An analytical model for estimation of internal erosion rate

ABSTRACT

Estimating erosion rate of solid particles in a porous medium is of interest to geotechnical engineers; which use analytical or numerical models for this purpose. Constitutive law of erosion is a key component in the development of such models. These models estimate the solid erosion rate as a function of various modelling parameters such as fluid velocity and time.

Using the principles of dimensional analysis, a constitutive law is proposed for assessment of the rate of erosion in relation to the fluid velocity and a dimensionless proportionality constant called the erosion coefficient,$\lambda$. Based on physics of the erosion process, experimental observations and approximation theory, $\lambda$ is expressed as a function of grain density, particle Reynolds number and porosity variation. Then, the proposed constitutive model is combined with the principle of conservation of mass to arrive at an analytical model for estimation of internal erosion rate.

The analytical model shows that the erosion rate has a non-linear direct relationship with fluid velocity and a non-linear inverse relationship with time. The proposed analytical model is calibrated and validated using experimental data available in the literature. The validation results show that the model estimations of erosion rate can closely reproduce experimental data.

KEYWORDS:

• Internal erosion rate
• analytical modeling
• dimensional analysis
• approximation theory

Nomenclature

$\mathrm{A}$, $B$	=	Parameters used to estimate critical fluid velocity
$A_{cs}$	=	Cross sectional area of an immersed body projected in the direction of flow
$c$	=	Concentration of solids in fluid phase
$c_{cr}$	=	Critical concentration of solids in the fluid phase
$C_D$	=	Drag coefficient
$D_p$	=	Particle diameter or average particle diameter
$F_{D,n}$	=	Drag force of fluid exerted on the particle in direction n
$F_{G,n}$	=	Resistive gravitational force in direction n
$F_{PG,n}$	=	Pressure gradient force exerted on the particle in direction n
$G_s$	=	Specific gravity of the solid particles
$i$	=	Hydraulic gradient
$k$	=	Permeability
$K$	=	Hydraulic conductivity
$m$	=	Eroded mass
$\dot{m}$	=	Tangent erosion rate per unit volume
$m_t$	=	Total mass of the test specimen
$n$	=	A symbol representing an arbitrary direction
$n_p$	=	Number of particles
$p$	=	Percentage of the eroded fine particles
$p_{if}$	=	Initial percentage of the fine particles
${\bar{q}}_i$	=	Flow flux of the mixture of solids and fluid
$Rp$	=	Particle Reynolds number
$R{p}_{cr}$	=	Particle Reynolds number calculated at critical fluid velocity
$t$	=	Time
$t_0$	=	Time at which erosion starts
$v_{A,n}$	=	Actual fluid velocity in direction n
$v_{f,n}$	=	Apparent (Darcy) fluid velocity in direction n
$V_{EB}$	=	Volume of the erosion boundary
$v_{cr,n}$	=	Critical fluid velocity in direction n
$V_p$	=	Volume of particle
$V_t$	=	Total volume of the sample
$\alpha$, $\beta$, ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$	=	Dimensionless calibration parameters
$\mathrm{\Delta }h$	=	Hydraulic head difference
$\mathrm{\Delta }L$	=	Sample length
${\theta }_n$	=	Angle between direction n and vertical upward direction
$\lambda$	=	Erosion coefficient
$\mu$	=	Fluid dynamic viscosity
${\rho }_f$	=	Fluid density
${\rho }_s$	=	Grain density
$\mathrm{\Phi }$	=	Fluid potential
$\phi$	=	Porosity
${\phi }_0$	=	Original porosity of the assembly of particles
$∥\hspace{0.17em}∥$	=	Norm of a vector

Acknowledgments

The authors would like to acknowledge the research funding for this study provided by NSERC through CRDPJ 387606-09.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Natural Sciences and Engineering Research Council of Canada [CRDPJ 387606-09].