6.4 Stochastic seepage and slope stability analysis
6.4.2 Stochastic seepage and slope stability analysis under steady seepage: Impact
Steady seepage solution does not require the SWCC term (Se−h) in Eq. 6.18 but only theK−h constitutive relationship. Although numerical codes can be used for steady seepage also, this study utilizes a close form solution for steady seepage derived by Yeh (1989). This expression utilizes the Eq. 6.20 for solving the one-dimensional form of Eq. 6.16 and is given as follows:
ln(K)de
-2 -1 0 1
Cumulative dennsity
0 0.2 0.4 0.6 0.8
1 (b)
p=0.463 Empirical GEV
ln(K)de
-2 -1 0 1
dennsity
0 0.5 1
1.5 (a)
GEV
Figure 6.13: Empirical and calculated PDF and CDF for the de-trended data ln(Kd) in Fig. 6.3(b).
Core No.
1 2 3 4 5 6 7 8 9 . . . 13 . . . 20
p value
0.05 0.2 0.4 0.6 0.8
1 (a)
Core No.
1 2 3 4 5 6 7 8 9 . . . 13
p value
0.05 0.2 0.4 0.6 0.8
1 (b)
GEV normal
Figure 6.14: Kolmogorov–Smirnov KS test p-values for the de-trended data ln(K)de at site (a) A−A0 (b)B−B0.
hxi = ln{e−αγwdx(eαγwhxi−1 +q/ksxi)−q/ksxi)}
αγw
(6.23) Since this section aims to investigate the potential impact of spatial dependence structure, only k is treated as a random field, and parameter α is treated as constant. Ideally, all the other parameters should also be represented by random fields, but onlyk was taken as the random field so that the results can completely reflect the difference only due to k and without interference from other parameters. This is a common practice to investigate the impact of some statistical assumption, e.g. Gaussian dependence structure (Tang et al., 2013, 2015, e.g.). The parameters for the random field and slope are summarized in Table 6.1 after Li et al. (2009) and Santoso et al.
(2011).
Fig. 6.16 presents 1000 realizations for h(x) using Gaussian copula in D-vine formulation. In Fig. 6.16, it can be noted that for some realizationshis positive, this is because, in layered soils, h not only depends upon conductivity but also the relative position of each other. If the lower layer is less conductive than the upper layer, then under certain realizations, positiveh may develop (Yeh, 1989; Santoso et al., 2011). The F S(x) realizations are also computed using the h profiles. It can be noted that some portions of theF S(x) <1 are observed at very shallow depths, e.g. x < 0.5.
Slope failures at such low depths are unrealistic (Santoso et al., 2011), therefore not considered in further calculations of Pf.
ln(K)
-12 -11 -10 -9 -8
Depth(m)
0
0.5
1
1.5
2
Gaussian Clayton Frank Gumbel
Figure 6.15: Simulated random field of hydraulic conductivityln(K) using different copulas.
To investigate the impact of spatial dependence structure ofk, the above exercise was conducted using all the other three copulas, and the resulting quantiles are presented in Fig. 6.17. It can be noted that there are certain differences among the quantiles of h(x) even though all the input parameters other than dependence structure are the same. Particularly the difference is more pronounced at lower and higher quantiles. For the F S(x) quantiles in Fig. 6.17, it can be noted that the same 2.5%Q can result in different failure events with the choice of copulas.
Fig. 6.18a presents the Pf corresponding to various copulas for the steady seepage case. It can be noted that thePf may vary almost 10 times (0.003−0.03) with the choice of copula, i.e. spatial dependence structure. This result is significant as this implies that not only spatial dependence can be non-Gaussian (shown in the previous section), but choosing an arbitrary spatial dependence may also significantly affect the Pf. Although Fig. 6.18a demonstrates the importance of spatial dependence structure, the Pf values are not very practical for the slopes. The general acceptable range ofPf for slopes as per Salgado and Kim (2014) is 2×10−4−10−2. The acceptable range of Pf for various cases are summarized in Table 1.1.
In Fig. 6.18a, the Pf are very close to the upper limit of Pf = 10−2. Therefore, a case for a lower Pf well within the acceptable range of 10−4 −10−2 was further considered. Fig. 6.18b presents the result of the analysis. The seepage flux was considered lower at q =−0.15m/day in comparison to theq =−0.20m/day (ref. Fig. 6.18a to reduce thePf. Three zones are marked in the Fig. 6.18b. The zones withPf ≤10−4 is marked as too conservative (too lowPf). Although Pf ≤10−4is not undesirable, it might result in unnecessary cost additions to the project. The zone withPf ≥0.01 is marked as risky as mostly temporary structures are considered to be constructed in this zone (Salgado and Kim, 2014). The well acceptable zone of 10−4−10−2 after Salgado and Kim (2014) is termed as acceptable. It can be noted that the practical importance of this study can be appreciated even more in Fig. 6.18b than Fig. 6.18a. This is because in Fig. 6.18b, thePf due to choice of copula vary close to 100 times (10−4−10−2) and span entirely over the acceptable zone.
h [m]
-2 -1.5 -1 -0.5 0 0.5
Depth [m]
0 0.5 1 1.5 2 2.5 3
FS
0.5 1 1.5 2
Depth [m]
0 0.5 1 1.5 2 2.5 3
Figure 6.16: N Realization of pressure head h and Factor of safety F S versus depth us- ing Gaussian copula for spatial dependence of hydraulic conductivity K. N = 1000.
h [m]
-1.5 -1 -0.5 0 0.5
Depth [m]
0 0.5 1 1.5 2 2.5 3
FS
0.5 1 1.5 2
Depth [m]
0 0.5 1 1.5 2 2.5 3
Gaussian Frank Clayton Gumbel 2.5%Q
50%Q
97.5%Q 50%Q
2.5%Q 97.5%Q
Figure 6.17: Quantiles of pressure head h and Factor of safety F S versus depth using various copulas for spatial dependence of hy- draulic conductivity K. N = 10000.
It should also be noted that only four copulas and one set of parameters mentioned in Table 6.1 were used for this case study. Further simulation experiments are needed to investigate the impact of spatial dependence structure with variation in the scale of fluctuation, boundary flux and other input parameters statistics. However, the same was not performed in this study for the sake of brevity. Nevertheless, this cases study for the first time demonstrated that the potential impact of spatial dependence structure of hydraulic parameters e.g. kis significant in the context of stochastic seepage and slope stability analysis and therefore deserves attention.