Incorporating Observations to Update the Piping
Reliability Estimate of the Francis Levee
Willem KANNING a, Carolyne BOCOVICH a, Timo SCHWECKENDIEK b,cand Michael A.MOONEY a a
Civil and Environmental Engineering, Colorado School of Mines, USA
b
Unit Geo-engineering, Deltares, the Netherlands
c
Department of Hydraulic Engineering, Delft University of Technology, the Netherlands
Abstract. Piping failure of levees is one of the main contributors to flood risk. Piping occurs when the head difference over a levee results in uplift of the inland blanket, followed by backward erosion in the aquifer due to high local flow velocities. Whether piping occurs, highly depends on local geo-hydrological circumstances which are uncertain due to a combination of high soil variability and limited site investigation. Fragility curves, which show the failure probability as a function of water level, are increasingly used for risk assessment of levees to address this uncertainty. The objective of this paper is to show how pore water pressure measurements and visual field observations reduce or increase the calculated failure probability of a levee. The Francis levee in Mississippi (USA) is used to show the effects since this levee is historically known for being very piping sensitive. The first step is to construct fragility curves based on analytical piping models. The second step consists of using historic piezometer data to update knowledge about aquifer and blanket properties. The final step is to use historic seepage observations and sand boils to update the fragility curve. This paper shows that measurements and observations can significantly reduce the geo-hydrological uncertainties regarding a piping-sensitive subsoil. The failure probability of the Francis levees increased with a factor 10 after incorporation of all the data. This increases the prediction accuracy of levee performance at high water levels and subsequently the estimation of whether it may require reinforcement or not.
Keywords. reliability, levee, piping, heave, Bayesian updating
1. The Lower Francis, MS, levee
1.1. Rationale
Piping is one of the most important failure modes of levees. It is also referred to as backward erosion or internal erosion and occurs when the head difference over a levee results in uplift of the inland blanket, followed by outflow and sand particles and backward erosion in the aquifer due to high local flow velocities (see section 2).
Soil typically exhibits high spatial variability. Together with limited data of the subsoil; this results in large uncertainties with respect to the subsoil parameters used to assess the reliability of a levee. The effects of these uncertainties on the levee performance are typically quantified in a reliability analysis (see e.g. Polanco and Rice, 2011).
In this paper, we demonstrate how to (1) reduce the uncertainty in the subsoil parameters by using observations of sand boils and pore
water pressure measurements, and (2) how to use this reduction in uncertainty to improve the estimated reliability for a specific case study, the Francis levee.
1.2. The Francis Levee
The Lower Francis levee is located along the Mississippi river, 20 miles southwest of Clarksdale, MS. The levee is very sensitive to seepage and sand boils, and is one of the levees analyzed for seepage in detail after the 1950 extreme Mississippi river levels (USACE, 1956A). During the 1950 event, heavy seepage and sand boils were observed. Furthermore, piezometers that had been installed in 1948 provide pore water pressure data for the 1950 event. Both these and the sand boil observations are used in this paper. During the extreme water levels of the Mississippi river in 2011, seepage and sand boils were observed and mitigated again, resulting in the installation of relief wells. © 2015 The authors and IOS Press.
This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License.
Only the historic data (USACE, 1956A and B) is used to construct cross-section and determine input parameters.
The cross-section of the Francis levee is shown in Figure 1. The levee consists of a levee core and wide berm (~ 100 m) on top of a 3.5 to 4 m thick blanket that is located on a thick aquifer. On the river side, the river likely connects with the aquifer at the location of an old borrow-pit. Whether this is the case will be investigated in this paper.
Figure 1. The Francis, MS, levee cross-section.
2. The Piping Failure Mechanism
2.1. Mechanism Description
Piping is usually described in the US by initiation, continuation, progression and breach (Fell and Fry, 2007), all of which need to occur for flooding to happen. In this paper, we use three sub-failure mechanisms to describe piping failure: uplift, heave and piping. The former two correspond to initiation and continuation, the latter to progression. The three sub-failure mechanisms and their corresponding field observation are shown in Figure 2. Uplift occurs if the uplift force of the water below the blanket is higher than the weight of the blanket, resulting in rupture of the blanket and the observation of a water boil or seepage. Heave occurs when the vertical hydraulic gradient over the blanket is larger than the critical heave gradient, resulting in the outflow of sand particles and the observation of a sand boil. Piping occurs when the horizontal gradient over the levee is large enough to cause erosion of sand to form a continuous channel in the sand aquifer below the blanket from the outer side to the inner side. When this happens, a breach usually occurs.
mechanism observation
1. uplift
2. heave
3. piping
Figure 2. The three sub-mechanisms of piping (www.enwinfo.nl – Technical Report Sand Boils)
2.2. Limit States
The three sub-mechanisms are described by the limit state functions Z (Zu for uplift, Zh for heave, Zp for piping), meaning that failure occurs when Z < 0: 0 , b w u c u w bL h Z i i z J J J (1) 0 ,h ,h h c c bL h Z i i i z (2) ,S p c ell Z 'h ' (3) h
where ic,u is the critical vertical gradient for uplift, i is the occurring vertical gradient, b is the blanket total unit weight, w is the water unit weight, h0 is the hydraulic head below the blanket at the toe of the levee, zbL is the thickness of the landside blanket ic,u is the critical heave hc,Sell is the critical head difference h is the occurring outside water level. For more information about the limit states of uplift, heave and piping, please refer to, e.g., Schweckendiek et al. (2014) or Sellmeijer (1988).
2.3. Blanket Theory
Blanket theory was developed after the 1950 event in order to predict and design for seepage (USACE, 1950A). The blanket equations determine analytically the hydraulic head (h0) below the blanket at the toe of the levee in order to determine the occurring gradients as a function of outside water level. The blanket configuration used in this paper consists of a river side blanket (L1) and an infinite landside blanket. For more information about blanket theory, see USACE (2000). The required input for the blanket theory is presented in section 3.3.
3. Piping Reliability
3.1. Failure Probability
Reliability is usually referred to as the probability of non-failure 1-P(F), where the probability of failure P(F) is generically given as:
( ) ( 0)
P F P Zd (4)
where Z could either be a single failure mechanism’s limit state or a combination of limit states. Eq. 4 is evaluated in this paper using crude Monte Carlo Sampling (MCS).The failure of the levee occurs when uplift and heave and piping occur:
( ) (Zu 0 Zh 0 Zp 0)
P F P (5)
3.2. Input Parameters
The Mississippi River level, being the load on the levee, is modeled with a Gumbel distribution with parameters (location = 49.7 and scale = 0.910), based on historic data. The other (strength) input parameters that are used for the reliability analysis are based on USACE (1956) and Kanning (2012) for the uncertainties and are summarized in Table 1. It should be noted that the critical gradient is lower than Terzaghi’s critical heave gradient (which is close to 1), and is based on the USACE (1956) findings.
Table 1. Prior input parameters
symbol description dist mean std
ic [m] Critical heave gradient L 0.7 0.1 zbL, zbR [m] Blanket thickness L 3.7 0.37 kbL, kbR [m/s] Blanket conductivity L 10-5 10-5 L1 [m] Entrance length L 91.6 9.2 L2 [m] Levee width L 176 17.6 kfL, kfR [m/s] Aquifer conductivity L 1.7 10-3 1.7 10-3 d [m] Aquifer thickness L 35 3.5 3.3. Subsoil Scenarios
The probability distributions as defined in Table 1 reflect our uncertainty in the values of the model input parameters of the statistically homogeneous units in the case study area. Due to the formation of the Mississippi basin, other units may be present as well. This can be modeled by defining multiple scenarios Ei with each scenario an associated (prior) probability of occurrence P(Ei) (see e.g. Schweckendiek, 2014). The first scenario is according to the USACE (1956A) data as presented in Table 1 with a probability of 0.9. The second scenario is a situation with a much thinner blanket (mean 2 m, standard deviation 2 m) and a probability of 0.1. 3.4. Fragility Curve Based on Combined Failure Mechanisms
The fragility curve shows the failure probability conditional to the water level P(F|h). The fragility curve for the main scenario (1) is shown in Figure 3. It is constructed by evaluating the limit states of uplift, heave and piping (eqs. 1,2 and 3) with the input of Table 1. The combined fragility curve is constructed by applying eq. 5. The total annual failure probability is obtained by integrating the fragility curve over the probability density of the water level; this is 0.024.
Figure 3. Fragility curve scenario 1 for uplift and heave
4. Updating with Sand Boil Observations
4.1. Observations During the 1950 Event
During the 1950 event, heavy seepage (uplift) and sand boils (heave) were observed at a water level of NAVD + 51.5 m. This provides valuable information to update the random variables and failure probability since only combinations of parameters that should theoretically result in uplift and heave are plausible.
4.2. Bayesian Updating Sand Boil Observations The observations are incorporated by following the methods developed by Schweckendiek et al. (2014). In general, the observations () are incorporated by using Bayes’ rule:
( | ) P( ) (F | ) ( ) P F F P P H H H (6)
where the probability of failure given the observation, is the likelihood of the observation and P(F) is the prior probability,. is referred to as the posterior throughout the paper. This can be rewritten into the following equation that is easier to implement in a Monte Carlo scheme (Schweckendiek, 2014):
(Z(X) 0 h(X) 0) (F | ) (h(X) 0) P P P H (7)
where h(X)<0 is the observational limit state expressed as a water level at which uplift and/or piping are observed. The observation of uplift and heave is so-called inequality information, as we know the mechanisms occurred at the given water level or a lower water level.
There is no observational limit state defined for piping, as it cannot be determined whether piping would have occurred without flood fighting or with longer event duration. However, the updated parameters due to uplift and heave observations, still can be used to update the piping (Sellmeijer) estimate.
It is assumed that the water level is aleatory uncertainty, which cannot be updated. The other (strength) parameter are assumed to be reducible, these can be updated by the observations. 4.3. Effects of Observations on Random Variables
The effects of the observation of uplift on the random variables for the main scenario (1) is shown in Figure 4. Only the variables that change significantly are shown. It can be seen in both cases that mainly the conductivity of the aquifer and blanket, as well as the blanket thickness are updated. The log-scale of the conductivity ‘hides’ the very significant effect.
Similar effects are seen in case of a heave observation. Mainly the conductivities and thicknesses are update. The critical gradient is updated for heave as well, while the blanket density is not (since this parameter is not in the heave equations).
Figure 4. Update main random variable after heave observation 45 50 55 60 65 0 0.2 0.4 0.6 0.8 1
water level h [m NAVD]
P(
F
|h
)
Combined fragility curve, scenario 1
combined uplift heave Sellmeijer 10-4 10-3 10-2 10-1 0 200 400 600 f(k -fL )
hydraulic conductivity aquifer k-f [m/s]
10 15 20 25 0 0.05 0.1 0.15 0.2 0.25 f( g a mma b )
blanket density gammab [kN/m3]
0 2 4 6 8 10
0 0.5 1 1.5
thickness landside blanket z-bL [m]
f(z -b L ) priorposterior 10-7 10-6 10-5 10-4 0 0.5 1 1.5 2x 10 5 f(k -b L )
hydraulic conductivity blanket k-b [m/s]
4.4. Effects of Observations on Fragility Curve The effects of the updated random variable on the fragility curve due to the uplift and heave observation are shown in Figure 5.
Figure 5. Updated fragility curve for heave, given uplift and heave observation
Note the fragility curve for uplift and heave is 1 above the water level at which the observation occurred since all their strength parameters are assumed to be reducible. The fragility curves of heave and piping are updated as well because their input variables are updated due to the uplift observations. A strong shift in the fragility curve is observed.
5. Updating with Measurements
5.1. Updating for Pore Water Pressure Measurements
The installed piezometers during the 1950 event measured a maximum head below (hobs) the blanket of NAVD + 49.5 m, which occurred with a corresponding water level of NAVD + 51.5 m The methods from Schweckendiek and Vrouwenvelder (2013) are used to include pore water pressure measurements. The same scheme as in section 4.1 may be used to include pore water pressure measurements; except that we have to deal with equality information (we know the head given the outer water level) instead of inequality information. This can be dealt with by defining measurement uncertainty em (mean 0 m, standard deviation 0.2 m) and following the
approach illustrated by Schweckendiek and Vrouwenvelder (2013) who apply the method developed by Straub (2011).
5.2. Effects of Measurements on Random Variables
The effects of the pore water pressure measurements on the distributions of the random variables are shown in Figure 6. Mainly the hydraulic conductivities of blanket and aquifer are updated.
Figure 6. Update main random variable after pore water pressure measurement
6. Combining Observations and Scenarios
6.1. Combined Effect Random Variables
The individual effects of uplift, heave and pwp observation, as well as the combined effect are shown in Figure 7.
Figure 7. Update aquifer conductivity for all measurements and observations 45 50 55 60 65 0 0.2 0.4 0.6 0.8 1
water level h [m NAVD]
P(
F
|h
)
update fragility curve uplift observation, scen 1
prior posterior uplift posterior heave posterior piping posterior combined observed WL 10-4 10-3 10-2 10-1 0 200 400 600 f(k -fL )
hydraulic conductivity aquifer k-f [m/s]
2 3 4 5 6 0 0.5 1 1.5 f(z -b L )
thickness landside blanket z-bL [m]
10-7 10-6 10-5 10-4 0 1 2 3x 10 5
hydraulic conductivity blanket k-b [m/s]
f(k -b L ) prior posterior 20 25 30 35 40 45 50 0 0.05 0.1 0.15 0.2 f(d ) thickness aquifer d [m]
Update variables for pwp observation, scenario 1
0 100 200 300 400 500 600
hydraulic conductivity aquifer k-f [m/s]
f( k -fL ) prior due to uplift due to heave due to pwp combined
It can be observed that all observations and measurements have a significant effect; the combined updated distribution is mainly determined by the pwp measurements for this case.
6.2. Combined Fragility Curve
This section determines the combination of both uplift and heave observations, the pore water pressure measurements and the probability of another subsoil scenario. The result is a combined fragility curve as shown in Figure 8. Figure 8 shows the posterior distributions of both scenarios and the combined posterior distribution. Both posteriors change significantly due to the measurements and observations, but due to its limited prior probability (0.1) of scenario 2, it does not impact the combined fragility curve too much and the combined fragility curve mainly follows the curve of Scenario 1.
The annual failure probability based on the prior input was determined to be 0.024 (Section 3). The annual failure probability based on the posterior analysis is determined to be 0.17, which is an order higher than the prior probability.
Figure 8. Updated fragility curve using all measurement, observations and scenarios.
6.3. Update Scenario Probabilities
Using the same methods, also the updated scenario probabilities may be found. Incorporating all observations and measurements results in an updated probability of scenario 1 of
0.92 (from 0.9), and scenario 2 of 0. 08 (from 0.1), indicating the likelihood of a (locally assumed) presence of a thinner blanket hardly changes. On the other hand, the small probability (0.1) of no connection between borrow-pit and aquifer reduces to 0.01.
7. Conclusions
This paper shows that levees with high uncertainties in the relevant variables, experience a significant change of the reliability estimate by incorporating information on the presence of seepage and sand boils or pore water pressure measurements. Especially the hydraulic conductivity of blanket and aquifer, as well as blanket thickness can be significantly updated. In case of the Francis levee, based on the 1950 historic data, a factor of 10 in failure probability increase is found.
References
Fell, R., and J. J. Fry, eds. (2007). Internal Erosion of Dams
and Their Foundations. London UK: Taylor & Francis.
Kanning, W. (2012). The weakest link – spatial variability in
the piping failure mechanism of dikes. PhD thesis Delft
University of Technology.
Polanco, L. and J. Rice (2011) Reliability Based
Underseepage Analysis in Levees Using Monte Carlo Simulation. Geo-Frontiers 2011.
U.S. Army Corps of Engineers – USACE (1956A),
Investigation of underseepage and its control, Lower Mississsippi River levee, Volume I. Vicksburg,
Mississippi, October 1956.
U.S. Army Corps of Engineers – USACE (1956B),
Investigation of underseepage and its control, Lower Mississsippi River levee, Volume II. Vicksburg,
Mississippi, October 1956.
U.S. Army Corps of Engineers – USACE (2000).
Engineering and Design – Design and construction of levees. EM 1110-2-1913. April 2000.
Schweckendiek, T. and Vrouwenvelder A.C.W.M. (2013). Reliability updating and decision analysis for head monitoring of levees. Georisk 7(2), p.110-121.
Schweckendiek, T., A.C.W.M. Vrouwenvelder, E.O.F. Calle (2014). Updating piping reliability with field performance observations. Structural Safety 47 (2014) 13–23
Sellmeijer, J. (1988). On the mechanism of piping under
impervious structures. PhD thesis, Delft University of
Technology.
Straub, D. (2011). Reliability Updating with Equality Information. Probabilistic Engineering Mechanics, 26 (2011), 254–258. 45 50 55 60 65 0 0.2 0.4 0.6 0.8 1
water level h [m NAVD]
P(
F
|h
)
Combined fragility curve
prior
posterior scen 1 posterior scen 2 posterior combined observed WL
