a method to
examine the small scale motion and then transfer mo-
ABSTRACT
We attempt to find the unified theory for the
prediction of the initiation of debris-flow by using
homogenization theory. In this study, we show the
leading order solution, which is the first step of this
derivation of unified theory. The derivation started
in the microscopic scale in the soil. The representa-
tive elementary volume (REV) in the soil is set to be
one order larger than the scale of porosity. Solids in
the REV are assumed to be rigid and adhesion-less.
The liquid velocity in the porosity is slow. By the no-
slip boundary condition and periodicity of REV, we
could obtain the microscopic flow conditions. Using
the assemble average with time dependence taken into
account, we obtain the macroscopic relation of water
content with the spatial and time variables from the
microscopic flow conditions. This macroscopic equa-
tion could be validated by the Richards’ equation.
K
ey
words
: homogenization theory, Richards’ equation
INTRODUCTION
It is well accepted concept that landslide together
with enough water can produce debris flows. But the
mechanism for landslides and occurrence of debris
flows are different. If we consider these as continuum,
physics involved is different. Landslide is a bulky mo-
tion of a soil where particle displacement is important.
However, debris flow is a flowing process where strain
rate is important. This means as the motion change
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Y.-H. wU & k.-F. LIU
108
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
scale in the pore level and under outer physical mech-
anism respectively. Using these two scales, we could
define the small parameter as below.
where l and L are micro- and macro-scopic character-
istic length scales respectively. The volume in micro-
scale is called representative element volume (REV
from here on). By this small parameter, we also define
the multiple-scale spatial and temporary independent
variables as
where x
1i
= εx
0i
,x
2i
= x
0i
, ... and t
1
= εt
0
, t
2
=εt
0
, ... and
so on. The physical dependent variable Φ(x
i
,t )rep-
resents velocity, pressure or other perturbed physical
variables in later derivation. They are expanded by the
small parameter in (1) as follow
In (3), for k
th
-order term, the (k+1)
th
or higher or-
der termspossess the property of periodicity in k-th
order REV. A compatibility condition exists between
equations of different order so as to assure the solu-
tions in different orders are independent. The condi-
tion (a
uRiault
, 1991) is
Substituting (2) and (3) into the governing equa-
tions of our problem, we could solve the micro-scopic
solution with the boundary conditions and compat-
ibility condition. Then, we use the spatial assemble
average in the REV to obtain the averaged physical
variable representing the macro-scopic property. The
assemble average is defined as
where |Ω
(k )
|is total volume of k
th
-order REV. Φ
(k )
is the
assemble averaged of Φ(k) in the kth-order REV, rep-
resenting the (k+1)
th
-order property and becomes the
tion of these small scale to that of continuum. In such
small scale, one should be able to visualize how parti-
cles start from stationary and then change to collision
based motion. As the first attempt, we shall use this
approach to examine if the well known equation such
as Richard’s equation which is based on experimental
results can be derived theoretically.
Therefore, we propose a new way to study the
initiation process. The initiation process starts from
static solids and flowing liquid in the pore. Then
gradually it develops to solid movement with strong
interaction of soil and liquid in the pore scale as well
as bulk motion of solid-liquid mixture. To study the
phenomenon, two drastically different concepts must
be used. Interaction between liquid and individual
solids is usually considered with Lagrangian coor-
dinates and bulk motion is usually considered with
Eulerian coordinates. In order to combine these two,
there must be two or more different characteristic
length scales involved in this initiation process. It is
reasonable to believe that the initiation of appreci-
able solid velocity has something to do with effects
from different scales. Homogenization theory (a
uRi
-
ault
, 1991) has been applied in this aspect and suc-
cessfully derived the flow condition of seepage in the
pore under saturated and static soil. Therefore, we
shall adopt similar approach to study the initiation
process. In this study, we show the leading order so-
lution which is the first step towards our goal.
Without any assumption of constitutive law of
the water-soil mixture, we begin to derive seepage
flow condition in the representative element volume
in the microscopic length scale -- the scale of the or-
der of pore. Then we use assemble average to obtain
the averaged flow condition in the macroscopic scale
-- the scale of total bulk soil-water mixture. In the
end, we can obtain the same result as Richards’ equa-
tion (R
iCHaRds
, 1931).
FUNDAMENTALS OF HOMOGENIZA-
TION THEORY
Homogenization theory is a method to obtain a
motion equation of interest by using a multiple-scale
perturbation method together with assembled aver-
ages in smaller scales.
The first step for homogenization is to decide the
two different characteristic length scales, micro- and
macro-scale which represent the characteristic length
(1)
(2)
(3)
(4)
(5)
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THE UNIFED THEORY OF DEBRIS FLOW INITIATION BY USING HOMOGENIZATION THEORY
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
109
u
i
is periodicity in micro-scale
where Γ is the solid boundary in REV, H the free sur-
face of water in the unsaturated REV, that is H =, z- h(
x, y, t ) ,δ
s
the surfacetension coefficient ranging from
0.019 at 0°C to 0 at 100°C (w
Hite
, 2006). k is the
curvature of the free surface. δ
ij
and e
ij
are Kronecker
delta the strain-rate tensor respectively. (8) is no-slip
condition on the solid surface.. Boundary conditions,
(9) and (10), would be only used in the unsaturated
micro-scopic REV. (11) is the condition of periodicity
in the saturated REV
From the definition, eq.(5), if we want to obtain
the averaged seepage flow condition, u should be aver-
aged as in (5), so.
where Ω
l(k )
is the liquid volume in k
th
-order REV. We
also define the porosity η and water content θ in k
th
-
order as
where Ω
s(k )
is the total solid volume in in k
th
-order
REV, and η
(k)
and θ(k) are all the function of x
(k +1)i
,
x
(k +2)i
,...,t0 ,t1,... . If the k
th
order REV is saturated, we
could have the relation that |Ω(k )| = Ω
l(k )
+ Ω
s(k )
. But
in unsaturated REV, Ω
l(k )
can vary in time. In most
soil. Water content θ ranges from 0 to η , and the po-
rosity η ranges from 0.25 to 0.75 (C
How
et alii, 1988)
NORMALIZATION
We define the microscopic characteristic length,
l , is one order larger than the characteristic length of
pores in soil; the macroscopic length, L , is the out-
er characteristic length of all bulk. Using these two
scales, the small parameter ε = l /L can be defined.
In our problem, the outer physical excitation in the
microscopic REV is the macroscopic pressure gradi-
function of macroscopic (higher order) independent
variables x
(k +1)i
, x
(k +2)i
….etc. The volume Ω
(k )
can
also be a function of temporary independent variable
in the unsaturated soil. Then the physical properties
of the bulk solid-liquid mixture can be found using
these averaged results with boundary conditions under
marco-scopic scale.
GOVERNING EQUATIONS
In this study, our problem is to derive the seep-
age flow condition in unsaturated and static soil. We
consider the pores in soil are large enough for water
to form a free surface interface of airliquid (Fig. 1).
If solid structure is stationary, we only have the
governing equations for pore water, which is the Na-
vier-Stokes equations
where ρ and μ are the density and dynamic viscosity of
water. p = p'+ ρgz is pressure with p' being dynamic
pressure. For boundary conditions, we need them for
different scales. At microscopic scale, there are kin-
ematic boundary conditions at the air-liquid and solid-
liquid interface. There is dynamic boundary condition
at free surface. At macroscopic scale, we use periodic-
ity of REV. These conditions are listed below.
(6)
(7)
(8)
(9)
Fi
g.1 - The definition of free surface of water and solid
boundary in the REV
(10)
(11)
(12)
(13)
(14)
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Y.-H. wU & k.-F. LIU
110
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
Eq (25a), (26a) and (27a) are continuity in differ-
ent orders of ε (25b), (26b) and (27b) are momentum
equations tensor. Besides, the boundary conditions of
k
th
-order of ε in saturated soil are
u
ki
is periodicity in micro-scale
ent. Due to the viscous effects dominate in the flow in
pores, we assume that the viscous term in micro-scale
is as important as the macro-scale pressure gradient.
So all scales are defined.
Substituting all scales in (15) to equation (6) to
(11), and omitting the primes, we obtain the dimen-
sionless equations as
With normalized boundary conditions.
u
i
is periodicity in micro-scale
Re is the Reynolds number of seepage flow in
pores. From the expression it is of O(ε) in our prob-
lem; β is the ratio of the effect fo surface tension
to shear stress at free surface in the pores within
εunsaturated REV.
Using small parameter ε , we expand the velocity,
pressure of water and free surface
Substituting (22) to (24) into eq. (16) to (21) and
collecting terms of the same order, we obtain
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25 a b)
(26 a b)
(28)
(29)
(30 a b)
(31 a b)
(32 a)
(32 b)
(32 c)
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THE UNIFED THEORY OF DEBRIS FLOW INITIATION BY USING HOMOGENIZATION THEORY
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
111
Equations (36) to (39) are a boundary-value-prob-
lem for solving k
ij
and A
j
in saturated REV. For spe-
cific sample of soil, it is possible to define Γ and then
solve the whole set of equations.
However, it is difficult to define the solid bound-
ary Γ in the microscopic REV in general without any
knowledge for the soil. Due to the complex composi-
tion of heterogeneous materials in nature, one often
obtain k
ij
and A
j
through experimental methods.
VERIFICATION
WITH
RICHARD’S
EQUATION
In this paper, we do not need the detail of k
ij
and
A
j
to verify our theory. We shall show that our theory
is equivalent to Richard’s equation with the same as-
sumption, i.e. isotropic and homogeneous
Substituting (35a) into (26a) and applying assem-
ble average in a saturated REV, we obtain
With periodic condition of u
1i
in 0
th
-order REV,
we could eliminate the first term in LHS of (42). So
where k
ij
is averaged hydraulic conductivity in mi-
croscopic REV, and it can be regarded as the repre-
senting hydraulic conductivity in macro-scale. If the
soil is isotropic and homogenous, we could simplify
k
i j
t o k
ij
, where k is a constant, and in this case. (43)
becomes
By applying the macroscopic boundary conditions
to (44), we get the pressure distribution p0 in saturated
soil. Furthermore, taking this solved p0 back to (35a),
we obtain velocity of seepage in the micro-scale. The
result is the same as m
ei
& a
uRiault
(1991).
Eq (25) to (32) are all the equations and boundary
conditions in our problem. In the following, we begin
to derive the flow condition in unsaturated soil.
DERIVATION OF UNSTEADY FLOW IN
UNSATURATED SOIL
In the bulk of solid-liquid mixture, there must ex-
ist saturated and unsaturated REV. We firstly derive
the flow condition insaturated REV, and then continue
to derive flow in unsaturated REV.From (25b), we find
This implies 0
th
-order pressure depends on macro-
scopic variables for 0
th
-order REV. To solve u
0i
, we
combine (33) and (25b) to get
Due to the linearity of (34), the solution form of
u
0i
and p1 are (m
ei
& a
uRiault
, 1991)
where kij and Aj are the 2
nd
and 1
st
-order tensors
representing the geometrical properties in the 0
th
-or-
der REV. p1 is the function of x
1i
, x
2i
,...,t , and is
a constant representing outer physical excitation for
0th-order pressure. Applying (35a,b) with (25a) and
(36b), we obtain
and boundary conditions for water in saturated
REV become

is periodicity in micro-scale
For unsaturated REVs, the free surface boundary
conditions become
(33)
(34)
(35 a b)
(36)
(37)
(38)
(40)
(41)
(42)
(43)
(44)
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Y.-H. wU & k.-F. LIU
112
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
For unsaturated REV, we apply assemble average
to (25a)
With Divergence theorem, we obtain
where S is the interface of water in 0
th
-order REV.
There are three different kinds of interfaces, water-
solid interface S
Γ
, water-air interface S
H
, and water
area on each surface of a REV SREV . Separate (46)
for different kinds of interface, we obtain
where n
i
is an unit normal vector of each surface, and
S0 is the element surface of 0
th
-order REV in integral.
With no-slip condition, the last term on RHS of (46)
is zero.
Dividing |VH
0
| from the free-surface kinematic
boundary condition (40), we obtain
the subscript zero of the V
0
H
0
means taking gradi-
ent with 0th-order spatial independent variables, x
0
, y
0
and z
0
. In (47), the second term is just u
0i
- n
i
. So this
is simply the normal flux at free surface of water.
Finally, the second term in RHS of (46) could be
expressed by following equation.
where S
x
, S
y
and S
z
are the areas of the surface normal
to yz-, xz and zy-plane respectively of a REV. Taking
(35a) into (50) together with each unit normal vectors
of its surface of REV and rearranging it, we obtain
We define the terms in brackets in RHS of (51)
as below.
where k
j
is the hydraulic conductivity of REV in each
direction of x, y and z. Then, (51) could be changed
into the form as below.
Finally, taking (52) and (49) into (46) to give
So far, we have obtained the averaged 0
th
-order
continuity. Before continuing the derivation, we need
to define water content first. As defined in (14), the
water content is
where A = A(z ) is the area of water with z variation
in a REV. Water content θ
0
depends on x1i , x2i ,... , t ,
and it can be regarded as the averaged water content in
macro-scale; and H0 is function of time in unsaturated
soil. Differentiating (54) with respect to t0 once, we
could obtain
where A ( H
0
) is the area at free surface of water in
REV. Using (55) and substituting it into (53), we ob-
tain
(45)
(46)
(47)
(48)
(49)
(50)
(51)
(52)
(52 b)
(53)
(54)
(55)
(56)
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THE UNIFED THEORY OF DEBRIS FLOW INITIATION BY USING HOMOGENIZATION THEORY
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
113
together with the hydraulic conductivity, from the ex-
periments or other calibrated data, in each direction of
soil, we could use (60) to obtain the water content in
soil. (60) is the same as the R
iCHaRds
’ equation(1931).
CONCLUSION AND DISCUSSION
In the problem of the seepage flow in unsaturated
static soil, without any constitutive assumption for
solid-liquid mixture, we successfully use homog-
enization theory to obtain the equation governs wa-
ter content which is proved to be the same as R
iCH
-
aRds
’ equation. From this result, we could conclude
that homogenization theory is adequate to be used in
the problem of unsteady and unsaturated solid-liquid
mixtures. However, the result in this paper is only the
leading order solution for fixed solid. We will con-
tinue using this theory in the problem of unsteady,
unsaturated and movable solid of solid-liquid mixture
to derive the macro-scopic motion of solid-liquid mix-
ture and study the process of debris-flow initiation.
ACKNOWLEDGEMENTS
We gratefully appreciate National Science Coun-
cil (Grant NSC 96-2625-Z-002-006-MY3) in Taiwan
for supporting this research.
The inner product of unit normal vector on the
surface of infinitesimal area, dS
0
, together with the
unit z-direction vector, e
z
= (0,0,1) , is
where θ is the angle between unit normal vector of
free surface and z-direction vector. (see Fig. 2)
Therefore, the integrand inside the integral of
LHS in (56) becomes
In (58), dS
0
cosθ is the projection area of free
surface of water on the z-plane. By using (58), (56)
becomes
(59) is another form of averaged continuity of 0
th
-
order REV. In the micro-scopic REV, the projection
area dS
0
cosθ approaches the original area, A ( H
0
) in
the macroscopic point of view. Therefore, integral on
LHS is very close to 1, and finally (59) becomes
k
i
could be regard as the REV-averaged hydraulic
conductivity in x, y and z-direction respectively. And
from (60), we could find that the time rate of change of
water content in the macro-scale is proportional to the
macroscopic pressure gradient in each direction. If we
have the macroscopic boundary conditions of pressure
(57)
Fig. 2 - Unit normal of infinitesimal surface and e
z
(58)
(59)
(60)
REFERENCES
a
nCey
C. (2007) - Plasticity and geophysical flows: A review. J. non-Newtonian Fluid Mech., 142: 4-35.
a
uRiault
J.-l. (1991) - Heterogenous medium. Is an equivalent macroscopic description possible? Int. J. eng. Sci.. 29:785-
795.
C
How
v.t., m
aidment
d.R. & m
ays
l.w. (1988) - Applied Hydrology. McGraw-Hill.
i
veRson
R.m., R
eid
m.e. & l
a
H
usen
R.G. (1997) - Debris-flow mobilization from landslide. Annu. Rev. Earth Planet. Sci. 25:
85-138.
i
veRson
R.m. (2000) - Landslide triggering by rain infiltration. water Resour. Res.. 36: 1897-1910.
m
ei
C.C. & a
uRiault
J.l. (1991) - The effect of weak inertia on flow through a porous medium. J. Fluid Mech. 222: 647-663.
R
iCHaRds
l.a. (1931) - Capillary conduction of liquids through porous medius. Physics 1: 318-333.
w
Hite
f.m. (2006) - Viscous Fluid Flow, 3
rd
ed.. McGraw Hill.
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