# IJEGE-11_BS-Wu-&-Liu

*Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza*

*DOI: 10.4408/IJEGE.2011-03.B-013*

**THE UNIFED THEORY OF DEBRIS FLOW INITIATION BY USING**

**HOMOGENIZATION THEORY**

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.

debris flow. An extensive review was given by a

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

*et alii*(1997)

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

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.

examine the small scale motion and then transfer mo-

**ABSTRACT**

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**

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

*Y.-H. wU & k.-F. LIU*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

anism respectively. Using these two scales, we could

define the small parameter as below.

*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

*x*

*x*

*x*

*x*

*t*

*εt*

*0*

*t*

*εt*

*x*

*t*)rep-

variables in later derivation. They are expanded by the

small parameter in (1) as follow

*k*

*(k+1)*

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

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

*k*

resenting the

*(k+1)*

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.

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

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.

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

**FUNDAMENTALS OF HOMOGENIZA-**

**TION THEORY**

perturbation method together with assembled aver-

ages in smaller scales.

macro-scale which represent the characteristic length

**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*

*u*

face of water in the unsaturated REV, that is H =, z- h(

x, y, t ) ,

*δ*

*k*is the

*δ*

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

aged as in (5), so.

*k*

*η*and water content

*θ*in

*k*

*th*

*k*

*η*

*θ*(k) are all the function of

*x*

*x*

*t*0

*,t*1,... . If the

*k*

*θ*ranges from 0 to

*η*, and the po-

rosity

*η*ranges from 0.25 to 0.75 (C

*et alii*, 1988)

*NORMALIZATION*

*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-

variables

*x*

*x*

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**

consider the pores in soil are large enough for water

to form a free surface interface of airliquid (Fig. 1).

vier-Stokes equations

*ρ*and

*μ*are the density and dynamic viscosity of

water.

*p*=

*p'+ ρ*gz 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.

*Fi*

*g.1 - The definition of free surface of water and solid*

*boundary in the REV*

*Y.-H. wU & k.-F. LIU*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

*ε*(25b), (26b) and (27b) are momentum

equations tensor. Besides, the boundary conditions of

*k*

*ε*in saturated soil are

*u*

pores, we assume that the viscous term in micro-scale

is as important as the macro-scale pressure gradient.

So all scales are defined.

sionless equations as

*u*

lem;

*β*is the ratio of the effect fo surface tension

to shear stress at free surface in the pores within

εunsaturated REV.

**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*

*k*

*A*

solve the whole set of equations.

knowledge for the soil. Due to the complex composi-

tion of heterogeneous materials in nature, one often

obtain

*k*

*A*

**VERIFICATION**

**WITH**

**RICHARD’S**

**EQUATION**

*k*

*A*

sumption, i.e. isotropic and homogeneous

*u*

*k*

senting hydraulic conductivity in macro-scale. If the

soil is isotropic and homogenous, we could simplify

*k*

*k*

*k*is a constant, and in this case. (43)

*p*0 in saturated

soil. Furthermore, taking this solved

*p*0 back to (35a),

we obtain velocity of seepage in the micro-scale. The

result is the same as m

to derive the flow condition in unsaturated soil.

**DERIVATION OF UNSTEADY FLOW IN**

**UNSATURATED SOIL**

the flow condition insaturated REV, and then continue

to derive flow in unsaturated REV.From (25b), we find

*u*

*u*

*p*1 are (m

*kij*and

*Aj*are the 2

*p*1 is the function of

*x*

*x*

*t*, and is

0th-order pressure. Applying (35a,b) with (25a) and

(36b), we obtain

*Y.-H. wU & k.-F. LIU*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

*S*is the interface of water in 0

solid interface S

for different kinds of interface, we obtain

*n*

is zero.

*x*

*y*

*z*

*u*

*0i*

*n*

*S*

*S*

*S*

(35a) into (50) together with each unit normal vectors

*k*

*x*,

*y*and

*z*. Then, (51) could be changed

into the form as below.

to define water content first. As defined in (14), the

water content is

in a REV. Water content

*θ*

*x1i*,

*x2i*,... ,

*t*,

macro-scale; and H0 is function of time in unsaturated

soil. Differentiating (54) with respect to t0 once, we

could obtain

tain

**THE UNIFED THEORY OF DEBRIS FLOW INITIATION BY USING HOMOGENIZATION THEORY**

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

**CONCLUSION AND DISCUSSION**

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

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**

for supporting this research.

*dS*

*e*

*θ*is the angle between unit normal vector of

free surface and z-direction vector. (see Fig. 2)

*dS*

*θ*is the projection area of free

becomes

area

*dS*

*θ*approaches the original area, A ( H

LHS is very close to 1, and finally (59) becomes

*k*

*i*

*could be regard as the REV-averaged hydraulic*

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

*Fig. 2 - Unit normal of infinitesimal surface and e*

*z*

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