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Modeling of Suffusion Considering the Influence of Soil Gradation

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Abstract

Suffusion is a type of internal erosion in soil that fine particles detach from the matrix and get transported in pores with the seepage flow. The internal erosion rate is a key factor influencing the development and resulting consequences of the erosion process. In this study, a new method to evaluate the internal erosion rate is proposed, introducing the concept of the representative elementary volume of soil gradation in the model. The maximum particle size under erosion is related to the seepage velocity. The remaining mass of particles is described by a decay function for any given particle size. The eroded particles are assumed to get transported with the seepage water without dispersion. The effects of key parameters in the new method are analyzed using a finite element model, and a method is introduced to determine the parameters based on test results. Unlike existing methods, the proposed method considers the influence of soil gradation. The proposed method can accurately predict the development of internal erosion under different hydraulic conditions.

Article Highlights

  • A method to calculate internal erosion rate is proposed.

  • The erodible mass is related to the seepage velocity.

  • The influence of soil gradation on internal erosion is considered.

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Data Availability

All data, models, and code generated or used during the study appear in the submitted article.

Abbreviations

c :

Volume fraction of eroded particles

c k :

Kozeny coefficient

C n :

Coefficients in Fr

C t :

Coefficient in characteristic time tc

C μ :

Coefficient in effective viscosity of fluid

d :

Particle size (m)

d 10 :

Particle size with a 10% cumulative mass percentage (m)

d e :

Maximum size of particles under erosion (m)

d gl :

Lower limit of the gapped interval in a gapped gradation soil (m)

d max :

Maximum particle size (m)

d min :

Minimum particle size (m)

d s :

Maximum size of erodible particles (m)

dV f :

Volume of fluid part in an REV of soil (m3)

dV fe :

Volume of eroded particles in an REV of soil (m3)

dV fw :

Volume of water in an REV of soil (m3)

dV r :

Volume of REV of soil (m3)

dV s :

Volume of solid part in an REV of soil (m3)

dV se :

Volume of erodible particles in an REV of soil (m3)

dV sk :

Volume of skeleton particles in an REV of soil (m3)

e :

Void ratio

F b :

Buoyancy force of particles (N)

F c :

Constraining force of particles from the skeleton (N)

F d :

Drag force of particles based on Stokes’ law (N)

F g :

Weight of particles (N)

F r :

Resistance force of particles (N)

h(v f):

Heaviside step function locating at vfd

i :

Hydraulic gradient

i cr :

Critical hydraulic gradient to erode minimum particles

K :

Hydraulic conductivity (m s−1)

k ed :

Rate constant of internal erosion

n :

Dummy index for summation

n :

Outer normal vector

p :

Excess pore fluid pressure (Pa)

p 0 :

Pressure applied on the inlet boundary (Pa)

P(d):

Function of the grading curve

P 0(d):

Function of the initial grading curve

Q e :

Internal erosion rate (kg m−3 s−1)

r(d):

A rectangle function with support of [dmin, ds]

s s :

Specific surface (m−1)

t :

Time (s)

t 0 :

Characteristic time in Sterpi’s erosion model (s)

t c, t c0 :

Characteristic time to determine the erosion rate (s)

t ed :

Duration of erosion of particles with size de (s)

u f :

Darcy’s velocity of fluid (m s−1)

u e :

Content of remaining erodible particles in Sterpi’s erosion model

u 0 :

Initial content of erodible particles in Sterpi’s erosion model

u f :

Darcy’s velocity magnitude of fluid (m s−1)

v s :

Velocity vector of solid part (m s−1)

v s :

Velocity magnitude of solid part (m s−1)

v cr :

Critical velocity to erode minimum particles (m s−1)

v f :

Velocity of fluid (m s−1)

v f :

Velocity magnitude of fluid (m s−1)

v fd :

Corresponding velocity to erode particles with size de (m s−1)

α, β, γ :

Nondimensional parameters in Sterpi’s erosion model

\(\overline{{\rho_{\text{e}} }}\) :

Partial density of eroded particles (kg m−3)

\(\overline{{\rho_{\text{f}} }}\) :

Partial density of fluid in pores (kg m−3)

\(\overline{{\rho_{\text{r}} }}\) :

Density of REV (kg m−3)

\(\overline{{\rho_{\text{s}} }}\) :

Partial density of the solid part (kg m−3)

\(\overline{{\rho_{\text{w}} }}\) :

Partial density of water (kg m−3)

∇p :

Pressure gradient (Pa m−1)

Δt i :

Erosion duration of particles (s)

ξ :

Exponent of vf in characteristic time tc

ζ :

Exponent in resistance force Fr

ψ :

Drag force coefficient in Stokes’ law

κ :

Permeability of soil (m2)

κ 0 :

Initial permeability of soil (m2)

μ :

Effective viscosity of fluid (Pa s)

μ 0 :

Viscosity of water (Pa s)

ρ f :

Density of fluid in pores (kg m−3)

ρ s :

Density of particles (kg m−3)

ρ w :

Density of water (kg m−3)

ϕ :

Porosity of soil

ϕ 0 :

Initial porosity of soil

ϕ e :

Volume ratio of eroded particles to the volume of REV

ϕ es :

Volume ratio of eroded particles to initial volume of particles

ϕ r :

Volume ratio of all remaining particles to the volume of REV

ϕ rs :

Volume ratio of all remaining particles to initial volume of particles

ϕ u :

Ultimate porosity of soil

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Acknowledgements

The research work was funded by the National Natural Science Foundation of China (NSFC) (Grant No. 51479112) and the National Dam Safety Research Center of China (Grant No. CX2019B08).

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Authors and Affiliations

  1. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Jinfeng Bi & Xianqi Luo

  2. Faculty of Civil Engineering and Urban Construction, Jiujiang University, Jiujiang, 332005, People’s Republic of China

    Hui Shen

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  1. Jinfeng Bi
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  2. Xianqi Luo
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Correspondence to Xianqi Luo.

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Bi, J., Luo, X. & Shen, H. Modeling of Suffusion Considering the Influence of Soil Gradation. Transp Porous Med 136, 765–790 (2021). https://doi.org/10.1007/s11242-020-01534-6

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