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Discrete numerical modeling of particle transport in granular filters
Computers and Geotechnics 47 (2013) 48–56
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Discrete numerical modeling of particle transport in granular filters Yu-Hua Zou a, Qun Chen b,⇑, Xiao-Qing Chen a, Peng Cui a a b
Key Laboratory of Mountain Hazards and Earth Surface Processes, Institute of Mountain Hazards and Environment, CAS, Chengdu 610041, China State Key Laboratory of Hydraulics and Mountain River Engineering, College of Hydraulic and Hydropower Engineering, Sichuan University, Chengdu 610065, China
a r t i c l e
i n f o
Article history: Received 15 December 2011 Received in revised form 5 June 2012 Accepted 6 June 2012 Available online 3 August 2012 Keywords: Internal erosion Base soil Filter Particle flow method Eroded mass Intruding depth
a b s t r a c t Granular filters are an essential component in earth dams to protect the dam core from seepage erosion. This paper uses the particle flow method (PFM) to study the mechanism of particle transport in a base soil–filter system. The distributions of the eroded base-soil particles in different filters are traced and analyzed. The eroded mass and intruding depth of the eroded particles into the filters are obtained under different times and hydraulic gradients. The simulation results show that the eroded mass and intruding depth of the base-soil particles into the filter are related to the representative particle size ratio of the base soil to the filter, hydraulic gradient and erosion time. The numerical predictions are also compared with the empirical filter design criterion. The results show that the particle flow model provides an effective approach for studying the filtration micro-property and the erosion mechanism in a base soil–filter system, which is useful for filter design. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Internal erosion of core soil is a key safety problem that threatens earth dams. Filters with appropriate properties placed downstream of the core have proven effective for preventing the progression of core soil erosion [1]. Empirical methods based on the representative particle-size ratio (the ratio of the characteristic particle sizes of the filter and base soil) have been commonly used to design granular filters [2–6]. These empirical criteria have been developed based on extensive laboratory tests, but they cannot describe the interface behavior mechanism and time-dependent processes that occur in the filter material. The interface behavior mechanism is extremely important because it can provide designers with detailed insight on what may occur throughout the design life of the structure [4]. Several analytical models have been established to describe the mechanism behind the movement of fine particles in the base soil through granular filters. Silveira [7] used probabilistic methods to examine the migration of base-soil particles through filters. An estimated infiltration depth through granular filters was obtained based on the observation that a particle could transport through a pore if the particle was smaller than the pore constriction [4,8]. Witt [9] established a three-dimensional filter void network model. Schuler [10] used Monte Carlo methods to study the infiltration depth of base-soil particles into filters based on a similar void net-
work. Indraratna et al. [11–13] proposed an analytical model that used finite-difference procedures, which was based on the conservation of mass and momentum to model particle movement, considering the number of elements at the base soil–filter interface. Compared the probabilistic methods with the mathematical model used in previous filtration studies, the particle flow method allows for the process of fine-grained particles eroding through a filter to be directly observed and recorded, providing a convenient approach to explain the fundamental mechanisms of particle erosion, transport and retention. Based on a particle flow numerical model, this paper traces and analyzes the distribution of eroded base-soil particles in a filter. The erosion ratio and intruding depth of the eroded particles into the filters at different times and under different hydraulic gradients are studied. The prediction results of the effective filters obtained from the numerical model and empirical filter criterion are then compared. 2. Basic modeling principles 2.1. Particle motion and interaction The particle flow method used in this paper solves the Newtonian equations of motion simultaneously for every particle in the system [14]: *
⇑ Corresponding author. Tel.: +86 28 85403351; fax: +86 28 85405604. E-mail address: [email protected] (Q. Chen). 0266-352X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compgeo.2012.06.002
*
@ u f mech * ¼ þg @t m
ð1Þ
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Y.-H. Zou et al. / Computers and Geotechnics 47 (2013) 48–56
(a) Particle-wall, particle-particle interaction
(b) Contact model
Fig. 1. Schematic diagram of the PFM used in the paper.
*
*
@x M ¼ @t I
ð2Þ *
*
where u is the particle velocity, f mech is the sum of additional forces (externally applied and contact forces) acting on the particle, m is * * the particle mass, g is * the acceleration due to gravity, x is the particle angular velocity, M is the moment acting on the particle, and I is the moment of inertia. As Fig. 1a shows, the wall boundary velocity x_ w causes displacement and overlap with particle a by dw a , which imparts force F w a and acceleration to particle a. The resolved velocity x_ a and subsequent displacement of particle a causes an overlap with particle b by da b , imparting contact force F a b with both normal and shear components. The resultant force, which includes the gravitational force, results in the linear and angular acceleration of both particles, and these accelerations are integrated to determine the * * instantaneous velocities x_ a , x_ b , h_ a and h_ b ðu and x in Eqs. (1) and (2), respectively). The mechanical contact model in the particle flow method (PFM) contains normal and tangential contact components, as shown in Fig. 1b. In the figure, ‘‘m1’’ and ‘‘m2’’ denote two particles. A normal contact resists the normal force, and a tangential contact resists the shear force. Both contacts include a spring (with normal stiffness Kn and shear stiffness Ks, respectively), which reflects the elastic behavior of the contact, and a dashpot (with normal viscosity gn and shear viscosity gs, respectively), which reflects energy dissipation and quasi-static deformations [15–17]. According to the linear spring-dashpot model, the spring and damping forces determine the normal contact force, and the tangential and damping forces give the tangential contact force [18,19]: *
*
*
*
qf
@ðn v Þ þ qf @t
*
*
*
*
v rðn v Þ ¼ nrp þ lr2 ðn v Þ þ f b
ð5Þ
*
* @ðn v Þ þ rðn v Þ ¼ 0 @t
ð6Þ
where qf is the density of the fluid, n is the porosity of the porous media, p is the fluid pressure, l is the dynamic viscosity of the fluid, * and f b is the body force per unit volume. The fluid velocity is de* noted as v in porous flow. It is the interstitial velocity, and the * quantity n v is the macroscopic or Darcy velocity. The simulation domain is discretized in elements which include many particles. The staggered grid scheme is used to solve the fluid velocities and pressures [21]. 2.2.2. Fluid-particle interaction force The preceding equations give the body force experienced by the fluid as a result of the moving particles. A force equal and opposite is distributed on the discrete particles for each fluid element. The * body force f b is applied to each particle proportional to the volume of the particle. If the particles are assumed to be spherical, the drag force applied to an individual discrete particle [14] is: * *
f drag ¼
4 3 fb pr 3 ð1 nÞ
ð7Þ
*
F n ¼ K n d n gn V rn
*
generalized form of the Navier–Stokes equation. In the PFM, it is modified to include the effect of a particulate solid phase mixed into the fluid [14]. The average effects expressed with the porosity * n and a coupling force f b over many particles (as opposed to attempting to model the details of the fluid flow between particles), can be characterized as [20]:
ð3Þ
where r is the radius of the particle. The total force applied by the fluid on the particle is the sum of the drag and buoyancy forces [14]:
ð4Þ
*
*
F s ¼ K s d s gs V rs
where Kn and Ks are the spring constant and shear stiffness, respectively; dn and ds are the normal and tangential displacements of invading points, respectively; gn and gs are the viscous damping * in the normal and tangential directions, respectively; and V rn and * r V s are the velocity in the normal direction and the sliding velocity in the tangential direction, respectively.
*
f fluid ¼ f drag þ
4 3 * pr q f g 3
ð8Þ
2.2.3. Particle motion in fluid Considering the motion of particles in fluid, Eq. (1) can be extended with an additional forcing term to account for interaction with the fluid [14]: *
*
*
2.2. Fluid–particle interaction
@ u f mech þ f fluid * ¼ þg @t m
2.2.1. Fluid motion or hydrodynamics equation The two-phase (fluid and solid) mass and momentum equations for the fluid velocities and pressures [20] can be considered a
To calculate the coupled fluid and particle motion, the first step is to mesh the simulation area to simulate a continuous fluid medium. The fluid boundary conditions must then be defined. Particle
ð9Þ
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Y.-H. Zou et al. / Computers and Geotechnics 47 (2013) 48–56
3 Filter Base soil
2 Wall boundary
z
5
y
x O(0, 0, 0) (a) The numerical model
Outflow section Inflow section
Fluid cell
z y
Wall
x
boundary
(b) Wall boundary and fluid cell Fig. 2. The numerical model for the filter test.
3. Numerical modeling for erosion 3.1. The numerical model The filter tests were modeled by cubic sample (0.12 m 0.06 m 0.06 m) within six walls (Fig. 2) in the Cartesian coordinate system. As Fig. 2a shows, the origin of coordinates (O(0, 0, 0)) is at on the front bottom left point of the model. The x-, yand z-directions of the coordinates are shown by three arrows pointing in three perpendicular directions (Fig. 2a and b). In the numerical model, the left (wall no. 4) and right walls (wall no. 2) simulate the permeable boundary for water inlet and outlet. Other four impermeable walls (1, 3, 5 and 6) are slip walls that particles can slip along the walls when the tangential force is greater than wall–particle friction. Spheres with specified size distributions were used to simulate particles of the base soil and filter. Gap-graded gravelly clay, which is higher in shear strength and lower in compression than clay, is increasingly used as a core in earth–rock fill dam, particularly in high earth–rock fill dams. The erosion of such a core with different
filters would be a useful reference for filter design. To simulate the gap-graded gravelly clay, four grain groups of spheres, marked r, s, t and u, were used to model particles having different sizes. The groups r, and t, respectively; simulated gravel, coarse sand and fine-sand particles, respectively, and the group u simulated clay particles. The white spheres on the right half of the model simulated the well-graded filter, which is traditionally used to protect different kinds of base soils. Both the base soil and filter particles occupy one half of the model to form a combined soil sample, shown in Fig. 2a. To investigate base-soil particle erosion in different combined samples, one base soil and four filters (F1, F2, F3 and F4) were used, and Fig. 3 shows their particle size distributions. The controlling particle size D15 (i.e., particle diameter at 15% passing) of the four filters gradually increases from F1 to F4.
100
Percent finer (%)
movement will also be a result of the differences in the water pressure. By solving the continuous fluid medium Navier–Stokes equation, the fluid pressure and flow velocity in each fluid grid are calculated. The fluid drag force moves the solid particles, which changes the porosity for different fluid cells; the porosity then affects the fluid flow rate of the grid. The continuing changing fluid flow drag force and porosity between particles ultimately cause the fluid–particle coupling balance to form.
Base F1 F2 F3 F4
80 60 40 20 0
1
10
100
Particle size (mm) Fig. 3. Particle size distribution of the base soil and the filters.
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Y.-H. Zou et al. / Computers and Geotechnics 47 (2013) 48–56 Table 1 Parameters used in numerical simulation for erosion. Parameters
Density (kg/m3) Normal stiffness, kn (N/m) Shear stiffness, ks (N/m) Friction coefficient Viscous coefficient (Pas) Parallel pond shear stiffness (N/m) Parallel pond normal stiffness (N/m) Parallel pond radius (m)
Material type Clay
Sand and gravel
Wall
Fluid
2600 1.0 106 1.0 106 0.5 – 2.6 108 2.6 108 1.0
2650 1.0 106 1.0 106 0.6 – – – –
– 1.0 106 1.0 106 0.5 – – – –
1000 – – – 1.0 103 – – –
As Fig. 2b shows, the entire space of the numerical model was meshed along the x-, y- and z-directions for the continuous fluid simulation. Hydraulic pressures were defined for the inflow and outflow sections of the simulation domain, and a pressure difference was created in the domain along the x-direction. Water can flow through the sample under the pressure difference. The spherical particles used in the model are different from the irregular shaped real soil particle. Published data have shown the influence particle shape on the engineering properties of granular soils [22–26] base soil–filter sphericity leads to more localized but vertically dilated deformation [23]. When spherical particles are used to model the real soil particles, it may be easier for finer particles to move in the filter–base combining system according to the study of liquefaction [25,26], because movement of spherical finer particles may not be blocked by the localized dense region. But the whole eroded mass of the base soil into the filter will not change much compared to the case modeled by irregular particles since the increasing sphericity leads to vertically dilated deformation. In this simulation, the cubic sample’s smallest dimension is 6 cm, the largest particle size of the base soil is 1.2 cm, the largest particle size of the filters is 1.2–2.0 cm, the ratio of the cubic sample’s smallest dimension to the largest particle size of the base soil is 5 which is satisfied with the standard specification [27]; the ratio of the cubic sample’s smallest dimension to the largest particle size of the filter is 3–5 that is indeed a bit small according to the standard, but in the base soil–filter combining system, it is the movement of finer grains of the base soil to be mainly focused on, so the results of the whole system would not be influenced so much by that. Table 1 lists the parameters used for the numerical simulation. The densities of the solid particles are based on laboratory test results, and the remaining soil property parameters are typical values used in the literature [28,29]. 3.2. Simulation cases The hydraulic gradient and erosion time, which directly affect filter functionality, are essential factors in practical engineering [4]. To investigate the internal erosion process of the combined soil sample for filters with different particle size distributions, four groups of analysis cases were designed, which are listed in Table 2. ‘‘Base-F1’’, ‘‘Base-F2’’, ‘‘Base-F3’’ and ‘‘Base-F4’’ represent four combined soil samples of which the representative particle size ra tio D15 =d85 gradually increases from Base-F1 to Base-F4. Here, d85 is the diameter of the particles at 85% passing of the finer-grain fraction of the base soil. F1 through F4 are the four filters in Fig. 3. In each analysis group, two cases were designed to investigate the impact of the hydraulic gradient and calculation time. The fluid calculation time was used to simulate the erosion time, which is important in practical projects. The four combined samples under different fluid calculation times with the same
Table 2 Analysis cases for numerical erosion study. Analysis cases
D15 =d85
Group I BaseF1
Case 1 Case 2
1.67
Group II BaseF2
Case 1 Case 2
3.47
Group III BaseF3
Case 1 Case 2
6.94
Group IV BaseF4
Case 1 Case 2
10.42
Hydraulic gradient
Fluid calculation time (s)
0.0025, 0.025, 0.5, 5, 8, 25, 50, 83, 250 83
0.02
0.0025, 0.025, 0.5, 5, 8, 25, 50, 83, 250 83 0.0025, 0.025, 0.5, 5, 8, 25, 50, 83, 250 83 0.0025, 0.025, 0.5, 5, 8, 25, 50, 83, 250 83
0.001, 0.005, 0.01, 0.015, 0.02, 0.03, 0.04 0.02 0.001, 0.005, 0.01, 0.015, 0.02, 0.03, 0.04 0.02 0.001, 0.005, 0.01, 0.015, 0.02, 0.03, 0.04 0.02 0.001, 0.005, 0.01, 0.015, 0.02, 0.03, 0.04
hydraulic gradient were designed to investigate the influence of infiltration time on the erosion of the base soil. The time step for the fluid calculation in the model is approximately 107 s. A longer fluid calculation time corresponds to a longer erosion time.
4. Simulation results and discussion Base soil erosion directly affects the erosion resistance of the combined soil. To further investigate the erosion process and the migration of the fine particles of the base soil in a filter, user-defined commands were used to track and record the position of each eroded particle through the filter to determine the intruding depth, quantities and distribution of the eroded particles in the filter. The detailed procedures are the following: (i) divide the filter into six regions along the flow direction (x-direction), (ii) count the number of particles and calculate the mass of the base-soil particles in different filter regions using the particle number and x-position and (iii) compare the radii of the intruding particles through the filter to determine the maximum particle size. 4.1. Influence of filter particle size distribution on erosion Fig. 4 shows the particle distributions of the base soil–filter systems obtained from the fluid–particle coupling analysis. The hydraulic gradient and fluid calculation time are 83 and 0.02 s, respectively. The potential of fine-grained particles intruding into the filter with different particle size distributions is not equivalent. The number of eroded soil particles increased as the controlling particle size (D15) of the filters increased. More and coarser particles eroded through the filter with coarser particles because the
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Y.-H. Zou et al. / Computers and Geotechnics 47 (2013) 48–56
(a) F1
(b) F2
(c) F3
(d) F4
Fig. 4. Particle distribution of base soil–filter systems at a time of 0.02 s under a gradient of 83.
15.0
5.0
12.0
Intruding depth (cm)
The erosion ratio (%)
F1 F2 F3 9.0
F4
6.0 3.0 0.0
1
2
3
4
5
6
j¼1
,
n0 X
qi r3i
0
2
4
6
8
10
Fig. 6. The relationship between intruding depth and particle size ratio D15 =d85 at 0.02 s under gradient 83.
coarser filter contains more void space that allows fine particles to continue moving. By tracking the position of each base-soil particle, the eroded particles into the filter can be identified and recorded. In this paper, the erosion ratio of the base soil Rer is defined to investigate the percent of eroded particles of the base soil, which can be calculated using the eroding mass of the base soil at a certain calculation time:
qj r3j
2.0
D 15 /d * 85
Fig. 5. Distributions of erosion ratios in different filters.
ner X
3.0
1.0 0
Distance from the interface d (cm)
Rer ¼
4.0
ð10Þ
i¼1
where ner and n0 are the number of the eroded base-soil particles and the original base-soil particles, respectively; qj and qi are the densities of the jth eroded particle and ith original base-soil particle, respectively; and rj and ri are the radii of the jth eroded particle and ith original base-soil particle, respectively.
To obtain a detailed distribution of the eroded particles in the filter, the filter was divided equally into six sections along the flow direction (x-direction). Within each section, the quantities of eroded particles and the base soil erosion ratio were calculated. Fig. 5 shows the distributions of the erosion ratios in different filters (at a time of 0.02 s under a gradient of 83). The intruding mass of the base-soil particles through the filters gradually reduced along the flow direction because the hydraulic gradient near the interface between the base soil and filter is greater than elsewhere in the filter. The intruding mass in each divided region also increased as the controlling particle size of the filter increased because a coarser filter has more void space that allows fine particles to continue moving. The intruding depths through different filters can be approximately obtained from the erosion ratio distributions shown in Fig. 5. Because the maximum intruding depth is an important factor for potential erosion, a specific program was used to record the
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Y.-H. Zou et al. / Computers and Geotechnics 47 (2013) 48–56
0.001s 0.005s
150
0.01s 0.03s
100 50 0
0
1
2
3
4
5
6
0.005s 1000
0.01s 0.03s
500
0
0
1
2
3
4
5
6
(a) F1
(b) F2
0.001s 0.005s
3000
0.01s 0.03s
2000 1000
0
0.001s
Distance from the interface d (cm)
4000
0
1500
Distance from the interface d (cm)
The number of intruding particles
The number of intruding particles
The number of intruding particles
The number of intruding particles
200
1
2
3
4
5
6
6000 0.001s 0.005s 4000
0.01s 0.03s
2000
0
0
1
2
3
4
5
6
Distance from the interface d (cm)
Distance from the interface d (cm)
(c) F3
(d) F4
Fig. 7. Distribution of intruding particles in the filters at different times of fluid calculation.
exact intruding depth, which is the maximum distance of the fine particles of the base soil moving into the filter from the interface between the base soil and filter. Fig. 6 shows the relationship between the intruding depth and particle size ratio D15 =d85 at 0.02 s under a gradient of 83. The intruding depth increased nonlinearly as the particle size ratio increased. When the particle size ratio was small, i.e., approximately 1, the intruding depth of the base-soil particles was approximately 1 cm, which indicates the base-soil particles moved into the filter under the water pressure difference and became clogged in a small region near the interface by the filter. When the particle size ratio was large, i.e., approximately 9, the intruding depth of the base-soil particles was greater than 4 cm. In this case, the eroded base-soil particles were not clogged in a small region of the filter but became dispersed in the larger region in the filter. Because the particle size ratio D15 =d85 reflects the potential of the fine particle to move through filter voids, a larger ratio value indicates that fine particles can move more easily and farther.
Intruding depth (cm)
6.0
0.005s 0.030s
0.010s 18.0
5.0 4.0 3.0 2.0 1.0 0.0
0
2
4
6
Fig. 7 shows the distributions of the numbers of intruding particles in the filters at different fluid calculation times. The maximum intruding depth of the eroded particles was not greater than 2 cm, and the distributions of the intruding particles in the filter were essentially the same after 0.005 s for F1. This indicates that, with the protection of F1, the eroded base-soil particles were kept within a small region (approximately 1 cm) of the filter, not far from the base soil–filter interface; erosion thus did not continue with time. Fig. 7b–d shows, the distributions of the intruding base particles in the filter changed with time. The quantities of the eroded base soil in different regions of the filter increased as the calculation time increased for F2, F3 and F4. The increasing rate of the number of eroded particles was greatly reduced with time for F2, which indicates that progressive erosion did not occur in this case. It can be further concluded that a coarser filter provides greater potential for fine particles to move, i.e., more fine particles can move through the filter, and they can move farther from the interface in the filter.
The erosion ratio Rer (%)
0.001s 0.020s
4.2. Influence of infiltration time on erosion
8
10
12.0 9.0 6.0 3.0 0.0
D 15 /d * 85
F1 F2 F3 F4
15.0
0
0.01
0.02
0.03
0.04
Time of fluid calculation t (s)
Fig. 8. The relationship between the intruding depth and particle size ratio D15 =d85 at different fluid calculation times.
Fig. 9. Changes in the erosion ratio with fluid calculation time.
3
2
1
0 0.00
x=3.0 cm x=4.5 cm x=6.0 cm 0.01
0.02
0.03
0.04
0.05
Interstitial velocity v (m/s)
Y.-H. Zou et al. / Computers and Geotechnics 47 (2013) 48–56
Interstitial velocity v (m/s)
54
3
2
1
0 0.00
x=3.0 cm x=4.5 cm x=6.0 cm 0.01
0.02
0.03
0.04
Time t (s)
Time t (s)
(a) D15/d*85 = 3.47 (Base-F2)
(b) D15/d*85=6.94 (Base-F3)
0.05
Fig. 10. The interstitial velocities in different elements in the soil sample.
4.3. Influence of hydraulic gradient on erosion The change of interstitial velocity can reflect the hydraulic gradient in the soil sample. Fig. 10 shows the interstitial velocities in different elements in two soil samples whose particle size ratios are 3.47 (Base-F2) and 6.94 (Base-F3), respectively. As mentioned
above, in the simulation model the entire model space is divided into a number of unit cells. The interstitial velocities in three elements along the x-direction are studied. The y-coordinate and zcoordinate of the center of the elements are 3.0 cm. The x-coordinate of the center of the elements are x = 3.0 cm, 4.5 cm and 6.0 cm, where the center x = 3.0 cm is at the middle of the base soil along the x-direction and the center x = 6.0 cm is at the interface between the base soil and the filter. Fig. 10a shows the velocity is almost the same in all elements before 0.003 s. After 0.003 s, the velocities at centers x = 3.0 cm and x = 4.5 cm do not change so much, which means there was no significant particle movement or erosion happened in the upstream regions far from the interface, and a stable pressure gradient was maintained. While at the interface (x = 6.0 cm) the velocity still fluctuated with time after 0.003 s because some finer particles moved across the interface section and changed the porosity of the soil. But after a longer time, the velocities at those different points do not change a lot, and become stable, which means the interstitial velocity of soil in the sample become stable after rearrangement of the particles and shows the effectiveness of filter F2. When the particle size ratio is 6.94 (Fig. 10b), the change in interstitial velocity with time differs from the case that the particle size ratio is 3.47. After 0.003 s, the velocities in all elements decrease as time increases and tend to decrease with time other than maintain at a stable value after a longer period, which shows the system is not stable under the protection of filter F3. Fig. 11 shows the base soil erosion ratios under different hydraulic gradients. The erosion ratio increased as the hydraulic gradient increased. When the hydraulic gradient is less than 9.6, the increase rate is relatively high. Afterwards, the increase rate is lower.
18.0
The erosion ratio R er (%)
Fig. 8 shows the relationships between the intruding depth and particle size ratio D15 =d85 at different fluid calculation times. The intruding depth increased as the particle size ratio increased at different fluid calculation times, and it also increased with the fluid calculation time. If the particle size ratio was large enough, i.e., approximately 9, the intruding depth was large even when the fluid calculation time was as short as 0.001 s, which indicates that it does not require much time for the base soil with a coarse filter to become eroded. This results in a more dangerous situation than using a finer filter. In the simulation, when the filter is coarse en ough, i.e., in the case the particle size ratio D15 =d85 is 6.94 or 10.42, a few fine grains in the base soil were washed out across the filter continuously after a longer period. Fig. 9 shows changes in the erosion ratio according to the fluid calculation time. The erosion ratios increased as the fluid calculation time increased. The increasing rates of the erosion ratio with time or the erosion rate in different filters were not equivalent. In this study, the erosion rate is defined as the changing rate of the erosion ratio with time. The initial increasing rates were larger before 0.01 s than those after 0.01 s for F1, F2 and F3. The rate reflects how erosion processes develop in filters. When a water pressure difference is applied to the combined soil, the fluid continuously drags the base-soil particles, which move gradually through the filter, thus increasing the number of eroded base-soil particles at the initial time stage in the filters. As time passes, the base particles become gradually clogged in the filter, which reduces the controlling diameter of the possible transport paths formed by the original filter particles. The increase rate of the eroded base-soil particle mass becomes lower, and the eroded mass does not change with time in the filter with grains that are fine enough. Compared with the results given above, the erosion ratio increased linearly with time with a large increase rate for F4, which indicates that the controlling diameters of the transport paths were large enough and the clogged eroded particles did not block the continuation of the erosion. In conclusion, the intruding mass hardly changed with time with the protection of F1 and F2. However, the eroded base soil mass increased with time for F3 and F4; this phenomenon is hereafter called ‘‘progressive erosion’’. Although the increase rate of the number of intruding particles was slightly lower when time increased, particularly for F3, the increasing trend did not change. F3 and F4 are thus considered ineffective filters for protecting base soils.
15.0 F1 F3
12.0
F2 F4
9.0 6.0 3.0 0.0
0
100
200
300
Hydraulic gradient i Fig. 11. Base soils erosion ratios under different hydraulic gradients.
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Y.-H. Zou et al. / Computers and Geotechnics 47 (2013) 48–56 Table 3 Prediction results from numerical simulations and empirical filter criterion. Empirical filter criteriaa
Calculation time (s)
Hydraulic gradient
1.67
E
0.02 0.02 0.02 0.03
6 30 83 83
0.24 0.26 0.26 0.26
E E E E
1.66
3.47
E
0.02 0.02 0.02 0.03
6 30 83 83
1.41 2.57 2.71 2.76
E E E E
6.42
1.26
6.94
I
0.02 0.02 0.02 0.03
6 30 83 83
4.28 7.08 7.43 8.29
I I I I
9.64
1.07
10.42
I
0.02 0.02 0.02 0.03
6 30 83 83
9.28 11.10 13.30 15.80
I I I I
Soil sample
D15 (mm)
Cuf
Base-F1
1.545
2.75
Base-F2
3.21
Base-F3
Base-F4
D15 =d85
Erosion rate (%)
The numerical modela
Cuf is the coefficient of uniformity of filter (d60/d10). a E = Effective, I = ineffective.
Furthermore, changes in the base soil erosion ratios differ for different filters. The erosion ratios mostly increase when the hydraulic gradient is lower than 60 for F3 and F4. The erosion ratio for F4 continues to increase almost linearly as the hydraulic gradient increase, particularly when the gradient become larger than 100, while that increase is little for F3. The erosion ratios essentially do not change at large hydraulic gradients when F1 and F2 protect the base soil. Progressive erosion did not occur in the base soil protected by F1 or F2, though the hydraulic gradient increase was large. However, the base soil eroded more with an increasing hydraulic gradient when protected by F3 or, especially, F4. 4.4. Comparison erosion results between the empirical filter criteria and numerical tests In current practices, empirical filter criteria have been widely used for evaluating the effectiveness of filter for a base soil. Because this study is for gap-graded base soil, the following filter design criterion for erosion could be used [30,31]:
D15 < 4 d85
ð11Þ
where d85 is the diameter of particles at 85% passing of finer-grain fraction of the base soil. Table 3 lists the representative particle size ratio D15 =d85 and simulation results. Based on the empirical filter criterion in Eq. (11), F1 and F2 can be considered as effective filters since the rep resentative particle size ratios D15 =d85 of soil samples Base-F1 and Base-F2 are 1.67 and 3.47 respectively, and are both less than 4.0. From our numerical model, filter effectiveness was identified based on the change in eroded particles with time and the hydraulic gradient. The erosion ratio or eroded mass does not increase under high hydraulic gradient and after longer time in F1 or F2, which differ from the increasing erosion ratio or quantity with time in F3 or F4. Thus it can be concluded that the filters F1 and F2 are effective filters that protect the base soil from progressive erosion. Conversely, filters F3 and F4 are ineffective because their erosion ratios continually increased with time. The results obtained using the numerical simulation agreed with the empirical filter criterion. The result that the erosion quantity and intruding depth of the base soil into filter increase with increasing particle size ratio D15 =d85 in this study agrees with the results from other researchers [5,6,32–35], which also verifies the theory of the constriction controlling size proposed by Kenney et al. [8].
5. Conclusions A granular filter is an essential component of earth–rock fill dams and protects the core soil from erosion and piping. This paper simulated the transient transport of eroded base-soil particles into a filter under different hydraulic gradients using the particle flow method and analyzed and discussed the erosion mechanism of the fine particles. By tracking and record the position of eroded particles in the filter, the distribution, intruding depth and mass of eroded particles in the filter were obtained and the erosion ratios in different cases were calculated. The intruding depth of eroded base-soil particles increases with the increasing particle size ratio at different fluid calculation time. The intruding depth of the eroded base-soil particles increased as the particle size ratio increased at different fluid calculation times. It took less time for the base soil protected by a coarse filter to become dangerously eroded than that protected by a fine filter. Furthermore, more fine particles could move into a coarser filter. The erosion ratio increased as the fluid calculation time increased. The erosion ratio also increased as the hydraulic gradient increased. When the particle size ratio D15 =d85 was small within a certain range, the erosion ratio essentially did not change under high hydraulic gradients, and the intruding mass did not increase at longer times. The simulation results were also used to identify effective filter for the base soil, which were compared with the results obtained from the empirical filter criterion. The results show that the PFM is an effective and practical approach to study the filtration micro-property and erosion mechanism in a base soil–filter system. Acknowledgments This research was substantially supported by Doctoral Fund of Ministry of Education of China (Approval No. 20100181110076), the National Natural Science Foundation of China (No. 41072270) and the Support Program for New Century Excellent Talent in China (Approval No. NCET-07-0569). References [1] Sherard JL. Sinkholes in dams of coarse, broadly graded soils. In: Sakhmander S, editor. Technology review, transactions: proceedings of 13th congress on large dams, vol. 2. New Delhi (India): ICOLD; 1979. p. 25–35. [2] Sherard JL, Dunnigan LP, Talbot J. Basic properties of sand and gravel filters. J Geotech Eng, ASCE 1984;110(6):684–700.
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[3] Richards KS, Reddy KR. Critical appraisal of piping phenomena in earth dams. Bull Eng Geol Environ 2007;66:381–402. [4] Locke M, Indraratna B, Adikari G. Time-dependent particle transport through granular filters. J Geotech Geoenviron Eng, ASCE 2001;127(6):621–9. [5] Lone MA, Hussain B, Asawa GL. Filter design criteria for graded cohesionless bases. J Geotech Geoenviron Eng, ASCE 2005;131(2):251–9. [6] Tomlinson SS, Vaid YP. Seepage forces and confining pressure effects on piping erosion. Can Geotech J 2000;37:1–13. [7] Silveira A. An analysis of the problem of washing through in protective filters. In: Proceedings of 6th international conference of soil mechanics and foundation engineering, Montreal, Canada, vol. 2; 1965. p. 551–5. [8] Kenney TC, Chahel R, Chiu E, et al. Controlling constriction sizes of granular filters. Can Geotech J 1985;22(1):32–43. [9] Witt K. Reliability study of granular filters. In: Brauns J, Heibaum M, Schuler U, editors. Filters in geo-technical and hydraulic engineering. Rotterdam (The Netherlands): Balkema; 1993. p. 35–42. [10] Schuler U. Scattering of the composition of soils. An aspect for the stability of granular filters. In: Lafleur J, Rollin AL, editors. Proceedings of Geofilters’96. Montreal (Canada): Bitech Publications; 1996. p. 21–34. [11] Indraratna B, Vafai F. Analytical model for particle migration within base soil– filter system. J Geotech Geoenviron Eng, ASCE 1997;123(2):100–9. [12] Indraratna B, Locke M. Analytical modeling and experimental verification of granular filter behaviour. In: Wolski W, Mlynarek J, editors. Filter sand drainage in geotechnical and geoenvironmental engineering. Rotterdam (The Netherlands): Balkema; 2000. p. 3–26. [13] Indraratna B, Trani LDO, Khabbaz H. A critical review on granular dam filter behaviour – from particle sizes to constriction-based design criteria. Geomech Geoeng 2008;3(4):279–90. [14] Patankar SV. Numerical heat transfer and fluid flow. Hemisphere Publishing; 1980. [15] Cundall PA, Strack ODL. The distinct element method as a tool for research in granular media, part II. Minnesota: University of Minnesota; 1979. [16] Cundall PA, Strack ODL. A discrete numerical model for granular assembles. Geotechnique 1979;29(1):47–65. [17] Jiang MJ, Yu HS, Harris D. A novel discrete model for granular material incorporating rolling resistance. Comput Geotech 2005;32(5):340–57. [18] Lu LS, Hsiau SS. DEM simulation of particle mixing in a sheared granular flow. Particuology 2008;6(6):445–54. [19] Tsuji Y, Tanaka T, Ishida T. Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technol 1992;71(3):239–50.
[20] Bouillard JX, Lyczkowski RW, Gidaspow D. Porosity distributions in a fluidized bed with an immersed obstacle. AIChE J 1989;35(6):908–22. [21] Tannehill JC, Anderson DA, Pletcher RH. Computational fluid mechanics and heat transfer. 2nd ed. US: Taylor and Francis; 1997. p. 594–6. [22] Mirghasemi AA, Rothenburg L, Matyas EL. Influence of particle shape on engineering properties of assemblies of two-dimensional polygon-shape particles. Geotechnique 2002;52(3):209–17. [23] Kock I, Huhn K. Influence of particle shape on the frictional strength of sediments – a numerical case study. Sediment Geol 2007;196: 217–33. [24] Vaid YP, Chern JC, Tumi H. Confining pressure, grain angularity, and liquefaction. J Geotech Eng, ASCE 1985;111(10):1229–35. [25] Hird CC, Hassona FAK. Some factors affecting the liquefaction and flow of saturated sands in laboratory tests. Eng Geol 1990;28:149–70. [26] Ashour M, Norris G. Liquefaction and undrained response evaluation of sands from drained formulation. J Geotech Geoenviron Eng, ASCE 1999;125(81):649–58. [27] The Ministry of Water Resources of the People’s Republic of China (MWRPRC). Specification of seepage and seepage deformation test for coarse grained soil (SL237-058-1999). Beijing: China Water Power Press; 1999 [in Chinese]. [28] Itasca. Itasca Consulting Group Inc. PFC3D (particle flow code in 3dimensions), version 4.0. ICG, Minneapolis, Minnesota; 2008. [29] Yang J. Simulation study on structural strength of dredger fill with high clay content during consolidation progress. Changchun: Jilin University; 2009 [in Chinese]. [30] ICOLD. Embankment dams granular filters and drains: Review and recommendations. In: Naudascher, editor. ICOLD bulletin, New Delhi, India; 1994(95). p. 25–60. [31] Liu J. The basic principles and experiences for seepage control of the earth dam. Beijing: China Water Power Press; 2006 [in Chinese]. [32] Hadj-Hamou T, Tavassoli MR, Sherman WC. Laboratory testing of filters and slot sizes for relief wells. J Geotech Eng, ASCE 1990;116(9): 1325–46. [33] Reddi LN, Lee IM, Bonala MVS. Comparison of internal and surface erosion using flow pump tests on a sand–kaolinite mixture. Geotech Test J 2000;23(1):116–22. [34] Fannin RJ, Moffat R. Observations on internal stability of cohesionless soils. Geotechnique 2006;56(7):497–500. [35] Sterpi D. Effects of the erosion and transport of fine particles due to seepage flow. Int J Geomech 2003;3(1):111–22.
Contents lists available at SciVerse ScienceDirect
Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Discrete numerical modeling of particle transport in granular filters Yu-Hua Zou a, Qun Chen b,⇑, Xiao-Qing Chen a, Peng Cui a a b
Key Laboratory of Mountain Hazards and Earth Surface Processes, Institute of Mountain Hazards and Environment, CAS, Chengdu 610041, China State Key Laboratory of Hydraulics and Mountain River Engineering, College of Hydraulic and Hydropower Engineering, Sichuan University, Chengdu 610065, China
a r t i c l e
i n f o
Article history: Received 15 December 2011 Received in revised form 5 June 2012 Accepted 6 June 2012 Available online 3 August 2012 Keywords: Internal erosion Base soil Filter Particle flow method Eroded mass Intruding depth
a b s t r a c t Granular filters are an essential component in earth dams to protect the dam core from seepage erosion. This paper uses the particle flow method (PFM) to study the mechanism of particle transport in a base soil–filter system. The distributions of the eroded base-soil particles in different filters are traced and analyzed. The eroded mass and intruding depth of the eroded particles into the filters are obtained under different times and hydraulic gradients. The simulation results show that the eroded mass and intruding depth of the base-soil particles into the filter are related to the representative particle size ratio of the base soil to the filter, hydraulic gradient and erosion time. The numerical predictions are also compared with the empirical filter design criterion. The results show that the particle flow model provides an effective approach for studying the filtration micro-property and the erosion mechanism in a base soil–filter system, which is useful for filter design. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Internal erosion of core soil is a key safety problem that threatens earth dams. Filters with appropriate properties placed downstream of the core have proven effective for preventing the progression of core soil erosion [1]. Empirical methods based on the representative particle-size ratio (the ratio of the characteristic particle sizes of the filter and base soil) have been commonly used to design granular filters [2–6]. These empirical criteria have been developed based on extensive laboratory tests, but they cannot describe the interface behavior mechanism and time-dependent processes that occur in the filter material. The interface behavior mechanism is extremely important because it can provide designers with detailed insight on what may occur throughout the design life of the structure [4]. Several analytical models have been established to describe the mechanism behind the movement of fine particles in the base soil through granular filters. Silveira [7] used probabilistic methods to examine the migration of base-soil particles through filters. An estimated infiltration depth through granular filters was obtained based on the observation that a particle could transport through a pore if the particle was smaller than the pore constriction [4,8]. Witt [9] established a three-dimensional filter void network model. Schuler [10] used Monte Carlo methods to study the infiltration depth of base-soil particles into filters based on a similar void net-
work. Indraratna et al. [11–13] proposed an analytical model that used finite-difference procedures, which was based on the conservation of mass and momentum to model particle movement, considering the number of elements at the base soil–filter interface. Compared the probabilistic methods with the mathematical model used in previous filtration studies, the particle flow method allows for the process of fine-grained particles eroding through a filter to be directly observed and recorded, providing a convenient approach to explain the fundamental mechanisms of particle erosion, transport and retention. Based on a particle flow numerical model, this paper traces and analyzes the distribution of eroded base-soil particles in a filter. The erosion ratio and intruding depth of the eroded particles into the filters at different times and under different hydraulic gradients are studied. The prediction results of the effective filters obtained from the numerical model and empirical filter criterion are then compared. 2. Basic modeling principles 2.1. Particle motion and interaction The particle flow method used in this paper solves the Newtonian equations of motion simultaneously for every particle in the system [14]: *
⇑ Corresponding author. Tel.: +86 28 85403351; fax: +86 28 85405604. E-mail address: [email protected] (Q. Chen). 0266-352X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compgeo.2012.06.002
*
@ u f mech * ¼ þg @t m
ð1Þ
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Y.-H. Zou et al. / Computers and Geotechnics 47 (2013) 48–56
(a) Particle-wall, particle-particle interaction
(b) Contact model
Fig. 1. Schematic diagram of the PFM used in the paper.
*
*
@x M ¼ @t I
ð2Þ *
*
where u is the particle velocity, f mech is the sum of additional forces (externally applied and contact forces) acting on the particle, m is * * the particle mass, g is * the acceleration due to gravity, x is the particle angular velocity, M is the moment acting on the particle, and I is the moment of inertia. As Fig. 1a shows, the wall boundary velocity x_ w causes displacement and overlap with particle a by dw a , which imparts force F w a and acceleration to particle a. The resolved velocity x_ a and subsequent displacement of particle a causes an overlap with particle b by da b , imparting contact force F a b with both normal and shear components. The resultant force, which includes the gravitational force, results in the linear and angular acceleration of both particles, and these accelerations are integrated to determine the * * instantaneous velocities x_ a , x_ b , h_ a and h_ b ðu and x in Eqs. (1) and (2), respectively). The mechanical contact model in the particle flow method (PFM) contains normal and tangential contact components, as shown in Fig. 1b. In the figure, ‘‘m1’’ and ‘‘m2’’ denote two particles. A normal contact resists the normal force, and a tangential contact resists the shear force. Both contacts include a spring (with normal stiffness Kn and shear stiffness Ks, respectively), which reflects the elastic behavior of the contact, and a dashpot (with normal viscosity gn and shear viscosity gs, respectively), which reflects energy dissipation and quasi-static deformations [15–17]. According to the linear spring-dashpot model, the spring and damping forces determine the normal contact force, and the tangential and damping forces give the tangential contact force [18,19]: *
*
*
*
qf
@ðn v Þ þ qf @t
*
*
*
*
v rðn v Þ ¼ nrp þ lr2 ðn v Þ þ f b
ð5Þ
*
* @ðn v Þ þ rðn v Þ ¼ 0 @t
ð6Þ
where qf is the density of the fluid, n is the porosity of the porous media, p is the fluid pressure, l is the dynamic viscosity of the fluid, * and f b is the body force per unit volume. The fluid velocity is de* noted as v in porous flow. It is the interstitial velocity, and the * quantity n v is the macroscopic or Darcy velocity. The simulation domain is discretized in elements which include many particles. The staggered grid scheme is used to solve the fluid velocities and pressures [21]. 2.2.2. Fluid-particle interaction force The preceding equations give the body force experienced by the fluid as a result of the moving particles. A force equal and opposite is distributed on the discrete particles for each fluid element. The * body force f b is applied to each particle proportional to the volume of the particle. If the particles are assumed to be spherical, the drag force applied to an individual discrete particle [14] is: * *
f drag ¼
4 3 fb pr 3 ð1 nÞ
ð7Þ
*
F n ¼ K n d n gn V rn
*
generalized form of the Navier–Stokes equation. In the PFM, it is modified to include the effect of a particulate solid phase mixed into the fluid [14]. The average effects expressed with the porosity * n and a coupling force f b over many particles (as opposed to attempting to model the details of the fluid flow between particles), can be characterized as [20]:
ð3Þ
where r is the radius of the particle. The total force applied by the fluid on the particle is the sum of the drag and buoyancy forces [14]:
ð4Þ
*
*
F s ¼ K s d s gs V rs
where Kn and Ks are the spring constant and shear stiffness, respectively; dn and ds are the normal and tangential displacements of invading points, respectively; gn and gs are the viscous damping * in the normal and tangential directions, respectively; and V rn and * r V s are the velocity in the normal direction and the sliding velocity in the tangential direction, respectively.
*
f fluid ¼ f drag þ
4 3 * pr q f g 3
ð8Þ
2.2.3. Particle motion in fluid Considering the motion of particles in fluid, Eq. (1) can be extended with an additional forcing term to account for interaction with the fluid [14]: *
*
*
2.2. Fluid–particle interaction
@ u f mech þ f fluid * ¼ þg @t m
2.2.1. Fluid motion or hydrodynamics equation The two-phase (fluid and solid) mass and momentum equations for the fluid velocities and pressures [20] can be considered a
To calculate the coupled fluid and particle motion, the first step is to mesh the simulation area to simulate a continuous fluid medium. The fluid boundary conditions must then be defined. Particle
ð9Þ
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Y.-H. Zou et al. / Computers and Geotechnics 47 (2013) 48–56
3 Filter Base soil
2 Wall boundary
z
5
y
x O(0, 0, 0) (a) The numerical model
Outflow section Inflow section
Fluid cell
z y
Wall
x
boundary
(b) Wall boundary and fluid cell Fig. 2. The numerical model for the filter test.
3. Numerical modeling for erosion 3.1. The numerical model The filter tests were modeled by cubic sample (0.12 m 0.06 m 0.06 m) within six walls (Fig. 2) in the Cartesian coordinate system. As Fig. 2a shows, the origin of coordinates (O(0, 0, 0)) is at on the front bottom left point of the model. The x-, yand z-directions of the coordinates are shown by three arrows pointing in three perpendicular directions (Fig. 2a and b). In the numerical model, the left (wall no. 4) and right walls (wall no. 2) simulate the permeable boundary for water inlet and outlet. Other four impermeable walls (1, 3, 5 and 6) are slip walls that particles can slip along the walls when the tangential force is greater than wall–particle friction. Spheres with specified size distributions were used to simulate particles of the base soil and filter. Gap-graded gravelly clay, which is higher in shear strength and lower in compression than clay, is increasingly used as a core in earth–rock fill dam, particularly in high earth–rock fill dams. The erosion of such a core with different
filters would be a useful reference for filter design. To simulate the gap-graded gravelly clay, four grain groups of spheres, marked r, s, t and u, were used to model particles having different sizes. The groups r, and t, respectively; simulated gravel, coarse sand and fine-sand particles, respectively, and the group u simulated clay particles. The white spheres on the right half of the model simulated the well-graded filter, which is traditionally used to protect different kinds of base soils. Both the base soil and filter particles occupy one half of the model to form a combined soil sample, shown in Fig. 2a. To investigate base-soil particle erosion in different combined samples, one base soil and four filters (F1, F2, F3 and F4) were used, and Fig. 3 shows their particle size distributions. The controlling particle size D15 (i.e., particle diameter at 15% passing) of the four filters gradually increases from F1 to F4.
100
Percent finer (%)
movement will also be a result of the differences in the water pressure. By solving the continuous fluid medium Navier–Stokes equation, the fluid pressure and flow velocity in each fluid grid are calculated. The fluid drag force moves the solid particles, which changes the porosity for different fluid cells; the porosity then affects the fluid flow rate of the grid. The continuing changing fluid flow drag force and porosity between particles ultimately cause the fluid–particle coupling balance to form.
Base F1 F2 F3 F4
80 60 40 20 0
1
10
100
Particle size (mm) Fig. 3. Particle size distribution of the base soil and the filters.
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Y.-H. Zou et al. / Computers and Geotechnics 47 (2013) 48–56 Table 1 Parameters used in numerical simulation for erosion. Parameters
Density (kg/m3) Normal stiffness, kn (N/m) Shear stiffness, ks (N/m) Friction coefficient Viscous coefficient (Pas) Parallel pond shear stiffness (N/m) Parallel pond normal stiffness (N/m) Parallel pond radius (m)
Material type Clay
Sand and gravel
Wall
Fluid
2600 1.0 106 1.0 106 0.5 – 2.6 108 2.6 108 1.0
2650 1.0 106 1.0 106 0.6 – – – –
– 1.0 106 1.0 106 0.5 – – – –
1000 – – – 1.0 103 – – –
As Fig. 2b shows, the entire space of the numerical model was meshed along the x-, y- and z-directions for the continuous fluid simulation. Hydraulic pressures were defined for the inflow and outflow sections of the simulation domain, and a pressure difference was created in the domain along the x-direction. Water can flow through the sample under the pressure difference. The spherical particles used in the model are different from the irregular shaped real soil particle. Published data have shown the influence particle shape on the engineering properties of granular soils [22–26] base soil–filter sphericity leads to more localized but vertically dilated deformation [23]. When spherical particles are used to model the real soil particles, it may be easier for finer particles to move in the filter–base combining system according to the study of liquefaction [25,26], because movement of spherical finer particles may not be blocked by the localized dense region. But the whole eroded mass of the base soil into the filter will not change much compared to the case modeled by irregular particles since the increasing sphericity leads to vertically dilated deformation. In this simulation, the cubic sample’s smallest dimension is 6 cm, the largest particle size of the base soil is 1.2 cm, the largest particle size of the filters is 1.2–2.0 cm, the ratio of the cubic sample’s smallest dimension to the largest particle size of the base soil is 5 which is satisfied with the standard specification [27]; the ratio of the cubic sample’s smallest dimension to the largest particle size of the filter is 3–5 that is indeed a bit small according to the standard, but in the base soil–filter combining system, it is the movement of finer grains of the base soil to be mainly focused on, so the results of the whole system would not be influenced so much by that. Table 1 lists the parameters used for the numerical simulation. The densities of the solid particles are based on laboratory test results, and the remaining soil property parameters are typical values used in the literature [28,29]. 3.2. Simulation cases The hydraulic gradient and erosion time, which directly affect filter functionality, are essential factors in practical engineering [4]. To investigate the internal erosion process of the combined soil sample for filters with different particle size distributions, four groups of analysis cases were designed, which are listed in Table 2. ‘‘Base-F1’’, ‘‘Base-F2’’, ‘‘Base-F3’’ and ‘‘Base-F4’’ represent four combined soil samples of which the representative particle size ra tio D15 =d85 gradually increases from Base-F1 to Base-F4. Here, d85 is the diameter of the particles at 85% passing of the finer-grain fraction of the base soil. F1 through F4 are the four filters in Fig. 3. In each analysis group, two cases were designed to investigate the impact of the hydraulic gradient and calculation time. The fluid calculation time was used to simulate the erosion time, which is important in practical projects. The four combined samples under different fluid calculation times with the same
Table 2 Analysis cases for numerical erosion study. Analysis cases
D15 =d85
Group I BaseF1
Case 1 Case 2
1.67
Group II BaseF2
Case 1 Case 2
3.47
Group III BaseF3
Case 1 Case 2
6.94
Group IV BaseF4
Case 1 Case 2
10.42
Hydraulic gradient
Fluid calculation time (s)
0.0025, 0.025, 0.5, 5, 8, 25, 50, 83, 250 83
0.02
0.0025, 0.025, 0.5, 5, 8, 25, 50, 83, 250 83 0.0025, 0.025, 0.5, 5, 8, 25, 50, 83, 250 83 0.0025, 0.025, 0.5, 5, 8, 25, 50, 83, 250 83
0.001, 0.005, 0.01, 0.015, 0.02, 0.03, 0.04 0.02 0.001, 0.005, 0.01, 0.015, 0.02, 0.03, 0.04 0.02 0.001, 0.005, 0.01, 0.015, 0.02, 0.03, 0.04 0.02 0.001, 0.005, 0.01, 0.015, 0.02, 0.03, 0.04
hydraulic gradient were designed to investigate the influence of infiltration time on the erosion of the base soil. The time step for the fluid calculation in the model is approximately 107 s. A longer fluid calculation time corresponds to a longer erosion time.
4. Simulation results and discussion Base soil erosion directly affects the erosion resistance of the combined soil. To further investigate the erosion process and the migration of the fine particles of the base soil in a filter, user-defined commands were used to track and record the position of each eroded particle through the filter to determine the intruding depth, quantities and distribution of the eroded particles in the filter. The detailed procedures are the following: (i) divide the filter into six regions along the flow direction (x-direction), (ii) count the number of particles and calculate the mass of the base-soil particles in different filter regions using the particle number and x-position and (iii) compare the radii of the intruding particles through the filter to determine the maximum particle size. 4.1. Influence of filter particle size distribution on erosion Fig. 4 shows the particle distributions of the base soil–filter systems obtained from the fluid–particle coupling analysis. The hydraulic gradient and fluid calculation time are 83 and 0.02 s, respectively. The potential of fine-grained particles intruding into the filter with different particle size distributions is not equivalent. The number of eroded soil particles increased as the controlling particle size (D15) of the filters increased. More and coarser particles eroded through the filter with coarser particles because the
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Y.-H. Zou et al. / Computers and Geotechnics 47 (2013) 48–56
(a) F1
(b) F2
(c) F3
(d) F4
Fig. 4. Particle distribution of base soil–filter systems at a time of 0.02 s under a gradient of 83.
15.0
5.0
12.0
Intruding depth (cm)
The erosion ratio (%)
F1 F2 F3 9.0
F4
6.0 3.0 0.0
1
2
3
4
5
6
j¼1
,
n0 X
qi r3i
0
2
4
6
8
10
Fig. 6. The relationship between intruding depth and particle size ratio D15 =d85 at 0.02 s under gradient 83.
coarser filter contains more void space that allows fine particles to continue moving. By tracking the position of each base-soil particle, the eroded particles into the filter can be identified and recorded. In this paper, the erosion ratio of the base soil Rer is defined to investigate the percent of eroded particles of the base soil, which can be calculated using the eroding mass of the base soil at a certain calculation time:
qj r3j
2.0
D 15 /d * 85
Fig. 5. Distributions of erosion ratios in different filters.
ner X
3.0
1.0 0
Distance from the interface d (cm)
Rer ¼
4.0
ð10Þ
i¼1
where ner and n0 are the number of the eroded base-soil particles and the original base-soil particles, respectively; qj and qi are the densities of the jth eroded particle and ith original base-soil particle, respectively; and rj and ri are the radii of the jth eroded particle and ith original base-soil particle, respectively.
To obtain a detailed distribution of the eroded particles in the filter, the filter was divided equally into six sections along the flow direction (x-direction). Within each section, the quantities of eroded particles and the base soil erosion ratio were calculated. Fig. 5 shows the distributions of the erosion ratios in different filters (at a time of 0.02 s under a gradient of 83). The intruding mass of the base-soil particles through the filters gradually reduced along the flow direction because the hydraulic gradient near the interface between the base soil and filter is greater than elsewhere in the filter. The intruding mass in each divided region also increased as the controlling particle size of the filter increased because a coarser filter has more void space that allows fine particles to continue moving. The intruding depths through different filters can be approximately obtained from the erosion ratio distributions shown in Fig. 5. Because the maximum intruding depth is an important factor for potential erosion, a specific program was used to record the
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Y.-H. Zou et al. / Computers and Geotechnics 47 (2013) 48–56
0.001s 0.005s
150
0.01s 0.03s
100 50 0
0
1
2
3
4
5
6
0.005s 1000
0.01s 0.03s
500
0
0
1
2
3
4
5
6
(a) F1
(b) F2
0.001s 0.005s
3000
0.01s 0.03s
2000 1000
0
0.001s
Distance from the interface d (cm)
4000
0
1500
Distance from the interface d (cm)
The number of intruding particles
The number of intruding particles
The number of intruding particles
The number of intruding particles
200
1
2
3
4
5
6
6000 0.001s 0.005s 4000
0.01s 0.03s
2000
0
0
1
2
3
4
5
6
Distance from the interface d (cm)
Distance from the interface d (cm)
(c) F3
(d) F4
Fig. 7. Distribution of intruding particles in the filters at different times of fluid calculation.
exact intruding depth, which is the maximum distance of the fine particles of the base soil moving into the filter from the interface between the base soil and filter. Fig. 6 shows the relationship between the intruding depth and particle size ratio D15 =d85 at 0.02 s under a gradient of 83. The intruding depth increased nonlinearly as the particle size ratio increased. When the particle size ratio was small, i.e., approximately 1, the intruding depth of the base-soil particles was approximately 1 cm, which indicates the base-soil particles moved into the filter under the water pressure difference and became clogged in a small region near the interface by the filter. When the particle size ratio was large, i.e., approximately 9, the intruding depth of the base-soil particles was greater than 4 cm. In this case, the eroded base-soil particles were not clogged in a small region of the filter but became dispersed in the larger region in the filter. Because the particle size ratio D15 =d85 reflects the potential of the fine particle to move through filter voids, a larger ratio value indicates that fine particles can move more easily and farther.
Intruding depth (cm)
6.0
0.005s 0.030s
0.010s 18.0
5.0 4.0 3.0 2.0 1.0 0.0
0
2
4
6
Fig. 7 shows the distributions of the numbers of intruding particles in the filters at different fluid calculation times. The maximum intruding depth of the eroded particles was not greater than 2 cm, and the distributions of the intruding particles in the filter were essentially the same after 0.005 s for F1. This indicates that, with the protection of F1, the eroded base-soil particles were kept within a small region (approximately 1 cm) of the filter, not far from the base soil–filter interface; erosion thus did not continue with time. Fig. 7b–d shows, the distributions of the intruding base particles in the filter changed with time. The quantities of the eroded base soil in different regions of the filter increased as the calculation time increased for F2, F3 and F4. The increasing rate of the number of eroded particles was greatly reduced with time for F2, which indicates that progressive erosion did not occur in this case. It can be further concluded that a coarser filter provides greater potential for fine particles to move, i.e., more fine particles can move through the filter, and they can move farther from the interface in the filter.
The erosion ratio Rer (%)
0.001s 0.020s
4.2. Influence of infiltration time on erosion
8
10
12.0 9.0 6.0 3.0 0.0
D 15 /d * 85
F1 F2 F3 F4
15.0
0
0.01
0.02
0.03
0.04
Time of fluid calculation t (s)
Fig. 8. The relationship between the intruding depth and particle size ratio D15 =d85 at different fluid calculation times.
Fig. 9. Changes in the erosion ratio with fluid calculation time.
3
2
1
0 0.00
x=3.0 cm x=4.5 cm x=6.0 cm 0.01
0.02
0.03
0.04
0.05
Interstitial velocity v (m/s)
Y.-H. Zou et al. / Computers and Geotechnics 47 (2013) 48–56
Interstitial velocity v (m/s)
54
3
2
1
0 0.00
x=3.0 cm x=4.5 cm x=6.0 cm 0.01
0.02
0.03
0.04
Time t (s)
Time t (s)
(a) D15/d*85 = 3.47 (Base-F2)
(b) D15/d*85=6.94 (Base-F3)
0.05
Fig. 10. The interstitial velocities in different elements in the soil sample.
4.3. Influence of hydraulic gradient on erosion The change of interstitial velocity can reflect the hydraulic gradient in the soil sample. Fig. 10 shows the interstitial velocities in different elements in two soil samples whose particle size ratios are 3.47 (Base-F2) and 6.94 (Base-F3), respectively. As mentioned
above, in the simulation model the entire model space is divided into a number of unit cells. The interstitial velocities in three elements along the x-direction are studied. The y-coordinate and zcoordinate of the center of the elements are 3.0 cm. The x-coordinate of the center of the elements are x = 3.0 cm, 4.5 cm and 6.0 cm, where the center x = 3.0 cm is at the middle of the base soil along the x-direction and the center x = 6.0 cm is at the interface between the base soil and the filter. Fig. 10a shows the velocity is almost the same in all elements before 0.003 s. After 0.003 s, the velocities at centers x = 3.0 cm and x = 4.5 cm do not change so much, which means there was no significant particle movement or erosion happened in the upstream regions far from the interface, and a stable pressure gradient was maintained. While at the interface (x = 6.0 cm) the velocity still fluctuated with time after 0.003 s because some finer particles moved across the interface section and changed the porosity of the soil. But after a longer time, the velocities at those different points do not change a lot, and become stable, which means the interstitial velocity of soil in the sample become stable after rearrangement of the particles and shows the effectiveness of filter F2. When the particle size ratio is 6.94 (Fig. 10b), the change in interstitial velocity with time differs from the case that the particle size ratio is 3.47. After 0.003 s, the velocities in all elements decrease as time increases and tend to decrease with time other than maintain at a stable value after a longer period, which shows the system is not stable under the protection of filter F3. Fig. 11 shows the base soil erosion ratios under different hydraulic gradients. The erosion ratio increased as the hydraulic gradient increased. When the hydraulic gradient is less than 9.6, the increase rate is relatively high. Afterwards, the increase rate is lower.
18.0
The erosion ratio R er (%)
Fig. 8 shows the relationships between the intruding depth and particle size ratio D15 =d85 at different fluid calculation times. The intruding depth increased as the particle size ratio increased at different fluid calculation times, and it also increased with the fluid calculation time. If the particle size ratio was large enough, i.e., approximately 9, the intruding depth was large even when the fluid calculation time was as short as 0.001 s, which indicates that it does not require much time for the base soil with a coarse filter to become eroded. This results in a more dangerous situation than using a finer filter. In the simulation, when the filter is coarse en ough, i.e., in the case the particle size ratio D15 =d85 is 6.94 or 10.42, a few fine grains in the base soil were washed out across the filter continuously after a longer period. Fig. 9 shows changes in the erosion ratio according to the fluid calculation time. The erosion ratios increased as the fluid calculation time increased. The increasing rates of the erosion ratio with time or the erosion rate in different filters were not equivalent. In this study, the erosion rate is defined as the changing rate of the erosion ratio with time. The initial increasing rates were larger before 0.01 s than those after 0.01 s for F1, F2 and F3. The rate reflects how erosion processes develop in filters. When a water pressure difference is applied to the combined soil, the fluid continuously drags the base-soil particles, which move gradually through the filter, thus increasing the number of eroded base-soil particles at the initial time stage in the filters. As time passes, the base particles become gradually clogged in the filter, which reduces the controlling diameter of the possible transport paths formed by the original filter particles. The increase rate of the eroded base-soil particle mass becomes lower, and the eroded mass does not change with time in the filter with grains that are fine enough. Compared with the results given above, the erosion ratio increased linearly with time with a large increase rate for F4, which indicates that the controlling diameters of the transport paths were large enough and the clogged eroded particles did not block the continuation of the erosion. In conclusion, the intruding mass hardly changed with time with the protection of F1 and F2. However, the eroded base soil mass increased with time for F3 and F4; this phenomenon is hereafter called ‘‘progressive erosion’’. Although the increase rate of the number of intruding particles was slightly lower when time increased, particularly for F3, the increasing trend did not change. F3 and F4 are thus considered ineffective filters for protecting base soils.
15.0 F1 F3
12.0
F2 F4
9.0 6.0 3.0 0.0
0
100
200
300
Hydraulic gradient i Fig. 11. Base soils erosion ratios under different hydraulic gradients.
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Y.-H. Zou et al. / Computers and Geotechnics 47 (2013) 48–56 Table 3 Prediction results from numerical simulations and empirical filter criterion. Empirical filter criteriaa
Calculation time (s)
Hydraulic gradient
1.67
E
0.02 0.02 0.02 0.03
6 30 83 83
0.24 0.26 0.26 0.26
E E E E
1.66
3.47
E
0.02 0.02 0.02 0.03
6 30 83 83
1.41 2.57 2.71 2.76
E E E E
6.42
1.26
6.94
I
0.02 0.02 0.02 0.03
6 30 83 83
4.28 7.08 7.43 8.29
I I I I
9.64
1.07
10.42
I
0.02 0.02 0.02 0.03
6 30 83 83
9.28 11.10 13.30 15.80
I I I I
Soil sample
D15 (mm)
Cuf
Base-F1
1.545
2.75
Base-F2
3.21
Base-F3
Base-F4
D15 =d85
Erosion rate (%)
The numerical modela
Cuf is the coefficient of uniformity of filter (d60/d10). a E = Effective, I = ineffective.
Furthermore, changes in the base soil erosion ratios differ for different filters. The erosion ratios mostly increase when the hydraulic gradient is lower than 60 for F3 and F4. The erosion ratio for F4 continues to increase almost linearly as the hydraulic gradient increase, particularly when the gradient become larger than 100, while that increase is little for F3. The erosion ratios essentially do not change at large hydraulic gradients when F1 and F2 protect the base soil. Progressive erosion did not occur in the base soil protected by F1 or F2, though the hydraulic gradient increase was large. However, the base soil eroded more with an increasing hydraulic gradient when protected by F3 or, especially, F4. 4.4. Comparison erosion results between the empirical filter criteria and numerical tests In current practices, empirical filter criteria have been widely used for evaluating the effectiveness of filter for a base soil. Because this study is for gap-graded base soil, the following filter design criterion for erosion could be used [30,31]:
D15 < 4 d85
ð11Þ
where d85 is the diameter of particles at 85% passing of finer-grain fraction of the base soil. Table 3 lists the representative particle size ratio D15 =d85 and simulation results. Based on the empirical filter criterion in Eq. (11), F1 and F2 can be considered as effective filters since the rep resentative particle size ratios D15 =d85 of soil samples Base-F1 and Base-F2 are 1.67 and 3.47 respectively, and are both less than 4.0. From our numerical model, filter effectiveness was identified based on the change in eroded particles with time and the hydraulic gradient. The erosion ratio or eroded mass does not increase under high hydraulic gradient and after longer time in F1 or F2, which differ from the increasing erosion ratio or quantity with time in F3 or F4. Thus it can be concluded that the filters F1 and F2 are effective filters that protect the base soil from progressive erosion. Conversely, filters F3 and F4 are ineffective because their erosion ratios continually increased with time. The results obtained using the numerical simulation agreed with the empirical filter criterion. The result that the erosion quantity and intruding depth of the base soil into filter increase with increasing particle size ratio D15 =d85 in this study agrees with the results from other researchers [5,6,32–35], which also verifies the theory of the constriction controlling size proposed by Kenney et al. [8].
5. Conclusions A granular filter is an essential component of earth–rock fill dams and protects the core soil from erosion and piping. This paper simulated the transient transport of eroded base-soil particles into a filter under different hydraulic gradients using the particle flow method and analyzed and discussed the erosion mechanism of the fine particles. By tracking and record the position of eroded particles in the filter, the distribution, intruding depth and mass of eroded particles in the filter were obtained and the erosion ratios in different cases were calculated. The intruding depth of eroded base-soil particles increases with the increasing particle size ratio at different fluid calculation time. The intruding depth of the eroded base-soil particles increased as the particle size ratio increased at different fluid calculation times. It took less time for the base soil protected by a coarse filter to become dangerously eroded than that protected by a fine filter. Furthermore, more fine particles could move into a coarser filter. The erosion ratio increased as the fluid calculation time increased. The erosion ratio also increased as the hydraulic gradient increased. When the particle size ratio D15 =d85 was small within a certain range, the erosion ratio essentially did not change under high hydraulic gradients, and the intruding mass did not increase at longer times. The simulation results were also used to identify effective filter for the base soil, which were compared with the results obtained from the empirical filter criterion. The results show that the PFM is an effective and practical approach to study the filtration micro-property and erosion mechanism in a base soil–filter system. Acknowledgments This research was substantially supported by Doctoral Fund of Ministry of Education of China (Approval No. 20100181110076), the National Natural Science Foundation of China (No. 41072270) and the Support Program for New Century Excellent Talent in China (Approval No. NCET-07-0569). References [1] Sherard JL. Sinkholes in dams of coarse, broadly graded soils. In: Sakhmander S, editor. Technology review, transactions: proceedings of 13th congress on large dams, vol. 2. New Delhi (India): ICOLD; 1979. p. 25–35. [2] Sherard JL, Dunnigan LP, Talbot J. Basic properties of sand and gravel filters. J Geotech Eng, ASCE 1984;110(6):684–700.
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