1. Introduction

2. Materials and Methods

2.2. Material properties

To understand the seepage flow behaviors inside the rock fill dam consisting of various materials, it is essential to define the appropriate parameter as every soil material has its own properties [16]. MMHPP dam shell consists of a mixture of silty sand, rip-rap consists of boulders, fine filter consists of sand, coarse filter consists of gravel, and clay-core consists of clay. The materials distinguished by the Unified Soil Classification System (USCS) were adopted for parameter values [22].

2.2.1. Soil Water Characteristic Curve (SWCC)

The SWCC describes the change in suction pressure due to the air entrainment causing water to drain out from the soil as a result of negative pore water pressure [23]. The negative pore water pressure changes until it reaches the residual water content, which is considered a fully unsaturated soil condition [24].

Figure 4. Soil Water Characteristic Curve, Rahardjo, H [24].

Figure 4. Soil Water Characteristic Curve, Rahardjo, H [24].

The Soil Water Characteristic Curves for different zonal materials of the dam were adopted based upon the volumetric water content defined by Khire, M.V. and Bosscher, P. J. [25], Figure 5. The sample functions based on the specified saturated and residual water content were used to plot a graph between the volumetric water content and matric suction for different soil materials using SEEP/W.

2.2.2. Hydraulic conductivity / Permeability ($k_w$)

The conductivity defines a fluid flowing through the porous medium under the influence of a hydraulic gradient, expressed in terms of travel distance per time required.

Figure 6. Hydraulic conductivity versus Matric suction, Khire, M.V., and Bosscher, P. J. [25].

Figure 6. Hydraulic conductivity versus Matric suction, Khire, M.V., and Bosscher, P. J. [25].

Based upon Khire, M.V. and Bosscher, P. J. [25], hydraulic conductivity was adopted for the different materials used in the dam of MMHPP. The hydraulic conductivity (Van Genuchten, 1980) in relation to the suction range was implemented using SEEP/W.

2.2.3. Angle of friction (${\mathit{\varphi }}^{\prime }$) and cohesion (${\mathit{c}}^{\prime }$)

The angle of internal friction characterizes the shear strength of the soil to slide on the failure plane. Whereas, the cohesive strength of the soil defines the internal bonding or attractive forces between the soil particles, which provides soil materials to retain their shape and resist deformation under loading conditions [24].

Table 1. Cohesion and internal angle of friction, Das B. M. [26].

Table 1. Cohesion and internal angle of friction, Das B. M. [26].

	Material	${\mathit{\varphi }}^{\prime }$ (∘)
Soils	Soft and firm clay of medium to high plasticity, silty clays, loose variable clayey fills, loose sandy silts (use $c^{\prime }=0-5\mathrm{kPa}$)	17-25
	Stiff sandy clays, gravelly clays, compacted clayey sands and sandy silts, compacted clay fill (use $c^{\prime }=0-10\mathrm{kPa}$)	26-32
	Gravelly sands, compacted sands, controlled crushed sandstone, and gravel fill, dense well-graded sands (use $c^{\prime }=0-5\mathrm{kPa}$)	32-37
	Weak weathered rock, controlled fills of roadbase, gravelly and recycled concrete (use $c^{\prime }=0-25\mathrm{kPa}$)	36-43
Rocks	Chalk	35
	Weathered granite	33
	Fresh basalt	37
	Weak sandstone	32
	Weak siltstone	28
	Weak mudstone	32

2.2.4. Modules of elasticity (E) and Poisson’s ratio (v)

Modules of elasticity help to determine the stiffness of the material, it shows how the material deforms elastically under applied stress and returns to its original position or deforms permanently [27]. Whereas, Poisson’s ratio defines the ratio of lateral strain and longitudinal strain [27], Figure 7, illustrates the relationship between the stress and strain under loading conditions.

The experimental lab test results, elastic constants of various soils modified after U.S. Department of the Navy (1982) and Bowels (1988) (AASHTO Table 10.6.2.2.3b-1) values were adopted to define the Modulus of elasticity and Poisson’s ratio of soil materials for the dam [28].

Table 2. Elastic constants of various soils materials, (AASHTO Table 10.6.2.2.3b-1) [28].

Table 2. Elastic constants of various soils materials, (AASHTO Table 10.6.2.2.3b-1) [28].

Soil type	Typical range of Young’s Modulus (MN/m²)	Poisson’s ratio
Clay: Soft sensitive	2.4-14.4	0.4-0.5 undrained
Medium stiff to stiff	14.4-48
Very stiff	48-95.7
Loess Silt	14.4-57.5	0.1-0.3
	2-19.2	0.3-0.35
Fine Sand: Loose	7.7-11.5	0.25
Medium Dense	11.5-19.2
Dense	19.2-28.7
Sand: Loose	9.6-28.7	0.25-0.35
Medium Dense	28.7-48	0.3-0.4
Dense	48-76.6
Gravel: Loose	28.7-76.6	0.2-0.35
Medium Dense	76.6-95.7	0.3-0.4
Dense	95.7-191.52

2.3. Governing equations

2.3.1. Darcy’s Law

SEEP/W has been formulated on the basis that the flow of water through the soil follows Darcy’s law [28,29]:

$$q=k·i$$

(1)

Where,

q= specific discharge

k = hydraulic conductivity and

i = gradient of the total hydraulic head

2.3.2. 2D Partial differential flow water equation

The governing partial differential equation for two-dimensional seepage implemented in SEEP/W:

$$\frac{\partial }{\partial x}\left(k_x\frac{\partial H}{dx}\right)+\frac{\partial }{\partial y}\left(k_y\frac{\partial H}{dy}\right)+Q=\frac{\partial \theta }{\partial t}$$

(2)

Where,

H = total head

$k_x$= hydraulic conductivity in the x- direction

$k_y$= hydraulic conductivity in the y-direction

Q = applied boundary flux

$\theta$ = volumetric water content

t = time

The GeoStudio solver uses Finite Element Method (FEM) to solve the partial differential equation considering the linear as well as quadratic mesh. For some analyses, such as slope stability and seepage analysis, linear meshes are sufficient to provide accurate results. However, for more complex analyses, such as deformation and stress analysis, quadratic meshes are implemented to capture the curvature and discontinuities of the solution.

2.3.3. Hydraulic conductivity/permeability ($k_w$)

The hydraulic conductivity (Van Genuchten, 1980) in relation to the suction range has been implemented in the model [25].

$$k_w=k_s\frac{{\left[1-\left(a{\psi }^{(n-1)}\right)\left(1+{\left(a{\psi }^n\right)}^{-m}\right)\right]}^2}{\left({\left({(1+a\psi )}^n\right)}^{\frac{m}{2}}\right)}$$

(3)

Where,

$k_w$= hydraulic conductivity

$k_s$ = saturated hydraulic conductivity

a,n,m = curve fitting parameters

n = express as 1/(1-m)

$\psi$ = required suction range

2.3.4. Coefficient of volume compressibility ($m_v$)

The coefficient of volume compressibility is expressed as the reciprocal of modules of elasticity often considered equivalent to the coefficient obtained from the 1D consolidation test [27].

$$m_v=\frac{1}{M}=\frac{a_v}{1+e_o}$$

(4)

Where,

M = modulus of elasticity

$a_v$ = coefficient of compressibility

$e_o$ = initial void ratio

2.3.5. Mohr-Coulomb theory

The shear strength is defined by the Mohr-Coulomb equation [27]:

$$\tau =c+\left(\sigma -u_w\right)tan\varphi$$

(5)

Where,

$\sigma$ = stress,

$u_w$= pore-water pressure,

c and $\varphi$ are the intercept on the shear stress axis and the slope of the Mohr-Coulomb failure envelope

Figure 8. Mohr-Coulomb circle with the failure envelope [27,30].

Figure 8. Mohr-Coulomb circle with the failure envelope [27,30].

The Mohr-Coulomb equation is modified to account for suction strength [31]:

$$\tau =c+\left(\sigma -u_a\right)tan\varphi +\left(u_a-u_w\right)\left[\frac{{\theta }_w-{\theta }_r}{{\theta }_s-{\theta }_r}\right]tan\varphi$$

(6)

Where,

$u_a$ = pore-air pressure,

($u_a$ – $u_w$) = soil suction,

${\theta }_w$ = volumetric water content,

${\theta }_r$ = residual water content,

${\theta }_s$= saturated water content.

2.3.6. Methods for Factor of Safety (FOS)

Several methods have been developed to evaluate the factor of safety against sliding over the past years. The difference between the methods depends upon the consideration of inter-slice normal and shear forces and relationships, as well as the force and moment equilibrium [31]. The typical forces acting upon the slice are represented in Figure 9.

Table 3. Inter-slice force characteristics and relationships (SLOPE/W) [31].

Table 3. Inter-slice force characteristics and relationships (SLOPE/W) [31].

Table 4. Equations of statics satisfied (SLOPE/W) [31].

Table 4. Equations of statics satisfied (SLOPE/W) [31].

2.3.7. Limit Equilibrium method:

The GLEM factory of safety for the horizontal force equilibrium (SLOPE/W) [31]:

$$F_f=\frac{\sum (c\beta cos\alpha +(N-u\beta )tan\varphi cos\alpha )}{\sum Nsin\alpha -\sum Dcos\omega }$$

(7)

The GLEM factory of safety with respect to the moment equilibrium (SLOPE/W) [31]:

$$F_m=\frac{\sum (c\beta R+(N-u\beta )Rtan\varphi )}{\sum Wx-\sum Nf\pm \sum Dd}$$

(8)

$$N=\frac{W+\left(X_R-X_L\right)-\frac{(c\beta sin\alpha +u\beta sin\alpha tan\varphi )}{F}}{cos\alpha +\frac{sin\alpha tan\varphi }{F}}$$

(9)

Where,

c = cohesion

$\varphi$= angle of friction

u = pore water pressure

W = slice weight

D = concentrated point load

$\beta$, R, x, f, d, $\omega$ = geometric parameters

$\alpha$= inclination of slice base

2.3.8. Stress-based FEM method:

The normal force at the base of the slice is primarily unknown in a Limit Equilibrium formulation, plotting the initial in-situ stress distribution provides the stress along the slip surface [8]. The FEM considers the stress-strain distribution within the soil mass considering the soil properties. Therefore, FOS in each slice is different in stress-based FEM whereas, the FOS is constant for all the slices along the slip circle in LEM [8].

2.4. Model parameters

The 2D unstructured meshes consisting of 1995 nodes and 3670 elements were generated with the provided Table 5 material properties.

Figure 10. 2D geometry section of the dam.

Figure 10. 2D geometry section of the dam.

Table 5. Material properties of different layers.

Table 5. Material properties of different layers.

Figure 11. Volumetric water content versus Matric suction curve

Figure 11. Volumetric water content versus Matric suction curve

Figure 12. Hydraulic conductivity versus Matric suction curve

Figure 12. Hydraulic conductivity versus Matric suction curve

2.4.1. Correlation matrix

Using a case study of October 2012, a linear draw-down of 8 hours for reservoir flushing was considered. The time-dependent critical FOS correlation matrix was plotted using Morgenstern-Price, Spencer, Crops of Engineers $\#1$, Lowe-Karafiath, Janbu, Bishop, and Ordinary for the Limit Equilibrium Method.

The Morgenstern-Price, Spencer, Janbu, Bishop, and Ordinary were highly correlated as plotted in Figure 13. The widely used Morgenstern-Price method was adopted for the LEM as it satisfies both the force and moment equilibrium conditions considering inter-slice normal and shear forces, Table 3 and Table 4.

3. Results

3.1. Steady-state

The steady-state FOS was evaluated using LEM and stress-based FEM to analyze the initial state stability of the dam considering the maximum water level in the reservoir.

The Limit Equilibrium Method applied by Morgenstern-Price was adopted to estimate the FOS against sliding at the upstream face of the dam. Whereas the Finite Element stress method applied with in-situ stress was adopted by considering the initial stress of the dam.

Figure 14. Upstream slope stability analysis using LEM.

Figure 14. Upstream slope stability analysis using LEM.

The critical slip circle with FOS of 1.92 was evaluated at the upstream side of the dam considering the maximum reservoir water level using LEM.

Figure 15. Upstream slope stability analysis using FEM

Figure 15. Upstream slope stability analysis using FEM

The critical slip circle with FOS of 1.89 was evaluated at the upstream side of the dam considering the maximum reservoir water level using stress-based FEM.

3.1.1. Stress distribution

Normal base stress acting upon the slice was evaluated using LEM and FEM. Each slice’s normal base stresses were plotted considering the equal radius i.e. 62.4m of slip circle.

The stress distribution at the base of each slice using LEM and FEM follows more or less the same distribution as illustrated by the results, Figure 16. The stress distribution-based FEM gives more realistic values than the LEM however, the true stress on the existing ground may not be the same.

3.1.2. Factor of safety (FOS)

The Limit Equilibrium Method assumed that the FOS is the same for all the slices, which is not realistic whereas, stress-based FEM captures all the local factors of safety of each slice assuming the initial in-situ stress distribution as illustrated by the plot, Figure 17. Therefore, FEM is suitable for small-scale slope stability to capture the local FOS within the slope considering true stress-strain distribution.

3.2. Transient state

The drawing down water level during the flushing of the reservoir is a critical task as the drawn rate directly affects the sliding stability of the slope [7]. The quick reduction of water load and the change in pore-water pressure inside the fill dam depending upon the soil parameter is necessary to access, to safely regulate the flushing without any instability of the slope.

The time-dependent critical FOS using LEM adopting Morgenstern-Price Method and Stress-based FEM were carried out for linear draw-down of 8h, 20h, 40h, 60h, and 80h as plotted in Figure 18.

3.2.1. Non-linear behavior

The stress-strain-based FEM was used to capture the non-linear behaviors adapting Mohr-Coulomb failure criteria during the draw-down of water level as water load on the upstream side of the dam varies with the time, depending upon the draw-down rate.

The non-linear behavior mesh shaded with a yellow region indicates an unstable zone at the upstream side of the dam during the 8 hours of draw-down with the rate of 1.91m/h in all the cases. The results indicate the maximum chances of movement of boulders rip-rap at the upstream side of the dam during the rapid draw-down, as it replicates the case of instability of the boulder rip-rap at the upstream side of the dam after the 8 hours of draw-down Figure 2.

3.2.2. Pressure and seepage discharge variation

The excess pore water pressure variation inside the dam highly dominates the FOS as it reduces the resisting shear force [3,4]. Upstream dam materials should safely release the seepage discharge from the dam during the draw-down of water level reducing the excess pore water pressure inside the dam. Therefore, three cases were established to evaluate the change in pressure and discharge at location A, Figure 19, Figure 20 and Figure 21. The result illustrated that less excess pore water pressure was achieved inside the dam with the increase in permeability at the upstream dam shell, as it quickly releases the pore water pressure from the dam.

Figure 19. At 8h with a draw-down rate of 1.91m/h- Upstream dam shell ($k_w=1\times {10}^{-4}$ m/s).

Figure 19. At 8h with a draw-down rate of 1.91m/h- Upstream dam shell ($k_w=1\times {10}^{-4}$ m/s).

Figure 20. At 8h with a draw-down rate of 1.91m/h - Upstream dam shell ($k_w=8\times {10}^{-4}$ m/s).

Figure 20. At 8h with a draw-down rate of 1.91m/h - Upstream dam shell ($k_w=8\times {10}^{-4}$ m/s).

Figure 21. At 8h with a draw-down rate of 1.91m/h - Three layer horizontal filters (Cross-section Length = 33m, 15m, 8.5m and Height = 1m with $k_w=1\times {10}^{-2}$ m/s).

Figure 21. At 8h with a draw-down rate of 1.91m/h - Three layer horizontal filters (Cross-section Length = 33m, 15m, 8.5m and Height = 1m with $k_w=1\times {10}^{-2}$ m/s).

Figure 22. Pore water pressure at location A with a draw-down rate of 1.91m/h.

Figure 22. Pore water pressure at location A with a draw-down rate of 1.91m/h.

The influence of change in hydraulic conductivity on seepage discharge and reduction of pore water pressure at location A was evaluated at 8 hours. The results illustrated that the higher hydraulic conductivity increases seepage discharge, Table 6 and reduces the excess pore water pressure inside the dam, Figure 19, Figure 20, Figure 21.

Table 6. Seepage discharge at location A.

Table 6. Seepage discharge at location A.

Upstream side of the dam at location A with a draw-down rate $1.91\mathbf{m}/\mathbf{h}$ after $8\mathbf{h}$
Hydraulic conductivity $\left({\mathrm{k}}_{\mathrm{w}}\right)$ m/s	Seepage discharge $\left({\mathrm{m}}^3/\mathrm{s}/{\mathrm{m}}^2\right)$
$1\times {10}^{-4}$	$2.81\times {10}^{-5}$
$8\times {10}^{-4}$	$8.86\times {10}^{-5}$
$Three$-layers horizontal filter $=1\times {10}^{-2}$ conductivity at location A is $1\times {10}^{-4}$	$3.96\times {10}^{-5}$

Figure 23. Left represents FOS and right illustrates pore water pressure at location A for different draw-down rate using LEM (Upstream dam shell ($k_w$= ${10}^{-4}m/s$)).

Figure 23. Left represents FOS and right illustrates pore water pressure at location A for different draw-down rate using LEM (Upstream dam shell ($k_w$= ${10}^{-4}m/s$)).

Figure 24. Left represents FOS and right illustrates pore water pressure at location A for different draw-down rate using LEM (Upstream dam shell ($k_w$= $8\times {10}^{-4}m/s$)).

Figure 24. Left represents FOS and right illustrates pore water pressure at location A for different draw-down rate using LEM (Upstream dam shell ($k_w$= $8\times {10}^{-4}m/s$)).

Figure 25. Left represents FOS and right illustrates pore water pressure at location A for different draw-down rate using LEM (Upstream three horizontal filter layers ($k_w$= $1\times {10}^{-2}m/s$)).

Figure 25. Left represents FOS and right illustrates pore water pressure at location A for different draw-down rate using LEM (Upstream three horizontal filter layers ($k_w$= $1\times {10}^{-2}m/s$)).

Figure 26. Left represents FOS and right illustrate pore water pressure at location A for different draw-down rate using FEM (Upstream dam shell ($k_w$= ${10}^{-4}m/s$)).

Figure 26. Left represents FOS and right illustrate pore water pressure at location A for different draw-down rate using FEM (Upstream dam shell ($k_w$= ${10}^{-4}m/s$)).

Figure 27. Left represents FOS and right illustrate pore water pressure at location A for different draw-down rate using FEM (Upstream dam shell ($k_w$= $8\times {10}^{-4}m/s$)).

Figure 27. Left represents FOS and right illustrate pore water pressure at location A for different draw-down rate using FEM (Upstream dam shell ($k_w$= $8\times {10}^{-4}m/s$)).

Figure 28. Left represents FOS and right illustrate pore water pressure at location A for different draw-down rate using FEM (Upstream three horizontal filter layers ($k_w$= $1\times {10}^{-2}m/s$)).

Figure 28. Left represents FOS and right illustrate pore water pressure at location A for different draw-down rate using FEM (Upstream three horizontal filter layers ($k_w$= $1\times {10}^{-2}m/s$)).

The overall FOS using the LEM and FEM was found to be more or less similar, the resulting different values of FOS might be due to various reasons such as LEM relies on the equilibrium equations and depends upon the predefined failure surface, does not consider the actual condition of the nature i.e. the behavior of the soil parameter[8]. On the other hand, true complex behaviors of the soil are hard to determine and might not be truly represented which has uncertainties in the analysis for FEM.

Table 7. Overall critical minimum FOS

Table 7. Overall critical minimum FOS

Overall minimum FOS
Draw-down	Draw-down rate	Dam shell $\left(k_w={10}^{-4}\right)$		Dam shell $\left(k_w=8\times {10}^{-4}\right)$		Upstream horizontal filter
		Using LEM	Using FEM	Using LEM	Using FEM	Using LEM	Using FEM
8 hours	$1.91\mathrm{m}/\mathrm{h}$	1.28	1.27	1.43	1.41	1.56	1.54
20 hours	$0.76\mathrm{m}/\mathrm{h}$	1.35	1.33	1.49	1.47	1.59	1.57
40 hours	$0.38\mathrm{m}/\mathrm{h}$	1.42	1.38	1.53	1.51	1.60	1.58
60 hours	$0.25\mathrm{m}/\mathrm{h}$	1.46	1.42	1.54	1.52	1.61	1.59
80 hours	$0.19\mathrm{m}/\mathrm{h}$	1.49	1.44	1.55	1.53	1.62	1.60

4. Discussion

5. Conclusion and Recommendation

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