CHEMICAL ENGINEERINGTRANSACTIONS 
 

VOL. 51, 2016 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors:Tichun Wang, Hongyang Zhang, Lei Tian 
Copyright © 2016, AIDIC Servizi S.r.l., 

ISBN978-88-95608-43-3; ISSN 2283-9216 

Discusion on Failure Mechanism of Piping 
Zhechao Fan*a,b, Jun Suna 
a Department of Civil Engineering, Chongqing Three Gorges University, Chongqing 404000, China 
b Key Laboratory of Hydraulic and Waterway Engineering of the Ministry of Education, Chongqing Jiaotong University, 
Chongqing 400074, China 
fanzhechao@163.com 

Supposing that the layer is consisted of two ideal elastic spheres with different diameters, the permeability 
coefficient of the layer can be calculated. After piping taking place and small grains being carried away, the 
permeability coefficient will change. The piping model is made to quantificationally calculate the permeability 
coefficient and hydraulic head of any point. The model explains the causes of piping and the development of 
concentrated passage, and it provides theoretical basis for prevention of dam piping. 

1. Introduction 

Dam break is mostly caused by permeability deformation, and slope sliding accidents are also caused by 
seepage, which are in the references of Evans et al.(2000), Liu (2012), Zhao et al. (2015), and Fread (1996). 
The dam of Yangtze River bursted in 1998, which were mostly caused by permeability deformation such as 
piping. The study of piping has been studied by many former, such as the literatures of Awal et al. (2012), 
Awal et al. (2008), and Meyer et al. (1994). But the damage cause of the layer which is caused by piping has 
been done little. The only way is to study the law of the grains moving in the layer after piping taking place. So 
the damage to the around layer which is caused by the piping can be studied further, and then reinforcing 
engineering for the dam can be guided. 
The mechanism of the seepage passage which is caused by piping has been studied qualitatively, and has 
been validated by Chen (2000). After the piping taking place, the groundwater well theory can be introduced to 
study the initial seepage field during piping. Piping can be approximately regarded as artesian well, and the 
river can be looked as half no-boundary with fixed water level. Steady flow theory or unsteady flow theory can 
be applied to study the distribution and the variety of the seepage field during the early period of piping. The 
complex interaction of the seepage water and soil determines the seepage time, which is a non-linear dynamic 
problem. The layer is predigested to be consisted of two ideal elastic spheres with different diameters, and the 
cohesion is ignore. The permeability coefficient can be equivalent as the average permeability coefficient of 
the layer consisting of two spheres with different diameters. So the variation rules of the permeability 
coefficient, critical surface, water grads, and so on can be calculated quantificationally after piping taking place.  

2. Model of permeability coefficient of the layer after piping taking place 

2.1 Permeability coefficient of the layer 
The actual layer is very complex, and the layer which may conduce piping is not well sorted. In order to 
establish the quantitative seepage model, we should start from simple model, and then induce the more 
complex sorted layer. Supposing the layer is consisted of two spheres with different diameters, and the small 
particles are all taken out of the big particles, the dam will be damaged because of the side force. We mainly 
discuss the form of the large particles liable to move while the small particles are carried away. 
The permeability coefficient K of the layer which is consisted of two kinds of grains can be shown: 

1 1 2 2
K J K J A K J A           

Two sides of the equation can be simplified: 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1651176

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Fan Z.C., Sun J., 2016, Discusion on failure mechanism of piping, Chemical Engineering Transactions, 51, 1051-
1056  DOI:10.3303/CET1651176   

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1 1 2 2
K K A K A                                                                                                                                               (1) 

Where K1 is the permeability coefficient of the layer consisted of large grains; K2 is the permeability coefficient 
of the layer consisted of small grains; A1 is the area ratio of the large grains to the layer; A2 is the area ratio of 
the small grains to the layer. Mao (2003) gave the expression of the permeability coefficient of the layer 
consisted of one kind of grain. 

2
2

(1 )

w
n

K d
n

 

 





                                                                                                                                              (2) 

Where β is form coefficient (β=π/6 for sphere); n is porosity; λ is the effect coefficient of of the adjoining 
grains; d is diameter of the sphere; γw is the density of water; μ is coefficient of viscosity of the fluid. 
When the porosity of large particles is filled with small particles, permeability coefficient is determined by the 
area ratio of small particles to the layer. When small particles are all carried away, permeability coefficient is 
determined by the area ratio of large particles to the layer. So permeability coefficient in any cell in the layer is 
determined by part of large particles and small particles, shown in Figure 1. 

 

Figure 1:  Equivalent permeability coefficient after small grains carried away 

When the overlay is penetrated by the hydraulic water after the piping taking place, the static hydraulic 
pressure releases, water effusing from the piping well. The streamlines and the equipotential lines around the 
piping well redistribute. We can use the theory about steady flowing of a well in confined aquifer to 
approximately determine the streamlines. We can regard the piping as artesian well. The length between the 
artesian well and the replenishment river is a, the discharge Q. During the process of piping, the hydraulic 
head of the replenishment boundary doesn’t change. We can replace the recharging boundary with a 
recharging image well. The river is regarded as recharging source, flow in the early period of the piping can be 
considered as laminar flow. The y coordination is parallel to the dam, and x coordination is through piping well 
and perpendicularity to y coordination. By the method of images applied to the well, Liu (2001) expressed the 
drawdown S at any point (x,y): 

2 2

2 2

( )
ln

2 ( )

x a yQ
S

T x a y

 


 
                                                                                                                                    (3) 

Where T is the coefficient of transmissibility. 

Let 
2 2

2 2

( ) 4
exp( )

( )

x a y TS
c

x a y Q

 
 

 
, so the following equation can be obtained: 

2 2
2 2

2

(1 )
( )

1 (1 )

c a c a
x y

c c


  

 
                                                                                                                                (4) 

Supposing different values to S, we can divide the flow field according to the values of S. 
First the discharge of the early period of the piping was calculated, and then overall discharge can be 
assigned to any cell. In the period time of △t, the permeability coefficient doesn’t change, and the permeability 
coefficient in the cell is constant. We can calculate flow velocity and actual velocity by permeability coefficient, 
hydraulic gradient and so on. If the flow velocity can carry small particles to move, small particles can be 
carried out of the cell. Supposing that the capacity of small grains transport relies on velocity in the porous 
medium, and the volume of the small grains transport from the no.i cell is △Vi, from Eq(1) the following 
equation can be expressed: 

1 2
(1 )i i

V t V t
K K K

S L M S L M


   
     

     
                                                                                       (5) 

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Where △S is the length of the cell; △L is the width of the cell; M is thickness of the layer. The discharge of the 
piping Q can be obtained according to water level of the river and ground elevation around the piping after 
piping taking place, and hydraulic loss △h can be calculated as follows: 

2
h C Q                                                                                                                                                           (6) 

Where C is well loss constant, and can be determined by experiment. When water is confused flow, hydraulic 
loss will be direct ratio to square of discharge. 
From H=H0-△h, hydraulic head of the piping well can be calculated, where H0 is static hydraulic head. We can 
calculate the initial discharge of the piping. 

0
2 ( )

2
ln( )

w

K H H M
Q

a

r

 


                                                                                                                                           (7) 

Where rw is radius of piping well, M is the thickness of the aquifer. 
In the early period of the piping the flow quality through each cell is equality. We take the radial i cells for 
example, shown in Figure 2. Flow quantity through this cell is q. There is no recharge from the adjoining cells, 
q1=q2=q. Lu (2006) and Chen et al. (1999) calculated the hydraulic gradient of the seepage passage in the 
early period by the image method. 

2 2 2 2 2 2

2 2

2 2 2 2

0

( ) 4
( , )

[( ) ][( ) ]
x y

a y x x yaQ
J x y J J

K M x a y x a y

  
  

   
                                                                                     (8) 

 

Figure 2:  The part of cell 

2.2 The permeability coefficient of the layer after piping taking place 
After piping taking place, hydraulic gradient near piping well is maximal. The critical hydraulic gradient can be 
calculated as follows: 

( 1)(1 )s
c

sk

d
J n

d





                                                                                                                                           (9) 

Where d is the diameter of the lost particle; dsk is equivalent diameter; γs is unit weight of soil grains; γ is unit 
weight of soil; α is shape coefficient of soil grains. When the velocity reaches starting velocity of the small 

grains, small grains will move, the starting velocity of small grains can be calculated as follows: 
2

0BV MV N                                                                                                                                               (10) 

Where 
6 (1 )

sk

Av n
M

d


 , 3

1
( 1)

6

S
s

N n gd





  . For velocity of laminar flow’s is small, B=0, so 
N

V
M

 . 

After small grains move, the quality of moving grains in each cell is different. Cell is nearer to the piping well, 
the more small grains are carried away. We can use the quality model to work out reduced small grains in the 
cell, and then calculate the permeability coefficient. In Figure 2, considering the no. i cell, in the period of △t 
the decrement of small grains in the cell can be calculated as follows: 

1i i i
V V V


                                                                                                                                                         (11) 

Where Vt is the volume of small grains carried away from i cell, Vi-1 is the volume of small grains carried into i 
cell. 
So we can calculate the small grains decrement in each cell in period of △t according to flow velocity. From 
Eq(1) and (2) we also can calculate permeability coefficient in each cell. For the permeability coefficient in 
each cell has changed, we can’t use Eq(7) or Eq(8) to calculate the discharge and hydraulic gradient in next 
△t. To calculate the discharge, we should calculate the radial equivalent permeability coefficient, and then 
sum up each permeability coefficient in each group. So the average permeability coefficient of the layer will be 

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got. The equivalent average permeability coefficient 
i

K  is used to replace the permeability coefficient of the 
layer whose permeability coefficient has changed. Considering one group in radial, the unit width discharge 
through the cell is qi:  

i i
i i

i

L
q K

S

 
 


                                                                                                                                            (12) 

Where △Hi is hydraulic head difference of the no.i cell; △Li is average length of no. i cell along equipotential 
line; △Si is average length of the seepage line. The difference of hydraulic head △Hi of the cell can be 
calculated: 

i i
i

i i

q S
H

K L


 


                                                                                                                                                  (13) 

For the discharge along the streamline through each cell is the same, the difference of hydraulic head 
 equals to the sum of the differences of hydraulic head of each cell: 

i
                                                                                                                                                        (14) 

Supposing the average permeability coefficient along the streamline is iK , so the discharge can be expressed 
approximately as follows: 

0
2 ( )

2
ln( )

w

K H H T
q

a
m

r

 



                                                                                                                                           (15) 

Where m is the number of cells around the circle. 
From Eq(12), (13)，average permeability coefficient 

i
K  can be expressed as follows: 

i

i
i

i

i i

S

L
K

S

K L








 




                                                                                                                                               (16) 

The average coefficient of the next streamline can be got using the same method, and then the total average 
permeability coefficient K in the field can be calculated.  
Supposing the average permeability coefficient along the no. i streamline is 

i
K , and discharge is qi, so 

i
i i i

i

q K L
S


  


                                                                                                                                           (17) 

The total discharge in the seepage field is Q. 

i
Q q K L

S


   


                                                                                                                                    (18) 

From Eq(15), (16) and 
1 2 i

        , the total average permeability coefficient K can be got: 

i
i

i

L
K

S
K

L

S











                                                                                                                                                (19) 

Supposing the hydraulic head of the river is constant, the second hydraulic head can be got by 
, 1, ,i t i t i t t

H H H
 

  , where hydraulic head has included kinematic hydraulic head. By this way, hydraulic 

head and permeability coefficient of each point in the field can be studied, and the total discharge can be 

calculated. The discharge of each cell qi can be distributed by the total discharge, 
i

i

K
q Q

K
  . So the iterative 

calculate can be performed, and the new discharge after the period of △t can be calculated. 

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3. Case Study 

Supposing the layer is consisted of two different diameters, the diameter of the large grain is 25mm, the 
diameter of small grain of 0.6mm. The permeability coefficient of the large grains is 38cm/s, and the 
permeability coefficient of the small grains is 0.019cm/s. The water head of the river is 150m. The ground 
elevation around the piping well is 145m, and the length between the piping well and the river is 10m. The 
radius of the piping well is 0.5m.  
3.1 Hydraulic head near the piping well after piping taking place 

The difference of hydraulic head before and after piping in the model can be calculated. Before piping taking 
place permeability coefficient in the layer is the same, and the distribution of hydraulic head is shown in Figure 
3(a). Hydraulic gradient near the river changes more significantly. With the development of piping, more and 
more small grains is carried away, permeability coefficient of the layer changes. The nearer the point to the 
piping is, the larger permeability coefficient is. The distribution of hydraulic head is shown in Figure 3(b). From 
Figure 3(b), hydraulic head near the piping is larger than before, and permeability coefficient near the river is 
significantly larger. When the discharge is steady after piping taking place, the distribution of hydraulic head is 
shown in Figure 3(c). Small grains near the river are all carried out, and form the piping passage eventually.  

 
 (a)                                        (b)                                             (c) 

Figure 3: Distribution of hydraulic head in different process of the piping  

(a) initial period of piping   (b) during the process of piping (c) the later period of the piping 

3.2 Relation among piping, permeability coefficient and flow velocity 

By the calculation, we find that there is no rigorous limit among piping and permeability coefficient. For 
cohesionless soil when the overlying is penetrated, small grains are all able to be carried away and piping well 
forms. When K1=36cm/s (D=25mm), K2=0.019cm/s (d=0.6mm), and a=10m, the permeability coefficient curve 
can be got, shown in Figure 4(a). The difference of permeability coefficient in the zone is large. The zone of 
gushing grains is like a circle whose center is piping well. There is a gap beside the river. Permeability 
coefficient ranges from 2.0-4.0cm/s. From Figure 4 we can find that there is an obvious critical surface. When 
hydraulic head in the piping well is rising during the development of the piping, hydraulic gradient descend, 
and the critical surface shrinks, shown in Figure 4(b). The gushing grain mostly come from the dam side. 
Permeability coefficient in the new critical surface is larger than before, and it descends toward to the dam. 
The piping passage forms eventually, shown in Figure 4(c).  

 
   (a)                                            (b)                                        (c) 

Figure 4: Distribution of permeability coefficient in different process of the piping  

(a) initial period of piping   (b) during the medium process of piping  (c) the period after the piping passage 

forms 

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4. Conclusion 

Piping is a random phenomenon, and it is affected by grain sort, grain composition, difference hydraulic head, 
etc. Permeability coefficient of the layer is determined mostly by small grains before piping occurring. Once 
piping happens, and small grains will be carried away, permeability coefficient of the layer will be asymmetry. 
Permeability coefficient of the layer becomes to be determined by large grains, and dangerous piping passage 
may eventually form. In this paper, the layer is supposed consisted of two spheres with different diameters. 
The flow field is be divided into cells under the assumption of the layer is homogenous. When the small grains 
are carried away, permeability coefficient becomes asymmetry. For the cell is little, permeability coefficient of 
the cell is constant. By the Darcy law, we can calculate the discharge, hydraulic head, etc. Though the model 
studies the two ideal sphere, it can be applied to actual stratum. This model has taken specific point in flow 
field into account, and so it reflects that absolute passage has effection on piping. From the analysis of this 
model we can conclude that piping is liable to happen in the normal direction to the dam. 

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