simulation of pipe progression in a levee foundation with coupled seepage and pipe flow domains

H O S T E D B Y The Japanese Geotechnical Society

Soils and Foundations

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Simulation of pipe progression in a levee foundation with coupled seepageand pipe flow domains

Da-yu Wang, Xu-dong Fun, Yu-xin Jie, Wei-jie Dong, Die Hu

State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China

Received 5 May 2012; received in revised form 3 May 2014; accepted 3 June 2014

Abstract

A very large percentage of piping cases have been brought about by internal erosion, which is the primary cause of dam failures. This studydeveloped a numerical model to simulate the pipe progression in a levee foundation by analyzing the inception and transportation of erodibleparticles from the soil fabric. An approach that considers the turbulent flow in an erodible pipe and the seepage flow in the remaining area of alevee foundation is employed to capture the main hydraulic characteristics of piping. The mechanical analysis of individual erodible particles isconsidered to quantify the critical condition for particle inception in an erodible pipe. In addition, physical piping model tests are numericallysimulated to examine the proposed approach. The simulation demonstrates that the flow in a pipe can progress backward from downstream toupstream when the upstream water head reaches a critical value. Furthermore, the function mechanism of a cut-off wall can be explained by thismodel. The results have revealed that this model can reproduce the experimental data, such as the critical water head and the progression time,which are obtained from the physical model. The relationship between the depth of a suspended cut-off wall and the critical water head isobtained; this relationship facilitates the practical design of the critical depth of a cut-off wall for a given water head.& 2014 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.

Keywords: Piping; Pipe flow; Seepage; Element free Galerkin method; Levee

1. Introduction

Most of the levees in China have a two-layer foundationcomposed of a nearly impervious surface layer and a perviousunderlying layer. Generally, the soil in the upper layer is clay,silty clay, or silt with mucky intercalations scattered in localareas; the underlying layer is primarily composed of pervioussoil such as sandy gravel, sandy cobble, or decomposedbedrock. In flood seasons with high water levels, many leveessuffer from severe piping. Some statistic research indicates thatpiping is a primary cause of the serious failure of embankment

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dams which brings about the risk of flooding in downstreamareas (Foster et al., 2000). As a consequence, it has become ofgreat interest to improve the understanding of the piping flowin such two-layer structures for effective seepage control.There has been much research work on piping within levees

and a general understanding of piping phenomena has beenachieved. Some researchers have proposed the internal stabilityof soil, which is based on the grain size and the distribution ofthe soil, to identify the susceptibility of soil to piping failure(Istomina, 1957; Kezdi, 1969; Kenney and Lau, 1985). Changand Zhang (2013) extended the criteria of internal stability tosoils containing a significant amount of fine particles. From theviewpoint of engineering applications, the critical water head orhydraulic gradient has been established; it provides an overalldescription of piping behavior that may result in the failure of a

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a levee foundation with coupled seepage and pipe flow domains. Soils and

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D.-y. Wang et al. / Soils and Foundations ] (]]]]) ]]]–]]]2

levee (Bligh, 1910; Sellmeijer, 1988; Weijers and Sellmeijer,1993; Ojha et al., 2003; Richards and Reddy, 2007). The processof pipe progression has been widely explored in laboratorymodel tests (Wit et al., 1981; Kohno et al., 1987; Yao et al.,2007; Zhou et al., 2007b), followed by studies on the transportbehavior and the mechanism of fine particles through the soilpores (Vardoulakis et al., 1996; Sterpi, 2003; Cividini and Gioda,2004). A few efforts to numerically simulate pipe progressionhave recently been reported (Sellmeijer and Koenders, 1991;Yin, 1998; Hagerty and Curini, 2004; Fujisawa et al., 2010;Zhou et al., 2012).

The numerical approach to piping simulations can beroughly grouped into three categories in terms of the repre-sentation of the piping phenomena. The first approach, atraditional approach, is simply to increase the permeabilitycoefficient of the piping zone, while conducting a routineseepage analysis based on continuum formulations andDarcy’s law (Yin, 1998; Hagerty and Curini, 2004; Li et al.,2005). It can be used to analyze the effect of the piping zoneon the remaining seepage field, but fails to reproduce theprocess of the pipe progression. The second approach is thediscrete element method (DEM) (Zhou et al., 2007a; El Shamyand Aydin, 2008a, 2008b). As an alternative approach, theporous soil medium is modeled as a mixture of water and solidparticulates. The water flow is described using an averagedform of the Navier–Stokes equations, while soil particles aremodeled at the microscale using DEM. The total number oftracked particles is too large to make it applicable toengineering practice. The third approach is the multi-phasesoil model (Stavropoulou et al., 1998; Bonelli et al., 2006; Luoet al., 2010; Fujisawa et al., 2010). The soils are divided intoseveral phases according to different erosion models. Thewater and fine soils in pores are distinguished from the soilfabric, and the erosion of the fine soils from the fabric isaccounted for. Such an approach enables the capturing of themain characteristics of piping, and its applicability to engineer-ing problems has been shown. For example, Fujisawa et al.(2010) successfully simulated the temporal alteration and thespatial distribution of porosity and predicted the typicaldevelopment of piping within an embankment. Zhou et al.(2012) conceptualized the computation domain as the coupling

Fig. 1. Sketch of piping in two-layer levee fo

Please cite this article as: Wang, D.-y., et al., Simulation of pipe progression inFoundations (2014), http://dx.doi.org/10.1016/j.sandf.2014.09.003

of two zones: an erodible piping zone and a zone withouterosion. They reproduced the process of the backward pipeprogression shown in their laboratory model tests. In suchmodels, however, there is no universally accepted mechanismof internal erosion which can depict just how fine particles areeroded from the soil fabric and transported out of the soil mass.It is possible that the water flow in the piping zone and the

water flow in the seepage zone follow different governinglaws. It is beneficial, therefore, to consider the computationdomain as coupled zones (Sellmeijer and Koenders, 1991;Koenders and Sellmeijer, 1992; Schmertmann, 2000; Zhouet al., 2012). The aim of this paper is to provide a practicalmethod for engineering applications by following Zhou et al.(2012). A new mechanism for soil inception and transportationin porous media has been proposed. The time for the pipingprogression has been estimated, which benefits the safetycontrol of levees.The method is firstly examined against laboratory model

tests and then applied to a real levee foundation. The behaviorof a suspended cut-off wall in controlling the pipe progressionis explored; this helps demonstrate the potential value of thepresent method in practical applications.

2. Model development

For a two-layer levee foundation, piping generally beginswith a soil flow at an unfiltered exit on the back side of thelevee, after which an erodible pipe is gradually developed. Theerodible pipe progresses backward along the interface betweenthe nearly impervious surface layer and the underlyingpervious layer, which may harm the safety of the levee.Artificial barriers, such as cut-off walls, may impede theprogression of the pipe, as shown in Fig. 1. They are usuallyadopted as practical seepage control measures.

2.1. Governing equations for flows

The flow in the piping zone may be either laminar orturbulent, depending on the flow velocity as well as the poresformulated by the soil skeleton. Soil particles are eroded fromboth the tip and the wall of the pipe and are carried away by

undation with hypothetical barrier.





Fig. 2. Forces acting on erodible particle in porous medium.

D.-y. Wang et al. / Soils and Foundations ] (]]]]) ]]]–]]] 3

the flowing water. Water exchanges between the piping zoneand the seepage zone, through their mutual boundaries, occurdue to the difference in flow pressure. Assuming that theconcentration of eroded fine particles in the pipe flow is not sohigh that the water–soil mixture is changed into a non-Newtonian fluid, the flow in the pipe may be considered as aviscous incompressible Newtonian fluid.

The pipe flow is probably unsteady, and thus, a transientmodel should be adopted (Fujisawa et al., 2010). Here, we usea quasi-steady model which is common in river engineering.That is, an unsteady process is divided into a number ofsuccessive steady phases which enables the applicability of asteady model. This approximation holds for cases where theflow does not change quickly, but facilitates the simplificationof the computation. The mass and momentum equations forsteady flows in an erodible pipe with lateral seepage read asfollows:

dQ

dx¼ q¼ A

dv

dxþv

dA

dx; ð1aÞ

∂∂x

Hþ v2

2g

� �þλ

1de

v2

2g¼ 0; ð1bÞ

where Q is the flow discharge, q is the lateral inflow per unitlength, v is the mean flow velocity, H is the water head,A¼πde2/4 is the cross-section area, de is the hydraulic radius, λis the coefficient of resistance, g is the acceleration due togravity, and x is the streamwise coordinate with its positivedirection identical to the mean velocity. The momentumequation, Eq. (1b), was adopted in Zhou et al. (2012). Itsuggests that the lateral inflow from seepage has a negligiblecontribution to the momentum of the pipe flow. This assump-tion is reasonable when the lateral seepage flow is comparablysmall or approximately perpendicular to the mean flow.

For a two-dimensional seepage domain Ω, the governingequation accounting for Darcy’s law is

kx∂2H∂x2

þky∂2H∂y2

¼ 0 in domain Ωð Þ ð2aÞ

with boundary conditions

HjΓ1¼H0 x; yð Þ; ð2bÞ

K∂H

∂n,

��Γ2

¼ �q H; x; yð Þ; ð2cÞ

where kx and ky are the hydraulic conductivity in the x and ydirections, respectively, Γ1 is the boundary with a given head,and Γ2 is the boundary with a given seepage rate q.

2.2. Inception of sediment

In the numerous studies on internal erosion, most research-ers attribute the erosion mechanism to fine particles beingdetached and transported from the soil fabric by a seepageflow, which leads to the increase in soil porosity and finally tothe occurrence of piping. Therefore, the inception of sediment


transport is important for quantifying the detachment oferodible particles from the soil fabric.The progression of the piping channel is also the progression

of erosion in the pipe tip. The particles along the progresseddirection will gradually be taken out of the soil fabric. In thisprocession, the forces acting on an individual particle areshowed in Fig. 2, which includes the seepage force, the gravityforce, and the friction force exerted by neighboring particles.Assuming that all forces act through the centroid of the

particle, at the critical condition of particle inception, the forcebalance can be written as follows:

f sx � sin θ� f sy � cos θ¼ f � f g � cos θ ð3Þf sx and f sy are the seepage forces acting on the particle in the x-direction and the y-direction, respectively, and f g is the gravityforce of the particle itself. f is the friction force.The seepage force acting on an individual particle can be

expressed as Howard and McLane (1988)

f s ¼ Cp � 3π � μ� d � u ð4Þwhere μ is the fluid dynamic viscosity, d is the diameter of theparticle, u is the local seepage velocity, and coefficient Cp

represents the ratio of the soil pore geometry to the localstreaming line configuration around the particle. For a singlesphere in an infinite fluid, Cp reaches its limiting value of 1.Local seepage velocity u can be expressed by Darcy’s law as

u¼ ki=n ð5Þwhere i is the local hydraulic gradient, n is the porosity of thesoil, and k is the hydraulic conductivity. Chapuis (2004) hasstudied the saturated hydraulic conductivity of sand using theeffective diameter and the void ratio, and proposed thefollowing relationship among hydraulic conductivity, soilgrading, and porosity:

k m=s� �¼ 1219:9� d1:56510 � n2:3475

1�nð Þ1:565 ð6Þ

where d10 is in m. The model tests by Zhou et al. (2007b) gavethe horizontal hydraulic conductivity of 9:5� 10�5 m=s,while the value calculated according to Eq. (6) is 9:23�10�5 m=s, quite similar to the test result.






Substituting Eqs. (5) and (6) into Eq. (4) yields the seepageforce acting on the particle, namely,

f sx ¼ 1219:9� Cp � 3π � μ� d � d1:56510 � n1:3475

1�nð Þ1:565 � ix

ð7aÞ

f sy ¼ 1219:9� Cp � 3π � μ� d � d1:56510 � n1:3475

1�nð Þ1:565 � iy

ð7bÞThe buoyant weight of a particle can be expressed as

f g ¼ Cs �π

6� γs�γ

� �� d3 ð8Þ

where Cs is a shape coefficient, and γs and γ are the unitweight of the sediment grain and the fluid, respectively.

Friction is produced by a moving particle, under the actionof seepage and gravity force, making contact with its neigh-boring particles. Lambe and Whitman (1979) studied thefriction between particles in various granular forms. Soil withdifferent particle shapes and size distributions has a differentfriction coefficient for f f .

f ¼ f f � f sy � sin θþ f sx � cos θ� f g � sin θ� � ð9Þ

Substituting Eqs. (7a,b)–(9) into Eq. (3), the critical balanceequation of a particle can be obtained. Therefore, we can getthe critical particle diameter that could incept. Therefore,combined with the particle-size distribution, shown in Fig. 3,the percentage pf of the particle weight that may be erodedaway from the soil fabric is determined.

2.3. Sediment transport and advance of pipe tip

The fine particles are eroded and transported, and then thepipe head advances. Thus, it is first necessary to accuratelyevaluate the developing direction of the pipe tip. Here, weassume that the pipe advances in the direction where theparticles are most easily eroded out. This effect could beexplained by the fact that a relatively fast flow erodes andtransports particles much more than a slow flow. Therefore, thedirection in which most particles are carried out (the maximum

0

20

40

60

80

100

0.0010.010.1110

Fine

r: %

Diameter: cm

P'fPc

Pf

d'10 dc (d10)

Fig. 3. Particle-size distribution of soil material.


Pc) is the next direction that the pipe will follow.

n0 ¼ n0þPc � 1�n0ð Þ ð10Þwhere n0 is the new porosity of the soil after the Pc percent ofparticles have eroded. n0 is the initial porosity. Meanwhile,d10 increases as the fine particles are taken out. The updatedd10 can be obtained by combining it with the particle-sizedistribution. We use d

010 to denote the new d10 after erosion,

and the corresponding percentage is p0f . The calculation

method for d010 is shown in Fig. 3 as follows:

p0f �pf100�pf

¼ 0:1 ð11Þ

then

p0f ¼ 10þ0:9pf ð12ÞAccording to Eqs. (6) and (9), the local hydraulic con-

ductivity will also increase with d10 and porosity n, whichleads to more and more soil particles being eroded out.The natural soil has its maximum void ratio for the loosest

condition. For the sand considered here, which has a certainparticle-size distribution and particle shape, the structurecannot exist beyond its maximum void ratio. The maximumvoid ratio may be determined experimentally using theAmerican Society for Testing and Material (ASTM) standardD4252 (ASTM, 2002). For most specimens of natural soil, themaximum void ratio is always in the range of 0.7–0.8(corresponding to porosity of 0.41–0.44). This paper adoptsthe maximum void ratio as the critical void ratio, which meanscritical porosity nc approximately equals 0.45 (correspondingto a critical void ratio of 0.82). When the porosity of the pipetip is beyond nc due to erosion, it is thought that the soilstructure in the pipe tip has already broken, which meansthe pipe tip has changed from the seepage domain to the pipeflow domain. The erosion in the pipe wall is not considered inthis paper. Therefore, the diameter of the eroded pipe is aconstant.The eroded particles are transported in the pipe flow, which

affects the rate of pipe progression. Although the concept ofthe sediment transport capacity has been widely established foropen channel flows and overland flows, there are only a fewworks addressing the transport due to seepage flows (Fox andWilson, 2010). The most universally accepted erosion equationis the excess shear stress equation, but some researchers haveused equations that are based on the excess hydraulic gradient,seepage velocity, or flow rate (Fox and Wilson, 2010; Midgleyet al., 2012).

Es ¼ K2 τ�τcð Þ ¼ K3 u�ucð Þ ð13Þwhere Es is the erosion rate (soil discharge per unit area); K2 andK3 are erosion coefficients, τ is the seepage shear stress, u is theseepage velocity,=ki/n, i is the local hydraulic gradient, and τcand uc are the corresponding critical values. Considering Darcy’slaw, i.e., u=ki/n, it can be seen that the above two equations forthe erosion rate are convertible to each other. Coefficients K2 andK3 are difficult to measure directly. Midgley et al. (2012) andFox et al. (2007) reported a range of 0.027–0.65 m3/m3 for K3 in





RtWater flow

Tip of channel

Additional nodes

Fig. 4. Addition of nodes near tip of erosion channel (elevation view).


their experiments (K3¼70–1700 g/L in the original literaturewhere Es is defined as the mass discharge. Here, the value for K3

was re-calculated using a density of 2600 kg/m3 for the soilparticles). In this paper, we adopt K3¼0.3.

For a given time (i.e., Δt), the length that the pipe progressesalong the piping direction, which also means the erodibleparticles are transported away, ΔL, may be estimated as

ΔL¼ Es � ΔtPf dcð Þ � 1�nð Þ ¼

K3 � u�ucð ÞPf dcð Þ � 1�nð Þ � Δt ð14Þ

where Pf is a portion of the erodible particles in the sizegradation curve by weight, and Eq. (13) has been involved.

2.4. General description of the scheme

According to the variational principles, the FEM scheme forsolving Eq. (2a–c) is

K½ � Hf gþ Ff g ¼ 0: ð15Þwhere

K½ � ¼∬Ω B½ �T k½ � B½ �dΩ; ð16aÞ

Kij ¼∬Ω kx∂ni∂x

∂nj∂x

þky∂ni∂y

∂nj∂y

� �dΩ; ð16bÞ

Fi ¼ZΓ2

qnidΓ; ð16cÞ

and

k½ � ¼kx 0

0 ky

" #; ð16dÞ

B½ � ¼∂n1∂x

∂n2∂x ⋯ ∂nm

∂x∂n1∂y

∂n2∂y ⋯ ∂nm

∂y

24

35; ð16eÞ

where ni (i¼1, 2, 3, …, m) is the shape function.Since the boundary changes with the development of the

erodible pipe, the element-free Galerkin (EFG) method (Nayroleset al., 1992; Belytschko et al., 1994, 1995; Lu et al., 1994; Liet al., 2003) is employed to facilitate the repartition of the domain.In the EFG method, the shape function, ni, is calculated using themoving least squares (MLS). For a given domain function, u(X),the MLS interpolation may be written as

Gu Xð Þ ¼ ∑n

i¼1ni Xð Þui; ð17Þ

where X=(x, y)T is the node in the domain, u(Xi)=ui, (i=1, 2, …,n) for the given n nodes, ni(X) is the value of the shape function ofnode i at X, and Gu(X) denotes the approximation of function u(X).Herein (Li et al., 2003),

ni Xð Þ ¼ wi Xð Þ ∑m

j ¼ 1cji Xð Þ; ð18aÞ

cji Xð Þ ¼ pj X;Xð Þpj Xi;Xð Þbj Xð Þ ; ð18bÞ


bj Xð Þ ¼ ∑n

i ¼ 1wi Xð Þp2j Xi;Xð Þ; ð18cÞ

where wi(X) is the weight function and pj is the orthogonal basisfunction. The usage of orthogonal basis functions facilitates thereduction in computational costs and improves the accuracy of theinterpolation.The weight function here follows Kou’s proposal (Li et al.,

2003), namely,

wi ¼r2mi

r2i þ r2mi1� r2i

r2mi

� �4riormið Þ

0 othersð Þ;

8<: ð19Þ

where ri is the distance from evaluation point X to node i, andrmi is the influence radius of node i. Zhou and Kou (1998) hasproposed an estimated equation for it.

rmi ¼ffiffiffiffiffiffiαn

πc

rð20Þ

If the computation uses the linear basis functions, n¼3. c isthe density of the nodes. α is a coefficient whose value rangesfrom 4 to 6.More nodes near the pipe tip are required at the beginning of

each step to improve precision. This operation is relativelyeasy when employing the EFG method instead of the routineFEM, since the nodes can be conveniently regenerated withoutmodifying the original integration mesh. We add new nodes tothe orbicular radicalized distribution around the pipe tip, asshown in Fig. 4. The node density in this area is three times theremainder of the domain, which meets the requirements of thenumerical precision. Meanwhile, the radius Rt of this area isalso an important factor affecting both the numerical precisionand the computational costs. Here, we set radius Rt as

Rt ¼ β � rm; ð21Þwhere rm is the influence radius of the node at the pipe tip (seealso Eq. (20)) and β is an experiential coefficient controllingthe density of the node distribution. Based on our experience, avalue from 0.5 to 1.5 for β proves effective and efficient. Thefurther increase in its value may add more nodes thannecessary and cause additional computational costs.Since the flow rate obtained by the two-dimensional analysis

is not in accordance with the actual rate in three dimensions, aconversion coefficient, K3D, is introduced to revise the compu-tational result for the total flow rate. The flow rate at node i inthe erosion channel is thought to be ~Qi ¼ K3DQi, where Qi isthe two-dimensional computational flow rate at node i.






The EFG method uses independent background meshes toconduct the numerical integration in Eq. (16a–e). Theinterpolating nodes are irrelevant to the meshes; therefore,they can be added, removed or changed conveniently withoutmodifying the integration meshes. This advantage makes itfeasible to deal with problems that have changeable bound-aries, such as the development of cracking, seepage with afree surface, and piping (Belytschko et al., 1994, 1995;Li et al., 2003; Zhou et al., 2012). The EFG method mayhave the disadvantage that the shape function lacks the deltafunction property, which complicates the treatment of theboundary with a given head (Γ1). However, according toHegen (1996), the finite element method can be considered asa special approach of the EFG method. Following Li et al.(2003), the Γ1 boundary conditions are treated as the finiteelement method. Li et al. (2003) have shown that the EFGmethod is appropriate for calculating the water head as wellas the flow rate of the seepage flow.

The erodible pipe and the seepage domain share the samewater head and have an equal value for the flow rate on theirmutual boundary. The coupling of the two domains is achievedas follows (Zhou et al., 2012):

100
(1)
80

PlFo

A section of the erodible pipe is pre-assigned. With amutual boundary and water head H¼H1 (we can simplymake it equal to the downstream water level), seepage flowdischarge q at the boundary can be computed using theEFG method.

60

ner
(2)
40% F

i

The distribution of water head H2 is computed according toEq. (1a,b) with the boundary condition of seepage flowdischarge q.

(3)

0

20

H1 and H2 are compared. If the error between H1 and H2

(ε¼ jH2�H1j=H1) is larger than tolerance value εc, H1 isreplaced with H0

1 ¼ ðH1þH2Þ=2 and the process is startedagain at Step 1.

0.0010.010.1110
(4) Diameter (mm)
Fig. 6. Particle-size distribution of soil material used in model test.

Procedures 1–3 are repeated until the stop criterion hasbeen met. Then, the computation moves to the nexttime step.

Fig. 5. Sketch of phy

ease cite this article as: Wang, D.-y., et al., Simulation of pipe progression inundations (2014), http://dx.doi.org/10.1016/j.sandf.2014.09.003

3. Model validation

In this section, the above model is tested against the resultsof a physical experiment performed by Zhou et al. (2007b,2012). The physical model was 230 cm long, 60 cm high, and80 cm wide. The levee and the impervious surface layer weremodeled with acrylic glass (see Fig. 5). An exit waspreinstalled by opening a hole in the acrylic glass. The distancefrom the exit to the upstream boundary was 145 cm. Tests fortwo cases, i.e., with and without a barrier, were carried out. Forthe case with a barrier, an acrylic glass barrier with a depth of6 cm was located 110 cm from the upstream, and it wassuspended in the subsoil. The size gradation curve of the soil isshown in Fig. 6. The coefficient of uniformity is Cu¼3.1.The computational domain is partitioned into triangular cells

for the integration, and seven Gaussian points are adopted in eachtriangle cell, which satisfies both the accuracy and the computa-tional cost requirements. The hydraulic conductivity of the soil is9.5� 10�3 cm/s in the horizontal direction and 3.2� 10�3 cm/sin the vertical direction; the diameter of the erodible pipe isassumed to be 0.5 cm; the other parameters are Cp¼1.35,Cs¼0.75, f¼0.5, and K3D¼0.7. The error tolerance for each

sical model test.






iteration here is set to εc¼0.001. The permeability coefficient ofthe acrylic glass is set to 1/1000 of that of the surrounding soil,which helps avoid singularity in the computation.

Fig. 7. Piping progression under water head H¼28.3 cm and 51 cm in casewithout a cutoff wall.

Fig. 8. Water head distribution and pipe progression un

Fig. 9. Water head distribution and pipe progression

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

3.00E-03

0

Prog

ress

ed r

ate

of p

ipe

tip: m

/s

Distance fro0.2 0.4 0.6

Fig. 10. Progressed rate of pipe tip under water h


For the case without a barrier, the simulation shows that ifthe water head exceeds the critical value, piping erosion willoccur and continue to progress until it reaches the upstreamboundary. The calculated critical water head Hc for theerodible pipe to break through the foundation is 28.3 cm,while the test result was 27.9 cm. As shown in Fig. 7, the pipeadvances increasingly fast as it approaches the upstream.Moreover, the higher the water head, the less time consumedin breaking through. When H¼28.3 cm, the computationalbreak-through time is 26 min, which conforms to the modeltests. When H¼51 cm, the break-through time is reduced to16 min.Fig. 8 presents the pipe progression before it breaks through

the foundation. It can be seen that the pipe progresses alongthe interface of the two layers.For the case with a barrier, according to the simulation, when the

upstream water head is small, piping will cease before the cut-offwall. If the water head exceeds the critical value, the erodible pipe

der water head H¼28.3 cm (without cut-off wall).

under water head H¼51 cm (with cut-off wall).

m upstream: m0.8 1 1.2 1.4

outline of upperboundary

vtip

ead H¼51 cm in case of cut-off wall.





Fig. 11. Cross section of real levee foundation.

Fig. 12. Particle-size-distribution of silty sand.

Fig. 13. Piping progression under water head Hup¼6 m and 3 m.


will extend over the cut-off wall and then reach the upstreamboundary (see Fig. 9). The calculated critical water head is 51 cm,while the physical model test result was 52.9 cm, and the resultcalculated by Zhou et al. (2012) was 46 cm. Fig. 10 presents theprogressed rate of the pipe tip under water head H¼51 cm. It isobvious that as the pipe tip approaches the cut-off wall, theprogressed rate decreases and reaches the minimum value when thepipe goes down to extend over the wall.

4. Application to real levee foundation

In this section, the numerical model is applied to a real leveefoundation to demonstrate its potential capability in practice.Fig. 11 shows the cross section of the levee foundation.According to the geotechnical investigation and a field survey,the levee body is constructed of silt and silty clay. Theoverlying soil of the levee foundation is silty clay and theunderlying layer is silty sand. The hydraulic conductivity ofthe silty clay is 3.7� 10�5 cm/s, while the hydraulic con-ductivity of the silty sand is 6.7� 10�3 cm/s. The sizegradation curve for the silty sand is shown in Fig. 12. Thecoefficient of uniformity is Cu¼2.9.

Since most of the information on piping in levee foundationsin China has been described in a qualitative rather thanquantitative manner, it is difficult to determine the shape ofthe erosion channel and other related parameters. Therefore,the calculation parameters here are determined according to ananalysis of the model test results and trial calculations. Theparameters are de¼10 cm, Cp¼1.35, Cs¼0.75, and f¼0.5.The error tolerance for the iteration is εc¼0.005.


4.1. Effect of water head

A numerical analysis was carried out to investigate the effectof the upstream water head on the channel development, wherethe preassigned exit is located at L¼0 m from the downstreamlevee toe (i.e., x¼200 m), while the upstream water head Hup

is set to 3 and 6 m. The time to failure under each water-headis presented in Fig. 13 (without a cut-off wall). The resultsindicate that the greater the water head, the faster the erodiblepipe progressed. The total times to failure under these twowater heads are 14 h and 8 h, respectively, which coincideswith the fact that the levee here would break in several hoursonce piping erosion in the foundation had occurred.

4.2. Effect of the location of the preassigned exit

The upstream water head is set to Hup¼2 m in thecalculation, while the location of the preassigned exit isassumed to be L¼0, 50, and 100 m from the upstream leveetoe; i.e., the abscissa are x¼200 m, x¼250 m, and x¼300 m,respectively. The simulation shows that when L¼0 m (x¼200m), the erosion channel continues to develop until the break-through occurs (Fig. 14), indicating that Hup¼2 m is largerthan or equal to the critical head. However, under the sameHup, the cases of the preassigned exit, located 100 m and150 m, respectively, from the downstream levee toe, are safe.

4.3. Effect of the suspended cut-off wall

For levees on pervious foundation soil, the traditional wayto prevent piping in China is to construct a cut-off wall.Obviously, piping can be completely prevented if the cut-offwall penetrates into the impervious underlying layer or the





Fig. 14. Water head contours before pipe break throughs the foundation in case of H¼2 m.

Fig. 15. Relationship between critical water head and depth of cut-off wall.

Fig. 16. Time consumption with piping progression under water head H¼5 min case of 5-m-deep cutoff wall.

0

10

20

30

40

50

60

70

80

90

100

0.0010.010.1110

The original soil

PSD-1

PSD-2

Diameter (mm)

% F

iner

Fig. 17. Variation in particle-size distribution of foundation soil.


bedrock. However, to reach the impervious layer or bedrock,the wall will often be too expensive and need to be much toodeep. Additionally, inland inundation and salinization may beinduced if the groundwater cannot communicate with the riverin low-water periods. Therefore, a suspended cut-off wall ismore desirable; i.e., the cut-off wall is simply constructed for acertain depth into the pervious layer so as not to totally blockthe seepage of groundwater, but to increase the critical head forthe pipe progression to an acceptable level. As a consequence,the determination of a suitable depth for the cut-off wall isimportant.

Generally, the more upstream the exit of the erosion channel islocated, the greater the risk of the pipe progression. Therefore, thefollowing analyses are carried out when the preassigned exit islocated at the downstream levee toe; i.e., L¼0 m. In thecalculations, the location of the exit remains unchanged, whilethe depth of the cut-off wall varies from 2.5 to 12.5 m. The cut-off wall is located in x¼180 m. The resulting relationship of thedepth of the cut-off wall and critical water head Hc is presented inFig. 15. In the case of a 5-m-deep cut-off wall, the critical waterhead is 5 m. The break-through time in this case is 14.8 h, and thetime consumed in getting over the cut-off wall is about 4 h (seeFig. 16).

As shown in Fig. 15, it is obvious that the deeper the cut-offwall, the greater the Hc required to trigger the pipe progression.However, as the depth of the cut-off wall increases, the increasingrate of the corresponding critical water head declines, whichindicates that the cut-off wall has an optimized depth.

This result is helpful in practice for determining the criticaldepth of the cut-off wall given an upstream water head; i.e., ifthe depth of the cut-off wall is less than the critical value, thepiping may run through the levee foundation and the levee willbe in danger. For the levee here, the maximum upstream waterhead is 6 m, so the levee’s safety can be guaranteed as long asthe depth of the cut-off wall is greater than 6–7 m, according toFig. 15. The design depth of a suspended cut-off wall isgenerally (1.0–1.5) Hup in this area according to experience.Thus, the experiential critical depth of the wall is 6–9 m, whichcoincides with the calculation result.

It should be noted that the design depth is also affected bythe gradation of the foundation soil. Fig. 17 shows the othertwo soil gradation curves that were calculated. The grain sizeof PSD-1 is coarser than that of the original soil, while that ofPSD-2 is finer. Fig. 18 shows that as the effective grain sizeincreases, the critical head increases for the same depth of thecut-off wall. This implies that the local empirical formula onlyapplies to the local levee foundation with a specific particle–size distribution of the soil, and is unlikely to be suitable forother regions with different foundation soils.


5. Conclusions

This paper has simulated the pipe progression in a leveefoundation by coupling the seepage domain and the pipe flowdomain. The method has been found to be more appropriatethan the ordinary seepage analysis method, in which the





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10111213

2 3 4 5 6 7 8 9 10 11 12 13

Crit

ical

wat

er h

ead:

m

Depth of cut-off wall: m

The original soil

PSD-1

PSD-2

Fig. 18. Critical water head Hc vs. depth of cut-off wall for different particle-size distributions of foundation soil.


permeability coefficient of the piping zone is simply increased,while being easier to apply than the discrete element method.Although the present simulation method is primitive, it obtainstolerably good results.

This paper has proposed a mechanism for the inception ofsoil particles in porous media, which was employed in thenumerical model and was able to obtain results that coincidewith those of the physical model. Additionally, the numericalmodel gave an appropriate prediction of the break-throughtime to the real situation.

It is known very well that the presence of a suspended cut-off wall provides a positive effect owing to the increase in theseepage path. In addition, the existence of a cut-off wall alsoforced the pipe to change its direction to one in which theparticle inception is relatively harder.

Generally, the proposed model is simple and potentiallypractical for application to real levee foundations. However,the model does not consider the expansion of the pipe diameterbrought about by erosion on the pipe wall. Furthermore, anextension to the three-dimensional analysis will indeedimprove the behavior of the present model, and this wouldbe a desirable topic for further research.

Acknowledgments

Support from the Natural Science Foundation of China(51039003), National Basic Research Program of China (973Program 2013CB036402), and State Key Laboratory of Hydro-science and Engineering (2013-KY-4) is gratefully acknowledged.The valuable comments and suggestions of anonymous reviewersand the editor also contributed greatly to the improvement ofthis paper.

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simulation of pipe progression in a levee foundation with coupled seepage and pipe flow domains

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